Perturbations of Dirac Operators

P erturbations of Dirac Op erators Steﬀen Sc hmidt Abstract. W e study perturbations of relativ e cubic Dirac op erators for basic classical Lie su- p eralgebras within the uniform formalism of the colour quan tum W eil algebra. This persp ective leads to three complemen tary classes of p erturbations and resulting in v arian ts. First, w e deﬁne semisimple p erturbations that assign to eac h ﬁnite-dimensional simple super- mo dule a ﬁnite collection of semisimple orbits, together with canonically deﬁned vector spaces measuring the degree of at ypicalit y . Second, we in tro duce nilp oten t p erturbations parametrized b y the self-comm uting v ariety of a quadratic Lie subsup eralgebra; the resulting family of co- homology theories combines Dirac cohomology and Duﬂo–Serganov a cohomology . Third, we deform the cubic Dirac operator b y a W eil-co v arian t diﬀerential built from the univ ersal 1-form in the colour quan tum W eil algebra and the W eil diﬀeren tial, producing a Chern-type in v ariant that assigns to eac h ﬁnite-dimensional module a natural class in the cohomology of the W eil complex. Contents 1. In tro duction 1 2. Preliminaries 7 3. The Cubic Dirac Op erator 12 4. Semisimple Perturbations 27 5. Nilp oten t Perturbations 38 6. Bism ut–Quillen’s Sup erconnection 43 References 53 1. Introduction 1.1. V ue d’Ensem ble. Symmetry serv es as a guiding principle in mathematics and ph ysics. The language of symmetry groups and their actions organizes a wide range of phenomena. In this setting, Dirac op erators form a natural bridge b et w een quantum mec hanics, geometry , and represen tation theory . In representation theory , Dirac op erators provide conceptual and computational to ols for ad- dressing basic problems concerning a (semisimple) Lie group G : they furnish explicit construc- tions, give eﬀective criteria for unitarizability , con tribute to classiﬁcation results, and relate the top ology of lo cally symmetric spaces to represen tation-theoretic data. A foundational instance is Parthasarath y’s use of a representation-theoretic Dirac op erator in the study of discrete series represen tations [ P ar72 ]. 1 2 STEFFEN SCHMIDT In terest in this circle of ideas intensiﬁed in the late 1990s, following Kostant’s introduction of the cubic Dir ac op er ator [ Kos99 ] and the subsequen t dev elopment of Dir ac c ohomolo gy , including Huang–P and ˇ zi ´ c’s pro of of V ogan’s conjecture [ HP02 ]. More generally , Kostan t deﬁned cubic Dirac op erators D g , u for a pair consisting of a quadratic Lie algebra g and a quadratic Lie subalgebra u . When g is semisimple and u has equal rank, these op erators yield, among other consequences, generalizations of the Bott–Borel–W eil theorem and of the W eyl character form ula. In [ AM00 ], Aleksee v and Meinrenken placed this construction in to a uniform framework by c haracterizing cubic Dirac op erators as the unique elements of the quantum W eil algebra whose induced diﬀerential is compatible with contractions and Lie deriv atives. In parallel with the represen tation-theoretic developmen ts, (cubic) Dirac op erators hav e also b een studied from a geometric viewp oint. F ollowing F reed–Hopkins–T eleman [ FHT13 ], one attac hes to a represen tation a family of Dirac-t yp e op erators, and the resulting Dirac fam- ily enco des the representation in K-theoretic terms. This construction ties in b eautifully with Kirillo v’s orbit metho d: for an irreducible represen tation the asso ciated K-class lo calizes on (essen tially) a single coadjoint orbit, together with its prequan tum line bundle (and corresp ond- ing twisting data). Meinrenk en reformulated this observ ation for unitarizable mo dules o v er reductiv e Lie algebras (see [ Mei13 , Section 8.6]). On the analytic side, Quillen introduced the notion of a sup er c onne ction as a framework for a lo cal family index theorem for Dirac op erators [ Qui85 ], a program that w as subsequen tly realized b y Bismut [ Bis86 ]. In Quillen’s principle, Dirac op erators are a quantization of the theory of connections, and the sup ertrace of the heat kernel of the square of a Dirac op erator quantizes the Chern character of the corresp onding connection. Sup erconnections hav e since pro ved fruitful in further contexts, including the deﬁnition of Chern characters on non-compact manifolds and for inﬁnite-dimensional vector bundles. W e use these ideas as a guiding principle throughout the present article, where w e consider extensions of the ab ov e theory of Dirac op erators to Lie sup eralgebras. 1.2. Dirac Op erators and Lie Sup eralgebras. Huang and P and ˇ zi ´ c introduced Dirac op- erators and Dirac cohomology for quadratic Lie sup eralgebras relative to the even subalgebra g ¯ 0 and prov ed that Dirac cohomology determines the inﬁnitesimal character of a simple mo d- ule [ HP05 ]. F urther asp ects and the relation to unitarit y were studied b y Xiao [ Xia17 ] and b y the author [ Sc h ]. Kostant-t yp e cubic Dirac op erators for quadratic Lie sup eralgebras were dev elop ed b y Kang–Chen [ KC21 ] and Meyer [ Mey22 ], and the corresp onding Dirac cohomology w as treated in [ NSS26 ]; in particular, muc h of the classical theory extends to the sup er setting. Despite these dev elopmen ts, the existing results are not yet organized within a uniform formal- ism in which cubic Dirac op erators app ear as canonical ob jects. The present article uses suc h a framew ork b y imp orting the Alekseev–Meinrenk en viewpoint: cubic Dirac operators are realized as distinguished elements of the colour quantum W eil algebra. W e implement this construction for quadratic Lie sup eralgebras and use it as a uniform language for the ensuing Dirac-theoretic statemen ts. W e now turn to the precise algebraic setting in whic h our construction takes place. Let g b e a Lie sup eralgebra equipp ed with a non-degenerate inv arian t sup ersymmetric bilinear form B Perturbations of Dirac Op erators 3 (a quadr atic Lie sup eralgebra). Let l ⊂ g b e a quadratic Lie subsup eralgebra with B l : = B | l ; we call ( g , l ) a quadr atic p air . Then (1.1) g = l ⊕ p , p : = l ⊥ , where p is ad( l )-stable and B p : = B | p is non-degenerate. Relativ e to a quadratic pair ( g , l ), w e study the cubic Dirac op erator in the uniform framew ork of the colour quantum W eil algebra, extending the approac h of Meinrenk en and Alekseev [ AM00 ] to Lie sup eralgebras. This formalism provides a natural construction of relativ e Dirac op erators and yields a streamlined pro of that the cubic Dirac op erator has a well-behav ed square. The colour quantum W eil algebra of g is the Z 2 -graded tensor pro duct (1.2) W ( g ) : = U ( g ) ⊗ Cl( g ) , where U ( g ) and Cl( g ) carry the Z 2 -grading induced by the parity on g . W e equip W ( g ) with the Z 2 × Z 2 -bigrading determined by placing the generators γ W ( x ) in bidegree ( ¯ 0 , p ( x )) and the generators 1 ⊗ x in bidegree ( ¯ 1 , p ( x )). The brack et on W ( g ) is deﬁned as the graded comm utator with resp ect to the asso ciated symmetric bicharacter. The algebra W ( g ) is naturally endo wed with contractions ι x and Lie deriv ativ es L x for x ∈ g . Inside W ( g ) there is a distinguished elemen t D g , the cubic Dir ac op er ator . F or a homogeneous basis { e a } of g with B -dual basis { e a } it is (1.3) D g : = X a e a ⊗ e a − 1 12 X a,b,c ( − 1) p ( e a ) p ( e b )+ p ( e c ) f abc e a e b e c . It induces the unique diﬀeren tial d W : = [D g , · ] W on W ( g ) compatible with contractions and Lie deriv atives. Moreov er, its square is central: (1.4) D 2 g = Ω g ⊗ 1 + 1 24 str g  ad g (Ω g )  (1 ⊗ 1) , where Ω g is the quadratic Casimir. The construction has a relativ e formulation in terms of a quadratic subalgebra l ⊂ g , which is viewed as an elemen t of W ( g ) under the canonical em b edding j : W ( l )  → W ( g ). The r elative cubic Dir ac op er ator is (1.5) D g , l : = D g − j (D l ) . It lies in the subalgebra of l -basic elemen ts W ( g , l ) of W ( g ) and squares to a central element. This places [ KC21 , Mey22 ] into a common framework. In this article we fo cus on the case where g is a basic classical Lie sup eralgebra. Let M ( p ) b e the oscillator sup ermo dule of Cl( p ). F or a g -sup ermo dule M , we consider the induced action of D g , l on M ⊗ M ( p ). The asso ciated Dirac cohomology is (1.6) H D g , l ( M ) : = k er(D g , l ) . k er(D g , l ) ∩ im(D g , l )  . It is a fundamental inv ariant: if H D g , l ( M )  = 0, then the central character of any simple l - submo dule o ccurring in H D g , l ( M ) determines the central character of M . Applications of Dirac cohomology in the sup er setting are developed systematically in [ NSS26 , Sch24 ]; for instance, H D g , l ( M ) v anishes unless M is a highest weigh t g -sup ermodule, and in that case it embeds into Kostan t’s cohomology , with explicit examples computed in lo c. cit. 4 STEFFEN SCHMIDT 1.3. Results. The main results are organized in to three parts, corresp onding to three families of deformations of the relativ e cubic Dirac op erator asso ciated with a quadratic pair. Throughout, ( g , l ) denotes a quadratic pair, and D g , l ∈ W ( g , l ) the corresp onding relative cubic Dirac op erator. Unless stated otherwise, g is assumed to b e basic classical. 1.3.1. Semisimple p erturb ations. The semisimple p erturbations extend to Lie sup eralgebras the lo calization ideas of F reed–Hopkins–T eleman [ FHT13 ], and single out the g ¯ 0 -constituen ts of ﬁnite-dimensional sup ermo dules. Comparing the resulting Laplace family with the v alue at the origin yields a natural energy op erator; together with the semisimple p erturbations it detects at ypicalit y , and the degree of atypicalit y at each g ¯ 0 -constituen t. Let D g ∈ W ( g ) b e the absolute cubic Dirac op erator and let h ∗ R b e the real span of the ro ot system. Using B to identify h ∗ R ∼ = h R , we obtain a semisimple p erturbation family (1.7) h ∗ R − → W ( g ) , ξ 7− → D g ( ξ ) : = D g +1 ⊗ h ξ , ∆ g ( ξ ) : = D g ( ξ ) 2 . If G ¯ 0 denotes the connected simply connected Lie group with Lie algebra g ¯ 0 , then these families are h R -in v ariant and G ¯ 0 -equiv ariant (coadjoint action on g ∗ , adjoint action on W ( g )). Fix a p ositive system in ∆ and let M b e a ﬁnite-dimensional simple g -sup ermo dule. The semisimple p erturbations ( 1.7 ) act on M ⊗ M ( n − ) and lead to a family of Laplace op erators whose kernel b eha viour is the main ob ject of study . In the sup er case the form B on the real span of ro ots is t ypically indeﬁnite, so the resulting Laplace equations do not lo calize a unique parameter. W e therefore pass to the corresp onding Laplace families for the even subalgebra g ¯ 0 , whic h reco ver a rigid parameter set and thereby detect the g ¯ 0 -decomp osition of M . Since g ¯ 0 is reductiv e, it decomp oses in to simple and ab elian factors, and for each factor one has the corresp onding family of Dirac and Laplace op erators. This p ermits the construction of a mo diﬁed family of non-negative op erators, denoted e ∆ g ¯ 0 ( ξ ), whose kernel is the intersection of the kernels of the factorwise Laplace op erators. The main result is that this mo diﬁed family detects precisely the G ¯ 0 -coadjoin t orbit attac hed to a ﬁnite-dimensional simple g ¯ 0 -mo dule. The follo wing theorem is Theorem 54 in the main text. Theorem 1. L et L (Λ) b e a ﬁnite-dimensional simple g -sup ermo dule with highest weight Λ such that L (Λ)   g ¯ 0 = L µ L 0 ( µ ) n ( µ ) . Then k er e ∆ g ¯ 0 ( ξ )  = { 0 } ⇔ ξ ∈ [ µ : n ( µ )  =0 Ad ∗ G ¯ 0 ( − µ − ρ ¯ 0 ) . If ξ = − µ − ρ ¯ 0 , then the kernel is L (Λ) µ ⊗ S ρ ¯ 0 ⊗ M ( n − ¯ 1 ) with S := S g , n − ¯ 0 . T o detect at ypicalit y , we compare the Laplace family with its v alue at the origin. This deﬁnes a canonical ener gy family T ( η ), parametrized by isotropic directions in h ∗ R and thus adapted to the defect of g . F or a ﬁnite-dimensional simple g -sup ermo dule L (Λ), we then consider the join t family ( T ( η ) , e ∆ g ¯ 0 ( ξ )), parametrized by ( η , ξ ) ∈ ( h iso R ) ∗ × h ∗ R , acting on the g ¯ 0 -submo dule of L (Λ) ⊗ M ( n − ) generated by L (Λ) ⊗ S ⊗ 1 M ( n − ¯ 1 ) with S := S g , n − ¯ 0 . The even parameter ξ lo calizes at the g ¯ 0 -constituen ts of L (Λ), while the isotropic parameter η detects atypicalit y at each suc h Perturbations of Dirac Op erators 5 constituen t; in particular, the joint kernel is non-zero precisely on the corresp onding pro duct of isotropic constraints and G ¯ 0 -orbits. The following theorem is Theorem 63 in the main text. Theorem 2. L et L (Λ) b e a ﬁnite-dimensional simple g -sup ermo dule. Then ker(( T ( η ) , e ∆ g ¯ 0 ( ξ )) = { 0 } unless b oth a) ξ = − µ − ρ ¯ 0 , wher e µ is the highest weight of a g ¯ 0 -c onstituent of L (Λ) , b) η ∈ X µ + ρ : = { η ′ ∈ ( h iso R ) ∗ : B ( µ + ρ, η ′ ) = 0 } . In p articular, k er( T ( η ) , e ∆ g ¯ 0 ( ξ ))  = { 0 } ⇔ ( η , ξ ) ∈ [ µ : n ( µ )  =0 X µ + ρ × Ad ∗ G ¯ 0 ( − µ − ρ ¯ 0 ) . 1.3.2. Nilp otent Perturb ations. F or a basic classical Lie sup eralgebra g with g ¯ 1  = { 0 } there are tw o complementary constructions attached to a g -sup ermo dule: Dirac cohomology , whic h con trols the inﬁnitesimal c haracter, and Duﬂo–Serganov a cohomology , a symmetric monoidal functor preserving the sup erdimension. W e construct a single family of relative cubic Dirac op erators whose cohomology combines these tw o theories. Fix a quadratic pair ( g , l ) and assume h ⊂ l and l ¯ 1  = { 0 } . Set (1.8) Y l : = { x ∈ l ¯ 1 : [ x, x ] = 2 x 2 = 0 } , the self-comm uting v ariet y of l . In contrast with the reductiv e (purely even) case, the bigrading of W ( g , l ) p ermits square-zero p erturbations of the relativ e cubic Dirac op erator while leaving its square unchanged. Let D g , l ∈ W ( g , l ) b e the relative cubic Dirac op erator. F or x ∈ Y l deﬁne (1.9) D x g , l : = D g , l + j  γ W ( x )  ∈ W ( g , l ) . This elemen t is o dd, in v ariant under the cen tralizer C l ( x ) : = { y ∈ l : [ x, y ] = 0 } , and it is L ¯ 0 -equiv ariant in the sense that Ad g (D x g , l ) = D Ad g ( x ) g , l for an y g ∈ L ¯ 0 . Here, L ¯ 0 denotes the connected simply connected Lie group with Lie algebra l ¯ 0 . Moreo v er, the square is indep endent of x : (1.10) (D x g , l ) 2 = D 2 g , l . F or a g -sup ermo dule M w e deﬁne the cohomology of D x g , l b y (1.11) H D x g , l ( M ) : = k er D x g , l  k er D x g , l ∩ im D x g , l  . Our aim is to describ e H D x g , l ( M ) in terms of Dirac cohomology (Section 3.5 ) and Duﬂo–Sergano v a cohomology (Section 5.1 ). F or the morphisms (1.12) η ∗ x : Hom  Z ( l x ) , C  − → Hom  Z ( l ) , C  , as well as the homomorphism η l : Z ( g ) → Z ( l ), which arise naturally in Duﬂo–Sergano v a coho- mology and Dirac cohomology , we establish the follo wing result in the unitarizable case ( cf. The- orem 73 in the main text): 6 STEFFEN SCHMIDT Theorem 3. Assume that M is a unitarizable highest weight g -sup ermo dule with highest weight Λ . Then for every x ∈ Y l one has (1.13) H D x g , l ( M ) = DS x  H D g , l ( M )  . In p articular, H D x g , l ( M ) is an l x -sup ermo dule. If V is an l x -sup ermo dule o c curring in H D x g , l ( M ) with inﬁnitesimal char acter χ l x ν , then (1.14) χ l x ν ∈ ( η ∗ x ) − 1  χ Λ ◦ η l  . If M is not unitary , ha ving a sp ectral sequence argument would lead to the following conjec- ture. Conjecture 4. Let x ∈ Y l . If DS x  H D g , l ( M )  = 0, then H D x g , l ( M ) = 0 . 1.3.3. Bismut–Quil len Sup er c onne ction. Let g b e a semisimple complex Lie algebra and let ( g , l ) b e a quadratic pair. F or a ﬁnite-dimensional g -mo dule M , consider the ﬁnite-dimensional W ( g , l )-sup ermo dule E : = M ⊗ S , where S is the spin mo dule asso ciated with the pair. A naive Chern–W eil-type expression formed directly from the relative cubic Dirac op erator D g , l v anishes iden tically . T o obtain a non-trivial inv ariant, w e therefore replace D g , l b y a Bism ut–Quillen-t yp e sup erconnection A M g , l ( t ), obtained by p erturbing D g , l b y a canonical term inv olving the universal connection 1-form of W ( g ) and the W eil diﬀerential. This yields an o dd, l -equiv ariant element of End C ( c W ( g , l ) ⊗ E ), and we deﬁne (1.15) c h M ( t ) : = str E  e − A M g , l ( t ) 2  ∈ c W ( g , l ) . W e pro ve that ch M ( t ) determines a cohomology class indep enden t of t . The following theorem app ears in the main text as Theorem 91 . Theorem 5. F or any ﬁnite-dimensional g -mo dule M the class (1.16) [c h M ( t )] ∈ H  c W ( g , l ) , d W g , l  is indep endent of t . W e ﬁnally in tro duce the appropriate replacement in the super setting. Since the oscillator mo dule is inﬁnite-dimensional, the sup ertrace on E is not av ailable in general. F or unitarizable M , ho w ev er, the “heat op erator” e − t D 2 g , l lo calizes as t → ∞ on ker D g , l , and the argument pro ducing the class [c h M ( t )] in the ﬁnite-dimensional case applies whenever a sup ertrace is a v ailable. This leads us to deﬁne a canonical substitute for ch M ( t ) in the sup er case. 1.4. Con v en tions and Notation. The ground ﬁeld is C . Let Z 2 : = Z / 2 Z b e the ring of integers mo dulo 2, and denote b y ¯ 0 and ¯ 1 the residue classes of ev en and o dd in tegers, resp ectiv ely . F or a sup er v ector space V = V ¯ 0 ⊕ V ¯ 1 , the parit y of an elemen t x ∈ V is written p ( x ) ∈ Z 2 . Expressions of the form ( − 1) p ( v ) p ( w ) , for general v , w ∈ V , are understo o d b y restricting ﬁrst to homogeneous elemen ts, inserting their parities in the exp onent, and extending linearly . Perturbations of Dirac Op erators 7 F or any tw o supe r vector spaces V and W , let Hom( V , W ) denote the space of all par- it y–preserving linear maps. The elements of Hom( V , W ) are called morphisms of sup er v ector spaces. F or any tw o sup er vector spaces V and W , deﬁne the Z 2 –graded tensor pro duct V ⊗ W by ( V ⊗ W ) ¯ 0 : = ( V ¯ 0 ⊗ W ¯ 0 ) ⊕ ( V ¯ 1 ⊗ W ¯ 1 ) , ( V ⊗ W ) ¯ 1 : = ( V ¯ 0 ⊗ W ¯ 1 ) ⊕ ( V ¯ 1 ⊗ W ¯ 0 ) . The assignment ( V , W ) 7→ V ⊗ W is additive in eac h v ariable. Moreo ver, ⊗ is asso ciative, and the map V ⊗ W → W ⊗ V , v ⊗ w 7→ ( − 1) p ( v ) p ( w ) w ⊗ v , is an isomorphism. If V and W are sup eralgebras, the pro duct in V ⊗ W is ( v 1 ⊗ w 1 )( v 2 ⊗ w 2 ) = ( − 1) p ( w 1 ) p ( v 2 ) ( v 1 v 2 ⊗ w 1 w 2 ) for an y homogeneous v 1 , v 2 ∈ V and w 1 , w 2 ∈ W . 