\section{$p$-adic Groups (notes by Kevin Childers)}

\subsection{Introduction}

We've seen the finite dimensional algebraic representation theory of compact Lie groups and reductive algebraic groups.
A natural next step in representation theory is to study the representation theory  of \textit{real} Lie groups (e.g. $\sl n\reals$) on Hilbert spaces (e.g. $L^2(\sl n\reals)$ under the regular representation).

In parallel, we could study the representation theory of groups such as $\sl n\Qp$.
Both the real representation theory and the $p$-adic representation theory arises naturally in number theory in the theory of modular forms.

A (semi-)classical perspective on modular forms, is to study the $\sl2\reals$-representation $L^2(\sl2\ints\backslash\sl2\reals)$.
A more modern approach is to package all completions of $\rats$ into the \textbf{ring of adeles}:
\[ \mathbb A_\rats := \prod_v^* \rats_v = \{(x_v)_v : x_v \in \ints_v \text{ for all but finitely many }v\}. \]
Here $v$ ranges through all places of $\rats$ and ``$\prod^*$'' denotes a restricted product with respect to $\ints_v \subset \rats_v$ for $v \neq \infty$ (the right hand side is the definition of restricted product in this case).
The ring of adeles is a locally compact group.
Now we can study the $\sl2{\mathbb A_\rats}$-representation $L^2(\sl2\rats\backslash\sl2{\mathbb A_\rats})$.

\subsection{Representations of $\gl n{\mathbb F_p}$}\label{finitereps}

To get a feel for what type of representations of $\gl n\Qp$ we will study, we will review the representation theory of $\gl 2{\mathbb F_p}$.

There are two types of easy representations of $\gl2{\mathbb F_p}$.
The first are the 1-dimensional representations factoring through the determinant map
\[ \gl2{\mathbb F_p} \xrightarrow{\ \det\ } \mathbb F_p^\times \to \cmplx^\times. \]
The second, are the representations induced from the Borel subgroup, which we now review.

Let $G = \gl2{\mathbb F_p}$ and 
\[ T = \bp * & 0 \\ 0 & * \ep \qquad\subset\qquad B = \bp * & * \\ 0 & * \ep \qquad\supset\qquad N = \bp 1 & * \\ 0 & 1 \ep \]
as subgroups of $G$.
Let $\chi: T \to \cmplx^\times$ be a character.
Such a $\chi$ can be described as a pair of characters $\chi_1,\chi_2: \mathbb F_p^\times \to \cmplx^\times$:
\[ \chi: \bp t_1 & 0 \\ 0 & t_2 \ep \mapsto \chi_1(t_1)\chi_2(t_2). \]
Such a $\chi$ extends to a character on the Borel $B$, and we obtain a $G$-representation
\[ \Ind_B^G\chi = \{ f: G \to \cmplx \mid f(bg) = \chi(b)f(g)\ \forall b \in B, g \in G\}. \]
Notice that the dimension of $\Ind_B^G\chi$ is $|G:B| = |\mathbb P^1(\mathbb F_p)| = p+1$.

The next question is: how do the representations induced in this way decompose into irreducible representations?
Let
\[ w = \bp 0 & 1 \\ 1 & 0 \ep,\qquad \psi^w\bp t_1 & 0 \\ 0 & t_2 \ep = \psi \bp t_2 & 0 \\ 0 & t_1 \ep \]
for any character $\psi : T \to \cmplx^\times$.
We have the following:

\begin{prop}\label{classify/gl2fp}
    With $G$, $B$, and $T$ as above, let $\chi,\psi: T \to \cmplx^\times$ be characters.
    Then
    \[ \dim\Hom_G(\Ind_B^G\chi,\Ind_B^G\psi) = \begin{cases} 0 & \text{if }\chi \neq \psi,\psi^w, \\
    1 & \text{if }\chi = \psi\text{ or }\chi = \psi^w \text{ but }\psi \neq \psi^w, \\ 2 & \text{if } \chi = \psi = \psi^w. \end{cases} \]
    In particular, $\Ind_B^G\chi$ is irreducible if and only if $\chi \neq \chi^w$.
    If $\chi = \chi^w$, then $\Ind_B^G\chi$ is the direct sum of two distinct irreducible representations.
\end{prop}

