Bruht-Tits Buildings and the Representation Theory of a Reductive P-adic Group

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AbstractMoyMoy will investigate two topics in the representation theory of reductive p-adic groups. In the first project, Moy will study the use of the Bruhat-Tits building of a reductive group in its representation theory and harmonic analysis. Moy will investigate by two different methods the very old fundamental conjecture that every irreducible supercuspidal representation is compactly induced from an open compact modulo center subgroup. The first of these two methods is based upon extension of earlier joint work of Moy and Prasad which proved the conjecture in the case the supercuspidal representation has depth zero. The second method would be to show that any irreducible smooth representation which does not contain a cuspidal representation of a parahoric subgroup must have a nonzero Jacquet functor. Both these methods involve the Bruhat-Tits building. Also as part of the first project, Moy will extend his joint work with Barbasch which gave a new proof of the Howe conjecture on orbital integrals to other twisted cases. For the second project, Moy will investigate explicit construction of G-invariant essentially compact distributions on the group G.