1.5. Leitfaden. The pap er is organized as follows. Section 2 collects the necessary bac kground on basic classical Lie superalgebras, highest weigh t supermo dules, inﬁnitesimal characters, at yp- icalit y , and unitarizable sup ermodules. In Section 3 we develop the cubic Dirac op erator in the required generality . After introducing the colour quantum W eil algebra, we deﬁne the cu- bic and relative cubic Dirac operators, discuss the oscillator supermo dule, and form ulate the corresp onding Dirac cohomology , with particular emphasis on the unitary setting. Section 4 studies semisimple p erturbations of the cubic Dirac op erator and introduces the asso ciated family of Laplace op erators, culminating in a detecting family . Section 5 treats nilp otent p er- turbations parametrized by the self-comm uting v ariety of a quadratic subalgebra and relates Duﬂo–Sergano v a cohomology to Dirac cohomology for unitarizable sup ermo dules. Finally , Sec- tion 6 places these p erturbations in the framew ork of Bismut–Quillen superconnections, ﬁrst for semisimple Lie algebras and then for basic classical Lie sup eralgebras. 2. Preliminaries This section ﬁxes conv en tions and notation used throughout the article. W e recall basic classical Lie sup eralgebras, highest weigh t sup ermo dules, inﬁnitesimal characters and atypicalit y , and then formulate unitarity . 2.1. Basic Classical Lie Sup eralgebra. Throughout, w e work with quadratic Lie sup er- algebras g , i.e. , Lie superalgebras admitting a non-degenerate in v ariant sup ersymmetric con- sisten t bilinear form B . Consistency means B ( x, y ) = 0 whenever p ( x )  = p ( y ), in v ariance means B ([ x, y ] , z ) = B ( x, [ y , z ]) for all x, y , z ∈ g , and sup ersymmetry means B ( x, y ) = ( − 1) p ( x ) p ( y ) B ( y , x ) for all homogeneous x, y ∈ g . Our main examples are basic classical Lie sup eralgebras g = g ¯ 0 ⊕ g ¯ 1 : g is simple, g ¯ 0 is reductiv e, and g carries such a form B , unique up to a nonzero scalar. According to Kac [ Kac77 ], the basic classical Lie sup eralgebras comprise the complex simple Lie algebras together with (2.1) A ( m | n ) , B ( m | n ) , C ( n ) , D ( m | n ) , F (4) , G (3) , D (2 , 1; α ) . W e assume throughout that α ∈ R for D (2 , 1; α ), so that g is contragredien t, unless otherwise stated. 8 STEFFEN SCHMIDT F or the simple Lie algebras we take the usual Killing form B , while for the basic classical Lie sup eralgebras A ( m | n ) with m  = n , B ( m | n ), C ( n + 1), D ( m | n ) with m  = n + 1, F (4), and G (3), w e tak e as B ( · , · ) the sup er Kil ling form (2.2) B ( x, y ) : = str(ad x ◦ ad y ) , x, y ∈ g . F or the remaining cases, the sup er Killing form v anishes identically; an alternative inv ariant consisten t form is then c hosen as in [ Mus12 , Section 5.4], and will again b e referred to simply as the sup er Killing form. Let h ⊂ g b e a Cartan subalgebra and let ∆ : = ∆( g , h ) b e the set of ro ots. Then (2.3) g = h ⊕ M α ∈ ∆ g α , g α : = { X ∈ g : [ H , X ] = α ( H ) X for all H ∈ h } . Since g is basic classical, one has h ⊂ g ¯ 0 . Moreo v er, w e ﬁx some p ositive system ∆ + = ∆ + ¯ 0 ⊔ ∆ + ¯ 1 and deﬁne the Weyl ve ctor to b e (2.4) ρ : = ρ ¯ 0 − ρ ¯ 1 , ρ ¯ 0 : = 1 2 X α ∈ ∆ + ¯ 0 α, ρ ¯ 1 : = 1 2 X α ∈ ∆ + ¯ 1 α. Asso ciated with ∆ + , g has a triangular de c omp osition (2.5) g = n − ⊕ h ⊕ n + , n ± : = M α ∈ ∆ + g ± α . The W eyl group W is the W eyl group of the Lie algebra g ¯ 0 . The asso ciated fundamental system Π is the set of ro ots in ∆ + not expressible as sums of t w o p ositive ro ots; its elements are called simple r o ots . 2.2. Highest W eight Sup ermo dules. The main class of g -sup ermo dules considered in this article is the class of highest weigh t sup ermo dules, which we now deﬁne. Fix a p ositiv e system ∆ + . This choice determines a triangular decomp osition g = n − ⊕ h ⊕ n + and the corresp onding Borel subsup eralgebra b : = h ⊕ n + . In what follows, ∆ + is kept ﬁxed and omitted from the notation. Deﬁnition 6. A g -sup ermo dule M is called a highest weight g -sup ermo dule with resp ect to a p ositiv e system ∆ + if there exists a nonzero vector v Λ ∈ M with Λ ∈ h ∗ suc h that the following holds: a) X v Λ = 0 for all X ∈ n + , b) H v Λ = Λ( H ) v Λ for all H ∈ h , and c) U ( g ) v Λ = M . The vector v Λ is referred to as the highest weight ve ctor of M , and Λ is referred to as the highest weight . Ev ery simple highest w eight g -sup ermo dule is the unique simple quotien t of a universal highest w eigh t sup ermo dule, the V erma sup ermo dule . Since U ( g ) is a right U ( b )-sup ermo dule by right m ultiplication, for λ ∈ h ∗ w e set (2.6) M b ( λ ) : = U ( g ) ⊗ U ( b ) C λ , Perturbations of Dirac Op erators 9 where C λ is the one-dimensional b -supermo dule on whic h n + acts trivially and h acts b y λ . Since b is ﬁxed, we omit the subscript. Then M ( λ ) is a highest w eight g -sup ermo dule with highest w eigh t λ and highest w eight vector [1 U ( g ) ⊗ 1], and U ( n − )[1 U ( g ) ⊗ 1] = M ( λ ). Moreo v er, if M is any g -sup ermo dule and v ∈ M has weigh t λ with n + v = 0, there is a unique g -morphism M ( λ ) → M sending [1 U ( g ) ⊗ 1] to v . The following prop erties of highest weigh t g -sup ermo dules are immediate from their realization as quotients of V erma sup ermo dules. Lemma 7. L et M b e a highest weight g -sup ermo dule with highest weight Λ ∈ h ∗ . a) M is a weigh t sup ermo dule , that is, M is h -semisimple. b) F or al l weights λ of M , we have dim( M λ ) < ∞ , while dim( M Λ ) = 1 . c) Any nonzer o quotient of M is again a highest weight sup ermo dule. d) M has a unique maximal subsup ermo dule and a unique simple quotient. e) Any two simple highest weight g -sup ermo dules with highest weight Λ ar e isomorphic. In what follows, we denote the unique simple quotien t of a highest weigh t g -sup ermodule of highest weigh t Λ by L (Λ). An important class of highest weigh t g -supermo dules is formed b y the ﬁnite-dimensional ones. F or our ﬁxed Borel subalgebra b = b ¯ 0 ⊕ b ¯ 1 with ∆ + = ∆ + ¯ 0 ⊔ ∆ + ¯ 1 , the ﬁnite-dimensional simple g -sup ermo dules are parametrized by dominant in tegral w eights λ ∈ h ∗ , i.e. , (2.7) B ( λ + ρ ¯ 0 , α ) > 0 for all α ∈ ∆ + ¯ 0 . Let P ++ denote the set of suc h weigh ts (with resp ect to our ﬁxed b ). F or λ ∈ P ++ , let L ( λ ) b e the simple highest weigh t g -sup ermo dule of highest weigh t λ with ev en highest weigh t vector. Then the full set of ﬁnite-dimensional simple g -sup ermo dules is (2.8) { L ( λ ) , Π L ( λ ) | λ ∈ P ++ } . 2.3. Inﬁnitesimal Characters and At ypicalit y. Any highest weigh t g -sup ermo dule admits an inﬁnitesimal char acter , i.e. , an algebra homomorphism χ : Z ( g ) → C . W e describ e these via the Harish–Chandra homomorphism. By PBW, (2.9) U ( g ) ∼ = U ( h ) ⊕  n − U ( g ) + U ( g ) n +  , and we let p : U ( g ) → U ( h ) b e the corresp onding pro jection. Its restriction p | Z ( g ) : Z ( g ) → U ( h ) ∼ = S is an algebra homomorphism. Deﬁne the twist ζ : S → S b y (2.10) λ ( ζ ( f )) = ( λ − ρ )( f ) ( f ∈ S , λ ∈ h ∗ ) , and set HC : = ζ ◦ p | Z ( g ) . The map HC : Z ( g ) → S is an injective ring homomorphism. Moreov er, its image can b e describ ed as follo ws. Let S( h ) W : = { f ∈ S( h ) : w ( λ )( f ) = λ ( f ) for all w ∈ W , λ ∈ h ∗ } and, for any λ ∈ h ∗ , deﬁne A λ : = { α ∈ ∆ + 1 : ( λ + ρ, α ) = 0 } . Then, the image of HC is given by ([ Kac84 , Ser87 , Ser88 ]): im(HC) =  f ∈ S( h ) W : ( λ + tα )( f ) = λ ( f ) for all t ∈ C , λ ∈ h ∗ , α ∈ A λ − ρ  . F or λ ∈ h ∗ deﬁne χ λ ( z ) : = ( λ + ρ )  HC( z )  for z ∈ Z ( g ). 10 STEFFEN SCHMIDT Deﬁnition 8. A g -sup ermo dule M has inﬁnitesimal char acter if Z ( g ) acts on M via χ λ for some λ ∈ h ∗ . In this case, χ λ is the inﬁnitesimal character of M . Examples include highest weigh t g -sup ermo dules M with highest weigh t Λ ∈ h ∗ . They hav e inﬁnitesimal character χ Λ . As a direct consequence of the description of im(HC), for any λ, λ ′ ∈ h ∗ one has χ λ = χ λ ′ if and only if λ ′ = w  λ + ρ + k X i =1 t i α i  − ρ, where w ∈ W , t i ∈ C , and α 1 , . . . , α k ∈ A λ : = { α ∈ ∆ + 1 : ( λ + ρ, α ) = 0 , B ( α, α ) = 0 } are linearly indep enden t o dd isotropic ro ots. This leads to the deﬁnition of typicalit y and atypicalit y . Deﬁnition 9. A w eigh t Λ ∈ h ∗ is called typic al if A Λ = ∅ , that is, (Λ + ρ, α )  = 0 for all α ∈ ∆ + ¯ 1 . Otherwise, Λ is called atypic al . The de gr e e of atypic ality of Λ, denoted by at(Λ), is the maximal n umber of linearly indep endent mutually orthogonal p ositiv e o dd isotropic ro ots α ∈ ∆ + ¯ 1 suc h that (Λ + ρ, α ) = 0. In brief, at(Λ) is the dimension of a maximal isotropic subspace of span C ( A Λ ) ⊂ h ∗ . W e call a highest weigh t g -sup ermo dule M with highest weigh t Λ typic al if at(Λ) = 0, and otherwise atypic al . R emark 10 . The degree of at ypicalit y is indep enden t of the c hoice of p ositiv e ro ot system. The defe ct of g , denoted def ( g ), is the dimension of a maximal isotropic subspace in the R - span of ∆. F or g of type A ( m − 1 | n − 1), B ( m | n ), or D ( m | n ) (so that ∆ R ∼ = R m,n ), one has def ( g ) = min( m, n ). F or g of type C ( n ), def ( g ) = 1. A simple Lie algebra and osp (1 | 2 n ) hav e defect 0. In all cases, (2.11) 0 ≤ at( λ ) ≤ def ( g ) for any λ ∈ h ∗ . 2.4. Unitarizable Sup ermo dules. In this subsection, we deﬁne unitarizable g -sup ermo dules, whic h are form ulated relative to real forms of g . Our notation follows [ CFV20 , CF15 , FG23 ]. Let V = V ¯ 0 ⊕ V ¯ 1 b e a complex super v ector space. A sup er Hermitian form on V is a sesquilinear map ⟨· , ·⟩ : V × V → C , linear in the ﬁrst v ariable and conjugate-linear in the second, satisfying (2.12) ⟨ v , w ⟩ = ( − 1) p ( v ) p ( w ) ⟨ w , v ⟩ for all homogeneous v , w ∈ V , and we assume it is consistent, i.e. , ⟨ v , w ⟩ = 0 whenev er p ( v )  = p ( w ). A sup er Hermitian form ⟨· , ·⟩ decomp oses as ⟨· , ·⟩ = ⟨· , ·⟩ ¯ 0 + i ⟨· , ·⟩ ¯ 1 , where (2.13) ⟨· , ·⟩ ¯ s : = ( − 1) s ⟨· , ·⟩| V ¯ s × V ¯ s , ¯ s ∈ Z 2 . Then ⟨· , ·⟩ ¯ 0 and ⟨· , ·⟩ ¯ 1 are ordinary Hermitian forms on V ¯ 0 and V ¯ 1 , resp ectively . W e call ⟨· , ·⟩ non-de gener ate (resp. p ositive deﬁnite ) if b oth ⟨· , ·⟩ ¯ 0 and ⟨· , ·⟩ ¯ 1 are non-degenerate (resp. p ositiv e deﬁnite). W e sa y that ⟨· , ·⟩ is sup er p ositive deﬁnite if ⟨· , ·⟩ ¯ 0 is positive deﬁnite and ⟨· , ·⟩ ¯ 1 is negativ e deﬁnite; in this case, ⟨· , ·⟩ is a Hermitian pr o duct on V . Perturbations of Dirac Op erators 11 F or T ∈ End C ( V ), we deﬁne the adjoint T † b y (2.14) ⟨ T v, w ⟩ = ( − 1) p ( v ) p ( T ) ⟨ v , T † w ⟩ for all homogeneous v , w ∈ V , and extend b y linearit y . Unitarit y for g -sup ermo dules is deﬁned relative to real forms of g , which we now set up. F or r ∈ { 2 , 4 } , let aut R 2 ,r ( g ) b e the set of automorphisms θ of g view ed as a real sup er vector space suc h that θ | g ¯ 0 ⊕ g ¯ 1  = id g ¯ 0 ⊕ g ¯ 1 and (2.15) θ 2   g ¯ 0 = id g ¯ 0 , θ 2   g ¯ 1 =    id g ¯ 1 , r = 2 , − id g ¯ 1 , r = 4 . W e set (2.16) aut 2 , 4 ( g ) : = { θ ∈ aut R 2 , 4 ( g ) : θ is C -linear } , aut 2 ,r ( g ) : = { θ ∈ aut R 2 ,r ( g ) : θ is conjugate-linear } . A r e al structur e on g is a conjugate-linear Lie sup eralgebra morphism φ : g → g such that φ ∈ aut 2 , 2 ( g ), i.e. , φ is a conjugate-linear inv olution. The ﬁxed-p oint subalgebra g ϕ is a r e al form of g . Real structures in aut 2 , 2 ( g ) are related to elements of aut 2 , 4 ( g ) as follows. Prop osition 11 ([ FG23 ]) . Ther e exists a unique ω ∈ aut 2 , 4 ( g ) , up to inner automorphisms of g , to gether with a p ositive system ∆ + and r o ot ve ctors e ± α for α ∈ ∆ + such that ω ( e ± α ) = − e ∓ α for al l even simple α, ω ( e ± α ) = ± e ∓ α for al l o dd simple α . Mor e over: a) ω induc es a bije ction aut 2 , 2 ( g ) \ { θ : θ | g ¯ 0 = ω | g ¯ 0 } − → aut 2 , 4 ( g ) , θ 7− → ω − 1 ◦ θ . b) F or the Kil ling form B ( · , · ) on g , one has B ( X, Y ) = B ( ω ( X ) , ω ( Y )) for al l X , Y ∈ g , and B ( · , ω ( · )) is p ositive deﬁnite. R emark 12 . The p ositive system ∆ + is the distinguishe d p ositive system , meaning that there is exactly one o dd p ositive ro ot that cannot b e expressed as the sum of tw o other p ositiv e ro ots. As a consequence, there is a one-to-one corresp ondence, up to equiv alence [ Ser83 , P el07 , F G23 , Ch u13 ] (2.17) { real forms g R of g } ← → { θ ∈ aut 2 , 4 ( g ) } . Here a real form is realized as the ﬁxed-p oin t subspace of a suitable inv olution, and t wo real forms are iden tiﬁed whenev er they are isomorphic (equiv alently , whenev er the corresp onding in v olutions are conjugate by an automorphism of g ). Fix a real form g R of a basic classical Lie sup eralgebra g , i.e. , g R = g θ for some θ ∈ aut 2 , 4 ( g ). Set σ : = ω ◦ θ ∈ aut 2 , 2 ( g ), the corresp onding conjugate-linear inv olution ( cf. Prop osition 11 ). W e call θ a Cartan automorphism of g (or of g R ) if (2.18) B θ ( · , · ) : = − B ( · , θ ( · )) 12 STEFFEN SCHMIDT is an inner pro duct on g R . F or each θ ∈ aut 2 , 4 ( g ) there is a unique real form g R for whic h θ restricts to a Cartan automorphism, and con versely each real form admits a unique Cartan automorphism [ Chu13 , Theorem 1.1]. Hence, we ma y assume throughout that the chosen θ is Cartan. With the notion of a real form established, we can now deﬁne unitarizable sup ermo dules. Deﬁnition 13. Let g R b e a real form of g , and let H b e a complex g R -sup ermo dule. The sup ermo dule H is called a unitarizable g -sup ermo dule (or unitarizable g R -sup ermo dule ) if there exists a Hermitian product ⟨· , ·⟩ on H such that X † = − X for all X ∈ g R . Explicitly , this means: ⟨ X v , w ⟩ = − ( − 1) p ( v ) p ( X ) ⟨ v , X w ⟩ for all v , w ∈ H and X ∈ g R . The following standard result is central. Prop osition 14. Unitarizable g -sup ermo dules ar e c ompletely r e ducible, that is, the ortho gonal c omplement of any g R -submo dule is again a g R -submo dule. 3. The Cubic Dirac Opera tor W e introduce the cubic Dirac op erator, its relativ e version, and Dirac cohomology for quadratic Lie sup eralgebras in the language of the colour quantum W eil algebra. Throughout this section, let g b e a quadratic Lie sup eralgebra with non-degenerate in v ariant sup ersymmetric bilinear form B . Via B , we identify g with g ∗ b y the m usical isomorphisms (3.1)  : g → g ∗ , x 7→ x ♭ , x ♭ ( y ) : = B ( x, y ) ,  : g ∗ → g , α 7→ α ♯ , B ( α ♯ , y ) = α ( y ) for all y ∈ g . This identiﬁcation will b e used without further commen t. 3.1. Colour Quantum W eil Algebra. The cubic Dirac op erators and their p erturbations tak e v alues in the colour quantum W eil algebra, which we now introduce. 