We want to show the following. Given a pinned reductive group $(G, B, T, e)$ inducing $$ LG \supseteq L^+G \supseteq I \supseteq I^+ \supseteq I^{++} \supseteq \dotsb $$ and an affine pinning $\phi \colon I^+/I^{++} \to \mathbb{G}_ a$, we can identify $\phi$ with $$ d\phi \colon V_I = \operatorname{Lie} I^+ / \operatorname{Lie} I^{++} \to k. $$ Recall we defined $M = N_{LG}(I, T)$ which acts on $V_I^\ast$ (which contains $d\phi$), and this sits in a short exact sequence $$ 1 \to T \to M \to \Omega \to 1. $$
Lemma 1. The action of $\Omega$ on $V_I^\ast // T \cong \mathbb{A}^1$ is trivial.
We write $d\phi_\alpha = d\phi \vert_{L\mathfrak{g}_ \alpha} \in (L\mathfrak{g}_ \alpha)^\ast$, so that $d\phi = \sum_\alpha d\phi_\alpha$. We can assume $k = \mathbb{C}$, and so we fix $\mathfrak{g}^\ast \cong \mathfrak{g}$. This induces an isomorphism $L\mathfrak{g} \cong L\mathfrak{g}^\ast$ where $(L\mathfrak{g}_ \alpha)^\ast \cong L\mathfrak{g}_ {-\alpha}$. Then we have $$ V_I^\ast \cong \bigoplus_\alpha L\mathfrak{g}_ {-\alpha}. $$
Recall that there are Kostant sections. We have a principal $\mathfrak{sl}_ 2$-triple $e, f, h \in \mathfrak{g}$, meaning that $f = \sum_a e_{-a}$. (For example, for $\mathfrak{sl}_ n$ we have $f$ the matrix with $1$ on the lower diagonal, and $h$ the diagonal matrix with entries $n-1, n-3, \dotsc, 1-n$.) Then we have a map $$ \mathfrak{s} = f + \mathfrak{g}^e \hookrightarrow \mathfrak{g} \twoheadrightarrow \mathfrak{c} = \mathfrak{g} // G, $$ which is an isomorphism. This is actually a graded isomorphism, and the grading on $\mathfrak{g}^e$ has highest degree $\mathfrak{g}_\theta$ in degree $h$ the Coxeter number, where $\theta$ is the highest root. Now this induces an isomorphism $$ L\mathfrak{s} \cong L\mathfrak{c}. $$
On the other hand, we may idenfity $$ \lbrace f + t^{-1} \lambda e_a : \lambda \in k \rbrace \subseteq \bigoplus_\alpha L\mathfrak{g}_ {-\alpha} \cong V_I^\ast \to V_I^\ast // T $$ in an $\Omega$-equivariant way. So we have an $\Omega$-equivariant embedding $$ V_I^\ast // T \cong \lbrace f + t^{-1} \lambda e_0 : \lambda \in k \rbrace \hookrightarrow L\mathfrak{s} \cong L\mathfrak{c}. $$ On the other hand, the $\Omega$-action on $L\mathfrak{c}$ is trivial, so we are done.
The one-dimensional module
We now come back to $$ {}_ \chi \mathcal{A}_ \phi = C_c \biggl( G(K) \backslash G(\mathbb{A}_ K) / (I_0, \chi) \times (I_\infty^{\mathrm{op},+}, \phi) \times \prod_{v \neq 0,\infty} G(\mathcal{O}_ v) \biggr). $$ This has commuting actions of $\mathcal{H}_ \chi$ and $M_\phi$, and has dimension equal to $\hash \Omega$. Now assuming that $Z_G$ is connected, we can extend the character $\phi$ to $$ (\tilde{\chi}, \phi) \colon M_\phi(k) \ltimes I_\infty^{\mathrm{op},+} \to \mathbb{C}^\times. $$ We can now add in this equivariance condition and define $$ {}_ \chi \mathcal{A}_ {\tilde{\chi}\phi}^\prime = C_c \biggl( G(K) \backslash G(\mathbb{A}_ K) / (I_0, \chi) \times (M_\phi \ltimes I_\infty^{\mathrm{op},+}, \tilde{\chi}\phi) \times \prod_{v \neq 0,\infty} G(\mathcal{O}_ v) \biggr). $$ This is the $1$-dimensional ${}_ \chi \mathcal{H}_ \chi$-module we wanted.
Theorem 2. If $Z_G$ is connected, then ${}_ \chi \mathcal{H}_ \chi$ is isomorphic to the Iwahori–Hecke algebra of $H$, the endoscopic group associated to $\chi$.
Note that ${}_ \chi \mathcal{A}_ {\tilde{\chi},\phi}^\prime$ also defines a $1$-dimensional module of the spherical Hecke algebra $\mathbb{T}^{0,\infty} = C_c(G(\mathcal{O}^{0,\infty}) \backslash G(\mathbb{A}^{0,\infty}) / G(\mathcal{O}^{0,\infty}))$. In particular, we get system of conjugacy classes $\lbrace \sigma_v \rbrace$ for $v \neq 0, \infty$. This indeed comes from a global Galois representation.
