Class polynomials for some affine Hecke algebras

TL;DR

This paper computes class polynomials for affine Hecke algebras of type d0A_2 and uses them to prove a conjecture related to affine Deligne-Lusztig varieties for certain groups, revealing interesting patterns.

Contribution

It introduces explicit computations of class polynomials for type d0A_2 affine Hecke algebras and proves a significant conjecture for specific groups using these computations.

Findings

Proved a conjecture of Gf6rtz-Haines-Kottwitz-Reuman for d0GL, unitary, and division algebra groups.

Identified patterns in affine Deligne-Lusztig varieties.

Computed class polynomials explicitly for type d0A_2 affine Hecke algebra.

Abstract

Class polynomials attached to affine Hecke algebras were first introduced by X.~He in \cite{He1}. They play an important role in the study of affine Deligne-Lusztig varieties. Motivated by \cite{He2}, we compute the class polynomials attached to an affine Hecke algebra of type (twisted) $\widetilde{A}_2$. Using these class polynomials we prove a conjecture of G\"{o}rtz-Haines-Kottwitz-Reuman for the general linear group, unitary group and division algebra of semisimple rank 2. Furthermore, we discuss some interesting patterns on affine Deligne-Lusztig varieties.