3.1.1. Exterior and Cliﬀor d Algebr a over g . W e brieﬂy recall the deﬁnition and prop erties of the exterior and Cliﬀor d algebr a o ver g . Let T ( g ) b e the tensor algebra of g with unit 1 T ( g ) . It carries the canonical Z -grading (3.2) T ( g ) = M k ≥ 0 T k ( g ) , T 0 ( g ) = C , T k ( g ) = span { x 1 ⊗ · · · ⊗ x k : x i ∈ g } . Reducing the degree mo d 2 yields a sup eralgebra structure with T ( g ) ¯ 0 : = L k ∈ 2 Z + T k ( g ) and T ( g ) ¯ 1 : = L k ∈ 2 Z + +1 T k ( g ) . Thus T ( g ) carries t wo Z 2 -gradings, hence tw o sup eralgebra struc- tures: one from tensor degree mo dulo 2, and one from the intrinsic parity of g . In this article w e use the latter, that is, we regard T ( g ) as the sup eralgebra induced by the g -parity . Exterior Algebra o v er g . Let I ∧ ⊂ T ( g ) b e the homogeneous ideal generated by (3.3) x ⊗ y + ( − 1) p ( x ) p ( y ) y ⊗ x, x, y ∈ g . Perturbations of Dirac Op erators 13 The quotient V ( g ) : = T ( g ) /I ∧ is the exterior algebr a of g , with unit 1 V ( g ) and pro duct ∧ (exterior m ultiplication). Equiv alen tly , it is generated b y g sub ject to (3.4) x ∧ y + ( − 1) p ( x ) p ( y ) y ∧ x = 0 , x, y ∈ g . The Z -grading of T ( g ) descends to V ( g ), so that V ( g ) = L k ≥ 0 V k ( g ), where V k ( g ) is spanned b y exterior pro ducts of k elements of g . Reducing the degree mo d 2 gives the natural sup eralgebra grading V ( g ) = V ( g ) ¯ 0 ⊕ V ( g ) ¯ 1 . W e also hav e the Z 2 -grading induced by the intrinsic parity of g , and we ﬁx this grading from now on. On V ( g ) w e use the standard op erators  , ι , and L . F or v ∈ g , let  ( v ) denote left exterior multiplic ation by v . Deﬁne a deriv ation ι v on T ( g ) by (3.5) ι v ( x 1 ⊗ · · · ⊗ x l ) : = l X k =1 ( − 1) k − 1 ( − 1) p ( v )( p ( x 1 )+ ··· + p ( x k − 1 )) B ( v , x k ) x 1 ⊗ · · · ⊗ b x k ⊗ · · · ⊗ x l . Then ι v ( I ∧ ) ⊂ I ∧ ( cf. [ CK16 , Proposition 4.5]), hence ι v descends to V ( g ); it is called c ontr action , has degree − 1 with resp ect to the Z -grading, and parity p ( v ). Finally , for homogeneous T ∈ End( g ) deﬁne a degree-0 deriv ation L T on T ( g ) by (3.6) L T ( x 1 ⊗ · · · ⊗ x l ) : = l X k =1 ( − 1) p ( T )( p ( x 1 )+ ··· + p ( x k − 1 )) x 1 ⊗ · · · ⊗ T ( x k ) ⊗ · · · ⊗ x l . It preserves I ∧ and therefore induces a degree-0 deriv ation on V ( g ), the Lie derivative . Moreo v er, B induces a non-degenerate bilinear form on V ( g ). F or k ≥ 0 and x i , y i ∈ g , deﬁne on T k ( g ) (3.7) ⟨ x 1 ⊗ · · · ⊗ x k , y 1 ⊗ · · · ⊗ y k ⟩ T k : = k − 1 Y i =0 B ( x k − i , y 1+ i ) , and set (3.8) ⟨ x 1 ⊗ · · · ⊗ x k , y 1 ⊗ · · · ⊗ y k ⟩ ∧ k : = X σ ∈ S k p ( σ ; x 1 , . . . , x k ) ⟨ x σ (1) ⊗ · · · ⊗ x σ ( k ) , y 1 ⊗ · · · ⊗ y k ⟩ T k , where p ( σ ; x 1 , . . . , x k ) = sgn( σ ) Q 1 ≤ iσ − 1 ( j ) ( − 1) p ( x i ) p ( x j ) . Then ⟨· , ·⟩ ∧ k v anishes on I ∧ ∩ T k ( g ) and hence descends to V k ( g ), yielding a bilinear form ⟨· , ·⟩ ∧ on V ( g ). Lemma 15 ([ CK16 , Section 5]) . The form ⟨· , ·⟩ ∧ is non-de gener ate and sup ersymmetric, and for homo gene ous x, y , z ∈ g one has ⟨ ι x y , z ⟩ ∧ = ( − 1) p ( x ) p ( y ) ⟨ y , x ∧ z ⟩ ∧ . W e fo cus on V 2 ( g ) and its relation to the orthosymplectic sup eralgebra osp ( g ). Deﬁne osp ( g ) ⊂ End( g ) to b e the subspace of endomorphisms T that are skew-supersymmetric with resp ect to B , that is, (3.9) B ( T x, y ) + ( − 1) p ( T ) p ( x ) B ( x, T y ) = 0 ∀ x, y ∈ g , with Lie sup erbrack et [ · , · ] osp ( g ) giv en b y the sup ercomm utator. Lemma 16. F or x ∈ g , one has ad x ∈ osp ( g ) . 14 STEFFEN SCHMIDT Pr o of. F or y , z ∈ g , in v ariance and sup ersymmetry of B give B (ad x y , z ) = B ([ x, y ] , z ) = − ( − 1) p ( x ) p ( y ) B ( y , [ x, z ]) = − ( − 1) p ( x ) p ( y ) B ( y , ad x z ) . □ With resp ect to ad : g → osp ( g ), deﬁne the moment map µ : g × g → osp ( g ) by (3.10) str  ad x ◦ µ ( y , z )  = B ([ x, y ] , z ) x, y , z ∈ g , where str is the sup ertrace on g . The map µ is skew-supersymmetric and satisﬁes (3.11) µ ( x, y )( z ) = − ι z ( x ∧ y ) = − B ( z , x ) y + ( − 1) p ( z ) p ( x ) B ( z , y ) x x, y , z ∈ g . F or ν ∈ V 2 ( g ) we also write A ν : = µ ( ν ). Prop osition 17 ([ Mey22 , Prop. 2.13]) . µ : V 2 ( g ) → osp ( g ) is an isomorphism of sup er ve ctor sp ac es, with inverse λ : = µ − 1 : osp ( g ) → V 2 ( g ) , given in a b asis { e a } of g , with B -dual b asis { e a } , by λ ( T ) : = µ − 1 ( T ) = − 1 2 X a T ( e a ) ∧ e a . Via the iden tiﬁcation g ∼ = g ∗ induced b y B , we can describ e λ in a basis-free notation. F or T ∈ osp ( g ) deﬁne ω T ∈ V 2 g ∗ b y ω T ( x, y ) : = B ( T x, y ); it is skew-supersymmetric. Lemma 18. Under g ∼ = g ∗ one has λ ( T ) = ω T for al l T ∈ osp ( g ) . Pr o of. Let  : g → g ∗ b e given by u ♭ = B ( u, · ) and extend to V 2 g → V 2 g ∗ b y ( u ∧ v ) ♭ = u ♭ ∧ v ♭ . F or homogeneous y , z we compute, using the conv en tion of Section 1.4 , (2 λ ( T )) ♭ ( y ∧ z ) = − X a ( − 1) p ( e a ) p ( y )  B ( T e a , y ) B ( e a , z ) − ( − 1) p ( y ) p ( z )+ p ( T ) p ( y ) B ( T e a , z ) B ( e a , y )  = − ( − 1) p ( y ) p ( z ) X a  B ( T e a , y ) B ( e a , z ) − ( − 1) p ( y ) p ( z ) B ( T e a , z ) B ( e a , y )  . Using P a B ( e a , · ) e a = id g , this b ecomes ( − 1) p ( y ) p ( z ) (2 λ ( T )) ♭ ( y ∧ z ) = − B ( T z , y ) + ( − 1) p ( y ) p ( z ) B ( T y , z ) . Since T ∈ osp ( g ), we ha ve B ( T z , y ) = − ( − 1) p ( T ) p ( z ) B ( z , T y ), and b y sup ersymmetry of B this yields (2 λ ( T )) ♭ ( y ∧ z ) = 2 B ( T y , z ) = 2 ω T ( y , z ) . Hence ( λ ( T )) ♭ = ω T , as claimed. □ A further direct calculation yields the following lemma. Lemma 19. F or any T 1 , T 2 ∈ osp ( g ) , one has L T 1 λ ( T 2 ) = λ ([ T 1 , T 2 ] osp ( g ) ) . Cliﬀord Sup eralgebra. Let B b e the ﬁxed inv arian t sup ersymmetric consistent bilinear form on g . Let I Cl ⊂ T ( g ) b e the tw o-sided ideal generated by (3.12) x ⊗ y + ( − 1) p ( x ) p ( y ) y ⊗ x − 2 B ( x, y ) 1 T ( g ) , x, y ∈ g . Perturbations of Dirac Op erators 15 The quotient Cl( g ) : = T ( g ) /I Cl is the Cliﬀor d algebr a of g . Identifying g with its image in Cl( g ), the deﬁning relations read (3.13) xy + ( − 1) p ( x ) p ( y ) y x = 2 B ( x, y ) , x, y ∈ g . F rom now on, we equip Cl( g ) with the Z 2 -grading induced from g . The op erations ( 3.5 ) and ( 3.6 ) preserve the ideal I Cl and hence induce c ontr actions ι x and Lie derivatives L T for any x ∈ g and T ∈ End( g ). W e relate V ( g ) and Cl( g ) via the usual Cliﬀord action. Deﬁne (3.14) τ : g → End  ^ ( g )  , τ ( x ) : =  ( x ) + ι x , where  ( x ) is left exterior multiplication and ι x is contraction. Then, for x, y ∈ g , (3.15) τ ( x ) τ ( y ) + ( − 1) p ( x ) p ( y ) τ ( y ) τ ( x ) = 2 B ( x, y ) . By the universal prop erty of Cl( g ) ( cf. [ CK16 , Prop osition 3.1]), τ extends uniquely to a sup er- algebra homomorphism Cl( g ) → End( V ( g )), hence makes V ( g ) a Cl( g )-mo dule. Theorem 20 ([ CK16 ]) . The map η : Cl( g ) → V ( g ) deﬁne d by η ( v ) : = τ ( v ) 1 V ( g ) is an isomor- phism of sup er ve ctor sp ac es. Its inverse is the quantization map q = P k q k : V ( g ) → Cl( g ) , wher e q k ( x 1 ∧ · · · ∧ x k ) : = 1 k ! X σ ∈ S k p ( σ ; x 1 , . . . , x k ) x σ (1) · · · x σ ( k ) . Mor e over, q intertwines c ontr actions, that is, ι x ◦ q = q ◦ ι x for al l x ∈ g . The Lie deriv ative on Cl( g ) by elemen ts of osp ( g ) is implemented b y a commutator. F or T ∈ osp ( g ), w e set (3.16) γ ′ ( T ) : = q ( λ ( T )) . In particular, γ ′ (ad x ) is deﬁned for every x ∈ g by Lemma 16 . In what follo ws, w e write γ ′ ( x ) for γ ′ (ad x ). Let [ · , · ] Cl( g ) denote the sup ercomm utator on Cl( g ), i.e. , [ v , w ] Cl( g ) = v w − ( − 1) p ( v ) p ( w ) w v for homogeneous v , w ∈ Cl( g ). Then a direct computation shows: Lemma 21. F or S, T ∈ osp ( g ) , one has: a) − 2 L T = [ γ ′ ( T ) , · ] Cl( g ) . b) [ γ ′ ( S ) , γ ′ ( T )] Cl( g ) = − 2 γ ′ ([ S, T ] osp ( g ) ) . 3.1.2. Colour Quantum Weil Algebr a. With the notation ﬁxed, w e introduce the colour quan tum W eil algebra. W e regard U ( g ) and Cl( g ) as Z 2 -graded algebras, with grading induced b y that of g . Deﬁnition 22. The c olour quantum Weil algebr a asso ciated with g is the Z 2 -graded tensor pro duct W ( g ) : = U ( g ) ⊗ Cl( g ) . 16 STEFFEN SCHMIDT W e take as generators of W ( g ) the elements 1 ⊗ x and γ W ( x ) : = x ⊗ 1 − 1 2 (1 ⊗ γ ′ ( x )), where x ∈ g . Recall that γ ′ ( x ) : = γ ′ (ad x ) = q ( λ (ad x )) for any x ∈ g . This choice is natural, since the map x 7→ x ⊗ 1 − 1 2 (1 ⊗ γ ′ ( x )) deﬁnes an embedding of g in to W ( g ). W e endo w W ( g ) with a Z 2 × Z 2 -grading by setting, for x ∈ g , (3.17) bideg(1 ⊗ x ) = ( ¯ 1 , p ( x )) , bideg ( γ W ( x )) = ( ¯ 0 , p ( x )) . R emark 23 . Another common c hoice of generators is giv en by the te nsor pro duct generators u x : = x ⊗ 1 , θ x : = 1 ⊗ x, x ∈ g . They are used only in Section 6 . They are of bidegree bideg ( u x ) = ( ¯ 0 , p ( x )) and bideg ( θ x ) = ( ¯ 1 , p ( x )). F or α = ( ¯ a, ¯ b ) and β = ( ¯ c, ¯ d ) in Z 2 × Z 2 w e use the standard symmetric bic haracter (3.18) χ ( α, β ) : = ( − 1) ¯ a ¯ c + ¯ b ¯ d . F or homogeneous A, B ∈ W ( g ) of bidegrees α and β , we set (3.19) [ A, B ] W : = AB − χ ( α, β ) B A, and extend C -linearly . The total degree is the sum of the tw o comp onen ts of the bidegree in Z 2 . In particular, if A ∈ W ( g ) has total degree ¯ 1, then A 2 = 1 2 [ A, A ] W . This endows W ( g ) with the structure of a colour Lie algebra ( cf. [ Ree60 ]), which we call the c olour quantum Weil algebr a . Lemma 24. F or x, y ∈ g one has [1 ⊗ x, 1 ⊗ y ] W = 2 B ( x, y )(1 ⊗ 1) , [ γ W ( x ) , γ W ( y )] W = γ W ([ x, y ] g ) , [ γ W ( x ) , 1 ⊗ y ] W = 1 ⊗ [ x, y ] g . Pr o of. Since bideg(1 ⊗ x ) = ( ¯ 1 , p ( x )) and bideg (1 ⊗ y ) = ( ¯ 1 , p ( y )), we ha v e χ (bideg(1 ⊗ x ) , bideg (1 ⊗ y )) = − ( − 1) p ( x ) p ( y ) . Hence [1 ⊗ x, 1 ⊗ y ] W = (1 ⊗ x )(1 ⊗ y ) + ( − 1) p ( x ) p ( y ) (1 ⊗ y )(1 ⊗ x ) = 1 ⊗  xy + ( − 1) p ( x ) p ( y ) y x  = 2 B ( x, y ) (1 ⊗ 1) , b y the Cliﬀord relation. Next, [ γ W ( x ) , 1 ⊗ y ] W = γ W ( x )(1 ⊗ y ) − ( − 1) p ( x ) p ( y ) (1 ⊗ y ) γ W ( x ) = − 1 2  (1 ⊗ γ ′ ( x ))(1 ⊗ y ) − ( − 1) p ( x ) p ( y ) (1 ⊗ y )(1 ⊗ γ ′ ( x ))  = − 1 2  1 ⊗ ( γ ′ ( x ) y − ( − 1) p ( x ) p ( y ) y γ ′ ( x ))  = 1 ⊗ ( − 1 2 [ γ ′ ( x ) , y )] Cl( g ) ) = 1 ⊗ L x ( y ) = 1 ⊗ [ x, y ] g using Lemma 21 in the last steps. The identit y [ γ W ( x ) , γ W ( y )] W = γ W ([ x, y ] g ) is prov ed simi- larly . □ Perturbations of Dirac Op erators 17 On W ( g ) there exist tw o canonical op erations. The ﬁrst is the contraction ι x , an o dd deriv ation c haracterized b y the relations (3.20) ι x (1 ⊗ y ) = B ( x, y )(1 ⊗ 1) , ι x ( γ W ( y )) = − 1 2 (1 ⊗ [ x, y ] g ) , for x, y ∈ g . The second is the Lie deriv ativ e, whic h is a combination of the Lie deriv ativ e L U on U ( g ) and L C on Cl( g ): (3.21) L x ( y ⊗ z ) : = L U x ( y ) ⊗ z + y ⊗ L C x ( z ) With these deﬁnitions, the following lemma is immediate by comparison with Lemma 24 . Lemma 25. F or x ∈ g one has: a) ι x = 1 2 [1 ⊗ x, · ] W . b) L x = [ γ W ( x ) , · ] W . Pr o of. a) This is immediate from the deﬁnition of ι x and the comm utator ( 3.19 ). b) It suﬃces to chec k the identit y on the generators of W ( g ). F or y ∈ g , L x (1 ⊗ y ) = 1 ⊗ L C x y = 1 ⊗  − 1 2 [ γ ′ ( x ) , y ] Cl( g )  = [ γ W ( x ) , 1 ⊗ y ] W , using [ x ⊗ 1 , 1 ⊗ y ] W = 0 and Lemma 24 . Moreo v er, L x ( γ W ( y )) = L x  y ⊗ 1 − 1 2 (1 ⊗ γ ′ ( y ))  = [ x, y ] g ⊗ 1 − 1 2  1 ⊗ L x γ ′ ( y )  = [ x, y ] g ⊗ 1 + 1 4  1 ⊗ [ γ ′ ( x ) , γ ′ ( y )] Cl( g )  = [ x, y ] g ⊗ 1 − 1 2  1 ⊗ γ ′ ([ x, y ] g )  = γ W ([ x, y ] g ) = [ γ W ( x ) , γ W ( y )] W , where we used Lemma 21 in the p enultimate step and Lemma 24 in the last step. Since the generators span W ( g ) as an algebra, this implies L x = [ γ W ( x ) , · ] W . □ 3.2. Cubic Dirac Op erator. W e construct the cubic Dirac op erator for quadratic Lie sup er- algebras, including, in particular, all basic classical Lie sup eralgebras. Recall that any quadr atic Lie sup er algebr a g is equipp ed with a non-degenerate inv ariant sup ersymmetric bilinear form B . In what follows, w e ﬁx such a pair ( g , B ). Then the cubic Dirac op erator is a distinguished ele- men t of the colour quantum W eil algebra W ( g ), obtained by adding to the canonical quadratic elemen t the Cliﬀord quantization of the structure-constants tensor asso ciated with ( g , B ). The resulting elemen t is inv arian t under the diagonal adjoint action of g and enco des the full Lie sup eralgebraic structure of g . 3.2.1. Structur e Constants T ensor. Let { e a } b e a homogeneous basis of g with B -dual basis { e a } , that is, B ( e a , e b ) = δ ab . The bilinear form B identiﬁes g ∼ = g ∗ and deﬁnes the Cartan 3-form (3.22) φ g ( x, y , z ) : = − 1 2 B  x, [ y , z ]  , x, y , z ∈ g . By inv ariance and sup ersymmetry of B , the form φ g is g -inv arian t and totally skew- sup ersymmetric. Via the identiﬁcation V ( g ) → Cl( g ) ( cf. 20 ), φ g determines a canonical cubic elemen t φ g ∈ V 3 ( g ) ¯ 0 c haracterized b y (3.23) ( φ g , x ∧ y ∧ z ) ∧ = − 1 2 B ([ x, y ] , z ) 18 STEFFEN SCHMIDT for all x, y , z ∈ g . Here ( · , · ) ∧ denotes the non-degenerate sup ersymmetric bilinear form on V ( g ) in tro duced in Section 3.1.1 . The element φ g is called the structur e c onstants tensor of g . If g is clear from the context, we omit the subscript g . Lemma 26. F or al l x, y , z ∈ g , the fol lowing hold: a) 2 ι x φ = − λ (ad x ) for al l x ∈ g . b) ι x ι y φ = B ([ x, y ] , · ) . In p articular, under the identiﬁc ation g ∼ = g ∗ , ι x ι y φ = [ x, y ] . c) In the homo gene ous b asis, φ = − 1 12 X a,b,c ( − 1) p ( e a ) p ( e b )+ p ( e c ) f abc e a ∧ e b ∧ e c . d) φ is invariant under the adjoint action of g . Pr o of. a) F or homogeneous x, y , z , we obtain using Lemma 18 ( ι x φ )( y ∧ z ) = φ ( x, y , z ) = − 1 2 B ([ x, y ] , z ) = − 1 2 ω ad x ( y , z ) = − 1 2 ( λ (ad x )) ♭ ( y ∧ z ) . Hence 2( ι x φ ) ♭ = − ( λ (ad x )) ♭ . Using the B -iden tiﬁcation V 2 ( g ) ∼ = V 2 ( g ∗ ), w e ﬁnally obtain the statemen t. b) One has by deﬁnition: ι x ι y φ ( z ) = ( ι x ι y φ, z ) ∧ = ( − 1) p ( x ) p ( y ) ( ι y φ, x ∧ z ) ∧ = ( − 1) p ( x ) p ( y ) ( φ, y ∧ x ∧ z ) ∧ = − ( − 1) p ( x ) p ( y ) B ([ y , x ] , z ) = B ([ x, y ] , z ) . c) is a straightforw ard computation and will b e omitted. d) One has φ ( x, y , z ) = − 1 2 B ([ x, y ] , z ). F or homogeneous w , x, y , z , the Lie deriv ativ e on forms giv es ( L w φ )( x, y , z ) = φ ([ w , x ] , y , z ) + ( − 1) p ( w ) p ( x ) φ ( x, [ w , y ] , z ) + ( − 1) p ( w )( p ( x )+ p ( y )) φ ( x, y , [ w , z ]) . Hence, using inv ariance of B , one has for an y w ∈ g − 2( L w φ )( x, y , z ) = B ([[ w, x ] , y ] , z ) + ( − 1) p ( w ) p ( x ) B ([ x, [ w , y ]] , z ) + ( − 1) p ( w )( p ( x )+ p ( y )) B ([ x, y ] , [ w , z ]) = B ([ w , x ] , [ y , z ]) + ( − 1) p ( w ) p ( x ) B ( x, [[ w , y ] , z ]) + ( − 1) p ( w )( p ( x )+ p ( y )) B ( x, [ y , [ w , z ]]) = ( − 1) p ( w ) p ( x )  − B ( x, [ w , [ y , z ]]) + B ( x, [[ w , y ] , z ]) + ( − 1) p ( w ) p ( y ) B ( x, [ y , [ w , z ]])  . By the Z 2 -graded Jacobi identit y [ w , [ y , z ]] = [[ w , y ] , z ] + ( − 1) p ( w ) p ( y ) [ y , [ w , z ]], the parentheses v anish. Therefore L w φ = 0 for all w , i.e. , φ is inv ariant under the adjoint action of g . □ W e are interested in the quan tization of φ , that is, (3.24) φ ′ : = q ( φ ) ∈ Cl( g ) . Theorem 27 ([ K C21 , Theorem 1.1]) . The squar e of φ ′ is a c onstant given by ( φ ′ ) 2 = 1 24 str(ad g (Ω g )) . Perturbations of Dirac Op erators 19 Whenev er g is basic classical, it is useful to rewrite the prop osition using the F reuden thal–de V ries formula (see [ Mey22 ]). Fix a p ositiv e ro ot system of g , and denote b y ρ the asso ciated W eyl vector, that is the half sum of the p ositiv e ro ots. Then, the F reudenthal–de-V ries form ula is (3.25) 1 24 str(ad g (Ω g )) = B ( ρ, ρ ) , and the square of φ ′ equals B ( ρ, ρ ). 3.2.2. Cubic Dir ac Op er ator. Let { e a } b e a homogeneous basis of g with B -dual basis { e a } . Note that p ( e a ) = p ( e a ) for all a . The cubic Dir ac op er ator of g is (3.26) D g : = X a e a ⊗ e a + 1 ⊗ φ ′ ∈ W ( g ) . Lemma 28. The cubic Dir ac op er ator D g satisﬁes the fol lowing pr op erties: a) The deﬁnition of D g is indep endent of the choic e of a b asis. b) D g is g -invariant, that is, L x D g = 0 . c) D g has bide gr e e ( ¯ 1 , ¯ 0) , i.e. , it is an o dd element of W ( g ) r elative to the total de gr e e. Pr o of. a) W e ﬁx tw o bases { e a } and { f a } of g with B -dual bases { e a } and { f a } , resp ectiv ely . W e can express any x ∈ g by x = P a B ( x, f a ) f a = P a B ( f a , x ) f a for any x ∈ g such that X a e a ⊗ e a = X a,b,c B ( f b , e a ) B ( e a , f c ) f b ⊗ f c = X b,c B X a B ( f b , e a ) e a , f c ! f b ⊗ f c = X b,c B ( f b , f c ) f b ⊗ f c = X b f b ⊗ f b . In particular, the deﬁnition D g is indep endent of the choice of a basis. b) One has for any x ∈ g L x X a e a ⊗ e a ! = X a [ x, e a ] ⊗ e a + X a ( − 1) p ( x ) p ( e a ) e a ⊗ [ x, e a ] = X a,b B ( e b , [ x, e a ]) e b ⊗ e a + X a,b ( − 1) p ( x ) p ( e a ) B ([ x, e a ] , e b ) e a ⊗ e b = X a,b B ( e b , [ x, e a ]) e b ⊗ e a − X a,b B ( e a , [ x, e b ]) e a ⊗ e b = 0 , where we used in v ariance and sup ersymmetry of B . Moreov er, φ ′ is g -inv arian t b y Lemma 26 and Theorem 20 . This completes the pro of of b). c) By Lemma 26 and the deﬁnition of q ( cf. Theorem 20 ), it is immediate that φ has bidegree ( ¯ 1 , ¯ 0) and total degree ¯ 1. Moreov er, parit y is additiv e on simple tensors, e.g. p ( e a ⊗ e a ) = p ( e a ⊗ 1) + p (1 ⊗ e a ), so P a e a ⊗ e a has bidegree ( ¯ 1 , ¯ 0) and hence total degree ¯ 1. Therefore D g , b eing a sum of homogeneous elements of total degree ¯ 1, is itself o dd. □ The cubic Dirac op erator D g has a particularly simple square. This theorem also app ears in [ Mey22 ]. W e give an alternative pro of using the quantum W eil algebra. 