On the category of finite-dimensional representations of OSp(r2n)(r|2n): Part I

  • Michael Ehrig

    Universität Bonn, Germany
  • Catharina Stroppel

    Universität Bonn, Germany
On the category of finite-dimensional representations of OSp$(r|2n)$: Part I cover

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We study the combinatorics of the category F\mathcal F of fi nite-dimensional modules for the orthosymplectic Lie supergroup OSp(r2n)(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\mathcal B we construct an algebra ABA_\mathcal B whose module category shares the combinatorics with B\mathcal B. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It turns out that ABA_\mathcal B is isomorphic to the endomorphism algebra of a minimal projective generator of B\mathcal B. This provides a direct link from F\mathcal F to parabolic categories O\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 F\mathcal F is not highest weight in general.