We were trying to understand the vector space structure of ${}_ \chi \mathcal{H}_ {\chi^\prime}$, which is 1-dimensional over each element of ${}_ \chi \tilde{W}_ {\chi^\prime} = \lbrace w \in \tilde{W} : w \chi^\prime = \chi \rbrace$. This has a surjection to the subgroup $$ {}_ \chi \Omega_{\chi^\prime} = \tilde{W}_ \chi^0 \backslash {}_ \chi \tilde{W}_ {\chi^\prime} = {}_ \chi \tilde{W}_ {\chi^\prime} / \tilde{W}_ {\chi^\prime}^0. $$ Here, $\tilde{W}_ \chi^0$ is the subgroup generated by $s_\alpha$ for $\alpha \in \Phi_{\mathrm{aff}, \chi}$.
Note that for each reflection $s_\alpha$, there exists a simple reflection $s_i \in W_\mathrm{aff}$ and $w \in W_\mathrm{aff}$ such that $\ell(s_\alpha) = 2 \ell(w) + 1$. Recall that for a reduced word we have $s_1 \dotsm s_\ell$ in $\tilde{W}$, we can calculate $\ell_\chi(s_\alpha)$ by counting the number of $i$ such that $s_i$ is in some ${}_ {\chi_i} \tilde{W}_ {\chi_{i+1}}^0$. It follows that $\ell_\chi(s_\alpha) = 1$ implies $w \in {}_ \chi \Omega_ {w^{-1} \chi}$ has length zero and $s_i \in {}_ {w^{-1} \chi} \tilde{W}_ {w^{-1} \chi}$. For each simple reflection $s_j$, we choose a lifting $\dot{s}_ j$, and then we set $$ \dot{s}_ \alpha = \dot{s}_ {i_1} \dotsm \dot{s}_ {i_r} \dot{s}_ i \dot{s}_ {i_r}^{-1} \dotsm \dot{s}_ {i_1}^{-1}. $$
Last time we computed
- $T_{\dot{s}_ {i_j}} T_{\dot{s}_ {i_j}^{-1}} = q T_1$,
- $T_{\dot{s}_ \alpha} = T_{\dot{s}_ {i_1}} \dotsm T_{\dot{s}_ {i_r}} T_{\dot{s}_ i} T_{\dot{s}_ {i_r}^{-1}} \dotsm T_{\dot{s}_ {i_1}^{-1}}$,
- $T_{\dot{s}_ i}^2 = (q-1) T_{\dot{s}_ i} + q T_1$.
It follows that we have $$ \Bigl( \frac{T_{\dot{s}_ \alpha}}{q^r} \Bigr)^2 = (q-1) \Bigl( \frac{T_{\dot{s}_ \alpha}}{q^r} \Bigr) + q T_1. $$
Construction of a 1-dimensional module
Once we construct a 1-dimensional module, we will be able to normalize all the generators and be able to conclude that $$ {}_ \chi \mathcal{H}_ \chi = {}_ \chi \mathcal{H}_ \chi^0 \rtimes \mathbb{C}[{}_ \chi \Omega_ \chi] $$ where ${}_ \chi \mathcal{H}_ \chi^0$ is some affine Hecke algebra.
Let us work with $F = \mathbb{F}_ q((\varpi))$. Consider the global field $K = \mathbb{F}_ q(\varpi)$ and look at $$ {}_ \chi M_\psi = C_c\biggl( G(K) \backslash G(\mathbb{A}_ K) / \prod_{v \neq 0, \infty} G(\mathcal{O}_ v) \times (I_0, \chi) \times (I_\infty^{\mathrm{opp},+}, \psi) \biggr) $$ where by $I^\mathrm{opp}$ we mean the one corresponding to the opposite Borel, and $I^{\mathrm{opp},+}$ is the pro-$p$ part. In general, we have $$ I \supseteq I^+ \supseteq I^{++} = [I^+, I^+] $$ where the first quotient is $T(k_F)$ and the second quotient is $\prod_\alpha U_\alpha(k_F)$ for $\alpha$ is over affine simple roots. Then $$ \psi \colon I_\infty^+ \twoheadrightarrow I_\infty^+ / I_\infty^{++} \to \mathbb{C}^\times $$ is an additive character, and we assume that $\psi$ restricted to each $k_F$ is nontrivial. (These are called affine generic characters.)
