On a finite type property of certain affine Deligne-Lusztig varieties

TL;DR

This paper provides a new proof that certain affine Deligne-Lusztig varieties are of finite type, using properties of Hecke algebras and affine Weyl group element lengths.

Contribution

It introduces a novel proof approach for the finiteness of affine Deligne-Lusztig varieties associated with superbasic elements.

Findings

Affine Deligne-Lusztig varieties are of finite type for algebraic groups of adjoint type.

The proof leverages properties of the associated Hecke algebra.

An estimate of lengths in the affine Weyl group is established.

Abstract

We give a new proof of the fact that affine Deligne-Lusztig varieties for an algebraic group of adjoint type, associated with superbasic elements, are of finite type. The proof uses a property of the associated Hecke algebra, which we reprove, and an estimate of the length of certain elements in the affine Weyl group.