Let $G/F$ be a reductive group, where $F$ is a non-archimedean local field. Then $G(F)$ is a locally compact topological group. Recall that a smooth representation of $G(F)$ (over $\mathbb{C}$) is a complex vector space $V$ together with a group homomorphism $$ \rho \colon G(F) \to \Aut_\mathbb{C}(V) $$ satisfying the property that for every $v \in V$ the stabilizer $\lbrace g \in G(F) : gv = v \rbrace \subseteq G(F)$ is open.
Example 1. For an open compact $K \subseteq G(F)$, we can take $\sigma$ a finite-dimensional representation of $K$ (with open kernel) and consider the compact induction $$ \operatorname{cInd}_ K^{G(F)} \sigma = \lbrace f \colon G(F) \to V \text{ with compact support} : f(kg) = \sigma(k)(f(g)) \rbrace. $$ This can be checked to be a smooth representation.
We want to understand them in terms of group algebras, but the abstract group algebra doesn’t take into account the topology on $G$.
Definition 2. For $K, \sigma$ as above, we define the Hecke algebra as $$ \mathcal{H}_ {(K,\sigma)} = \End_{G(F)}(\operatorname{cInd}_ K^{G(F)} \sigma). $$
We can then check that $$ \mathcal{H}_ {(K,\sigma)} = \lbrace \Phi \colon G(F) \to \End V \text{ with compact support} : \Phi(k_1 g k_2) = \sigma(k_1) \Phi(g) \sigma(k_2) \rbrace, $$ where the action is given by $$ (\Phi \ast f)(g) = \sum_{g^\prime \in G/K} \Phi(g g^\prime) (f(g^{\prime-1})) = \sum_{g^\prime \in G/K} \Phi(g^\prime) (f(g^{\prime-1} g)). $$ The algebra structure also can be written out as $$ (\Phi_1 \ast \Phi_2)(g) = \sum_{g^\prime \in G/K} \Phi_1(g g^\prime) \Phi_2(g^{\prime-1}). $$
Lemma 3. For every representation $V$ of $G(F)$ (not necessarily smooth) we have an isomorphism $$ \Hom_{G(F)}(\operatorname{cInd}_ K^{G(F)} \sigma, V) = \Hom_K(\sigma, V \vert_K). $$
So for $V$ smooth, every $v \in V$ is stabilized by some $K$, and then we have a map $\operatorname{cind}_ K^{G(F)} 1 \to V$. If $V$ is irreducible and $v$ is nonzero, then this map must be surjective.
Remark 4. The category $\mathsf{Rep}(G(F))$ of smooth representations form an abelian category, and moreover these $\operatorname{cInd}_ K^{G(F)} \sigma$ are projective.
Corollary 5. The functor $$ D(\mathcal{H}_ {(K,\sigma)}^\mathrm{op}) \to D(\mathsf{Rep}(G(F))); \quad M \mapsto M \otimes^\mathbb{L} \operatorname{cInd}_ K^{G(F)} \sigma $$ is fully faithful.
Remark 6. We can define some $$ C_c^\infty(G(F)) = \bigcup_K C_c^\infty(K \backslash G(F) / K) $$ and then $\mathsf{Rep}(G(F))$ is just the category of non-degenerate $C_c^\infty(G(F))$-modules.
So it is important to study $\mathcal{H}_ {(K,\sigma)}$, but this is actually pretty complicated. So let us start with some simple examples.
Example 7. Let $G/F$ be a split reductive group, so that $K = G(\mathcal{O})$ makes sense. Let $\sigma$ be the trivial representation of $K$ and let $T \subseteq B \subseteq G$ be as usual. Then the Satake isomorphism states that $$ \mathcal{H}_ K \cong \mathbb{C}[X_\ast(T)]^W $$ as algebras. In particular, $\mathcal{H}_ K$ is commutative. To see this, we note that there is a Cartan decomposition $$ K \backslash G(F) / K \cong X_\ast(T)^+ $$ where $\lambda \colon \mathbb{G}_ m \to T$ corresponds to $K \lambda(\varpi) K$. But $1_{K\varpi^\lambda K}$ doesn’t just correspond to $\sum_{w \in W} e^{w(\lambda)}$.
The Iwahori–Hecke algebra
Let us now consider the case of $K = I$ a Iwahori subgroup. For us, let us focus on the case when $G$ is split. Then we can define the Iwahori $I \subseteq G(\mathcal{O})$ as the preimage of $B(k) \subseteq G(k)$ under the reduction map $G(\mathcal{O}) \twoheadrightarrow G(k)$. Then we again look at the Hecke algebra $$ \mathcal{H}_ I \cong C_c(I \backslash G(F) / I). $$ This is more complicated than the previous case, but it is still well-understood. This time, we have a bijection $$ I \backslash G(F) / I \cong \tilde{W}, $$ where $\tilde{W} = N_G(T)(F) / T(\mathcal{O})$ is the extended affine Weyl group or the Iwahori–Weyl group. This sits in a short exact sequence $$ 1 \to T(F)/T(\mathcal{O}) \cong X_\ast(T) \to \tilde{W} \to N_G(T)(F) / T(F) = W \to 1. $$ This also has a canonical splitting because $$ W = N_G(T)(\mathcal{O})/T(\mathcal{O}) \hookrightarrow N_G(T)(F) / T(\mathcal{O}). $$ So we can write $$ \tilde{W} = X_\ast(T) \rtimes W. $$
Remark 8. If $G = G_\mathrm{sc}$ is simply connected, then $\tilde{W}$ agrees with the affine Weyl group $W_\mathrm{aff}$, which is a Coxeter group.
Fact 9. Assuming $G$ is split, the size of $IwI / I \cong I / (I \cap wIw^{-1})$ is $q^{\ell(w)}$.
In general, we can let $$ \Omega = \lbrace w \in \tilde{W} : \ell(w) = 0 \rbrace \cong N_{G(F)}(I) / I $$ which is a subgroup of $\tilde{W}$. Then we have a short exact sequence $$ 1 \to W_\mathrm{aff} \to \tilde{W} \to \tilde{W}/W_\mathrm{aff} \to 1, $$ where $\Omega$ maps isomorphically to $\tilde{W}/W_\mathrm{aff}$.
We now define $$ T_w = 1_{IwI} \in C_c(I \backslash G(F) / I) $$ so that $T_w$ form a basis of $\mathcal{H}_ I$.
Theorem 10. For $G = G_\mathrm{sc}$ so that $\tilde{W} = W_\mathrm{aff}$, the Hecke algebra $\mathcal{H}_ I$ has the following relations:
- $(T_s - q) (T_s + 1) = 0$ for $s$ a simple reflection,
- $T_s T_t \dotsb = T_t T_s \dotsb$ for $s, t \in S$, where the number of terms on both sides are $n_{s,t}$.
In this case, $T_{wv} = T_w T_v$ when $\ell(wv) = \ell(w) + \ell(v)$. In general, when $G$ is not necessarily simply connected, we have $$ \mathcal{H}_ I \cong \mathcal{H}_ \mathrm{aff} \rtimes \mathbb{C}[\Omega], $$ where $T_v T_\omega = T_\omega T_{\omega^{-1} v \omega}$ for $\omega \in \Omega$.