20 STEFFEN SCHMIDT Theorem 29. The squar e of the cubic Dir ac op er ator satisﬁes D 2 g = Ω g ⊗ 1 + 1 24 str g  ad g (Ω g )  (1 ⊗ 1) wher e Ω g denotes the quadr atic Casimir element of g . Pr o of. W e ﬁrst sho w that D 2 g ∈ U ( g ) ⊗ 1. It suﬃces to c heck ι x D 2 g = 0 for all x ∈ g . Indeed, ι x D 2 g = 1 2 [ ι x D g , D g ] W = − 1 4 [ γ ′ ( x ) , D g ] W = 1 2 L x D g = 0 , since D g is g -inv ariant. Next we identify D 2 g inside U ( g ). Let π : W ( g ) → U ( g ) b e the pro jection id U ( g ) ⊗  , where  : Cl( g ) = C · 1 ⊕ Cl > 0 ( g ) → C · 1. F or brevity write s ( x ) : = x ⊗ 1. By Prop osition 27 , D 2 g mo d ker π = X a,b ( − 1) p ( e a ) p ( e b ) s ( e a ) s ( e b ) e a e b + ( φ ′ ) 2 mo d k er π = X a,b ( − 1) p ( e b ) s ( e a ) s ( e b ) B ( e a , e b ) + 1 24 str g  ad g (Ω g )  mo d k er π = Ω g ⊗ 1 + 1 24 str g  ad g (Ω g )  mo d k er π , where we used the graded pro duct in W ( g ) and the Cliﬀord identit y e a e b = 1 2  e a e b + ( − 1) p ( e a ) p ( e b ) e b e a  + 1 2 [ e a , e b ] = B ( e a , e b ) + 1 2 [ e a , e b ] in Cl( g ) , together with P a B ( e a , e b ) e a = e b . This completes the pro of. □ R emark 30 . Using the F reuden thal–de V ries form ula ( 3.25 ), for a basic classical Lie sup eralgebra g with W eyl vector ρ one can rewrite the square as D 2 g = Ω g ⊗ 1 + B ( ρ, ρ ) . The following lemma will b e used in subsequent constructions. Lemma 31. In W ( g ) , the fol lowing c ommutation r elations hold: [D g , D g ] W = 2(Ω g ⊗ 1 + B ( ρ, ρ )(1 ⊗ 1)) , [ γ W ( x ) , D g ] W = 0 for al l x ∈ g . In addition, if x ∈ g is even, one has [1 ⊗ x, D g ] W = 2 γ W ( x ) . Pr o of. The cubic Dirac op erator D g ∈ W ( g ) has bidegree ( ¯ 1 , ¯ 0), that is, total degree ¯ 1. Hence [D g , D g ] W = 2 D 2 g = 2  Ω g ⊗ 1 + B ( ρ, ρ )(1 ⊗ 1)  . The commutation relation [ γ W ( x ) , D g ] W = 0 for all x ∈ g follows from the iden tit y L x = [ γ W ( x ) , · ] W in W ( g ) (see Lemma 25 ). Since D g is g -in v ariant, the claim follo ws. Next, for x ∈ g ¯ 0 the element 1 ⊗ x is o dd with resp ect to the total degree. Since b oth D g and 1 ⊗ x are o dd, we obtain [1 ⊗ x, D g ] W = 2 ι x D g = 2  ι x ( X a e a ⊗ e a ) + 1 ⊗ q ( ι x φ )  = 2  x ⊗ 1 − 1 2  1 ⊗ q ( λ (ad x ))   = 2 γ W ( x ) , Perturbations of Dirac Op erators 21 where we used ι x  X a e a ⊗ e a  = X a ( − 1) p ( e a ) p ( x ) e a ⊗ B ( x, e a ) = X a ( − 1) p ( e a ) B ( x, e a ) e a ⊗ 1 = x ⊗ 1 , since B ( x, e a ) = 0 unless p ( x ) = p ( e a ). □ 3.3. Relativ e Cubic Dirac Op erator and W eil Diﬀerential. W e deﬁne relativ e cubic Dirac op erators with resp ect to quadratic Lie subsup eralgebras l of basic classical Lie sup eralgebras g . This generalizes the constructions in the literature in the case where l is a Levi subsup eralgebra; see [ HP05 , NSS26 ]. Let g b e a basic classical Lie sup eralgebra and let l ⊂ g b e a quadratic Lie subsup eralgebra, with non-degenerate sup ersymmetric inv ariant bilinear form B l : = B | l . W e refer to ( g , l ) as a quadr atic p air . F or a quadratic pair ( g , l ), there is an orthogonal decomp osition (3.27) g = l ⊕ p , p = l ⊥ , with resp ect to B . The restriction B p : = B | p is non-degenerate. In particular, one can form the Cliﬀord sup eralgebra Cl( p ) : = Cl( p ; B p ). F or simplicit y , assume that dim p = 2( p 0 + p 1 ) is even, where 2 p 0 = dim p ¯ 0 and 2 p 1 = dim p ¯ 1 . By inv ariance of B , one has [ l , p ] ⊆ p . As ab o v e, denote b y D g ∈ W ( g ) and D l ∈ W ( l ) the cubic Dirac op erators asso ciated with g and l , resp ectively . The relative cubic Dirac op erator is obtained by combining these in to a single op erator via a natural embedding j : W ( l ) − → W ( g ) , induced by the adjoint action of l on p . F or an y x ∈ l , ad x ∈ osp ( g ) decomp oses as the sum of ad l x ∈ osp ( l ) and ad p x ∈ osp ( p ). Set γ ′ l ( x ) : = − 1 2 q ( λ (ad l x )) and ν ∗ ( x ) : = γ ′ p ( x ) : = − 1 2 q ( λ (ad p x )) ∈ Cl( p ). The latter deﬁnes a Lie sup eralgebra homomorphism ν ∗ : l → Cl( p ). This map has an explicit description. As the dimension of p is even, we can decomp ose p in isotropic sup er subspaces p = u ⊕ u compatible with the c hoice of a ﬁxed p ositiv e system, that is u ⊂ n + and u ⊂ n − . Let ∆ l ⊂ ∆ denote the asso ciated ro ot system, and set ∆ + l : = ∆ l ∩ ∆ + . The w eigh ts of u and u then b elong to ∆ + p : = ∆ + ∩ ∆ \ ∆ l and ∆ − p : = ∆ − ∩ ∆ \ ∆ l . Set ∆ + p , ¯ 0 : = ∆ + p ∩ ∆ ¯ 0 and ∆ + p , ¯ 1 : = ∆ + p ∩ ∆ ¯ 1 . Finally , set ρ u : = 1 2 P α ∈ ∆ + p , ¯ 0 α − 1 2 P α ∈ ∆ + p , ¯ 1 α . Lemma 32. The map ν ∗ : l → Cl( p ) is a Lie sup er algebr a homomorphism. In p articular, if u 1 , . . . , u p is a b asis of u and u 1 , . . . , u p is a b asis of u , then ν ∗ ( X ) = 1 2 p X j,k =1 ( − 1) p ( u j ) B ( X , [ u j , u k ]) u k u j +    ρ u ( X ) , if X ∈ h , 0 , else . Pr o of. The pro of is verbatim the pro of of Lemma 2.3.7 in [ NSS26 ]. □ T o distinguish the generators of W ( l ) from those of W ( g ), we write γ W ( l ) instead of γ W when necessary . Consider the natural embedding j : W ( l ) → W ( g ) deﬁned on generators by (3.28) j (1 ⊗ x ) = 1 ⊗ x, j ( γ W ( l ) ( x )) = γ W ( x ) , x ∈ l . Prop osition 33. The fol lowing hold: a) The map j : W ( l ) → W ( g ) is a Lie sup er algebr a homomorphism. 22 STEFFEN SCHMIDT b) j : W ( l ) → W ( g ) intertwines with c ontr actions and Lie derivatives, that is, ι x ◦ j = j ◦ ι x , L x ◦ j = j ◦ L x ∀ x ∈ l . Pr o of. a) This is immediate from [ γ W ( x ) , γ W ( y )] W = γ W ([ x, y ] g ) for x, y ∈ g (Lemma 24 ). b) W e prov e the statement for Lie deriv ativ es L x for x ∈ g . It is enough to prov e the statemen t on the generators. One has ( L x ◦ j )(1 ⊗ y ) = L x (1 ⊗ y ) = 1 ⊗ L x ( y ) = ( j ◦ L x )(1 ⊗ y ) . Moreo v er, using that γ W and ν ∗ are Lie sup eralgebra homomorphisms, and that L x = [ γ W ( x ) , · ] W b y Lemma 25 , one obtains with Lemma 24 for all x ∈ l : ( L x ◦ j )( γ W ( l ) ( y )) = [ γ W ( x ) , j ( γ W ( l ) ( y ))] W = [ γ W ( x ) , γ W ( y )] W = γ W ([ x, y ] g ) = ( j ◦ L x )( γ W ( l ) ( x )) . That j intert wines with contractions is another direct calculation and will b e omitted. □ Ha ving ﬁxed the notation, deﬁne the r elative cubic Dir ac op er ator by (3.29) D g , l : = D g − j (D l ) ∈ W ( g ) . In W ( g ), set W ( g , l ) : = ( U ( g ) ⊗ Cl( p )) l , the l -basic subalgebra of the colour quan tum W eil algebra, that is, the subalgebra consisting of all elements annihilated by L x and ι x for all x ∈ l . Theorem 34. The fol lowing hold: a) The op er ator D g , l lies in W ( g , l ) . b) One has D 2 g , l = Ω g − j (Ω l ) + 1 24 str g (Ω g ) − 1 24 str l (Ω l ) . Pr o of. a) Note γ ′ ( x ) = − 1 2 q ( λ (ad x )), so that γ ′ ( x ) = γ ′ l ( x ) + γ ′ p ( x ). One ﬁrst veriﬁes that ι x D g = γ W ( x ) . Indeed, ι x D g = X a e a ⊗ ( − 1) p ( x ) p ( e a ) B ( x, e a ) + 1 ⊗ ι x φ g = X a ( − 1) p ( e a ) B ( x, e a ) e a ⊗ 1 + q ( ι x φ g ) = x ⊗ 1 − 1 2  1 ⊗ q ( λ (ad x ))  , where the identit y 2 ι x φ g = − λ (ad x ) from Lemma 26 is used. By Lemma 33 , the Lie sup eralgebra homomorphism j : W ( l ) → W ( g ) intert wines con tractions and Lie deriv ativ es. Hence, by deﬁnition of j , ι x D g , l = ι x D g − j ( ι x D l ) = γ W ( x ) − j ( γ W ( l ) ( x )) = 0 . Moreo v er, L x D g , l = 0 for all x ∈ l follows directly from the l -inv ariance of the cubic Dirac op erators; see Lemma 28 . b) In a), w e ha v e prov en that D g , l lies in W ( g , l ). In particular, it comm utes with the image of j since [ j (1 ⊗ x ) , D g , l ] W = 2 ι x D g , l = 0 , [ j ( γ W ( l ) ( x )) , D g , l ] W = [ γ W ( x ) , D g , l ] W = L x D g , l = 0 Perturbations of Dirac Op erators 23 where we used ι x = 1 2 [1 ⊗ x, · ] W b y the Cliﬀord relations, and Lemma 25 . Thus, [D g , j (D l )] W = 0. As D g , l is an o dd element in W ( g , l ), w e conclude D 2 g , l = 1 2 [D g , l , D g , l ] W = 1 2 [D g , D g ] W − 1 2 j ([D l , D l ] W ) = D 2 g − j (D 2 l ) . The statement now follows with Theorem 29 . □ 3.3.1. Weil Diﬀer ential. W e grade U ( g ) ⊗ Cl( p ) by the Z 2 -grading of Cl( p ), so that ( U ( g ) ⊗ Cl( p )) ¯ s : = U ( g ) ⊗ Cl( p ) ¯ s . Via the diagonal embedding l → U ( g ) ⊗ Cl( p ), the adjoint action mak es U ( g ) ⊗ Cl( p ) into an l -sup ermo dule. The l -in v ariants then coincide with W ( g , l ). The space W ( g , l ) is endo wed with the obvious Z 2 × Z 2 -grading from W ( g ), making it into a colour Lie algebra. Since D g , l has a simple square, it induces a diﬀerential on W ( g , l ). F or homogeneous A ∈ U ( g ) ⊗ Cl( p ) set (3.30) ( d ′ ) W ( A ) : = [D g , l , A ] = D g , l A − ( − 1) p ( A ) A D g , l . As D g , l is l -inv arian t, ( d ′ ) W preserv es W ( g , l ) and restricts to d W : W ( g , l ) → W ( g , l ) . This op erator is nilp oten t, ( d W ) 2 = 0 (see [ K C21 ]), and compatible with the Lie deriv ative and con traction of W ( g ) deﬁned in ( 3.20 ) and ( 3.21 ). Lemma 35. The diﬀer ential d W is c omp atible with L x and ι x for any x ∈ l , that is, on W ( g , l ) d W L x − ( − 1) p ( x ) L x d W = 0 , d W ι x + ( − 1) p ( x ) ι x d W = L x . Mor e over, d W is unique with this pr op erty. Pr o of. W e note that L x = [ γ W ( x ) , · ] W . If we write d W = ad W D g , l and L x = ad W γ W ( x ) , then d W L x − ( − 1) p ( x ) L x d W = ad W D g , l ad W γ W ( x ) − ( − 1) p ( x ) ad W γ W ( x ) ad W D g , l = ad W [ γ W ( x ) , D g , l ] W = ad W L x D g , l = 0 where we used l -inv ariance of D g , l in the last equalit y . Similarly , one pro v es d W ι x + ( − 1) p ( x ) ι x d W = L x . □ Since ( d W ) 2 = 0, we may form its cohomology ker d W / im d W . Recall that Z ( l ) denotes the cen ter of the univ ersal env eloping subalgebra of l . Let Z ( l ∆ ) denote the image of Z ( l ) in W ( g , l ) under the diagonal embedding. Then: Theorem 36 ([ K C21 , Theorem 6.2]) . One has ker d W = Z ( l ∆ ) ⊕ im d W . Equivalently, the c ohomolo gy of d W is isomorphic to Z ( l ∆ ) ∼ = Z ( l ) . 3.4. Oscillator Sup ermo dule. At this stage the Dirac op erator is only a ” Dir ac element “ D g , l in the W eil algebra W ( g ). T o extract represen tation-theoretic information, we view it as an o dd endomorphism of a suitable represen tation space. F or this, one must c ho ose a Cl( p )-supermo dule on which the Cliﬀord generators act. The oscil lator sup ermo dule provides the canonical choice (after ﬁxing a p olarization p = u ⊕ u ), realizing the Cliﬀord action by creation and annihilation op erators. W e brieﬂy introduce it and refer to [ NSS26 ] for all details. Let g b e a basic classical Lie sup eralgebra. W e adapt the notation from Section 3.3 and ﬁx a quadratic Lie subsup eralgebra ( l , B l : = B | l ) ⊂ ( g , B ) suc h that g = l ⊕ p , h ⊂ l and p is 24 STEFFEN SCHMIDT ev en-dimensional. Since we work o ver C and B p is non-degenerate, there exist subsup erspaces u , u ⊂ p with u ⊂ n − and u ⊂ n + suc h that (3.31) p = u ⊕ u , u = u ⊥ , u = u ⊥ , with resp ect to B p , i.e. , u and u are complementary Lagrangian subsup erspaces. Moreov er, p decomp oses in to its ev en and o dd parts as p = p ¯ 0 ⊕ p ¯ 1 , and this splitting is compatible with the p olarization: (3.32) p ¯ 0 = u ¯ 0 ⊕ u ¯ 0 , p ¯ 1 = u ¯ 1 ⊕ u ¯ 1 . Note that these decomp ositions are not orthogonal direct sums with resp ect to B . Recall the deﬁnitions of ∆ + l and ρ u in Section 3.3 . W e consider the Cliﬀord sup eralgebra Cl( p ) = Cl( p ¯ 0 ) ⊗ Cl( p ¯ 1 ) . T o construct the oscillator sup ermo dule, we treat the Cliﬀord algebra Cl( p ¯ 0 ) and the W eyl algebra Cl( p ¯ 1 ) separately . First, consider the Cliﬀord algebra Cl( p ¯ 0 ). The subspaces u ¯ 0 and u ¯ 0 are isotropic and com- plemen tary in p ¯ 0 with resp ect to B p . Fix a basis u 1 , . . . , u p 0 of u ¯ 0 and u 1 , . . . , u p 0 of u ¯ 0 suc h that (3.33) B ( u j , u i ) = B ( u i , u j ) = δ ij (1 ≤ i, j ≤ p 0 ) . Via B p w e iden tify u ¯ 0 ∼ = u ∗ ¯ 0 , so that the dual basis element u ∗ i corresp onds to u i . W e consider the exterior (spinor) mo dules (3.34) S g , l : = ^ u ¯ 0 , S g , l : = ^ u ¯ 0 . W e fo cus on S g , l ; the discussion for S g , l is analogous. On V u 0 , w e hav e a natural action of p ¯ 0 , where u ∈ u ¯ 0 acts as the left exterior m ultiplication  ( u ), and u ∈ u ¯ 0 acts as the con traction ι ( u ) deﬁned in Equation ( 3.5 ). By the universal prop ert y of the Cliﬀord sup eralgebra Cl( p ¯ 0 ), this extends to an action of Cl( p ¯ 0 ) on S g , l , whic h realizes S g , l as a Cl( p ¯ 0 )-mo dule, called the spin mo dule . The follo wing lemma is standard. Lemma 37. The spin mo dule S g , l is the unique simple Cl( p ¯ 0 ) -mo dule, up to isomorphism. A dditional ly, S g , l c ontains a highest weight ve ctor with r esp e ct to ∆ + l , whose weight is ρ u ¯ 0 . W e endow S g , l with the Z 2 -grading induced from the tensor algebras T ( u ¯ 0 ), so that S g , l = S g , l ¯ 0 ⊕ S g , l ¯ 1 . Moreov er, w e equip the spin module S g , l with a non-degenerate Hermitian form ⟨· , ·⟩ S g , l for which u i and u i are mutually adjoint. T o this end, ﬁx a θ -real form g R suc h that B θ ( x, y ) : = − B ( x, θ ( y )) is p ositive deﬁnite on g R , and extend B θ sesquilinearly to g , hence to p . W e may choose the dual bases u 1 , . . . , u p 0 ∈ u ¯ 0 and u 1 , . . . , u p 0 ∈ u ¯ 0 orthonormal for B θ , so that θ ( u i ) = − u i and θ ( u i ) = − u i . The induced Hermitian form on tensor p ow ers T n ( u ¯ 0 ) descends to a Hermitian form on S g , l = V u ¯ 0 , denoted ⟨· , ·⟩ S g , l . In particular: Lemma 38 ([ NSS26 , Lemma 3.2.3]) . The fol lowing holds: Perturbations of Dirac Op erators 25 a) ⟨· , ·⟩ S g , l is a sup er p ositive deﬁnite sup er Hermitian form, that is, ⟨ v , w ⟩ S g , l = ( − 1) p ( v ) p ( w ) ⟨ w , v ⟩ S g , l , v , w ∈ S g , l , ⟨ v , w ⟩ S g , l = 0 whenever p ( v )  = p ( w ) , and ⟨· , ·⟩ S g , l is p ositive deﬁnite on S g , l ¯ 0 and − i -times p ositive deﬁnite on S g , l ¯ 1 . b) The adjoint of u i with r esp e ct to ⟨· , ·⟩ S g , l is θ ( u i ) = − u i , and the adjoint of u i is θ ( u i ) = − u i . W e next consider the W eyl algebra Cl( p ¯ 1 ). The bilinear form B | p ¯ 1 is symplectic, and p ¯ 1 = u ¯ 1 ⊕ u ¯ 1 is a polarization in to maximal isotropic subspaces. Accordingly , let ( X , Y ) b e either ( u ¯ 1 , u ¯ 1 ) or ( u ¯ 1 , u ¯ 1 ). Fix dual bases e 1 , . . . , e p 1 of X and f 1 , . . . , f p 1 of Y , and set W ( g ¯ 1 ) := Cl( p ¯ 1 ). Then W ( g ¯ 1 ) acts on C [ X ] = Sym( X ) by multiplication by elements of X and by contraction by elemen ts of Y , determined by (3.35) f i · e j = B p ( f i , e j ) (1 ≤ i, j ≤ p 1 ) . The W ( g ¯ 1 )-mo dule C [ X ] is called the oscillator mo dule. If v ∈ C [ X ] is annihilated by all f i , then v ∈ C . Hence C [ X ] is simple. W e write (3.36) M ( p ¯ 1 ) := C [ u ¯ 1 ] = Sym( u ¯ 1 ) , M ( p ¯ 1 ) := C [ u ¯ 1 ] = Sym( u ¯ 1 ) . In the sequel, M ( p ¯ 1 ) will b e used. The corresp onding statements for M ( p ¯ 1 ) are obtained in the same wa y . W e endo w M ( p ¯ 1 ) with the Z 2 -grading by p olynomial degree mo dulo 2. It is useful to give an equiv alen t formulation of the W eyl algebra and the oscillator mo dule. Fix bases ∂ 1 , . . . , ∂ p 1 of u ¯ 1 and x 1 , . . . , x p 1 of u ¯ 1 suc h that (3.37) B ( x k , ∂ l ) = 1 2 δ kl . Iden tifying ∂ k with ∂ /∂ x k , the W eyl algebra W ( g ¯ 1 ) = Cl( p ¯ 1 ) b ecomes the algebra of p olynomial diﬀeren tial op erators in the v ariables x 1 , . . . , x p 1 , with relations (3.38) [ x k , x l ] W = 0 , [ ∂ k , ∂ l ] W = 0 , [ x k , ∂ l ] W = δ kl . Hence W ( g ¯ 1 ) acts naturally on M ( p ¯ 1 ) = C [ x 1 , . . . , x p 1 ]. The space M ( p ¯ 1 ) = C [ x 1 , . . . , x p 1 ] carries the (Bar gmann–/Fischer–)F o ck Hermitian form ⟨· , ·⟩ M ( p ¯ 1 ) uniquely determined by (3.39) D p 1 Y k =1 x p k k , p 1 Y k =1 x q k k E M ( p ¯ 1 ) =    Q p 1 k =1 p k ! if p k = q k for all k , 0 otherwise . It is p ositive deﬁnite and consisten t, i.e. , ⟨ M ( p ¯ 1 ) 0 , M ( p ¯ 1 ) 1 ⟩ M ( p ¯ 1 ) = 0. Moreo ver, for v , w ∈ M ( p ¯ 1 ) and 1 ≤ k ≤ p 1 one has (3.40) ⟨ ∂ k v , w ⟩ M ( p ¯ 1 ) = ⟨ v , x k w ⟩ M ( p ¯ 1 ) , ⟨ x k v , w ⟩ M ( p ¯ 1 ) = ⟨ v , ∂ k w ⟩ M ( p ¯ 1 ) , that is, the adjoint of x k is ∂ k , and the adjoint of ∂ k is x k for all 1 ≤ k ≤ p 1 . Finally , we deﬁne the oscil lator sup ermo dules ov er Cl( p ) by (3.41) M ( p ) : = M ( p ¯ 1 ) ⊗ S g , l , M ( p ) : = M ( p ¯ 1 ) ⊗ S g , l 26 STEFFEN SCHMIDT and consider them as Cl( p )-sup ermo dules, with the Z 2 -grading induced by that of S g , l and S g , l . W e conclude by recording basic prop erties of M ( p ) (the mo dule M ( p ) is analogous). Prop osition 39 ([ NSS26 ]) . The fol lowing hold: a) The Cl( p ) -sup ermo dule M ( p ) is simple. b) The set of weights of M ( p ) is P M ( p ) = { ρ u − Z + [ A ] : A ⊂ ∆ + \ ∆ + l } . c) M ( p ) c arries a p ositive deﬁnite sup er Hermitian form. d) Under the action of l on M ( p ) induc e d by L emma 32 , M ( p ) is a unitarizable l - sup ermo dule. Mor e over, ther e ar e l -sup ermo dule isomorphisms M ( p ) ∼ = ^ u ⊗ C ρ u ∼ = ^ u ¯ 0 ⊗ S( u ¯ 1 ) ⊗ C ρ u ; i.e. , the action of l on M ( p ) induc e d by ν ∗ and the adjoint action of l on M ( p ) diﬀers by a shift of ρ u . Let M b e a g -sup ermo dule. The l -relativ e cubic Dirac op erator D g , l is naturally an element of W ( g , l ), and hence, via the induced action, an o dd op erator acting on M ⊗ M ( p ), where w e equip M ⊗ M ( p ) with the Z 2 -grading of M ( p ). 