Invariant generalized functions supported on an orbit

TL;DR

This paper investigates the structure of invariant generalized functions supported on orbits under algebraic group actions, introducing methods to compute their dimension generating functions and illustrating these with the example of ext{GL}_3( ext{C}).

Contribution

It introduces a novel approach using algebraic stacks to analyze invariant generalized functions and provides explicit computations for specific group actions.

Findings

Derived formulas for the generating functions of the Bruhat filtration dimensions.

Applied the methods to compute these functions for the adjoint action of ext{GL}_3( ext{C}).

Showed the effectiveness of algebraic stacks in this context.

Abstract

We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of $\mathrm{GL}_3(\mathbb{C})$. Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently by Sakellaridis.