Jessica Fintzen: Representation of p-adic Groups

Definition 1. A smooth representation of $G$ is a pair $(\pi,V)$ consisting of

• $V$: a $\mathbb{C}$-vector space
• $\pi: G\rightarrow \operatorname{Aut}(V)$ group homomorphism such that for any $v\in V$, there exists open compact subgroup $K$ of $G$, such that $\pi(k)v=v$ for any $k\in K, v\in V$.

Remark 2. A basis of open neighborhood of $1\in \operatorname{GL}_ {n}(F)$ is given by $1+\varpi^{N}\operatorname{Mat}_ {n\times n}(\mathscr{O}).$

Example 3. Let $V=\mathbb{C}$ and $\operatorname{triv}:=\pi: \mathbb{G}\rightarrow 1\in \mathbb{C}^{\times}=\operatorname{Aut}(V)$, called trivial representation.

Definition 4. Let $P=M\ltimes N$ be a parabolic subgroup with Levi subgroup $M$ and unipotent radical $N$ and a smooth representation $(\sigma,V_ {\sigma})$ of $M$. Then the parabolic induction $(\operatorname{Ind}_ {P}^{G}\sigma, \operatorname{Ind}_ {P}^{G} V_ {\sigma})$ is the smooth representation $$\operatorname{Ind}_ {P}^{G}V_ {\sigma}:=\{f: G\rightarrow V_ {\sigma}: f(mng)=\sigma(m)f(g), \forall m\in M,n\in N,g\in G; \exists K_ {f}\subseteq G \text{ compact open, s.t. }f(gk)=f(g)\quad \forall k\in K_ {f}\}.$$

Definition 5. A supercuspidal representation $(\pi, V)$ of $G$ is an irreducible smooth representation such that $(\pi,V)$ does not embed into any $(\operatorname{Ind}_ {P}^{G}\sigma, \operatorname{Ind}_ {P}^{G}V_ {\sigma})$ for any proper parabolic subgroup $P=M\ltimes N$ and any irreducible smooth representation $(\sigma,V_ {\sigma})$ of $M$.

Remark 6. Over mod $\ell$ coefficients, cuspidal (not sub of any proper parabolic induction) and supercuspidal (not subquotient of any proper parabolic induction) are different notions.

Example 7. For $G=\operatorname{SL}(F)$,

1. $M=G$, and $\operatorname{Rep}(G)_ {[G,\sigma]} = \{\sigma,\sigma\oplus\sigma,\dots\}.$
2. $M=T=\begin{pmatrix}\ast & 0 \\ 0 &\ast\end{pmatrix}$, $B=\begin{pmatrix}\ast & \ast \\ 0 &\ast\end{pmatrix},$ $\operatorname{Rep}(G)_ {[T,\text{triv}]}\ni \operatorname{Ind}_ {B}^{G}(\text{triv}), \text{triv},\mathrm{St}$. This is called principal block. $$0\rightarrow \text{triv} \rightarrow \operatorname{Ind}_ {B}^{G}(\text{triv})\rightarrow \mathrm{St}\rightarrow 1.$$

Definition 8. A BT triple is a triple $x=(T,\{X_ {\alpha}\}_ {\alpha\in\Phi(G,T)},\chi_ {\text{BT}})$

1. $T\subseteq G$ is a split maximal torus, e.g. $g\begin{pmatrix} \ast & & 0 \\ &\ddots & \\ 0 & &\ast \end{pmatrix}g^{-1}\subseteq \operatorname{GL}_ {n}(F)$,
2. $X_ {\alpha}\in \operatorname{Lie}(G)_ {\alpha}$ and $\{X_ {\alpha}\}$ form a Chevalley system,
3. $\chi_ {\text{BT}}\in \mathbb{X}_ {\ast}(T)\otimes_ {\mathbb{Z}}\mathbb{R}:=\operatorname{Hom}(\mathbb{G}_ {m},T)=\operatorname{Hom}_ {F}(F^{\times},T).$

Example 9. Take $G=\operatorname{SL}_ {2}(F)$,

1. $x_ {1}=(T=\begin{pmatrix} \ast & 0 \\ 0 & \ast \end{pmatrix}, \{X_ {\alpha}= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},X_ {-\alpha}=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\},\chi_ {\text{BT}}=0)$.

