When is compact induction in GL(2) from an open compact group admissible?

Question

Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional representation $\rho$ of $K$:

• the compact induction $ind_K^G \rho$ is admissible

• compact induction is isomorphic to the usual (analytic) induction $Ind_K^G \rho$

• the compact induction decomposes into a finite sum of irreducible representations

My main question is the following:

What are necessary and sufficient condition on $\rho$ for $ind_K^G \rho$ to be admissible?

I am mostly interested in the case, where $K$ maximal compact subgroup (mod the center) in $GL(2,F)$ for a non archimedean local field $F$.

I am little bit familiar with Bushnell-Henniart's monograph on Langlands for $GL(2)$ and also much with the work of Kutzko and others, and understand the classification of supercuspidal representation of $GL(2,F)$. So, I am aware of some sufficient conditions for those representation arising from cuspidal type strata.

I also am aware of the basic strategy for proving admissibility with the restriction-induction formula and Frobenius reciprocity by proving finite-dimensionality of the algebra $$ Hom_G( ind_K^G \rho, ind_K^G \rho) = \bigoplus_{\gamma \in G//K} Hom_{K \cap K^\gamma}( \rho^\gamma, \rho),$$ and using Iwahori decomposition to spell out $K \cap K^\gamma$, but I do not see what to require about $\rho$ to make this work. So I hope somebody cant hint me to the relevant section in the literature, or give a hint.

Is is necessary and sufficient that $Res_{N \cap K} \rho$ does not contain the trivial representation, for any unipotent group $N \subset G$?

On a related matter, consider the distinct two maximal compact groups (mod the center) $K$ and $K'$ with $\rho \in Rep(K)$ and $\rho' \in Rep(K')$ such that the compact induction is admissible.

Is it true that $$ Hom_G( ind_K^G \rho, ind_{K'}^G \rho') = \bigoplus_{\gamma \in K' \backslash G/K} Hom_{K' \cap K^\gamma} ( \rho^\gamma, \rho) = 0?$$

Thanks a lot.

Answer 1

The answer to the second question is no. Example: $G=GL_2(F)$, $K$ stabilizer of an edge $e$ in the Bruhat-Tits tree, and $\rho$ such that the induction is irreducible $=: \pi$. Now let $K'$ be the stabilizer of a vertex of $e$, and $\rho'$ induction to $K'$ of the restriction of $\rho$ to the pointwise stabilizer of an edge. Then the second induction is isomorphic to $\pi\oplus \pi \otimes \chi\circ\det$.

I think the answer to the first question is yes. It seems to me that the condition you state is equivalent to the assertion that the Jacquet module of the compact induction is zero for all parabolic subgroups of $G$. Bernstein's centre then implies that all irreducible subquotients of the induction are supercuspidal. (here $G$ is reductive). Now since $K$ contains the center supercuspidals are projective, and hence the induction is a direct sum of irreducible supercuspidal reps. This sum is finite iff $End_G( ind_K^G \rho)$ is finite dimensional. So if it is infinite then the induction cannot be admissible by the projection formula you state above. If it is finite then it is automatically admissible.