Galois actions on l-adic local systems and their nearby cycles: a geometrization of fourier eigendistributions on the p-adic lie algebra sl(2)

Abstract

In this thesis, two Qe-local systems, and 0 £1 (Definition 3.2.1) on the regular unipotent subvariety Uo,I< of p-adic SL(2)1< are constructed. Making use of the equiv­alence between Qe-local systems and £-adic representations of the etale fundamental group, we prove that these local systems are equivariant with respect to conjugation by SL(2)1< (Proposition 3.3.5) and that their nearby cycles, when taken with respect to appropriate integral models, descend to local systems on the regular unipotent subvariety of SL(2)k, k the residue field of K (Theorem 4.3.1). Distributions on SL(2, K) are then associated to 0 £ and 0 £1 (Definition 5.1.4) and we prove properties of these distributions. Specifically, they are admissible distributions in the sense of Harish-Chandra (Proposition 5.2.1) and, after being transferred to the Lie algebra, are linearly independent eigendistributions of the Fourier transform (Proposition 5.3.2). Together, this gives a geometrization of important admissible invariant distributions on a nonabelian p-adic group in the context of the Local Langlands program.