Galois Actions on $\ell$-adic Local Systems and Their Nearby Cycles: A Geometrization of Fourier Eigendistributions on the $p$-adic Lie Algebra $\mathfrak{sl}(2)$

TL;DR

This thesis constructs and analyzes specific local systems on the unipotent variety of p-adic SL(2), linking geometric objects to invariant distributions and Fourier eigendistributions, advancing the geometric understanding within the Local Langlands program.

Contribution

It introduces new local systems on the unipotent variety of p-adic SL(2) and relates their nearby cycles to invariant distributions, providing a geometric perspective on Fourier eigendistributions.

Findings

Local systems are equivariant under conjugation by SL(2)

Nearby cycles descend to the residue field

Distributions are admissible and form a linearly independent eigenspace

Abstract

In this thesis, two $\bar{\mathbb{Q}}_\ell$-local systems, $\vphantom{\mathcal{E}}^\circ \mathcal{E}$ and $\vphantom{E}^\circ \mathcal{E}^\prime$ on the regular unipotent subvariety $\mathcal{U}_{0,K}$ of $p$-adic $\operatorname{SL}_2(K)$ are constructed. Making use of the equivalence between $\bar{\mathbb{Q}}_\ell$-local systems and $\ell$-adic representations of the \'etale fundamental group, we prove that these local systems are equivariant with respect to conjugation by $\operatorname{SL}_2(K)$ and that their nearby cycles, when taken with respect to appropriate integral models, descend to local systems on the regular unipotent subvariety of $\operatorname{SL}_2(k)$, $k$ the residue field of $K$. Distributions on $\operatorname{SL}_2(K)$ are then associated to $\vphantom{E}^\circ \mathcal{E}$ and $\vphantom{E}^\circ \mathcal{E}^\prime$ and we prove properties of these distributions. Specifically, they are admissible distributions in the sense of Harish-Chandra and, after being transferred to the Lie algebra, are linearly independent eigendistributions of the Fourier transform. Together, this gives a geometrization of important admissible invariant distributions on a nonabelian $p$-adic group in the context of the Local Langlands program.