On the full set of unitarizable supermodules over $\mathfrak{sl}(m\vert n)$

On the F ull Set of Unitarizable Sup ermo dules o v er sl ( m | n ) Steﬀen Sc hmidt Abstract. W e presen t a nov el classiﬁcation of unitarizable supermo dules ov er special linear Lie superalgebras using an algebraic quadratic Dirac operator in tro duced by Huang and P and ˇ zi ´ c and a corresp onding Dirac inequalit y . Contents 1. In tro duction 1 2. Sp ecial Linear Lie Superalgebras sl ( m | n ) 8 3. Unitarizable Sup ermodules and Dirac Op erators 12 4. Classifying the F ull Set 25 References 42 1. Introduction The sp e cial line ar Lie sup er algebr a sl ( m | n ) is deﬁned as the Lie subsuperalgebra of the general linear Lie superalgebra gl ( m | n ) consisting of all blo c k matrices of the form (1.1) X = A B C D ! , where A is an m × m complex matrix, B an m × n complex matrix, C an n × m complex matrix, and D an n × n complex matrix. These matrices are sub ject to the condition that the supertrace v anishes, i.e. , (1.2) str( X ) : = tr( A ) − tr( D ) = 0 . Its natural Z 2 -grading is given by sl ( m | n ) = sl ( m | n ) ¯ 0 ⊕ sl ( m | n ) ¯ 1 , where sl ( m | n ) ¯ 0 consists of all blo c k matrices with B = C = 0, and sl ( m | n ) ¯ 1 consists of all blo ck matrices with A = D = 0 (still sub ject to the v anishing sup ertrace condition). In what follo ws, w e write g = sl ( m | n ) and assume m + n > 2, th us excluding the nilp oten t case m = n = 1. Since s l ( m | n ) ∼ = s l ( n | m ), w e ma y further assume m ≤ n . In this w ork w e giv e a classiﬁcation of unitarizable supermo dules o v er g . They are semisimple, and therefore constitute a fundamental class of sup ermo dules from which one may approac h the general theory . They also appear naturally in mathematical ph ysics, in particular in the study of sup erconformal quantum ﬁeld theories (see [ ES15 , CDI19 ] and the references cited therein). Unitarizable g -sup ermo dules are deﬁned with resp ect to a conjugate-linear anti-in v olution ω . The possible choices of ω (equiv alently , the real forms of g ) are classiﬁed in [ P ar80 , Ser83 ]. A 1 2 STEFFEN SCHMIDT g -sup ermo dule M is said to be ω -unitarizable if there exists a p ositive deﬁnite Hermitian form ⟨· , ·⟩ on M such that (1.3) ⟨ xv , w ⟩ = ⟨ v , ω ( x ) w ⟩ ( x ∈ g , v , w ∈ M ) . Unitarizable g -sup ermo dules are rare. In [ NS11 ], Neeb and Salmasian prov ed that an y ω -unitarizable g -supermo dule is trivial unless ω corresp onds to a real form su ( p, q | 0 , n ) or su ( p, q | n, 0), and in these cases the sup ermo dule is either a highest or a lo west w eight sup er- mo dule [ FN91 ]. F or p = 0 or q = 0, the unitarizable g -sup ermo dules are ﬁnite-dimensional; for p, q  = 0, they are inﬁnite-dimensional. The treatmen t of highest weigh t and lo west w eigh t g -sup ermo dules is analogous. This reduces the classiﬁcation to the follo wing problem: Problem. Describ e all unitarizable simple g -supermo dules; equiv alen tly , determine all highest w eigh ts Λ ∈ h ∗ for which there exists an ω -unitarizable simple g -sup ermo dule of highest w eigh t Λ. Our approac h is based on the algebraic Dirac op erator D, in tro duced b y Huang and Pand ˇ zi ´ c [ HP05 ] for Lie sup eralgebras of Riemannian type. The sup ertrace deﬁnes a sup ersymmetric in v ariant bilinear form on gl ( m | n ), whic h is non-degenerate if and only if m  = n . Accordingly , for m  = n we w ork in the quadratic Lie sup eralgebra g , whereas in the remaining case w e w ork in gl ( m | n ). In either case, the restriction of the form to g ¯ 1 is symplectic and hence gives rise to a W eyl algebra W ( g ¯ 1 ). After choosing a symplectic basis { x i , ∂ i } of g ¯ 1 , compatible with a ﬁxed conjugate-linear an ti-in volution ω , one obtains the distinguished element (1.4) D = 2 mn X i =1 ( ∂ i ⊗ x i − x i ⊗ ∂ i ) ∈ U ( g ) ⊗ C W ( g ¯ 1 ) . It comm utes with the adjoin t action of g ¯ 0 and its square is giv en b y a sum of quadratic Casimir elemen ts and a scalar. These prop erties make D the basic to ol in our analysis. Let M ( g ¯ 1 ) be the W eyl module ov er W ( g ¯ 1 ), and let M b e a highest w eigh t g -sup ermodule. The W eyl mo dule M ( g ¯ 1 ) is naturally a unitarizable g ¯ 0 -mo dule, called the ladder module, and D acts on M ⊗ M ( g ¯ 1 ). If M is ω -unitarizable, then D 2 is either negative- or p ositive-deﬁnite, according to whether M is ﬁnite- or inﬁnite-dimensional. This yields a Dirac inequalit y for ev ery g ¯ 0 -constituen t L 0 ( µ ) of M ⊗ M ( g ¯ 1 ): (1.5) ( µ + 2 ρ, µ )    < (Λ + 2 ρ, Λ) , if M is ﬁnite-dimensional , > (Λ + 2 ρ, Λ) , if M is inﬁnite-dimensional . Con v ersely , [ Sch24 ] prov es that if Λ is the highest w eigh t of a unitarizable g ¯ 0 -mo dule satisfying suitable unitarity conditions, and if this inequality is strict for all g ¯ 0 -comp osition factors in a g ¯ 0 -ﬁltration of M , then L (Λ) is unitarizable. This is the criterion on whic h our classiﬁcation rests. 1.1. The Classiﬁcation Metho d. W e classify all unitarizable highest weigh t sup ermo dules o v er g , equiv alently all su ( p, q | 0 , n )-unitarizable sup ermo dules. The cases p = 0 or q = 0 giv e exactly the ﬁnite-dimensional unitarizable sup ermo dules, whereas the case p, q  = 0 giv es the On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 3 gen uinely inﬁnite-dimensional part of the classiﬁcation, apart from the trivial sup ermo dule. Accordingly , we consider these cases separately . In each case w e ﬁx a certain p ositiv e system with Borel b and consider highest w eight sup ermo dules whose highest weigh t vectors are even. Accordingly , all classiﬁcation results are understo od up to application of the parity reversion functor Π. The classiﬁcation of unitarizable highest weigh t g -sup ermo dules L (Λ) starts from the imme- diate consequences of the deﬁnition of unitarity . Since unitarizability with resp ect to g implies unitarizabilit y with resp ect to g ¯ 0 , we ma y assume that Λ is the highest weigh t of a unitarizable g ¯ 0 -mo dule L 0 (Λ). The classiﬁcation of such mo dules is well kno wn. W e then pro ve that L b (Λ) has a g ¯ 0 -ﬁltration with simple comp osition factors, eac h a unitarizable highest w eigh t g ¯ 0 -mo dule. In the ﬁnite-dimensional case, L b (Λ) decomp oses in to a direct sum of these comp osition factors. In addition, the unitarizability of L b (Λ) forces certain conditions on Λ and on the relations among its en tries in the standard realization; see Lemma 6 . W e refer to these as the unitarity conditions. These conditions ﬁrst reduce the p ossible highest weigh ts of unitarizable highest w eigh t sup ermo dules to a manageable class. Within this class, the Dirac inequality coming from suitable g ¯ 0 -comp osition factors of L b (Λ) then singles out exactly those highest w eigh ts that are unitarizable. 1.1.1. The Finite-Dimensional Case. In the case p = 0 or q = 0, w e work with the distinguished Borel subalgebra and assume that Λ satisﬁes the unitarit y conditions. W e then obtain necessary and suﬃcient conditions for the unitarizabilit y of L (Λ) in terms of the Dirac inequalities. Once an explicit parametrization of Λ is av ailable, this criterion b ecomes completely explicit. In parallel, we describ e the metho d b oth in terms of the Dirac inequalities and in terms of an explicit realization. More concretely , the highest weigh t Λ admits tw o equiv alent descriptions. In standard co or- dinates, Λ has general form (1.6) Λ = ( λ 1 , . . . , λ m | µ 1 , . . . , µ k 0 − 1 , µ, . . . , µ ) , with µ k 0 − 1  = µ . Equiv alently , using the real form of the even Lie subalgebra, ev ery such Λ may b e written in the form (1.7) Λ = Λ 0 + x 0 2 (1 , . . . , 1 | 1 , . . . , 1) , x 0 ∈ R , where Λ 0 is the highest weigh t of a unitarizable g ¯ 0 -mo dule. Th us, for ﬁxed Λ 0 , one obtains a one-parameter family of highest weigh ts (1.8) Λ( x ) = Λ 0 + x 2 (1 , . . . , 1 | 1 , . . . , 1) , of which Λ is the member corresp onding to the v alue x = x 0 and each Λ( x ) is the highest weigh t of a ﬁnite-dimensional unitarizable g ¯ 0 -mo dule. The Dirac inequalities then give necessary and suﬃcien t conditions on x for unitarizability . The classiﬁcation of all unitarizable sup ermo dules may b e view ed either in terms of the Dirac inequalities for Λ or, for ﬁxed Λ 0 , as a three-step analysis of the parameter x . The analysis rests on a substantial simpliﬁcation of the Dirac inequalities. The key ob jects are the g ¯ 0 -constituen ts 4 STEFFEN SCHMIDT of the form L 0 (Λ( x ) − α ), with α ∈ ∆ + ¯ 1 , together with the asso ciated Dirac inequalities (Λ( x ) + ρ, α ) > 0. (1) The ﬁrst step is to determine all v alues of x such that (Λ( x ) + ρ, α ) > 0 for ev ery α ∈ ∆ + ¯ 1 . In this case, the Dirac inequality holds automatically on eac h g ¯ 0 - constituen t. This deﬁnes a threshold x max ∈ R with the prop ert y that the simple highest w eigh t g -sup ermo dule L (Λ( x )) is unitarizable for all x ∈ ( x max , ∞ ). Equiv alently , L (Λ) is unitarizable whenever (Λ + ρ, α ) > 0 for ev ery α ∈ ∆ + ¯ 1 . (2) The next step is to determine a threshold x min ∈ R , using the unitarity conditions, the Dirac inequality , and the Kac–Shap o v alov determinant form ula, b elow whic h unitarity fails. Namely , x min is the largest real num b er such that for every x < x min there exists α ∈ ∆ + ¯ 1 for which L (Λ( x )) has a g ¯ 0 -constituen t of highest w eigh t Λ( x ) − α and the Dirac inequalit y fails (Λ( x ) + ρ, α ) < 0 . F or all such x , the mo dule L (Λ( x )) is not unitarizable. By the unitarity conditions, this threshold is determined b y the v alue of (Λ( x ) + ρ, ϵ m − δ k 0 ). Equiv alently , L (Λ) for Λ ∈ h ∗ of the form ( 1.6 ) is not unitarizable if (Λ + ρ, ϵ m − δ k 0 ) < 0 . (3) It remains to analyze the in terv al I : = [ x min , x max ] . W e sho w that, on this interv al, L (Λ( x )) is unitarizable precisely when x is in tegral, or equiv alently , precis ely when (Λ( x ) + ρ, ϵ m − δ k ) = 0 for some k = k 0 , . . . , n. If x is non-integral, then Λ( x ) fails to satisfy the unitarity conditions. If x is integral, then atypicalit y implies that the g ¯ 0 -constituen ts for which the Dirac inequalit y fails do not o ccur in the g ¯ 0 -decomp osition of L (Λ( x )). As a consequence, we obtain the following classiﬁcation of all ﬁnite-dimensional unitarizable highest w eight g -supermo dules. The following theorem app ears as Theorem 27 in the main text. Theorem 1. Λ ∈ h ∗ is the highest weight of a unitarizable highest weight g -sup ermo dule if and only if the fol lowing c onditions hold: a) Λ satisﬁes the unitarity c onditions. b) If k 0 is the smal lest inte ger such that (Λ , δ k 0 − δ n ) = 0 , then one of the fol lowing holds: (i) (Λ + ρ, ϵ m − δ k ) = 0 for some k = k 0 , . . . , n , or (ii) (Λ + ρ, ϵ m − δ n ) > 0 . 1.1.2. Inﬁnite-Dimensional Case. In the case p, q  = 0, w e work with the standard p ositiv e system for ∆ ¯ 0 and with the non-standard p ositive system ∆ + ¯ 1 = A ⊔ B , where (1.9) A : = { ϵ i − δ k : 1 ≤ i ≤ p, 1 ≤ k ≤ n } , B : = {− ϵ j + δ k : p + 1 ≤ j ≤ m, 1 ≤ k ≤ n } , On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 5 as this choice is adapted to the study of unitarity and the Dirac op erator. The classiﬁcation for the distinguished p ositive system then follows b y applying a sequence of o dd reﬂections. As in the ﬁnite-dimensional case, w e obtain necessary and suﬃcient conditions for the unitarizability of L (Λ) in terms of the Dirac inequalities. Once a suitable parametrization is ﬁxed, these conditions tak e an explicit form. Assume that Λ satisﬁes the unitarity conditions. It has in standard co ordinates the general form (1.10) Λ = ( λ, . . . , λ, λ i 0 +1 , . . . , λ p , λ p +1 , . . . , λ m − j 0 − 1 , λ ′ , . . . , λ ′ | µ 1 , . . . , µ n ) , with λ i 0 +1  = λ and λ m − j 0 − 1  = λ ′ . Equiv alently , (1.11) Λ = Λ 0 + x 0 2 (1 , . . . , 1 | 1 , . . . , 1) , x 0 ∈ R , where Λ 0 is the highest w eight of an inﬁnite-dimensional unitarizable highest weigh t g ¯ 0 -mo dule. Th us, for ﬁxed Λ 0 , one obtains a one-parameter family of highest weigh ts (1.12) Λ( x ) = Λ 0 + x 2 (1 , . . . , 1 | 1 , . . . , 1) , of which Λ is the member corresp onding to the v alue x = x 0 and each Λ( x ) is the highest weigh t of a unitarizable g ¯ 0 -mo dule. As in the ﬁnite-dimensional case, the Dirac inequalities provide necessary and suﬃcient conditions for the unitarizability of L (Λ( x )). The analysis is gov erned b y the g ¯ 0 -comp osition factors L 0 (Λ − α ) and L 0 (Λ − β ), where α ∈ A and β ∈ B , together with the corresp onding Dirac inequalities (Λ + ρ, α ) ≤ 0 and (Λ + ρ, β ) ≤ 0. This leads naturally to a separate treatment of the tw o families. F rom this p oin t of view, the classiﬁcation of all unitarizable sup ermo dules may b e describ ed either in terms of the Dirac inequalities for Λ or, for ﬁxed Λ 0 , as a three-step analysis of the parameter x . (1) A ﬁrst step is to determine the range of v alues of x for which (Λ( x ) + ρ, α ) < 0 for all α ∈ A, (Λ( x ) + ρ, β ) < 0 for all β ∈ B . When these inequalities imp ose non-trivial restrictions, the Dirac inequalit y holds au- tomatically on all g ¯ 0 -constituen ts. Indeed, they deﬁne tw o thresholds: a left threshold x L max , determined by the maximal v alue of (Λ( x ) + ρ, β ), and a righ t threshold x R min , deter- mined b y the maximal v alue of (Λ( x ) + ρ, α ). It then follows that L (Λ( x )) is unitarizable for all x ∈ ( x L max , x R min ) . Equiv alently , L (Λ) is unitarizable whenever (Λ + ρ, α ) < 0 for all α ∈ A, (Λ + ρ, β ) < 0 for all β ∈ B . (2) The next step is to determine the extremal v alues x L min and x R max , using the unitarit y conditions, the Dirac inequalit y , and the Kac–Shap ov alov determinant formula, where x L min is the minimal v alue such that L (Λ( x )) is not unitarizable for all x < x L min , and x R max is the maximal v alue suc h that L (Λ( x )) is not unitarizable for all x > x R max . These thresholds are gov erned b y the o ccurrence of g ¯ 0 -comp osition factors for which the Dirac inequalities fail. By the unitarit y conditions, x L min is determined by (Λ( x )+ ρ, − ϵ m − j 0 + δ n ) 6 STEFFEN SCHMIDT and x R max b y (Λ( x ) + ρ, ϵ i 0 − δ 1 ). Equiv alently , for Λ of the form ( 1.10 ), L (Λ) is not unitarizable whenev er (Λ + ρ, ϵ i 0 − δ 1 ) > 0 or (Λ + ρ, − ϵ m − j 0 + δ 1 ) > 0. (3) It remains to consider the residual in terv als I L : = [ x L min , x L max ] , I R : = [ x R min , x R max ] , and, if x L max < x R min , then I : = [ x L min , x R max ] . As these interv als hav e integral length, they admit a natural notion of integralit y . Unitarizability on these interv als o ccurs precisely at the integral p oints, or equiv alen tly , precisely when (Λ + ρ, − ϵ m − j + δ n ) = 0 or (Λ + ρ, ϵ i − δ 1 ) = 0 for some 0 ≤ j ≤ j 0 or 1 ≤ i ≤ i 0 . The non-in tegral p oint s are excluded by the unitarit y conditions, whereas at the in tegral p oints the Dirac inequalities are satisﬁed on all g ¯ 0 - comp osition factors. Equiv alently , let Λ ∈ h ∗ b e a weigh t of the form ( 1.10 ) such that (Λ + ρ, α )  < 0 for some α ∈ A or (Λ + ρ, β )  < 0 for some β ∈ B . Then L (Λ) is unitarizable if and only if one of the following conditions holds: (i) (Λ + ρ, β ) < 0 for all β ∈ B and (Λ + ρ, ϵ i − δ 1 ) = 0 for some 1 ≤ i ≤ i 0 ; (ii) (Λ + ρ, − ϵ m − j + δ n ) = 0 and (Λ + ρ, ϵ i − δ 1 ) = 0 for some 0 ≤ j ≤ j 0 and some 1 ≤ i ≤ i 0 ; (iii) (Λ + ρ, − ϵ m − j + δ n ) = 0 for some 0 ≤ j ≤ j 0 and (Λ + ρ, α ) < 0 for all α ∈ A . As a consequence, we obtain the following classiﬁcation of all unitarizable highest weigh t g - sup ermo dules in the case p, q  = 0. The following theorem app ears as Theorem 34 in the main text. Theorem 2. R epr esent any weight Λ ∈ h ∗ in the form ( 1.10 ) . Then Λ is the highest weight of a unitarizable highest weight g -sup ermo dule if and only if the fol lowing holds: a) Λ satisﬁes the unitarity c onditions of L emma 6 , and b) one of the fol lowing c onditions hold: (i) (Λ + ρ, − ϵ m + δ n ) < 0 and (Λ + ρ, ϵ i − δ 1 ) = 0 for 1 ≤ i ≤ i 0 ; (ii) (Λ + ρ, − ϵ m − j + δ n ) = 0 and (Λ + ρ, ϵ i − δ 1 ) = 0 for 0 ≤ j ≤ j 0 and 1 ≤ i ≤ i 0 ; (iii) (Λ + ρ, − ϵ m − j + δ n ) = 0 and (Λ + ρ, ϵ 1 − δ 1 ) < 0 for 0 ≤ j ≤ j 0 ; (iv) (Λ + ρ, − ϵ m + δ n ) < 0 and (Λ + ρ, ϵ 1 − δ 1 ) < 0 . 