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

Abstract

In this thesis, two ${\bar{\mathbb{Q}}}_{\ell }$-local systems, ${}^{\circ }\mathcal{E}$ and ${}^{\circ }{\mathcal{E}}^{\prime }$ on the regular unipotent subvariety ${\mathcal{U}}_{0,K}$ of $p$-adic ${\mathrm{SL}}_2(K)$ are constructed. Making use of the equivalence between ${\bar{\mathbb{Q}}}_{\ell }$-local systems and $\ell$-adic representations of the étale fundamental group, we prove that these local systems are equivariant with respect to conjugation by ${\mathrm{SL}}_2(K)$ and that their nearby cycles, when taken with respect to appropriate integral models, descend to local systems on the regular unipotent subvariety of ${\mathrm{SL}}_2(k)$, $k$ the residue field of $K$. Distributions on ${\mathrm{SL}}_2(K)$ are then associated to ${}^{\circ }\mathcal{E}$ and ${}^{\circ }{\mathcal{E}}^{\prime }$ 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.

arXiv comment117 pages, Thesis