Support varieties of $(\frak g, \frak k)$-modules of finite type

TL;DR

This paper studies $(rak g, rak k)$-modules of finite type over a reductive Lie algebra, proving they are holonomic and characterized by specific subvarieties and local systems, with a finite classification list.

Contribution

It establishes that finitely generated $(rak g, rak k)$-modules of finite type are holonomic and describes their governing subvarieties and local systems, providing a finite classification list.

Findings

$(rak g, rak k)$-modules of finite type are holonomic.

Modules are governed by subvarieties and local systems.

A finite list of possible governing subvarieties is provided.

Abstract

Let $\frak g$ be a reductive Lie algebra over an algebraically closed field of characteristic 0 and $\frak k$ be a reductive in $\frak g$-subalgebra. Let $M$ be a finitely generated (possibly, infinite-dimensional) $\frak g$-module. We say that $M$ is a $(\frak g, \frak k)$-module if $M$ is a direct sum of a (possibly, infinite) amount of simple finite-dimensional $\frak k$-modules. We say that $M$ is of finite type if $M$ is a $(\frak g, \frak k)$-module and Hom$_\frak k(V, M)<\infty$ for any simple $\frak k$-module $V$. Let $X$ be a variety of all Borel subalgebras of $\frak g$. Let $M$ be a finitely generated $(\frak g, \frak k)$-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.