1.2. Comparison to the Literature. The classiﬁcation of unitarizable sl ( m | n )-sup ermo dules has b een addressed in three main works. The ﬁrst is the w ork of F urutsu–Nishiy ama [ FN91 ], whic h classiﬁes the unitarizable supermo d- ules with integral highest w eigh t. Their method is based on a realization of the Lie superalgebras su ( p, q | 0 , n ) inside suitable orthosymplectic Lie sup eralgebras. They then study the restriction to su ( p, q | 0 , n ) of the oscillator sup ermodule, namely the unique unitarizable simple sup ermo d- ule of the am bient orthosymplectic Lie sup eralgebra. This construction is the sup er-analogue of the classical oscillator representation, also known as the Segal–Shale–W eil representation. In this wa y , their work extends to the sup er setting a theorem of Kashiw ara and V ergne, which states that every unitarizable highest weigh t su ( p, q )-mo dule with in tegral highest weigh t o ccurs inside a tensor pro duct of oscillator representations. On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 7 The second approach w as developed b y Jakobsen [ Jak94 ], using his principle of the last p ossible plac e of unitarity . The idea is to study one-parameter families of highest weigh t mo dules and to determine the last parameter v alue for which unitarity can o ccur. In practice, this amounts to lo cating the reducibilit y p oints where singular vectors app ear and identifying the ﬁrst p oint at which the inv ariant Hermitian form fails to remain p ositive semideﬁnite. The classiﬁcation problem is thus reduced to determining this threshold and describing the mo dules on the unitary side of it. Later, G ¨ una ydin–V olin observed that Jak obsen’s analysis leav es out certain cases; in partic- ular, for the ph ysically imp ortan t Lie sup eralgebras su (2 , 2 | N ), his list do es not coincide with the classiﬁcation used in the physics literature, due to Dobrev and Petk ov a [ DP85 ], whereas these cases are included in the work of G ¨ una ydin–V olin. Their classiﬁcation is obtained by ﬁrst deriving linear inequalities on the highest weigh t from p ositivity of the norms of o dd-ro ot de- scendan ts, expressed in terms of plaquette constraints, and then showing that these inequalities are suﬃcient as w ell. T o do so, they realize ev ery admissible highest weigh t in a γ -deformed os- cillator construction on a F o c k space in whic h determinant op erators are allow ed to app ear with nonin tegral p o w ers. F or this reason, the results of G ¨ una ydin–V olin provide the most natural p oin t of comparison for our work. The classiﬁcation obtained b y G ¨ una ydin–V olin may b e view ed as an extension of the F u- rutsu–Nishiy ama classiﬁcation. When their conditions are reformulated in terms of our Dirac inequalities, one ﬁnds that their classiﬁcation agrees with ours. In particular, our results recov er the classiﬁcation of G ¨ unaydin–V olin, while recasting it in the language of Dirac inequalities. 1.3. Notation. W e denote by Z + the set of p ositiv e int egers. Let Z 2 : = Z / 2 Z b e the ring of in tegers mo dulo 2. W e denote the elemen ts of Z 2 b y 0 (the residue class of ev en integers) and 1 (the residue class of o dd integers). The ground ﬁeld is C , unless otherwise stated. If V : = V ¯ 0 ⊕ V ¯ 1 is a sup er v ector space and v ∈ V is a homogeneous element, then p ( v ) denotes the parit y of v , meaning p ( v ) = ¯ 0 if v ∈ V ¯ 0 and p ( v ) = ¯ 1 if v ∈ V ¯ 1 . F or the Lie sup eralgebra g = g ¯ 0 ⊕ g ¯ 1 , we denote its universal en v eloping sup eralgebra by U ( g ). The universal en v eloping algebra of the Lie subalgebra g ¯ 0 is denoted by U ( g ¯ 0 ). Their cen ters are denoted b y Z ( g ) and Z ( g ¯ 0 ), resp ectiv ely . An y g -sup ermo dule M restricts to a g ¯ 0 -sup ermo dule, where eac h g ¯ 0 -mo dule is view ed as concen trated in a single parity . In what follows, this Z 2 -grading is left implicit, and we simply refer to them as g ¯ 0 -mo dules. 1.4. Leitfaden. Section 2 presen ts the structure of the Lie sup eralgebra sl ( m | n ) and its relev ant real forms. Section 3 in tro duces unitarizable sup ermo dules, realized as quotients of V erma and generalized V erma sup ermodules, and describes the asso ciated (unique) Hermitian form. The Dirac operator is deﬁned and its relation to unitarity is recalled. Section 4 giv es the classiﬁcation of unitarizable highest w eight g -sup ermo dules, including parametrizations of highest w eigh ts and represen tativ e examples. A cknow le dgments. W e extend sp ecial thanks to Rainer W eissauer and Johannes W alcher. This w ork is partially funded b y the Deutsc he F orsc h ungsgemeinschaft (DFG, German Researc h 8 STEFFEN SCHMIDT F oundation) under pro ject n um b er 517493862 (Homological Algebra of Sup ersymmetry: Lo cal- it y , Unitary , Dualit y), and by the Deutsche F orsch ungsgemeinsc haft (DFG, German Research F oundation) under Germany’s Excellence Strategy EXC 2181/1 — 390900948 (the Heidelb erg STR UCTURES Excellence Cluster). 2. Special Linear Lie Superalgebras sl ( m | n ) W e brieﬂy review the structure theory of g : = sl ( m | n ), with a fo cus on the real forms su ( p, q | r, s ). These will later prov e central in the study of unitarizable sup ermo dules ov er g . 2.1. Structure Theory. The Lie sup eralgebra sl ( m | n ) is simple whenev er m  = n and m + n ≥ 2. In this case, the extension to gl ( m | n ) admits a splitting, realized b y sending 1 ∈ C to the identit y matrix E m + n ∈ gl ( m | n ). If m = n , then E 2 n already b elongs to s l ( n | n ). As a result, gl ( n | n ) admits no splitting, and s l ( n | n ) is no longer simple, while still remaining indecomp osable. The corresp onding simple quotient of co dimension one is the pr oje ctive sp e cial line ar Lie sup er algebr a , psl ( n | n ) : = s l ( n | n ) / C E 2 n . W e consider sl ( m | n ) as a subalgebra of gl ( m | n ). The ab elian Lie subalgebra d : = { H = diag( h 1 , . . . , h m + n ) } of diagonal matrices in gl ( m | n ) is a Cartan sub algebr a for gl ( m | n ), that is, a maximal ad-diagonalizable subalgebra. As Cartan subalgebra of sl ( m | n ), we tak e the subspace h ⊂ d of diagonal matrices with v anishing sup ertrace. The dual space d ∗ is equipp ed with the standard basis ( ϵ 1 , . . . , ϵ m , δ 1 , . . . , δ n ), deﬁned by (2.1) ϵ i ( H ) = h i , δ k ( H ) = h m + k , for H ∈ d , with 1 ≤ i ≤ m and 1 ≤ k ≤ n . Accordingly , an y weigh t λ ∈ h ∗ can b e written as (2.2) λ = λ 1 ϵ 1 + · · · + λ m ϵ m + µ 1 δ 1 + · · · + µ n δ n , and w e iden tify λ with the tuple ( λ 1 , . . . , λ m | µ 1 , . . . , µ n ). Note that shifting by (1 , . . . , 1 | − 1 , . . . , − 1) leav es the weigh t unc hanged. Since h ⊂ g ¯ 0 is a Cartan subalgebra for g ¯ 0 , its action on any ﬁnite-dimensional simple g ¯ 0 -mo dule is diagonalizable. Hence the adjoint action of h on g is diagonalizable, and g admits a ro ot space decomp osition: (2.3) g = h ⊕ M α ∈ h ∗ \{ 0 } g α , g α : = { X ∈ g : [ H, X ] = α ( H ) X for all H ∈ h } . The set of r o ots is ∆ = ∆ ¯ 0 ⊔ ∆ ¯ 1 where ∆ ¯ 0 = {± ( ϵ i − ϵ j ) , ± ( δ k − δ l ) : 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n } , ∆ ¯ 1 = {± ( ϵ i − δ k ) : 1 ≤ i ≤ m, 1 ≤ k ≤ n } , (2.4) are the even and o dd r o ots , resp ectively . Eac h ro ot space has sup erdimension either (1 | 0) or (0 | 1). The set ∆ ¯ 0 decomp oses as the disjoin t union of the ro ot systems of s l ( m ) and s l ( n ). F or the even part, we choose once and for all the standard system of p ositive ro ots (2.5) ∆ + ¯ 0 : = { ϵ i − ϵ j , δ k − δ l : 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n } , so that the ro ot vectors asso ciated with ϵ i − ϵ j for i < j are realized as strictly upp er triangular matrices in s l ( m ), while those asso ciated with δ k − δ l for k < l are strictly upp er triangular On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 9 matrices in s l ( n ); both embeddings are taken diagonally inside g ¯ 0 . The o dd p ositive ro ots ∆ + ¯ 1 will b e sp eciﬁed in the next section ( cf. ( 2.21 )). In total, the set of p ositive ro ots is ∆ + : = ∆ + ¯ 0 ⊔ ∆ + ¯ 1 , and w e deﬁne the Weyl ve ctor ρ as ρ = ρ ¯ 0 − ρ ¯ 1 where (2.6) ρ ¯ 0 : = 1 2 X α ∈ ∆ + ¯ 0 α = 1 2   m X i =1 ( m − 2 i + 1) ϵ i + n X j =1 ( n − 2 j + 1) δ j   , ρ ¯ 1 : = 1 2 X α ∈ ∆ + ¯ 1 α. F or a ﬁxed p ositive system ∆ + , w e deﬁne the fundamental system π ⊂ ∆ + to b e the set of all α ∈ ∆ + whic h cannot b e written as the sum of tw o ro ots in ∆ + . Elements of π are called simple ro ots. F or π = { α 1 , . . . , α r } , an y α ∈ ∆ can be uniquely represented as a linear combination α = P r i =1 k i α i , where either all k i ∈ Z ≥ 0 or all k i ∈ Z ≤ 0 . With resp ect to a c hosen system of p ositiv e ro ots ∆ + , the Lie sup eralgebra g admits the triangular de c omp osition (2.7) g = n − ⊕ h ⊕ n + , n ± : = M α ∈ ∆ + g ± α , where n ± are nilp oten t Lie subsup eralgebras, and the even part repro duces the usual triangular decomp osition of g ¯ 0 . The corresp onding Bor el sub algebr a is b = h ⊕ n + . The general linear Lie sup eralgebra gl ( m | n ) is equipp ed with a natural bilinear form (2.8) ( X , Y ) : = str( X Y ) , X , Y ∈ gl ( m | n ) , whic h is even, sup ersymmetric, and inv ariant. Explicitly , the form is symmetric on gl ( m | n ) ¯ 0 , sk ew-symmetric on gl ( m | n ) ¯ 1 , and gl ( m | n ) ¯ 0 is orthogonal to gl ( m | n ) ¯ 1 . Inv ariance means that ([ X , Y ] , Z ) = ( X, [ Y , Z ]) holds for all X , Y , Z ∈ gl ( m | n ). The form ( · , · ) is alw a ys non-degenerate on gl ( m | n ). On sl ( m | n ), how ev er, it remains non-degenerate only if m  = n ; in the case m = n , the one-dimensional center of s l ( n | n ) coincides with the radical. Its restriction to the diagonal subalgebra d is still non-degenerate, and the induced bilinear form on d ∗ will b e denoted by the same sym b ol. With resp ect to the standard basis, we obtain for 1 ≤ i, j ≤ m and 1 ≤ k , l ≤ n : (2.9) ( ϵ i , ϵ j ) = δ ij , ( δ k , δ l ) = − δ kl , ( ϵ i , δ k ) = 0 . F or m  = n , the bilinear form ( · , · ) restricts to a non-degenerate form on h . In this situation, ev ery ro ot α ∈ ∆ corresp onds to a uniquely determined element h α ∈ h , c haracterized b y the iden tit y α ( H ) = ( H, h α ) for all H ∈ h . W e refer to h α as the dual r o ot of α . In the exceptional case m = n , w e ﬁx the dual ro ots as elemen ts of h by requiring α ( H ) = ( H , h α ) for all H ∈ d . Extending the assignment by linearity gives a bilinear form on h ∗ , deﬁned by (2.10) ( α, β ) : = ( h α , h β ) , α, β ∈ ∆ , whic h is non-degenerate exactly when m  = n . It follo ws immediately that all o dd ro ots are isotropic, that is, ( α , α ) = 0 for every α ∈ ∆ ¯ 1 . The ro ot system ∆ admits an action of the W eyl group W of the even part g ¯ 0 . The group W is isomorphic to S m × S n , the pro duct of the symmetric groups on m and n letters. It is generated b y the reﬂections with resp ect to the ev en ro ots, (2.11) r α ( β ) = β − 2 ( α, β ) ( α, α ) α, α ∈ ∆ ¯ 0 , β ∈ ∆ . 10 STEFFEN SCHMIDT This action extends linearly to h ∗ and preserv es the bilinear form ( · , · ). The W eyl group of g is b y deﬁnition that of g ¯ 0 . The dot action of W on h ∗ is deﬁned by (2.12) w · λ = w ( λ + ρ ) − ρ, λ ∈ h ∗ , w ∈ W. Tw o weigh ts λ, µ ∈ h ∗ are W -linked if µ = w · λ for some w ∈ W ; this deﬁnes an equiv alence relation on h ∗ . The equiv alence class { w · λ : w ∈ W } is called the W -link age class of λ . Not all fundamental systems can b e transformed into one another through the action of W . F or this, we need a sequence of o dd r eﬂe ctions . Given an o dd, isotropic simple ro ot θ ∈ π , we recall that an o dd r eﬂe ction satisﬁes: (2.13) r θ ( α ) =          α + θ if ( α, θ )  = 0 , α if ( α, θ ) = 0 , − θ if α = θ for an y α ∈ π . Then, ∆ + θ = {− θ } ∪ (∆ + \ { θ } ) forms a new p ositive system with the fundamen tal system π θ : = r θ ( π ). If π and π ′ are tw o fundamental systems such that ∆ + ¯ 0 = (∆ ′ ) + ¯ 0 , then π ′ can b e obtained from π by a sequence of o dd reﬂections [ Ser17 , Prop osition 1]. 