# Lecture 6: Representations of $\GL_n(k)$ and the Local Langlands Conjecture

:::{.remark}
The LLC will take the following form:
\[
\correspond{
  \text{$n\dash$dimensional $F\dash$semisimple } \\
  \text{Weil-Deligne representations of } W_k
}/\sim
&\mapstofrom
\correspond{
  \text{Irreducible admissible reps} \\
  \text{ of } \GL_n(K)
}
,\]
where we're working toward describing the RHS.
:::

:::{.warnings}
Most representations here will be infinite dimensional.
:::

:::{.remark}
Setup: 

- $E\in \Top\Field$ with the discrete topology and any characteristic,
- $V\in \mods{E}$ an infinite-dimensional $E\dash$vector space,
- $K/\QQpadic$ a finite extension,
- $\GL_n(K)$ has the \(p\dash \)adic topology, described by a bases of open neighborhoods of the identity:
\[
\mcb \da \ts{M \da \matt a b c d \in \GL_n(\OO_K) \st M \equiv I_n \mod \mfp_K^m\, \forall m\in \ZZ_{\geq 1}}
.\]

- $\pi\in \Grp(\GL_n(K), \Aut_E(V))$, where we're not using the topology on $E$ yet.

We want a sensible notion of continuity for $\pi$.
:::

:::{.definition title="Smooth representations"}
A representations $\pi$ is **smooth** iff stabilizers are open, i.e. 
\[
\forall v\in V, \, \stab_\pi(v) \da \ts{g\in \GL_n(K) \st g.v \da \pi(g)(v) = v} \text{ is open in } \GL_n(K)
.\]
:::

:::{.definition title="Admissible representations"}
A smooth representation $\pi$ is **admissible** iff for all $U \leq \GL_n(K)$ open subgroups, the fixed set $V^U$ is finite dimensional.
:::

:::{.remark}
On terminology: admissible will always imply smooth.
:::

:::{.example title="?"}
If $\dim V = 1$ and $\pi(g) = 1$, then $\pi$ is admissible.
If $\dim V = \infty$ instead, this is smooth but not admissible, and in some sense is an infinite direct sum of trivial 1-dimensional reps.
:::

:::{.fact}
Irreducible and smooth implies admissible.
:::

:::{.definition title="Irreducible representations"}
A representation $\pi: G\to \GL(V)$ for $G = \GL_n(K)$ is **irreducible** iff there are only 2 $G\dash$invariant subspaces: $0$ and $V$ itself.
Note that this still holds in infinite-dimensions in this case, i.e. we don't have to require the invariant subspaces to be closed.
:::

:::{.remark}
By LCFT, we know $K\units \iso W_k^\ab$, but how do we understand $W_K$?
Langlands' insight: reinterpreting this isomorphism in terms of representations.
If the groups are isomorphic, their reps are isomorphic, so
\[
\correspond{
  \text{Irreducible 1-dimension reps} \\
  \text{of }K\units = \GL_1(K)
}/\sim
&\mapstofrom
\correspond{
  \text{Irreducible 1-dimensional reps} \\
  \text{ of } W_K
}/\sim
\]

So the conjecture is that there should be a canonical bijection, say of sets, of the following form:
\[
\Irr\Rep^{\dim = n} \GL_1(K) \mapstofrom \Irr\Rep^{\dim = n} W_k
.\]
Next time we'll see how to generalize this to $n=2$, where the $n=1$ case is class field theory, and we'll see how to match things up.
:::