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Let normalG denote a connected, quasi‐split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)‐conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p‐adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.

Let k be a finite extension of Q p , let 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 G K , Moy and Prasad have attached a filtration of 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 G(k).

Let k be a non-archimedean local field of residual characteristic p = 2. Let G be a (connected) reductive group that splits over a tamely ramified field extension of k. We revisit Yu's construction of smooth complex representations of G(k) from a slightly different perspective and provide a proof that the resulting representations are supercuspidal.Moreover, we show that an analogous construction yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p and that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.