The Bernstein projector determined by a weak associate class of good cosets (2007.02396v1)

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.