A classification of irreducible unitary modules over $\mathfrak{u}(p,q|n)$

A CLASSIFICA TION OF IRREDUCIBLE UNIT AR Y MODULES O VER u p p, q | n q MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG Abstract. W e classify all irreducible highest-weigh t unitary modules ov er the non- compact real form u p p, q | n q of the general linear Lie superalgebra gl p ` q | n . The classi- fication is giv en b y explicit necessary and sufficient conditions on the highest w eights. Our approach combines the How e duality for gl p ` q | n with a quadratic in v arian t of the maximal compact subalgebra. As consequences, w e classify all irreducible lo w est-w eight unitary mo dules ov er u p p, q | n q via dualit y , and all irreducible unitary mo dules ov er u p n | q , p q via an isomorphism of Lie superalgebras. Contents 1. In tro duction 1 2. Star-op erations and unitarity 3 2.1. Basics and duals 3 2.2. Compatibilit y with the sup er Killing form 5 3. The general linear Lie sup eralgebra 5 3.1. Algebraic structure 6 3.2. Real form u p p, q | n q of gl p ` q | n 7 3.3. Unitary highest-w eigh t mo dules 8 3.4. Finite-dimensional unitary mo dules 10 4. Main classification result 11 5. Necessit y 12 6. A unitarit y criterion 14 7. Ho w e duality 17 7.1. Oscillator sup eralgebra 17 7.2. Decomp osition of sup ersymmetric algebra 18 8. Unitary mo dules with integral highest weigh ts 19 9. Sufficiency 20 10. F urther classifications of unitary mo dules 24 10.1. Dual unitary mo dules 24 10.2. Unitary mo dules ov er gl n | q ` p 24 10.3. Unitary mo dules ov er gl p ` q | r ` s 26 References 27 1. Intr oduction The theory of Lie sup eralgebras [Kac77a] and their represen tations pla ys a fundamen tal role in the understanding and exploitation of sup ersymmetry in physical systems. The notion of sup ersymmetry first arose in elemen tary particle physics and quan tum field theory but has since found applications in a v ariety of areas, including n uclear physics, in tegrable mo dels, and string theory [Jun96, DF04, Din07]. 1 2 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG The representation theory of the simple basic classical Lie sup eralgebras was first in- v estigated b y Kac [Kac77b], who introduced the now familiar dic hotom y b et w een t ypical and at ypical finite-dimensional irreducible represen tations. In this pap er, we are con- cerned with the unitary represen tations of the basic classical Lie sup eralgebra u p p, q | n q , and as the corresp onding mo dules admit contra v ariant p ositiv e-definite Hermitian forms, they are amenable to ph ysical applications where unitarit y is a basic requiremen t. Unitary represen tations of Lie sup eralgebras are natural generalisations of those ap- p earing in the theory of ordinary Lie algebras, and arise naturally from so-called star- op erations. Such op erations w ere first in tro duced in the sup er setting by Sc heunert, Nahm and Rittenberg [SNR77], who show ed that a basic classical Lie sup eralgebra admits at most t wo t yp es of finite-dimensional unitary irreducible representations. These were sub- sequen tly classified b y Gould and Zhang [GZ90b]. Crucially , such unitary represen tations arise from star-op erations corresp onding to a compact real form of the underlying even subalgebra, and only exist for the t yp e-I basic classical Lie sup eralgebras osp 2 | 2 n and gl m | n . Apart from any physical applications, irreducible representations of Lie sup eralgebras are mathematically interesting in their o wn right. In particular, the category of finite- dimensional unitary mo dules (of a giv en type) is closed under tensor pro ducts, so the tensor pro duct of tw o such mo dules is completely reducible. Moreo v er, there are t w o distinct t yp es of unitary representations related by dualit y [GZ90b]. They include the so-called co v ariant and con tra v ariant tensor represen tations, whic h explains the applica- bilit y of Y oung diagram metho ds to this class of mo dules. In fact, there exists a m uc h larger class of non-tensorial typical unitary mo dules. Indeed, corresp onding to every ir- reducible tensor mo dule is a one-parameter family of typical unitary mo dules which are non-tensorial. It is particularly in teresting that such represen tations underlie in tegrable electron mo dels (and corresp onding link p olynomials) where the parameter lab elling the mo dules has physical significance [GHLZ96]. Unlik e the finite-dimensional case, infinite-dimensional unitary highest-weigh t repre- sen tations exist for all basic classical Lie sup eralgebras. Here, we pro vide a classification of such u p p, q | n q -representations that arise from a star-op eration for whic h u p p, q q ‘ u p n q is the real form of the ev en complex subalgebra gl p ` q ‘ gl n . As noted ab o v e, the compact case ( q “ 0) w as treated in [GZ90b]. In the non-compact case ( p, q ‰ 0), p ositive-energy unitary irreducible representations of su p p, q | n q hav e b een studied for small n and for p “ q “ 2, motiv ated b y superconformal field theory; see [GZ90a, GV19] and references therein. F urutsu and Nishiyama [FN91] subsequen tly classified the highest-weigh t unitary irreducible su p p, q | n q -mo dules with in teger highest w eigh ts. By design, this classification do es not cov er the large class of unitary mo d- ules with non-in teger highest weigh ts. More general and systematic classification results for highest-w eigh t unitary irreducible su p p, q | n q -mo dules were later prop osed b y Jak ob- sen [Jak94] and, more recently , by G ¨ una ydin and V olin [GV19]. As noted in [GV19], Jak obsen’s classification do es not include the p ositiv e-energy unitary representations of su p 2 , 2 | n q . In the present pap er, w e extend the approac h of [GZ90b] and classify b oth highest- w eigh t and low est-w eigh t unitary u p p, q | n q -mo dules. The latter are related to the former b y duality , a fact that, to the b est of our knowledge, has not b een explicitly noted in the literature. W e giv e detailed pro ofs of b oth necessity and sufficiency of the classification, based on an induced mo dule construction and an application of Ho we dualit y to the super setting [CW01, CLZ04]. A k ey feature of our approach is that the classification is form ulated directly with resp ect to the standar d Borel subalgebra, enhancing the applicability in both physics and CLASSIFICA TION OF UNIT AR Y MODULES 3 mathematics. This also distinguishes our work from that of G ¨ una ydin and V olin [GV19], who obtained unitarity conditions using Y oung diagrams and oscillator techniques for a non-standar d Borel subalgebra. Their classification w as recen tly recov ered by Sc hmidt [Sc h26] using an algebraic Dirac op erator and corresp onding Dirac inequalities, but still form ulated in the non-standard setting. 1 In addition, since u p p, q | n q has a nontrivial cen tre, its unitary mo dules in v olv e a twist parameter not presen t at the lev el of su p p, q | n q . The pap er is set up as follo ws. Section 2 sp ecifies our notation and conv en tions and presen ts some general results in the area. An in-depth discussion of star-op erations on gl m | n and the corresponding unitary mo dules is given in Section 3. Theorem 4.2 in Section 4 presen ts our main result, with Section 6-Section 9 devoted to the pro of of this classification. Necessary conditions for unitarity are thus derived in Section 5, while a new criterion for unitarity is deriv ed in Section 6. Section 7 outlines the relev ant How e dualit y in our setting. F ollowing a discussion of unitary mo dules with in tegral highest w eigh ts in Section 8, the proof of sufficiency of the conditions deriv ed in Section 5 is giv en in Section 9. This then completes the pro of of the classification result in Theorem 4.2. In Section 10, using the classification in Theorem 4.2, w e classify the unitary low est-w eigh t u p p, q | n q -mo dules and unitary u p n | q , p q -mo dules. Con v en tion. W e denote by C (resp ectiv ely R ) the field of complex (resp ectiv ely real) n um b ers, b y Z ` (resp ectiv ely Z ´ ) the set of non-negative (resp ectiv ely non-p ositiv e) in- tegers, and b y Z 2 the ring of in tegers mo dulo 2. W e write b for the tensor pro duct o v er C , denote the imaginary unit by i , and let t x u denote the in teger part of x P R . F or an y Lie sup eralgebra l , we denote b y U p l q the corresp onding univ ersal env eloping alge- bra. When characterising mo dules and representations, w e use the terms “simple” (and “semisimple”) and “irreducible” (and “completely reducible”) in terc hangeably . Through- out, w e w ork o v er C , unless otherwise stated. All homomorphisms b etw een sup er v ector spaces are assumed to b e even. Giv en a sup er v ector space V “ V ¯ 0 ‘ V ¯ 1 , the parity of homogeneous v P V is given b y r v s “ ¯ 0 (resp ectiv ely ¯ 1) if v P V ¯ 0 (resp ectiv ely V ¯ 1 ). If a sup er v ector space V is finite-dimensional, we let V ˚ : “ Hom p V , C q denote the usual dual. If V is infinite-dimensional, we assume that V “ À r P Z V r is Z -graded, with each V r finite-dimensional, and b y abuse of notation, w e write V ˚ for the graded dual space defined b y V ˚ : “ À r P Z V ˚ r . 