Parabolic Multiplicative Affine Springer Fibers

TL;DR

This paper introduces parabolic multiplicative affine Springer fibers and their global counterparts, establishing their geometric properties and dimensions, and relating them to affine Deligne Lusztig varieties and Hitchin fibers.

Contribution

It defines new geometric objects called parabolic multiplicative affine Springer fibers and studies their properties, including equidimensionality and dimension, linking them to existing structures.

Findings

Parabolic multiplicative affine Springer fibers are equidimensional.

The dimension of these fibers is explicitly determined.

Global counterparts called parabolic multiplicative Hitchin fibers are introduced.

Abstract

We introduce parabolic multiplicative affine Springer fibers, which resemble the admissible union of affine Deligne Lusztig varieties in the affine flag variety. We also study their global counterparts called parabolic multiplicative Hitchin fibers. Their associated fibration is a global analogue of the Grothendieck simultaneous resolution for monoids. Using this fibration, we show that the parabolic multiplicative affine Springer fibers are equidimensional and find their dimension.