Euler characteristics of the generalized Kloosterman sheaves for symplectic and orthogonal groups

TL;DR

This paper investigates the monodromy and Euler characteristics of generalized Kloosterman sheaves for symplectic and orthogonal groups, providing evidence for conjectural Swan conductors related to Langlands parameters.

Contribution

It computes Euler characteristics of these sheaves and supports conjectures on Swan conductors and Langlands parameters for specific algebraic groups.

Findings

Calculated Euler characteristics for symplectic and orthogonal groups

Provided evidence for Reeder-Yu's conjectural Swan conductors

Linked monodromy properties to Langlands parameters

Abstract

We study the monodromy of certain $\ell$-adic local systems attached to the generalized Kloosterman sheaves constructed by Yun and calculate their Euler characteristics under standard representations in the cases of symplectic and split/quasi-split orthogonal groups. This provides evidence for the conjectural description of their Swan conductors at $\infty$ which is predicted by Reeder-Yu on the Langlands parameters attached to the epipelagic representations.