2. St ar-opera tions and unit arity 2.1. Basics and duals. Largely following [SNR77, GZ90b, CLZ04], here we recall some basic facts ab out star-sup eralgebras and their unitary mo dules, and ho w these carry o v er to Lie sup eralgebras. W e also consider dual star-op erations and the corresp onding unitarit y mo dules. First, a star-sup er algebr a ov er C is an asso ciativ e sup eralgebra A “ A ¯ 0 ‘ A ¯ 1 equipp ed with an ev en an ti-linear an ti-inv olution ϕ : A Ñ A ; i.e., ϕ p ca q “ ¯ ca, ϕ p ab q “ ϕ p b q ϕ p a q , c P C , a, b P A, where ¯ c denotes the complex conjugate of c . A star-sup eralgebra homomorphism f : p A, ϕ q Ñ p A 1 , ϕ 1 q is a sup eralgebra homomorphism satisfying f ˝ ϕ “ ϕ 1 ˝ f . Second, let p A, ϕ q b e a star-sup eralgebra and V a Z 2 -graded A -mo dule. A Hermitian form x´ , ´y on V is said to b e p ositive-definite if x v , v y ą 0 for all nonzero v P V , and c ontr avariant if x av , w y “ x v , ϕ p a q w y for all a P A and all v , w P V . An A -mo dule 1 The authors only b ecame a w are of Schmidt’s w ork just prior to submission of the presen t pap er; all results rep orted here w ere obtained before the preprin t [Sch26] appeared. 4 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG equipp ed with a p ositiv e-definite contra v arian t Hermitian form is called a unitary A - mo dule. Third, a star-op er ation on a complex Lie sup eralgebra g is an ev en an ti-linear map ‹ suc h that r X , Y s ‹ “ r Y ‹ , X ‹ s , p X ‹ q ‹ “ X , X, Y P g . As this star-op eration extends to an ev en an ti-linear an ti-in volution on U p g q , it equips U p g q with the structure of a star-sup eralgebra. The star-induced notion of unitarity for star-sup eralgebras can thus b e applied to U p g q -mo dules, and hence to g -mo dules. F ourth, a real Lie sup eralgebra u is said to b e a r e al form of a complex Lie superalgebra g if g – u b R C . A real form u is called c omp act if u ¯ 0 is a compact Lie algebra; otherwise, it is called non-c omp act . A real form can b e induced from a star-op eration on g , setting u ¯ 0 “ span R t X P g ¯ 0 | X ‹ “ ´ X u , u ¯ 1 “ span R t X P g ¯ 1 | X ‹ “ i X u . W e now define the dual star-op er ation ✩ by X ✩ : “ p´ 1 q r X s X ‹ , X P g ¯ 0 Y g ¯ 1 , extended linearly to all of g . The unitary g -mo dules with resp ect to a star-op eration and its dual are related b y dualit y as follo ws. Prop osition 2.1. If V is a unitary g -mo dule with r esp e ct to a star-op er ation, then the dual mo dule V ˚ is unitary with r esp e ct to the dual star-op er ation. Pr o of. Let V b e a unitary g -module and recall our conv en tion that, if V is infinite- dimensional, then V is assumed Z -graded with finite-dimensional graded comp onen ts, while V ˚ denotes the graded dual. Let t v i u i P I b e a homogeneous orthonormal V -basis with resp ect to the p ositiv e-definite Hermitian form x´ , ´y , and let t v ˚ i u i P I denote the dual basis of V ˚ , so that v ˚ i p v j q “ δ ij for all i, j P I . The Hermitian form on V induces a p ositiv e-definite Hermitian form on V ˚ suc h that t v ˚ i u i P I is an orthonormal basis. In particular, for an y homogeneous X P g and all i, j P I , w e ha v e v ˚ i p X v j q “ x v i , X v j y . Moreo v er, the dual g -mo dule structure on V ˚ is giv en b y p X v ˚ i qp v j q “ ´p´ 1 q r X sr v i s v ˚ i p X v j q . Since ev ery f P V ˚ can b e written as f “ ř i P I f p v i q v ˚ i , it follo ws that x v ˚ i , X v ˚ j y “ ÿ k P I x v ˚ i , v ˚ k y p X v ˚ j qp v k q “ ´p´ 1 q r X sr v j s ÿ k P I x v ˚ i , v ˚ k y v ˚ j p X v k q “ ´p´ 1 q r X sr v j s v ˚ j p X v i q “ ´p´ 1 q r X sr v j s x v j , X v i y . Using this together with the unitarity of V and the conjugation symmetry of the Her- mitian form, w e obtain x v ˚ i , X ✩ v ˚ j y “ x v ˚ i , p´ 1 q r X s X ‹ v ˚ j y “ ´p´ 1 q r X s p´ 1 q r X sr v j s x v j , X ‹ v i y “ ´p´ 1 q r X s p´ 1 q r X sr v j s x X v j , v i y “ ´p´ 1 q r X s p´ 1 q r X sr v j s x v i , X v j y “ ´p´ 1 q r X sr v i s x v i , X v j y “ x v ˚ j , X v ˚ i y “ x X v ˚ i , v ˚ j y . Note that this is indep endent of the choice of homogeneous orthonormal basis of V . It follo ws that the unitary mo dules with resp ect to a star-op eration are related to those with resp ect to the dual star-op eration via dualit y . □ CLASSIFICA TION OF UNIT AR Y MODULES 5 2.2. Compatibilit y with the sup er Killing form. Here, w e note a compatibilit y b et w een star-op erations and the sup er Killing form on g . Although this will not b e used in the remainder of the pap er, it may b e of indep enden t in terest. T o set the stage, recall that the sup er Kil ling form on g is defined by p X , Y q : “ Str p ad X ˝ ad Y q , X , Y P g , where Str denotes the sup ertrace and ad the adjoint represen tation of g . W e now let t x a u denote a homogeneous basis for g and write r x a , x b s “ ÿ c Γ c ab x c . By sup er skew-symmetry , the structur e c onstants Γ c ab satisfy Γ c ab “ ´p´ 1 q r a sr b s Γ c ba , where r a s : “ r x a s . Note that r a s ` r b s “ r c s for an y index c in Γ c ab . It follo ws that r x a , r x b , x c ss “ ÿ d Γ d bc r x a , x d s “ ÿ d,s Γ s ad Γ d bc x s , hence p x a , x b q “ Str p ad x a ˝ ad x b q “ ÿ c,d p´ 1 q r c s Γ c ad Γ d bc . Prop osition 2.2. L et ‹ b e a star-op er ation on g . Then, for al l X , Y P g ¯ 0 Y g ¯ 1 , p X ‹ , Y ‹ q “ p´ 1 q r X s p X , Y q . Pr o of. Using the basis and structure-constant notation from ab o ve, we hav e r x a , x b s ‹ “ r x ‹ b , x ‹ a s , r x a , x b s ‹ “ ÿ c p Γ c ab x c q ‹ , so r x ‹ a , x ‹ b s “ ÿ c p Γ c ba x c q ‹ “ ÿ c Γ c ba x ‹ c , hence r x ‹ a , r x ‹ b , x ‹ c ss “ ÿ d Γ d cb r x ‹ a , x ‹ d s “ ÿ d,s Γ d cb Γ s da x ‹ s . It follo ws that p x ‹ a , x ‹ b q “ Str p ad x ‹ a ˝ ad x ‹ b q “ ÿ c,d p´ 1 q r c s Γ d cb Γ c da “ p´ 1 q r a s ÿ c,d p´ 1 q r c s Γ c ad Γ d bc “ p´ 1 q r a s p x a , x b q . Extending b y linearit y , we obtain the desired result. □ 3. The general linear Lie superalgebra Here, we discuss the general linear Lie sup eralgebra gl p ` q | n , its real form u p p, q | n q , and its unitary mo dules. 6 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG 3.1. Algebraic structure. Let t e 1 , . . . , e m ` n u denote the standard (ordered) basis for the complex sup erspace C m | n of dimension m | n . That is, t e 1 , . . . , e m u is an ordered basis for the ev en subspace p C m | n q ¯ 0 “ C m , while t e m ` 1 , . . . , e m ` n u is an ordered basis for the o dd subspace p C m | n q ¯ 1 “ C n . Relativ e to the standard basis, the generators of the gener al line ar Lie sup er algebr a gl m | n : “ gl p C m | n q are p m ` n q ˆ p m ` n q matrices: gl m | n “ ! ˆ A B C D ˙ ˇ ˇ ˇ A P M m,m , B P M m,n , C P M n,m , D P M n,n ) , where M r,s , r , s P Z ` , denotes the complex space of p r ˆ s q -matrices. The ev en subalgebra p gl m | n q ¯ 0 “ gl m ‘ gl n consists of the matrices for which B “ 0 and C “ 0, while the o dd subspace p gl m | n q ¯ 1 consists of the matrices for which A “ 0 and D “ 0. The Lie brack et is defined for homogeneous X , Y P gl m | n b y r X, Y s “ X Y ´ p´ 1 q r X sr Y s Y X , extended linearly to all of gl m | n . F or each pair a, b P t 1 , . . . , m ` n u , let E ab denote the matrix unit of gl m | n suc h that, for all c P t 1 , . . . , m ` n u , E ab e c “ δ b,c e a , where δ b,c is the Kronec ker sym b ol. The set t E ab | a, b “ 1 , . . . , m ` n u is then a basis for gl m | n , and r E ab , E cd s “ δ b,c E ad ´ p´ 1 q pr a s`r b sqpr c s`r d sq δ d,a E cb , where r a s “ r e a s . A basis for the Cartan sub algebr a h m | n of gl m | n is t E aa | a “ 1 , . . . , m ` n u , while t E ab | 1 ď a ď b ď m ` n u is a basis for the corresp onding standar d Bor el sub algebr a b m | n . Let t ϵ a | a “ 1 , . . . , m ` n u b e a basis for the dual space h ˚ m | n “ Hom C p h m | n , C q such that ϵ a p E bb q “ δ a,b for all a, b . The set of p ositiv e ro ots relative to b m | n is Φ ` “ t ϵ a ´ ϵ b | 1 ď a ă b ď m ` n u . W riting δ µ “ ϵ m ` µ for µ “ 1 , . . . , n , the sets of ev en resp ectively o dd p ositive ro ots are giv en b y Φ ` ¯ 0 “ t ϵ i ´ ϵ j | 1 ď i ă j ď m u Y t δ µ ´ δ ν | 1 ď µ ă ν ď n u , Φ ` ¯ 1 “ t ϵ i ´ δ µ | 1 ď i ď m, 1 ď µ ď n u , while the simple ro ot system is given by ∆ “ t ϵ i ´ ϵ i ` 1 , ϵ m ´ δ 1 , δ µ ´ δ µ ` 1 | 1 ď i ď m ´ 1 , 1 ď µ ď n ´ 1 u . F or eac h α P Φ ` , the corresp onding ro ot space is the 1-dimensional vector space spanned b y E α “ E ab , where α “ ϵ a ´ ϵ b . Let p´ , ´q : h ˚ m | n ˆ h ˚ m | n Ñ C denote the symmetric bilinear form defined by p ϵ i , ϵ j q “ δ i,j , p ϵ i , δ µ q “ 0 , p δ µ , δ ν q “ ´ δ µ,ν , for i, j P t 1 , . . . , m u and µ, ν P t 1 , . . . , n u . The graded half-sum of p ositiv e ro ots, ρ : “ 1 2 ´ ÿ α P Φ ` ¯ 0 α ´ ÿ α P Φ ` ¯ 1 α ¯ “ 1 2 m ÿ i “ 1 p m ´ n ´ 2 i ` 1 q ϵ i ` 1 2 n ÿ µ “ 1 p m ` n ´ 2 µ ` 1 q δ µ , satisfies p ρ, ϵ i ´ ϵ j q “ j ´ i, p ρ, ϵ i ´ δ µ q “ m ` 1 ´ i ´ µ, p ρ, δ µ ´ δ ν q “ µ ´ ν . (3.1) CLASSIFICA TION OF UNIT AR Y MODULES 7 3.2. Real form u p p, q | n q of gl p ` q | n . Let p, q b e p ositiv e in tegers suc h that m “ p ` q . On gl p ` q | n , w e define the star-op eration p E ab q ‹ : “ # E ba , a, b ď p or a, b ą p, ´ E ba , otherwise , (3.2) and w e denote b y u p p, q | n q the real form of gl p ` q | n induced b y this star-op eration. By construction, it is a real Lie sup eralgebra, and as its ev en subalgebra is u p p, q q ‘ u p n q , it is non-compact. F ollowing the discussion in Section 2, a gl p ` q | n -mo dule is unitary with resp ect to (3.2) if it is unitary as a mo dule ov er the corresp onding star-sup eralgebra U p gl p ` q | n q ; that is, if it carries a p ositive-definite Hermitian form x´ , ´y satisfying x E ab v , w y “ x v , p E ab q ‹ w y , v , w P V , a, b P t 1 , . . . , p ` q ` n u . (3.3) The dual star-op eration ✩ , defined by p E ab q ✩ : “ p´ 1 q r a s`r b s p E ab q ‹ , and the corresp ond- ing unitary mo dules will b e treated in Section 10. Key to our work, u p p, q | n q is exactly the real form of gl p ` q | n asso ciated with the star- op eration app earing in (3.3). It follo ws that (i) ev ery unitary represen tation of u p p, q | n