The Bernstein projector determined by a weak associate class of good cosets

Abstract

Let $G$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p\gg 0$. In [Kim04], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for $G$ in the sense of A. Moy and G. Prasad. Following [Kim04], we attach to the set \(\overline{\mathfrak s}\) of good \(K\)-types in a weak associate class of positive-depth unrefined minimal $K$-types a $G(F)$-invariant open and closed subset $\mathfrak{g}(F{)}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g}(F)$ of $G(F)$, and a subset ${\tilde{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual \(\tilde G\) of \(G(F)\) consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then \(\tilde G_{\overline{\mathfrak s}}\) is the union of finitely many Bernstein components for $G$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy--Prasad $G(F)$-domain $G(F{)}_r\subset G(F)$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to $G(F{)}_r$, pushed forward via the logarithm to the Moy--Prasad $G(F)$-domain $\mathfrak{g}(F{)}_r\subset \mathfrak{g}(F)$, agrees on $\mathfrak{g}(F{)}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}(F{)}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan and Y. Varshavsky in arXiv:1504.01353 for the depth-$r$ Bernstein projector.

16 pages