Recall we are working with a reductive group $G/F$ with $B, T$, which are actually all defined over $\mathcal{O}$. Then we had the Iwahori $I$ which is the preimage of $B(k_F)$ under $G(\mathcal{O}) \to G(k_F)$. Then we looked at the Iwahori–Hecke algebra $$ \mathcal{H}_ I = C_c(I \backslash G(F) / I, \mathbb{C}). $$ To study this we looked at the $\tilde{W} = N_G(T)(F) / T(\mathcal{O})$ and the decomposition $$ G(F) = \coprod_{w \in \tilde{W}} I w I. $$ Given a cocharacter $\lambda \in X_\ast(T)$, we get a map $F^\times \to T(F)$ so we get $\lambda(\varpi) \in T(F)$, which defines an element $t_\lambda \in T(F) / T(\mathcal{O})$. We then get a short exact sequence $$ 1 \to X_\ast(T) \to \tilde{W} \to W \to 1. $$ This also has a splitting using $W \cong N_G(T)(\mathcal{O}) / T(\mathcal{O})$. Then $\tilde{W} = X_\ast(T) \rtimes W$. We also have $$ \begin{CD} 1 @>>> Q^\vee @>>> W_\mathrm{aff} @>>> W @>>> 1 \br @. @VVV @VVV @| \br 1 @>>> X_\ast(T) @>>> \tilde{W} @>>> W @>>> 1 \br @. @VVV @VVV \br @. X_\ast(T)/Q^\vee @>{\cong}>> \pi_0(\tilde{W}) \cong \Omega. \end{CD} $$ where $Q^\vee$ is the coroot lattice and $W_\mathrm{aff}$ is the affine Weyl group, which is a Coxeter group. Here $\Omega$ is also a subgroup of $\tilde{W}$ by identifying $\Omega = N_{G(F)}(I) / I$. It is also the subgroup of length zero elements.
In terms of Bruhat–Tits theory
In general, let $\Phi(G, T) \subseteq X^\ast(T)$ be the set of roots. We then write $$ \Phi_\mathrm{aff}(G, T) = \lbrace a + k : a \in \Phi(G, T), k \in \mathbb{Z} \rbrace $$ viewed as functions on $$ A(G, T) = X_\ast(T^\mathrm{ad}) \otimes \mathbb{R} \cong Q^\vee \otimes \mathbb{R}. $$ For each $a \in \Phi(G, T)$ there is an isomorphism $U_a \cong \mathbb{G}_ a$ up to multiplication by $\mathcal{O}^\times$. Then we can defined for $\alpha = a + k \in \Phi_\mathrm{aff}(G, T)$ the subgroup $$ U_\alpha \cong \varpi^k \mathcal{O} \subseteq F \cong U_a(F). $$ This is well-defined and independent of choices. Then we can also write $$ I = \prod_{\alpha = a+1, a \in \Phi^-} U_\alpha \times T(\mathcal{O}) \times \prod_{\alpha = a, a \in \Phi^+} U_\alpha. $$
Inside $A(G, T)$ we can look at all the hyperplanes $H_\alpha = \lbrace x \in A(G, T) : \alpha(x) = 0 \rbrace$ and take the complement of these. This is a disjoint union of convex connected components, which are called alcoves. There the fundamental alcove, which is the unique one $C$ satisfying $0 \in \bar{C}$ and $a(C) \ge 0$ for all $a \in \Phi^+$. For each $v \in A(G, T)$, we can define the parahoric $$ P_v = \langle T(\mathcal{O}), U_\alpha \rangle_{\alpha(v) \ge 0}. $$ We check that $P_v = I$ for $v \in C$ and $P_0 = G(\mathcal{O})$.
We can define an action of $\tilde{W}$ on $A(G, T)$ by affine transformations. Writing $\bar{\lambda}$ for the image of $\lambda \in X_\ast(T)$ under $X_\ast(T) \to X_\ast(T^\mathrm{ad})$, we can use the semidirect decomposition $\tilde{W} = X_\ast(T) \rtimes W$ and use the actions $$ t_\lambda(v) = v - \bar{\lambda}, \quad w(v) = w(v). $$ Then the $\tilde{W}$-action preserves $\Phi_\mathrm{aff}(G, T)$.
Lemma 1. For $w \in \tilde{W}$, if we lift it to $\dot{w} \in N_G(T)(F)$ then $\dot{w} U_\alpha \dot{w}^{-1} = U_{w(\alpha)}$.
Now we can check that $\Omega$ is the collection of $w \in \tilde{W}$ for which $w(C) = C$, and $\ell(w)$ is the number of $\alpha \in \Phi_\mathrm{aff}$ such that $\alpha \in \Phi_\mathrm{aff}^+$ and $w(\alpha) \in \Phi_\mathrm{aff}^-$. Here, $\Phi_\mathrm{aff}^+$ is the set of $\alpha$ satisfying $\alpha(C) \gt 0$. Then $w I w^{-1} = P_{w(v)}$.
For every $\alpha = a + k \in \Phi_\mathrm{aff}$, we have the reflection $$ s_\alpha = t_{ka^\vee} s_a, \quad s_\alpha(v) = v - \alpha(v) a^\vee. $$ Then $s_\alpha^2 = \id$ and is an affine reflection with fixed points $H_\alpha = \lbrace \alpha = 0 \rbrace$. We define the affine Weyl group as $W_\mathrm{aff} = \langle s_\alpha \rangle_{\alpha \in \Phi_\mathrm{aff}}$.
