Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$. Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$. We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference.

Let $k$ be a finite extension of $\mathbb{Q}_{p}$, let ${\mathcal{G}}$ be an absolutely simple split reductive group over $k$, and let $K$ be a maximal unramified extension of $k$. To each point in the Bruhat–Tits building of ${\mathcal{G}}_{K}$, Moy and Prasad have attached a filtration of ${\mathcal{G}}(K)$ by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy–Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. 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 ${\mathcal{G}}(k)$.

In a recent paper, Gross and Reeder study the arithmetic properties of discrete Langlands parameters for semi-simple -adic groups, and they conjecture that a special class of these – the simple wild parameters – should correspond to -packets consisting of simple supercuspidal representations. We provide a construction of this correspondence, and show that the simple wild -packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group, and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

We describe the supercuspidal representations of Sp4(F), for F a non-archimedean local field of residual characteristic different from two, and determine which are generic.

Let $F$ be a non-archimedean local field of residual characteristic different from $2$, and let $G$ be a unitary, symplectic or orthogonal group, considered as the fixed point subgroup in $\tilde G={\rm GL}(N,F)$ of an involution $\sigma$. We generalize the notion of a {\it simple character} for $\tilde G$, which was introduced by Bushnell and Kutzko [Annals of Mathematics Studies 129 (Princeton University Press, 1993)], to define semisimple characters. Given a semisimple character $\theta$ for $\tilde G$ fixed by $\sigma$, we transfer it to a character $\theta_{-}$ for $G$ and calculate its intertwining. If the torus associated to $\theta_{-}$ is maximal compact, we obtain supercuspidal representations of $G$, which are new if the torus is split only over a wildly ramified extension. 2000 Mathematics Subject Classification: 22E50.