The root $$\alpha: \begin{pmatrix} t & 0 \\ 0 & t^{-1} \end{pmatrix}\mapsto t^{2}.$$ Then $$x_ {\alpha}: x\mapsto \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$$ and $$U_ {\alpha,x,r}=\begin{pmatrix} 1 & \varpi^{\lceil r \rceil}\mathscr{O} \\ 0 & 1 \end{pmatrix}$$ and $$ U_ {-\alpha,x,r}=\begin{pmatrix} 1 & 0\\ \varpi^{\lceil r \rceil}\mathscr{O} & 1 \end{pmatrix} $$

2. $x_ {2}:=(T, \{X_ {\alpha},X_ {-\alpha}\}, \chi_ {\text{BT}}=\frac{1}{4}\alpha^{\vee})$. Then $\alpha(\chi_ {\text{BT}})=\frac{1}{2}.$ Then $$U_ {\alpha,x,r}=\begin{pmatrix} 1 & \varpi^{\lceil r-\frac{1}{2}\rceil}\mathscr{O} \\ 0 & 1 \end{pmatrix}$$ and $$U_ {\alpha,x,r}=\begin{pmatrix} 1 & 0 \\ \varpi^{\lceil r+\frac{1}{2}\rceil}\mathscr{O} & 1 \end{pmatrix}$$

Example 10. For $G=\operatorname{SL}_ {2}(F)$,

1. $G_ {x_ {1},0}=\operatorname{SL}_ {2}(\mathscr{O})$, and $G_ {x,r}= \begin{pmatrix} 1+\varpi^{\lceil r\rceil}\mathscr{O} & \varpi^{\lceil r\rceil}\mathscr{O} \\ \varpi^{\lceil r\rceil}\mathscr{O} & 1+\varpi^{\lceil r\rceil}\mathscr{O} \end{pmatrix}$

2. $G_ {x_ {2},0}=\begin{pmatrix} \mathscr{O} & \mathscr{O} \\ \varpi \mathscr{O} & \mathscr{O} \end{pmatrix}$ and similarly for $G_ {x,r}$.

Example 11. Take $G=\operatorname{SL}_ {2}$ as before. Then $G_ {x_ {1},0}/G_ {x_ {1},0+}= \operatorname{SL}_ {2}(\mathbb{F}_ {q})$ and $G_ {x_ {2},0}/G_ {x_ {2},0+}\cong \begin{pmatrix} \mathbb{F}_ {q} & 0 \\ 0 & \mathbb{F}_ {q} \end{pmatrix}_ {\text{det}=1}$

Definition 12. The (reduced) Bruhat-Tits buiding $\mathscr{B}(G,F)$ is as a set $\{\text{BT triples}\}/\sim$, where $x_ {1}\sim x_ {2}$ if and only if $G_ {x_ {1},r}=G_ {x_ {2},r}$ for any $r\geq 0$. Therefore, we can write $G_ {x,r}$ for $x\in \mathscr{B}(G,F)$.

Remark 13. Fix $T$, let $\mathscr{A}(T,F):=(T,\{X_ {\alpha}\},\chi_ {\text{BT}})/\sim$ with $T$ fixed, then $\mathscr{A}(T,F)=(T,\{X_ {\alpha}\},\chi_ {\text{BT}})/\sim$ with $T$ and $X_ {\alpha}$ fixed, and $\chi_ {\text{BT},1}\sim \chi_ {\text{BT},2}$ if and only if $$\chi_ {\text{BT},1}-\chi_ {\text{BT},2}\in \mathbb{X}_ {\ast}(Z(G))\otimes\mathbb{R}.$$

In particular, one can equip $\mathscr{A}(T,F)$ with structure of an affine space over $\mathbb{X}_ {\ast}(T)\otimes_ {\mathbb{Z}}\mathbb{R}/X_ {\ast}(Z(G))\otimes_ {\mathbb{Z}}\mathbb{R}$.

