Euler characteristic of analogues of a Deligne-Lusztig variety for $GL_n$

TL;DR

This paper provides a combinatorial formula for calculating the Euler characteristic of a generalized Deligne-Lusztig variety for GL_n, depending on the unipotent part of the element and Weyl group conjugacy.

Contribution

It introduces a new combinatorial approach to compute Euler characteristics for analogues of Deligne-Lusztig varieties with conjugation, extending Springer fiber formulas.

Findings

Euler characteristic depends only on unipotent part and Weyl conjugacy class

Generalizes Springer fiber Euler characteristic formula for type A

Provides a combinatorial calculation method

Abstract

In this paper we give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne-Lusztig variety if we replace Frobenius morphism with conjugation by an element for $GL_n$. The main theorem states that it only depends on the unipotent part of the Jordan decomposition of an element and the conjugacy class in the Weyl group. Also it generalizes the formula of the Euler characteristic of a Springer fiber for type A.