Bernstein Gel'fand Gel'fand reciprocity and indecomposable projective modules for classical algebraic supergroups

Bernstein-Gel'fand-Gel'fand reciprocity and indecomposable projective modules for classical algebraic supergroups Caroline Gruson 1 and Vera Serganova 2 To the Memory of I. M. Gelfand Abstract: We prove a BGG type reciprocity law for the category of finite dimensional modules over algebraic supergroups satisfying certain conditions. The equivalent of a standard module in this case is a virtual module called Euler characteristic due to its geometric interpretation. In the orthosymplectic case, we also describe indecomposable projective modules in terms of those Euler characteristics. Key words: Finite dimensional representations of algebraic supergroups, Flag variety, BGG reciprocity law. Introduction In many representation theories, there exist reciprocity laws. Roughly speaking, if the category in question has enough projective modules, one defines in a natural way a family of so-called standard modules such that every projective indecomposable module has a filtration with standard quotients. The reciprocity law states that the multiplicity of a standard module in the projective cover of a simple module equals the multiplicity of this simple module in the standard module. Those standard modules are usually easy to describe, in particular, their characters are given by simple formulae. For instance, Brauer discovered such a law in the case of finite groups representations in positive characteristic, [5]. Another example is a result of Humphreys, [13], for repre- sentations of semi-simple Lie algebras in positive characteristic.

• gel'fand


• lie superalgebras


• projectives inside


• h? ?


• all finite-dimensional


• finite dimensional