Definition 14. Let $(\pi,V)$ be an irreducible smooth representation of $p$-adic group $G$. The depth of $(\pi,V)$ is the smallest $r\in \mathbb{R}_ {\geq 0}$ such that $V^{G_ {x,r+}}\neq 0$ for some $x\in \mathscr{B}(G,F)$.

Theorem 15 (Moy-Prasad). Let $x\in \mathscr{B}(G,F)$ be a vertex, $G_ {x}:=\operatorname{Stab}_ {G}(x)$. Let $(\rho,V_ {\rho})$ be an irreducible representation of $G_ {x}$, such that

1. $\rho|_ {G_ {x,0+}}=\text{triv}$,
2. $\rho|_ {G_ {x,0}}$ is a cuspidal representation of $G_ {x,0}/G_ {x,0+}$. Then $\text{c-ind}_ {G_ {x}}^{G}\rho$ is an irreducible supercuspidal representation of $G$ of depth 0 and all depth 0 supercuspidal representations arise in this way.

Remark 16. If $G$ is simply connected, then $G_ {x}=G_ {x,0}$.

Example 17. For $G=\operatorname{SL}_ {2}(\mathbb{Q}_ {p})$, $p\neq 2$ and $n=1$, take $G^{0}=T^{\text{an}}=\{\begin{pmatrix} a & b \\ pb & a \end{pmatrix}\in \operatorname{SL}_ {2}(\mathbb{Q}_ {p}) \}$ and splits over $E=\mathbb{Q}_ {p}(\sqrt{p})$ (conjugate to $\begin{pmatrix} a + b\sqrt{p} & 0 \\ 0 & a-b\sqrt{p} \end{pmatrix}_ {\operatorname{det}=1}$).

We can take $x$ to be “the mid point” in the Bruhat-Tits tree.

For $r=\frac{1}{2}$, can take $\phi$ to be $$\begin{pmatrix} a & b \\ pb & a \end{pmatrix}\mapsto \varphi(2ab)$$ where $\varphi: \mathbb{Z}_ {p}\rightarrow \mathbb{C}^{\times }$ sending $p\mathbb{Z}_ {p}$ to $1$.

Theorem 18 (Yu 2001, Fintzen 2021 for $p\neq 2$, Fintzen-Schwein 2025 for $p=2,q\neq 2$). The representation $\operatorname{c-ind}_ {\widetilde {K} }^{G}\widetilde {\rho} $ is irreducible supercuspidal representation.

Theorem 19 (Kim 2007 for $p>>0$, $\text{char}(F)=0$, Fintzen 2021). If $p\nmid |\text{Wely group of $G$}|$, then all supercuspidal representations arise in this way.

Remark 20. Same for $\overline{\mathbb{F}_ {\ell}}$-representations for $\ell\neq p$.

Definition 21. A pair $(K,\rho)$ consisting of a compact open subgroup $K\subseteq G$ and an irreducible representation $(\rho,V_ {\rho})$ of $K$ is an $[M,\sigma]$-type, if for all irreducible representation $(\pi,V)$ of $G$, the following are equivariant:

1. $\pi\in \operatorname{Rep}(G)_ {[M,\sigma]}$,
2. $\rho\hookrightarrow \rho|_ {K}$, i.e. $\operatorname{Hom}_ {G}(\operatorname{c-ind}_ {K}^{G}\rho,\pi)\cong \operatorname{Hom}_ {K}(\rho,\pi)\neq 0$.

Example 22. For $G=\operatorname{SL}_ {2}(F)$, $T=\begin{pmatrix}\ast & 0 \\ 0 & \ast \end{pmatrix}$, $(I_ {w},\text{triv})$ is a $[T,\text{triv}]$-type, where $I_ {w}$ is the standard Iwahori subgroup.