Theorem 3 (Heinloth–Ngo–Yun, V. Lafforgue). There exists a $\rho \colon \pi_1(\mathbb{P}_ k^1 - \lbrace 0, \infty \rbrace) \to \hat{G}$ such that $\rho(\mathrm{Frob}_ v) \sim \sigma_v$ for all $v \neq 0, \infty$.
Example 4. For $G = \mathrm{GL}_ n$ and $k \in \mathbb{F}_ q$, we can compute that for $a \in \mathbb{G}_ m(\mathbb{F}_ q)$ we have $$ \tr(\mathrm{Frob}_ a) = \sum_{(x_i) \in (\mathbb{F}_ q^\times)^n, \chi_1 \dotsc \chi_n = a} \prod_i \chi_i(x_i) \phi({\textstyle \sum x_i}). $$ These are called generalized Kloosterman sums. There are some bounds due to Weil.
There are two possible directions we can try to generalize this.
- (Roche) We might work with more general characters $T(\mathcal{O}) \to \mathbb{C}^\times$. This still factors through some $T(\mathcal{O}/\varpi^r) \to \mathbb{C}^\times$, and then there exists a $I^+ \supseteq J_\chi \twoheadrightarrow T(\mathcal{O}/\varpi^r) \xrightarrow{\chi} \mathbb{C}^\times$ and consider $$ {}_ \chi \mathcal{H}_ \chi = \End(\operatorname{cInd}_ {J_\chi}^{G(F)} \chi). $$ Then we can try to look at the double cosets where it can be supported and so on. The problem is that these double cosets $J_\chi \backslash G(F) / J_\chi$ are quite complicated.
- (Morris) We can also look at parahorics $P \subseteq G(F)$ with Levi quotients $P \to L_P$. Now given an irreducible cuspidal representation $\sigma \colon L_P \to \GL(V)$, we may consider the Hecke algebra of the corresponding block $$ \mathcal{H}(P, \sigma) = \End(\operatorname{cInd}_ P^G \sigma). $$ Here, $\mathcal{H}(P, \sigma)^0$ is again an affine Hecke algebra for an endoscopic group, and the actual Hecke algebra is something like $\mathcal{H}(P, \sigma)^0 \rtimes \mathbb{C}[\Omega(P, \sigma)]$.
Geometrizing the picture
Let us start with the character $\chi \colon T(\mathbb{F}_ q) \to \mathbb{C}^\times$. Let $H$ be a connected algebraic group over $k$. We let $$ \tilde{H} = \varprojlim_{H^\prime \to H} H^\prime $$ where $H^\prime$ is a connected algebraic group and $H^\prime \to H$ is finite '{e}tale. Since is a cofiltered limit and $\tilde{H}$ is a pro-algebraic group. Then $\ker(\tilde{H} \to H) = \pi_1^\mathrm{c}(H)$ is a profinite group over $k$. Note that this is a central extension because the conjugation action can only act trivially, and so $\pi_1^\mathrm{c}(H)$ is abelian. There is also a surjective map $$ \pi_1^\mathrm{et}(H) \twoheadrightarrow \pi_1^\mathrm{c}(H). $$
Remark 5. For $H$ commutative, this was introduced by Serre, but he denoted it as just $\pi_1(H)$. The $\mathrm{c}$ stands for both “central” and “character”.
Example 6. For $k$ of characteristic $p$, for $H = \mathrm{PGL}_ p$ we have $\pi_1^\mathrm{c}(H) = 1$ but $\pi_1^\mathrm{alg}(H) \neq 1$ because $\mathrm{SL}_ p \to \mathrm{PGL}_ p$ is not étale.
Lemma 7. If $H$ is commutative and $k = \bar{\mathbb{F}}_ p$ then we have $$ \pi_1^\mathrm{c}(H) = \varprojlim_n H(\mathbb{F}_ {q^n}), $$ where the transition maps are the norm maps.
Proof.
Given any short exact sequence $$ 1 \to \Gamma \to H^\prime \to H \to 1, $$ we can choose $q$ large enough so that $\Gamma \subseteq H^\prime(\mathbb{F}_ q)$. Then we have the finite '{e}tale homomorphism $$ H^\prime \to H^\prime; \quad g \mapsto g^{-1} \mathrm{Frob}_ q(g), $$ whose kernel is $H^\prime(\mathbb{F}_ q)$. So $\Gamma$ is contained in the kernel, so that we have a factorization $$ H^\prime \to H \to H^\prime. $$ This allows us to regard $H^\prime$ as a quotient of $H$ by some subgroup of $H(\mathbb{F}_ q)$. Now we can work out the transition maps.
It follows that $\pi_1^\mathrm{c}(H)$ does not depend on the rational structure of $H$.