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

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

2011-01-03

arxiv.org/abs/1101.0472v1

Mathematics

Representation Theory

The notion of $(\frak g, \frak k)$-module have been introduced by I. Penkov, V. Serganova, G. Zuckerman. Main ingredients are

Scientific paper

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.

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