2.2. Real F orms su ( p, q | r, s ) . Let V = C m | n . F or p, q , r, s ∈ Z + with p + q = m , r + s = n , w e deﬁne the Hermitian form (2.14) ⟨ v , w ⟩ : = v T J ( p,q | r ,s ) w , J ( p,q | r ,s ) : = I p,q 0 0 I r,s ! , where · denotes complex conjugation, and vectors of V are taken as columns. The matrix I k,l is the diagonal matrix having the ﬁrst k entries equal to 1 follo w ed by the last l entries equal to − 1. This Hermitian form is c onsistent , that is, ⟨ V ¯ 0 , V ¯ 1 ⟩ = 0. The unitary Lie sup er algebr as u ( p, q | r , s ) are deﬁned by (2.15) u ( p, q | r , s ) ¯ k = { X ∈ gl ( m | n ) ¯ k : ⟨ X v , w ⟩ + ⟨ v , X w ⟩ = 0 , ∀ v , w ∈ V } , and u ( p, q | r, s ) = u ( p, q | r, s ) ¯ 0 ⊕ u ( p, q | r, s ) ¯ 1 . The sp e cial unitary Lie sup er algebr as are (2.16) su ( p, q | r , s ) = u ( p, q | r, s ) ∩ sl ( m | n ) = { X ∈ sl ( m | n ) : J − 1 ( p,q | r ,s ) X † J ( p,q | r ,s ) = − X } , with X † b eing the conjugate transp ose. W e think of su ( p, q | r , s ) as the real form of sl ( m | n ) deﬁned by the ﬁxed p oint Lie subalegebra of the conjugate-linear anti-in v olution ω on g , where (2.17) ω ( X ) = J − 1 ( p,q | r ,s ) X † J ( p,q | r ,s ) , X ∈ g . Among these real forms, only su ( p, q | n, 0) and su ( p, q | 0 , n ), whic h are isomorphic, admit non- trivial unitarizable sup ermo dules (Theorem 5 ). A direct computation, distinguishing the com- pact case ( p = 0 or q = 0) and the non-compact case ( p, q  = 0), shows that ω admits exactly t wo matrix realizations, yielding the explicit conjugate-linear anti-in volutions according to whether p = 0, q = 0, r = 0 or s = 0. W e express a general element X ∈ g as (2.18) X =    a b c d P 1 P 2 Q 1 Q 2 E    , On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 11 where P 1 is a p × n matrix, P 2 is a q × n matrix, Q 1 is a n × p matrix, Q 2 is a n × q -matrix, a b c d ! ∈ su ( p, q ) C , E ∈ su ( n ) C and tr a b c d ! − tr E = 0. In the case p = 0 or q = 0 we use the notation ( 1.1 ) for the standard blo ck form. Lemma 3 ([ Jak94 , Lemma 4.1], [ Sc h24 ]) . Ther e ar e exactly two c onjugate-line ar anti-involutions on g c omp atible with the standar d or dering, which pr o duc e the even r e al forms su ( p, q | 0 , n ) ¯ 0 ∼ = su ( p, q | n, 0) ¯ 0 : a) If p = 0 or q = 0 , then ω ± A B C D ! = A † ± C † ± B † D † ! . The r e al Lie sup er algebr a su ( m, 0 | n, 0) ∼ = su (0 , m | 0 , n ) c orr esp onds to the c onjugate-line ar anti-involution ω + , while the r e al Lie algebr a su ( m, 0 | 0 , n ) ∼ = su (0 , m | n, 0) b elongs to the c onjugate-line ar anti-involution ω − . b) If p, q  = 0 , then ω ( − , +) ( X ) =    a † − c † − b † d † − Q † 1 Q † 2 − P † 1 P † 2 E †    , ω (+ , − ) ( X ) =    a † − c † − b † d † Q † 1 − Q † 2 P † 1 − P † 2 E †    , The r e al Lie sup er algebr a su ( p, q | 0 , n ) b elongs to ω ( − , +) while su ( p, q | n, 0) b elongs to ω (+ , − ) . In the compact case ( p = 0 or q = 0), there exist tw o con v enien t systems of odd p ositiv e ro ots, (2.19) ∆ + ¯ 1 , st = { ϵ i − δ j | 1 ≤ i ≤ m, 1 ≤ j ≤ n } , ∆ + ¯ 1 , − st = {− ϵ i + δ j | 1 ≤ i ≤ m, 1 ≤ j ≤ n } , whic h diﬀer only b y sign. In the literature, ∆ + ¯ 1 , st is known as the distinguished (or standard) p ositiv e system, while ∆ + ¯ 1 , − st is called the anti-distinguished p ositive system. The corresp ond- ing Borel subalgebras b st and b − st are referred to as distinguishe d and anti-distinguishe d , re- sp ectiv ely . Under the canonical isomorphism s l ( m | n ) ∼ = s l ( n | m ), the an ti-distinguished Borel subalgebra b − st of s l ( m | n ) is mapp ed to the distinguished one b st of s l ( n | m ). In this article, we choose ∆ + ¯ 1 : = ∆ + ¯ 1 , st , so that n + consists of upp er blo ck matrices in g and b = b st is the distinguished Borel. In particular, the o dd part of the W eyl vector ( 2.6 ) is (2.20) ρ ¯ 1 = 1 2   n m X i =1 ϵ i − m n X j =1 δ j   . T aking into account the real structure, in the non-compact case ( p, q  = 0) let p i (resp. q i ) denote the subspace of g in whic h only P i (resp. Q i ) in ( 2.18 ) is nonzero. One obtains three relev ant systems of o dd p ositiv e ro ots describ ed by: (2.21) n + ¯ 1 , st = p 1 ⊕ p 2 , n + ¯ 1 , − st = q 1 ⊕ q 2 , n + ¯ 1 , nst = p 1 ⊕ q 2 , called standar d , minus standar d , and non-standar d , resp ectiv ely . W e ﬁx the non-standard sys- tem, denoted ∆ + ¯ 1 , since this choice is adapted to the study of unitarit y and the Dirac op erator 12 STEFFEN SCHMIDT [ Sc h24 ]. The relev ance of this choice for the classiﬁcation will b e discussed in Section 3.2 . Accordingly , ∆ + = ∆ + ¯ 0 ⊔ ∆ + ¯ 1 , and the corresp onding half-sum of p ositiv e o dd ro ots is (2.22) ρ ¯ 1 = 1 2   n p X i =1 ϵ i − n m X j = p +1 ϵ j + ( q − p ) n X k =1 δ k   . 3. Unit arizable Supermodules and Dirac Opera tors In this section, we introduce unitarizable sup ermo dules ov er g and summarize their basic prop- erties. W e also presen t the (algebraic) quadratic Dirac op erator D and formulate the Dirac inequalit y , whic h serves as the main tool for classifying the full set of unitarizable sup ermo dules. 3.1. F undamen tals. A sup er Hilb ert sp ac e is a Z 2 -graded complex Hilb ert space ( H = H ¯ 0 ⊕ H ¯ 1 , ⟨· , ·⟩ H ) such that H ¯ 0 and H ¯ 1 are mutually orthogonal subspaces of H with resp ect to the inner pro duct ⟨· , ·⟩ H . 1 The inner pro duct is conjugate-linear in the ﬁrst and linear in the second argumen t. With this in place, we deﬁne unitarizable sup ermo dules relative to a real form of g . W e realize the real forms as the ﬁxed p oin t Lie subalgebras of conjugate-linear anti-in v olutions ω , that is, g ω : = { x ∈ g : ω ( x ) = − x } . Deﬁnition 4 ([ Jak94 , Deﬁnition 2.3]) . Let H b e a g -sup ermo dule, and let ω b e a conjugate- linear anti-in v olution on g . The sup ermo dule H is called an ω -unitarizable g -sup ermo dule if H is a sup er Hilb ert space such that for all v , w ∈ H and all X ∈ g , the Hermitian pro duct ⟨· , ·⟩ H is ω -con tra v ariant: ⟨ X v , w ⟩ H = ⟨ v , ω ( X ) w ⟩ H . If we w ork with U ( g )-sup ermo dules instead, w e extend ω to U ( g ) in the natural wa y , using the same notation. In this context, a g -sup ermo dule H is unitarizable if and only if it is a Hermitian represen tation o ver ( U ( g ) , ω ), meaning that ⟨ X v , w ⟩ H = ⟨ v , ω ( X ) w ⟩ H holds for all v , w ∈ H and X ∈ U ( g ). When ω is to b e implied from context, we just say “unitarizable”. W e ﬁx a real form g ω . Let H b e an ω -unitarizable g -sup ermo dule. By a standard argument, H is completely reducible, i.e. , the orthogonal complement of an y inv ariant subspace is again in v ariant. Consider the underlying g ¯ 0 -mo dule H ev , obtained from H by restriction to g ¯ 0 and forgetting the Z 2 -grading. Then H ev is a unitarizable g ¯ 0 -mo dule with resp ect to the real form g ω ¯ 0 , and in particular completely reducible as a g ¯ 0 -mo dule in the sense ab ov e. This complete reducibilit y , together with the following theorem, motiv ates our fo cus on highest and low est w eigh t sup ermodules; we therefore brieﬂy recall their deﬁnition. Fix a triangular decomp osition g = n − ⊕ h ⊕ n + . A g -sup ermo dule M is called highest (resp. lowest weight ) if there exist Λ ∈ h ∗ and a nonzero vector v Λ ∈ M suc h that: a) X v Λ = 0 for all X ∈ n + (resp. X ∈ n − ), b) H v Λ = Λ( H ) v Λ for all H ∈ h , and c) U ( g ) v Λ = M . 1 Sup er Hilb ert spaces hav e an equiv alen t description in terms of even sup er p ositiv e deﬁnite sup er Hermitian forms. F or a detailed comparison, we refer to [ Jak94 , Sc ha ]. On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 13 The v ector v Λ is called a highest (resp. lowest ) w eigh t vector of M . W e call M even or o dd according to the parit y of v Λ . In this article, we place v Λ in even degree. Accordingly , all our results are understo o d up to application of the parity reversion functor Π. Theorem 5 ([ FN91 , NS11 ]) . The sp e cial line ar Lie sup er algebr a g admits non-trivial ω - unitarizable sup ermo dules if and only if the c onjugate-line ar anti-involution ω is asso ciate d with one of the r e al forms su ( p, q | n, 0) or su ( p, q | 0 , n ) , p + q = m. Mor e over, any simple ω -unitarizable sup ermo dule is either a highest weight or a lowest weight g -sup ermo dule. F or the remainder, w e restrict atten tion to ω -unitarizable highest w eigh t g -sup ermo dules, since the treatmen t of the lo west weigh t case is entirely analogous. It is imp ortant to note that non-trivial unitarizable highest weigh t g -sup ermo dules for p, q  = 0 exist only with resp ect to ω ( − , +) , while ω (+ , − ) giv es rise to unitarizable low est w eigh t g -sup ermodules (see [ Jak94 , Sc ha ]). Accordingly , we work with the real forms su ( p, q | 0 , n ), and for conv enience of notation w e abbreviate su ( p, q | 0 , n ) by su ( p, q | n ). If e ither p = 0 or q = 0, we set su ( m | n ) : = su ( m, 0 | n, 0) and ﬁx the conjugate-linear anti-in v olution ω + . The discussion for ω − is entirely analogous, and w e therefore state only the corresp onding results. In the following, the preﬁx ω + /ω ( − , +) will b e omitted, and we shall simply sp eak of unitarizable highest w eigh t g -sup ermo dules. W e are now in a p osition to state the main problem of this w ork. Problem. Describ e the full set of unitarizable simple g -sup ermo dules; equiv alen tly , determine all highest weigh ts Λ ∈ h ∗ for which there exists a unitarizable simple g -sup ermo dule of highest w eigh t Λ. A ﬁrst step tow ards a complete description consists in restricting the p ossible form of the o ccurring highest w eigh ts, which follows directly from the deﬁnition of unitarit y . W e refer to these as the unitarity c onditions . Lemma 6. L et H b e a unitarizable highest weight g -sup ermo dule with highest weight Λ = ( λ 1 , . . . , λ m | µ 1 , . . . , µ n ) ∈ h ∗ . Then the fol lowing unitarit y conditions hold: a) If p = 0 or q = 0 and ω = ω + , then Λ is the highest weight of a unitarizable highest weight g ¯ 0 -mo dule and (i) λ 1 ≥ · · · ≥ λ m ≥ − µ n ≥ · · · ≥ − µ 1 , (ii) (Λ + ρ, ϵ m − δ k ) = 0 ⇒ (Λ + ρ, ϵ m − δ j ) > 0 for al l j = 1 , . . . , k − 1 and (Λ , δ k − δ n ) = 0 . (iii) (Λ + ρ, ϵ m − δ k )  = 0 for al l k = 1 , . . . n ⇒ (Λ + ρ, ϵ m − δ k ) > 0 for k = 1 , . . . , n . b) If p, q  = 0 and ω = ω ( − , +) , then Λ is the highest weight of a unitarizable highest weight g ¯ 0 -mo dule, and (i) λ p +1 ≥ · · · ≥ λ m ≥ − µ n ≥ · · · ≥ − µ 1 ≥ λ 1 ≥ · · · ≥ λ p , (ii) (Λ + ρ, − ϵ i + δ n ) = 0 for p + 1 ≤ i ≤ m implies (Λ , ϵ i − ϵ m ) = 0 , (iii) (Λ + ρ, ϵ i − δ 1 ) = 0 for 1 ≤ i ≤ p implies (Λ , ϵ 1 − ϵ i ) = 0 . (iv) If (Λ + ρ, ϵ i − δ 1 )  = 0 for 1 ≤ i ≤ i 0 , then (Λ + ρ, ϵ i − δ 1 ) < 0 for i = 1 , . . . , i 0 . 14 STEFFEN SCHMIDT (v) If (Λ + ρ, ϵ m − j + δ n )  = 0 for 0 ≤ j ≤ j 0 , then (Λ + ρ, − ϵ m − j + δ n ) < 0 for j = 0 , . . . , j 0 . Pr o of. The pro ofs of a), and b) are analogous; for simplicity , we restrict to a). Let v Λ b e the highest weigh t vector of H . Then v Λ generates, in particular, a unitarizable highest weigh t g ¯ 0 -mo dule of highest w eigh t Λ. F or an o dd ro ot α , let g α b e the asso ciated ro ot space of sup erdimension (0 | 1). If e α ∈ g α is a ro ot v ector corresponding to α , w e ha v e ω ( e α ) = e − α ∈ g − α since ω = ω + . Fix an o dd p ositive ro ot α : = ϵ i − δ j ∈ ∆ + ¯ 1 . If ⟨· , ·⟩ denotes the p ositive-deﬁnite Hermitian form on H , w e ha v e b y p ositiv e-deﬁniteness 0 ≤ ⟨ e − α v Λ , e − α v Λ ⟩ = ⟨ v Λ , ω ( e − α ) e − α v Λ ⟩ = Λ([ ω ( e − α ) , e − α ]) ⟨ v Λ , v Λ ⟩ = ( λ i + µ j ) ⟨ v Λ , v Λ ⟩ . Since ⟨ v Λ , v Λ ⟩ > 0, it follo ws that λ i ≥ − µ j for all 1 ≤ i ≤ m and 1 ≤ j ≤ n . Moreo v er, since Λ is the highest w eigh t of a ﬁnite-dimensional g -sup ermo dule, it is dominan t in tegral, i.e. , λ i − λ i +1 ∈ Z ≥ 0 and µ i − µ j ∈ Z ≥ 0 . Th us, λ 1 ≥ · · · ≥ λ m ≥ − µ n ≥ · · · ≥ − µ 1 . Assume now that (Λ + ρ, ϵ m − δ k ) = 0 for some 1 ≤ k ≤ n . Then (Λ + ρ, ϵ m − δ l ) > 0 for all 1 ≤ l ≤ k − 1. Set v k − 1 : = e − ϵ m + δ 1 · · · e − ϵ m + δ k − 1 v Λ , and note that ⟨ v k − 1 , v k − 1 ⟩ = (Λ − k − 2 X i =1 ( ϵ m − δ i ) , ϵ m − δ k − 1 ) ⟨ v k − 2 , v k − 2 ⟩ = (Λ + ρ, ϵ m − δ k − 1 ) ⟨ v k − 2 , v k − 2 ⟩ = . . . = k − 1 Y l =1 (Λ + ρ, ϵ m − δ l ) ⟨ v Λ , v Λ ⟩ > 0 , where w e used ( ρ, ϵ m − δ l ) = − l + 1 and (Λ + ρ, ϵ m − δ l ) > 0 for all 1 ≤ l ≤ k . By p ositiv e-deﬁniteness, one has 0 ≤ ⟨ e − ϵ m + δ n v k − 1 , e − ϵ m + δ n v k − 1 ⟩ = (Λ − k − 1 X i =1 ( ϵ m − δ i ) , ϵ m − δ n ) ⟨ v k − 1 , v k − 1 ⟩ = [(Λ , ϵ m − δ k ) − k + 1 + (Λ , δ k − δ n )] ⟨ v k − 1 , v k − 1 ⟩ = [(Λ + ρ, ϵ m − δ k ) + (Λ , δ k − δ n )] ⟨ v k − 1 , v k − 1 ⟩ = (Λ , δ k − δ n ) ⟨ v k − 1 , v k − 1 ⟩ . Since (Λ , δ k − δ n ) = − µ k + µ n ≤ 0 and ⟨ v k − 1 , v k − 1 ⟩ > 0, we conclude that (Λ , δ k − δ n ) = 0. Finally , a similar line of reasoning yields, in the case (Λ + ρ, ϵ m − δ k )  = 0 for all k = 1 , . . . , n , that ⟨ v k , v k ⟩ = k Y l =1 (Λ + ρ, ϵ m − δ l ) ⟨ v Λ , v Λ ⟩ > 0 , k = 1 , . . . , n. Inductiv ely , starting from ⟨ v 1 , v 1 ⟩ > 0, we conclude that (Λ + ρ, ϵ m − δ k ) > 0 for all k = 1 , . . . , n . This establishes (iii) and completes the pro of. □ The notion of unitarit y , and in particular the c haracterization of the full set of unitarizable su- p ermo dules, is closely connected to the (algebraic) Dirac op erator. This op erator w as introduced in [ HP05 , HP06 ], and its connection to unitarity w as studied in [ Sch24 ]. Before outlining the main ideas, w e recall an explicit realization of simple highest w eigh t g -supermo dules as quotien ts of V erma sup ermo dules, a construction that will b e useful for the subsequent classiﬁcation. On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 15 3.2. In termezzo: Highest W eigh t Sup ermo dules. W e study the structure of (unitarizable) highest weigh t sup ermo dules o v er g and identify them as Harish-Chandra supermo dules. W e also in tro duce the Kac–Shap ov alov form and record the restriction prop erties of the g ¯ 0 -constituen ts needed later. 3.2.1. Unitarizable Highest Weight g ¯ 0 -Mo dules. Any unitarizable highest weigh t g -sup ermo dule of highest w eigh t Λ decomposes as a direct sum of unitarizable highest weigh t g ¯ 0 -mo dules. Hence Λ is itself the highest w eigh t of a unitarizable highest w eight g ¯ 0 -mo dule. The structure of these sup ermo dules is as follo ws. One has the decomp osition g ¯ 0 = L ⊕ R ⊕ u (1) C , with L = su ( p, q ) C and R = su ( n ) C . W e denote the asso ciated ro ot systems of ( L , h | L ) and ( R , h | R ) by ∆ L and ∆ R , resp ectively . In particular, ∆ ¯ 0 = ∆ L ⊔ ∆ R . The p ositive systems are ∆ + L : = ∆ + ¯ 0 ∩ ∆ L and ∆ + R : = ∆ + ¯ 0 ∩ ∆ R . The highest weigh t of a unitarizable simple g -sup ermo dule is of the form ν = ( ν L | ν R ), where, in standard co ordinates, ν L = ( λ 1 , . . . , λ m ) and ν R = ( µ 1 , . . . , µ n ). Here ν L is the highest w eigh t of a unitarizable L -mo dule, while ν R is the highest w eigh t of a ﬁnite-dimensional simple R -mo dule. In particular, one obtains the following description. Lemma 7. Any unitarizable simple highest weight g ¯ 0 -mo dule L 0 ( µ ) is given by the outer tensor pr o duct of unitarizable simple highest weight su ( p, q ) C -, su ( n ) C -, and u (1) C -mo dules, r esp e ctively, i.e. , L 0 ( µ ) ∼ = L 0 ( µ L ; L ) ⊠ L 0 ( µ R ; R ) ⊠ C µ . All suc h mo dules will b e listed in Section 4 , pro viding a parametrization of the highest w eigh ts of unitarizable highest weigh t g ¯ 0 -mo dules. A characteristic prop ert y of unitarizable highest weigh t sup ermo dules is that they b elong to the class of Harish-Chandr a mo dules . W e b egin by recalling the deﬁnition. Set k : = su ( p ) ⊕ su ( q ) ⊕ u (1) ⊕ su ( n ) ⊕ u (1). Then k is a maximal compact subalgebra of su ( p, q | n ) ¯ 0 , with k = su ( p, q | n ) ¯ 0 if p = 0 or q = 0. One has the e qual r ank c ondition (3.1) h ⊂ k C ⊂ g ¯ 0 ⊂ g , where k C denotes the complexiﬁcation of k . The ro ot system ∆ c asso ciated with ( h , k C ) is (3.2) ∆ c : = {± ( ϵ i − ϵ j ) : 1 ≤ i < j ≤ p, p + 1 ≤ i < j ≤ m } ∪ {± ( δ i − δ j ) : 1 ≤ i < j ≤ n } , whic h forms a subset of ∆ ¯ 0 . Thus, a ro ot α ∈ ∆ ¯ 0 is termed c omp act if α ∈ ∆ c , or equiv alently , if the corresp onding ro ot vector b elongs to k C ; otherwise, it is called non-c omp act . The W eyl group asso ciated with ∆ c will b e denoted b y W c , which is indeed a subgroup of W . The set ∆ n : = ∆ ¯ 0 \ ∆ c will b e called the set of non-c omp act r o ots , so that w e ha v e the decomp osition: (3.3) ∆ ¯ 0 = ∆ c ⊔ ∆ n . Moreo v er, we deﬁne the set of p ositive c omp act and p ositive non-c omp act r o ots b y ∆ + c,n : = ∆ + ∩ ∆ c,n . The asso ciated W eyl v ectors are given b y ρ c,n : = 1 2 P α ∈ ∆ + c,n α , and with resp ect to ∆ + c , an element λ ∈ h ∗ is called ∆ + c -dominant inte gr al if it satisﬁes the following conditions: (3.4)    ( λ + ρ c , α ) ∈ Z ≥ 0 for all α ∈ { ϵ i − ϵ j : 1 ≤ i < j ≤ p or p + 1 ≤ i < j ≤ m } , ( λ + ρ c , α ) ∈ Z ≤ 0 for all α ∈ { δ i − δ j : 1 ≤ i < j ≤ n } . 