# On the EKL-Degree of a Weyl Cover

## TL;DR

This paper computes the EKL-degree for specific finite covers induced by Weyl group actions, providing explicit examples in $A^1$-enumerative geometry using cohomology of flag varieties.

## Contribution

It offers the first explicit calculations of the EKL-degree for Weyl covers, linking algebraic topology and $A^1$-enumerative geometry.

## Key findings

- Explicit EKL-degree computations for Weyl group quotients
- Connections established between cohomology rings and EKL-degree
- New examples illustrating $A^1$-enumerative geometry

## Abstract

More than four decades ago, Eisenbud, Khim\v{s}ia\v{s}vili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers $f\colon \mathbb{A}^n\to \mathbb{A}^n$ induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of $\mathbb{A}^1$-enumerative geometry.