3.5. Dirac Cohomology. W e deﬁne Dirac cohomology for the relative cubic Dirac op erator D g , l , where l is the quadratic subalgebra from Section 3.3 , and we record the main results of [ NSS26 ]. Although stated there for parab olic subalgebras, the arguments carry ov er verbatim to our setting. Giv en an endomorphism T ∈ End C ( V ) of a sup er vector space V , we deﬁne its cohomology b y (3.42) H( T ; V ) : = k er T / (im T ∩ ker T ) . W e apply this to the (relativ e) cubic Dirac op erator. Let M b e a g -supermo dule and let M ( p ) b e the oscillator sup ermo dule from Section 3.4 . On the U ( g ) ⊗ Cl( p )-sup ermo dule M ⊗ M ( p ), the op erator D g , l acts comp onen t wise. Moreo ver, D g , l sup ercomm utes with l , so ker D g , l is naturally an l -sup ermo dule. Deﬁnition 40. The Dir ac c ohomolo gy of a g -sup ermo dule M (with resp ect to D g , l ) is the l -sup ermo dule H D g , l ( M ) : = H(D g , l ; M ⊗ M ( p )) = ker D g , l  k er D g , l ∩ im D g , l  . W e are particularly in terested in admissible ( g , l )-sup ermo dules, i.e. , g -sup ermo dules that are l -semisimple. F or these sup ermo dules, the Dirac cohomology is l -semisimple by Section 2.4 , as D g , l comm utes with the action of l . Lemma 41. L et M b e an admissible ( g , l ) -sup ermo dule. Then H D g , l ( M ) is a semisimple l - sup ermo dule, that is, H D g , l ( M ) is c ompletely r e ducible as an l -sup ermo dule. Perturbations of Dirac Op erators 27 Giv en the square form ula of Theorem 34 , Dirac cohomology is most eﬀective for sup ermo dules with inﬁnitesimal character. The following sup er-analogue of the Casselman–Osb orne theorem is due to [ NSS26 ]. Theorem 42 ([ NSS26 ]) . L et HC g , l denote the Harish–Chandr a monomorphism and let res : S W g → S W l b e r estriction. Then ther e exists an algebr a homomorphism η l : Z ( g ) → Z ( l ) such that for every z ∈ Z ( g ) one c an write (3.43) z ⊗ 1 = η l ( z ) + D g , l A + A D g , l for some A ∈ W ( g , l ) , and the diagr am Z ( g ) Z ( l ) S W g S W l η l H C g H C l res c ommutes. In p articular, if M has inﬁnitesimal char acter χ λ , then z ⊗ 1 acts on H D g , l ( M ) as η l ( z ) ⊗ 1 . Conse quently, if V ⊂ H D g , l ( M ) is an l -subsup ermo dule with inﬁnitesimal char acter χ l µ , then χ λ = χ l µ ◦ η l . Moreo v er, H D g , l ( M ) is trivial unless M is a highest w eigh t supermo dule, and H D g , l ( M ) embeds in to Kostan t-type cohomology; see [ NSS26 ]. Explicit computations app ear in [ Sc h24 , NSS26 ]. 3.6. The Unitary Case. Of particular interest are the unitarizable highest weigh t sup ermo d- ules o ver basic classical Lie sup eralgebras. As shown in [ CFV20 ], these admit geometric real- izations as spaces of sections of holomorphic sup er vector bundles ov er Hermitian sup erspaces. F or such sup ermo dules the Dirac op erator is selfadjoin t, and the resulting Dirac cohomology is esp ecially simple. W e record the main statements and refer to [ NSS26 , Section 5] for pro ofs. W e retain the notation introduced ab ov e. Theorem 43. L et ( H , ⟨· , ·⟩ H ) b e a unitarizable g -sup ermo dule. Then: a) D g , l is selfadjoint with r esp e ct to ⟨· , ·⟩ H⊗ M ( p ) ; in p articular ker D g , l = k er D k g , l for al l k ∈ Z + . b) If H is simple, then H ⊗ M ( p ) = ker D g , l ⊕ im D g , l ; in p articular H D g , l ( H ) = ker D g , l . 4. Semisimple Per turba tions W e introduce a family of cubic Dirac op erators and asso ciated Laplace op erators, parametrized b y elements of the real span of the ro ot system, extending the standard cubic Dirac op erator. F or eac h ﬁnite-dimensional sup ermo dule, the Laplace op erator asso ciated to g ¯ 0 determines a ﬁnite collection of G ¯ 0 -orbits enco ding the g ¯ 0 -decomp osition. On the other hand, comparison of a family of Laplace op erators for g with the square of the cubic Dirac op erator gives energy op erators detecting the degree of at ypicalit y . T ogether, these op erators detect b oth the g ¯ 0 - decomp osition and the atypicalit y . 28 STEFFEN SCHMIDT 4.1. F amily of Cubic Dirac Op erators and Laplace Op erators. W e construct a family of cubic Dirac operators parametrized b y the real span of the root system and study their prop erties. This family sp ecializes to the ordinary cubic Dirac op erator at the origin. W e b egin with the case in which g is a complex simple Lie algebra. Our results generalize those of [ FHT11a , FHT13 , FHT11b ]. Next, we deﬁne analogous families for basic classical Lie sup eralgebras with non trivial o dd part, as w ell as for the ev en part of basic classical Lie sup eralgebras. W e study these families as elemen ts in the color quantum W eil algebra W ( g ) and operators acting on M ⊗ M ( n − ) , where M is a ﬁnite-dimensional simple (sup er)mo dule. 4.1.1. Complex Simple Lie Algebr as. Let g b e a ﬁnite-dimensional simple complex Lie algebra; that is, g is isomorphic to sl ( n ; C ), so (2 n ; C ), sp (2 n ; C ), so (2 n + 1; C ), or one of the exceptional Lie algebras g 2 , f 4 , e 6 , e 7 , e 8 . W e adapt the notation from ab ov e. In particular, B denotes the Killing form on g . It restricts to a non-degenerate form on h , and allows us to iden tify h with h ∗ . The induced non-degenerate bilinear form on h ∗ will b e denoted b y the same symbol by abuse of notation. Let h ∗ R ⊂ h ∗ denote the real span of ∆. The induced form B is p ositive deﬁnite on h ∗ R . Using B , w e identify h R with h ∗ R and denote the induced p ositive deﬁnite form on h ∗ R b y the same symbol. In particular, for ξ ∈ h ∗ R , we write h ξ ∈ h R for the corresp onding element, and con v ersely . Let G b e the connected simply connected Lie group with Lie algebra g . Then G acts naturally on g by the adjoint represen tation Ad and on g ∗ b y the coadjoint represen tation Ad ∗ . The musical isomorphism  : g → g ∗ in tert wines the adjoint and coadjoin t represen tations, that is, (4.1) (Ad g x ) ♯ = Ad ∗ g ( x ♯ ) , g ∈ G, x ∈ g . This identiﬁcation yields a canonical bijection b etw een adjoint and coadjoin t orbits. Deﬁne (4.2) D g ( ξ ) : = D g +1 ⊗ h ξ ∈ W ( g ) , ξ ∈ h ∗ R . Via the identiﬁcation h R ∼ = h ∗ R , this gives a family of cubic Dir ac op er ators parametrized by h ∗ R . F or ξ = 0, one reco vers the cubic Dirac op erator D g , whic h w e shall refer to as the absolute cubic Dirac op erator. Prop osition 44. F or any ξ ∈ h ∗ R , the fol lowing hold: a) The element D g ( ξ ) is h -invariant. In p articular, it pr eserves al l weight sp ac es. b) F or every ξ ∈ h ∗ R , the op er ator D g ( ξ ) is an o dd element in W ( g ) . c) One has D g ( ξ ) 2 = D 2 g + h ξ ⊗ 1 + 2  1 ⊗ ν ∗ ( h ξ )  + B ( ξ , ξ )(1 ⊗ 1) . Pr o of. a) The cubic Dirac op erator D g is g -in v arian t, so in particular h -inv ariant. The statemen t no w follo ws as h is ab elian; so in particular [ h, ξ ] = 0 for all h ∈ h R . b) By deﬁnition of the Z 2 -grading on W ( g ), the element 1 ⊗ x is o dd for every x ∈ g ; see ( 3.17 ). In particular, 1 ⊗ h ξ is o dd. Since D g is o dd by Lemma 28 , it follows that D g ( ξ ) is o dd. Perturbations of Dirac Op erators 29 c) Using b), we compute D g ( ξ ) 2 = 1 2 [D g ( ξ ) , D g ( ξ )] W = D 2 g +[ D g , 1 ⊗ h ξ ] W + 1 2 [1 ⊗ h ξ , 1 ⊗ h ξ ] W = D 2 g +2 γ W ( h ξ ) + B ( ξ , ξ )(1 ⊗ 1) , where we used the Cliﬀord relations for [1 ⊗ h ξ , 1 ⊗ h ξ ] W = 2 B ( h ξ , h ξ )(1 ⊗ 1) = 2 B ( ξ , ξ )(1 ⊗ 1) and Lemma 31 . □ Fix a ﬁnite-dimensional simple g -sup ermo dule V . Suc h supermo dules are highest weigh t mo dules, parametrized b y dominant in tegral weigh ts Λ ∈ P ++ b ; see Section 2.2 . Thus V ∼ = L b (Λ) for some Borel subalgebra b . F rom now on, w e ﬁx some b = h ⊕ n + and omit the sup erscript. Recall the oscillator mo dule M ( p ). Since g is even, one has M ( p ) ∼ = S g , n − ; see ( 3.34 ). As the p ositiv e system is ﬁxed, w e write S : = S g , n − . Then the op erators D g ( ξ ) deﬁne a family of cubic Dir ac op er ators (4.3) h ∗ R → End C  L (Λ) ⊗ S  , ξ 7→ D g ( ξ ) = D g +1 ⊗ h ξ . Our ob ject of study is the subset { ξ ∈ h ∗ R | ker D g ( ξ )  = 0 } . In general, ker D g ( ξ ) is diﬃcult to analyze directly , whereas Prop osition 44 giv es eﬀectiv e con trol of D g ( ξ ) 2 . F or D g , one may also use unitarity: every ﬁnite-dimensional g -mo dule is unitarizable, and D g is selfadjoint on L (Λ) ⊗ S . This usual argument is not av ailable for D g ( ξ ), since D g ( ξ ) is not selfadjoin t in general. Lemma 45. L et r b e the c omp act r e al form of g . If h ξ ∈ h R ∩ r , then D g ( ξ ) is selfadjoint. In p articular, k er D g ( ξ ) = ker D g ( ξ ) 2 . In general, h ξ / ∈ h R ∩ r . Accordingly , one considers D g ( ξ ) 2 instead of D g ( ξ ). This is reﬂected in the notation (4.4) h ∗ R − → End  L (Λ) ⊗ S  , ξ 7− → ∆ g ( ξ ) : = D g ( ξ ) 2 . Throughout the remainder of this article, the family ∆ g is referred to as the family of L aplac e op er ators . W e compute the kernel of ∆ g ( ξ ) on L (Λ). The sets of weigh ts of L (Λ) and of S are contained in h ∗ R . Since the restriction of B to h ∗ R is p ositive deﬁnite, we set ∥ µ ∥ : = B ( µ, µ ) for µ ∈ h ∗ R . Lemma 46. ∆ g ( ξ ) acts on any weight sp ac e ( L (Λ) ⊗ S ) µ of weight µ ∈ h ∗ R as the sc alar ∥ Λ + ρ ∥ 2 − ∥ µ ∥ 2 + ∥ µ + ξ ∥ 2 . Pr o of. By Prop osition 44 , D g ( ξ ) 2 = D 2 g +2  h ξ ⊗ 1 + 1 ⊗ ν ∗ ( h ξ )  + B ( ξ , ξ )(1 ⊗ 1) . By Theorem 29 , D 2 g = Ω g ⊗ 1 + B ( ρ, ρ )(1 ⊗ 1), and Ω g acts on L (Λ) b y the scalar B (Λ + ρ, Λ + ρ ) − B ( ρ, ρ ). Hence D 2 g acts on L (Λ) ⊗ S by ∥ Λ + ρ ∥ 2 . Let µ = µ 1 + µ 2 with µ 1 , µ 2 ∈ h ∗ R suc h that L (Λ) µ 1  = 0 and S µ 2  = 0. Then h ξ acts on ( L (Λ) ⊗ S ) µ b y B ( µ 1 , ξ ) and ν ∗ ( h ξ ) by B ( µ 2 , ξ ) (Prop osition 39 ). Consequen tly , D g ( ξ ) 2 = ∥ Λ + ρ ∥ 2 + 2 B ( µ, ξ ) + B ( ξ , ξ ) = ∥ Λ + ρ ∥ 2 − ∥ µ ∥ 2 + ∥ µ + ξ ∥ 2 . □ 30 STEFFEN SCHMIDT Theorem 47. The kernel of ∆ g ( ξ ) is trivial unless ξ = − Λ − ρ . In this c ase the kernel is one-dimensional and given by L (Λ) Λ ⊗ S ρ . Pr o of. Using Lemma 46 , the kernel is the direct sum of all weigh t spaces ( L (Λ) ⊗ S ) µ suc h that ∥ Λ + ρ ∥ 2 − ∥ µ ∥ 2 + ∥ µ + ξ ∥ 2 = 0 . W e ﬁrst note that ∥ µ + ξ ∥ 2 ≥ 0 since µ + ξ ∈ h ∗ R b y assumption. Moreo ver, ∥ Λ + ρ ∥ 2 − ∥ µ ∥ 2 ≥ 0 and equality holds precisely when µ = Λ + ρ [ Kac90 , Prop osition 11.4]. Thus, the w eigh t space b elongs to the k ernel precisely when µ = Λ + ρ and ∥ Λ + ρ + ξ ∥ 2 = 0. Since B is p ositive deﬁnite, w e conclude ξ = − Λ − ρ . As Λ is the highest weigh t of L (Λ) and ρ is the highest weigh t of S , the weigh t space is one-dimensional and precisely given by L (Λ) Λ ⊗ S ρ . □ The adjoin t action Ad : G → GL( g ) extends canonically to actions Ad U on U ( g ) and Ad C on Cl( g ); the latter uses the inv ariance of B . This induces an action Ad W of G on W ( g ): (4.5) Ad W g ( x ⊗ y ) = Ad U g ( x ) ⊗ Ad C g ( y ) . View h R and h ∗ R as subspaces of g and g ∗ resp ectiv ely , such that Ad g ( h R ) ⊂ g or Ad ∗ g ( h ∗ R ) ⊂ g ∗ . Although our family was deﬁned only on h ∗ R , it may b e viewed as the restriction of a more general family on g ∗ . More precisely , regarding h ∗ R as a subspace of g ∗ , one has families (4.6) g ∗ → W ( g ) , ξ 7→ D g ( ξ ) and ξ 7→ ∆ g ( ξ ) , whose restrictions to h ∗ R reco v er the families considered ab o v e. Lemma 48. The maps D g ( ξ ) , ∆ g ( ξ ) : g ∗ − → W ( g ) ar e G -e quivariant with r esp e ct to the c o ad- joint action Ad ∗ on g ∗ and the adjoint action Ad W on W ( g ) , i.e. , Ad W g  D g ( ξ )  = D g  Ad ∗ g ξ  , Ad W g  ∆ g ( ξ )  = ∆ g  Ad ∗ g ξ  g ∈ G, ξ ∈ g ∗ . Pr o of. Since D g is g -inv ariant, it is ﬁxed b y Ad W g . Hence Ad W g  D g ( ξ )  = D g +1 ⊗ Ad g ( h ξ ) . Under the iden tiﬁcation g ∼ = g ∗ via B , the adjoint action of G corresp onds to the coadjoin t action Ad ∗ , so Ad g ( h ξ ) = h Ad ∗ g ξ . Analogously , for the Laplace op erator one has Ad W g ( ∆ g ( ξ )) = D g +2(Ad g ( h ξ ) ⊗ 1 + 1 ⊗ q ( λ (ad Ad ∗ g h ξ ))) + B ( h ξ , h ξ )(1 ⊗ 1) = ∆ g (Ad ∗ g ξ ) , where we used G -equiv ariance of q and λ and inv ariance of B , that is, B ( h ξ , h ξ ) = B (Ad g ( h ξ ) , Ad g ( h ξ )) = B ( h Ad ∗ g ξ , h Ad ∗ g ξ ) . □ Since G is connected and simply connected with Lie algebra g , an y ﬁnite-dimensional g -mo dule M in tegrates to a representation π M : G → GL( M ) whose deriv ed represen tation coincides with the giv en g -action. Let π Λ and π S denote the corresp onding representations on L (Λ) and S . Then L (Λ) ⊗ S carries the representation (4.7) π : G → GL  L (Λ) ⊗ S  , π ( g ) : = π Λ ( g ) ⊗ π S ( g ) . Using Lemma 48 and the compatibility of the g - and G -mo dule structures on L ( λ ) ⊗ S , one obtains the following. Perturbations of Dirac Op erators 31 Lemma 49. F or g ∈ G and ξ ∈ h ∗ R , k er D g (Ad ∗ g ξ ) = π ( g )  k er D g ( ξ )  , k er ∆ g (Ad ∗ g ξ ) = π ( g )(ker ∆ g ( ξ )) . In p articular, for al l g ∈ G and ξ ∈ h ∗ R , ther e ar e ve ctor sp ac e isomorphisms k er D g (Ad ∗ g ξ ) ∼ = k er D g ( ξ ) , ker ∆ g (Ad ∗ g ξ ) ∼ = k er ∆ g ( ξ ) . F or x ∈ g ∗ , denote by Ad ∗ G ( x ) ⊂ g ∗ its coadjoint orbit. A coadjoint orbit is called semisimple if it con tains a semisimple element, i.e. , an element lying in the dual of a Cartan subalgebra of g . Equiv alently , Ad ∗ G ( x ) ∩ t ∗  = { 0 } for any Cartan subalgebra t ⊂ g . A semisimple coadjoint orbit is called R -split if Ad ∗ G ( x ) ∩ h ∗ R  = { 0 } . Note that for any other Cartan subalgebra t ⊂ g with corresp onding real span of ro ots t R , the spaces h R and t R are conjugate under the adjoint action. In particular, for µ ∈ h ∗ R , the coadjoint orbit Ad ∗ G ( µ ) is R -split and semisimple. Com bining Lemma 49 with Theorem 47 , one obtains the following result. Theorem 50. L et L (Λ) b e a ﬁnite-dimensional simple g -mo dule. Then ker ∆ g ( ξ ) = { 0 } unless ξ ∈ Ad ∗ G ( − Λ − ρ ) . In p articular, this assignment asso ciates to e ach ﬁnite-dimensional simple g -mo dule a unique R -split semisimple c o adjoint orbit. 4.1.2. Basic Classic al Lie Sup er algebr a. Let g b e a basic classical Lie sup eralgebra with g ¯ 1  = { 0 } . Fix the sup er Killing form B on g , which identiﬁes h with h ∗ ; in particular, h ⊂ g ¯ 0 . W e denote b y the same sym b ol the non-degenerate form on h ∗ . As ab o v e, let h ∗ R denote the real span of the ro ot system ∆. Unlike the ﬁnite-dimensional simple complex Lie algebra case, the restriction of B to h R is not p ositiv e deﬁnite. As an example, consider g = psl (2 | 2) and µ : = (1 , 0 | 0 , − 1) ∈ h ∗ R , whic h is isotropic, that is, B ( µ, µ ) = 0. The even subalgebra g ¯ 0 is reductiv e; let G ¯ 0 b e the connected simply connected Lie group with Lie algebra g ¯ 0 . As in the complex simple even case, deﬁne a family of cubic Dirac op erators and Laplace op erators by (4.8) h ∗ R → W ( g ) , ξ 7→    D g ( ξ ) : = D g +1 ⊗ h ξ , ∆ g ( ξ ) : = D g ( ξ ) 2 . By the same argument as in the pro of of Prop osition 44 , one obtains the following. Prop osition 51. F or any ξ ∈ h ∗ R , the fol lowing hold: a) D g ( ξ ) is h -invariant. b) F or every ξ ∈ h ∗ R , the op er ator D g ( ξ ) has bide gr e e ( ¯ 1 , ¯ 0) ; in p articular, it is o dd with r esp e ct to the total de gr e e. c) ∆ g ( ξ ) = D 2 g + h ξ ⊗ 1 − 1 2 (1 ⊗ ν ∗ ( h ξ )) + B ( ξ , ξ )(1 ⊗ 1) . d) Ad W g D g ( ξ ) = D g (Ad ∗ g ξ ) for al l g ∈ G ¯ 0 . e) Ad W g ∆ g ( ξ ) = ∆ g (Ad ∗ g ξ ) for al l g ∈ G ¯ 0 . As in the simple complex even case, the natural question is whether, for each ﬁnite-dimensional simple g - supermo dule, the kernel of ∆ g ( ξ ) determines a unique G ¯ 0 -coadjoin t orbit. Recall from 32 STEFFEN SCHMIDT Section 2.2 that suc h sup ermo dules are parametrized by dominan t integral weigh ts λ ∈ h ∗ with resp ect to a Borel subalgebra b = b ¯ 0 ⊕ b ¯ 1 , that is, λ satisﬁes (4.9) B ( λ + ρ ¯ 0 , α ) > 0 for all α ∈ ∆ + ¯ 0 , where ∆ + = ∆ + ¯ 0 ⊔ ∆ + ¯ 1 . W e ﬁx a Λ ∈ P ++ for a ﬁxed Borel b and consider the ﬁnite-dimensional simple sup ermo dule L ( λ ) with even highest w eight v ector such that w e view the family as (4.10) h ∗ R → End C ( L (Λ) ⊗ M ( p )) , ξ 7→    D g ( ξ ) : = D g +1 ⊗ h ξ , ∆ g ( ξ ) : = D g ( ξ ) 2 . Since B restricted to h R is not p ositive deﬁnite, we cannot conclude as in the pro of of Theo- rem 47 . Indeed, we only hav e the following partial result which is pro v ed as ab o ve: Lemma 52. The fol lowing hold: a) F or al l g ∈ G ¯ 0 and ξ ∈ h ∗ R , k er D g (Ad ∗ g ξ ) ∼ = k er D g ( ξ ) , ker ∆ g (Ad ∗ g ξ ) ∼ = k er ∆ g ( ξ ) as sup er ve ctor sp ac es. b) ∆ g ( ξ ) acts on any weight sp ac e ( L (Λ) ⊗ S ) µ of weight µ ∈ h ∗ R as the sc alar B (Λ + ρ, Λ + ρ ) − B ( µ, µ ) + B ( µ + ξ , µ + ξ ) . F or a ﬁxed weigh t µ ∈ h ∗ R of L (Λ) ⊗ S , the expression (4.11) B (Λ + ρ, Λ + ρ ) − B ( µ, µ ) + B ( µ + ξ , µ + ξ ) ma y admit many solutions in ξ as B is indeﬁnite. T o o vercome this, we consider appropriate mo diﬁcations of families of Laplace op erators deﬁned with resp ect to the even subalgebra g ¯ 0 , using Theorem 47 . This allows one to detect the g ¯ 0 -decomp osition of a ﬁnite-dimensional simple g -sup ermo dule. 4.1.3. Even Lie Sub algebr a g ¯ 0 . Let g = g ¯ 0 ⊕ g ¯ 1 b e a basic classical Lie sup eralgebra with g ¯ 1  = { 0 } . The Lie algebra g ¯ 0 is reductive. Consequen tly , g ¯ 0 admits a decomp osition (4.12) g ¯ 0 = g 0 ¯ 0 ⊕ g 1 ¯ 0 ⊕ · · · ⊕ g r ¯ 0 , where g 0 ¯ 0 is either trivial or an ab elian Lie algebra, and g i ¯ 0 is simple for all i = 1 , . . . , r . The p ossible cases are summarized in T able 1 . The summands g i ¯ 0 are mutually orthogonal with resp ect to restriction of the sup er Killing form B . Moreov er, for any i = 1 , . . . , r , the restriction of B to g i ¯ 0 is a nonzero scalar m ultiple of