Support varieties of $(\mathfrak{g},\mathfrak{k})$-modules of finite type

Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed field of characteristic 0 and $\mathfrak{k}$ be a reductive in $\mathfrak{g}$-subalgebra. Let $M$ be a finitely generated (possibly, infinite-dimensional) $\mathfrak{g}$-module. We say that $M$ is a $(\mathfrak{g},\mathfrak{k})$-module if $M$ is a direct sum of a (possibly, infinite) amount of simple finite-dimensional $\mathfrak{k}$-modules. We say that $M$ is of finite type if $M$ is a $(\mathfrak{g},\mathfrak{k})$-module and Hom${}_{\mathfrak{k}}(V,M)<\infty$ for any simple $\mathfrak{k}$-module $V$. Let $X$ be a variety of all Borel subalgebras of $\mathfrak{g}$. Let $M$ be a finitely generated $(\mathfrak{g},\mathfrak{k})$-module of finite type. In this article we prove that $M$ is holonomic, i.e. $M$ is governed by some subvariety $L_M\subset X$ and some local system $S_M$ on it. Furthermore we provide a finite list in which L${}_M$ necessarily appear.

The notion of $(\mathfrak{g},\mathfrak{k})$-module have been introduced by I. Penkov, V. Serganova, G. Zuckerman. Main ingredients are Hilbert-Mumford Criterion, A. Beilinson- J. Bernstein localization theorem, O. Gabber theorem