On the category of finite-dimensional representations of OSp$(r|2n)$: Part I

Abstract

We study the combinatorics of the category $\mathcal{F}$ of finite-dimensional modules for the orthosymplectic Lie supergroup OSp$(r|2n)$. In particular we present a positive counting formula for the dimension of the space of homomorphisms between two projective modules. This refines earlier results of Gruson and Serganova. For each block $\mathcal{B}$ we construct an algebra $A_{\mathcal{B}}$ whose module category shares the combinatorics with $\mathcal{B}$. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It turns out that $A_{\mathcal{B}}$ is isomorphic to the endomorphism algebra of a minimal projective generator of $\mathcal{B}$. This provides a direct link from $\mathcal{F}$ to parabolic categories $\mathcal{O}$ of type B/D, with maximal parabolic of type A, to the geometry of isotropic Grassmannians of types B/D and to Springer fibres of type C/D. We also indicate why $\mathcal{F}$ is not highest weight in general.