q extends by complexification to a gl p ` q | n -mo dule satisfying (3.3), and conv ersely , that (ii) restricting the action of a unitary gl p ` q | n -mo dule to its real part yields a unitary u p p, q | n q - mo dule. Utilizing this, we shall study unitary u p p, q | n q -mo dules via the corresp onding gl p ` q | n -mo dules. Relativ e to the real form u p p, q | n q , we define the non-c omp act p ositive r o ot system of gl p ` q | n b y Φ ` nc : “ Φ ` nc , ¯ 0 Y Φ ` nc , ¯ 1 , where Φ ` nc , ¯ 0 : “ t ϵ i ´ ϵ j | 1 ď i ď p ă j ď p ` q u , Φ ` nc , ¯ 1 : “ t ϵ i ´ δ µ | 1 ď i ď p, 1 ď µ ď n u . Asso ciated with this, we hav e the triangular decomp osition gl p ` q | n “ k ´ ‘ k ‘ k ` , where k ´ : “ span t E α | α P ´ Φ ` nc u , k – gl p ‘ gl q | n , k ` : “ span t E α | α P Φ ` nc u . W e also define the corresponding c omp act p ositive r o ot system by Φ ` c : “ Φ ` c , ¯ 0 Y Φ ` c , ¯ 1 , where Φ ` c , ¯ 0 : “ Φ ` ¯ 0 ´ Φ ` nc , ¯ 0 , Φ ` c , ¯ 1 : “ Φ ` ¯ 1 ´ Φ ` nc , ¯ 1 , and note that Φ ` c is the p ositiv e root system of the gl p ` q | n -subalgebra k . A unitary gl p ` q | n -mo dule is called admissible if its restriction to k decomp oses as a direct sum of finite-dimensional simple k -mo dules, each o ccurring with finite multiplicit y . Ev ery admissible unitary mo dule admits a weigh t-space decomp osition with resp ect to the Cartan subalgebra h of gl p ` q | n . Throughout, we w ork with the category of admissible unitary gl p ` q | n -mo dules, which is the sup er analogue of the framework used by Enright, Ho w e and W allach in their classification of unitary highest-weigh t mo dules [EHW83]. The following gives necessary w eight conditions for admissible unitary gl p ` q | n -mo dules (cf. [FN91, Lemma 2.1]). Lemma 3.1. L et V b e an admissible unitary gl p ` q | n -mo dule, and let λ “ ř p ` q i “ 1 λ i ϵ i ` ř n µ “ 1 ω µ δ µ b e any weight of V . Then, λ is r e al, and λ i ď ´ ω µ ď λ j , for al l 1 ď i ď p , p ` 1 ď j ď p ` q , and 1 ď µ ď n . 8 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG Pr o of. Let v b e a nonzero w eigh t vector with w eigh t λ . Then H v “ λ p H q v for any H P h . Since H ‹ “ H and the Hermitian form is an ti-linear in the first argumen t, w e hav e λ p H qx v , v y “ x H v , v y “ x v , H v y “ λ p H qx v , v y , Since x v , v y ą 0, w e ha ve λ p H q “ λ p H q for an y H P h , so λ is a real weigh t. F or 1 ď i ď p and 1 ď µ ď n , w e ha v e r E i,m ` µ , E m ` µ,i s “ E ii ` E m ` µ,m ` µ and p E i,m ` µ q ‹ “ ´ E m ` µ,i . It follo ws that p λ i ` ω µ qx v , v y “ xr E i,m ` µ , E m ` µ,i s v , v y “ x E i,m ` µ E m ` µ,i v , v y ` x E m ` µ,i E i,m ` µ v , v y “ ´x E m ` µ,i v , E m ` µ,i v y ´ x E i,m ` µ v , E i,m ` µ v y ď 0 . Therefore, w e obtain λ i ď ´ ω µ . Similarly , one can prov e ´ ω µ ď λ j for p ` 1 ď j ď m and 1 ď µ ď n using r E j,m ` µ , E m ` µ,j s “ E j j ` E m ` µ,m ` µ and p E j,m ` µ q ‹ “ E m ` µ,j . □ In contrast, when n “ 0 (the non-compact real form u p p, q q ), Lemma 3.1 do es not apply: for any weigh t λ of V , the commutation relations for the non-compact ro ot vectors do not yield the inequalities λ i ď λ j for 1 ď i ď p and p ` 1 ď j ď p ` q . An imp ortan t consequence of Lemma 3.1 is the follo wing (cf. [FN91, Prop osition 2.2]). Prop osition 3.2. L et V b e an admissible unitary simple gl p ` q | n -mo dule. Then, V is a highest-weight mo dule with highest weight Λ “ ř p ` q i “ 1 λ i ϵ i ` ř n µ “ 1 ω µ δ µ satisfying λ p ` 1 ě ¨ ¨ ¨ ě λ m ě ´ ω n ě ¨ ¨ ¨ ě ´ ω 1 ě λ 1 ě ¨ ¨ ¨ ě λ p . Pr o of. Since V is simple and admits a w eigh t-space decomp osition, we choose a nonzero w eigh t vector v P V with weigh t λ such that V “ U p gl m | n q v . By Lemma 3.1, we hav e λ i ´ λ j ď 0 for all 1 ď i ď p ă j ď p ` q . As the action of each non-compact even ro ot v ector E ij on v increase the weigh t λ i ´ λ j b y 2, there must b e a non-negative integer s ij suc h that E s ij ij v ‰ 0 and E s ij ` 1 ij v “ 0 for 1 ď i ď p ă j ď m . Therefore, there exists a Φ ` nc , ¯ 0 -highest-w eigh t v ector w such that E α w “ 0 for all α P Φ ` nc , ¯ 0 . Since E 2 β “ 0 and r E α , E β s “ 0 for any α P Φ ` nc , ¯ 0 and β P Φ ` nc , ¯ 1 , one can similarly apply the ro ot v ectors E β to w , thus obtaining a Φ ` nc -highest-w eigh t vector w 1 suc h that E α w 1 “ 0 for all α P Φ ` nc . Since V is admissible unitary , the Φ ` nc -highest-w eigh t vector w 1 generates a finite- dimensional k -mo dule. Hence there exists X P U p k q such that u “ X w 1 is a Φ ` c -highest- w eigh t v ector. The vector u is also a Φ ` nc -highest-w eigh t v ector, as r k ` , k s Ď k ` . Therefore, u is a Φ ` -highest-w eigh t vector, and V is a highest-w eigh t mo dule. □ 3.3. Unitary highest-w eigh t mo dules. Let D ` p,q | n denote the set of Φ ` c -dominant in- te gr al weights ; that is, weigh ts of the form p λ 1 , . . . , λ p ` q , ω 1 , . . . , ω n q “ p ` q ÿ i “ 1 λ i ϵ i ` n ÿ µ “ 1 ω µ δ µ , (3.4) where λ i , ω µ P R for all i, µ , and suc h that λ i ´ λ i ` 1 P Z ` , i P t 1 , . . . , p ´ 1 u Y t p ` 1 , . . . , p ` q ´ 1 u , ω µ ´ ω µ ` 1 P Z ` , µ P t 1 , . . . , n ´ 1 u . (3.5) The unique simple k -mo dule of highest weigh t Λ P D ` p,q | n is denoted b y L 0 p Λ q . W e can turn L 0 p Λ q into a p k ‘ k ` q -mo dule by letting k ` act by zero, and we use this to define the highest-w eigh t gl p ` q | n -mo dule V p Λ q : “ U p gl p ` q | n q b U p k ‘ k ` q L 0 p Λ q . (3.6) CLASSIFICA TION OF UNIT AR Y MODULES 9 The quotient L p Λ q of V p Λ q b y its unique maximal prop er submo dule thus yields an irreducible gl p ` q | n -mo dule with highest weigh t Λ. By the PBW theorem for gl p ` q | n , we ha v e V p Λ q “ U p k ´ q b L 0 p Λ q . T ogether with the con v ention deg p E ai q “ 1 for a P t p ` 1 , . . . , p ` q ` n u and i P t 1 , . . . , p u , this yields the Z ` -grading V p Λ q “ à k P Z ` V k p Λ q , (3.7) where V 0 p Λ q “ L 0 p Λ q and eac h V k p Λ q is a finite-dimensional k -mo dule. Prop osition 3.3. L et Λ “ p λ 1 , . . . , λ p ` q , ω 1 , . . . , ω n q P D ` p,q | n , and supp ose V p Λ q is uni- tary. Then, the fol lowing ine qualities hold: (1) p Λ , α q ě 0 for al l α P Φ ` c . (2) p Λ , β q ď 0 for al l β P Φ ` nc . (3) λ p ` 1 ě ¨ ¨ ¨ ě λ p ` q ě ´ ω n ě ¨ ¨ ¨ ě ´ ω 1 ě λ 1 ě ¨ ¨ ¨ ě λ p . Pr o of. Let v Λ b e a highest-w eight vector of V p Λ q . If α P Φ ` c , ¯ 0 , then (i) α “ ϵ i ´ ϵ j for some i, j P t 1 , . . . , p u or i, j P t p ` 1 , . . . , p ` q u such that i ă j , in whic h case p Λ , α q “ λ i ´ λ j ě 0 b y (3.5), or (ii) α “ δ µ ´ δ ν for some µ, ν P t 1 , . . . , n u such that µ ă ν , in which case p Λ , α q “ ω µ ´ ω ν ě 0, again b y (3.5). If α P Φ ` c , ¯ 1 , then α “ ϵ i ´ δ µ for some i P t p ` 1 , . . . , p ` q u and µ P t 1 , . . . , n u , so p E µi q ‹ “ E iµ and 0 ď x E µi v Λ , E µi v Λ y “ x v Λ , E iµ E µi v Λ y “ x v Λ , p E ii ` E µµ q v Λ y “ p λ i ` ω µ q x v Λ , v Λ y , hence 0 ď λ i ` ω µ “ p Λ , ϵ i ´ δ µ q “ p Λ , α q . If β P Φ ` nc , ¯ 0 , then β “ ϵ i ´ ϵ j for some i P t 1 , . . . , p u and j P t p ` 1 , . . . , p ` q u , so p E ij q ‹ “ ´ E j i and 0 ď x E j i v Λ , E j i v Λ y “ ´p λ i ´ λ j q x v Λ , v Λ y , hence 0 ě λ i ´ λ j “ p Λ , ϵ i ´ ϵ j q “ p Λ , β q . If β P Φ ` nc , ¯ 1 , then β “ ϵ i ´ δ µ for some i P t 1 , . . . , p u and µ P t 1 , . . . , n u , so p E iµ q ‹ “ ´ E µi and 0 ď x E µi v Λ , E µi v Λ y “ ´p λ i ` ω µ q x v Λ , v Λ y , hence 0 ě λ i ` ω µ “ p Λ , ϵ i ´ δ µ q “ p Λ , β q . P art (3) follo ws by com bining (3.5) with 0 ď λ i ` ω µ for i P t p ` 1 , . . . , p ` q u and 0 ě λ i ` ω µ for i P t 1 , . . . , p u . □ By Prop osition 3.3, L p Λ q is an infinite-dimensional gl p ` q | n -mo dule unless λ p ` 1 “ ¨ ¨ ¨ “ λ p ` q “ ´ ω n “ ¨ ¨ ¨ “ ´ ω 1 “ λ 1 “ ¨ ¨ ¨ “ λ p . (If these equalities all hold, then L p V q is a 1-dimensional unitary mo dule.) Moreov er, it inherits the Z ` -grading (3.7): L p Λ q “ à k P Z ` L k p Λ q , (3.8) where eac h summand L k p Λ q is a finite-dimensional k -mo dule. F or eac h nonzero s P R , we also introduce the 1-dimensional gl p ` q | n -mo dule C s with action giv en b y E ab ¨ 1 “ δ a,b p´ 1 q r a s s ¨ 1 for all a, b P t 1 , . . . , p ` q ` n u . Accordingly , C s has the unique w eigh t Λ s “ p s, . . . , s loomoon p ` q , ´ s, . . . , ´ s lo ooo omo ooo on n q and is clearly a unitary mo dule. F or an y Λ P D ` p,q | n , the shifted w eigh t Λ p s q : “ Λ ` Λ s “ p λ 1 ` s, . . . , λ p ` q ` s, ω 1 ´ s, . . . , ω n ´ s q is the highest w eigh t of the gl p ` q | n -mo dule V p Λ q b C s . 