BOTT–BOREL–WEIL THEORY AND BERNSTEIN–GEL’FAND–GEL’FAND RECIPROCITY FOR LIE SUPERALGEBRAS

Abstract

The main focus of this paper is Bott–Borel–Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova becomes an actual reciprocity in the relatively generic region. We also obtain a complete solution of BBW theory for $$ \mathfrak{o}\mathfrak{s}\mathfrak{p} $$ (m|2), D(2, 1; α), F(4) and G(3) with distinguished Borel subalgebra. Furthermore, we derive information about the category of finite-dimensional $$ \mathfrak{o}\mathfrak{s}\mathfrak{p} $$ (m|2)-modules, such as BGG-type resolutions and Kostant homology of Kac modules and the structure of projective modules.