Generic supercuspidal representations of $\operatorname{GL}_n$ can be defined by integrals over $U$

Question

Let $(V,\pi)$ be an irreducible, admissible, supercuspidal representation of $G = \operatorname{GL}_n(F)$ for $F$ a $p$-adic field. Let $B = TU$ be the usual Borel subgroup, maximal torus, and unipotent radical of $G$. Let $f_{v^{\ast},v}(g) = \langle v^{\ast}, \pi(g)v \rangle$ be a matrix coefficient for $v \in V$ and $v^{\ast} \in V^{\ast \infty}$. Then $f = f_{v^{\ast},v}$ is locally constant and compactly supported modulo the center $Z$ of $G$. If $\chi$ is a generic character of $U$, the integral

$$\int\limits_U f(ug)\chi(u^{-1}) du $$

can be shown to converge absolutely for every $g \in G$. Also, a change of variables shows that for fixed $v^{\ast}$,

$$v \mapsto \int\limits_U f_{v^{\ast},v}(u)\chi(u^{-1})du \tag{1} $$

defines a linear functional $\lambda: V \rightarrow \mathbb{C}$ which satisfies $\lambda(\pi(u_1)v) = \chi(u_1)\lambda(v)$ for all $u_1 \in U$. However, this linear functional might be the zero functional.

In general, $\pi$ is called generic if there exists a nonzero linear functional satisfying the property of the previous paragraph for $\chi$. If $\pi$ is generic for $\chi$, it is also generic for every other generic character.

1 . Is every irreducible, admissible supercuspidal representation of $G$ generic?

2 . If $\pi$ is generic, does there exist a smooth linear functional $v^{\ast}$ such that the map (1) is not the zero map? In other words, if there is a nonzero Whittaker functional, can it always be defined by an integral?

Answer 1

The answer to both questions is yes.

1. All irreducible supercuspidal representations of ${\rm GL}(N,F)$ are generic. See e.g. I. M. Gelfand and D. A. Kajdan, Representations of the group ${\rm GL}(n,K)$ where $K$ is a local field, Lie groups and their representations.

2. All ingredients to prove 2. are in

Paskunas, Vytautas; Stevens, Shaun On the realization of maximal simple types and epsilon factors of pairs. Amer. J. Math. 130 (2008), no. 5, 1211–1261.

First it is known by Bushnell and Kutzko that any irreducible supercuspidal representation of $G$ is of the form $\pi ={\rm ind}_J^G \Lambda$ (compactly induced representation), where $(J,\Lambda )$ is a maximal simple type in the sense of B. and K. Paskunas and Stevens prove that one can arrange the data so that $Hom_{U\cap J} (\Lambda , \chi ) \not= 0$ (1). It is a classical fact that if $c$ is a coefficient of $\Lambda$, then viewed as a fonction on $G$ (extend by $0$ off $J$), $c$ is a coefficient of $\pi$ (of the form $f_{v* ,v}$) (cf. e.g. Carayol Ann. Scient. ENS). Now writing condition (1) as the non-vanishing of an integral, you get exactly what you want.

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