10 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG 3.4. Finite-dimensional unitary mo dules. F or later use, here w e recall the classifi- cation of finite-dimensional unitary gl m | n -mo dules. First, we ha v e the triangular decomposition gl m | n “ g ´ 1 ‘ g 0 ‘ g 1 , where g 0 “ gl m ‘ gl n is the ev en subalgebra, while g ˘ 1 “ span t E α | α P ˘ Φ ` ¯ 1 u . Second, every finite-dimensional simple gl m | n -mo dule is uniquely characterised by its highest weigh t Λ “ p λ 1 , . . . , λ m , ω 1 , . . . , ω n q , which must satisfy λ i ´ λ i ` 1 P Z ` and ω µ ´ ω µ ` 1 P Z ` for all i P t 1 , . . . , m ´ 1 u and all µ P t 1 , . . . , n ´ 1 u . That is, Λ is a dominan t in tegral w eigh t of gl m ‘ gl n , and w e denote the corresp onding simple mo dule b y L p Λ q . F ollowing [Kac77a], L p Λ q and the corresp onding w eigh t Λ are said to b e typic al if ź α P Φ ` ¯ 1 p Λ ` ρ, α q ‰ 0; they are called atypic al otherwise. Asso ciated to the same weigh t Λ, w e also hav e the so-called Kac mo dule K p Λ q : “ U p gl m | n q b U p g 0 ‘ g 1 q L 0 p Λ q , where L 0 p Λ q is a simple g 0 -mo dule equipp ed with trivial g 1 -action. If Λ is t ypical, then L p Λ q is isomorphic to K p Λ q , while if Λ is at ypical, then K p Λ q is non-simple and L p Λ q is isomorphic to K p Λ q{ M p Λ q where M p Λ q is the unique maximal prop er submo dule of K p Λ q . F ollowing [SNR77, GZ90b], there exists an induced non-degenerate Hermitian form x´ , ´y on L p Λ q (unique up to a scalar m ultiple) which is p ositive-definite on L 0 p Λ q and suc h that for all v , w P L p Λ q and all a, b P t 1 , . . . , m ` n u , x E ab v , w y “ p´ 1 q θ pr a s`r b sq x v , E ba w y , for some fixed θ P t 0 , 1 u . F ollo wing [GZ90b], w e say that L p Λ q is a typ e-1 (resp ectiv ely typ e-2 ) unitary mo dule if the Hermitian form is p ositiv e-definite on L p Λ q for θ “ 0 (resp ectiv ely θ “ 1). Let D ` m | n denote the set of real dominan t in tegral w eigh ts of gl m ‘ gl n . Theorem 3.4 ([GZ90b]) . L et Λ P D ` m | n . Then, L p Λ q is typ e-1 unitary if and only if one of the fol lowing c onditions holds: (1) p Λ ` ρ, ϵ m ´ δ n q ą 0 . (2) Ther e exists µ P t 1 , . . . , n u such that p Λ ` ρ, ϵ m ´ δ µ q “ p Λ , δ µ ´ δ n q “ 0 . R emark 3.5 . If p Λ ` ρ, ϵ m ´ δ n q ą 0, then for ev ery i P t 1 , . . . , m u and µ P t 1 , . . . , n u , p Λ ` ρ, ϵ i ´ δ µ q “ p Λ ` ρ, ϵ i ´ ϵ m ` ϵ m ´ δ n ` δ n ´ δ µ q “ p λ i ´ λ m ` m ´ i q ` p Λ ` ρ, ϵ m ´ δ n q ` p ω µ ´ ω n ` n ´ µ q . As this is strictly p ositiv e, the conditions (1) and (2) in Theorem 3.4 are m utually ex- clusiv e, so any simple unitary mo dule L p Λ q with Λ P D ` m | n is either t ypical or atypical, dep ending on whether condition (1) or (2) is satisfied. T yp e-1 and t yp e-2 unitary mo dules are related by duality , although this relationship is not immediately evident from their definitions. How ev er, it w as shown in [GZ90b] (see also Prop osition 2.1) that a simple gl m | n -mo dule L p Λ q is type-1 unitary if and only if the dual mo dule L p Λ q ˚ is t yp e-2 unitary . Note that L p Λ q has a natural Z -grading L p Λ q “ À d Λ k “ 0 L k p Λ q inherited from the Kac mo dule, where 0 ď d Λ ď mn and each L k p Λ q is a g 0 -mo dule. In particular, L 0 p Λ q is simple with highest weigh t Λ. F ollo wing [GZ90b], the g 0 -highest weigh t of the minimal Z -graded comp onen t L d Λ p Λ q is determined as follo ws. If Λ is typical, we set µ “ n ` 1; CLASSIFICA TION OF UNIT AR Y MODULES 11 otherwise we set µ to b e the o dd index satisfying condition (2) in Theorem 3.4, and in tro duce µ m “ µ ´ 1 , µ i “ min t n, µ m ` p Λ , ϵ i ´ ϵ m qu , i “ 1 , . . . , m ´ 1 . Then, the g 0 -highest w eigh t of L d Λ p Λ q is ¯ Λ “ Λ ´ m ÿ i “ 1 µ i ÿ ν “ 1 p ϵ i ´ δ ν q , with corresp onding g 0 -highest-w eigh t vector given by Ω m | n “ ´ m ź i “ 1 µ i ź ν “ 1 E m ` ν,i ¯ v Λ . (3.9) A classification of the t yp e-2 unitary mo dules now follows. Theorem 3.6 ([GZ90b]) . L et Λ P D ` m | n . Then, L p Λ q is typ e-2 unitary if and only if one of the fol lowing c onditions holds: (1) p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0 . (2) Ther e exists k P t 1 , . . . , m u such that p Λ ` ρ, ϵ k ´ δ 1 q “ p Λ , ϵ 1 ´ ϵ k q “ 0 . R emark 3.7 . As in Theorem 3.4, the tw o conditions in Theorem 3.6 are mutually exclusive. W e conclude this section with a well-kno wn fact (cf. Lemma 3.1). Let L p Λ q b e a finite-dimensional simple gl m -mo dule. Then, the mo dule L p Λ q is unitary if and only if Λ is a real dominant integral w eigh t. A Hermitian form on L p Λ q is p ositiv e-definite and con tra v arian t if it satisfies the following tw o conditions: (i) x v Λ , v Λ y ą 0, where v Λ is a highest-w eigh t v ector of L p Λ q ; (ii) x E ab v , w y “ x v , E ba w y for all v , w P L p Λ q and all a, b P t 1 , . . . , m u . Moreo v er, a p ositiv e-definite contra v ariant Hermitian form on L p Λ q is unique up to scal- ing. 4. Main classifica tion resul t Here, we present our main result: a classification of all unitary simple highest-weigh t u p p, q | n q -mo dules of the form L p Λ q with Λ P D ` p,q | n ; see Theorem 4.2 b elo w. Lemma 4.1. L et Λ P D ` p,q | n and m “ p ` q . Then, the fol lowing six c onditions ar e mutual ly exclusive: (U1) p Λ ` ρ, ϵ m ´ δ n q ą 0 and p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0 . (U2) p Λ ` ρ, ϵ m ´ δ n q ą 0 , and ther e exists i P t 1 , . . . , p u such that p Λ ` ρ, ϵ i ´ δ 1 q “ p Λ , ϵ i ´ ϵ 1 q “ 0 . (U3) p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0 , and ther e exists µ P t 2 , . . . , n u such that p Λ ` ρ, ϵ m ´ δ µ q “ p Λ , δ µ ´ δ n q “ 0 . (U4) Ther e exists µ P t 2 , . . . , n u such that p Λ ` ρ, ϵ m ´ δ µ q “ p Λ , δ µ ´ δ n q “ 0 , and ther e exists i P t 1 , . . . , p u such that p Λ ` ρ, ϵ i ´ δ 1 q “ p Λ , ϵ i ´ ϵ 1 q “ 0 . (U5) p Λ ` ρ, ϵ m ´ δ 1 q “ p Λ , δ 1 ´ δ n q “ 0 , and ther e exists j P t p, . . . , m ´ 1 u such that p Λ , ϵ 1 ´ δ 1 q ă 1 ´ j and p Λ , ϵ j ` 1 ´ ϵ m q “ 0 . 12 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG (U6) p Λ ` ρ, ϵ m ´ δ 1 q “ p Λ , δ 1 ´ δ n q “ 0 , and ther e exist i P t 1 , . . . , p u and j P t p, . . . , m ´ 1 u such that p Λ , ϵ i ´ ϵ 1 q “ p Λ , ϵ j ` 1 ´ ϵ m q “ 0 and p Λ , ϵ i ´ δ 1 q “ i ´ j . Pr o of. Throughout, w e use the identities (3.1) and the dominance condition (3.5). If p Λ ` ρ, ϵ m ´ δ n q ą 0, then for an y µ P t 1 , . . . , n u , p Λ ` ρ, ϵ m ´ δ µ q “ p Λ ` ρ, ϵ m ´ δ n q ` p Λ ` ρ, δ n ´ δ µ q “ p Λ ` ρ, ϵ m ´ δ n q ` p Λ , δ n ´ δ µ q ` n ´ µ ą 0 . Consequen tly , (U1) and (U2) are each disjoint from (U3) – (U6) . Similarly , if p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0, then for an y i P t 1 , . . . , p u , p Λ ` ρ, ϵ i ´ δ 1 q “ p Λ ` ρ, ϵ i ´ ϵ 1 q ` p Λ ` ρ, ϵ 1 ´ δ 1 q “ p Λ , ϵ i ´ ϵ 1 q ` p 1 ´ i q ` p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0 . Therefore, (U1) and (U2) are disjoin t, and (U3) and (U4) are disjoint. If p Λ ` ρ, ϵ m ´ δ 1 q “ p Λ , δ 1 ´ δ n q “ 0, then, for an y µ P t 2 , . . . , n u , p Λ ` ρ, ϵ m ´ δ µ q “ p Λ ` ρ, ϵ m ´ δ 1 q ` p Λ ` ρ, δ 1 ´ δ µ q “ p Λ , δ 1 ´ δ µ q ` 1 ´ µ “ 1 ´ µ ă 0 . Hence, (U5) and (U6) are eac h disjoin t from (U3) – (U4) . Finally , suppose that (U6) is satisfied, so there exist i P t 1 , . . . , p u and j P t p, . . . , m ´ 1 u suc h that p Λ , ϵ i ´ ϵ 1 q “ p Λ , ϵ j ` 1 ´ ϵ m q “ 0 and p Λ , ϵ i ´ δ 1 q “ i ´ j . Then, p Λ , ϵ 1 ´ δ 1 q “ p Λ , ϵ 1 ´ ϵ i q ` p Λ , ϵ i ´ δ 1 q “ i ´ j ě 1 ´ j, whic h con tradicts the condition in (U5) . It follows that (U5) and (U6) are disjoin t. Com bining the ab ov e, the conditions (U1) – (U6) are seen to b e m utually exclusiv e. □ Theorem 4.2. L et Λ P D ` p,q | n . Then, the u p p, q | n q -mo dule L p Λ q is unitary if and only if Λ satisfies one of the c onditions (U1) – (U6) in L emma 4.1. Sections 5 – 9 are devoted to the pro of of Theorem 4.2. In the course of the pro of, for eac h of the conditions (U1) – (U6) , we will establish the existence of a highest weigh t Λ satisfying the condition. Theorem 4.2 also reco v ers the classical criterion [EHW83] for the unitarity of the u p p, q q - mo dule L p Λ q with Λ P D ` m , m “ p ` q . Sp ecifically , this mo dule is unitary if and only if there exist i P t 1 , . . . , p u and j P t 1 , . . . , q u such that p Λ , ϵ 1 ´ ϵ i q “ p Λ , ϵ m ´ j ` 1 ´ ϵ m q “ 0 , and suc h that one of the follo wing t w o conditions holds: (C1) λ m ´ λ 1 “ m ´ j ´ i . (C2) λ m ´ λ 1 P R and λ m ´ λ 1 ą min t m ´ i, m ´ j u ´ 1. By eliminating the o dd co ordinates, condition (C1) is obtained from (U6) , while (C2) follo ws from a com bination of (U2) and (U5) . 5. Necessity In this section, w e sho w that one of the conditions (U1) – (U6) ne c essarily holds if the u p p, q | n q -mo dule L p Λ q is unitary . Our arguments are based on the unitarity conditions for the finite-dimensional simple mo dules ov er the Lie sup eralgebras gl p | n and gl q | n . Here and in the remainder of the paper, we will use the notation m “ p ` q . CLASSIFICA TION OF UNIT AR Y MODULES 13 Clearly , gl p | n em b eds in to gl m | n , ha ving simple ro ot system ∆ p | n “ t ϵ i ´ ϵ i ` 1 , ϵ p ´ δ 1 , δ µ ´ δ µ ` 1 | i “ 1 , . . . , p ´ 1; µ “ 1 , . . . , n ´ 1 u and graded half-sum of p ositiv e ro ots ρ p | n “ 1 2 p ÿ i “ 1 p p ´ n ´ 2 i ` 1 q ϵ i ` 1 2 n ÿ µ “ 1 p p ` n ´ 2 µ ` 1 q δ µ . Similarly , gl q | n has simple ro ot system ∆ q | n “ t ϵ i ´ ϵ i ` 1 , ϵ m ´ δ 1 , δ µ ´ δ µ ` 1 | i “ p ` 1 , . . . , m ´ 1; µ “ 1 , . . . , n ´ 1 u and graded half-sum of p ositiv e ro ots ρ q | n “ 1 2 m ÿ i “ p ` 1 p q ´ n ´ 2 p i ´ p q ` 1 q ϵ i ` 1 2 n ÿ µ “ 1 p q ` n ´ 2 µ ` 1 q δ µ . Lemma 5.1. L et Λ P D ` p,q | n . Then, the k -mo dule L 0 p Λ q is typ e-1 unitary if and only if one of the fol lowing c onditions holds: (1) p Λ ` ρ, ϵ m ´ δ n q ą 0 . (2) Ther e exists µ P t 1 , . . . , n u such that p Λ ` ρ, ϵ m ´ δ µ q “ p Λ , δ µ ´ δ n q “ 0 . Pr o of. Λ decomp oses into gl p - and gl q | n -w eigh ts, as Λ “ p Λ 1 , Λ 2 q . If L 0 p Λ q is a simple k -mo dule, then L 0 p Λ q – L p p Λ 1 q b L q | n p Λ 2 q , where L p p Λ 1 q and L q | n p Λ 2 q are gl p - and gl q | n - mo dules with highest w eights Λ 1 and Λ 2 . As p er the remarks at the end of Section 3.4, L p p Λ 1 q is unitary if and only if Λ 1 is real and gl p -dominan t integral. Theorem 3.4 implies that L p Λ 2 q is type-1 unitary if and only if either p Λ 2 ` ρ q | n , ϵ m ´ δ n q ą 0 or there exists µ P t 1 , . . . , n u suc h that p Λ 2 ` ρ q | n , ϵ m ´ δ µ q “ p Λ 2 , δ µ ´ δ n q “ 0. As p Λ ` ρ, ϵ m ´ δ µ q “ p Λ 2 ` ρ q | n , ϵ m ´ δ µ q , µ “ 1 , . . . , n, the result follo ws. □ Let v Λ b e a highest-weigh t vector of L p Λ q . F ollo wing the construction in (3.9), in accordance with the em b edding gl q | n Ă gl p ` q | n , we define the p gl q ‘ gl n q -highest-w eigh t v ector Ω q | n “ ´ m ź j “ p ` 1 µ j ź ν “ 1 E m ` ν,j ¯ v