\begin{exmpl}
    We give an example of the reducible case.
    The induction of the trivial representation is just
    \[ \Ind_B^G(1_B) = \{f:B\backslash G \to \cmplx\}. \]
    The constant functions give a subrepresentation, isomorphic to the trivial representation of $G$.
    We have an exact sequence
    \[ 0 \to 1_G \to \Ind_B^G(1_B) \to St \to 0. \]
    The quotient is an irreducible representation called the \textit{Steinberg representation}.
\end{exmpl}

\begin{proof}
    \begin{align*}
        \Hom_G(\Ind_B^G\chi,\Ind_B^G(\psi)) 
        &= \Hom_B(\Ind_B^G\chi|_B,\psi) \\
        &= \Hom_B(\textstyle\bigoplus\limits_{w' \in \{1,w\}} \Ind_{B\cap B^{w'}}^B\chi^{w'},\psi) \\
        &= \Hom_B(\chi,\psi)\oplus\Hom_B(\Ind_T^B\chi^w,\psi) \\
        &= \Hom_T(\chi,\psi)\oplus\Hom_T(\chi^w,\psi).
    \end{align*}
    The first and last equality are Frobenius reciprocity, and the second inequality is the induction-restriction formula.
    Here we are using the fact that $1,w$ are a set of representatives for $B\backslash G/B$, and $B\cap B^w = T$ ($B^w$ consists of lower triangular matrices).
\end{proof}

We will be studying the analogue of these representations on $p$-adic groups (e.g. $\gl2\Qp$).

\begin{rmk}\label{non-split}
    We've now classified the easy representations.
    For the rest, we take a \textit{non-split} maximal torus in $G$, induce the representations, and decompose the result.
    These are called the \textit{cuspidal} representations.
\end{rmk}

\begin{exmpl}
    As an example of a non-split maximal torus, as in Remark \ref{non-split}, choose a basis of $\mathbb F_{p^2}$ over $\mathbb F_p$.
    Then we get a map $\mathbb F_{p^2} \to M_2(\mathbb F_p)$ by mapping $\alpha$ to the matrix representing ``multiplication by $\alpha$'' with respect to the chosen basis.
    Taking units gives a map $\mathbb F_{p^2}^\times \hookrightarrow \gl2{\mathbb F_p}$, which is the inclusion of a maximal torus.
\end{exmpl}

\subsection{Locally profinite groups}

Let $K$ denote a non-archimedean local field (i.e. a finite extension of $\mathbb F_p((t))$ or $\Qp$).
Our focus will be the representation theory of connected reductive groups $G/K$.
Of course, some of the results hold more generally.

\begin{defn}
    A \textbf{locally profinite group} is a Hausdorff topological group, such that every open neighborhood of the identity contains a compact open subgroup.
\end{defn}

\begin{ex}\label{addnormal}
    Let $G$ be a Hausdorff topological group.
    Show that $G$ is profinite if and only if every open neighborhood of the identity contains a compact open \textit{normal} subgroup.
\end{ex}

\begin{exmpl}
    $G = \gl n\Qp$ is locally profinite.
    A basis of compact open subgroups at the identity is given by
    \[ \{1 + p^NM_n(\Zp) \mid N \geq 1\}. \]
\end{exmpl}

The basic class of representations of a locally profinite group $G$ is the following.

\begin{defn}\ 
    \begin{enumerate}
        \item Let $G$ be a locally profinite group.
        A \textbf{smooth representation} of $G$ is a homomorphism $\pi: G \to \aut_\cmplx(V)$ for some $\cmplx$-vector space $V$ such that $V = \bigcup_K V^K$, where $K$ ranges over compact open subgroups $K$ of $G$.
        In other words, every vector $v \in V$ has open stabilizer.
        \item A smooth representation is \textbf{admissible} if $V^K$ is finite dimensional for all compact open subgroups $K$ of $G$.
    \end{enumerate}
\end{defn}

Let $\mathcal M(G)$ denote the category of smooth representations of a locally profinite group $G$.

\begin{ex}
    Show that $\mathcal M(G)$ is an abelian category.
\end{ex}

\begin{exmpl}
    A 1-dimensional smooth representation of a locally profinite group $G$ is a character $\chi: G\to \cmplx^\times$ such that $\chi$ is trivial on a compact open subgroup (i.e. $\ker\chi$ is open).
    For instance, if $G = \Qp^\times$, characters $\chi: \Qp^\times \to \cmplx^\times$ with $\chi|_{1+p^n\Zp}$ trivial for some $n\geq 1$ are smooth.
    If $\chi: \Qp^\times \to \cmplx^\times$ is trivial on $\Zp^\times$, then $\chi$ is completely determined by $\chi(p)$.
    These characters are called \textbf{unramified}.
\end{exmpl}