the Killing form of g i ¯ 0 , and the restriction to g 0 ¯ 0 is non-trivial. W e denote by B i the induced bilinear form on g i ¯ 0 and ( g i ¯ 0 ) ∗ , resp ectively . Fix a Cartan subalgebra h of g ; then h ⊂ g ¯ 0 . Accordingly , h decomp oses compatibly with g ¯ 0 as (4.13) h = h 0 ⊕ h 1 ⊕ · · · ⊕ h r , Perturbations of Dirac Op erators 33 g g ¯ 0 T yp e of g ¯ 0 sl ( m | n ) , m  = n sl ( m ) ⊕ sl ( n ) ⊕ C reductiv e, not semisimple psl ( n | n ) , n ≥ 2 sl ( n ) ⊕ sl ( n ) semisimple osp (2 | 2 n ) so (2) ⊕ sp (2 n ) ∼ = C ⊕ sp (2 n ) reductiv e, not semisimple osp ( m | 2 n ) , m  = 2 so ( m ) ⊕ sp (2 n ) semisimple D (2 , 1; α ) , α  = 0 , − 1 sl (2) ⊕ sl (2) ⊕ sl (2) semisimple F (4) so (7) ⊕ sl (2) semisimple G (3) g 2 ⊕ sl (2) semisimple T able 1. Even Lie subalgebras of basic classical Lie sup eralgebras where h i is a Cartan subalgebra of g i ¯ 0 for i  = 0, and h 0 = g 0 ¯ 0 . The restriction of B i to h i induces a non-degenerate symmetric bilinear form on ( h i ) ∗ , deﬁned analogously to the form obtained by restricting B to h ; by abuse of notation, this form is again denoted by B i . Consequently , h ∗ = ( h 0 ) ∗ ⊕ ( h 1 ) ∗ ⊕ · · · ⊕ ( h r ) ∗ , and for i  = 0 the space ( h i ) ∗ is identiﬁed with h i via B i . With resp ect to this decomp osition, eac h λ ∈ h ∗ admits a unique decomp osition λ = ( λ 0 , λ 1 , . . . , λ r ), where λ i is the restriction of λ to h i . F or λ, µ ∈ h ∗ , the bilinear form B satisﬁes (4.14) B ( λ, µ ) = r X i =0 B i ( λ i , µ i ) . In the sequel, the subscript i is omitted whenever no ambiguit y arises. Let h ∗ R denote the real span of ∆ with dual space h R . Then h R = h 1 R ⊕ · · · ⊕ h r R . In addition, let G ¯ 0 b e the connected simply connected Lie group with Lie algebra g ¯ 0 . Then (4.15) G ¯ 0 = G 0 ¯ 0 × G 1 ¯ 0 × · · · × G r ¯ 0 , where G i ¯ 0 is the connected simply connected Lie group with Lie algebra g i ¯ 0 . W e hav e a natural coadjoin t action of G ¯ 0 on g ¯ 0 ∗ . Let ∆ b e the ro ot system of g with resp ect to h , and let ∆ ¯ 0 denote the ro ot system of g ¯ 0 . The ro ot systems ∆ 1 ¯ 0 , . . . , ∆ r ¯ 0 are identiﬁed with subsets of ∆ ¯ 0 suc h that ∆ ¯ 0 = ∆ 1 ¯ 0 ⊔ . . . ⊔ ∆ r ¯ 0 . A c hoice of p ositiv e system ∆ + ⊂ ∆ induces p ositive systems on each ∆ i ¯ 0 . With this choice, the p ositiv e system of ∆ ¯ 0 is the disjoint union of the p ositive systems of the ∆ i ¯ 0 . The even W eyl v ector ρ ¯ 0 is deﬁned accordingly , and decomp oses as ρ ¯ 0 = (0 , ρ 1 ¯ 0 , . . . , ρ r ¯ 0 ) . Fix a Borel subalgebra b ¯ 0 ⊂ g ¯ 0 . Then b ¯ 0 decomp oses as b ¯ 0 = b 0 ¯ 0 ⊕ b 1 ¯ 0 ⊕ · · · ⊕ b r ¯ 0 with b 0 ¯ 0 = g 0 ¯ 0 . In particular, the set of dominant integral weigh ts parameterizing ﬁnite-dimensional simple g ¯ 0 - mo dules is (4.16) P ++ b ¯ 0 = P ++ b 0 ¯ 0 × . . . × P ++ b r ¯ 0 , where P ++ b 0 ¯ 0 is iden tiﬁed with ( b 0 ¯ 0 ) ∗ : = Hom C ( b 0 ¯ 0 , C ) and each P + b i ¯ 0 for i = 1 , . . . , r is describ ed in Section 2.2 . 34 STEFFEN SCHMIDT Let L 0 (Λ) b e a ﬁnite-dimensional simple g ¯ 0 -mo dule. Then L 0 (Λ) is a highest w eight mo dule with resp ect to b ¯ 0 with highest w eigh t Λ ∈ P ++ b ¯ 0 . Moreo ver, L 0 (Λ) decomp oses as an outer tensor pro duct (4.17) L 0 (Λ) = L (Λ 0 ; g 0 ¯ 0 ) ⊠ L (Λ 1 ; g 1 ¯ 0 ) ⊠ · · · ⊠ L (Λ r ; g r ¯ 0 ) , where each L (Λ i ; g i ¯ 0 ) is a ﬁnite-dimensional simple mo dule ov er g i ¯ 0 . Since each ( g i ¯ 0 , B i ) is a quadratic Lie algebra, there is a cubic Dirac op erator D g i ¯ 0 ∈ W ( g i ¯ 0 ). As ab ov e, this deﬁnes families (4.18) ( h i R ) ∗ → W ( g i ¯ 0 ) , ξ 7→    D g i ¯ 0 ( ξ i ) : = D g i ¯ 0 +1 ⊗ h ξ i , ∆ g i ¯ 0 ( ξ i ) : = D g i ¯ 0 ( ξ i ) 2 . F or i  = j , the elements D g i ¯ 0 ( ξ i ) and D g j ¯ 0 ( ξ j ) an ti-commute in W ( g ¯ 0 ), via the embedding ( 3.28 ). Hence, for ξ = ( ξ 0 , . . . , ξ r ) ∈ h ∗ R , we deﬁne (4.19) ∆ g ¯ 0 ( ξ ) : = r X i =0 ∆ g i ¯ 0 ( ξ i ) ∈ W ( g ¯ 0 ) , whic h equals  P r i =0 D g i ¯ 0 ( ξ i )  2 . W e are interested in the action of ∆ g ¯ 0 ( ξ ) on L 0 (Λ) ⊗ S , where now S : = M ( n − ¯ 0 ). In general, the k ernel of ∆ g ¯ 0 ( ξ ) do es not coincide with the in tersection of the kernels of the op erators ∆ g i ¯ 0 ( ξ i ). As a consequence, Theorem 50 is not directly applicable. W e address this b y introducing a mo diﬁed family of op erators, using that S is unitarizable and that L 0 (Λ) is ﬁnite-dimensional. Since L 0 (Λ) is ﬁnite-dimensional, we can ﬁx a Hermitian inner product on ⟨· , ·⟩ L 0 (Λ) on L 0 (Λ). Let ⟨· , ·⟩ L 0 (Λ) ⊗ S : = ⟨· , ·⟩ L 0 (Λ) ⟨· , ·⟩ S b e the induced Hermitian inner pro duct on L 0 (Λ) ⊗ S . With resp ect to this pro duct, we denote b y ( · ) † the adjoint of an endomorphism. W e then deﬁne (4.20) e ∆ g ¯ 0 ( ξ ) : = r X i =0 ( ∆ g i ¯ 0 ( ξ i )) † ∆ g i ¯ 0 ( ξ i ) ∈ End C ( L 0 (Λ) ⊗ S ) . Lemma 53. The op er ator e ∆ g ¯ 0 ( ξ ) satisﬁes the fol lowing pr op erties: a) e ∆ g ¯ 0 ( ξ ) is p ositive semi-deﬁnite, that is, ⟨ e ∆ g ¯ 0 ( ξ ) v , v ⟩ L 0 (Λ) ⊗ S ≥ 0 for al l v ∈ L 0 (Λ) ⊗ S. b) One has k er e ∆ g ¯ 0 ( ξ ) = r \ i =0 k er ∆ g i ¯ 0 ( ξ i ) . Pr o of. F or v ∈ L 0 (Λ) ⊗ S , one has ⟨ e ∆ g ¯ 0 ( ξ ) v , v ⟩ L 0 (Λ) ⊗ S = r X i =0  ( ∆ g i ¯ 0 ( ξ i )) † ∆ g i ¯ 0 ( ξ i ) v , v  L 0 (Λ) ⊗ S = r X i =0  ∆ g i ¯ 0 ( ξ i ) v , ∆ g i ¯ 0 ( ξ i ) v  L 0 (Λ) ⊗ S , hence ⟨ e ∆ g ¯ 0 ( ξ ) v , v ⟩ L 0 (Λ) ⊗ S ≥ 0 . Perturbations of Dirac Op erators 35 This prov es a). F or b), p ositivit y of the Hermitian inner pro duct implies that v ∈ ker e ∆ g ¯ 0 ( ξ ) ⇐ ⇒ ⟨ e ∆ g ¯ 0 ( ξ ) v , v ⟩ L 0 (Λ) ⊗ S = 0 ⇐ ⇒ ∆ g i ¯ 0 ( ξ i ) v = 0 for all i = 0 , . . . , r , whic h is equiv alent to v ∈ T r i =0 k er ∆ g i ¯ 0 ( ξ i ) . □ Theorem 54. L et L 0 (Λ) b e a ﬁnite-dimensional simple g ¯ 0 -mo dule with highest weight Λ ∈ h ∗ R . Then e ∆ g ¯ 0 ( ξ ) has trivial kernel unless ξ = − Λ − ρ ¯ 0 . In p articular, k er e ∆ g ¯ 0 ( ξ )  = { 0 } ⇐ ⇒ ξ ∈ Ad ∗ G ( − Λ − ρ ¯ 0 ) : = Ad ∗ G ¯ 0 ( − Λ − ρ ¯ 0 ) . Pr o of. By Lemma 53 is the kernel of e ∆ g ¯ 0 ( ξ ) the intersection of the kernels of ∆ g i ¯ 0 ( ξ i ). By Theorem 47 , ∆ g i ¯ 0 ( ξ i ) has trivial k ernel unless ξ i = − Λ i − ρ i ¯ 0 for i = 1 , . . . , r , and ker ∆ g i ¯ 0 ( ξ i )  = { 0 } iﬀ ξ ∈ Ad G i ¯ 0 ( − Λ i − ρ i ¯ 0 ). F or the ab elian factor g 0 ¯ 0 , one has ρ 0 ¯ 0 = 0 and L (Λ 0 ; g 0 ¯ 0 ) is one-dimensional, with ∆ g 0 ¯ 0 ( ξ 0 ) acting by the scalar B 0 (Λ 0 + ξ 0 , Λ 0 + ξ 0 ). Since B 0 is non-degenerate, this scalar v anishes only for ξ 0 = − Λ 0 . The claim follo ws. □ R emark 55 . Let r b e the compact real form of g ¯ 0 . If h ξ ∈ h R ∩ r , then D g i ¯ 0 ( ξ ) is self-adjoint with resp ect to the Hermitian form. Consequently , ∆ g i ¯ 0 ( ξ ) is non-negative for all i = 0 , . . . , r . Therefore Theorem 54 remains v alid with e ∆ g ¯ 0 ( ξ ) replaced by ∆ g ¯ 0 ( ξ ). W e now apply the preceding result to ﬁnite-dimensional g -sup ermo dules. W e th us embed W ( g ¯ 0 ) into W ( g ). Fix a ﬁnite-dimensional simple g -sup ermo dule L (Λ) with resp ect to a ﬁxed p ositiv e Borel b . Up on restriction to the ev en subalgebra g ¯ 0 , the mo dule L (Λ) decomp oses as a ﬁnite direct sum of simple ﬁnite-dimensional g ¯ 0 -mo dules, (4.21) L (Λ)    g ¯ 0 = M µ L 0 ( µ ) n ( µ ) , where the sum runs ov er the highest weigh ts µ of the g ¯ 0 -constituen ts and n ( µ ) denotes their m ultiplicities. Under the embedding W ( g ¯ 0 )  → W ( g ), the mo diﬁed family e ∆ g ¯ 0 ( ξ ) is viewed as a family in W ( g ) acting on L (Λ) ⊗ M ( n − ) by (4.22) e ∆ g ¯ 0 ( ξ ) ⊗ 1 M ( n − ¯ 1 ) ∈ End C (( L (Λ) ⊗ S ) ⊗ M ( n − ¯ 1 )) , where we use the Z 2 -graded tensor pro duct decomp osition M ( n − ) = S ⊗ M ( n − ¯ 1 ) and identify S with S g , n − ¯ 0 . This family then detects precisely the g ¯ 0 -constituen ts of L (Λ) via their highest w eigh ts, as describ ed in Theorem 54 . Corollary 56. L et L (Λ) b e a ﬁnite-dimensional simple g -sup ermo dule with highest weight Λ such that L (Λ)   g ¯ 0 = L µ L 0 ( µ ) n ( µ ) . Then k er e ∆ g ¯ 0 ( ξ )  = { 0 } ⇔ ξ ∈ [ µ : n ( µ )  =0 Ad ∗ G ¯ 0 ( − µ − ρ ¯ 0 ) . If ξ = − µ − ρ ¯ 0 , then the kernel is L (Λ) µ ⊗ S ρ ¯ 0 ⊗ M ( n − ¯ 1 ) . Example 57 . W e consider the pro jective sp ecial linear Lie sup eralgebra psl (2 | 2). Fix the distin- guished p ositiv e system suc h that the W eyl vector takes the form ρ = (1 , 0 | 0 , − 1). W e consider 36 STEFFEN SCHMIDT L (Λ) with highest w eigh t Λ = (1 , 0 | 0 , − 1). Let ξ ∈ h ∗ R , where h is the Cartan subalgebra of diagonal matrices. Then k er e ∆ g ¯ 0 ( ξ )  = { 0 } ⇔ − ξ ∈ Ad ∗ G ¯ 0 ( X + ρ ¯ 0 ) , X = { Λ , (0 , 0 | 1 , − 1) , (1 , − 1 | 0 , 0) , (0 , − 1 | 1 , 0) } . 4.2. A Detecting F amily. Let g = g ¯ 0 ⊕ g ¯ 1 b e a basic c lassical Lie sup eralgebra with g ¯ 1  = { 0 } , and retain the notation in tro duced ab o v e. In the previous section, a family of cubic Dirac op erators (4.23) D g ( ξ ) : = D g +1 ⊗ h ξ , ξ ∈ h ∗ R , w as introduced; its inv ariance prop erties w ere established; and the k ernel of the associated family of Laplace op erators ∆ g ( ξ ) = D 2 g ( ξ ) was analyzed. When restricted to g ¯ 0 , a resulting Laplace family e ∆ g ¯ 0 ( ξ ) detects the g ¯ 0 -decomp osition of a ﬁnite-dimensional simple g -sup ermo dule, but do es not provide further information. F or g with g ¯ 1  = { 0 } , the represen tation theory diﬀers from the classical case b ecause of the presence of isotropic o dd ro ots. This is reﬂected in Kac’s distinction b etw een typical and at ypical simple ﬁnite-dimensional sup ermo dules, deﬁned in terms of the relation b et w een the highest weigh t and the isotropic ro ots; see Section 2.3 . Typical represen tations retain many of the features familiar from ordinary semisimple Lie algebras, whereas atypical represen tations do not. In particular, the even subalgebra no longer determines all relev ant structure. This leads to additional inv arian ts, and in this pap er we in tro duce an energy op erator for the detection of at ypicalit y . Recall that a weigh t Λ is typic al if A Λ = ∅ and atypic al otherwise, where A Λ = { α ∈ ∆ + ¯ 1 : B (Λ + ρ, α ) = 0 , B ( α, α ) = 0 } . The de gr e e of atypic ality of Λ, denoted b y at(Λ), is the maximal cardinalit y of a linearly independent set of mutually orthogonal isotropic ro ots α ∈ ∆ + ¯ 1 suc h that B (Λ + ρ, α ) = 0. T o detect this information, w e in tro duce an additional family of op erators measuring the deviation from the Laplace op erator ∆ g , called ener gy op er ators . These enco de atypicalit y and will b e combined with e ∆ g ¯ 0 ( ξ ) b elow. W e recall that (4.24) D g ( ξ ) : = D g +1 ⊗ h ξ , ∆ g ( ξ ) : = D g ( ξ ) 2 , ξ ∈ h ∗ R . Deﬁnition 58. Let η ∈ h ∗ R . The diﬀerence of the family ∆ g ( η ) and the Laplace op erator ∆ g is called ener gy op er ator , that is, T ( η ) : = ∆ g ( η ) − ∆ g = 2 γ W ( h η ) + B ( η , η ) . R emark 59 . The op erator T ( η ) is called ener gy op er ator , since it generates the Lie deriv ative in W ( g ) by Lemma 31 . Note that T (0) = 0. Let h iso R ⊂ h R b e a maximal isotropic subspace. Using the form B , w e identify h iso R with its dual ( h iso R ) ∗ . The latter is spanned by pairwise distinct, mutually orthogonal o dd isotropic ro ots of g . In particular, by deﬁnition, the dimension of h iso R coincides with the defect of g . Since B is inv arian t, it is preserved under the adjoin t action of G ¯ 0 . In particular, for λ ∈ ( h iso R ) ∗ and g ∈ G ¯ 0 , the element Ad ∗ g ( h ) ∈ g ∗ is isotropic. Perturbations of Dirac Op erators 37 The op erators T ( η ) are regarded as a family of ener gy op er ators parametrized by ( h iso R ) ∗ , deﬁned by (4.25) ( h iso R ) ∗ → W ( g ) , η 7→ T ( η ) : = 2 γ W ( h η ) , where we note that B ( η , η ) = 0 since η ∈ ( h iso R ) ∗ . If w e ﬁx a Borel subalgebra b and a ﬁnite- dimensional simple sup ermo dule L (Λ) with Λ ∈ P ++ , we consider (4.26) ( h iso R ) ∗ − → End C ( L (Λ) ⊗ M ( n − )) , η 7− → T ( η ) . The set of weigh ts of L (Λ) is P L (Λ) = { Λ − Z + [ A ] : A ⊂ ∆ + } and the set of weigh ts of M ( p ) is P M ( p ) = { ρ − Z + [ B ] : B ⊂ ∆ + } . Both are subsets of h ∗ R . A general w eight of L (Λ) ⊗ M ( n − ) tak es the form Λ − µ 1 + ρ − µ 2 suc h that L (Λ) Λ − µ 1 and M ( n − ) ρ − µ 2 are non-trivial, and µ 1 , µ 2 are p ositive sums of p ositive ro ots. As in the pro of of Lemma 46 , one obtains the following. Lemma 60. F or any weight µ ∈ h ∗ R , the op er ator T ( η ) acts on the weight sp ac e ( L (Λ) ⊗ M ( n − )) µ by the sc alar 2 B ( µ, η ) . In p articular, k er T ( η ) = k er T ( η ) 2 . An y elemen t η ∈ ( h iso R ) ∗ can b e written as (4.27) η = a 1 α 1 + · · · + a r α r , a i ∈ R , where α i ∈ ∆ ¯ 1 are pairwise orthogonal o dd isotropic ro ots. The integer 0 ≤ r ≤ def ( g ) is called the r ank of η and is denoted by rk( η ). F or ﬁxed µ ∈ h ∗ R , deﬁne the real vector subspace (4.28) X µ : = { η ∈ ( h iso R ) ∗ | B ( µ, η ) = 0 } , called the isotr opic annihilator of µ . Corollary 61. L et µ = Λ − µ 1 − µ 2 + ρ b e a weight of L (Λ) ⊗ M ( n − ) and η ∈ ( h iso R ) ∗ . If ( L (Λ) ⊗ M ( n − )) µ ⊂ ker T ( η ) , then Λ − µ 1 − µ 2 is atypic al of de gr e e at le ast rk( η ) . Conversely, if Λ − µ 1 − µ 2 is atypic al of de gr e e k , then ( L (Λ) ⊗ M ( n − )) µ ⊂ k er T ( η ) for al l η ∈ X µ . In p articular, k er T ( η )   ( L (Λ) ⊗ M ( n − )) µ  = { 0 } ⇐ ⇒ η ∈ X µ . F or a ﬁnite-dimensional g -sup ermo dule L (Λ), Corollary 61 sho ws that the kernel of T on L (Λ) ⊗ M ( n − ) is to o large to yield a useful lo calization. It is therefore natural to replace L (Λ) ⊗ M ( n − ) by a smaller subspace. By construction ( cf. ( 3.41 )) the oscillator sup ermo dule M ( n − ) is a highest weigh t mo dule with highest weigh t vector 1 S ⊗ 1 M ( n − ¯ 1 ) . Set E Λ := L (Λ) ⊗ S and write 1 := 1 M ( n − ¯ 1 ) . W e then consider the g ¯ 0 -submo dule ⟨ E Λ ⊗ 1 ⟩ g ¯ 0 , where parity is disregarded and g ¯ 0 acts diagonally . This submo dule already suﬃces for the application of the Dirac operator to the Dirac inequality; see [ Sch24 ]. Lemma 62 ([ Sc h24 ]) . If L (Λ)   g ¯ 0 ∼ = L µ L 0 ( µ ) n ( µ ) , then ⟨ E Λ ⊗ 1 ⟩ g ¯ 0 ∼ = M µ L 0 ( µ + ρ ) n ( µ ) , and any g ¯ 0 -highest weight ve ctor is of the form v µ ⊗ 1 S ⊗ 1 M ( n − ¯ 1 ) for some v µ ∈ L (Λ) µ . 38 STEFFEN SCHMIDT By Lemma 62 , b oth e ∆ g ¯ 0 ( ξ ) and T ( η ) act on ⟨ E Λ ⊗ 1 ⟩ g ¯ 0 . F or T ( η ), this is immediate from the deﬁnition and the h -semisimplicity of the mo dule. W e combine these op erators into a single map (4.29) ( T , e ∆ g ¯ 0 ) : ( h iso R ) ∗ × h ∗ R → End C  ⟨ E Λ ⊗ 1 ⟩ g ¯ 0 , ( ⟨ E Λ ⊗ 1 ⟩ g ¯ 0 ) ⊕ 2  , called the dete cting family . Theorem 63. L et L (Λ) b e a ﬁnite-dimensional g -sup ermo dule with L (Λ) | g ¯ 0 ∼ = L µ L 0 ( µ ) n ( µ ) . Then k er(( T ( η ) , e ∆ g ¯ 0 ( ξ )) = { 0 } unless b oth of the fol lowing c onditions hold: a) ξ = − µ − ρ ¯ 0 , wher e µ is the highest weight of a g ¯ 0 -c onstituent of L (Λ) , b) η ∈ X µ + ρ . In p articular, k er( T ( η ) , e ∆ g ¯ 0 ( ξ ))  = { 0 } ⇔ ( η , ξ ) ∈ [ µ : n ( µ )  =0 X µ + ρ × Ad ∗ G ¯ 0 ( − µ − ρ ¯ 0 ) . Pr o of. By construction, k er( T ( η ) , e ∆ g ¯ 0 ( ξ )) = ker T ( η ) ∩ ker e ∆ g ¯ 0 ( ξ ) . It therefore suﬃces to determine ker e ∆ g ¯ 0 ( ξ ) and the action of T ( η ) on this space. By ( 4.22 ), Lemma 62 , and Theorem 54 , the kernel of e ∆ g ¯ 0 ( ξ ) is non-zero precisely for ξ ∈ [ µ : n ( µ )  =0 Ad ∗ G ¯ 0 ( − µ − ρ ¯ 0 ) , and in this case k er e ∆ g ¯ 0 ( ξ ) ∼ = M µ : ξ ∈ Ad ∗ G ¯ 0 ( − µ − ρ ¯ 0 ) ( L 0 ( µ ) µ ⊗ S ρ ¯ 0 ⊗ M ( n − ¯ 1 ) − ρ ¯ 1 ) ⊕ n ( µ ) . Here M ( n − ¯ 1 ) − ρ ¯ 1 = C (1 M ( n − ¯ 1 ) ), so the summand indexed by µ has dimension n ( µ ). Now T ( η ) preserv es eac h such summand, and it acts on ( L (Λ) µ ⊗ S ρ ¯ 0 ) ⊗ 1 M ( n − ¯ 1 ) b y the scalar 2 B ( µ + ρ, η ). Consequen tly , the joint kernel ker( T ( η ) , e ∆ g ¯ 0 ( ξ )) is obtained by restricting the ab ov e direct sum to those µ for which b oth conditions hold: ξ ∈ Ad ∗ G ¯ 0 ( − µ − ρ ¯ 0 ) and B ( µ + ρ, η ) = 0 . The assertion follows from Corollary 61 . □ 5. Nilpotent Per turba tions In this section, let g b e a basic classical Lie sup eralgebra with g ¯ 1  = { 0 } . Tw o standard to ols for studying g -sup ermo dules are: (1) Dir ac c ohomolo gy , which detects the inﬁnitesimal character. (2) Duﬂo–Ser ganova c ohomolo gy (DS cohomology), a symmetric monoidal functor preserv- ing the sup erdimension of ﬁnite-dimensional sup ermo dules. W e construct a family of relativ e cubic Dirac op erators D x g , l , parametrized b y the self-commuting v ariety , whose cohomology interpolates b et w een Dirac cohomology and Duﬂo–Serganov a coho- mology . W e start b y recalling DS cohomology . Perturbations of Dirac Op erators 39 5.1. Duﬂo–Sergano v a Cohomology. The Duﬂo–Ser ganova functor (DS functor), in troduced b y Duﬂo and Serganov a [ DS05 ], is a symmetric monoidal tensor functor asso ciated with an o dd square-zero elem en t. In what follows, we brieﬂy recall the DS functor, mainly following [ DS05 , GHSS22 ]; see also [ Ser11 ]. The DS functor dep ends on a choice of element in the self-c ommuting variety (5.1) Y : = { x ∈ g ¯ 1 : [ x, x ] = 0 } . Let G ¯ 0 b e the connected simply connected Lie group with Lie algebra g ¯ 0 . It acts on Y by the adjoin t action, and Y is a G ¯ 0 -stable Zariski-closed cone in g ¯ 1 . Its G ¯ 0 -orbits are in bijection with W -orbits of subsets of mutually orthogonal, linearly indep enden t, o dd isotropic ro ots; in particular, Y has ﬁnitely many G ¯ 0 -orbits. Concretely , ﬁx x ∈ Y . Then there exist g ∈ G ¯ 0 and m utually orthogonal, linearly indep enden t isotropic ro ots α 1 , . . . , α k suc h that (5.2) Ad g ( x ) = x 1 + · · · + x k , x i ∈ g α i . The in teger rk( x ) : = k is indep enden t of the choices and is called the r ank of x ; equiv alently , it is the rank of x in the standard representation. T o each x ∈ Y w e asso ciate a Lie sup eralgebra (5.3) g x : = k er(ad x ) / im(ad x ) , whic h is well-deﬁned since im(ad x ) = [ x, g ] is an ideal in k er(ad x ). Moreov er, g x has Cartan subalgebra (5.4) h x : =  k \ i =1 k er( α i ) .  