Λ . Clearly , Ω q | n is a gl p -highest-w eigh t vector, and as E i,m ` µ Ω q | n “ 0 for all i P t 1 , . . . , p u and all µ P t 1 , . . . , n u , Ω q | n is also a p gl p | n ‘ gl q q -highest-w eigh t v ector. If L p Λ q is unitary , then ev ery gl p | n -submo dule of L p Λ q is type-2 unitary . Prop osition 5.2. L et Λ P D ` p ` q | n , and supp ose L p Λ q is unitary. Then, Λ satisfies one of the c onditions (U1) – (U6) . Pr o of. If L p Λ q is unitary , then the k -submo dule L 0 p Λ q is t yp e-1 unitary . Consequen tly , the weigh t Λ satisfies one of the conditions in Lemma 5.1. If condition (1) in that lemma holds, or condition (2) holds with µ ą 1, then Ω q | n is a p gl p | n ‘ gl q q -highest-w eigh t vector of w eigh t Λ q | n “ Λ ´ m ÿ i ą p µ i ÿ ν “ 1 p ϵ i ´ δ ν q , 14 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG where µ i ě µ . The highest weigh t Λ q | n remains dominant in tegral for gl p ‘ gl q ‘ gl n . Since L p Λ q is unitary and hence semisimple as a p gl p | n ‘ gl q q -mo dule, it con tains a finite- dimensional t yp e-2 unitary simple p gl p | n ‘ gl q q -submo dule generated b y Ω q | n . By Theo- rem 3.6, w e ha v e either p Λ q | n ` ρ p | n , ϵ 1 ´ δ 1 q “ p Λ , ϵ 1 ´ δ 1 q ` m ´ 1 “ p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0 , in whic h case Λ satisfies (U1) or (U3) , or there exists i P t 1 , . . . , p u suc h that p Λ q | n ` ρ p | n , ϵ i ´ δ 1 q “ p Λ , ϵ i ´ δ i q ` m ´ i “ p Λ ` ρ, ϵ i ´ δ 1 q “ 0 , p Λ q | n , ϵ 1 ´ ϵ i q “ p Λ , ϵ 1 ´ ϵ i q “ 0 , in whic h case Λ satisfies (U2) or (U4) . If condition (2) in Lemma 5.1 holds with µ “ 1, then there exists j P t p, . . . , m ´ 1 u suc h that n ě µ p ` 1 ě ¨ ¨ ¨ ě µ j ą µ j ` 1 “ ¨ ¨ ¨ “ µ m “ µ ´ 1 “ 0 . It follo ws that p Λ , ϵ j ` 1 ´ ϵ m q “ 0 and that Ω q | n is a p gl p | n ‘ gl q q -highest-w eigh t vector of w eigh t Λ q | n “ Λ ´ j ÿ i ą p µ i ÿ ν “ 1 p ϵ i ´ δ ν q . Again using Theorem 3.6, w e ha v e either p Λ q | n ` ρ p | n , ϵ 1 ´ δ 1 q “ p Λ , ϵ 1 ´ δ 1 q ` j ´ 1 ă 0 , in whic h case Λ satisfies (U5) , or there exists i P t 1 , . . . , p u suc h that p Λ q | n ` ρ p | n , ϵ i ´ δ 1 q “ p Λ , ϵ i ´ δ 1 q ` j ´ i “ 0 , p Λ q | n , ϵ 1 ´ ϵ i q “ p Λ , ϵ 1 ´ ϵ i q “ 0 , in whic h case Λ satisfies (U6) . □ 6. A unit arity criterion Here, we introduce a quadratic inv arian t of the subalgebra k that acts by scalar mul- tiplication on eac h simple k -submo dule of L p Λ q for L p Λ q unitary . This yields a useful criterion for the unitarit y of L p Λ q that w e will use later to pro ve the sufficiency of the conditions (U1) – (U6) . W e let Π k p Λ q denote the set of all k -highest weigh ts of L p Λ q , and recall the notation m “ p ` q . W e denote by ρ k the graded half-sum of p ositiv e ro ots of k , and note that ρ “ ρ k ` 1 2 ´ p ÿ i “ 1 m ÿ j “ p ` 1 p ϵ i ´ ϵ j q ´ p ÿ i “ 1 n ÿ µ “ 1 p ϵ i ´ δ µ q ¯ . W e also note that the Casimir elemen t of k is giv en by C k “ p ÿ i,j “ 1 E ij E j i ` m ` n ÿ a,b “ p ` 1 p´ 1 q r b s E ab E ba and set Γ : “ p ÿ i “ 1 m ` n ÿ a “ p ` 1 E ai E ia . By construction, Γ P U p g q , and it is straigh tforward to verify that Γ is k -inv ariant in the sense that r Γ , X s “ 0 for all X P k . Recall that V p Λ q “ U p k ´ q b L 0 p Λ q , where L 0 p Λ q is a simple k -mo dule and U p k ´ q is isomorphic to the sup ersymmetric algebra S pp C p q ˚ b C q | n q as a k -mo dule. Note that CLASSIFICA TION OF UNIT AR Y MODULES 15 S pp C p q ˚ b C q | n q is a t yp e-1 unitary k -mo dule and hence k -semisimple (cf. [CLZ04] and [Ser01, Theorem 2.1]), while V p Λ q need not b e k -semisimple. Lemma 6.1. L et v ξ b e a k -highest-weight ve ctor in V p Λ q of weight ξ . Then, Γ acts on v ξ as multiplic ation by the sc alar γ “ 1 2 p Λ ´ ξ , Λ ` ξ ` 2 ρ q . Pr o of. The Casimir element of gl m | n is giv en b y C “ m ` n ÿ a,b “ 1 p´ 1 q r b s E ab E ba “ C k ` p ÿ i “ 1 m ` n ÿ a “ p ` 1 p E ai E ia ` p´ 1 q r a s E ia E ai q “ C k ` 2Γ ` p ÿ i “ 1 m ` n ÿ a “ p ` 1 p´ 1 q r a s r E ia , E ai s , and acts on L p Λ q as multiplication b y the scalar p Λ , Λ ` 2 ρ q , while C k acts on v ξ as m ultiplication b y the scalar p ξ , ξ ` 2 ρ k q . As r E ia , E ai s v ξ “ p E ii ´ p´ 1 q r a s E aa q v ξ “ p ξ , ϵ i ´ ϵ a q v ξ for all i P t 1 , . . . , p u and all a P t p ` 1 , . . . , m ` n u , Γ acts on v ξ as m ultiplication b y γ “ 1 2 ´ p Λ , Λ ` 2 ρ q ´ p ξ , ξ ` 2 ρ k q ´ p ÿ i “ 1 m ` n ÿ a “ p ` 1 p´ 1 q r a s p ξ , ϵ i ´ ϵ a q ¯ “ 1 2 pp Λ , Λ ` 2 ρ q ´ p ξ , ξ ` 2 ρ qq “ 1 2 p Λ ´ ξ , Λ ` ξ ` 2 ρ q . □ T o construct a unitary structure on V p Λ q , w e use (3.7) and start with a unitary k -mo dule V 0 p Λ q equipp ed with a fixed p ositiv e-definite con tra v ariant Hermitian form x´ , ´y . This form can then b e extended to a con trav ariant Hermitian form on V p Λ q suc h that (3.3) is satisfied, with V k p Λ q and V ℓ p Λ q orthogonal whenever k ‰ ℓ , cf. [GZ90a, Lemma 1]. The follo wing result app ears in [GL96, Lemma 3.1] in the context of quantum sup er- groups. Lemma 6.2. L et V b e a k -mo dule e quipp e d with a c ontr avariant Hermitian form x´ , ´y . (1) If v µ and v ν ar e weight ve ctors of V of weights µ ‰ ν , then x v µ , v ν y “ 0 . (2) If L p µ q and L p ν q ar e simple submo dules of V of highest weights µ ‰ ν , then x L p µ q , L p ν qy “ 0 . Pr o of. F or part (1), since µ ‰ ν , there exists a P t 1 , . . . , m ` n u suc h that µ p E aa q ‰ ν p E aa q . As the star-op eration fixes the Cartan subalgebra of k , we hav e µ p E aa q x v µ , v ν y “ x E aa v µ , v ν y “ x v µ , p E aa q ‹ v ν y “ ν p E aa q x v µ , v ν y , hence x v µ , v ν y “ 0. F or part (2), let v µ and v ν b e highest-w eight vectors of L p µ q and L p ν q , resp ectiv ely . Let b 1 “ b X k b e the standard Borel subalgebra of k , and let n 1 ` (resp ectiv ely n 1 ´ ) b e the nilpotent radical (respectively opposite nilp oten t radical) of b 1 . Since the Cartan elements are fixed under the star-op eration and act as scalars on highest-w eigh t vectors, it follows from 0 “ x v µ , v ν y that 0 “ x U p b 1 q v µ , v ν y “ x v µ , U p n 1 ´ q v ν y “ x v µ , L p ν qy “ x v µ , U p k q L p ν qy “ x U p k q v µ , L p ν qy “ x L p µ q , L p ν qy . 16 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG □ As the simple quotient of V p Λ q , the mo dule L p Λ q inherits the extended contra v arian t Hermitian form on V p Λ q . As demonstrated in the next prop osition, a simple criterion ensures that this form on L p Λ q is p ositive-definite. Prop osition 6.3. L et Λ P D ` p,q | n , and supp ose L 0 p Λ q is a typ e-1 unitary simple k -mo dule. Then, L p Λ q is unitary if and only if p Λ ´ ξ , Λ ` ξ ` 2 ρ q ď 0 , @ ξ P Π k p Λ q . Pr o of. W e first assume L p Λ q is unitary . Then, L p Λ q is semisimple as a k -module, and there exists a k -highest-w eigh t v ector v ξ of w eigh t ξ for eac h ξ P Π k p Λ q . W e then hav e x Γ v ξ , v ξ y “ ´ p ÿ i “ 1 m ` n ÿ a “ p ` 1 x E ia v ξ , E ia v ξ y ď 0 , so p Λ ´ ξ , Λ ` ξ ` 2 ρ q ď 0 b y Lemma 6.1. As to the rev erse implication, we use induction on the Z -grading (3.8) of L p Λ q , and let L p,q | n p ξ q Ă V p Λ q denote a simple k -mo dule generated b y a highest-w eigh t vector v ξ of w eigh t ξ P Π k p Λ q . As L 0 p Λ q is a type-1 unitary k -mo dule equipp ed with a p ositiv e-definite con tra v arian t Hermitian form x´ , ´y , V k p Λ q “ S k pp C p q ˚ b C q | n q b L 0 p Λ q is also a type-1 unitary k -mo dule and thus k -semisimple. It follo ws that L k p Λ q is a semisimple k -mo dule for an y k P Z ` , so w e ha v e a finite k -mo dule decomp osition of the form L k p Λ q – à r K p,q | n p ξ r q , where K p,q | n p ξ r q is a direct sum of simple k -mo dules isomorphic to L p,q | n p ξ r q . T o show that the inherited con trav arian t Hermitian form x´ , ´y on L k p Λ q is p ositiv e-definite for k ě 1, we supp ose it is p ositiv e-definite on L k ´ 1 p Λ q for k ą 1. Now, every nonzero v ector v r P K p,q | n p ξ r q is a linear com bination of vectors of the form E ai w , where i P t 1 , . . . , p u , a P t p ` 1 , . . . , m ` n u , and w P L k ´ 1 p Λ q . Moreo v er, there exist j P t 1 , . . . , p u and b P t p ` 1 , . . . , m ` n u such that E j b v r ‰ 0; otherwise, w e would hav e a contradiction with v r P L k p Λ q . As Γ acts as scalar multiplication by γ on v r , the induction hypothesis implies that γ x v r , v r y “ ´ p ÿ i “ 1 m ` n ÿ a “ p ` 1 x E ia v r , E ia v r y ă 0 , and since γ “ 1 2 p Λ ´ ξ , Λ ` ξ ` 2 ρ q ď 0 , w e hav e x v r , v r y ą 0. By Lemma 6.2, K p,q | n p ξ i q and K p,q | n p ξ j q are orthogonal for i ‰ j , so for an y v ector v “ ř r v r with v r P K p,q | n p ξ r q , x v , v y “ ÿ r x v r , v r y ą 0 . Hence, the con tra v arian t Hermitian form x´ , ´y on L k p Λ q is p ositive-definite, and L p Λ q is a unitary mo dule. □ CLASSIFICA TION OF UNIT AR Y MODULES 17 7. Ho we duality In this section, w e recall from [CLZ04] (see also [CW01]) the action of gl d ˆ gl p ` q | n on the sup ersymmetric algebra S pp C d q ˚ b p C p q ˚ ‘ C d b C q | n q for any fixed p ositiv e integer d . This giv es rise to the p gl d , gl p ` q | n q -Ho w e duality whic h yields a m ultiplicity-free decom- p osition of the sup ersymmetric algebra in to simple p gl d ‘ gl p ` q | n q -mo dules. In Section 8, this enables an explicit construction of infinite-dimensional unitary gl p ` q | n -mo dules with in tegral highest w eights as classified in Section 4. W e recall the notation m “ p ` q . 