On the local Langlands correspondence

Abstract

This thesis contributes to the proof of the conjectural local Langlands correspondence in the case of small residue characteristic. Let G be an absolutely simple split reductive group over a finite extension k of ℚ_p. To each point in the Bruhat-Tits building of G(K), Moy and Prasad have attached a filtration of G(K) by bounded subgroups. In the first main result of this thesis we give necessary and sufficient conditions for the first Moy-Prasad filtration quotient to have stable functionals for the action of the reductive quotient (this result is joint with Jessica Fintzen). Our work extends earlier results by Reeder and Yu, who gave a classification in the case when p is sufficiently large. By passing to a finite unramified extension of k if necessary, we obtain new supercuspidal representations of G(k) when p is small. Next we consider G = G₂. For this case we explicitly describe the locus of stable functionals on the first Moy-Prasad filtration quotient for every point in the Bruhat-Tits building. Our description is in terms of the invariant theory of SL₂ x SL₂. This allows us to construct a previously unknown representation π of G₂(ℚ₂) using the construction of Reeder-Yu. We then prove that there exists a unique Langlands parameter that satisfies the local degree conjecture of Hiraga, Ichino, and Ikeda with respect to π. We give an explicit construction of this parameter.