h α 1 ⊕ · · · ⊕ h α k  , h α : = [ g α , g − α ] , and the corresp onding ro ot system is (5.5) ∆ x = { α ∈ ∆ : B ( α, α i ) = 0 for i = 1 , . . . , k , α  = α i } . W e now deﬁne the DS functor . Given a highest weigh t g -sup ermo dule M , eac h x ∈ Y acts on M b y an endomorphism x M ∈ End C ( M ) with x 2 M = 0. Deﬁne (5.6) M x : = k er( x M ) / im( x M ) . Then M x is naturally a g x -sup ermo dule, since ker( x M ) is ker(ad x )-stable and [ x, g ] · ker( x M ) ⊆ im( x M ). Thus M 7→ M x deﬁnes a functor from g -sup ermo dules to g x -sup ermo dules, denoted (5.7) DS x ( M ) : = M x . This is the Duﬂo–Ser ganova functor (DS functor). F or ﬁnite-dimensional g -sup ermo dules, it has the following basic prop erties. Lemma 64 ([ GHSS22 , Section 2]) . L et g -smo d denote the c ate gory of ﬁnite-dimensional g - sup ermo dules. F or any x ∈ Y , the functor DS x : g -smo d → g x -smo d is additive and symmetric monoidal. Mor e over, DS x pr eserves sup er dimension. 40 STEFFEN SCHMIDT F or the remainder of this section, ﬁx a highest weigh t g -sup ermo dule M . W e recall a con- struction from [ DS05 , Section 6] (see also [ Ser11 ]). Let U ( g ) ad x b e the subsup eralgebra of ad x -in v ariants in U ( g ), and set I x : = [ x, U ( g )] ⊂ U ( g ) . Consider φ = π ◦ ι in the sequence (5.8) U ( g x ) ι − → U ( g ) ad x π − − → U ( g ) ad x /  I x ∩ U ( g ) ad x  , where ι is the inclusion and π the pro jection. Both are morphisms of g x -sup ermo dules (adjoint action), and φ is an isomorphism of sup er vector spaces [ DS05 , Lemma 6.6]. Deﬁne (5.9) η : = φ − 1 ◦ π : U ( g ) ad x → U ( g x ) . Then u · m = η ( u ) · m for u ∈ U ( g ) ad x and m ∈ M x , since ker( x M ) is U ( g ) ad x -stable and I x k er( x M ) ⊆ im( x M ). Let Z ( g x ) b e the cen ter of U ( g x ). Since Z ( g ) ⊂ U ( g ) ad x and η is a morphism of g x - sup ermo dules, we hav e η  Z ( g )  ⊂ Z ( g x ) , and the dual map (5.10) η ∗ : Hom( Z ( g x ) , C ) → Hom( Z ( g ) , C ) is injective [ DS05 , Theorem 6.11]. Let χ b e the inﬁnitesimal character of M . F or z ∈ Z ( g ) and m ∈ ker( x M ), (5.11) χ ( z ) m = z m ≡ η ( z ) m (mo d xM ) , so if M x con tains a submo dule with inﬁnitesimal character ξ , then η ∗ ( ξ ) = χ . Moreov er, (5.12) at  η ∗ ( ξ )  = at( ξ ) + rk( x ) , and hence at( ξ ) = at( χ ) − rk( x ). Th us w e obtain: Theorem 65 ([ Ser11 , Theorem 2.1]) . L et M b e a simple highest weight g -sup ermo dule with inﬁn- itesimal char acter χ . Then M x de c omp oses as a dir e ct sum of g x -sup ermo dules whose gener alize d inﬁnitesimal char acters lie in ( η ∗ ) − 1 ( χ ) , and e ach ξ ∈ ( η ∗ ) − 1 ( χ ) satisﬁes at( ξ ) = at( χ ) − rk( x ) . Corollary 66. L et x ∈ Y \ { 0 } have r ank l , and let M b e a highest weight g -sup ermo dule of atypic ality k . If k < l , then DS x ( M ) = 0 . 5.2. A F amily of Nilp otent Perturbations. Fix a quadratic Lie subsup eralgebra l ⊂ g such that B l : = B | l is non-degenerate and (5.13) g = l ⊕ p , p = l ⊥ , where the decomp osition is orthogonal with resp ect to B . In what follows, w e embed W ( l ) in to W ( g ) using the embedding ( 3.28 ). Assume h ⊂ l and l ¯ 1  = { 0 } . Let (5.14) Y l : = { x ∈ l ¯ 1 : [ x, x ] = 2 x 2 = 0 } b e the self-commuting v ariety of l . If L ¯ 0 is a connected simply connected Lie group with Lie algebra l ¯ 0 , then Y l is an L ¯ 0 -stable Zariski-closed cone in l ¯ 1 . W e will deﬁne a family of cubic Dirac op erators parametrized by x ∈ Y l . Recall that the quantum W eil algebra W ( l ) is generated by 1 ⊗ x and γ W ( x ) = x ⊗ 1 + 1 ⊗ ν ∗ ( x ), with bidegrees ( ¯ 1 , p ( x )) and ( ¯ 0 , p ( x )) (hence total degrees p ( x ) + ¯ 1 and p ( x ), resp ectively). W e Perturbations of Dirac Op erators 41 write deg for the total degree on W ( l ). Unlik e the reductive (purely ev en) case, this grading allo ws square-zero p erturbations of the relative cubic Dirac op erator while keeping a nice square. Deﬁnition 67. F or x ∈ Y l set D x g , l : = D g , l + j ( γ W ( x )) ∈ W ( g ) . R emark 68 . Since j : W ( l )  → W ( g ) is a Lie sup eralgebra homomorphism, we ha v e j ( γ W ( x )) 2 = 0 for all x ∈ Y l . Indeed, x is o dd, hence γ W ( x ) is o dd for the total degree, and therefore j ( γ W ( x )) 2 = 1 2 [ j ( γ W ( x )) , j ( γ W ( x ))] W = 1 2 j ([ γ W ( x ) , γ W ( x )] W ) = 1 2 j ( γ W ([ x, x ] g )) = 0 . Let C l ( x ) : = { y ∈ l : [ x, y ] = 0 } b e the c ommutant of x in l . The following lemma summarizes the main prop erties D x g , l . Lemma 69. F or x ∈ Y l the fol lowing hold: a) D x g , l is o dd in W ( g ) with r esp e ct to the total de gr e e. b) D x g , l is C l ( x ) -invariant. c) F or g ∈ L ¯ 0 , one has Ad g (D x g , l ) = D Ad g ( x ) g , l . d) (D x g , l ) 2 = D 2 g , l . Pr o of. a) Both D g , l and j ( γ W ( x )) ha ve total degree ¯ 1, hence so does D x g , l . b) Let y ∈ C l ( x ). Since D g , l is l -inv ariant, it comm utes with y under the embedding l  → W ( g ). Moreo v er, [ γ W ( x ) , γ W ( y )] W = γ W ([ x, y ] g ) = 0 , so D x g , l is C l ( x )-in v ariant. c) This follows from Ad-equiv ariance of D g , l and of j ◦ γ W . d) Since D x g , l is o dd, (D x g , l ) 2 = 1 2 [D x g , l , D x g , l ] W = 1 2 [D g , l , D g , l ] W + [D g , l , j ( γ W ( x ))] W + 1 2 [ j ( γ W ( x )) , j ( γ W ( x ))] W . The middle term v anishes by l -in v ariance of D g , l , and the last term is zero by Remark 68 . Hence (D x g , l ) 2 = D 2 g , l . □ Fix a highest weigh t simple g -sup ermo dule M such that we can consider the family (5.15) Y l → End( M ⊗ M ( p )) , x 7→ D x g , l . Recall that w e assigned to any morphism T of a sup er v ector space V a cohomology H( T ; V ) (Section 3.5 ), which yields the Dirac cohomology H D g , l ( M ) for T = D g , l and V = M ⊗ M ( p ). Analogously , we deﬁne (5.16) H D x g , l ( M ) : = H(D x g , l ; M ⊗ M ( p )) = k er D x g , l / (k er D x g , l ∩ im D x g , l ) . Our goal is to describ e this cohomology . First, by Lemma 69 , we hav e the following lemma Lemma 70. F or any x ∈ Y l is H D x g , l ( M ) is a C l ( x ) -sup ermo dule. Computing the cohomology on M ⊗ M ( p ) reduces to the Laplacian ∆ g , l : = D 2 g , l . W e need: Lemma 71. The fol lowing hold: a) M ⊗ M ( p ) ∼ = k er ∆ g , l ⊕ im ∆ g , l . 42 STEFFEN SCHMIDT b) H ∆ g , l ( M ) : = H( ∆ g , l ; M ⊗ M ( p )) = ker ∆ g , l . Pr o of. Part a) is prov ed in [ NSS26 , Prop osition 4.1.2]. Part b) follo ws immediately from a). □ Lemma 72. One has H D x g , l ( M ) = H(D x g , l ; ker(D x g , l ) 2 ) = H(D x g , l ; ker D 2 g , l ) = H(D x g , l ; H ∆ g , l ( M )) . Pr o of. By deﬁnition, D x g , l = D g , l + j ( γ W ( x )), and Lemma 69 giv es (D x g , l ) 2 = D 2 g , l = ∆ g , l . T ogether with Lemma 71 , this yields M ⊗ M ( p ) ∼ = k er ∆ g , l ⊕ im ∆ g , l . Decomp ose M ⊗ M ( p ) into eigenspaces of ∆ g , l . Since ∆ g , l comm utes with D g , l and with j ( γ W ( x )), it commutes with D x g , l , hence ker D x g , l , im D x g , l , and H(D x g , l ; M ⊗ M ( p )) split as direct sums ov er these eigenspace s. Let λ  = 0 and let V λ b e the λ -eigenspace of ∆ g , l . It suﬃces to show that ker(D x g , l | V λ ) ⊂ im(D x g , l | V λ ). T ake v ∈ V λ with D x g , l v = 0. Then D g , l v = − j ( γ W ( x )) v , and hence D x g , l D g , l v = (D g , l + j ( γ W ( x ))) D g , l v = D 2 g , l v + j ( γ W ( x )) D g , l v = ∆ g , l v − j ( γ W ( x )) 2 v = 2 ∆ g , l v = 2 λv , where w e used j ( γ W ( x )) 2 = 0 and D g , l j ( γ W ( x )) = − j ( γ W ( x )) D g , l (b oth are o dd). Th us v = 1 2 λ D x g , l (D g , l v ) ∈ im(D x g , l | V λ ), pro ving the claim. Therefore only the λ = 0 eigenspace con tributes, i.e. , k er(D x g , l ) 2 = k er ∆ g , l = H ∆ g , l ( M ), and the stated equalities follow. □ F or a unitarizable highest w eigh t sup ermodule M (Section 2.4 ), the previous lemma yields an explicit description of H D x g , l ( M ) in terms of Dirac cohomology (Section 3.5 ) and Duﬂo–Serganov a cohomology (Section 5.1 ). Recall η x from ( 5.9 ) and its dual injectiv e map η ∗ x : Hom( Z ( l x ) , C ) → Hom( Z ( l ) , C ) for x ∈ Y l . Also let η l : Z ( g ) → Z ( l ) b e the homomorphism from Theorem 42 . W e then hav e: Theorem 73. Assume M is a unitarizable highest weight sup ermo dule with highest weight Λ . Then for any x ∈ Y l one has H D x g , l ( M ) = DS x (H D g , l ( M )) . In p articular, H D x g , l ( M ) is a l x -sup ermo dule a nd de c omp oses as a dir e ct sum of l x -sup ermo dules. If V is a l x -sup ermo dule with inﬁnitesimal char acter χ l x ν , then χ l x ν ∈ ( η ∗ x ) − 1 ( χ Λ ◦ η l ) . Pr o of. If M is unitarizable, then ker D 2 g , l = ker D g , l b y Lemma 43 and H D g , l ( M ) = ker D g , l b y unitarit y . Hence, by Lemma 72 , H D x g , l ( M ) = H(D x g , l ; M ⊗ M ( p )) = H(D x g , l ; ker D 2 g , l ) = H(D g , l + j ( γ W ( x )); ker D g , l ) = H( j ( γ W ( x )); ker D g , l ) . Perturbations of Dirac Op erators 43 The action of l on M ⊗ M ( p ) is induced by its embedding in to W ( l ) via γ W . Hence, by the em b edding ( 3.28 ), one has H D x g , l ( M ) = DS x (k er D g , l ) = DS x  H D g , l ( M )  . This prov es the ﬁrst part of the theorem. The remaining claims follo w b y combining Theo- rem 42 with Theorem 65 . □ R emark 74 . The pro of of Theorem 73 relies on the identit y k er D g , l = k er D 2 g , l , whic h holds for unitarizable M . In general, this is equiv alen t to eac h of the following conditions: (a) ker D g , l = k er D 2 g , l ; (b) ker D g , l ∩ im D g , l = { 0 } ; (c) M ⊗ M ( p ) = k er D g , l ⊕ im D g , l ; (d) H D g , l ( M ) = k er D g , l . Th us, the assumption is equiv alent both to the direct sum decomp osition in c) and to the iden tiﬁcation of Dirac cohomology with the full kernel in d). The following corollary is immediate. Corollary 75. L et M b e a unitarizable highest weight sup ermo dule with highest weight Λ . Then: a) H D x g , l ( M ) = 0 if rk( x ) > at(Λ) . b) F or g ∈ L ¯ 0 , ther e is a c anonic al isomorphism H D x g , l ( M ) ∼ = H D Ad g ( x ) g , l ( M ) . If M is not unitarizable, we can only formulate a v anishing conjecture. By Lemma 72 , H D x g , l ( M ) = H(D x g , l , ker D 2 g , l ) . Moreo v er, on V : = k er D 2 g , l , both D g , l and j ( γ W ( x )) are an ti-commuting diﬀeren tials, since [D g , l , j ( γ W ( x ))] W = 0 . This suggests the following v anishing statement from the p ersp ectiv e of sp ectral sequences. Conjecture 76. Let x ∈ Y l . If DS x  H D g , l ( M )  = 0, then H D x g , l ( M ) = 0 . 6. Bismut–Quillen’s Superconnection W e p erturb the relative cubic Dirac op erator by the cov arian t W eil diﬀeren tial asso ciated to a ﬁnite-dimensional (sup er)module M , obtained from the univ ersal connection 1-form of the colour W eil algebra. The resulting op erator deﬁnes an elemen t of End C ( W ( g ) ⊗ E ), where E : = M ⊗ S , and admits a well-deﬁned exp onential after completing W ( g ). T aking the relative sup ertrace yields a cohomology class in the completed colour W eil algebra, thereb y assigning to each ﬁnite-dimensional mo dule a canonical class. W e ﬁrst consider reductiv e complex Lie algebras, and then adapt the construction to ﬁnite-dimensional sup ermo dules o v er basic classical Lie sup eralgebras. Moreov er, Section 6.1.4 contains an explicit example of the construction in the case of the complex simple Lie algebra sl (2 , C ). 44 STEFFEN SCHMIDT 6.1. Semisimple Complex Lie Algebra. Let g b e a ﬁnite-dimensional semisimple complex Lie algebra with Killing form B . In the following, w e use B to iden tify g and g ∗ . Fix a Cartan subalgebra h ⊂ g with ro ot system ∆, and c ho ose a p ositiv e system ∆ + . This yields a triangular decomp osition (6.1) g = n − ⊕ h ⊕ n + , n ± : = M α ∈ ∆ + g ± α . Fix a basis { e a } of g and its B -dual basis { e a } . W e consider the quantum W eil algebra W ( g ), and ﬁx the tensor pro duct generators ( cf. Remark 23 ) (6.2) u a : = e a ⊗ 1 ∈ W 2 ( g ) , θ a : = 1 ⊗ e a ∈ W 1 ( g ) , where the u a are even and the θ a are o dd. W e deﬁne the g -value d quantum Weil algebr a to b e W ( g ) ⊗ g , and we equip it with the Lie brac ket (6.3) [ A, B ] : = X i,j ( a i b j ) ⊗ [ x i , y j ] g , where A = P i a i ⊗ x i , B = P j b j ⊗ y j ∈ W ( g ) ⊗ g . W e regard g as purely even; this induces a natural Z 2 -grading on W ( g ) ⊗ g from W ( g ). Henceforth, let ( ρ M , M ) b e a ﬁxed ﬁnite-dimensional simple g -mo dule; when M is clear from con text, we suppress ρ M . Recall the spin mo dule S from Section 3.4 . Note that S : = V n − carries a natural Z 2 -grading S = S ¯ 0 ⊕ S ¯ 1 , making it a g -sup ermo dule, i.e. , the g -action preserves the grading. Set (6.4) E : = M ⊗ S. Equipp ed with the induced grading from S , the g -mo dule ( ρ E , E ) b ecomes a g -sup ermo dule. Recall that W ( g ) is Z 2 -graded. Using the Z 2 -graded tensor pro duct ⊗ , the space W ( g ) ⊗ E is a sup er vector space. Hence End C ( W ( g ) ⊗ E ) inherits a Z 2 -grading: its even part consists of parity-preserving endomorphisms and its o dd part of parity-rev ersing endomorphisms. With the canonical sup ercommutator [ · , · ] End , it is a Lie sup eralgebra. W e hav e moreov er a natural action of g on W ( g ) ⊗ E : (6.5) α ( x )( v ⊗ w ) : = L x v ⊗ w + v ⊗ ρ E ( x ) w , x ∈ g , v ⊗ w ∈ W ( g ) ⊗ E , where L x denotes the Lie deriv ativ e ( 3.21 ) on W ( g ). 6.1.1. The Weil Covariant Diﬀer ential. The Bism ut–Quillen sup erconnection ma y b e viewed as a p erturbation of the cubic Dirac op erator b y the Weil c ovariant diﬀer ential ∇ M asso ciated with M , which we now deﬁne. The universal c onne ction 1 -form is the o dd element (6.6) Θ : = X a θ a ⊗ e a ∈ W ( g ) ⊗ g . It is indep enden t of the c hoice of basis. In trinsically , Θ is the canonical element corresp onding to id g under the identiﬁcation g ∗ ⊗ g ≃ End( g ), viewed in W 1 ( g ) ⊗ g . Moreov er, it is g -inv ariant for the diagonal action L x on the ﬁrst factor and ad x on the second factor. Perturbations of Dirac Op erators 45 Lemma 77. One has for al l x ∈ g ( L x ⊗ 1 + 1 ⊗ ad x )Θ = 0 , ( ι x ⊗ 1)Θ = x. Pr o of. W e ha ve b y deﬁnition ( L x ⊗ 1)Θ = X a L x θ a ⊗ e a = X a (1 ⊗ [ x, e a ]) ⊗ e a = X a (1 ⊗ X b B ([ x, e a ] , e b ) e b ) ⊗ e a = X b θ b ⊗ X a B ([ x, e a ] , e b ) e a = − X b θ b ⊗ X a B ([ x, e b ] , e a ) e a = − (1 ⊗ ad x )Θ , where we used in v ariance of B , that is, B ([ x, e a ] , e b ) = B ( x, [ e a , e b ]) = − B ( x, [ e b , e a ]) = − B ([ x, e b ] , e a ). The second prop ert y ( ι x ⊗ 1)Θ = x follows by deﬁnition. □ F urthermore, Θ satisﬁes a Maurer–Cartan curv ature t yp e equation. The following lemma is a straightforw ard computation. Lemma 78. One has ( d W ⊗ 1)Θ + 1 2 [Θ , Θ] = X a u a ⊗ e a . The universal connection 1-form Θ induces a C -linear endomorphism (6.7) Θ M : = X a θ a ⊗ ρ E ( e a ) ∈ End C ( W ( g ) ⊗ E ) , acting on W ( g ) ⊗ E comp onen t wise. Lemma 79. The fol lowing hold: a) Θ M ∈ End C ( W ( g ) ⊗ E ) ¯ 1 . b) Θ M is g -e quivariant, that is, [ α ( x ) , Θ M ] End = 0 for al l x ∈ g . c) ( d W ⊗ 1)Θ M + 1 2 [Θ M , Θ M ] End = F M with F M : = P a u a ⊗ ρ E ( e a ) . d) ( ι x ⊗ 1)Θ M = ρ E ( x ) for al l x ∈ g . Pr o of. a) follows from the deﬁnition of the Z 2 -grading. F or b), a direct calculation gives [ α ( x ) , Θ M ] End = X a L x θ a ⊗ ρ E ( e a ) + X a θ a ⊗ ρ E ([ x, e a ]) . Using Lemma 77 and applying ρ E to the g -factor, one gets X a ( L x θ a ) ⊗ ρ E ( e a ) + X a θ a ⊗ ρ E ([ x, e a ]) = 0 , hence b) follo ws. Finally , c) and d) f ollow from Lemma 77 and Lemma 78 b y applying 1 ⊗ ρ E . □ Let D g b e the cubic Dirac ope rator of g , and let d W : = [D g , · ] denote the asso ciated W eil diﬀeren tial on W ( g ). The Weil c ovariant diﬀer ential com bines b oth, Θ M and d W in to a single op erator acting on W ( g ) ⊗ E . Deﬁnition 80. The Weil c ovariant diﬀer ential is ∇ M : = d W ⊗ 1 + Θ M ∈ End C ( W ( g ) ⊗ E ) . The following lemma summarizes the main prop erties of ∇ M . 