Let $F$ be a non-archimedean local field of residual characteristic $p$. $\mathbb{G}$ reductive group defined over which splits tamely ramified extension and set $G=\mathbb{G}(F)$. We assume that $p$ does not divide the order Weyl $\mathbb{G}$. Given closed connected $F$-subgroup $\mathbb{H}$ contains derived subgroup $\mathbb{G}$, we study restriction to $H$ an irreducible supercuspidal repres...

Let G be any connected reductive group defined over a nonarchimedean local field F of residual characteristic p. Under some tameness assumptions on G, we construct families of positive-depth supercuspidal representations of G = G(F ). In particular, we classify (§2.7) the representations of G that contain any anisotropic unrefined minimal K-type (in the sense of MoyPrasad [28]) that satisfies a...

Let F be a nonarchimedean local field of residual characteristic p≠2. G (connected) reductive group that splits over tamely ramified extension F. We show construction analogous to Yu’s complex supercuspidal representations yields smooth, irreducible, cuspidal an arbitrary algebraically closed R different from p. Moreover, we prove this provides all R-representations if p does not divide the ord...

We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These satisfy multiplicativity, and are expressible as products Gauss sums. The tensor product related to the Rankin-Selberg gamma factors, by which we prove that can be written exterior square relate Jacquet-Shalika Langlands-Shahidi level zero supercuspidal representatio...

Let K/F be a quadratic extension of p-adic fields. We show that a generic irreducible representation of GL(n, K) is distinguished if and ony if its Rankin-Selberg Asai L-function has an exceptional pole at zero. We use this result to compute Asai L-functions of ordinary irreducible representations of GL(2, K). In the appendix, we describe supercuspidal dihedral representations of GL(2, K) in te...

We recall here the basic definitions needed to construct simple types, with no proofs given of the many claims that we make. For a much more detailed account, see [1] and the many other sources cited therein. Note that most of the statements made here could be proven without much difficulty for the reader who has the time and inclination. Any statements requiring a much more elaborate proof are...