16 STEFFEN SCHMIDT The dominan t in tegral w eigh ts are in one-to-one corresp ondence with simple k C -mo dules. By the equal rank condition ( 3.1 ), ev ery g ¯ 0 -mo dule is a k C -mo dule. A ( g ¯ 0 , k C )-mo dule is called a Harish-Chandr a mo dule if it is ﬁnitely generated and lo cally ﬁnite as a k C -mo dule. Lemma 8 ([ Nee00 , Lemma IX.3.10]) . A ny unitarizable highest weight g ¯ 0 -mo dule is a Harish- Chandr a mo dule. 3.2.2. Highest Weight Sup ermo dules. Let b = h ⊕ n + b e the Borel subalgebra corresp onding to a p ositive system ∆ + of Section 2.2 . The env eloping sup eralgebra U ( g ) is viewed as a right U ( b )-sup ermo dule b y right multiplication. F or Λ ∈ h ∗ , let C Λ denote the one-dimensional U ( b )- sup ermo dule of w eight Λ, with trivial action of n + . The V erma sup ermo dule of highest weigh t Λ is then deﬁned by (3.5) M b (Λ) = U ( g ) ⊗ U ( b ) C Λ . The V erma sup ermo dule M b (Λ) is a highest weigh t g -supermo dule generated by the vector [1 ⊗ 1] of w eight Λ, with M b (Λ) = U ( n − )[1 ⊗ 1]. F or any g -sup ermodule M and an y b -eigen v ector v Λ ∈ M of w eight Λ, there exists a unique surjective morphism M b (Λ) → M sending [1 ⊗ 1] to v Λ . The mo dule M b (Λ) admits a unique maximal subsup ermo dule, hence a unique simple quotien t L (Λ); it is therefore indecomp osable [ Mus12 , Chap. 8]. If Λ is the highest weigh t of a unitarizable g ¯ 0 -mo dule, M b (Λ) is an example of a Harish-Chandra sup ermo dule ( cf. [ CFV20 ]), that is, M b (Λ) is ﬁnitely generated and lo cally ﬁnite as a k C -sup ermo dule. If b is ﬁxed, or is clear from the context, we omit the sup erscript in M b (Λ) and L b (Λ). The deﬁnition of the V erma sup ermodule dep ends on the choice of a p ositiv e system. In Section 2.2 w e introduced tw o p ositive systems ( cf. ( 2.21 )) relev ant for unitarity , namely the standard one ∆ + st and the non-standard one ∆ + nst , whic h share the same even part ∆ + ¯ 0 ﬁxed in ( 2.5 ). W e ﬁx ∆ + st in the case p = 0 or q = 0, and ∆ + nst in the case p, q  = 0. The corresp onding Borel subalgebras are denoted b st and b nst . The tw o systems are connected b y a sequence of o dd reﬂections deﬁned in ( 2.13 ). Under these reﬂections one has [ CW12 , Lemma 1.40] (3.6) M b st (Λ) ∼ = M b nst (Λ ′ ) , L b st (Λ) ∼ = L b nst (Λ ′ ) , up to parity , where the highest w eights Λ , Λ ′ ∈ h ∗ are related by the o dd reﬂection op erations. The sup erscript indicates the c hoice of Borel subalgebra. The mo dules L b st (Λ) and L b nst (Λ ′ ) ha v e the same g ¯ 0 -constituen ts. Moreov er, Λ ′ is typical if and only if Λ is t ypical ( cf. [ Mus12 , Cor. 9.3.5]). The choice of ∆ + st is conv enient, since it yields a natural Z -grading of g compatible with the Z 2 -grading. By contrast, ∆ + nst is b etter suited to unitarit y and inﬁnite-dimensional sup ermo dules, i.e. , to the real forms su ( p, q | n ) with p, q  = 0. Accordingly , w e work with ∆ + nst whenev er p, q  = 0; the classiﬁcation for ∆ + st follo ws b y a sequence of o dd reﬂections. F or the standard system ∆ + st it is a classical result of Kac [ Kac77 ] that M b (Λ) is simple if and only if 2(Λ + ρ, α )  = m ( α, α ) for all α ∈ ∆ + ¯ 0 ⊔ ∆ + ¯ 1 , st and m ∈ Z > 0 . A w eight Λ satisfying this condition for all α ∈ ∆ + ¯ 1 is called typic al ; otherwise it is called atypic al . Using ( 3.6 ), together with the fact that Λ is typical with resp ect to ∆ + ¯ 1 , st if and only if Λ ′ is typical with resp ect to ∆ + ¯ 1 , nst , one obtains the analogous statement for b oth p ositiv e systems. On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 17 Theorem 9. Fix either the standar d or the non-standar d p ositive system. Then the V erma sup ermo dule M b (Λ) is simple if and only if 2(Λ + ρ, α )  = m ( α, α ) for al l α ∈ ∆ + ¯ 1 and m ∈ Z > 0 . W e are concerned with the structure of M b (Λ), or more precisely with that of its unique simple quotient, viewed as g ¯ 0 -mo dules neglecting parit y . T o this end, w e recall a Jordan–H¨ older g ¯ 0 -ﬁltration due to Musson [ Mus12 , C hap. 10]. Let Γ denote the set of sums of distinct o dd p ositiv e ro ots. F or each γ ∈ Γ, let p ( γ ) b e the num b er of distinct partitions of γ into o dd p ositiv e ro ots. F or a subset S ⊆ ∆ + ¯ 1 , set Γ S = P γ ∈ S γ and X − S : = Q α ∈ S X − α , where X − α is the ro ot v ector corresp onding to − α ∈ ∆ − ¯ 1 , with ∆ − ¯ 1 : = − ∆ + ¯ 1 . Cho ose an ordering S 1 , . . . , S N of the subsets of ∆ + ¯ 1 suc h that Γ S i < Γ S j implies i < j . Deﬁne (3.7) M k : = M 1 ≤ j ≤ k U ( n − ¯ 0 ) X − S j [1 ⊗ 1] . By the PBW theorem this sum is direct. A direct computation shows that the image of X − S i +1 [1 ⊗ 1] in M / M i is a highest w eight vector, and the submo dule it generates is isomorphic to M b ¯ 0 0 (Λ − Γ S i +1 ), the V erma (sup er)mo dule ov er g ¯ 0 with resp ect to b ¯ 0 . Prop osition 10 ([ Mus12 , Theorem 10.4.5]) . The sup ermo dule M b (Λ) has a ﬁltr ation as a g ¯ 0 - mo dule: 0 = M 0 ⊂ M 1 ⊂ M 2 ⊂ . . . ⊂ M r = M b (Λ) , such that e ach factor M i +1 / M i is isomorphic to a V erma mo dule M b ¯ 0 0 (Λ − γ ) , wher e γ ∈ Γ . This mo dule app e ars with multiplicity p ( γ ) in the ﬁltr ation. In the same wa y one obtains a g ¯ 0 -ﬁltration of L b (Λ), whose simple highest weigh t g ¯ 0 - comp osition factors are L 0 (Λ − γ ) with γ ∈ Γ (see for instance [ Jak94 , Theorem 2.5]). Note that each comp osition factor is a g -sup ermo dule. Hence, if L b (Λ) is unitarizable, the complete reducibilit y of unitarizable sup ermodules implies that the g ¯ 0 -ﬁltration is in fact a direct sum. Corollary 11. L et H b e a unitarizable simple g -sup ermo dule with highest weight Λ . Then, viewe d as a g ¯ 0 -mo dule, H de c omp oses as H = H 1 ⊕ · · · ⊕ H r , wher e e ach c onstituent H i is isomorphic to a unitarizable highest weight g ¯ 0 -mo dule of the form L 0 (Λ − γ ) for some γ ∈ Γ . Moreo v er, if L b (Λ) is ﬁnite-dimensional, then it is a direct sum of simple g ¯ 0 -mo dules of the form L 0 ( µ ), since Ext 1 g ¯ 0  L 0 ( µ ) , L 0 ( ν )  = 0 for any g ¯ 0 -comp osition factors L 0 ( µ ) and L 0 ( ν ) o ccurring in the g ¯ 0 -ﬁltration of L b (Λ). Lemma 12. Fix either the standar d or non-standar d p ositive system. If L b (Λ) is ﬁnite- dimensional, then it de c omp oses as a ﬁnite dir e ct sum of simple g ¯ 0 -mo dules, e ach of highest weight Λ − γ for some γ ∈ Γ . In general, ho w ever, L (Λ) is not semisimple as a g ¯ 0 -mo dule, although eac h simple composition factor is a unitarizable highest weigh t g ¯ 0 -mo dule. Lemma 13. Fix the non-standar d p ositive system. L et Λ b e the highest weight of a unitarizable g ¯ 0 -mo dule. Then every g ¯ 0 -c omp osition factor L 0 ( µ ) app e aring in the g ¯ 0 -ﬁltr ation of L (Λ) is a unitarizable highest weight g ¯ 0 -mo dule. 18 STEFFEN SCHMIDT Pr o of. Any g ¯ 0 -comp osition factor is of the form L 0 (Λ − γ ), where γ is a sum of distinct o dd p ositiv e ro ots. Let L 0 ( ν ) b e such a g ¯ 0 -comp osition factor, and supp ose that L 0 ( ν ) is a unitariz- able highest w eigh t g ¯ 0 -mo dule. The claim follows if one shows that, for ev ery α ∈ ∆ + ¯ 1 suc h that L 0 ( ν − α ) o ccurs as a comp osition factor, the mo dule L 0 ( ν − α ) is itself a unitarizable highest w eigh t g ¯ 0 -mo dule. Assume that L 0 ( ν − α ) o ccurs as a comp osition factor. Since ν is the highest weigh t of a unitarizable g ¯ 0 -mo dule, we may write, in standard co ordinates, ν = ( λ 1 , . . . , λ m | µ 1 , . . . , µ n ) , satisfying λ p +1 ≥ · · · ≥ λ m ≥ λ 1 ≥ · · · ≥ λ p , µ 1 ≥ · · · ≥ µ n . F or α ∈ ∆ + ¯ 1 , one has either α = ϵ i − δ k or α = − ϵ j + δ l , where 1 ≤ i ≤ p < j ≤ m and 1 ≤ k , l ≤ n . Consequen tly , ν − α is k C -dominan t integral, since ν is. Moreo v er, by the form of α , it satisﬁes the same inequalities and changes the parameter λ 1 − λ m b y 0 or − 1. Hence, according to the classiﬁcation of unitarizable su ( p, q )-modules ( cf. ( 4.28 )), L 0 ( ν − α ) is the highest weigh t of a unitarizable g ¯ 0 -mo dule. □ W e ﬁnally state a non-existence result for certain g ¯ 0 -comp osition factors in the atypical case. F or this purp ose, w e consider the sup ermo dule M b (Λ). W e ﬁrst require the following lemma. Lemma 14. If (Λ + ρ, α ) = 0 for some o dd p ositive r o ot α , then ther e exists an emb e dding M b (Λ − α ) ⊂ M b (Λ) . F or the pro of, we refer to [ DS05 , Lemma 10.3] in the case of the standard p ositive system, and to [ Mus23 , Theorem 3.7] in the non-standard case, where an explicit construction of the Shap o v alov elements is giv en for all Borel subalgebras obtained from the standard one b y a sequence of o dd reﬂections. F or w eigh ts Λ satisfying the unitarity conditions of Lemma 6 , this yields the following imp ortant implication. Lemma 15. Assume Λ satisﬁes the unitarity c onditions. Then the fol lowing holds: a) Assume p = 0 or q = 0 and Λ = ( λ 1 , . . . , λ m | µ 1 , . . . µ k 0 − 1 , µ, . . . µ ) with µ k 0 − 1  = µ . If (Λ + ρ, ϵ i − δ k ) = 0 for some k 0 ≤ k ≤ n , then any g ¯ 0 -c omp osition factor of L b (Λ) has a de c omp osition of the form Λ − γ for γ ∈ Γ such that γ = γ 1 + . . . + γ r and none of the γ s is of the form ϵ i − δ l for l = k , . . . , n . b) Assume p, q  = 0 and Λ = ( λ, . . . , λ, λ i 0 +1 , . . . , λ p , λ p +1 , . . . , λ m − j 0 − 1 , λ ′ , . . . , λ ′ | µ 1 , . . . µ n ) with λ i 0 +1  = λ and λ m − j 0 − 1  = λ ′ . (i) If (Λ + ρ, ϵ i − δ k ) = 0 for some 1 ≤ i ≤ i 0 , then any g ¯ 0 -c omp osition factor of L b (Λ) has a de c omp osition of the form Λ − γ for γ ∈ Γ such that γ = γ 1 + . . . + γ r and none of the γ s is of the form ϵ i ′ − δ k for i ′ = 1 , . . . , i . (ii) If (Λ + ρ, − ϵ m − j + δ k ) = 0 for some 0 ≤ j ≤ j 0 , any g ¯ 0 -c omp osition factor of L b (Λ) has a de c omp osition of the form Λ − γ for γ ∈ Γ such that γ = γ 1 + . . . + γ r and none of the γ s is of the form − ϵ m − j ′ + δ k for j ′ = 0 , . . . , j . Pr o of. W e prov e only a), since b) is analogous. Consider ﬁrst α : = ϵ i − δ k . By the g ¯ 0 -ﬁltration of M b (Λ − α ), its comp osition factors ha ve highest weigh ts of the form Λ − α − γ , where α + γ is a sum of pairwise distinct p ositive o dd ro ots. Since (Λ + ρ, α ) = 0, Lemma 14 yields an On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 19 em b edding M b (Λ − α ) ⊂ M b (Λ). Thus M b (Λ − α ) is contained in the Shap ov alo v radical of M b (Λ). Consequently , every g ¯ 0 -comp osition factor of highest weigh t Λ − α − γ arising from a decomp osition inv olving α is already con tained in this radical. Moreov er, the multiplicit y of suc h a factor in M b (Λ − α ) is exactly the num b er of distinct o dd partitions of α + γ inv olving α . Next, ﬁx β : = ϵ i − δ l for k < l ≤ n . W e consider the g ¯ 0 -comp osition factors of L b (Λ) inv olving β . Every suc h factor is of the form L 0 (Λ − γ ), where γ = β + ˜ γ is a sum of pairwise distinct p ositiv e o dd ro ots. Since M b (Λ − α ) ⊂ M b (Λ), we may assume that no o dd ro ot o ccurring in ˜ γ equals α . W e claim that none of these factors o ccurs in L b (Λ). Set η : = δ k − δ l for k < l ≤ n . Then r η (Λ) = Λ by assumption, hence r η · Λ = Λ − η . By V erma’s theorem, there is an em b edding M b ( r η · Λ) ⊂ M b (Λ). By the g ¯ 0 -ﬁltration, the g ¯ 0 -comp osition factors of M b ( r η · Λ) are exactly the mo dules M 0 ( r η · Λ − ν ), where ν is a sum of pairwise distinct p ositiv e o dd ro ots. Since r η · Λ = Λ − η and (Λ − η ) − α = Λ − β , ev ery factor M 0 ( r η · Λ − ν ) for whic h some decomp osition of ν inv olves α is of the form M 0 (Λ − γ ), where some decomp osition of γ inv olv es β . Indeed, if ν = α + ν 1 , then r η · Λ − ν = ( r η · Λ − α ) − ν 1 = (Λ − β ) − ν 1 = Λ − ( β + ν 1 ) . In L b (Λ), let L 0 (Λ − γ ) b e a g ¯ 0 -comp osition factor with γ = β + ˜ γ . By the preceding argument, w e may assume that no decomposition of γ in v olv es α . In particular, ν : = α + ˜ γ is a sum of pairwise distinct p ositiv e o dd ro ots. Hence M 0 (Λ − γ ) o ccurs as a g ¯ 0 -comp osition factor of M b ( r η · Λ). Therefore M b ( r η · Λ) con tains every g ¯ 0 -comp osition factor of the form M 0 (Λ − γ ) for whic h some decomp osition of γ inv olves β . Since M b ( r η · Λ) ⊂ M b (Λ) is a prop er submo dule, all g ¯ 0 -factors arising from it lie in the Shap ov alov radical of M b (Λ). Consequently , M 0 (Λ − β ), and more generally ev ery g ¯ 0 -factor arising from a decomp osition inv olving β , do es not survive in the irreducible quotient L b (Λ). □ 3.2.3. Shap ovalov F orm. Ev ery highest weigh t g -sup ermo dule is the unique simple quotient of a V erma sup ermo dule. V erma sup ermo dules carry a natural ω -con trav ariant Hermitian form, the Shap ovalov form . A highest weigh t g -sup ermo dule is unitarizable precisely when the induced form on its simple quotient is p ositive deﬁnite. W e now deﬁne the Shap ovalov form . T o introduce the Shap ov alo v form we recall the notion of inﬁnitesimal characters. An inﬁni- tesimal char acter is an algebra homomorphism χ : Z ( g ) → C , obtained from the Harish-Chandra homomorphism. Concretely , one has the decomp osition of sup er vector spaces (3.8) U ( g ) = U ( h ) ⊕  n − U ( g ) + U ( g ) n +  , whic h is a direct consequence of the PBW theorem for g . This decomp osition is preserved by ω , extended to U ( g ). The pro jection p : U ( g ) → U ( h ) is the Harish-Chandr a pr oje ction ; its restriction to Z ( g ) deﬁnes an algebra homomorphism (3.9) p | Z ( g ) : Z ( g ) → U ( h ) ∼ = S ( h ) . The Harish-Chandr a homomorphism HC : Z ( g ) → S ( h ) is deﬁned as the comp osition of p | Z ( g ) with the automorphism ζ : S ( h ) → S ( h ) given by (3.10) λ ( ζ ( f )) = ( λ − ρ )( f ) λ ∈ h ∗ , f ∈ S ( h ) , 20 STEFFEN SCHMIDT where ρ is the W eyl v ector of the c hosen system ∆ + . F or Λ ∈ h ∗ , the map (3.11) χ Λ ( z ) = (Λ + ρ )(HC( z )) , deﬁnes an algebra homomorphism χ Λ : Z ( g ) → C . A g -supermo dule M is said to admit an inﬁnitesimal char acter if there exists Λ ∈ h ∗ suc h that ev ery z ∈ Z ( g ) acts on M as the scalar χ Λ ( z ). The map χ Λ is then the inﬁnitesimal char acter of M . Highest weigh t g -sup ermo dules provide particular examples of sup ermo dules with an inﬁnitesimal c haracter. A V erma sup ermo dule carries a contra v arian t Hermitian form whenever its highest w eigh t Λ ∈ h ∗ satisﬁes Λ( ω ( H )) = Λ( H ) for any H ∈ h . Suc h a w eigh t is called symmetric . Every highest w eigh t of a unitarizable highest weigh t g -sup ermo dule is symmetric (see [ Sc h24 ]). Let χ Λ : Z ( g ) → C b e the inﬁnitesimal character of M b (Λ). If Λ is symmetric, deﬁne (3.12) F ( X , Y ) : = χ Λ (p( X ω ( Y ))) , X , Y ∈ U ( g ) , where p : U ( g ) → U ( h ) is the Harish-Chandra pro jection from ab o v e. Recall M b (Λ) = U ( n − )[1 ⊗ 1]. Then (3.13) ⟨ X [1 ⊗ 1] , Y [1 ⊗ 1] ⟩ : = F ( X , Y ) , X , Y ∈ U ( n − ) , deﬁnes an ω -contra v ariant Hermitian form on M b (Λ), denoted ⟨· , ·⟩ . ω -con tra v ariance means ⟨ X v , w ⟩ = ⟨ v , ω ( X ) w ⟩ for X ∈ g and v , w ∈ M b (Λ) . This form is called the Shap ovalov form on M b (Λ). The form is unique up to scalar multiples and satisﬁes (3.14) ⟨ M b (Λ) µ , M b (Λ) ν ⟩ = 0 unless µ = ν . The Shap o v alo v form induces on L b (Λ) a non-degenerate ω -contra v arian t form, denoted again b y ⟨· , ·⟩ . W e call the induced form ⟨· , ·⟩ on L b (Λ) again Shap ovalov form . W e use the Shapov alov form as a to ol to test the existence of g ¯ 0 -constituen ts in the g ¯ 0 -ﬁltration of L b (Λ), when Λ is the highest weigh t of a unitarizable g ¯ 0 -mo dule, i.e. , Λ is symmetric. Lemma 16 ([ Mus12 , Chap. 8]) . Supp ose Λ ∈ h ∗ is symmetric. Then the r adic al of the Shap o- valov form ⟨· , ·⟩ on M b (Λ) is the lar gest pr op er subsup ermo dule of M b (Λ) . F or Y ∈ U ( g ) the fol lowing ar e e quivalent: a) Y [1 ⊗ 1] lies in a pr op er submo dule of M b (Λ) . b) F ( X , Y ) = 0 for al l X ∈ U ( g ) . W e consider the equation F ( X , Y ) = 0. By ( 3.14 ), it suﬃces to tak e X ∈ U ( n + ) η and Y ∈ U ( n − ) − η , where U ( n ± ) is decomp osed under the action of h , i.e. , (3.15) U ( n − ) = M η ∈ h ∗ U ( n − ) − η , U ( n + ) = M η ∈ h ∗ U ( n + ) η . Let F η b e the restriction of F to the weigh t space U ( n − ) − η . Since this space is ﬁnite-dimensional, F η ma y b e represen ted by a square matrix, and its determinant can b e considered. If this determinan t is nonzero, no nonzero v ector b elongs to the radical of ⟨· , ·⟩ , and consequen tly no elemen t Y [1 ⊗ 1] for Y ∈ U ( n − ) − η b elongs to a prop er submo dule of M b (Λ). On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 21 The determinant is given b y an explicit form ula, the Kac–Shap ovalov determinant formula . Denote by P ( γ ) the Kostant p artition function , i.e. , the num b er of decomp ositions γ = P i γ i in to p ositive ro ots, sub ject to the condition that if γ i ∈ ∆ + ¯ 1 , then γ j  = γ i for all j  = i . Denote b y P γ ( γ ) the n um b er of partitions of γ in which γ itself do es not o ccur. Theorem 17 ([ Kac86 , Gor04 ]) . Up to a nonzer o c onstant factor, det F η = D 1 D 2 , wher e D 1 = Y γ ∈ ∆ + ¯ 0 ∞ Y r =1 ( h γ + ( ρ, γ ) − r ) P ( η − r γ ) , D 2 = Y γ ∈ ∆ + ¯ 1 ( h γ + ( ρ, γ )) P γ ( η − γ ) . 3.3. The Relativ e Dirac Op erator D g , g ¯ 0 . W e brieﬂy construct the relativ e (quadratic) Dirac op erator D g , g ¯ 0 , follo wing [ HP05 , Xia17 ]. W e emphasize, how ev er, that it arises naturally from the quan tum W eil sup eralgebra; see [ Sc h24 , Sc h b ]. Since ( · , · ) is non-degenerate if and only if m  = n , we distinguish the cases m  = n and m = n , and b egin with m  = n . Fix B ( · , · ) : = 1 2 ( · , · ) on g ; this normalization will b e conv enien t later. Its restriction to g ¯ 1 is symplectic. W e then ﬁx tw o complementary Lagrangian subspaces of g ¯ 1 with bases { ∂ i } and { x i } , for 1 ≤ i ≤ mn , s uc h that B ( ∂ i , x j ) = − B ( x j , ∂ i ) = 1 2 δ ij and ω ( x i ) = − ∂ i . Here, ω = ω ( − , +) in the inﬁnite-dimensional case and ω + in the ﬁnite-dimensional case. This c hoice is compatible with our ﬁxed ∆ + ( cf. Section 2.2 ) in the sense that the basis { ∂ i } spans the o dd p ositive ro ots n + ¯ 1 and the basis { x i } spans the o dd negative ro ots n − ¯ 1 . Let T ( g ¯ 1 ) denote the tensor algebr a of the v ector space g ¯ 1 , regarded as concen trated in ev en degree. The Weyl algebr a is then deﬁned as the quotient W ( g ¯ 1 ) = T ( g ¯ 1 ) /I , where I is the tw o-sided ideal generated by the relations v ⊗ w − w ⊗ v − 2 B ( v, w ) for all v , w ∈ g ¯ 1 . Equiv alently , W ( g ¯ 1 ) can b e realized as the algebra of diﬀerential op erators with p olynomial co eﬃcients in the v ariables x 1 , . . . , x mn , under the identiﬁcation of the generators ∂ i with the partial deriv ativ es ∂ ∂ x i for i = 1 , . . . , mn . W e equip W ( g ¯ 1 ) with the canonical Lie brack et [ · , · ] W . Using the relations [ x i , x j ] W = 0, [ ∂ i , ∂ j ] W = 0, and [ ∂ i , x j ] W = δ ij for all 1 ≤ i, j ≤ mn , one c hec ks that the Lie algebra g ¯ 0 em b eds as a subalgebra of W ( g ¯ 1 ). The corresp onding Lie algebra homomorphism α : g ¯ 0 → W ( g ¯ 1 ) is given explicitly in [ Xia17 , Equation 2]: (3.16) α ( X ) = mn X k,j =1 ( B ( X , [ ∂ k , ∂ j ]) x k x j + B ( X, [ x k , x j ]) ∂ k ∂ j ) − mn X k,j =1 2 B ( X , [ x k , ∂ j ]) x j ∂ k − mn X l =1 B ( X , [ ∂ l , x l ]) . Let Ω g and Ω g ¯ 0 denote the quadratic Casimir op erators of g and g ¯ 0 , resp ectiv ely . W e write Ω g ¯ 0 , ∆ for the image of Ω g ¯ 0 under the diagonal embedding (3.17) g ¯ 0 − → U ( g ) ⊗ W ( g ¯ 1 ) , X 7→ X ⊗ 1 + 1 ⊗ α ( X ) . 22 STEFFEN SCHMIDT The Dir ac element D for g is deﬁned as the o dd elemen t (3.18) D : = D g , g ¯ 0 = 2 mn X i =1  ∂ i ⊗ x i − x i ⊗ ∂ i  ∈ U ( g ) ⊗ W ( g ¯ 1 ) . It is indep endent of the c hoice of basis of g ¯ 1 and is g 0 -in v ariant with resp ect to the g ¯ 0 -action on U ( g ) ⊗ W ( g ¯ 1 ) induced by the adjoin t action on b oth factors [ HP06 , Lemma 10.2.1]; that is, [ g ¯ 0 , D] = 0. In analogy with the case of reductiv e Lie algebras, the Dirac element admits a particularly nice square. Prop osition 18 ([ HP06 , Prop osition 10.2.2]) . The Dir ac element D ∈ U ( g ) ⊗ W ( g ¯ 1 ) satisﬁes D 2 = − Ω g ⊗ 1 + Ω g ¯ 0 , ∆ − C , wher e C is a c onstant that e quals 1 / 8 of the tr ac e of Ω g ¯ 0 on g ¯ 1 . An y Dirac element D may b e realized as a Dirac op erator. F or a g -sup ermo dule M , it acts comp onen t wise on M ⊗ M ( g ¯ 1 ), where M ( g ¯ 1 ) = C [ x 1 , . . . , x mn ] is the oscillator mo dule for W ( g ¯ 1 ) in tro duced b elo w. W e refer to this op erator as the Dir ac op er ator and denote it by the same sym b ol D. Finally , consider the case m = n . W e then w ork in gl ( n | n ), since its sup ertrace form is non-degenerate, and deﬁne the corresp onding Dirac element D. As we are mainly interested in highest w eight sup ermo dules ( cf. Theorem 5 ) and every highest weigh t s l ( n | n )-sup ermo dule is the restriction of a highest weigh t gl ( n | n )-sup ermodule, the same elemen t D acts on s l ( n | n )- sup ermo dules via restriction. W e refer to the resulting op erator as the Dirac op erator for s l ( n | n ), and w e denote it by the same symbol. 3.4. Dirac Operators and Unitarit y . The simple W ( g ¯ 1 )-mo dule M ( g ¯ 1 ) carries a unique Her- mitian form ⟨· , ·⟩ M ( g ¯ 1 ) , kno wn as the Bar gmann–F o ck Hermitian form or Fischer–F o ck Hermit- ian form [ Bar61 , Fis11 , F o c28 ], suc h that ∂ k and x k are adjoint to eac h other and the following orthogonalit y relations hold: ⟨ mn Y k =1 x p k k , mn Y k =1 x q k k ⟩ M ( g ¯ 1 ) =    Q mn k =1 p k ! if p k = q k for all k , 0 otherwise . (3.19) If  H , ⟨· , ·⟩ H  is a unitarizable g -sup ermo dule, w e equip the U ( g ) ⊗ W ( g ¯ 1 )-sup ermo dule H ⊗ M ( g ¯ 1 ) with the p ositive deﬁnite Hermitian form (3.20) ⟨· , ·⟩ H⊗ M ( g ¯ 1 ) = ⟨· , ·⟩ H ⊗ ⟨· , ·⟩ M ( g ¯ 1 ) . Up to multiplication b y a real scalar, ⟨· , ·⟩ H⊗ M ( g ¯ 1 ) is the unique Hermitian form that is g -anti- con tra v ariant in the ﬁrst factor, and satisﬁes x † i = ∂ i in the second. F or our choice of p ositiv e system, this is compatible with the extension of the conjugate-linear anti-in v olution ω to U ( g ), whic h implies that the Dirac op erator is self-adjoint on H ⊗ M ( g ¯ 1 ). W e arriv e at the follo wing Dir ac ine quality : (3.21) ⟨ v , D 2 v ⟩ H⊗ M ( g ¯ 1 ) ≥ 0 On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 23 for all v ∈ H ⊗ M ( g ¯ 1 ). The Dirac inequalit y , together with Prop osition 18 , pro vides a complete c haracterization of unitarit y in terms of the highest w eights of the simple comp osition factors in the g ¯ 0 -ﬁltration of M describ ed in Section 3.2.2 . Theorem 19. L et M b e a highest weight g -sup ermo dule with highest weight Λ . Then M is unitarizable if and only if the fol lowing two c onditions ar e satisﬁe d: a) Λ is the highest weight of a unitarizable highest weight g ¯ 0 -mo dule. b) If L 0 ( µ ) is a simple g ¯ 0 -c omp osition factor in the g ¯ 0 -ﬁltr ation of M with highest weight µ , then ( µ + 2 ρ, µ )    > (Λ + 2 ρ, Λ) p, q  = 0 , < (Λ + 2 ρ, Λ) p = 0 or q = 0 . Theorem 19 is the basis of the classiﬁcation of unitarity . The crucial p oin t is the system of inequalities imp osed b y the weigh ts Λ − α with α ∈ ∆ + ¯ 1 . A direct computation yields the follo wing lemma. Lemma 20. L et µ = Λ − α with α ∈ ∆ + ¯ 1 . Then ( µ + 2 ρ, µ ) ≥ (Λ + 2 ρ, Λ) ⇐ ⇒ (Λ + ρ, α ) ≤ 0 . As a ﬁrst consequence, we obtain the following prop osition, which will b e our main to ol for pro ving unitarit y . Prop osition 21. L et M b e a simple highest weight g -sup ermo dule of highest weight Λ ∈ h ∗ , and assume that Λ is the highest weight of a unitarizable highest weight g ¯ 0 -mo dule. Supp ose that for every highest weight µ = Λ − γ of a g ¯ 0 -c omp osition factor of M , one of the fol lowing holds: a) p = 0 or q = 0 , M is ﬁnite-dimensional, and γ = γ 1 + · · · + γ k with (Λ + ρ, γ i ) > 0 for al l i = 1 , . . . , k ; b) p, q  = 0 , M is inﬁnite-dimensional, and γ = γ 1 + · · · + γ k with (Λ + ρ, γ i ) < 0 for al l i = 1 , . . . , k . Then M is unitarizable. In p articular, M is unitarizable if (Λ + ρ, α ) > 0 for al l α ∈ ∆ + ¯ 1 in the ﬁnite-dimensional c ase, and if (Λ + ρ, α ) < 0 for al l α ∈ ∆ + ¯ 1 in the inﬁnite-dimensional c ase. Pr o of. W e show that for every highest w eight µ of any comp osition factor L 0 ( µ ) of M the Dirac inequalit y holds, that is, ( µ + 2 ρ, µ ) < (Λ + 2 ρ, Λ) if M is ﬁnite-dimensional and ( µ + 2 ρ, µ ) > (Λ + 2 ρ, Λ) if M is inﬁnite-dimensional. The prop osition then follo ws from Theorem 19 . An y highest weigh t µ is of the form µ = Λ − γ , where γ is a sum of pairwise distinct p ositive o dd ro ots. Fix an arbitrary decomp osition γ = γ 1 + · · · + γ k , γ i ∈ ∆ + ¯ 1 . W e call k the length of γ . It is unique. By assumption, (Λ + ρ, γ i )    > 0 , if M is ﬁnite-dimensional , < 0 , if M is inﬁnite-dimensional , for all i . 24 STEFFEN SCHMIDT W e ﬁrst consider the case where M is ﬁnite-dimensional. Here the p ositive system ∆ + is the standard one ∆ + ¯ 1 , st . The v alues (Λ + ρ, α ) for α ∈ ∆ + ¯ 1 arrange as follows: (Λ + ρ, ϵ 1 − δ 1 ) > . . . > (Λ + ρ, ϵ 1 − δ k ) > . . . > (Λ + ρ, ϵ 1 − δ n ) ∨ ∨ ∨ . . . . . . . . . ∨ ∨ ∨ (Λ + ρ, ϵ m − δ 1 ) > . . . > (Λ + ρ, ϵ m − δ k ) > . . . > (Λ + ρ, ϵ m − δ n ) F or any µ = Λ − P k i =1 γ i , w e compute (3.22) (Λ + 2 ρ, Λ) − ( µ + 2 ρ, µ ) = 2(Λ + ρ, k X i =1 γ i ) −  k X i =1 γ i , k X j =1 γ j  . Note that ( γ i , γ j ) ∈ {− 1 , 0 , 1 } . In terpreting the diagram ab o ve as a matrix, we hav e, for i  = j , that ( γ i , γ j ) = 1 if γ i and γ j lie in the same ro w, ( γ i , γ j ) = − 1 if they lie in the same column, and ( γ i , γ j ) = 0 otherwise. By assumption (Λ + ρ, γ i ) > 0 for all i . In view of ( 3.22 ), it therefore suﬃces to consider subsets { γ i } lying in a single row. Fix such a subset. Assume the subset has cardinality k ′ and denote the elements for clarity b y γ ′ 1 , . . . , γ ′ k ′ . Without loss of generalit y , order the γ ′ 1 , . . . , γ ′ k ′ so that (Λ + ρ, γ ′ i ) < (Λ + ρ, γ ′ j ) ⇒ i < j. This is p ossible by the diagram ab o v e. Set ξ : = (Λ + ρ, γ ′ 1 ) with γ ′ 1 : = ϵ r − δ s . Since ξ is minimal, the only ro ots ϵ r − δ j that ma y app ear among the γ ′ i are those with j ∈ { 1 , . . . , s − 1 } . Assume that ϵ r − δ s 1 , . . . , ϵ r − δ s l are pairwise distinct elemen ts of { γ ′ 1 , . . . , γ ′ k } for some l ≤ k ′ and s 1 , . . . , s l ∈ { 1 , . . . , s − 1 } . Then (Λ + ρ, ϵ r − δ s i ) − ξ = (Λ + ρ, δ s − δ s i ) = µ s i − µ s + s − s i ≥ 1 . Altogether, w e obtain for the single row we obtain the estimate 2(Λ + ρ, k ′ X i =1 γ ′ i ) −  k ′ X i =1 γ ′ i , k ′ X j =1 γ ′ j  ≥ 2(Λ + ρ, l X i =1 ( ϵ r − δ s i )) −          0 , l = 1 , 1 , l = 2 , 2 l, else , ≥ 2 ξ + 2 l −          0 , l = 1 , 1 , l = 2 , 2 l, else , > 0 , since ξ > 0. This completes the pro of for the ﬁnite-dimensional case. F or the inﬁnite-dimensional case, set A : = { ϵ i − δ k : 1 ≤ i ≤ p, 1 ≤ k ≤ n } and B : = {− ϵ j + δ k : p + 1 ≤ j ≤ m, 1 ≤ k ≤ n } . W e reﬁne the decomp osition of γ to γ = k 1 X i =1 α i + k 2 X j =1 β j , α i ∈ A, β j ∈ B , k = k 1 + k 2 . On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 25 Using ( α i , β j ) ∈ { 0 , 1 } for all i = 1 , . . . , k 1 and j = 1 , . . . , k 2 , w e obtain ( µ + 2 ρ, µ ) − (Λ + 2 ρ, Λ) = − 2 k 1 X i =1 (Λ + ρ, α i ) − 2 k 2 X j =1 (Λ + ρ, β j ) +  k 1 X i =1 α i + k 2 X j =1 β j , k 1 X i =1 α i + k 2 X j =1 β j  ≥ − 2 k 1 X i =1 (Λ + ρ, α i ) +  k 1 X i =1 α i , k 1 X j =1 α j  − 2 k 2 X j =1 (Λ + ρ, β j ) +  k 2 X i =1 β i , k 2 X j =1 β j  , so it remains to show − 2 k 1 X i =1 (Λ + ρ, α i ) +  k 1 X i =1 α i , k 1 X j =1 α j  > 0 , − 2 k 2 X j =1 (Λ + ρ, β j ) +  k 2 X i =1 β i , k 2 X j =1 β j  > 0 . The pro of of these inequalities is identical to the ﬁnite-dimensional case, where w e now work with the inequalities (Λ + ρ, ε 1 − δ 1 ) > · · · > (Λ + ρ, ε 1 − δ n ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, ε i − δ 1 ) > · · · > (Λ + ρ, ε i − δ n ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, ε p − δ 1 ) > · · · > (Λ + ρ, ε p − δ n ) and (Λ + ρ, − ε m + δ n ) > · · · > (Λ + ρ, − ε m + δ 1 ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, − ε m − j + δ n ) > · · · > (Λ + ρ, − ε m − j + δ 1 ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, − ε p +1 + δ n ) > · · · > (Λ + ρ, − ε p +1 + δ 1 ) This ﬁnishes the pro of. □ 4. Classifying the Full Set In this section, we classify the unitarizable highest w eigh t g -sup ermodules, follo wing the metho d of Section 1.1 . W e distinguish tw o cases: p, q  = 0 and p = 0 or q = 0. Any unitarizable simple g -sup ermodule decomp oses as a ﬁnite direct sum of unitarizable simple g ¯ 0 -mo dules, where g ¯ 0 ≃ su ( p, q ) C ⊕ su ( n ) C ⊕ u (1) C . Each g ¯ 0 -constituen t is an outer tensor pro duct of unitarizable simple su ( p, q )-, su ( n )-, and u (1)-mo dules. Since su ( p, q ) admits non trivial ﬁnite- dimensional unitarizable mo dules only when p = 0 or q = 0, it follows that g has no nontrivial 26 STEFFEN SCHMIDT unitarizable ﬁnite-dimensional sup ermo dules if p, q  = 0. If p = 0 or q = 0, then ev ery simple g ¯ 0 - mo dule is ﬁnite-dimensional, and hence every unitarizable g -sup ermo dule is ﬁnite-dimensional. In particular, if p, q  = 0, then every nontrivial unitarizable g -sup ermo dule is necessarily inﬁnite- dimensional. 