Pro-reductive groups attached to irreducible representations of the General Linear Supergroup

Abstract

We study tensor product decompositions of representations of the General Linear Supergroup Gl(m|n). We show that the quotient of Rep(Gl(m|n),\epsilon)$ by the tensor ideal of negligible representations is the representation category of a pro-reductive supergroup G red. In the Gl(m|1)-case we show G red = Gl(m-1) \times Gl(1) \times Gl(1). In the general case we study the image of the canonical tensor functor Fmn from Deligne's interpolating category Rep (Gl m-n) to Rep(Gl(m|n),\epsilon). We determine the image of indecomposable elements under Fmn. This implies tensor product decompositions between projective modules and between certain irreducible modules, including all irreducible representations in the Gl(m|1)-case. Using techniques from Deligne's category we derive a closed formula for the tensor product of two maximally atypical irreducible Gl(2|2)-representations. We study cohomological tensor functors DS : Rep(Gl(m|m), epsilon) -> Rep(Gl(m-1|m-1)) and describe the image of an irreducible element under DS. At the end we explain how these results can be used to determine the pro-reductive group G L \hookrightarrow Gl(m|m) red corresponding to the subcategory Rep(G L, epsilon) generated by the image of an irreducible element L in Rep(Gl(m|m) red, epsilon).