7.1. Oscillator sup eralgebra. Fix a positive in teger d P Z ` , and let C d denote the nat- ural gl d -mo dule with standard basis t v 1 , . . . , v d u . Let p C d q ˚ b e the dual mo dule spanned b y the dual basis t v 1 , . . . , v d u suc h that v a p v b q “ δ a,b for all a, b P t 1 , . . . , d u . F or each pair a, b P t 1 , . . . , d u , denote by e ab the matrix unit suc h that e ab v c “ δ b,c v a for all c P t 1 , . . . d u . Also, let t e 1 , . . . , e m ` n u denote the standard homogeneous basis for the natural gl p ` q | n - mo dule C m | n , and let t e 1 , . . . , e m ` n u b e the basis for p C m | n q ˚ suc h that e a p e b q “ δ a,b for all a, b P t 1 , . . . , m ` n u . W e identify p C p q ˚ with the subspace of p C m | n q ˚ spanned by t e 1 , . . . , e p u , and C q | n with the subspace of C m | n spanned b y t e p ` 1 , . . . , e m ` n u . The sup ersymmetric algebra S pp C d q ˚ b p C p q ˚ ‘ C d b C q | n q is isomorphic to the p oly- nomial sup eralgebra C d p,q | n r x , y , η s in the following v ariables: x a k : “ v a b e p ` k , y a i : “ v a b e i , η a µ : “ v a b e m ` µ , (7.1) with a P t 1 , . . . , d u , k P t 1 , . . . , q u , i P t 1 , . . . , p u , µ P t 1 , . . . , n u . Note that x a k and y a i are ev en v ariables, while η a µ are o dd. W riting B x a k , B y a i , B η a µ for the partial deriv ativ es with resp ect to these v ariables, we let D d p,q | n r x , y , η s denote the oscilla- tor sup eralgebra generated by the v ariables x a k , y a i , η a µ and their deriv atives B x a k , B y a i , B η a µ . Then, C d p,q | n r x , y , η s is a simple mo dule o ver D d p,q | n r x , y , η s . Let ρ b e the asso ciative sup eralgebra homomorphism ρ : U p gl d ‘ gl p ` q | n q Ñ D r x , y , η s , defined b y ρ p e ab q “ q ÿ k “ 1 x a k B x b k ´ p ÿ i “ 1 y b i B y a i ` n ÿ µ “ 1 η a µ B η b µ , a, b P t 1 , . . . , d u , and ρ p E i,j q “ ´ d ÿ a “ 1 B y a i y a j , ρ p E i,p ` ℓ q “ d ÿ a “ 1 B y a i B x a ℓ , ρ p E i,p ` q ` ν q “ d ÿ a “ 1 B y a i B η a ν , ρ p E p ` k,j q “ ´ d ÿ a “ 1 x a k y a j , ρ p E p ` k,p ` ℓ q “ d ÿ a “ 1 x a k B x a ℓ , ρ p E p ` k,p ` q ` ν q “ d ÿ a “ 1 x a k B η a ν , ρ p E p ` q ` µ,j q “ ´ d ÿ a “ 1 η a µ y a j , ρ p E p ` q ` µ,p ` ℓ q “ d ÿ a “ 1 η a µ B x a ℓ , ρ p E p ` q ` µ,p ` q ` ν q “ d ÿ a “ 1 η a µ B η a ν . It is straigh tforw ard to verify that the differen tial op erators ρ p e ab q (resp ectiv ely ρ p E ij q ) satisfy the comm utation relations of gl d (resp ectiv ely gl p ` q | n ), and that ρ p e ab q comm utes with ρ p E ij q for all relev an t a, b, i, j . This gives a realisation of gl d ˆ gl p ` q | n in terms of differen tial op erators, yielding a linear action on C d p,q | n r x , y , η s . 18 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG The oscillator sup eralgebra D p,q | n d r x , y , η s admits the star-sup eralgebra structure ψ de- fined b y ψ p z q “ B z , ψ pB z q “ z , for all v ariables z given in (7.1). There exists a unique Hermitian form x´ , ´y on C d p,q | n r x , y , η s satisfying x 1 , 1 y “ 1 and x f g , h y “ x g , ψ p f q h y , f , g , h P C d p,q | n r x , y , η s . (7.2) Consequen tly , x M , M y ą 0 for ev ery nonzero monomial M P C d p,q | n r x , y , η s , and x´ , ´y is p ositiv e-definite. Similarly , U p gl d ‘ gl p ` q | n q has a star-op eration σ giv en b y σ p e ab q “ e ba , σ p E ij q “ # E j i , i, j ď p or i, j ą p, ´ E j i , otherwise, for all applicable a, b, i, j . These t w o star-structures are compatible in the sense that ρσ p X q “ ψ ρ p X q , X P U p gl d ‘ gl p ` q | n q , and w e conclude that C d p,q | n r x , y , η s is a unitary U p gl d ‘ gl p ` q | n q -mo dule with resp ect to the Hermitian form (7.2), cf. [CLZ04, Theorem 3.2]. 7.2. Decomp osition of sup ersymmetric algebra. A p artition λ “ p λ 1 , . . . , λ k q of length k is a non-increasing sequence of non-negative in tegers: λ 1 ě ¨ ¨ ¨ ě λ k ě 0. W e denote by P k the set of partitions of length k . The c onjugate p artition of λ P P k is λ 1 “ p λ 1 1 , . . . , λ 1 ℓ q , where ℓ “ λ 1 and λ 1 i “ # t j | λ j ě i u for i “ 1 , . . . , ℓ . If λ 1 “ 0, w e set λ 1 “ p 0 q . A gener alise d p artition of length k is a non-increasing sequence of k in tegers (some of whic h could b e negativ e). By construction, each generalised partition λ “ p λ 1 , . . . , λ k q can b e written as λ “ λ ` ` λ ´ , where λ ` : “ p max t λ 1 , 0 u , . . . , max t λ k , 0 uq , λ ´ : “ p min t λ 1 , 0 u , . . . , min t λ k , 0 uq . W e also introduce λ ˚ ´ : “ p´ min t λ k , 0 u , . . . , ´ min t λ 1 , 0 uq , and note that λ ` , λ ˚ ´ P P k . W e adopt the conv en tion that λ i “ 0 if the index i is non-p ositiv e or greater than k . Denote by P k p,q | n the set of generalised partitions λ “ p λ 1 , . . . , λ k q of length k suc h that λ q ` 1 ď n and λ k ´ p ě 0. Asso ciated to a generalised partition λ “ p λ 1 , . . . , λ d q P P d p,q | n , w e define a sequence λ 5 of length p ` q ` n b y setting λ 5 : “ p´ d, . . . , ´ d, ´ d ` λ r , . . . , ´ d ` λ d , p λ ` q 1 , . . . , p λ ` q q , max tp λ 1 ` q 1 ´ q , 0 u , . . . , max tp λ 1 ` q n ´ q uq . (7.3) Here, r P t d ´ p ` 1 , . . . , d u is the smallest index such that λ r ă 0, if such an index exists; otherwise, the first p en tries of λ 5 are all ´ d . T o each generalised partition λ P P d p,q | n , we associate the unique dominan t integral gl d -w eigh t λ “ d ÿ i “ 1 λ i ε i , where t ε 1 , . . . , ε d u is the standard basis for the dual space of the Cartan subalgebra of gl d . Let L d p λ q denote the corresp onding simple highest-w eight mo dule. Similarly , we identify λ 5 with a Φ ` c -dominan t integral w eight of gl p ` q | n via (3.4). The following theorem is deriv ed from [CLZ04, Lemma 3.2, Theorem 3.3] and establishes the p gl d , gl p ` q | n q -Ho w e dualit y . CLASSIFICA TION OF UNIT AR Y MODULES 19 Theorem 7.1. Under the action of gl d ˆ gl p ` q | n describ e d in Se ction 7.1, C d p,q | n r x , y , η s de c omp oses into a multiplicity-fr e e dir e ct sum of simple mo dules, as C d p,q | n r x , y , η s – à λ P P d p,q | n L d p λ q b L p ` q | n p λ 5 q . Corollary 7.2. Every infinite-dimensional gl p ` q | n -mo dule L p λ 5 q with λ P P p ` q | n is uni- tary. 8. Unit ar y modules with integral highest weights Let P ` p,q | n denote the subset of D ` p,q | n that consists of the weigh ts whose comp onen ts are all in tegers. Using How e dualit y , we ha ve the following classification result. W e recall the notation m “ p ` q . Theorem 8.1. L et Λ “ p λ 1 , . . . , λ p ` q , ω 1 , . . . , ω n q P P ` p,q | n . Then, L p Λ q is unitary if and only if one of the fol lowing c onditions holds: (1) Either λ 1 ` ω 1 ă 1 ´ m or ther e exists i P t 1 , . . . , p u such that λ 1 “ ¨ ¨ ¨ “ λ i , λ 1 ` ω 1 “ i ´ m, and either λ m ` ω n ą n ´ 1 or ther e exists µ P t 2 , . . . , n u such that ω µ “ ¨ ¨ ¨ “ ω n , λ m ` ω n “ µ ´ 1 . (2) Ther e holds ω 1 “ ¨ ¨ ¨ “ ω n , λ m ` ω n “ 0 , and ther e exists j P t p, . . . , m ´ 1 u such that λ j ` 1 “ ¨ ¨ ¨ “ λ m , and either λ 1 ` ω 1 ă 1 ´ j or ther e exists i P t 1 , . . . , p u such that λ 1 “ ¨ ¨ ¨ “ λ i , λ 1 ` ω 1 “ i ´ j. Pr o of. The necessity follo ws from Prop osition 5.2, as the conditions (U1) – (U6) can b e expressed in terms of in tegral w eigh t comp onen ts as follo ws. Using (3.1), w e ha ve p Λ ` ρ, ϵ m ´ δ n q “ λ m ` ω n ` 1 ´ n, p Λ ` ρ, ϵ 1 ´ δ 1 q “ λ 1 ` ω 1 ` m ´ 1 . Hence, condition (U1) is equiv alent to the inequalities λ m ` ω n ą n ´ 1 and λ 1 ` ω 1 ă 1 ´ m . Similarly , condition (U2) requires the existence of i P t 1 , . . . , p u suc h that λ 1 “ ¨ ¨ ¨ “ λ i and λ 1 ` ω 1 “ i ´ m . Com bining conditions (U1) – (U4) , w e obtain part (1) of the theorem. Part (2) follo ws from the combination of (U5) and (U6) . T o establish sufficiency , w e first supp ose Λ satisfies condition (1) and sho w that L p Λ q is, up to tensoring with a 1-dimensional mo dule, a submodule of S pp C d q ˚ b p C p q ˚ ‘ C d b C q | n q for some p ositiv e in teger d . T o this end, we define the generalised partitions λ 1 p Λ q “ p λ p ` 1 ` ω n , . . . , λ m ` ω n q , λ 2 p Λ q “ p ω 1 ´ ω n , . . . , ω µ ´ 1 ´ ω n q 1 , λ 3 p Λ q “ p λ i ` 1 ´ λ 1 , . . . , λ p ´ λ 1 q , where the prime in the second line denotes conjugation of partition, and we set µ : “ n if λ m ` ω n ą n ´ 1. Here, λ 1 p Λ q and λ 2 p Λ q are partitions, while λ 3 p Λ q is a generalised partition of non-p ositiv e in tegers. W riting ℓ p S q for the length of any finite sequence S , w e find 3 ÿ k “ 1 ℓ p λ k p Λ qq ď m ´ i ` ω 1 ´ ω n ď ´p λ 1 ` ω 1 q ` ω 1 ´ ω n “ ´ λ 1 ´ ω n , 20 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG where the first inequalit y holds as i is not required to b e the maximal index that satisfies λ 1 “ ¨ ¨ ¨ “ λ i , and similarly for µ . Let d “ ´ λ 1 ´ ω n . W e define a generalised partition of length d b y setting λ “ p λ 1 p Λ q , λ 2 p Λ q , 0 , . . . , 0 , λ 3 p Λ qq , where 0 app ears d ´ ř 3 i “ 1 ℓ p λ i p Λ qq times. Clearly , λ q ` 1 “ µ ´ 1 ď n and λ d ´ p ě 0, so λ P P p ` q | n X P d . It follo ws from Theorem 7.1 that L p ` q | n p λ 5 q app ears as a submo dule of S pp C d q ˚ b p C p q ˚ ‘ C d b C q | n q and is therefore unitary . By (7.3), the highest w eigh t λ 5 is giv en b y λ 5 “ p´ d, . . . , ´ d lo ooo omo ooo on i , ´ d ` λ i ` 1 ´ λ 1 , . . . , ´ d ` λ p ´ λ 1 , λ p ` 1 ` ω n , . . . , λ m ` ω n , ω 1 ´ ω n , . . . , ω µ ´ 1 ´ ω n , 0 , . . . , 0 lo omo on n ´ µ ` 1 q . As gl p ` q | n -mo dules, L p Λ q – L p ` q | n p λ 5 q b C d ` λ 1 , and since b oth L p ` q | n p λ 5 q and C d ` λ 1 are unitary , so is L p Λ q . The situation when Λ satisfies condition (2) is similar, so w e only sk etc h the proof here. Let λ 1 p Λ q “ p λ p ` 1 ` ω n , . . . , λ j ` ω n q , λ 3 p Λ q “ p λ i ` 1 ´ λ 1 , . . . , λ p ´ λ 1 q . Then, ℓ p λ 1 p Λ qq ` ℓ p λ 3 p Λ qq “ j ´ i ď ´ λ 1 ´ ω 1 “ ´ λ 1 ´ ω n . Let d “ ´ λ 1 ´ ω n , and define a generalised partition λ of length d