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

### Michael Ehrig

Universität Bonn, Germany### Catharina Stroppel

Universität Bonn, Germany

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## Abstract

We study the combinatorics of the category $F$ of finite-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 $B$ we construct an algebra $A_{B}$ whose module category shares the combinatorics with $B$. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It turns out that $A_{B}$ is isomorphic to the endomorphism algebra of a minimal projective generator of $B$. This provides a direct link from $F$ to parabolic categories $O$ 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 $F$ is not highest weight in general.