4.1. Finite-Dimensional Unitarizable Sup ermo dules. Assume that p = 0 or q = 0 and ﬁx the standard p ositiv e system. W e b egin by parametrizing the p ossible highest weigh ts of unitarizable g -sup ermo dules. Since any such highest weigh t is, in particular, the highest w eigh t of a unitarizable highest weigh t g ¯ 0 -mo dule, the description from Section 3.2.1 applies. On this basis, we formulate the classiﬁcation metho d and illustrate it by tw o explicit examples, namely s l (2 | 1) and s l (2 | 2). The general classiﬁcation is then obtained by extending the same argument. 4.1.1. Par ametrization of the Weight Sp ac e. The ﬁnite-dimensional simple sup ermodules are classiﬁed by the dominant in tegral weigh ts λ ∈ h ∗ , tak en with resp ect to a Borel subalgebra b = b ¯ 0 ⊕ b ¯ 1 determined by a p ositive system ∆ + = ∆ + ¯ 0 ⊔ ∆ + ¯ 1 . Recall that ∆ + ¯ 0 w as ﬁxed in Section 2.1 . There are tw o natural choices of Borel subalgebras, namely the distinguishe d Bor el sub algebr a b st and the anti-distinguishe d Bor el sub algebr a b − st , corresp onding resp ectively to (4.1) ∆ + ¯ 1 , st = { ϵ i − δ j | 1 ≤ i ≤ m, 1 ≤ j ≤ n } , ∆ + ¯ 1 , − st = {− ϵ i + δ j | 1 ≤ i ≤ m, 1 ≤ j ≤ n } . Recall that under the canonical isomorphism s l ( m | n ) ∼ = s l ( n | m ), the an ti-distinguished Borel subalgebra b − st of s l ( m | n ) maps to the distinguished one b st of s l ( n | m ). W e w ork with the distinguished Borel subalgebra b st and hence write b : = b st and ∆ + ¯ 1 : = ∆ + ¯ 1 , st in the sequel. A w eigh t λ is dominant inte gr al if and only if (4.2) ( λ + ρ ¯ 0 , α ) ∈ Z > 0 for all α ∈ ∆ + ¯ 0 , equiv alently , if there exists a ﬁnite-dimensional simple g ¯ 0 -mo dule of highest w eigh t λ with resp ect to b ¯ 0 . If we denote the highest weigh t in terms of standard co ordinates on d ∗ b y Λ = ( λ 1 , . . . , λ m | µ 1 , . . . , µ n ) (mo dulo shifts by (1 , . . . , 1 |− 1 , . . . , − 1)), this imp oses a classical sequence of standard conditions on Λ, which are (4.3) λ 1 ≥ · · · ≥ λ m , µ 1 ≥ · · · ≥ µ n and the diﬀerences in the tw o c hains hav e to b e in tegral. W e denote the set of suc h b -dominant in tegral w eights b y P ++ , and refer to them as ∆ + -dominan t in tegral w eights. F or λ ∈ P ++ , w e let L ( λ ) denote the simple supermo dule of highest weigh t λ with resp ect to b , whose highest weigh t vector is even. With this notation, the simple ﬁnite-dimensional g -sup ermo dules are parameterized by (4.4) { L ( λ ) , Π L ( λ ) : λ ∈ P ++ } . By deﬁnition, ﬁnite-dimensional unitarizable g -sup ermodules are precisely those that are uni- tarizable with resp ect to one of the conjugate-linear an ti-in volutions ( cf. Lemma 3 ) (4.5) ω ± A B C D ! = A † ± C † ± B † D † ! , A B C D ! ∈ g . On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 27 These conjugate-linear an ti-in volutions corresp ond to the compact real form of g ¯ 0 , namely su ( m ) C ⊕ su ( n ) C ⊕ u (1) C . In the sequel, we consider only ω : = ω + , as the ω − -case is analo- gous. W e ﬁnd it conv enient to describ e Λ ∈ P ++ more explicitly , so that Λ may b e regarded as a one-parameter family in h ∗ . Assume that the ﬁnite-dimensional simple g -sup ermo dule L (Λ) is ω -unitarizable. Then Λ is the highest weigh t of a ﬁnite-dimensional unitarizable simple g ¯ 0 - mo dule L 0 (Λ). Suc h a mo dule is (isomorphic to) the outer tensor pro duct of simple su ( m ) C -, su ( n ) C -, and u (1) C -mo dules. The simple su ( m ) C -mo dules hav e highest weigh ts of the form (4.6) ( − a 1 , − a 2 , . . . , − a m | 0 , . . . , 0) , with a 1 , . . . , a m ∈ Z ≥ 0 and 0 = − a 1 ≥ − a 2 ≥ · · · ≥ − a m . Analogously , the simple su ( n ) C - mo dules hav e highest weigh ts of the form (4.7) (0 , . . . , 0 | b 1 , . . . , b n ) , with b 1 , . . . , b n ∈ Z ≥ 0 and b 1 ≥ · · · ≥ b n = 0. Since u (1) is ab elian, Sch ur’s lemma implies that its simple mo dules are one-dimensional and, by unitarit y , uniquely determined b y a p ositive real n um b er. F or conv enience we write (4.8) Λ = (0 , − a 2 , . . . , − a m | b 1 , . . . , b n − 1 , 0) + x 0 2 (1 , . . . , 1 | 1 , . . . , 1) , x 0 ∈ R . T o trac k the nontrivial comp onents among the a i and b j , we deﬁne i 0 as the largest integer suc h that a i 0 = 0, and k 0 as the smallest integer such that b k 0 = 0. In general, a highest w eigh t g -supermo dule L (Λ) with Λ as in ( 4.8 ) need not be ω -unitarizable. T o classify the ω -unitarizable simple g -sup ermo dules, we regard any Λ as a particular v alue of a one-parameter family in h ∗ , indexed by x ∈ R , that is (4.9) Λ( x ) = Λ 0 + x 2 (1 , . . . , 1 | 1 , . . . , 1) , where Λ 0 = (0 , − a 2 , . . . , − a m | b 1 , . . . , b n − 1 , 0) with a i , b j ∈ Z ≥ 0 , 0 ≥ − a 2 ≥ · · · ≥ − a m , and b 1 ≥ · · · ≥ b n − 1 ≥ 0 as ab ov e. Note that ( α, (1 , . . . , 1 | 1 , . . . , 1)) = 0 for all α ∈ ∆ ¯ 0 . In standard co ordinates, we write (4.10) Λ( x ) = ( λ 1 ( x ) , . . . , λ m ( x ) | µ 1 ( x ) , . . . , µ n ( x )) , µ k 0 ( x ) = · · · = µ n ( x ) . Throughout, w e assume the unitarity conditions from Lemma 6 : (i) λ 1 ( x ) ≥ · · · ≥ λ m ( x ) ≥ − µ n ( x ) ≥ · · · ≥ − µ 1 ( x ); (ii) if (Λ( x ) + ρ, ϵ m − δ k ) = 0, then (Λ( x ) + ρ, ϵ m − δ j ) > 0 for all j = 1 , . . . , k − 1, and (Λ( x ) , δ k − δ n ) = 0; (iii) if (Λ( x ) + ρ, ϵ m − δ k )  = 0 for all k = 1 , . . . , n , then (Λ( x ) + ρ, ϵ m − δ k ) > 0 for all k = 1 , . . . , n . In what follows, w e write Λ = Λ( x ) and suppress the dep endence on x whenev er no confusion can arise. The classiﬁcation is obtained by applying the metho d of Section 1.1.1 . 28 STEFFEN SCHMIDT 4.1.2. Examples. W e illustrate our classiﬁcation metho d b y considering tw o examples: the real Lie sup eralgebras su (2 | 1) and su (2 | 2). su (2 | 1): Set g : = su (2 | 1). The set of positive ro ots is ∆ + = { ϵ 1 − ϵ 2 , ϵ 1 − δ 1 , ϵ 2 − δ 1 } . In particular, ρ = (0 , − 1 | 1). F or con v enience, set α : = ϵ 1 − δ 1 and β : = ϵ 2 − δ 1 . The family Λ has general form (4.11) Λ = Λ( x ) =  x 2 , − a + x 2   x 2  , x ∈ R for some a ∈ Z ≥ 0 . By Lemma 12 , L (Λ) decomp oses as a direct sum of simple g ¯ 0 -mo dules. The only p ossibilities are L 0 (Λ) , L 0 (Λ − α ), L 0 (Λ − β ), and L 0 (Λ − α − β ). The Dirac inequalities for the g ¯ 0 -constituen ts L 0 (Λ − α ) for α ∈ ∆ + ¯ 1 read (Lemma 20 ) (4.12) 0 ≤ (Λ + ρ, α ) = x + 1 ⇔ x ≥ − 1 0 ≤ (Λ + ρ, β ) = x − a ⇔ x ≥ a. W e now obtain a complete classiﬁcation, parametrized by x , follo wing the metho d of Sec- tion 1.1.1 . W e b egin b y determining x max suc h that L (Λ) is unitarizable for all x > x max . If x > a , then Λ is t ypical, and all g ¯ 0 -constituen ts listed ab o ve o ccur. By ( 4.12 ), the Dirac inequality is strict on each of them. Hence Prop osition 21 implies that Λ is the highest weigh t of a unitarizable g -sup ermo dule. Therefore x max = a . W e next determine x min . Let x < a . Then the g ¯ 0 -constituen t L 0 (Λ − β ) o ccurs in L (Λ), but the Dirac inequality fails on it. Indeed, β = ϵ 2 − δ 1 is a simple ro ot, and the weigh t space M b (Λ) Λ − β is one-dimensional, spanned b y e − β v Λ where e − β is the ro ot vector asso ciated with the ro ot − β . Since x < a , this vector do es not lie in the radical of the Shap ov alov form, b ecause (4.13) ⟨ e − β v Λ , e − β v Λ ⟩ = (Λ + ρ, β ) ⟨ v Λ , v Λ ⟩ = ( x − a ) ⟨ v Λ , v Λ ⟩ < 0 , where ⟨ v Λ , v Λ ⟩ = 1. Hence L 0 (Λ − β ) o ccurs in the g ¯ 0 -ﬁltration of L (Λ). Since the Dirac inequalit y do es not hold on this constituen t, L (Λ) is not unitarizable for x < a by Theorem 19 . Therefore x min = a . It remains to study unitarity on I = [ x min , x max ] = { a } . Assume x = a . By Theorem 19 , the mo dule L (Λ) is unitarizable if and only if ( µ + 2 ρ, µ ) − (Λ + 2 ρ, Λ) > 0 for ev ery g ¯ 0 -constituen t of L (Λ). Since x = a , one has (Λ + ρ, β ) = 0, so Λ is atypical. Hence the g ¯ 0 -constituen ts of highest weigh ts Λ − β and Λ − α − β do not o ccur; see Lemma 15 . Since (Λ + ρ, α ) > 0, L (Λ) is unitarizable. Altogether, the full set of unitarizable highest weigh t su (1 , 1 | 1)-sup ermo dules is (4.14) { Λ = ( x 2 , − a + x 2 | x 2 ) : a ∈ Z ≥ 0 , x ∈ [ a, ∞ ) } . Equiv alently , Λ is the highest weigh t of a unitarizable highest weigh t g -sup ermodule if and only if a) Λ satisﬁes the unitarity conditions; b) either (Λ + ρ, ϵ 2 − δ 1 ) = 0 or (Λ + ρ, ϵ 2 − δ 1 ) > 0. On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 29 su (2 | 2): Set g : = su (2 | 2). The p ositiv e ro ot system is ∆ + = { ϵ 1 − ϵ 2 , δ 1 − δ 2 , ϵ 1 − δ 1 , ϵ 1 − δ 2 , ϵ 2 − δ 1 , ϵ 2 − δ 2 } . In particular, ρ =  − 1 2 , − 3 2   3 2 , 1 2  . The family Λ = Λ( x ) has general form (4.15) Λ =  x 2 , − a + x 2   b + x 2 , x 2  , x ∈ R , with a, b ∈ Z ≥ 0 . The unitarit y conditions are: (4.16) λ 1 ≥ λ 2 ≥ − µ 2 ≥ − µ 1 , (Λ + ρ, ϵ 2 − δ k ) = 0 ⇒ (Λ , δ k − δ 2 ) = 0 , (Λ + ρ, ϵ 2 − δ k )  = 0 for k = 1 , 2 ⇒ (Λ + ρ, ϵ 2 − δ k ) > 0 for k = 1 , 2 . Moreo v er, the Dirac inequalities are (4.17) 0 ≤ (Λ + ρ, ϵ 1 − δ 1 ) = x + b + 1 ⇔ x ≥ − b − 1 , 0 ≤ (Λ + ρ, ϵ 1 − δ 2 ) = x ⇔ x ≥ 0 , 0 ≤ (Λ + ρ, ϵ 2 − δ 1 ) = x + b − a ⇔ x ≥ − b + a, 0 ≤ (Λ + ρ, ϵ 2 − δ 2 ) = x − a − 1 ⇔ x ≥ a + 1 . As in the case of sl (2 | 1), we now obtain a complete classiﬁcation parametrized by x , follo wing the metho d of Section 1.1.1 . W e ﬁrst determine x max . The most restrictiv e condition is imp osed by α : = ϵ 2 − δ 2 , namely x ≥ a + 1. If x > a + 1, then (Λ + ρ, α ) > 0 for all α ∈ ∆ + ¯ 1 . Hence the Dirac inequalities are strict for all g ¯ 0 -constituen ts, and Prop osition 21 implies that Λ is the highest weigh t of a unitarizable g -sup ermo dule. Therefore x max = a + 1. Next, we identify x min . T o treat the cases b = 0 and b  = 0 uniformly , we in tro duce k 0 deﬁned b y k 0 = 1 if b = 0 and k 0 = 2 if b  = 0. The idea is to consider the g ¯ 0 -constituen t of L (Λ) that imp oses the strongest condition on x . If b  = 0, this is L 0 (Λ − ϵ 2 + δ 2 ), and if b = 0, this is L 0 (Λ − ϵ 2 + δ 1 ) according to dominance; equiv alently , we write L 0 (Λ − ϵ 2 + δ k 0 ). W e sho w that L 0 (Λ − ϵ 2 + δ k 0 ) exists as a g ¯ 0 -constituen t whenever (Λ + ρ, ϵ 2 − δ k 0 ) < 0, using the Kac–Shap o v alov determinant formula (Theorem 17 ). In that case, L (Λ) cannot b e unitarizable, since the Dirac inequality fails on this constituent. The determinan t of the Shap ov alo v form restricted to M (Λ) Λ − ϵ 2 + δ k 0 is (4.18) ((Λ + ρ, δ 1 − δ k 0 ) − 1) · (Λ + ρ, k 0 Y k =1 ϵ 2 − δ k ) . The ﬁrst factor equals − 1 if k 0 = 1 and − b − 1 < 0 if k 0 = 2, hence never v anishes. The second factor is zero only if (Λ + ρ, ϵ 2 − δ k ) = 0 for some k = 1 , . . . , k 0 . Since by assumption (Λ + ρ, ϵ 2 − δ k 0 ) < 0, zeros may o ccur only for k < k 0 . How ever, by the unitarity conditions (see ( 4.16 )), this would imply (Λ , δ k − δ n ) = 0, contradicting the minimality of k 0 . W e conclude that the Kac–Shap o v alov determinant do es not v anish on M b (Λ) Λ+ ϵ 2 − δ k 0 , and L 0 (Λ + ϵ 2 − δ k 0 ) cannot lie in the radical. Since the Dirac inequalit y fails on this constituen t, L (Λ) is not unitarizable for x < a + k 0 − 1. Therefore x min = a + k 0 − 1. Set I : = [ x min , x max ] = [ a + k 0 − 1 , a + 1]. It remains to consider x ∈ I . The integral p oints of I are { a + k 0 − 1 , a + 1 } , and non-integral p oints o ccur precisely when k 0 = 1. W e ﬁrst treat 30 STEFFEN SCHMIDT this case. Let k 0 = 1 and x ∈ ( a, a + 1). Then (Λ + ρ, ϵ 2 − δ j )  = 0 for j = 1 , 2, and (4.19) (Λ + ρ, ϵ 2 − δ 2 ) < 0 < (Λ + ρ, ϵ 2 − δ 1 ) , whic h con tradicts the unitarit y conditions for Λ. Hence L (Λ) is not unitarizable. The in tegral p oin ts of this interv al are precisely the p oints at which Λ is at ypical, that is, (4.20) (Λ + ρ, ϵ 2 − δ k ) = 0 ⇐ ⇒ x = a + k − 1 , k = k 0 , . . . , 2 . W e treat only the case k 0 = 2; the remaining cases are analogous. Then (4.21) (Λ + ρ, ϵ 1 − δ 1 ) > (Λ + ρ, ϵ 1 − δ 2 ) ∨ ∨ (Λ + ρ, ϵ 2 − δ 1 ) > 0 = (Λ + ρ, ϵ 2 − δ 2 ) and, by Lemma 15 , the g ¯ 0 -constituen t L 0 (Λ − ϵ 2 + δ 2 ) do es not o ccur in L (Λ). Moreov er, for any g ¯ 0 -constituen t with highest weigh t µ = Λ − γ one can choose a decomp osition γ = γ 1 + · · · + γ r that do es not inv olv e the ro ot ϵ 2 − δ 2 . By the diagram ab o v e, this implies (Λ + ρ, γ i ) > 0 for all i , so we are in the situation of the pro of of Prop osition 21 . Consequen tly , L (Λ) is unitarizable for x = a + 1. Altogether, the full set of unitarizable highest weigh t su (1 , 1 | 2)-sup ermo dules is (4.22)  Λ =  x 2 , − a + x 2   b + x 2 , x 2  : a, b ∈ Z ≥ 0 , x ∈ { a + k 0 − 1 , . . . , a + 1 } ⊔ ( a + 1 , ∞ )  . Equiv alently , Λ is the highest weigh t of a unitarizable highest weigh t g -sup ermodule if and only if a) Λ satisﬁes the unitarity conditions; b) one of the following holds: (i) (Λ + ρ, ϵ 2 − δ k ) = 0 for some k = k 0 , . . . , 2, or (ii) (Λ + ρ, ϵ 2 − δ 2 ) > 0. 