The jet schemes of the nilpotent cone of $\mathfrak{gl}_n$ over $\mathbb{F}_\ell$ and analytic properties of the Chevalley map

TL;DR

This paper establishes dimension bounds for jet schemes of nilpotent matrices over fields of positive characteristic, leading to new insights into the analytic behavior of the Chevalley map and its implications for representation theory.

Contribution

It provides the first dimension bounds for jet schemes in positive characteristic and links these bounds to the analytic properties of the Chevalley map, advancing understanding in algebraic and representation theory.

Findings

Dimension bounds on jet schemes of nilpotent matrices

Analytic properties of the Chevalley map in positive characteristic

Connection to Harish-Chandra's integrability theorem

Abstract

We prove dimension bounds on the jet schemes of the variety of nilpotent matrices (and of related varieties) in positive characteristic. This result has applications to the analytic properties of the Chevalley map that sends a matrix to its characteristic polynomial. We show that our dimension bound implies, under the assumption of existence of resolution of singularities in positive characteristic, that the Chevalley map pushes a smooth compactly supported measure to a measure whose density function is $L^t$ for any $t<\infty$. We also prove this analytic property of the Chevalley map, unconditionally, when the characteristic of the field exceeds $n/2$. The zero characteristic counterpart of this result is an important step in the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKSb], we show that also in positive characteristic, this analytic statement implies Harish-Chandra's integrability theorem for cuspidal representations of the general linear group.