However, the typical examples of interest to us will be infinite dimensional, since $G$ will typically be non-abelian.

\begin{ex}
    What are the smooth representations when $G$ is a compact group, for example $G = \gl2\Zp$?
\end{ex}

Representations $\pi \in \mathcal M(G)$ are not necessarily semi-simple.
However, we do have the following.

\begin{lem}
    Let $\pi \in \mathcal M(G)$ and let $K$ be a compact open subgroup of $G$.
    Then
    \begin{enumerate}
        \item $\pi|_K$ is semi-simple, and 
        \item $\pi \mapsto \pi^K$ is an exact functor.
    \end{enumerate}
\end{lem}

\begin{proof}[Proof of (1)]
    Let $v \in V$.
    Since $V$ is smooth, exercise \ref{addnormal} implies $v \in V^{K'}$ for some compact open normal subgroup $K'$.
    By compactness, $K'$ has finite index in $K$.
    Thus the $K$-representation generated by $v$ is finite dimensional (and semisimple, since it is a representation of the finite group $K/K'$).
    So $V|_K$ is the sum of irreducible $K$-representations.
    A standard Zorn's Lemma argument shows this implies that $V|_K$ is a \textit{direct} sum of irreducible $K$-representations.
\end{proof}

\subsection{The induction functor}

In order to construct representations of a profinite group $G$, we want an induction functor from representations of a closed subgroup $H$.

\begin{lem}[Frobenius reciprocity]
    The restriction functor $\mathcal M(G) \to \mathcal M(H)$ has a right adjoint, denoted by $\Ind_H^G:\mathcal M(H) \to \mathcal M(G)$.
\end{lem}

In finite group theory, the definition of the induction functor for an $H$-representation $\pi$ is
\[ \Ind_H^G\pi = \{ f:G \to \pi \mid f(hg) = \pi(h)f(g)\ \forall h \in H,g \in G\}. \]
The problem with this definition in our setting is that this will not in general produce smooth representations.
However, there is a smoothing functor from the category of all $G$-representations to the category $\mathcal M(G)$, which sends a representation $V$ to $\bigcup_K V^K$, where $K$ ranges over compact open subgroups.
Applying the smoothing functor to the above definition gives the

\begin{defn}
    Let $G$ be a locally profinite group and $H$ a closed subgroup.
    Let $\pi \in \mathcal M(G)$.
    We define $\Ind_H^G\pi \in \mathcal M(H)$ to be
    \[ \Ind_H^G\pi = \left\{ f: G \to \pi \left| 
    \begin{array}{ll} f(hg) = \pi(h)f(g)\ \forall h\in H,g\in G,\ \text{and} \forall g \in G \exists \text{ a compact open}\\ \text{subgroup $K$ such that } f(gk) = f(g)\ \forall k\in K \end{array} \right.\right\}. \]
\end{defn}

With this definition, the proof of Frobenius reciprocity is the same as in the finite group case, with an additional check of the topological conditions.

\begin{ex}
    Show that $\Ind_H^G: \mathcal M(H) \to \mathcal M(G)$ is an exact functor.
\end{ex}

\begin{exmpl}
    We will now parallel the constructions given in subsection \ref{finitereps} on the locally profinite group $G = \gl2\Qp$.
    Let $B$ and $T$ denote the borel and maximal torus, parallel to the finite example.
    
    For any smooth character $\chi : T \to \cmplx^\times$ we form a smooth $G$-representation $\Ind_B^G\chi$ (this will be infinite dimensional).
    A similar argument to Proposition \ref{classify/gl2fp} shows
    \begin{enumerate}
        \item If $\chi \neq \chi^w$, $\Ind_B^G\chi$ is irreducible.
        \item If $\chi = \chi^w$, $\Ind_B^G\chi$ fits in an exact sequence with two smooth irreducible representations.
        For example, we have an exact sequence
        \[ 0 \to 1_G \to \Ind_B^G(1_B) \to St \to 0. \]
    \end{enumerate}
    The sequence in (2) does \textit{not} split.
    The problem we run into is that $\Ind_B^G$ is \textit{not} a left adjoint to restriction, as opposed to the finite group setting.
\end{exmpl}