4.1.3. Classiﬁc ation. W e start with a family of highest w eights Λ of unitarizable highest w eight g ¯ 0 -mo dules of the form (4.23) Λ = (0 , − a 2 , . . . , − a m | b 1 , . . . , b n − 1 , 0) + x 2 (1 , . . . , 1 | 1 , . . . , 1) , x ∈ R , where a 2 ≤ · · · ≤ a m and b 1 ≥ · · · ≥ b n are non-negative in tegers. W e assume throughout that for eac h Λ satisﬁes for each x the unitarity conditions of Lemma 6 . W e now apply the metho d of Section 1.1.1 step by step. The analysis rests on the follo wing computation. The ﬁxed standard system of p ositive o dd ro ots is ∆ + ¯ 1 = { ϵ i − δ j : 1 ≤ i ≤ m, 1 ≤ j ≤ n } . Moreov er, (4.24) (Λ + ρ, ϵ i − δ j ) = λ i + µ j + m − i − j + 1 = − a i + b j + x + m − i − j + 1 , where a 1 = 0 and b n = 0. Recall that k 0 is the smallest index such that b k 0 = 0. T ogether with the form of the highest weigh t ( 4.8 ), this immediately gives the follo wing lemma. On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 31 Lemma 22. The numb ers (Λ + ρ, α ) , with α ∈ ∆ + ¯ 1 , arr ange as fol lows: (Λ + ρ, ϵ 1 − δ 1 ) > · · · > (Λ + ρ, ϵ 1 − δ k ) > · · · > (Λ + ρ, ϵ 1 − δ n ) ∨ ∨ ∨ . . . . . . . . . ∨ ∨ ∨ (Λ + ρ, ϵ m − δ 1 ) > · · · > (Λ + ρ, ϵ m − δ k ) > · · · > (Λ + ρ, ϵ m − δ n ) As a direct consequence, one can determine x max , that is, the v alue suc h that L (Λ) is unita- rizable for all x > x max . Lemma 23. If (Λ + ρ, ϵ m − δ n ) > 0 , e quivalently if x > a m + n − 1 , then L (Λ) is unitarizable. Pr o of. By Lemma 22 , the minimum of all (Λ + ρ, ϵ i − δ j ) o ccurs at (Λ + ρ, ϵ m − δ n ), i.e. , at x = a m + n − 1. Hence, if (Λ + ρ, ϵ m − δ n ) > 0, we hav e (Λ + ρ, α ) > 0 for all α ∈ ∆ + ¯ 1 , and the statemen t follo ws with Prop osition 21 . □ The lemma yields x max = a m + n − 1. On the other hand, the unitarity conditions show that the weak est constraint is imp osed by the Dirac inequalit y for L 0 (Λ − ϵ m + δ k 0 ). This determines x min , namely the largest v alue such that L (Λ) is not unitarizable for all x < x min . The follo wing lemma mak es this precise. Lemma 24. If (Λ + ρ, ϵ m − δ k 0 ) < 0 , that is, x < a m + k 0 − 1 , then L (Λ) is not unitarizable. Pr o of. W e show that the g ¯ 0 -constituen t L 0 (Λ − ϵ m + δ k 0 ) o ccurs in L (Λ), hence L (Λ) is not unitarizable as the Dirac inequality do es not hold. It suﬃces to pro v e that the w eigh t space M (Λ) Λ − ϵ m + δ k 0 do es not lie in the radical of the Shap ov alov form ⟨· , ·⟩ on M (Λ); equiv alently , the Kac–Shap o v alov determinant on this w eigh t space is nonzero. Set η : = ϵ m − δ k 0 . Consider the factor from ∆ + ¯ 0 in Theorem 17 , D 1 = Y γ ∈ ∆ + ¯ 0 ∞ Y r =1 ( h γ + ( ρ, γ ) − r ) P ( η − r γ ) . No w η − r γ is a s um of p ositive ro ots only if γ = δ k − δ k 0 and r = 1 with 1 ≤ k ≤ k 0 . F or these γ one has (Λ + ρ, δ k − δ k 0 ) = ( µ k 0 − µ k ) − k 0 + k < 0, so the factor is nev er zero. Consider the factor from ∆ + ¯ 1 , D 2 = Y γ ∈ ∆ + ¯ 1 ( h γ + ( ρ, γ )) P γ ( η − γ ) . Here η − γ is a sum of p ositiv e ro ots if and only if η = ϵ m − δ k for 1 ≤ k ≤ k 0 . Hence the zeros of the Kac–Shap ov alo v determinant on M (Λ) Λ − ϵ m + δ k 0 o ccur precisely when (Λ + ρ, ϵ m − δ k ) = 0 , 1 ≤ k ≤ k 0 . Since k 0 is the minimal p ositive integer with b k = 0, these v alues are excluded by the unitarity conditions (Lemma 6 ). Th us D 2  = 0, and the Kac–Shap ov alo v determinant is nontrivial. This completes the pro of. □ 32 STEFFEN SCHMIDT The lemma yields x min = a m + k 0 − 1. It remains to consider the range x ∈ [ x min , x max ]. The in teger v alues in this interv al corresp ond to at ypicality of Λ, i.e. , (4.25) (Λ + ρ, ϵ m − δ k ) = 0 ⇐ ⇒ x = a m + k − 1 , k 0 ≤ k ≤ n. These p oin ts corresp ond to p oints of unitarit y , that is, unitarizable sup ermo dules. Lemma 25. Assume (Λ + ρ, ϵ m − δ k ) = 0 for some k 0 ≤ k ≤ n . Then L (Λ) is unitarizable. Pr o of. By Prop osition 21 , it suﬃces to show that ev ery g ¯ 0 -constituen t L 0 (Λ − γ ) admits a decomp osition γ = γ 1 + · · · + γ r in to pairwise distinct p ositiv e o dd roots such that (Λ + ρ, γ i ) > 0 for all i . Consider the diagram of Lemma 22 . By monotonicit y , negative v alues can o ccur only for ro ots of the form ϵ i − δ j with j > k . F or such ro ots, (Λ + ρ, ϵ i − δ j ) = (Λ + ρ, ϵ i − ϵ m ) + (Λ + ρ, ϵ m − δ j ) = a m − a i + m − i + (Λ + ρ, ϵ m − δ k ) + (Λ + ρ, δ k − δ j ) = a m − a i + m − i + k − j. Hence the relev an t part of the diagram has the form (Λ + ρ, ϵ 1 − δ 1 ) > · · · > (Λ + ρ, ϵ 1 − δ k ) > ( a m − a 1 ) + m − 1 > · · · > ( a m − a 1 ) + m − 1 − ( n − k ) ∨ ∨ ∨ ∨ · · · · · · · · · · · · ∨ ∨ ∨ ∨ (Λ + ρ, ϵ m − 1 − δ 1 ) > · · · > (Λ + ρ, ϵ m − 1 − δ k ) > ( a m − a m − 1 ) > · · · > ( a m − a m − 1 ) − ( n − k ) ∨ ∨ ∨ ∨ (Λ + ρ, ϵ m − δ 1 ) > · · · > 0 > − 1 > · · · > − ( n − k ) Since (Λ + ρ, ϵ m − δ k ) = 0, Lemma 15 implies that ev ery g ¯ 0 -constituen t of L (Λ) admits a decomp osition which do es not inv olve any of the ro ots ϵ m − δ k , . . . , ϵ m − δ n . No w assume that (Λ + ρ, ϵ i − δ j ) < 0 for some 1 ≤ i < m and k < j ≤ n . Since (Λ + ρ, ϵ i − δ k ) > 0 and (Λ + ρ, ϵ i − δ k ) − (Λ + ρ, ϵ i − δ j ) = j − k , monotonicit y implies that there exists s with k ≤ s < j suc h that (Λ + ρ, ϵ i − δ s ) = 0. By Lemma 15 , every comp osition factor L 0 (Λ − γ ) of L (Λ) therefore admits a decomp osition into pairwise distinct p ositive o dd ro ots whic h do es not inv olv e any of the ro ots ϵ i − δ s , . . . , ϵ i − δ n . Applying this argument to each i , one obtains that ev ery such g ¯ 0 -constituen t factor admits a decomp osition γ = γ 1 + · · · + γ r in to pairwise distinct p ositiv e o dd roots such that (Λ + ρ, γ i ) > 0 for all i . This pro v es the claim. □ It remains to consider the non-integral p oints of the in terv al I . Lemma 26. If (Λ + ρ, ϵ m − δ k ) < 0 < (Λ + ρ, ϵ m − δ k − 1 ) , then L (Λ) is not unitarizable. On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 33 Pr o of. If (Λ + ρ, ϵ m − δ k ) < 0 < (Λ + ρ, ϵ m − δ k − 1 ), then (Λ + ρ, ϵ m − δ k )  = 0 for all k = 1 , . . . , n . Y et the condition (Λ + ρ, ϵ m − δ k ) < 0 is incompatible with the unitarit y conditions imp osed on Λ, and L (Λ) is not unitarizable. □ Com bining the preceding lemmas, w e obtain a complete classiﬁcation of the ﬁnite-dimensional unitarizable g -sup ermodules. Theorem 27. The ful l set of unitarizable g -sup ermo dules is p ar ameterize d by the union over al l  Λ( x ) = Λ 0 + x 2 (1 , . . . , 1 | 1 , . . . , 1) : x ∈ { a m + k 0 − 1 , . . . , a m + n − 1 } ⊔ ( a m + n − 1 , ∞ )  with Λ 0 = (0 , − a 2 , . . . , − a m | b 1 , . . . , b n − 1 , 0) with a i , b j ∈ Z ≥ 0 , 0 ≥ − a 2 ≥ · · · ≥ − a m , and b 1 ≥ · · · ≥ b k 0 − 1 > b k 0 = . . . = b n = 0 . Equivalently, Λ ∈ h ∗ is the highest weight of a unitarizable highest weight g -sup ermo dule if and only if it satisﬁes the fol lowing two c onditions: a) Λ satisﬁes the unitarity c onditions. b) one of the fol lowing holds: (i) (Λ + ρ, ϵ m − δ k ) = 0 for some k = k 0 , . . . , n , or (ii) (Λ + ρ, ϵ m − δ n ) > 0 . R emark 28 . W e assume highest w eight v ectors to b e ev en. Accordingly , the full set of unitarizable highest w eigh t g -sup ermo dules is obtained up to application of the parit y rev ersion functor Π. 4.2. Inﬁnite-Dimensional Unitarizable Sup ermo dules. W e assume p, q  = 0 and, without loss of generalit y , p ≤ q . W e ﬁx the non-standard system of p ositiv e ro ots. Recall the ex- pression for the W eyl vector from ( 2.22 ). W e ﬁrst describ e the highest weigh ts that can o ccur for unitarizable g -sup ermo dules, in direct analogy with the ﬁnite-dimensional case. W e then state the classiﬁcation strategy and illustrate it in the examples sl (2 | 1) and sl (2 | 2). The general classiﬁcation follo ws b y extending the same argument. 4.2.1. Par ametrization of the Weight Sp ac e. Let L (Λ) b e a simple ω -unitarizable g -sup ermo dule, where w e recall that ω : = ω ( − , +) . It follo ws from the deﬁnition of unitarity that a necessary condition for L (Λ) to b e unitarizable as a g -sup ermo dule is that Λ is the highest weigh t of a unitarizable g ¯ 0 -mo dule, denoted by L 0 (Λ). Recall g ¯ 0 = su ( p, q ) C ⊕ su ( n ) C ⊕ u (1) C to em- phasize the real form. This imp oses a classical sequence of standard conditions on the high- est weigh t, whic h we recall is parameterized in terms of the standard co ordinates on h ∗ as Λ = ( λ 1 , . . . , λ m | µ 1 , . . . , µ n ) , mo dulo shifts by (1 , . . . , 1 |− 1 , . . . , − 1). W e b egin by recalling the parametrization of unitarizable highest w eight g ¯ 0 -mo dules. First, we consider the restriction to the maximal compact subalgebra k : = su ( p ) ⊕ su ( q ) ⊕ u (1) ⊕ su ( n ) ⊕ u (1) of su ( p, q | n ). If L 0 (Λ) is unitarizable as a g ¯ 0 -mo dule, then as a k C -mo dule it is semisimple with ﬁnite m ultiplicities. In particular, Λ is the highest weigh t of a unitarizable simple (hence ﬁnite-dimensional) k C -mo dule, whic h app ears with multiplicit y one. Namely , Λ must b e in tegral and dominant with resp ect to the p ositiv e system induced from g , that is (4.26) λ p +1 ≥ · · · ≥ λ m ≥ λ 1 · · · ≥ λ p , µ 1 ≥ · · · ≥ µ n 34 STEFFEN SCHMIDT F ollowing [ Jak94 ], we parameterize the solution to these constraints by writing (4.27) Λ = (0 , a 2 , . . . , a m − 1 , 0 | b 1 , . . . , b n − 1 , 0) + λ 2 (1 , . . . , 1 , − 1 , . . . , − 1 | 0 , . . . , 0) + x 2 (1 , . . . , 1 | 1 , . . . , 1) , with integers a i satisfying a p +1 ≥ · · · ≥ a m − 1 ≥ 0 ≥ a 2 ≥ · · · ≥ a p , integers b k satisfying b 1 ≥ · · · ≥ b n − 1 ≥ b n : = 0 and real num bers x ∈ R and λ ∈ R ≥ 0 . In order to obtain this parameterization, w e use the shift-inv ariance to imp ose the relation λ 1 + λ m = 2 µ n =: x , and set λ : = λ 1 − λ m . The conditions on the a i and b k then follow from the preceding discussion. It is known [ EHW83 , Jak94 ] that, for ﬁxed a i and b k sub jec t to these conditions, the w eight Λ in ( 4.27 ) is the highest weigh t of a unitarizable simple highest w eight g ¯ 0 -mo dule if and only if (4.28) λ ∈  −∞ , − m + max( i 0 , j 0 ) + 1  ∪  − m + max( i 0 , j 0 ) + 1 , − m + max( i 0 , j 0 ) + 2 , . . . , − m + i 0 + j 0  , where i 0 is the largest index for whic h a i = 0, and j 0 is the largest integer for which a m − j = 0 (if a p +1 = 0 then j 0 = q ). Moreov er, we deﬁne k 0 to b e the smallest integer suc h that b k 0 = 0. The v alues i 0 , j 0 and k 0 are part of the dominance data with resp ect to the maximal compact subalgebra. W e supp ose that Λ satisﬁes the unitarity conditions (Lemma 6 ), i.e. , (i) λ p +1 ≥ · · · ≥ λ m ≥ − µ n ≥ · · · ≥ − µ 1 ≥ λ 1 ≥ · · · ≥ λ p , (ii) (Λ + ρ, − ϵ i + δ n ) = 0 for p + 1 ≤ i ≤ m implies (Λ , ϵ i − ϵ m ) = 0, (iii) (Λ + ρ, ϵ i − δ 1 ) = 0 for 1 ≤ i ≤ p implies (Λ , ϵ 1 − ϵ i ) = 0. (iv) If (Λ + ρ, ϵ i − δ 1 )  = 0 for 1 ≤ i ≤ i 0 , then (Λ + ρ, ϵ i − δ 1 ) < 0 for i = 1 , . . . , i 0 . (v) If (Λ + ρ, ϵ m − j + δ n )  = 0 for 0 ≤ j ≤ j 0 , then (Λ + ρ, − ϵ m − j + δ n ) < 0 for j = 0 , . . . , j 0 . In general, a highest weigh t g -sup ermo dule L (Λ) with Λ satisfying (i)-(v) need not b e ω - unitarizable. T o classify the ω -unitarizable simple g -sup ermo dules, it is conv enien t to in tro duce one-parameter families in h ∗ , as in the ﬁnite-dimensional case, indexed by x ∈ R , of the form (4.29) Λ( x ) = Λ 0 + x 2 (1 , . . . , 1 | 1 , . . . , 1) , where Λ 0 = (0 , a 2 , . . . , a m − 1 , 0 | b 1 , . . . , b n − 1 , 0) + λ 2 (1 , . . . , 1 , − 1 , . . . , − 1 | 0 , . . . , 0) with λ satisfying ( 4.28 ), and in tegers a p +1 ≥ · · · ≥ a m − 1 ≥ a m = 0 = a 1 ≥ a 2 ≥ · · · ≥ a p and b 1 ≥ · · · ≥ b n − 1 ≥ 0 as ab o v e. W e assume that the unitarity conditions hold for all x ∈ R . In what follows, w e write Λ = Λ( x ) and suppress the dep endence on x whenever no confusion can arise. The classiﬁcation is obtained by applying the metho d of Section 1.1.2 . 4.2.2. Examples. W e no w illustrate the classiﬁcation pro cedure on the Lie superalgebras su (1 , 1 | 1) and su (1 , 1 | 2). These are sp eciﬁc instances of sup ersymmetry algebras of sup erconfor- mal quan tum mec hanics [ FR84 , ES15 ]. su (1 , 1 | 1). Set g : = su (1 , 1 | 1). The non-standard p ositive system is ∆ + = { ϵ 1 − ϵ 2 , ϵ 1 − δ 1 , − ϵ 2 + δ 1 } , the corresp onding W eyl vector is ρ = (0 , 0 | 0), and w e set α : = ϵ 1 − δ 1 and β : = − ϵ 2 + δ 1 . In particular, A = { α } and B = { β } . On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 35 A general family Λ of highest weigh ts of a unitarizable g ¯ 0 -mo dule is of the form (4.30) Λ =  λ 2 + x 2 , − λ 2 + x 2   x 2  , x ∈ R , with λ ∈ R ≤ 0 . The p ossible simple g ¯ 0 -comp osition factors are L 0 (Λ) , L 0 (Λ − α ) , L 0 (Λ − β ), and L 0 (Λ − α − β ). Since ρ = (0 , 0 | 0), the Dirac inequalities for the g ¯ 0 -constituen ts L 0 (Λ − γ ), γ ∈ ∆ + ¯ 1 , are (4.31) 0 ≥ (Λ + ρ, α ) = λ 2 + x, 0 ≥ (Λ + ρ, β ) = λ 2 − x, whic h is equiv alent to (4.32) λ 2 ≤ x ≤ − λ 2 . W e now obtain a complete classiﬁcation, follo wing the classiﬁcation metho d in Section 1.1.2 . W e ﬁrst determine the interv al on which unitarit y holds automatically . Set x L max : = max { (Λ + ρ, α ) : α ∈ A } and x R min : = min { (Λ + ρ, β ) : β ∈ B } . Then x L max = λ 2 and x R min = − λ 2 . Hence, if x L max < x < x R min , then (Λ + ρ, γ ) < 0 for all γ ∈ ∆ + ¯ 1 , so the Dirac inequality is strict for ev ery o dd p ositive ro ot. Therefore Prop osition 21 implies that L (Λ) is unitarizable on this interv al. Next, one has x L min = x L max and x R max = x R min , and we set x L : = x L min = x L max and x R : = x R max = x R min . W e show that L (Λ) is not unitarizable for x < x L or x > x R . Indeed, if x < x L , then L (Λ) has the non-trivial g ¯ 0 -comp osition factor L 0 (Λ − α ), and if x > x R , then it has the non-trivial g ¯ 0 -comp osition factor L 0 (Λ − β ). In b oth cases, the Dirac inequality fails on this factor. T o sho w that these g ¯ 0 -comp osition factors do indeed o ccur, it suﬃces to prov e that under these conditions the corresp onding w eigh t spaces in M (Λ) are not con tained in the radical; equiv alently , that the Shap ov alo v form restricted to these weigh t spaces is non-degenerate. This follo ws once the Kac–Shap ov alov determinant (see Theorem 17 ) is shown to b e nonzero. Set η ∈ { ϵ 1 − δ 1 , − ϵ 2 + δ 1 } . The Kac–Shap o v alov determinan t on M (Λ) Λ − η is (Λ + ρ, η ), which is nonzero for x < x L = λ 2 or x > x R = − λ 2 . Hence the Dirac inequality fails on these g ¯ 0 - constituen ts, and L (Λ) is not unitarizable. Hence, one can assume x L ≤ x R , since for x L > x R the mo dule L (Λ) is not unitarizable. It remains to consider the b oundary p oints x = x L and x = x R max , equiv alen tly the at ypicalit y conditions (Λ + ρ, α ) = 0 and (Λ + ρ, β ) = 0. W e treat only the case x = x L < x R , since the cases x = x L = x R and x L < x = x R are analogous. Assume therefore that x = x L , so that (Λ + ρ, α ) = 0. By Lemma 15 , every g ¯ 0 -comp osition factor L 0 (Λ − γ ) admits a decomp osition of γ into pairwise distinct p ositiv e odd ro ots which do es not inv olv e α . Hence the only possible non-trivial g ¯ 0 -comp osition factors are L 0 (Λ) and L 0 (Λ − β ). Since x L < x R , the Dirac inequality is strict on the factor L 0 (Λ − β ). Prop osition 21 therefore implies that L (Λ) is unitarizable at x = x L if x L < x R . Hence, the set of all unitarizable highest weigh ts for su (1 , 1 | 1) is (4.33) n  λ 2 + x 2 , − λ 2 + x 2   x 2  : λ ∈ R ≤ 0 , 2 x ∈ [ λ, − λ ] o . Equiv alently , Λ is the highest weigh t of a unitarizable highest weigh t g -sup ermodule if and only if a) Λ satisﬁes the unitarity conditions of Lemma 6 . 