The geometry of some generalized affine Springer fibers

TL;DR

This paper explores the geometric structure of generalized affine Springer fibers associated with reductive groups, proposing a conjecture linking their irreducible components to dual group weight multiplicities, and proves it in specific cases.

Contribution

It introduces a new conjecture connecting the geometry of generalized affine Springer fibers with Langlands dual group representations and verifies it for unramified conjugacy classes.

Findings

Conjecture relating irreducible components to dual group weight multiplicities.

Proof of the conjecture for unramified conjugacy classes.

Comparison between group and Lie algebra affine Springer fibers.

Abstract

We study basic geometric properties of some group analogue of affine Springer fibers and compare with the classical Lie algebra affine Springer fibers. The main purpose is to formulate a conjecture that relates the number of irreducible components of such varieties for a reductive group $G$ to certain weight multiplicities defined by the Langlands dual group $\hat{G}$. We prove our conjecture in the case of unramified conjugacy class.