Bernstein–Gelfand–Gelfand Reciprocity and Indecomposable Projective Modules for Classical Algebraic Supergroups
Moscow Mathematical Journal
We prove a BGG type reciprocity law for the category of finite dimensional modules over algebraic supergroups satisfying certain conditions. The equivalent of a standard module in this case is a virtual module called Euler characteristic due to its geometric interpretation. In the orthosymplectic case, we also describe indecomposable projective modules in terms of those Euler characteristics. 1 2 by a highest weight, and if λ is a highest weight, we denote by L λ the corresponding irreducible
... nding irreducible representation. Every L λ has an indecomposable projective cover P λ in the category of finite-dimensional representations of G,  . However, in this situation there is no direct analogue of the so-called standard modules. Hence we introduce a family of virtual modules E(µ), living in the Grothendieck group of the category: we call those modules Euler characteristics because they come from the cohomology of line bundles on flag supervarieties. It turns out that in the Grothendieck ring, the class [P λ ] of P λ are linear combinations of E(µ)-s and we denote the coefficient of E(µ)i n[ P λ ]b ya(λ, µ). In general the coefficients a(λ, µ) may be negative. The reciprocity law (Theorem 1) states that a(λ, µ) is exactly the multiplicity of L λ in E(µ). The key argument in the proof is a Z/2Z-graded analogue of the Bott reciprocity result, , see Proposition 1. All the constructions above depend on the choice of a Borel subgroup in G:i nt h es u p e r case, this choice is not unique up to conjugation, and the result is true for every possible choice. In particular, in the case of GL(m, n) our result generalizes Zou's result. In this case the modules E(µ) are not virtual -they coincide with the so-called Kac modules (see the example at the end of Section 2). It is worth mentioning that in general the weights λ (labeling L λ and P λ ) and µ (labeling E(µ)) do not belong to the same set. For instance, in the orthosymplectic case (Section 4) the µ-s must have tailless weight diagrams. Finally, let us emphasize on the fact that this category has infinite cohomological dimension and the subgroup generated by [P λ ]-s is a proper subgroup in the whole Grothendieck group. Probably the simplest example of such situation is the category of finite-dimensional representations of the algebra C[z]/(z 2 ) with a unique simple module L and a unique indecomposable projective module P related by [P ] = 2[L] in the Grothendieck group. The rest of the paper deals with the computation of the coefficients a(λ, µ) for the orthosymplectic supergroup SOSP(m, 2n). The first computation of those coefficients in the GL(m, n) case was made in  . In , J. Brundan used another method, relating this representation theory with the one of gl 1 . He interpreted the translation functors for gl(m, n) as linear operators of gl 1 acting on Λ n (V ) ⌦ Λ m (V ⇤ ), where V is the standard representation of gl 1 . Later on, in [6, 7, 8] Brundan and Stroppel introduced weight diagrams, which give a clear picture of the translation functors action. Thus the category of finite dimensional GL(m, n)-modules is very well understood now, including the projective modules. We adopt Brundan's categorification approach. Here we have to separate in two cases depending on the parity of m.I fm is odd, we identify the lattice in the Grothendieck group generated by E(µ)-s with a natural lattice in the tensor representation Λ m (V ⇤ ) ⌦ Λ n (V ) of the infinite-dimensional Lie algebra gl 1/2 with Dynkin diagram As in the case of GL(m, n) certain translation functors correspond to the Chevalley generators of gl 1/2 . However, there is another translation functor, which we call the switch functor which does not have such interpretation. We compute the coefficients a(λ, µ)i n Section 8 (see Theorem 2, Theorem 3 and Theorem 4) via a comparison between the action of the translation functors on P λ -s and E(µ)-s. We start with a typical λ (in this case P λ , E(λ) and L λ coincide) and then obtain an arbitrary P λ by application of translation functors. If m is even, the corresponding infinite-dimensional Lie algebra is gl 1/2 ⊕ gl 1/2 (see Section 7).