Admissible representation

Remarks

The multiplicity of $\tau$ is the dimension of $\Hom_K(V_\tau, V)$, the space of continuous $K$-equivariant linear maps from the space of $\tau$ to $V$.

Example: let $G = \mathrm{SL}_2(\R)$ and $K = \mathrm{SO}(2)$, and let $G$ act on functions on $\R^2 \setminus \set{0}$ that satisfy $f(tx) = \abs{t}^{-1-\nu} f(x)$ for $t \ne 0$, with $\nu \in i\R$, by $(\pi(g) f)(x) = f(g^{-1} x)$. Such an $f$ is determined by its values on the unit circle, and the norm is the $L^2$ norm there. The characters of $K$ are $e^{i\theta} \mapsto e^{in\theta}$ with $n \in \Z$. Each one with $n$ even occurs exactly once, spanned by the function $e^{in\theta}$ on the circle, and the odd ones do not occur, so $\pi$ is admissible.

Non-example: the right regular representation of $G = \mathrm{SL}_2(\R)$ on $L^2(G)$ is not admissible. Its vectors fixed by $K$ are the functions invariant under right translation by $K$, which form the infinite-dimensional space $L^2(G/K)$. So clause 2 fails for the trivial representation of $K$.

Over a non-archimedean local field $F$, a representation of $G(F)$ on a complex vector space $V$ is admissible under two conditions. Every vector is fixed by some compact open subgroup, and the space $V^{K'}$ is finite-dimensional for each compact open subgroup $K'$.