46 STEFFEN SCHMIDT Lemma 81. The fol lowing hold: a) ∇ M ∈ End C ( W ( g ) ⊗ E ) ¯ 1 . b) ∇ M is g -e quivariant, that is, [ α ( x ) , ∇ M ] End = 0 for al l x ∈ g . c) ∇ M and 1 ⊗ D g anti-c ommute, that is, [ ∇ M , 1 ⊗ D g ] End = 0 . d) ( ∇ M ) 2 = F M with F M : = P a u a ⊗ ρ E ( e a ) . e) F or any x ∈ g one has ( ι x ⊗ 1) ∇ M = − d W ( ι x ⊗ 1) + L x ⊗ 1 + 1 ⊗ ρ E ( x ) . Pr o of. a) follows from Lemma 28 , the deﬁnition of d W in Section 3.3.1 and Lemma 79 . Next, b) follows if w e show that [ α ( x ) , d W ⊗ 1] End =0 since [ α ( x ) , Θ M ] End = 0 by Lemma 79 . How ev er, b y Lemma 35 , one has [ α ( x ) , d W ⊗ 1] End = ( L x d W − ( − 1) p ( x ) d W L x ) ⊗ 1 = 0 . This prov es b). F or c), we compute [ ∇ M , 1 ⊗ D g ] End = [ d W ⊗ 1 , 1 ⊗ D g ] End + [Θ M , 1 ⊗ D g ] End = X a θ a ⊗ [ ρ E ( e a ) , D g ] = 0 b y the g -in v ariance of D g . Next, we pro ve d). Since ∇ M , d W ⊗ 1 and Θ M are o dd, one has ( ∇ M ) 2 = 1 2 [ ∇ M , ∇ M ] = ( d W ) 2 ⊗ 1 + Θ M ( d W ⊗ 1) + ( d W ⊗ 1)Θ M + (Θ M ) 2 = Θ M ( d W ⊗ 1) + ( d W ⊗ 1)Θ M + (Θ M ) 2 , where we used ( d W ) 2 = 0. The statement follo ws by computing the three app earing summands on any elementary tensor w ⊗ v b y linearity since the action is comp onent wise. One has: ( d W ⊗ 1)Θ M ( w ⊗ v ) = X a d W ( θ a w ) ⊗ v = X a (( d W θ a ) w − θ a d W w ) ⊗ e a v Θ M ( d W ⊗ 1)( w ⊗ v ) = X a ( θ a d W w ) ⊗ e a v , (Θ M ) 2 ( w ⊗ v ) = X a,b θ a θ b w ⊗ e a e b v = 1 2 X a,b θ a θ b w ⊗ [ e a , e b ] v No w, the statemen t follows b y using d W θ a = u a − 1 2 X b,c B ([ e b , e c ] , e a ) θ b θ c . Finally , e) follows by [ ι x , d W ] W = ι x d W + d W ι x = L x and Lemma 79 . □ R emark 82 . The prop erties of ∇ M justify the name cov ariant W eil diﬀerential. 6.1.2. Bismut–Quil len Sup er c onne ction. The Bismut–Quillen sup erconnection couples the cu- bic Dirac op erator acting on E to the quan tum W eil algebra W ( g ) using the W eil co v ariant diﬀeren tial. Deﬁnition 83. The Bismut–Quil len sup er c onne ction asso ciated with M is A M g ( t ) : = ∇ M + √ t (1 ⊗ D g ) ∈ End C ( W ( g ) ⊗ E ) , t > 0 . Perturbations of Dirac Op erators 47 The follo wing prop osition collects imp ortan t prop erties of the Bismut–Quillen sup erconnec- tion. Prop osition 84. The fol lowing hold: a) A M g ( t ) ∈ End C ( W ( g ) ⊗ E ) ¯ 1 . b) A M g ( t ) is g -e quivariant, that is, [ α ( x ) , A M g ( t )] End = 0 for al l x ∈ g . c) A M g ( t ) has squar e ( A M g ( t )) 2 = F M + t (1 ⊗ D 2 g ) . Pr o of. Since D g is an o dd op erator on E and ∇ M is o dd, the Bismut–Quillen sup erconnection A M g ( t ) is o dd. This prov es a). Statemen t b) is a direct consequence of the g -inv ariance of D g and Lemma 81 . It remains to compute the square. Since A M g ( t ) , 1 ⊗ D g and ∇ M are o dd, one has ( A M g ( t )) 2 = 1 2 [ A M g ( t ) , A M g ( t )] End = 1 2 [ ∇ M , ∇ M ] End + √ t 2 [ ∇ M , 1 ⊗ D g ] End + √ t 2 [1 ⊗ D g , ∇ M ] End + t 2 [1 ⊗ D g , 1 ⊗ D g ] End = ( ∇ M ) 2 + √ t [ ∇ M , 1 ⊗ D g ] End + t (1 ⊗ D 2 g ) = F M + t (1 ⊗ D 2 g ) , where we used in the last equality that [ ∇ M , 1 ⊗ D g ] End = 0 and F M : = ( ∇ M ) 2 . This ﬁnishes the pro of. □ Next, we reﬁne the deﬁnition by in volving certain quadratic subalgebras yielding a relative v ersion of the Bism ut–Quillen sup erconnection. As in Section 3.3 , ﬁx a quadratic Lie subalgebra l of g with non-degenerate sup ersymmetric inv arian t bilinear form B l : = B | l suc h that we hav e an orthogonal decomp osition (6.8) g = l ⊕ p , p = l ⊥ , with resp ect to B . Moreov er, assume h ⊂ l . Let D l ∈ W ( l ) denote the asso ciated cubic Dirac op erator. An y g -mo dule is in particular a l -mo dule. Under the natural embedding j : W ( l )  → W ( g ) deﬁned in ( 3.28 ), we consider j (D l ) as an endomorphism of E . Deﬁnition 85. The l -r elative Bismut–Quil len sup er c onne ction is A M g , l ( t ) : = A M g ( t ) − √ t (1 ⊗ j (D l )) = ∇ M + √ t (1 ⊗ D g , l ) . Recall that the space of l -basic elements is the subalgebra W ( g , l ) of elements annihilated b y L x and ι x for all x ∈ l . Inside W ( g ) ⊗ E , we consider the space of l -inv arian t elements (6.9) ( W ( g , l ) ⊗ E ) basic : = { v ⊗ w ∈ W ( g , l ) ⊗ E : α ( x )( v ⊗ w ) = 0 for all x ∈ l } . The l -relative Bismut–Quillen sup erconnection A M g , l ( t ) has the following prop erties: Prop osition 86. The fol lowing hold: a) A M g , l ( t ) ∈ End C ( W ( g ) ⊗ E ) ¯ 1 . b) A M g , l ( t ) is l -e quivariant, that is, [ α ( x ) , A M g , l ( t )] End = 0 for al l x ∈ l . 48 STEFFEN SCHMIDT c) A M g , l ( t ) r estricts to an endomorphism of ( W ( g , l ) ⊗ E ) b asic . d) A M g , l ( t ) has squar e ( A M g , l ( t )) 2 = F M + t (1 ⊗ D 2 g , l ) . Pr o of. a) follo ws from Lemma 81 and Prop osition 84 and b) follows from Lemma 28 and Prop o- sition 84 . Moreov er, c) is a direct consequence of b). Finally , d) follo ws by a similar line of argumen t as in Prop osition 84 using that ∇ M , D g and j (D l ) anti-comm ute. □ 6.1.3. Chern Char acter-T yp e Invariant. Using the Bismut–Quillen sup erconnection, w e assign to any ﬁnite-dimensional mo dule M ov er a complex semisimple Lie algebra g a cohomology class. W e use the notation from ab ov e. W e ﬁrst introduce a suitable completion of W ( g ). Consider the canonical em b edding ι : g  → U ( g ). There exists a unique unital algebra morphism  : U ( g ) → C , the augmentation , satisfying  ( ι ( x )) = 0 for all x ∈ g . Its k ernel I : = ker(  ) = U ( g ) g is the augmentation ide al of U ( g ). F or W ( g ) = U ( g ) ⊗ Cl( g ), deﬁne  W : =  ⊗ id Cl : W ( g ) → Cl( g ) and set J : = ker(  W ). Then J = I ⊗ Cl( g ). F or n ≥ 1, let J n denote the n -th p o w er of the t w o-sided ideal J ; explicitly , J n = I n ⊗ Cl( g ) for all n ≥ 1. The J -adic completion of W ( g ) is deﬁned as the pro jectiv e limit (6.10) c W ( g ) : = lim ← − n ≥ 1 W ( g ) /J n , where the transition maps are induced by the canonical surjections W ( g ) /J n +1 → W ( g ) /J n . In particular, (6.11) c W ( g ) ∼ = b U ( g ) ⊗ Cl( g ) , b U ( g ) : = lim ← − n ≥ 1 U ( g ) /I n . The diﬀeren tial d W of W ( g ) naturally induces a diﬀerential on c W ( g ), denoted b y the same sym b ol, since it preserves the J -adic ﬁltration. W e denote by c W ( g , l ) the subalgebra of l -basic elemen ts. F or any x ∈ g , iden tiﬁed with its image in W ( g ), the exp onential series exp( x ) : = P n ≥ 0 x n n ! con v erges in c W ( g ). Since E is ﬁnite-dimensional, algebraic exp onentials of endomorphisms of E are well-deﬁned. Hence: Lemma 87. One has e − ( A M g , l ( t )) 2 = ∞ X k =0 ( − 1) k  ( A M g , l ( t )) 2  k k ! ∈ End C  ( c W ( g , l ) ⊗ E ) basic  . Since E is ﬁnite-dimensional, w e can form the relative sup ertrace str E . Lemma 88. One has str E ( e − A M g , l ( t ) 2 ) ∈ c W ( g , l ) . Pr o of. Consider a general element T ∈ End C (( c W ( g , l ) ⊗ E ) basic ). In particular, ( ι x ⊗ 1) T = 0 and α ( x ) T = 0 for all x ∈ l . W e sho w that str E ( T ) ∈ c W ( g , l ), that is, it suﬃces to prov e that ι x str E ( T ) = 0 and L x str E ( T ) = 0 for all x ∈ l . Then the statement follows. Perturbations of Dirac Op erators 49 Fix x ∈ l . By linearity , it suﬃces to assume that T is a simple tensor of the form T = v ⊗ w . Then, using ( ι x ⊗ 1) T = 0, one has ι x str E ( T ) = ι x ( v str E ( w )) = ( ι x v ) str E ( w ) = str E (( ι x v ) ⊗ w ) = str E (( ι x ⊗ 1) T ) = 0 . Next, using α ( x ) T = 0, one has str E (( L x ⊗ 1) T ) = − str E ((1 ⊗ ad ρ E ( x ) ) T ) . Consequen tly , L x str E ( T ) = ( L x v ) str E ( w ) = str E (( L x ⊗ 1) T ) = − str E ((1 ⊗ ad ρ E ( x ) ) T ) = v str E ([ ρ E ( x ) , w ]) = 0 . This ﬁnishes the pro of. □ This leads to the following deﬁnition: (6.12) c h M ( t ) := str E  e − ( A M g , l ( t )) 2  ∈ c W ( g , l ) . In general, c h M ( t ) is non-zero, although str E ( e − D 2 g , l ) = 0 ( cf. Section 6.1.4 ). Since E is ﬁnite- dimensional, ch M ( t ) is well deﬁned for every t > 0 by Prop osition 86 and Lemma 87 . W e will sho w that c h M ( t ) is indep endent of t and determines a class in (6.13) H( c W ( g , l ) , d W g , l ) ∼ = b Z ( l ) , where b Z ( l ) denotes the completion of the center of U ( l ). Lemma 89. F or any T ∈ End C (( c W ( g , l ) ⊗ E ) b asic ) one has d W g , l str E ( T ) = str E ([ ∇ M , T ] End ) Pr o of. By linearit y of str E , it suﬃces to consider T = v ⊗ w for v ∈ c W ( g , l ) and w ∈ End C ( E ). Then str E ([ ∇ M , v ⊗ w ] End ) = str E ( d W g , l v ⊗ w + X a ( θ a v ⊗ [ ρ E ( e a ) , w ] End )) = ( d W g , l v ) str E ( w ) + X a ( θ a v ) str E ([ ρ E ( e a ) , w ] End ) = ( d W g , l v ) str E ( w ) , since str E ([ ρ E ( e a ) , w ]) = 0 for all a b y cyclicit y of the sup ertrace. □ Lemma 90. F or al l t > 0 , one has d W g , l c h M ( t ) = 0 . Pr o of. By Lemma 89 , one has d W g , l str E ( e − ( A M g , l ( t )) 2 ) = str E ([ ∇ M , e − ( A M g , l ( t )) 2 ] End ) = str E (0) = 0 since ∇ M comm utes with ( ∇ M ) 2 and D 2 g , l , and A M g , l ( t ) 2 = ( ∇ M ) 2 + t (1 ⊗ D 2 g , l ) by Prop osition 86 . □ 50 STEFFEN SCHMIDT Theorem 91. F or any ﬁnite-dimensional g -mo dule M one has [c h M ( t )] ∈ H( c W ( g , l ) , d W g , l ) . Mor e over, the class [ch M ( t )] is indep endent of t . Pr o of. By H( c W ( g , l ) , d W g , l ) = H 0 ( c W ( g , l ) , d W g , l ) ∼ = b Z ( U ( l )), and Lemma 90 , it follows that [c h M ( t )] ∈ H ( c W ( g , l ) , d W g , l ). It remains to sho w that it is indep enden t of t . F or this purp ose, w e consider A M g , l ( t ) as a smo oth family of endomorphisms in the v ariable t . Then, using that A M g , l ( t ) is o dd for all t , one has d d t ( A M g , l ( t ) 2 ) = A M g , l ( t ) d d t A M g , l ( t ) + d d t A M g , l ( t ) A M g , l ( t ) = [ A M g , l ( t ) , d d t A M g , l ( t )] End and thus d d t e − ( A M g , l ( t )) 2 = − ˆ 1 0 e − s ( A M g , l ( t )) 2 ( d d t ( A M g , l ( t )) 2 ) e − (1 − s )( A M g , l ( t )) 2 d s = − ˆ 1 0 e − s ( A M g , l ( t )) 2 [ A M g , l ( t ) , d d t A M g , l ( t )] End e − (1 − s )( A M g , l ( t )) 2 d s. Applying the sup ertrace and using [ A M g , l ( t ) , e − ( A M g , l ( t )) 2 ] End = 0, one has d d t c h M ( t ) = − str E ([ A M g , l ( t ) , d d t A M g , l ( t )] End e − ( A M g , l ( t )) 2 ) = − str E ([ A M g , l ( t ) , d d t A M g , l ( t )) e − ( A M g , l ( t )) 2 ] End ) = − str E ([ ∇ M , ( d d t A M g , l ( t )) e − ( A M g , l ( t )) 2 ] End ) = − d W g , l str E (( d d t A M g , l ( t )) e − ( A M g , l ( t )) 2 ) . This ﬁnishes the pro of. □ Altogether, we hav e assigned to eac h ﬁnite-dimensional simple mo dule a cohomology class in the colour quan tum W eil algebra, which can b e seen as a coarse Chern–W eil type in v ariant of g -mo dules. W e close this section with an example. 6.1.4. Example: sl (2 , C ) and the Bismut–Quil len Sup er c onne ction. W e consider g : = sl (2 , C ). Fix a standard g -triple ( e, f , h ) with (6.14) [ h, e ] = 2 e, [ h, f ] = − 2 f , [ e, f ] = h. Let h : = C h b e the Cartan subalgebra and let B ( X, Y ) : = tr C (ad X ad Y ) b e the Killing form (standard normalization). Set n + : = C e and n − : = C f , so that g = n − ⊕ h ⊕ n + . F or n ∈ Z ≥ 0 , let V n b e the simple g -mo dule of highest weigh t n . It has basis v 0 , . . . , v n with action hv k = ( n − 2 k ) v k , f v k =    v k +1 , 0 ≤ k < n, 0 , k = n, ev k =    0 , k = 0 , k ( n − k + 1) v k − 1 , 1 ≤ k ≤ n. Perturbations of Dirac Op erators 51 Th us the weigh ts are n, n − 2 , . . . , − n , eac h with multiplicit y one. The cen ter Z ( g ) of the universal en v eloping algebra is generated by the quadratic Casimir element (6.15) Ω g = ef + f e + 1 2 h 2 , whic h acts on V n b y the scalar n ( n +2) 2 . Let S := V n − b e the ρ -shifted spin mo dule for the Cliﬀord algebra of p := n + ⊕ n − , with resp ect to the pairing induced by B . It is generated by s ¯ 0 := 1 ∈ V 0 n − and s ¯ 1 := f ∈ V 1 n − , of parity ¯ 0 and ¯ 1, resp ectively . The action of g is giv en by e · s ¯ 0 = s ¯ 1 , e · s ¯ 1 = 0 , f · s ¯ 0 = 0 , f · s ¯ 1 = s ¯ 0 , h · s ¯ 0 = s ¯ 0 , h · s ¯ 1 = − s ¯ 1 . Hence S ¯ 0 has weigh t − ρ and S ¯ 1 has weigh t ρ . Therefore S is the simple g -mo dule of highest w eigh t ρ , that is, S ∼ = V 1 . Its Z 2 -grading is S ¯ 0 = V 0 n − = C s ¯ 0 and S ¯ 1 = V 1 n − = C s ¯ 1 . As a quadratic subalgebra w e tak e l : = h . It has quadratic Casimir Ω h = 1 8 h 2 . It acts on a one-dimensional h -mo dule C λ (where h acts by λ ) by λ 2 8 . As an h -sup ermo dule, V n ⊗ S decomposes into one-dimensional w eight spaces, with parity determined by the S -factor: (6.16) ( V n ⊗ S ) ¯ 0 = n M k =0 C ( v k ⊗ s ¯ 0 ) , ( V n ⊗ S ) ¯ 1 = n M k =0 C ( v k ⊗ s ¯ 1 ) , and (6.17) h ( v k ⊗ s ¯ 0 ) = ( n − 2 k + 1)( v k ⊗ s ¯ 0 ) , h ( v k ⊗ s ¯ 1 ) = ( n − 2 k − 1)( v k ⊗ s ¯ 1 ) . Equiv alently , writing C λ for the one-dimensional h -mo dule of weigh t λ , (6.18) V n ⊗ S ∼ = n M k =0 C n − 2 k +1 ! ¯ 0 ⊕ n M k =0 C n − 2 k − 1 ! ¯ 1 . Th us the o dd weigh ts are n − 1 , n − 3 , . . . , − n − 1, while the even weigh ts are n +1 , n − 1 , . . . , − n + 1. Let D g , h b e the h -relative cubic Dirac op erator. It squares to (6.19) D 2 g , h = Ω g ⊗ 1 − 1 8 ( h ⊗ 1 + 1 ⊗ γ W ( h )) − 1 8 (1 ⊗ 1) . Consequen tly , if C µ is a h -weigh t space of V n ⊗ S , it acts as scalar multiplication by (6.20) ( n + 1) 2 − µ 2 8 . W e conclude that C µ b elongs to ker D 2 g , h iﬀ µ = ± ( n + 1). Because eac h ﬁnite-dimensional highest weigh t mo dule V n carries a su (2)-inv ariant positive Hermitian form (so the compact real form acts b y skew-Hermitian op erators), the basis-independent cubic Dirac op erator D g , h is therefore selfadjoint, and for a selfadjoint op erator one has k er D g , h = k er D 2 g , h . Hence (6.21) k er D g , h = k er D 2 g , h = C n +1 ⊕ C − ( n +1) . Note that C n +1 is spanned b y v 0 ⊗ s ¯ 0 and C − ( n +1) is spanned b y v n ⊗ s ¯ 1 , that is, C n +1 is ev en while C − ( n +1) is o dd. In particular, sdim C (k er D g , h ) = 1 − 1 = 0. A naive construction of an inv arian t w ould b e (6.22) str V n ⊗ S ( e − D 2 g , h ) . 52 STEFFEN SCHMIDT Ho w ever, this is alwa ys trivial. Indeed, b y Lemma 71 , we hav e V n ⊗ S ∼ = k er D 2 g , h ⊕ im D 2 g , h , and since D 2 g , h is p ositive, all its eigenv alues are p ositiv e. Let λ > 0 b e an eigenv alue, and set E : = V n ⊗ S . Then (6.23) D g , h : k er(D 2 g , h − λ ) ∩ E ¯ 0 → ker(D 2 g , h − λ ) ∩ E ¯ 1 deﬁnes an isomorphism. W e conclude (6.24) str E ( e − D 2 g , h ) = str E (id ker D 2 g , h ) = sdim(k er D 2 g , h ) = 1 − 1 = 0 . If, instead, w e couple D g , h to the co v ariant W eil diﬀerential coming from the universal con- nection 1-form, the resulting class is non-trivial. F or the standard triple { e, f , h } , denote the corresp onding ev en and o dd generators in W ( g ) b y u e , u f , u h and θ e , θ f , θ h . Then (6.25) ∇ V n = 1 4 θ e ⊗ ρ E ( f ) + 1 4 θ f ⊗ ρ E ( e ) + 1 8 θ h ⊗ ρ E ( h ) , F V n = 1 4 u e ⊗ ρ E ( f ) + 1 4 u f ⊗ ρ E ( e ) + 1 8 u h ⊗ ρ E ( h ) . The g -action on E is giv en by (6.26) ρ E ( e ) = ρ V n ( e ) ⊗ 1 , ρ E ( f ) = ρ V n ( f ) ⊗ 1 , ρ E ( h ) = ρ V n ( h ) ⊗ 1 + 1 ⊗ γ ′ ( h ) , where the last formula is sp eciﬁed in ( 6.17 ). If n = 0, so that V 0 ∼ = C is trivial, then ρ V 0 ( e ) = ρ V 0 ( f ) = ρ V 0 ( h ) = 0, hence (6.27) ∇ V 0 = 1 8 u h ⊗ ρ E ( h ) , F V 0 = 1 8 u h ⊗ ρ E ( h ) , D 2 g , h = 0 , and therefore (6.28) c h V 0 ( t ) = str S ( e − F V 0 ) . Using ρ E ( h ) s ¯ 0 = s ¯ 0 and ρ E ( h ) s ¯ 1 = − s ¯ 1 , one obtains (6.29) c h V 0 ( t ) = e 1 8 u h − e − 1 8 u h = 2 sinh  1 8 u h  ∈ c W ( g , l ) , whic h deﬁnes a class in H( c W ( g , l ) , d W g , l ) ∼ = C [[ h ]]. F or n ≥ 1, equip V n with a Hermitian form suc h that D g , h is selfadjoin t and D 2 g , h is non- negativ e. Decomp osing E into D 2 g , h -eigenspaces E ( λ ) gives (6.30) e − t D 2 g , h t →∞ − − − →    1 , λ = 0 , 0 , λ  = 0 . Since D g , h and ∇ V n comm ute and [c h V n ( t )] is indep endent of t , (6.31) [c h V n ( t )] = [str E ( e − F V n e − t D 2 g , h ] = [str ker D g , h  e − F V n  ] , and by ( 6.21 ), (6.32) [c h V n ( t )] =  e n +1 8 u h − e − n +1 8 u h  =  2 sinh  n +1 8 u h  ∈ H( c W ( g , h ) , d W g , h ) ∼ = C [[ h ]] . 6.2. A W ord on Basic Classical Lie Sup eralgebras. Let g b e a basic classical Lie sup eral- gebra and let l ⊆ g b e a quadratic Lie subsup eralgebra as in Section 3.3 , so that g = l ⊕ p . F or ev ery ﬁnite-dimensional g -sup ermo dule M we may form the Bism ut–Quillen sup erconnection A M g , l ( t ) introduced ab o ve; with the usual sign conv en tions, the construction and its formal prop- erties extend to the sup er setting. In contrast to the case of a complex semisimple Lie algebra, the oscillator mo dule M ( p ) is inﬁnite-dimensional, and hence the sup ertrace on E : = M ⊗ M ( p ) is not a priori a v ailable. Nevertheless, for unitarizable ﬁnite-dimensional g -sup ermo dules M (Section 2.4 ) one can extract a canonical substitute. In this case D g , l is selfadjoin t, hence D 2 g , l is p ositiv e. W riting E = L λ ≥ 0 E ( λ ) for the D 2 g , l -eigenspace decom position, one has on E ( λ ) e − t D 2 g , l − − − → t →∞    id E (0) , λ = 0 , 0 , λ  = 0 . Moreo v er k er D 2 g , l = k er D g , l = H D g , l ( M ) and k er D g , l is ﬁnite-dimensional b y [ NSS26 ]. Hence, whenev er a relative sup ertrace on E can b e deﬁned so that the transgression argument applies, the class [ χ M ( t )] is indep endent of t , and letting t → ∞ yields the substitute (6.33) e c h M : = str ker D g , l  e F M  . 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Dirac op erators and cohomology for Lie superalgebra of t ype I. J. Lie The ory , 27(1):111–121, 2017. – cited on p. 2 Center for Quantum Ma thema tics, University of Southern Denmark, DK-5230 Odense, Denmark Email addr ess : stschmidt@imada.sdu.dk

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