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

TL;DR

This paper introduces a geometric framework for analyzing (g, k)-modules of finite type, classifies K-spherical partial flag varieties, and characterizes bounded modules via sphericity, revealing finiteness and structural properties.

Contribution

It establishes a finite set of invariants for simple (g, k)-modules, classifies K-spherical partial flag varieties, and characterizes bounded modules through geometric sphericity.

Findings

Finite set of invariants V(M), 𝓥(M), L(M) for simple (g, k)-modules.

Classification of all K-spherical partial W-flag varieties.

Characterization of bounded modules via K-sphericity of associated varieties.

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.