| In this thesis, two Qℓ -local systems, °E and °E′ (Definition 3.2.1) on the regular unipotent subvariety U0,K of p-adic SL(2)K are constructed. Making use of the equivalence between Qℓ -local systems and ℓ-adic representations of the étale fundamental group, we prove that these local systems are equivariant with respect to conjugation by SL(2)K (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 °E and °E′ (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. |
