On the category of finite-dimensional representations of $\OSPrn$: Part I

Michael Ehrig, Catharina Stroppel

Published 2016-07-14Version 1

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 homomorphism between two projective modules. This refines earlier results of Gruson and Serganova. Moreover, for each block B of F 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 will turn 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 or D, with maximal parabolic of type A, to the geometry of isotropic Grassmannians of types B/D and to Springer fibres of types C/D. We also indicate why F is not highest weight in general.