Fact 2. There exists a subset $\Delta_\mathrm{aff} \subseteq \Phi_\mathrm{aff}^+$ called the affine simple roots with the property that every element of $\Phi_\mathrm{aff}^+$ is a unique linear combination with nonnegative integral coefficients.
When $G$ is simple, we can take $\alpha_0 = 1 - \theta$ and $\alpha_i = a_i$ where $a_i$ are the simple roots and $\theta$ is the highest root. But when $G$ is not simple, we can have $\lvert \Delta \rvert$ bigger than $\dim T + 1$.
The Iwahori–Matsumoto presentation
Let us write $T_w = 1_{TwT} \in \mathcal{H}_I$. Then we have
- $T_w T_v = T_{wv}$ for $\ell(w) + \ell(v) = \ell(wv)$,
- $T_s^2 = (q-1) T_s + q$, and this gives the Iwahori–Matsumoto presentation of $\mathcal{H}_ I$.
Why do these things satisfy this relation? For the first part, we need to check that the multiplication $$ IwI \times^I IvI \to G(F) $$ actually lands in $IwvI$, and moreover the it is a bijection onto it. This can be checked using our definition of length. For the second part, we similarly need to show that $$ IsI \times^I IsI \to G(F) $$ has preimage of size $q-1$ over $s$ and preimage of size $q$ over $1$. This will later have a geometric interpretation in terms of $\mathbb{G}_ m$ and $\mathbb{G}_ a$.
Corollary 3. We have $$ \mathcal{H}_ I \cong H_\mathrm{aff} \rtimes \mathbb{C}[\Omega] \cong \mathbb{C}[X_\ast(T)] \rtimes H_w. $$
The second presentation is called the Bernstein presentation. But again we need to be careful because $\lambda \in \mathbb{C}[X_\ast(T)]$ corresponds to $T_{t_\lambda}$ only when $\lambda$ is dominant (or anti-dominant depending on convention).
We have an equivalence of categories $$ \mathcal{H}_ I\mathsf{-mod} \xrightarrow{\cong} \mathsf{Rep}(G(F))^{[I]}; \quad M \mapsto M \otimes_{\mathcal{H}_ I} \operatorname{cInd}_ I^{G(F)} 1. $$ In particular, if $W \subseteq V$ and $V$ is generated by $V^I$ then $W$ is generated by $W^I$. This is not true if we replace $I$ by $G(\mathcal{O})$. This is also saying that $\mathsf{Rep}(G(F))^{[I]}$ is a block of $\mathsf{Rep}(G(F))$.
Twisted Iwahori–Hecke algebra
We now consider $$ I^+ \subseteq I \subseteq G(\mathcal{O}) $$ the preimage of $U(k_F) \subseteq B(k_F) \subseteq G(k_F)$. This satisfies $I / I^+ \cong T(k_F)$. Now we can look at the Hecke algebra $$ \mathcal{H}_ {I^+} = C_c(I^+ \backslash G(F) / I^+) = \End_{G(F)}(\operatorname{cInd}_ {I^+}^{G(F)} 1). $$ But this has a further decomposition. We note that $$ \operatorname{cInd}_ {I^+}^{G(F)} 1 = \operatorname{cInd}_ I^{G(F)} \operatorname{Ind}_ {I^+}^I 1, $$ and then because $I / I^+$ is an abelian group, we have $$ \operatorname{Ind}_ {I^+}^I 1 = \bigoplus_{\chi \colon T(k_F) \to \mathbb{C}^\times} \chi, \quad \operatorname{cInd}_ {I^+}^{G(F)} = \bigoplus_\chi \operatorname{cInd}_ I^{G(F)} \chi. $$ So we might want to look at $$ \mathcal{H}_ {(I,\chi)} = \operatorname{End}(\operatorname{cInd}_ I^{G(F)} \chi) = \lbrace \Phi \colon G \to \mathbb{C} : \Phi(k g k^\prime) = \chi(k) \Phi(g) \chi(k^\prime) \rbrace. $$ But not all elements $\tilde{W}$ can support a function.
Lemma 4. Let $\sigma$ be a representation of $K$ and consider $$ \mathcal{H}_ {(K,\sigma)} = \lbrace \Phi \colon G(F) \to \End(V) : \Phi(k_1 g k_2) = \sigma(k_1) \Phi(g) \sigma(k_2) \rbrace. $$ We can decompose $\mathcal{H}_ {(K,\sigma)} = \bigoplus_{g \in K \backslash G / K} \mathcal{H}_ {K,\sigma,g}$ corresponding to the supports. Let $K_g = K \cap gKg^{-1}$. Then we have an isomorphism $$ \mathcal{H}_ {K,\sigma,g} \cong \Hom_{K_g}(V \vert_{K_{g^{-1}}}, V \vert_{K_g}), $$ where we use $K_g \cong K_{g^{-1}}$ given by conjugation by $g^{-1}$.
Now in our case, we have $\mathcal{H}_ {I,\chi,w} \neq 0$ if and only if $$ \Hom_{I_w}(\chi \vert_{I_{w^{-1}}}, \chi \vert_{I_w}) \neq 0. $$ On the other hand, we see that $I_w \hookrightarrow I$ surjects onto $T(k_F)$.
Corollary 5. We have $$ \mathcal{H}_ {I,\chi} = \bigoplus_{w \in \tilde{W}, {}^w \chi = \chi} \mathcal{H}_ {I,\chi,w}, $$ where ${}^w \chi$ corresponds to the action of $\tilde{W} \twoheadrightarrow W$ on $T(k_F)$.
We will later see that $\tilde{W}_ \chi = \lbrace w \in \tilde{W} : {}^w \chi = \chi \rbrace$ also has the structure of a quasi-Coxeter group.