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

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.