by λ “ p λ 1 p Λ q , 0 , . . . , 0 lo omo on d ´ j ` i , λ 3 p Λ qq . Clearly , λ P P p ` q | n X P d , and the asso ciated highest gl p ` q | n -w eigh t is λ 5 “ p´ d, . . . , ´ d lo ooo omo ooo on i , ´ d ` λ i ` 1 ´ λ 1 , . . . , ´ d ` λ p ´ λ 1 , λ p ` 1 ` ω n , . . . , λ j ` ω n , 0 , . . . , 0 lo omo on m ` n ´ j q . It follo ws that L p Λ q – L p ` q | n p λ 5 q b C d ` λ 1 , whic h again implies that L p Λ q is unitary . □ R emark 8.2 . Unitary su p p, q | n q -mo dules with in tegral highest or lo w est w eigh ts are clas- sified in [FN91] using an oscillator representation of an orthosymplectic Lie sup eralgebra. Theorem 8.1 agrees with [FN91, Theorem 5.5] up to tensoring with a 1-dimensional mo d- ule. 9. Sufficiency W e are now in a p osition to complete the pro of of the sufficiency p ortion of Theorem 4.2. The sufficiency of condition (U1) is thus established in Prop osition 9.1, and of condition (U3) in Prop osition 9.2. Using Lemma 9.3 b elo w, the sufficiency of condition (U2) is established in Prop osition 9.4, and of condition (U5) in Prop osition 9.5. Using Theorem 8.1 based on Ho w e dualit y , the sufficiency of condition (U4) , and independently of (U6) , is established in Proposition 9.6. F or Λ P D ` p,q | n , we write Λ “ p λ 1 , . . . , λ m , ω 1 , . . . , ω n q and recall that L p Λ q is the unique simple quotien t of the highest-weigh t gl m | n -mo dule V p Λ q defined in (3.6), where, as usual, m “ p ` q . F or Λ P P ` p,q | n and an y real n um b er s P r 0 , 1 s , it is con v enient to introduce Λ p s `q : “ p λ 1 , . . . , λ p , λ p ` 1 ` s, . . . , λ m ` s, ω 1 , . . . , ω n q , CLASSIFICA TION OF UNIT AR Y MODULES 21 Λ p s ´q : “ p λ 1 ´ s, . . . , λ p ´ s, λ p ` 1 , . . . , λ m , ω 1 , . . . , ω n q . Prop osition 9.1. L et Λ P D ` p,q | n such that p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0 ă p Λ ` ρ, ϵ m ´ δ n q . (9.1) Then, L p Λ q is unitary. Pr o of. By Lemma 5.1, the k -mo dule L 0 p Λ q is type-1 unitary , so according to Prop osi- tion 6.3, it suffices to sho w that p Λ ´ ξ , Λ ` ξ ` 2 ρ q ď 0 for all ξ P Π k p Λ q . As ev ery ξ P Π k p Λ q is of the form ξ “ Λ ´ θ with θ “ p ÿ i “ 1 θ i , θ i “ m ÿ k “ p ` 1 a ik p ϵ i ´ ϵ k q ` n ÿ µ “ 1 b iµ p ϵ i ´ δ µ q , (9.2) where a ik P Z ` and b iµ P t 0 , 1 u , it follo ws that p Λ ´ ξ , Λ ` ξ ` 2 ρ q “ 2 p Λ ` ρ, θ q ´ p θ , θ q . (9.3) W e now turn to estimating p Λ ` ρ, θ q and p θ , θ q . F or each i P t 1 , . . . , p u , w e ha ve p Λ ` ρ, θ i q “ m ÿ k “ p ` 1 a ik p Λ ` ρ, ϵ i ´ ϵ k q ` n ÿ µ “ 1 b iµ p Λ ` ρ, ϵ i ´ δ µ q , p θ i , θ i q “ m ÿ k,ℓ “ p ` 1 p a ik a iℓ ` δ kℓ a ik a iℓ q ` m ÿ k “ p ` 1 n ÿ ν “ 1 a ik b iν ` m ÿ ℓ “ p ` 1 n ÿ µ “ 1 b iµ a iℓ ` n ÿ µ,ν “ 1 µ ‰ ν b iµ b iν , where p Λ ` ρ, ϵ i ´ ϵ k q “ p Λ ` ρ, ϵ i ´ ϵ 1 q ` p Λ ` ρ, ϵ 1 ´ δ 1 q ` p Λ ` ρ, δ 1 ´ δ n q ` p Λ ` ρ, δ n ´ ϵ m q ` p Λ ` ρ, ϵ m ´ ϵ k q , p Λ ` ρ, ϵ i ´ δ µ q “ p Λ ` ρ, ϵ i ´ ϵ 1 q ` p Λ ` ρ, ϵ 1 ´ δ 1 q ` p Λ ` ρ, δ 1 ´ δ µ q , while for all i, j P t 1 , . . . , p u suc h that i ă j , p θ i , θ j q “ m ÿ k “ p ` 1 a ik a j k ´ n ÿ µ “ 1 b iµ b j µ ě ´ n ÿ µ “ 1 b iµ b j µ . Using (3.1), (3.5), (9.1), and p Λ ` ρ, ϵ i ´ ϵ 1 q “ p Λ , ϵ i ´ ϵ 1 q ´ p i ´ 1 q , it follo ws that p Λ ` ρ, θ i q ď ´ n ÿ µ “ 1 b iµ p i ´ 1 q , hence p Λ ` ρ, θ q “ p ÿ i “ 1 p Λ ` ρ, θ i q ď ´ p ÿ i “ 1 n ÿ µ “ 1 b iµ p i ´ 1 q . (9.4) As p θ i , θ i q is seen to b e non-negative, we also hav e p θ , θ q “ p ÿ i “ 1 p θ i , θ i q ` 2 ÿ 1 ď i ă j ď p p θ i , θ j q ě ´ 2 ÿ 1 ď i ă j ď p n ÿ µ “ 1 b iµ b j µ . (9.5) By com bining (9.3), (9.4), and (9.5), it follows that p Λ ´ ξ , Λ ` ξ ` 2 ρ q ď 2 p ÿ j “ 2 n ÿ µ “ 1 b j µ ˆ j ´ 1 ÿ i “ 1 b iµ ´ j ` 1 ˙ . 22 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG As b iµ P t 0 , 1 u , the expression in brac kets is non-p ositive for j ě 2, so p Λ ´ ξ , Λ ` ξ ` 2 ρ q ď 0 and the unitarit y of L p Λ q follo ws from Prop osition 6.3. □ Prop osition 9.2. L et Λ P D ` p,q | n such that p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0 and p Λ ` ρ, ϵ m ´ δ µ q “ p Λ , δ µ ´ δ n q “ 0 for some µ P t 2 , . . . , n u . Then, L p Λ q is unitary. Pr o of. The pro of is the same as that of Prop osition 9.1, except the justification for p Λ ` ρ, ϵ i ´ ϵ k q ă 0. W e now hav e p Λ ` ρ, ϵ i ´ ϵ k q “ p Λ ` ρ, ϵ i ´ ϵ 1 q ` p Λ ` ρ, ϵ 1 ´ δ 1 q ` p Λ ` ρ, δ 1 ´ ϵ m q ` p Λ ` ρ, ϵ m ´ ϵ k q . W e analyse each term as follows. By assumption, p Λ ` ρ, ϵ 1 ´ δ 1 q ă 0, and b y (3.1) and (3.5), p Λ ` ρ, ϵ i ´ ϵ 1 q ď 0 , p Λ ` ρ, ϵ m ´ ϵ k q ď 0 . The condition p Λ ` ρ, ϵ m ´ δ µ q “ 0 is equiv alent to λ m ` ω µ “ µ ´ 1. Consequen tly , p Λ ` ρ, δ 1 ´ ϵ m q “ ´ ω 1 ´ λ m “ ´ ω 1 ´ p µ ´ 1 ´ ω n q ă 0 , so p Λ ` ρ, ϵ i ´ ϵ k q ă 0. □ Lemma 9.3. L et Λ P P ` p,q | n . (1) If λ m ` ω n ě n ´ 1 and ther e exists i P t 1 , . . . , p u such that λ 1 “ ¨ ¨ ¨ “ λ i , λ 1 ` ω 1 “ i ´ m, then L p Λ p s `q q is unitary for every s P r 0 , 1 s . (2) If ω 1 “ ¨ ¨ ¨ “ ω n , λ m ` ω n “ 0 , and ther e exists j P t p, . . . , m ´ 1 u such that λ j ` 1 “ ¨ ¨ ¨ “ λ m , λ 1 ` ω 1 ď 1 ´ j, then L p Λ p s ´q q is unitary for every s P r 0 , 1 s . Pr o of. F or con venience, let Λ p s q denote Λ p s `q or Λ p s ´q . Recall from (3.6) that V p Λ p s q q “ U p k ´ q b V 0 p Λ p s q q , and that its unique simple quotien t is denoted by L p Λ p s q q . According to Lemma 6.1, the k -in v ariant Γ acts on eac h simple k -submo dule L p,q | n p ξ p s q q of V p Λ p s q q as multiplication by the scalar γ s “ 1 2 p Λ p s q ´ ξ p s q , Λ p s q ` ξ p s q ` 2 ρ q . Note that ξ p s q “ Λ p s q ´ θ , where θ is of the form (9.2). It follo ws that γ s “ p Λ p s q ` ρ, θ q ´ p θ , θ q . Th us, γ s is an affine linear function of s ; that is, γ s “ as ` b for some constan ts a, b P R . By Lemma 5.1, the k -mo dule V 0 p Λ p s q q is t yp e-1 unitary for every s P r 0 , 1 s . If s equals 0 or 1, then Λ p s q is an in tegral highest weigh t and Theorem 8.1 implies that L p Λ p s q q is unitary . In this case, b y Prop osition 6.3, γ s ď 0. Since γ s dep ends linearly on s , this means that γ s ď 0 for all s P r 0 , 1 s . Another application of Prop osition 6.3 yields the unitarit y of L p Λ p s q q for all s P r 0 , 1 s . □ Prop osition 9.4. L et Λ P D ` p,q | n such that p Λ ` ρ, ϵ m ´ δ n q ą 0 , and supp ose ther e exists i P t 1 , . . . , p u such that p Λ ` ρ, ϵ i ´ δ 1 q “ p Λ , ϵ i ´ ϵ 1 q “ 0 . Then, L p Λ q is unitary. CLASSIFICA TION OF UNIT AR Y MODULES 23 Pr o of. The assumptions imply that λ m ` ω n ą n ´ 1 and that there exists i P t 1 , . . . , p u suc h that λ 1 “ ¨ ¨ ¨ “ λ i and λ i ` ω 1 “ i ´ m . Clearly , ω µ ´ ω n P Z ` for µ P t 1 , . . . , n ´ 1 u . Also, for eac h k P t 1 , . . . , p u , λ k ` ω n “ p λ k ´ λ i q ` p λ i ` ω 1 q ` p ω 1 ´ ω n q P Z ´ , while for eac h k P t p ` 1 , . . . , m u , λ k ` ω n “ λ k ´ λ m ` λ ` ω n ą n ´ 1 . Let Υ : “ p λ 1 ` ω n , . . . , λ m ` ω n , ω 1 ´ ω n , . . . , ω n ´ 1 ´ ω n , 0 q , (9.6) and set s “ λ n ` ω n ´ t λ n ` ω n u . Then, r Υ : “ p λ 1 ` ω n , . . . , λ p ` ω n , λ p ` 1 ` ω n ´ s, . . . , λ m ` ω n ´ s, ω 1 ´ ω n , . . . , ω n ´ 1 ´ ω n , 0 q is an integral w eigh t satisfying r Υ p s `q “ Υ, so b y Lemma 9.3, the simple gl m | n -mo dule L p Υ q is unitary . It follows that L p Λ q – L p Υ q b C ´ ω n is unitary . □ Prop osition 9.5. L et Λ P D ` p,q | n such that p Λ , ϵ 1 ´ δ 1 q ă 1 ´ j , p Λ , ϵ j ` 1 ´ ϵ m q “ p Λ ` ρ, ϵ m ´ δ 1 q “ p Λ , δ 1 ´ δ n q “ 0 , for some j P t p, . . . , m ´ 1 u . Then, L p Λ q is unitary. Pr o of. F or eac h k P t 1 , . . . , p u , λ k ` ω 1 “ λ k ´ λ 1 ` λ 1 ` ω 1 ă 1 ´ j while for eac h k P t p ` 1 , . . . , j u , λ k ` ω 1 “ λ k ´ λ m ` λ m ` ω 1 “ λ k ´ λ m P Z ` . Let Υ : “ p λ 1 ` ω 1 , . . . , λ j ` ω 1 , 0 , . . . , 0 q , (9.7) and set s “ t λ 1 ` ω 1 u ´ p λ 1 ` ω 1 q . Then, r Υ : “ p λ 1 ` ω 1 ` s, . . . , λ p ` ω 1 ` s, λ s ` ω 1 , . . . , λ j ` ω 1 , 0 , . . . , 0 q is an integral w eigh t satisfying r Υ p s ´q “ Υ, so b y Lemma 9.3, the simple gl m | n -mo dule L p Υ q is unitary . It follows that L p Λ q – L p Υ q b C ´ ω 1 is unitary . □ Prop osition 9.6. L et Λ P D ` p,q | n , and supp ose ther e exist µ P t 2 , . . . , n u and i P t 1 , . . . , p u such that p Λ ` ρ, ϵ m ´ δ µ q “ p Λ , δ µ ´ δ n q “ p Λ ` ρ, ϵ i ´ δ 1 q “ p Λ , ϵ i ´ ϵ 1 q “ 0 , or ther e exist j P t p, . . . , m ´ 1 u and i P t 1 , . . . , p u such that p Λ , ϵ j ` 1 ´ ϵ m q “ p Λ ` ρ, ϵ m ´ δ 1 q “ p Λ , δ 1 ´ δ n q “ p Λ , ϵ i ´ ϵ 1 q “ 0 , p Λ , ϵ i ´ δ 1 q “ i ´ j . Then, L p Λ q is unitary. Pr o of. If Λ satisfies the first c hain of equalities in the prop osition, then there exists µ P t 2 , . . . , n u such that ω µ “ ¨ ¨ ¨ “ ω n and λ m ` ω n “ µ ´ 1. Additionally , there is i P t 1 , . . . , p u such that λ 1 “ ¨ ¨ ¨ “ λ i and λ 1 ` ω 1 “ i ´ m . It is straightforw ard to v erify that Υ (9.6) lies in P ` p,q | n and satisfies condition (1) of Theorem 8.1. Consequently , the mo dule L p Υ q is unitary . Since L p Λ q – L p Υ q b C ´ ω n , it follows that L p Λ q is also unitary . Similarly , the w eight (9.7) Υ P P ` p,q | n satisfies condition (2) of Theorem 8.1. Th us, L p Υ q is unitary , and so is L p Λ q – L p Υ q b C ´ ω 1 . □ 24 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG 10. Fur ther classifica tions of unit ar y modules Using the classification of unitary gl p ` q | n -mo dules in Theorem 4.2, we will classify dual unitary gl p ` q | n -mo dules with resp ect to the dual star-operation ✩ , as w ell as unitary gl n | q ` p -mo dules. 