Action of Hecke algebra on the double flag variety of type AIII

TL;DR

This paper explicitly describes how the Hecke algebra acts on the space of orbits in a double flag variety of type AIII, revealing its structure through combinatorial graph methods and connecting it to Weyl group representations.

Contribution

It provides a combinatorial description of the Hecke algebra action on the double flag variety of type AIII, including the explicit structure and Weyl group representation decomposition.

Findings

Explicit Hecke algebra action determined via graphs

Description of Weyl group representation as induced representations

Connection between Hecke module structure and combinatorial data

Abstract

Consider a connected reductive algebraic group $ G $ and a symmetric subgroup $ K $. Let $ \mathfrak{X} = K/B_K \times G/P $ be a double flag variety of finite type, where $ B_K $ is a Borel subgroup of $ K $, and $ P $ a parabolic subgroup of $ G $. A general argument shows that the orbit space $ \mathbb{C}\,\mathfrak{X}/K $ inherits a natural action of the Hecke algebra $ \mathscr{H} = \mathscr{H}(K, B_K) $ of double cosets via convolutions. However, to find out the explicit structure of the Hecke module is a quite different problem. In this paper, we determine the explicit action of $ \mathscr{H} $ on $ \mathbb{C}\,\mathfrak{X}/K $ in a combinatorial way using graphs for the double flag variety of type AIII. As a by-product, we also get the description of the representation of the Weyl group on $ \mathbb{C}\,\mathfrak{X}/K $ as a direct sum of induced representations.