Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, II

TL;DR

This paper establishes a connection between geometric structures and algebraic ideals in affine Hecke algebras, confirming a conjecture by Ginzburg and advancing understanding of their structure.

Contribution

It proves that certain geometric ideals correspond to algebraic two-sided ideals defined via two-sided cells in affine Weyl groups, confirming Ginzburg's conjecture.

Findings

Two-sided ideals correspond to closures of nilpotent orbits

The geometric ideals match algebraic two-sided ideals after tensoring with rationals

Supports Ginzburg's conjecture from 1987

Abstract

An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by rational numbers field. This proves a weak form of a conjecture of Ginzburg proposed in 1987.