Proposition 1 (Gross, Heinloth–Ngo–Yun). There is an isomorphism of $\mathbb{C}$-vector spaces ${}_ \chi M_\psi \cong \mathbb{C}[\Omega]$. In particular, if $G$ is simple and simply connected then this is 1-dimensional.
The point is that every $G$-torsor over $\mathbb{A}^1$ is trivial, and so we get a uniformization $$ G(K) \backslash G(\mathbb{A}_ K) / \prod_{v \neq 0,\infty} G(\mathcal{O}_ v) \times I_0 \times I_\infty^{\mathrm{opp},+} \cong I \backslash G(F) / I_{\infty,\mathrm{pol}}^{\mathrm{opp},+} $$ where we have $$ I_{\infty,\mathrm{pol}} = \lbrace f \colon \mathbb{P}^1 - 0 \to G : f(\infty) \in B \rbrace \subseteq G(\mathbb{P}^1 - 0) \subseteq G(F). $$ Now it is a fact that we have a decomposition $$ G(F) = \coprod_{w \in \tilde{W}} I w I_{\infty,\mathrm{pol}}^{\mathrm{op},+}, $$ where $I I_{\infty,\mathrm{pol}}^{\mathrm{op},+}$ is the open cell where $I \cap I_{\infty,\mathrm{pol}}^{\mathrm{op},+} = 1$.
Lemma 2. For any function $f \in C_c((I, \chi) \backslash G(F) / (I_{\infty,\mathrm{pol}}^{\mathrm{op},+}, \psi))$ the support can be on $I w I_{\infty,\mathrm{pol}}^{\mathrm{op},+}$ if and only if $w \in \Omega$.
Pinning the loop group
For $G/F$ we have a loop group $LG/k_F$ given by $$ LG(R) = G(R((\varpi))), \quad L^+G(R) = G(R[[\varpi]]) $$ This is represented by an ind-affine scheme $LG = \varinjlim_i X_i$ where each $X_i$ is affine (of infinite type) and $X_i \hookrightarrow X_{i+1}$ are finitely presented closed embeddings. We can also define $L^+G \twoheadrightarrow G$ that restricts to $I \twoheadrightarrow B$. We then have $$ I \supseteq I^+ \supseteq I^{++} = [I^+, I^+] $$ where $I^+$ is the pro-unipotent radical. Then we have $I = T \ltimes I^+$.
Definition 3. An affine pinning of $LG$ is a triple $(I, T, \psi)$ where $I \subseteq LG$ is Iwahori, $T \subseteq I$ is a maximal torus, and $\psi \colon I^+ \to \mathbb{G}_ a$ which necessarily factor through $I^+ / I^{++} = \prod_{\alpha} \mathbb{G}_ a$ where $\psi$ satisfies the property that $\psi \vert_{\mathbb{G}_ a}$ is an isomorphism on each factor.
We will now consider $$ N_{LG}(I, T, \psi)(k_F), $$ which will have an action on ${}_ \chi M_\psi$. Then we will be able to cut down the space to something that is one-dimensional. One observation is that all Iwahori are conjugate to each other, and then all maximal tori are conjugate to each other, but there is an $\mathbb{A}^1$-space of pinnings. So this group $M_\psi$ depends only on $\psi$.
Proposition 4. There is a short exact sequence $$ 1 \to Z_G \to M_\psi \to \Omega \to 1 $$ of commutative groups.
Now there is a commuting action of ${}_ \chi \mathcal{H}_ \chi$ and $M_\psi(k_F)$ on ${}_ \chi M_\psi$. Using this, we will be able to take any character of $\Omega$ and cut down the dimension to $1$.