Smooth representation of a locally profinite group

A complex group representation $\pi :G\to \mathrm{GL}(V)$ is smooth when every vector is fixed by some compact-open subgroup. Equivalently each vector's stabilizer is open. Smoothness is local constancy of orbit maps with the vector space regarded discretely, not ordinary continuity into a finite-dimensional complex topology.

A smooth vector in any abstract group representation of a locally profinite group has an open stabilizer, equivalently is fixed by some compact-open subgroup. Smooth vectors form an invariant linear subspace: use intersections for addition and conjugation for translation. A smooth representation is one in which every vector has this property.

A smooth linear character is a one-dimensional smooth representation, equivalently a homomorphism whose kernel is open. On a local multiplicative group, each smooth linear character is trivial on some sufficiently deep principal-unit subgroup. Its inverse has the same kernel.

The unnormalized Jacquet module is the space of coinvariants of a unipotent subgroup $U$. If a subgroup $T$ normalizes $U$, it acts on this quotient. Every linear functional transforming by a linear character trivial on $U$ factors through it. Normalized Jacquet functors include an additional modulus twist; that convention must be specified.

The right action on compactly supported locally constant functions is a smooth representation. For a compact-open subgroup $K$, its fixed-vector space has basis the characteristic functions of cosets in $G/K$, since compact support on that discrete quotient is finite support. Consequently this representation is admissible exactly when $G$ is compact.

An admissible representation is a smooth representation for which every fixed-vector space $V^K$ for a compact-open subgroup has finite dimension. Admissibility can hold for an infinite-dimensional representation; it bounds each local fixed level, not their union.

For the upper-triangular subgroup of ${\mathrm{GL}}_n(F)$, conjugation by $\mathrm{diag}({\varpi }^{-(n-1)},\dots ,1)$ expands the upper-unipotent coordinates. For a decomposed compact-open subgroup $K=T_0{U}_0$, its conjugates eventually contain $K$, while their fixed-vector spaces are isomorphic. Finite dimension therefore forces those spaces to be equal. Inverse conjugation contracts any fixed unipotent element into $U_0$, proving that the unipotent radical acts trivially in every admissible representation of this triangular group.

The fixed-vector space is $V^K=\{v:\pi (k)v=v\text{ for all }k\in K\}$. Conjugation gives an isomorphism $V^K\to V^{gK{g}^{-1}}$, $v\mapsto \pi (g)v$. If $K\subseteq L$, then $V^L\subseteq V^K$.