10.1. Dual unitary mo dules. Recall from Section 3.2 that the dual star-op eration ✩ is defined b y p E ab q ✩ “ p´ 1 q r a s`r b s p E ab q ‹ , a, b P t 1 , . . . , m ` n u . A gl p ` q | n -mo dule is dual unitary if it carries a p ositive-definite Hermitian form satisfying (3.3) with ‹ replaced with ✩ . W e consider only admissible mo dules, i.e., those whose restriction to gl p ‘ gl q | n decom- p oses as a direct sum of finite-dimensional simple mo dules with finite multiplicities. All suc h mo dules are w eigh t mo dules. W e hav e the follo wing analogues of Lemma 3.1 and Prop osition 3.2; the pro ofs are similar. Lemma 10.1. L et V b e an admissible dual unitary gl p ` q | n -mo dule, and let λ “ ř p ` q i “ 1 λ i ϵ i ` ř n µ “ 1 ω µ δ µ b e any weight of V . Then λ is r e al, and λ j ď ´ ω µ ď λ i , for al l i P t 1 , . . . , p u , j P t p ` 1 , . . . , p ` q u , and µ P t 1 , . . . , n u . Prop osition 10.2. L et V b e an admissible dual unitary simple gl p ` q | n -mo dule. Then V is a lowest-weight mo dule with lowest weight Λ “ ř p ` q i “ 1 λ i ϵ i ` ř n µ “ 1 ω µ δ µ satisfying λ p ` 1 ď ¨ ¨ ¨ ď λ m ď ´ ω n ď ¨ ¨ ¨ ď ´ ω 1 ď λ 1 ď ¨ ¨ ¨ ď λ p . Indeed, these t wo kinds of unitary mo dules are related b y duality , as in the finite- dimensional case [GZ90a]. Let L p Λ q “ À k P Z ` L k p Λ q b e a unitary simple gl p ` q | n -mo dule with highest w eigh t Λ, the graded dual is defined b y L ˚ p Λ q : “ à k P Z ` L ˚ k p Λ q . Prop osition 10.3. L et Λ P D ` p ` q | n . Then, the simple highest-weight gl p ` q | n -mo dule L p Λ q is unitary if and only if the simple lowest-weight mo dule L ˚ p Λ q is dual unitary with lowest weight Λ ˚ “ ´ Λ . Pr o of. Since taking duals negates the weigh ts, the low est weigh t of L ˚ p Λ q is Λ ˚ “ ´ Λ. The result no w follo ws from Prop osition 2.1. □ The follo wing classification is a consequence of Theorem 4.2 and Proposition 10.3. Theorem 10.4. The simple gl p ` q | n -mo dule L p Υ q is dual unitary if and only if the lowest weight Υ satisfies one of the c onditions (U1) – (U6) with al l Λ r eplac e d by ´ Υ . 10.2. Unitary mo dules o v er gl n | q ` p . F or later use, it is con v enient to introduce some notation for gl n | q ` p . W e thus set ˜ I n | q ` p : “ ˜ I n Y ˜ I q Y ˜ I p , where ˜ I n : “ t 1 , . . . , n u , ˜ I q : “ t n ` 1 , . . . , n ` q u , ˜ I p : “ t n ` q ` 1 , . . . , m ` n u , noting that the indices in ˜ I n are ev en, while those in ˜ I q and ˜ I p are o dd. F or a, b P ˜ I n | q ` p , w e let r E ab denote the matrix unit of gl n | q ` p , so that r r E ab s “ r a s ` r b s . W e also set ˜ a : “ m ` n ` 1 ´ a, a P ˜ I n | q ` p . CLASSIFICA TION OF UNIT AR Y MODULES 25 The standard Borel subalgebra of gl n | q ` p is spanned by the elemen ts r E ab with a ď b , and its Cartan subalgebra is spanned b y the diagonal elemen ts r E aa , where a P ˜ I n | q ` p . Note that there is an ev en isomorphism of Lie sup eralgebras τ : gl n | q ` p Ñ gl p ` q | n , τ p r E ab q “ E ˜ a, ˜ b , a, b P ˜ I n | q ` p . W e further define a sign function ˜ s : ˜ I n | q ` p Ñ t˘ 1 u b y ˜ s p a q : “ # 1 , a P ˜ I n Y ˜ I q , ´ 1 , a P ˜ I p . Using this, w e define a star-op eration ‹ on gl n | q ` p b y p r E ab q ‹ : “ ˜ s p a q ˜ s p b q r E ba , a, b P ˜ I n | q ` p . The corresp onding dual star-op eration ✩ is giv en by p r E ab q ✩ : “ p´ 1 q r a s`r b s p r E ab q ‹ , a, b P ˜ I n | q ` p . Lemma 10.5. The isomorphism τ : gl n | q ` p Ñ gl p ` q | n pr eserves the star-op er ations in the sense that, for al l a, b P ˜ I n | q ` p , τ ` p r E ab q ‹ ˘ “ τ p r E ab q ‹ , τ ` p r E ab q ✩ ˘ “ τ p r E ab q ✩ . Pr o of. F or the star-op eration ‹ , we hav e τ pp r E ab q ‹ q “ ˜ s p a q ˜ s p b q E ˜ b, ˜ a “ τ p r E ab q ‹ . The relation for the dual star-op eration follows similarly , using that r ˜ a s “ r a s ` ¯ 1. □ W e are only concerned with admissible mo dules whose restriction to gl n | q ‘ gl p decom- p oses into a direct sum of finite-dimensional simple mo dules with finite multiplicities. It is straigh tforw ard to establish the follo wing result. Prop osition 10.6. L et V b e an admissible simple gl n | q ` p -mo dule. (1) If V is unitary, then V is a lowest-weight mo dule with lowest weight Λ “ ř n µ “ 1 ω µ ϵ µ ` ř m i “ 1 λ i δ i satisfying λ q ` 1 ď ¨ ¨ ¨ ď λ m ď ´ ω n ď ¨ ¨ ¨ ď ´ ω 1 ď λ 1 ď ¨ ¨ ¨ ď λ q . (2) If V is dual unitary, then V is a highest-weight mo dule with highest weight Λ “ ř n µ “ 1 ω µ ϵ µ ` ř m i “ 1 λ i δ i satisfying λ q ď ¨ ¨ ¨ ď λ 1 ď ´ ω 1 ď ¨ ¨ ¨ ď ´ ω n ď λ m ď ¨ ¨ ¨ ď λ q ` 1 . As a consequence of Lemma 10.5, there is a bijectiv e corresp ondence b etw een uni- tary mo dules o v er gl n | q ` p and gl p ` q | n . More precisely , let L p Λ q b e a unitary simple gl p ` q | n -mo dule with highest weigh t Λ “ p λ 1 , . . . , λ p , λ p ` 1 , . . . , λ m , ω 1 , . . . , ω n q , and let π : gl p ` q | n Ñ End C p L p Λ qq denote the corresp onding linear representation. Then, L p Λ q , when view ed as a gl n | q ` p -mo dule via π ˝ τ , is a unitary simple gl n | q ` p -mo dule with low est w eigh t Λ τ : “ p ω n , . . . , ω 1 , λ m , . . . , λ p ` 1 , λ p , . . . , λ 1 q . Indeed, one can verify that Λ τ satisfies the low est-w eigh t conditions in part (1) of Prop o- sition 10.6. Therefore, exc hanging the even and o dd indices interc hanges highest-w eight unitary gl p ` q | n -mo dules and lo w est-weigh t unitary gl n | q ` p -mo dules. A similar corresp on- dence holds for dual unitary mo dules. 26 MARK D. GOULD, AR TEM PULEMOTO V, JØRGEN RASMUSSEN, Y ANG ZHANG Theorem 10.7. The simple gl n | q ` p -mo dule L p Υ q is unitary if and only if the lowest weight Υ “ p ω 1 , . . . , ω n , λ 1 , . . . , λ q , λ q ` 1 , . . . , λ m q satisfies one of the c onditions (U1) – (U6) with al l Λ r eplac e d by Υ τ : “ p λ m , . . . , λ q ` 1 , λ q , . . . , λ 1 , ω n , . . . , ω 1 q . Similarly , we hav e the follo wing classification of dual unitary simple gl n | q ` p -mo dules. This follows from the fact that taking dual interc hanges unitary and dual unitary gl n | q ` p - mo dules. Theorem 10.8. The simple gl n | q ` p -mo dule L p Υ q is dual unitary if and only if the highest weight Υ “ p ω 1 , . . . , ω n , λ 1 , . . . , λ q , λ q ` 1 , . . . , λ m q satisfies one of the c onditions (U1) – (U6) with al l Λ r eplac e d by ´ Υ τ : “ p´ λ m , . . . , ´ λ q ` 1 , ´ λ q , . . . , ´ λ 1 , ´ ω n , . . . , ´ ω 1 q . 10.3. Unitary mo dules o v er gl p ` q | r ` s . Let m “ p ` q and n “ r ` s . It is w ell-kno wn that, if pq ‰ 0 and r s ‰ 0, then the real form su p p, q | r, s q admits only trivial unitary simple mo dules; see [GV19, Prop osition 1] and [FN91, Lemma 2.1]. F or u p p, q | r , s q , the only unitary simple mo dules are 1-dimensional; see Prop osition 10.9 b elo w. In preparation, w e define I p ` q | r ` s : “ I p Y I q Y I r Y I s , where I p : “ t 1 , . . . , p u , I q : “ t p ` 1 , . . . , p ` q u , I r : “ t m ` 1 , . . . , m ` r u , I s : “ t m ` r ` 1 , . . . , m ` n u , noting that I p and I q are ev en, while I r and I s are o dd. W e also define the function s : I p ` q | r ` s Ñ t˘ 1 u b y s p a q “ # 1 , a P I p Y I s , ´ 1 , a P I q Y I r . W e let the star-op eration ‹ and its dual star-op eration ✩ on gl p ` q | r ` s act as p E ab q ‹ “ s p a q s p b q E ba , p E ab q ✩ “ p´ 1 q r a s`r b s s p a q s p b q E ba , where a, b P I p ` q | r ` s . Accordingly , we ha ve unitary gl p ` q | r ` s -mo dules and dual unitary gl p ` q | r ` s -mo dules. Such a mo dule is said to b e admissible if its restriction to the subal- gebra gl p ‘ gl q | r ‘ gl s decomp oses in to a direct sum of finite-dimensional simple mo dules with finite m ultiplicities. All such mo dules are w eigh t mo dules. Prop osition 10.9. If pq ‰ 0 and r s ‰ 0 , the only admissible unitary or dual unitary simple gl p ` q | r ` s -mo dules ar e the 1 -dimensional mo dules. Pr o of. Let V b e an admissible unitary simple gl p ` q | r ` s -mo dule, and let v P V b e a weigh t v ector with w eight given by λ “ p ÿ i “ 1 λ i ϵ i ` p ` q ÿ j “ p ` 1 λ j ϵ j ` r ÿ µ “ 1 ω µ δ µ ` r ` s ÿ ν “ r ` 1 ω ν δ ν . F or i P t 1 , . . . , p u and µ P t 1 , . . . , r u , w e hav e p λ i ` ω µ qx v , v y “ xr E i,m ` µ , E m ` µ,i s v , v y “ x E i,m ` µ E m ` µ,i v , v y ` x E m ` µ,i E i,m ` µ v , v y “ ´x E m ` µ,i v , E m ` µ,i v y ´ x E i,m ` µ v , E i,m ` µ v y ď 0 . It follo ws that λ i ` ω µ ď 0 for all relev ant i, µ . Similarly , w e hav e λ i ` ω ν ě 0 , λ j ` ω µ ě 0 , λ j ` ω ν ď 0 . CLASSIFICA TION OF UNIT AR Y MODULES 27 Com bining these inequalities, w e obtain λ i ď ´ ω µ ď λ j ď ´ ω ν ď λ i , forcing the inequalities to b e equalities. Hence, all w eight spaces of V are 1-dimensional. Since V is simple, it m ust b e 1-dimensional. The same holds for the dual unitary simple mo dules. □ References [CLZ04] S.J. Cheng, N. Lam, R.B. Zhang, Char acter formula for infinite-dimensional unitarizable mo dules of the gener al line ar sup er algebr a , J. Algebra 273 (2004) 780–805. [CW01] S.J. Cheng, W. W ang, Howe duality for Lie sup er algebr as , Compositio Math. 128 (2001) 55–94. [DF04] E. D’Hok er, D.Z. F reedman, Sup ersymmetric gauge the ories and the ADS/CFT c orr esp on- denc e , W orld Scientific Publishing Co., Inc., Riv er Edge, NJ, 2004. [Din07] M. Dine, Sup ersymmetry and String the ory: Beyond the Standar d Mo del , Cambridge Univer- sit y Press, Cambridge, 2007. [EHW83] T. Enright, R. 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Schmidt, On the ful l set of unitarizable sup ermo dules over sl p m | n q , [math.R T]. [Ser01] A. Sergeev, An analo g of the classic al invariant the ory for Lie sup er algebr as, I , Michigan Math. J. 49 (2001) 113–146. School of Ma thema tics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia Email addr ess : m.gould1 @ uq.edu.au Email addr ess : a.pulemotov @ uq.edu.au Email addr ess : j.rasmussen @ uq.edu.au Email addr ess : yang.zhang @ uq.edu.au