Geometry of Kottwitz-Viehmann Varieties

TL;DR

This paper investigates the geometric structure of Kottwitz-Viehmann varieties, establishing their equidimensionality, deriving their dimensions, and proposing a conjecture relating their irreducible components to Langlands dual group weight multiplicities.

Contribution

It provides the first proof of equidimensionality and a dimension formula for Kottwitz-Viehmann varieties, and formulates and verifies a conjecture on their irreducible components.

Findings

Kottwitz-Viehmann varieties are equidimensional.

A precise dimension formula is established.

A conjecture relating irreducible components to Langlands dual group weights is proposed and proved in special cases.

Abstract

We study basic geometric properties of Kottwitz-Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.