Definitions: Let G be a topological group. Consider a representation $\rho:G\to GL(V)$. We call $\rho$ to be smooth if for any $v \in V$, stabilizer of v in G is an open subgroup.
$\rho$ is admissible if for any compact open subgroup K of G, $V^{K}=\{v\in V|\sigma(v)=v$ $\forall$ $\sigma \in K$} is finite dimensional.
I'll also mention the Local Langlands correspondence for $GL(n,F)$ :
$\Bigg\{$Irreducible smooth,admissble representations of $GL(n,F) \Bigg\}$ $\leftrightarrow$ $\Bigg\{$ n-dimensional representations of $W_{F}$ $\Bigg\}$, where $F$ is a non-archemedian field and $W_{F}$ is the Weil group.
Context: I've been trying to understand the local langlands correspondence and in that the aim is studying these irreducible smooth and admissible representations.
Now I know that studying representations gives us various insights about the group and that's what we are aiming for in the Langlands program but why these irreducible admissible smooth ones? Are the other representations are sort of composition of these representations? Or is it because there is a correspondence between only these sort of representations on both sides of the correspondence? Any insights on these would be great.
Best Answer
The group $\mathrm{GL}_n(F)$ has a topology coming from the non-archimedean topology on $F$, so you better leverage that. So (much as in the case of Lie group representations), instead of considering abstract representations of $\mathrm{GL}_n(F)$, it is better to consider representations with certain continuity conditions.
The most naive way to do so would be to ask the homomorphism $\pi\colon \mathrm{GL}_n(F)\to\mathrm{GL}(V)$ to be continuous. The problem here is that $V$ is usually not finite-dimensional, so there is no clear choice of a topology on $\mathrm{GL}(V)$. So we rephrase this, by fixing a $v\in V$ and asking the function $G\to V:g\mapsto \pi(g)v$ to satisfy a continuity condition. Again, the correct topology on $V$ is unclear since it is infinite-dimensional. So you just give it the discrete topology. Thus: a representation $(\pi,V)$ is smooth if for any $v\in V$ the function $G\to V:g\mapsto \pi(g)v$ is locally constant.
Remark 1: One indication that the above is a good definition, is that when $V$ is finite-dimensional $(\pi,V)$ being smooth is in fact equivalent to $\pi\colon\mathrm{GL}_n(F)\to \mathrm{GL}(V)$ being continuous, where $\mathrm{GL}(V)$ carries the usual archimedean topology.
A way to rephrase the above condition is to say $V=\bigcup_{K\subset G}V^K$, where $K\subset G$ are compact opens subgroups of $G$. Now, admissibility asks each $V^K$ to be finite-dimensional, which is the natural finiteness condition here. Admissibility tells you that most questions about the representation $V$ can be somehow be reduced to a question about finite-dimensional spaces.
Remark 2: In fact an irreducible smooth representation of $\mathrm{GL}_n(F)$ is automatically admissible. So you could get rid of admissible as an adjective in your statement of local Langlands.