Admissible representation of a p-adic group

Definition

Let $G$ be a locally profinite group. A smooth complex representation $V$ of $G$ is admissible if

\dim_{\mathbb C}V^K<\infty

for every compact open subgroup $K\leq G$ . For a connected reductive group $\mathbf G$ over a nonarchimedean local field, local Langlands concerns irreducible admissible representations of $G=\mathbf G(F)$ .

Why compact-open fixed spaces appear

Each $V^K$ is a finite-dimensional module over the Hecke algebra $\mathcal H(G,K)$ . Admissibility is therefore the nonarchimedean analogue of finite $K$ -multiplicities for representations of real reductive groups, but the two definitions live in different representation categories.

Scope

For reductive $p$ -adic groups, every irreducible smooth complex representation is admissible. This theorem uses reductivity and is false for arbitrary locally profinite groups, so admissibility remains part of the standard statement of the local correspondence.

References

Joseph Bernstein and Andrei Zelevinsky, “Induced representations of reductive $p$ -adic groups I,” Annales scientifiques de l’École Normale Supérieure 10 (1977), 441–472. Numdam.
Tasho Kaletha, “Representations of reductive groups over local fields,”
arXiv.