A geometric approach to (g, k)-modules of finite type

Abstract

Let $g$ be a semisimple Lie algebra over $\mathbb{C}$ and $k$ be a reductive in $g$ subalgebra. We say that a simple $g$-module $M$ is a $(g;k)$-module if as a $k$-module $M$ is a direct sum of finite-dimensional $k$-modules. We say that a simple $(g;k)$-module $M$ is of finite type if all $k$-isotypic components of $M$ are finite-dimensional. To a simple $g$-module $M$ one assigns interesting invariants V$(M)$, $\EuScript V(M)$ and L$(M)$ reflecting the 'directions of growth of M'. In this work we prove that, for a given pair $(g;k)$, the set of possible such invariants is finite. Let $K$ be a reductive Lie group with Lie algebra $k$. We say that a $K$-variety $X$ is $K$-spherical if $X$ has an open orbit of a Borel subgroup of $K$. Let $W$ be a finite-dimensional $K$-module. The set of flags ($W_1,...,W_s)$ of $W$ with fixed dimensions $(n_1;...;ns)$ is a homogeneous space of the group GL(W). We call such a variety partial $W$-flag variety. In this work we classify all $K$-spherical partial $W$-flag varieties. We say that a simple $(g;k)$-module is bounded if there exists constant C${}_M$ such that, for any simple $k$-module $E$, the isotypic component of $E$ in $M$ is a direct sum of not more than C${}_M$-copies of $E$. To any simple sl$(W)$-module one assigns a partial $W$-flag variety. In this thesis we prove that a simple (sl$(W);k$)-module is bounded if and only if the corresponding partial $W$-flag variety is $K$-spherical. Moreover, we prove that the pair (sl$(W);k$) admits an infin? ite-dimensional simple bounded module if and only if P$(W)$ is a $K$-spherical variety. For four particular case we say more about category of bounded modules and the set of simple bounded modules.

arXiv commentPh.D. thesis, 73 pages, peacewise published

Cited by in corpus (1)

• On annihilators of bounded $(\mathfrak{g},\mathfrak{k})$-modules