Kostant Modules in Blocks of Category 𝒪_S

Abstract

In this article, the authors investigate infinite-dimensional representations L in blocks of the relative (parabolic) category 𝒪 _S for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in L “looks like” the cohomology with coefficients in a finite-dimensional module, as in Kostant's theorem. A complete classification of these “Kostant modules” in regular blocks for maximal parabolics in the simply laced types is given. A complete classification is also given in arbitrary (singular) blocks for Hermitian symmetric categories.

Key Words:

• BGG resolution
• Hermitian symmetric pair
• Kostant module
• Minimal free resolution
• Parabolic category 𝒪
• Rationally smooth Schubert variety

2000 Mathematics Subject Classification:

• Primary 17B10
• Secondary 17B56, 14M15

ACKNOWLEDGMENTS

The authors thank Daniel Nakano and Jonathan Kujawa for helpful discussions, and the referee for numerous useful comments. Research of the first author was partially supported by NSA grant H98230-04-1-0103.

Notes

Communicated by D. K. Nakano.