\subsection{Representation theory of reductive groups over local fields}

Let $F$ denote a non-archimedean local field, and let $\mathcal O$ denote the ring of integers of $F$.
Let $G/\mathcal O$ denote a split, connected, reductive group. We will also abusively write $G$ for $G(F)$.
Let $\pi \in \mathcal M(G)$ be irreducible and admissible.
Consider the condition that $\pi^{G(\mathcal O)} \neq 0$.

\begin{rmk}
    In the global setting this happens almost everywhere locally.
    For example, if $F$ is a number field, an \textbf{automorphic representation} of $\text{GL}_1$ over $F$ is a continuous character $\chi: \mathbb A_F^\times/F^\times \to \cmplx^\times$.
    For any such $\chi$, there exists a finite set $S$ of primes such that $\chi$ is trivial on $\mathcal O_{F_v}^\times$ for all $v \not\in S$.
    The reason for this is that $\prod_{v\nmid\infty}\mathcal O_{F_v}^\times$ is a profinite group, and any continuous homomorphism from a profinite group to $\cmplx^\times$ must factor through a finite quotient.
    There exists an analogous result for automorphic representations on higher rank groups.
\end{rmk}

\begin{defn}
    Back in the setting described before the remark, fix a compact open subgroup $K$ of $G$.
    We say $\pi \in \mathcal M(G)$ is \textbf{$K$-spherical} if $\pi$ is irreducible and $\pi^K \neq 0$.
\end{defn}

If we let $K = G(\mathcal O)$, do $K$-spherical representations exist?
The answer is yes, and to construct them we use the Iwasawa decomposition.

\begin{lem}[Iwasawa decomposition]
    With notation as above, if $G$ is split, then $G(F) = B(F)G(\mathcal O)$.
\end{lem}

\begin{proof}[Proof for $\text{GL}_n$]
    I claim it suffices to show that $G(\mathcal O)$ surjects onto $G/B(\mathcal O)$.
    Recall that $G/B$ is a projective variety, and as such we can ``clear denominators'' in projective coordinates to get $G/B(\mathcal O) = G/B(F)$.
    Now $G(F) \to G/B(F)$ is surjective with kernel $B(F)$, so surjection of $G(\mathcal O)$ onto $G/B(\mathcal O) = G/B(F)$ will imply $G(F) = B(F)G(\mathcal O)$.
    Now a flag in $G/B(\mathcal O)$ must look like 
    \[ 0 \subset L_1 \subset L_2 \subset \cdots \subset L_n = R^n, \]
    where $L_i$ is a free $R$-module of rank $i$, which is a direct summand of $L_n$.
    In particular, such a flag is in the image of the map $G(\mathcal O) \to G/B(\mathcal O)$.
\end{proof}

Consider a character $\chi:T(F)/T(\mathcal O) \to \cmplx^\times$.
I claim that $\Ind_B^G(\chi)^K \neq 0$.
To see this, define $\varphi_\chi \in \Ind_B^G\chi$ by $\varphi_\chi(bk) = \chi(b)$, where $bk \in B(F)G(\mathcal O) = G(F)$.
In fact, this definition is forced upon us.
Note that this is well-defined, since $\chi|_{B\cap K}$ is trivial.

\begin{rmk}
    This example does not quite produce $K$-spherical representations, since $\Ind_B^G\chi$ is not necessarily irreducible.
    But as for the finite group case of subsection \ref{finitereps}, $\Ind_B^G\chi$ will be irreducible if and only if $\chi \neq \chi^w$.
    We will not provide a proof of this.
\end{rmk}

If we pursued this remark further, we would arrive at the 

\begin{thm}[Borel-Matsumoto-Casselman]\label{ups} With notation as above,
    \begin{enumerate}
        \item Any unramified principal series $\Ind_B^G\chi$ with $\chi$ unramified has a unique $K = G(\mathcal O)$-spherical subquotient.
        \item Any $K$-spherical representation embeds into an unramified principle series.
    \end{enumerate}
\end{thm}

This theorem provides one way to classify $K$-spherical representations for $K = G(\mathcal O)$.
Later we will give a different classification, which is more directly related to the Satake isomorphism.