fixed vector of a generic representation of GL(n,F)

Question

Let $F$ be a locally compact non-archimedean field and $G_{n}$ the locally profinite group $GL(n,F)$. Let $\Gamma_{n,k}$ be the subgroup of $G_{n}$ whose elements are the matrices of the form $$ \begin{pmatrix} A_{1,1} & A_{1,2} \\ \\ A_{2,1} & A_{2,2} \\ \end{pmatrix} $$ where $A_{1,1}\in GL(n-1,O_{F})$, $A_{1,2}\in M_{n-1,1}(O_{F})$, $A_{2,1}\in M_{1,n-1}(p_{F}^{k})$ and $A_{2,2}\in 1+p_{F}^{k}$. Here, $p_F$ denotes the maximal ideal in the ring of integers $O_F$ of $F$.

Let $(\pi,V)$ be a generic representation of $G_{n}$. We know that the space $V^{\Gamma_{n,k}}$ of fixed vectors is non-zero for $k$ large enough. Moreover, if $c(\pi) = \min\{ k\in\mathbb{N} : V^{\Gamma_{n,k}}\neq 0 \}$ ($c(\pi)$ is the conductor of $\pi$) then $\dim(V^{\Gamma_{n,c(\pi)}})=1$. Reference: Jacquet, Piatetski-Shapiro, Shalika, "Conducteur des représentations du groupe linéaire", Math. Ann. 256 (1981).

My question concerns replacing the subgroups $\Gamma_{n,k}$ by small subgroups $P_{n,k}$ whose elements are the upper-triangular matrices mod $p_{F}^{k}$. More precisely, if $\varphi:GL(n,O_{F})\longrightarrow GL(n,O_{F}/p_{F}^{k})$ is the morphism of reduction mod $p_{F}^{k}$, define $P_{n,k}=\varphi^{-1}(B)$, where $B$ is the standard Borel subgroup of $GL(n,O_{F}/p_{F}^{k})$.

It is clear that $V^{P_{n,k}}\neq 0$ for $k$ large enough. Denote $u(\pi) = \min\{ k\in\mathbb{N} : V^{P_{n,k}}\neq 0 \}$.

Question 1: Is it true that $\dim(V^{P_{n,u(\pi)}})=1$?

Question 2: If that is false for a generic representation, does it hold for only a supercuspidal representation of $GL(n,F)$?

Answer 1

To my knowledge, it is not known whether $Ind_{P_{n,k}}^{GL_n(o)} 1$ decomposes with single multiplicity.

This is certainly necessary by Frobenius reciprocity $$ dim Hom_{P_{n,k}}( 1 , Res_{P_{n,k}} \pi) = dim Hom_{GL_n(F)}( Ind^{GL_n(F)} Ind_{P_{n,k}}^{GL_n(o)} 1, \pi).$$

For more information, see Parabolic induction GL(n,Zp)

Answer 2

The group $P_{n,k}$ is not "smaller" than $\Gamma_{n,k}$ as claimed in the question. The $A_{n,n}$ for $\Gamma_{n,k}$ must be in $1+\mathfrak{p}^k$ whereas for $P_{n,k}$ it can be arbitrary element of $\mathcal{O}^\times$. Regardless, the question makes sense. The case $n=2$ is treated in Casselman's paper on the method of Atkin and Lehner. He shows that the answer to Question 1 is positive, and he defines the conductor that way.