Theorem 23 (Bushnell-Kuszko,1998). If $(K,\rho)$ is an $[M,\sigma]$-type, then $$\operatorname{Rep}(G)_ {[M,\sigma]}\cong \operatorname{Mod}\text{-}\mathscr{H}(G,K,\rho),$$ where the Hecke algebra $$\mathscr{H}(G,K,\rho) = \{f: G\rightarrow \operatorname{End}(V_ {\rho}): f(kgk^{\prime})=\rho(k)f(g)\rho(k^{\prime}),\forall k,k^{\prime}\in K, g\in G, f\, \text{compactly supported}\}$$ equipped with convolution product.

Example 24. Take $G=\operatorname{SL}_ {2}(F)$,

• $M=G$, and $\sigma= \operatorname{c-ind}_ {\widetilde {K}^{G}\rho}$, then $\mathscr{H}(G,\widetilde {K},\rho)\cong \operatorname{End}(\operatorname{c-ind}_ {\widetilde {K}^{G}\rho})$ and $(\widetilde {K},\rho)$ is a type for $[G,\sigma]$.
• Take $[T,\sigma=\text{triv}]$, then $\mathscr{H}(G,I_ {w},\text{triv})\cong \mathscr{H}_ {\text{aff}}(W_ {\text{aff}},q)$ is the Iwahori-Hecke algebra.

Theorem 25 (Kim-Yu 2017, Fintzen 2021). Assume $G$ splits over a tame extension as before. A construction analagous to Yu’s construction (but with more general input) yields an $[M,\sigma]$-type. If $p\nmid |\text{Weyl group of $G$}|$, then for any $[M,\sigma]$, there exists a type as above.

Theorem 26 (Adler-Fintzen-Mishra-Ohara, 2024). There exists a subgroup $N^{\heartsuit}\subseteq N_ {G^{0}}(M^{0},(M^{0}_ {x})_ {\text{cpt}})$, such that $$K^{0}\backslash \operatorname{supp}(\mathscr{H}(G^{0},K^{0},\rho^{0}))/K^{0}\xleftarrow {\cong}N^{\heartsuit}/N^{\heartsuit}\cong (M_ {x}^{0})_ {\text{cpt}}=:W^{\heartsuit}.$$

Theorem 27 (Adler-Fintzen-Mishra-Ohara, 2024). There is a representation $\widetilde {K}: N^{\heartsuit}(K\cap M)\rightarrow \operatorname{End}(V_ {K})$, such that $\widetilde {K}|_ {K\cap M}=K$, and there is an isomorphism of algebras $$J:\mathscr{H}(G^{0},K^{0},\rho^{0})\xrightarrow{\cong}\mathscr{H}(G,K,\rho),$$ given by the following:

• If $\varphi\in \mathscr{H}(G^{0},K^{0},\rho^{0})$ is supported on $K^{0}nK^{0}$ with $n\in N^{\heartsuit}$, then $J(\varphi)$ is supported on $KnK$, and $J(\varphi)(n) = d_ {n}\varphi(n)\otimes \widetilde {K}(n),$ where $d_ {n}=\sqrt{\frac{|K^{0}/nK^{0}n^{-1}\cap K^{0}|}{|K/nKn^{-1}\cap K|}}.$

Corollary 28. There is an equivalence $$\operatorname{Rep}(G)_ {[M,\sigma]}\cong \operatorname{Rep}(G^{\circ})_ {[M^{0},\sigma^{0}]},$$ where $[M,\sigma]$ corresponds to $(K,\rho)$.

Theorem 29. There is an isomorphism $W^{\heartsuit}\cong W(\rho)_ {\text{aff}}\rtimes \Omega(\rho)$, and $\mathscr{H}(G,K,\rho) \cong \mathscr{H}_ {\text{aff}}((N(\rho))_ {\text{aff}}, \{qs\})\rtimes \mathbb{C}[\Omega(\rho),\mu]$ for some 2-cocycle $\mu$ and some $qs\in \mathbb{Q}_ {>1}$, and $s\in $set of simple refecltions of $W(\rho)_ {\text{aff}}$.