36 STEFFEN SCHMIDT b) (Λ + ρ, ϵ 1 − δ 1 ) ≤ 0 and (Λ + ρ, − ϵ 2 + δ 1 ) ≤ 0. su (1 , 1 | 2): Set g : = su (1 , 1 | 2). The non-standard positive ro ot system is ∆ + = { ϵ 1 − ϵ 2 , δ 1 − δ 2 , ϵ 1 − δ 1 , ϵ 1 − δ 2 , − ϵ 2 + δ 1 , − ϵ 2 + δ 2 } . In particular, ρ =  − 1 2 , 1 2   1 2 , − 1 2  . A general family Λ of highest w eigh t of a unitarizable highest weigh t g ¯ 0 -mo dule has the general form (4.34) Λ = ( λ 1 , λ 2 | µ 1 , µ 2 ) =  λ 2 + x 2 , − λ 2 + x 2   b + x 2 , x 2  , with b ∈ Z ≥ 0 , λ ∈ R ≤ 0 and x ∈ R . W e assume Λ satisﬁes the additional unitarity conditions of Lemma 6 . The Dirac inequalities are (4.35) 0 ≥ (Λ + ρ, ϵ 1 − δ 1 ) = λ 2 + x + b ⇔ x ≤ − λ 2 − b, 0 ≥ (Λ + ρ, ϵ 1 − δ 2 ) = λ 2 + x − 1 ⇔ x ≤ − λ 2 + 1 , 0 ≥ (Λ + ρ, − ϵ 2 + δ 1 ) = λ 2 − x − b − 1 ⇔ x ≥ λ 2 − b − 1 , 0 ≥ (Λ + ρ, − ϵ 2 + δ 2 ) = λ 2 − x ⇔ x ≥ λ 2 . These constraints on x yield all necessary information to classify all unitarizable highest w eigh t g -sup ermo dules. W e use the metho d of Section 1.1.2 . By deﬁnition, x L = λ 2 and x R = − λ 2 − b . Hence, if x L < x R , then (Λ + ρ, γ ) < 0 for all γ ∈ ∆ + ¯ 1 whenev er λ 2 < x < − λ 2 − b . Therefore Prop osition 21 implies that L (Λ) is unitarizable on this in terv al. Note that i 0 = 1 and j 0 = 0. Hence x R min = x R max , and x L min = x L max , whic h we will denote b y x R and x L , resp ectiv ely . In particular, unitarity do es not hold for all x > x R and x < x L . Indeed, for x > x R , the mo dule L (Λ) has the g ¯ 0 -comp osition factor L 0 (Λ − ϵ 1 + δ 1 ), and the Dirac inequality fails on this factor. Its o ccurrence follows from the non-v anishing of the Kac– Shap o v alov form on M Λ − ϵ 1 + δ 1 . Namely , for γ ∈ ∆ + , the diﬀerence ( − ϵ 1 + δ 1 ) − γ is equal to 0 or a sum of p ositive ro ots only for γ = ϵ 1 − δ 1 . Thus the determinant v anishes only at (Λ + ρ, ϵ 1 − δ 1 ) = 0. If x > x R , it follows that this g ¯ 0 -comp osition factor occurs, and hence unitarit y fails. Analogously , one sho ws that L (Λ( x )) is not unitarizable for x < x L , since for suc h x the Dirac inequality fails on the non-v anishing comp osition factor L 0 (Λ + ϵ 2 − δ 2 ). It remains to pro v e unitarity in the degenerate case where one of the residual in terv als collapses to a single p oint, namely when (4.36) x = x L ≤ x R , or x = x R ≥ x L . It suﬃces to pro ve unitarity in the case x = x L = x R , since the other cases are analogous. Then (4.37) (Λ + ρ, − ϵ 2 + δ 2 ) = 0 , (Λ + ρ, ϵ 1 − δ 1 ) = 0 . By Lemma 15 , ev ery g ¯ 0 -comp osition factor L 0 (Λ − γ ), where γ = γ 1 + · · · + γ r , admits a decomp osition in whic h neither ϵ 1 − δ 1 nor − ϵ 2 + δ 2 app ears. Since ( 4.35 ) gives (Λ + ρ, γ i ) < 0 for all other p ositive o dd ro ots γ i , Prop osition 21 shows that L (Λ) is unitarizable. Altogether, the full set of unitarizable highest weigh t su (1 , 1 | 2)-sup ermo dules is (4.38) { Λ =  λ 2 + x 2 , − λ 2 + x 2   b + x 2 , x 2  : b ∈ Z + , λ ∈ R ≤ 0 , 2 x ∈ [ λ, − λ + 2 b ] } On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 37 Equiv alently , Λ is the highest weigh t of a unitarizable g -sup ermo dule if and only if a) Λ satisﬁes the unitarity conditions of Lemma 6 , and b) (Λ + ρ, ϵ 1 − δ 1 ) ≤ 0 and (Λ + ρ, − ϵ 2 + δ 2 ) ≤ 0. 4.2.3. Gener al Case. W e start with a highest weigh t Λ of a unitarizable highest w eight g ¯ 0 -mo dule of the form ( 4.29 ), parametrized by x , namely (4.39) Λ( x ) = Λ 0 + x 2 (1 , . . . , 1 | 1 , . . . , 1) , where Λ 0 = (0 , a 2 , . . . , a m − 1 , 0 | b 1 , . . . , b n − 1 , 0) + λ 2 (1 , . . . , 1 , − 1 , . . . , − 1 | 0 , . . . , 0), with λ satisfy- ing ( 4.28 ), a p +1 ≥ · · · ≥ a m − 1 ≥ a m = 0 = a 1 ≥ a 2 ≥ · · · ≥ a p , and b 1 ≥ · · · ≥ b n − 1 ≥ 0. Recall that i 0 is the largest index such that a i = 0, and that j 0 is the largest integer such that a m − j = 0; if a p +1 = 0, then j 0 = q . Assume moreov er that Λ satisﬁes the unitarity conditions of Lemma 6 . The classiﬁcation of unitarizable highest weigh t g -sup ermodules is then obtained by analyzing L (Λ( x )) as a function of x . In what follows, we suppress the parameter x and write simply Λ for Λ( x ). Our classiﬁcation metho d is indep endent of the explicit c hoice in ( 4.29 ); we retain it only for the sak e of an explicit realization. Our result will also b e form ulated entirely in terms of Dirac inequalities as seen in the examples ab ov e. W e apply the metho d of Section 1.1.2 . Recall the decomp osition ∆ + ¯ 1 = A ⊔ B . The analysis reduces to the v alues (Λ + ρ, γ ) for γ ∈ ∆ + ¯ 1 . F or ϵ i − δ k ∈ A , with 1 ≤ i ≤ p and 1 ≤ k ≤ n , one has (4.40) (Λ + ρ, ϵ i − δ k ) = λ i + µ k + p − i − k + 1 = x + λ 2 + a i + b k + p − i − k + 1 . F or − ϵ j + δ k ∈ B , with p + 1 ≤ j ≤ m and 1 ≤ k ≤ n , one has (4.41) (Λ + ρ, − ϵ j + δ k ) = − λ j − µ k − n − p + j + k − 1 = − x + λ 2 − a j − b k − n − p + j + k − 1 . By the form of Λ in ( 4.29 ) and the dominance relations, one obtains the following lemma. Lemma 29. One has: a) The numb ers (Λ + ρ, α ) , with α ∈ A , arr ange as fol lows: (Λ + ρ, ε 1 − δ 1 ) > · · · > (Λ + ρ, ε 1 − δ n ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, ε i − δ 1 ) > · · · > (Λ + ρ, ε i − δ n ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, ε p − δ 1 ) > · · · > (Λ + ρ, ε p − δ n ) 38 STEFFEN SCHMIDT b) The numb ers (Λ + ρ, β ) , with β ∈ B , arr ange as fol lows: (Λ + ρ, − ε m + δ n ) > · · · > (Λ + ρ, − ε m + δ 1 ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, − ε m − j + δ n ) > · · · > (Λ + ρ, − ε m − j + δ 1 ) ∨ ∨ . . . . . . ∨ ∨ (Λ + ρ, − ε p +1 + δ n ) > · · · > (Λ + ρ, − ε p +1 + δ 1 ) Note that the maxim um of (Λ + ρ, α ) for α ∈ A is attained at (Λ+ ρ, ϵ 1 − δ 1 ), while the maximum of (Λ + ρ, β ) for β ∈ B is attained at (Λ + ρ, − ϵ m + δ n ). In particular, if (Λ + ρ, ϵ 1 − δ 1 ) < 0 and (Λ + ρ, − ϵ m + δ n ) < 0, then Λ is typical. By the classiﬁcation metho d, x R min is determined b y (Λ + ρ, ϵ 1 − δ 1 ) = 0, that is, (4.42) (Λ + ρ, ϵ 1 − δ 1 ) = 0 ⇐ ⇒ x = − λ 2 − b 1 − p + 1 , hence x R min = − λ 2 − b 1 − p + 1. Likewise, x L max is determined by (Λ + ρ, − ϵ m + δ n ) = 0, that is, (4.43) (Λ + ρ, − ϵ m + δ n ) = 0 ⇐ ⇒ x = λ 2 + q − 1 , hence x L max = λ 2 + q − 1. This yields an interv al on whic h unitarit y is immediate. Lemma 30. If (Λ + ρ, ϵ 1 − δ 1 ) < 0 and (Λ + ρ, − ϵ m + δ n ) < 0 , then L (Λ) is unitarizable. Pr o of. If the conditions hold, one has (Λ + ρ, γ ) < 0 for all γ ∈ ∆ + ¯ 1 b y Lemma 29 . The statemen t follo ws b y Prop osition 21 . □ If (Λ + ρ, ϵ 1 − δ 1 ) < 0 and (Λ + ρ, − ϵ m + δ n ) > 0, then (4.44) x ∈ ( x L max , x R min ) =  λ 2 + q − 1 , − λ 2 − b 1 − p + 1  . If these inequalities are not satisﬁed, this interv al may b e empt y . F or this reason, we formulate the result in terms of Dirac inequalities rather than the in terv al itself. Recall that Λ satisﬁes the unitarit y conditions; see Lemma 6 . In particular, in standard co ordinates, (4.45) λ p +1 ≥ · · · ≥ λ m ≥ − µ n ≥ · · · ≥ − µ 1 ≥ λ 1 ≥ · · · ≥ λ p . T ogether with ( 4.40 ) and ( 4.41 ), this yields the deﬁnitions (4.46) x L min : = λ 2 + q − j 0 − 1 , x R max : = − λ 2 − b 1 − p + i 0 , equiv alently (Λ + ρ, − ϵ m − j 0 + δ n ) = 0 and (Λ + ρ, ϵ i 0 − δ 1 ) = 0. Moreov er, x L min ≤ x L max and x R min ≤ x R max . These are the w eak est b ounds. Lemma 31. If one of the fol lowing two e quivalent c onditions holds, then L (Λ) is not unitariz- able: On the F ull Set of Unitarizable Sup ermodules o ver sl ( m | n ) 39 a) (Λ + ρ, ϵ i 0 − δ 1 ) > 0 or (Λ + ρ, − ϵ m − j 0 + δ n ) > 0 . b) x > x R max or x < x L min . Pr o of. W e show that if (Λ + ρ, ϵ i 0 − δ 1 ) > 0 or (Λ + ρ, − ϵ m − j 0 + δ n ) > 0, then L (Λ) contains the g ¯ 0 -constituen t L 0 (Λ − ϵ i 0 + δ 1 ) or L 0 (Λ + ϵ m − j 0 − δ n ), resp ectiv ely . Since the Dirac inequality fails on these g ¯ 0 -constituen ts, L (Λ) is not unitarizable. W e sho w that L 0 (Λ + ϵ m − j 0 − δ n ) is a g ¯ 0 -constituen t if (Λ + ρ, − ϵ m − j 0 + δ n ) > 0. The remaining case is analogous. It suﬃces to prov e that the w eigh t space M b (Λ) Λ − η for η : = − ϵ m − j 0 + δ n do es not lie in the radical of the Shap o v alov form of M b (Λ), equiv alen tly , the Kac–Shap ov alov determinan t (see Theorem 17 ) do es not v anish. Then L 0 (Λ − η ) do es not b elong to the radical and is a non-trivial g ¯ 0 -constituen t of L (Λ). Consider the factor from ∆ + ¯ 0 , D 1 = Y γ ∈ ∆ + ¯ 0 ∞ Y r =1 ( h γ + ( ρ, γ ) − r ) P ( η − r γ ) . Here η − r γ is a sum of p ositiv e ro ots only if γ = ϵ i − ϵ m − j 0 and r = 1 for p + 1 ≤ i < m − j 0 . Then h γ (Λ) + ( ρ, γ ) − 1 = (Λ + ρ, ϵ i − ϵ m − j 0 ) − 1 = a i + ( m − j 0 ) − i − 1 > 0 , since a i ≥ 1 and m − j 0 − i ≥ 1. This shows D 1  = 0. F or the factor from ∆ + ¯ 1 , D 2 = Y γ ∈ ∆ + ¯ 1 ( h γ + ( ρ, γ )) P γ ( η − γ ) , one has η − γ a sum of p ositiv e ro ots if and only if γ = − ϵ i + δ n for p + 1 ≤ i ≤ m − j 0 . The p ossible zeros of the Kac–Shap ov alov determinant are (Λ + ρ, − ϵ i + δ n ) = 0 p + 1 ≤ i ≤ m − j 0 , By the unitarity conditions none of the v alues (Λ + ρ, − ϵ i + δ n ) can v anish for p + 1 ≤ i < m − j 0 , since j 0 is the largest integer with a m − j 0 = 0, implying (Λ , ϵ i − ϵ n )  = 0. F or i = m − j 0 w e hav e (Λ + ρ, − ϵ m − j 0 + δ n ) > 0 by assumption. Hence D 2  = 0, completing the pro of. □ Com bining Lemma 30 and Lemma 31 , it remains to analyze the ranges (4.47) x ∈ I L : = [ x L min , x L max ] = h λ 2 + q − j 0 − 1 , λ 2 + q − 1 i , x ∈ I R : = [ x R min , x R max ] = h − λ 2 − b 1 − p + 1 , − λ 2 − b 1 − p + i 0 i . By Lemma 31 , unitarity can o ccur only if x L min ≤ x R max . W e call the sets of in tegers (4.48) I L ∩ Z = { x L min , x L min + 1 , . . . , x L max } , I R ∩ Z = { x R min , x R min + 1 , . . . , x R max } , the step 1 p artition p oints of I L and I R . They are precisely the follo wing at ypicalit y p oin ts: (4.49) I L ∩ Z =  x ∈ I L   (Λ + ρ, ϵ i − δ 1 ) = 0 for some 1 ≤ i ≤ i 0  , I R ∩ Z =  x ∈ I R   (Λ + ρ, − ϵ m − j + δ n ) = 0 for some 1 ≤ j ≤ j 0  . W e will see that, whenever x ∈ I L ∪ I R , the mo dule L (Λ) is unitarizable if and only if x ∈ ( I L ∪ I R ) ∩ Z . If x ∈ I L ∪ I R but x / ∈ ( I L ∪ I R ) ∩ Z , we obtain the follo wing lemma. 40 STEFFEN SCHMIDT Lemma 32. If one of the fol lowing two e quivalent c onditions holds, then L (Λ) is not unitariz- able. a) (Λ + ρ, ϵ i +1 − δ 1 ) < 0 < (Λ + ρ, ϵ i − δ 1 ) or (Λ + ρ, − ϵ m − j +1 + δ n ) < 0 < (Λ + ρ, − ϵ m − j + δ n ) for 1 ≤ i < i 0 or 1 ≤ j < j 0 . b) The p oint x ∈ I L ∪ I R is not a step 1 p artition p oint of I L or I R , i.e. , x / ∈ I L ∩ Z and x / ∈ I R ∩ Z . Pr o of. Assume that L (Λ) is unitarizable. By Lemma 6 , if (Λ + ρ, ϵ i − δ 1 )  = 0 for some 1 ≤ i ≤ i 0 , or (Λ + ρ, − ϵ m − j + δ n )  = 0 for some 0 ≤ j ≤ j 0 , then (Λ + ρ, ϵ i − δ 1 ) < 0 ( i = 1 , . . . , i 0 ) , (Λ + ρ, − ϵ m − j + δ n ) < 0 ( j = 0 , . . . , j 0 ) . If x is not a step 1 partition p oin t, then (Λ + ρ, ϵ i − δ 1 )  = 0 for all i = 1 , . . . , i 0 and (Λ + ρ, − ϵ m − j + δ n )  = 0 for all j = 1 , . . . , j 0 ; moreo v er, b y the deﬁnition of I L and I R , one has (Λ + ρ, ϵ i − δ 1 ) > 0 for some i or (Λ + ρ, − ϵ m − j + δ n ) > 0 for some j. This con tradicts Lemma 6 . Hence L (Λ) is not unitarizable. □ Lemma 33. If one of the fol lowing c onditions holds, then L (Λ) is unitarizable: a) (Λ + ρ, − ϵ m + δ n ) < 0 and (Λ + ρ, ϵ i − δ 1 ) = 0 for 1 ≤ i ≤ i 0 . b) (Λ + ρ, − ϵ m − j + δ n ) = 0 and (Λ + ρ, ϵ i − δ 1 ) = 0 for 0 ≤ j ≤ j 0 and 1 ≤ i ≤ i 0 . c) (Λ + ρ, − ϵ m − j + δ n ) = 0 and (Λ + ρ, ϵ 1 − δ 1 ) < 0 for 0 ≤ j ≤ j 0 . Pr o of. W e pro ve a); the argumen ts for b) and c) are analogous. W e sho w that for any g ¯ 0 -comp osition factor L 0 (Λ − γ ) of L (Λ), the elemen t γ admits a decomp osition into pairwise distinct p ositive o dd ro ots γ s suc h that (Λ + ρ, γ s ) < 0 for all s . The statement then follows from Prop osition 21 . Indeed, in each of the cases a)–c) the pro of is analogous to the pro of of Prop osition 21 , using Lemma 15 (b). As sho wn in the pro of of Prop osition 21 , it suﬃces to prov e that (Λ + ρ, α ) < 0 whenever α ∈ A o ccurs as a direct summand of some γ , and separately that (Λ + ρ, β ) < 0 whenev er β ∈ B o ccurs as a direct summand of some γ . By assumption (Λ + ρ, − ϵ m + δ n ) < 0, hence (Λ + ρ, β ) < 0 for all β ∈ B by Lemma 29 ; it remains to treat the case α ∈ A . Consider the diagram of Lemma 29 . By monotonicit y , non-negative v alues can o ccur only for ro ots of the form ϵ i ′ − δ k with i ′ ≥ i and k = 1 , . . . , n . F or suc h ro ots one has (4.50) (Λ + ρ, ϵ i ′ − δ k ) = − ( b 1 − b k ) − ( k − 1) + ( i ′ − i ) . Hence the diagram takes the form i − 1 > − ( b 1 − b 2 ) + i − 2 > · · · > − ( b 1 − b n ) − n + i − 1 ∨ ∨ ∨ . . . . . . . . . ∨ ∨ ∨ 1 > − ( b 1 − b 2 ) > · · · > − ( b 1 − b n ) − n + 2 ∨ ∨ ∨ 0 > − ( b 1 − b 2 ) − 1 > · · · > − ( b 1 − b n ) − n + 1 ∨ ∨ ∨ (Λ + ρ, ϵ i − δ 1 ) > (Λ + ρ, ϵ i − δ 2 ) > · · · > (Λ + ρ, ϵ i − δ n ) ∨ ∨ ∨ . . . . . . . . . ∨ ∨ ∨ (Λ + ρ, ϵ p − δ 1 ) > (Λ + ρ, ϵ p − δ 2 ) > · · · > (Λ + ρ, ϵ p − δ n ) Since (Λ + ρ, ϵ i − δ 1 ) = 0, Lemma 15 implies that ev ery g ¯ 0 -comp osition factor of L (Λ) admits a decomp osition that do es not in volv e an y of the ro ots ϵ i − δ 1 , . . . , ϵ 1 − δ 1 . Now assume that (Λ + ρ, ϵ i ′ − δ k ) > 0 for some i ′ > i . Since (Λ + ρ, ϵ i − δ k ) < 0, monotonicity in the i -direction (step 1) implies that there exists i ′′ with i ≤ i ′′ ≤ i ′ suc h that (Λ + ρ, ϵ i ′′ − δ k ) = 0. But then Lemma 15 shows that none of the ro ots ϵ i ′′ − δ 1 , ϵ i ′′ − 1 − δ 1 , . . . , ϵ 1 − δ 1 can app ear. Applying this argumen t for each i ′ > i , we conclude that every g ¯ 0 -comp osition factor admits a decomp osition γ = γ 1 + · · · + γ r in to pairwise distinct p ositive o dd ro ots with (Λ + ρ, γ s ) < 0 for all s . This pro v es the claim. □ Com bining the preceding lemmas, we obtain the complete complete classiﬁcation of unitariz- able g -sup ermo dules, formulated in Theorem 34 b elo w. The statemen t is basis-indep endent; the explicit parameters are given by the discussion ab o ve. Theorem 34. A weight Λ is the highest weight of a unitarizable highest weight g -sup ermo dule if and only if a) Λ satisﬁes the unitarity c onditions of L emma 6 , and b) one of the fol lowing holds: (i) (Λ + ρ, − ϵ m + δ n ) < 0 and (Λ + ρ, ϵ i − δ 1 ) = 0 for 1 ≤ i ≤ i 0 ; (ii) (Λ + ρ, − ϵ m − j + δ n ) = 0 and (Λ + ρ, ϵ i − δ 1 ) = 0 for 0 ≤ j ≤ j 0 and 1 ≤ i ≤ i 0 ; (iii) (Λ + ρ, − ϵ m − j + δ n ) = 0 and (Λ + ρ, ϵ 1 − δ 1 ) < 0 for 0 ≤ j ≤ j 0 . (iv) (Λ + ρ, − ϵ m + δ n ) < 0 and (Λ + ρ, ϵ 1 − δ 1 ) < 0 . R emark 35 . All highest w eigh t supermo dules are understoo d to ha v e ev en highest w eight v ector. The cases with o dd highest weigh t vector are obtained by applying the parity reversion functor Π. This giv es the full set of all unitarizable highest weigh t g -sup ermo dules. 4.3. The Case psl ( n | n ) . If m = n , the special linear Lie sup eralgebra sl ( n | n ) is not simple, as it con tains a non-trivial ideal generated b y the identit y matrix E 2 n . The quotient Lie superalgebra 42 STEFFEN SCHMIDT psl ( n | n ) : = sl ( n | n ) / C E 2 n is simple and is called the pr oje ctive sp e cial line ar Lie sup er algebr a . Our classiﬁcation result for p, q  = 0 pro vides a complete classiﬁcation of unitarizable psl ( n | n )- sup ermo dules, after imp osing the additional condition that Λ = ( λ 1 , . . . , λ n | µ 1 , . . . , µ n ) satisﬁes P n i =1 λ i − P n j =1 µ j = 0. References [Bar61] V. Bargmann. 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On the full set of unitarizable supermodules over $\mathfrak{sl}(m\vert n)$

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