Last time we had the decomposition $$ \mathcal{H}_ {I^+} = \End \operatorname{cInd}_ {I^+}^{G(F)} 1 = \bigoplus_{\chi,\chi^\prime} \Hom(\operatorname{cInd}_ I^{G(F)} \chi, \operatorname{cInd}_ I^{G(F)} \chi^\prime), $$ where we have $$ {}_ \chi \mathcal{H}_ {\chi^\prime} = \lbrace f \colon G(F) \to \mathbb{C} : f(k g k^\prime) = \chi(k) f(g) \chi^\prime(k^\prime) \rbrace. $$ We still have the Iwahori decomposition here, and so its support corresponds to $$ {}_ \chi \tilde{W}_ {\chi^\prime} = \lbrace w \in \tilde{W} : \chi = w \chi^\prime \colon T(k_F) \to \mathbb{C}^\times \rbrace. $$ When $\chi = \chi^\prime$, we just write this by $\tilde{W}_ \chi$. This then fits in a short exact sequence $$ 1 \to X_\ast(T) \to \tilde{W}_ \chi \to W_\chi \to 1. $$
Finer structure of the restricted Weyl group
If we define $$ \Phi_\chi^\vee = \lbrace a^\vee \in \Phi^\vee : \chi \circ a^\vee = 1 \rbrace \subseteq \Phi $$ then this is a subroot system of $\Phi^\vee$. But note that $\Phi_\chi$ the dual of $\Phi_\chi^\vee$ need not be a subroot system in $\Phi$. (Consider the case of $B_2$ for example, where the sum of things in $\Phi_\chi$ need not be in $\Phi_\chi$.) Write $$ \Phi_{\chi,\mathrm{aff}} = \lbrace a + k : a \in \Phi_\chi, k \in \mathbb{Z} \rbrace \subseteq \Phi_\mathrm{aff}. $$ Then inside the above short exact sequence we have $$ \begin{CD} 1 @>>> X_\ast(T) @>>> \tilde{W}_ \chi @>>> W_\chi @>>> 1 \br @. @| @AAA @AAA @. \br 1 @>>> X_\ast(T) @>>> \tilde{W}_ \chi^1 @>>> W_ \chi^0 @>>> 1 \br @. @AAA @AAA @| \br 1 @>>> \mathbb{Z}\Phi_\chi^\vee @>>> \tilde{W}_ \chi^0 @>>> W_\chi^0 @>>> 1 \end{CD} $$ where we write $$ W_\chi^0 = \langle s_a : a \in \Phi_\chi \rangle, \quad \tilde{W}_ \chi^0 = \langle s_\alpha : \alpha \in \Phi_{\chi,\mathrm{aff}} \rangle. $$
For $G$, we consider the $G^\vee$, which naturally comes with a pinning $(\hat{G}, \hat{B}, \hat{T}, \hat{e} \colon \hat{U} \to \mathbb{G}_ a)$. Then we naturally have $$ X_\ast(T) = X^\ast(\hat{T}), \quad X^\ast(T) = X_\ast(\hat{T}), \quad \Phi(G, T) = \Phi(\hat{G}, \hat{T})^\vee. $$ We can then think of $\chi$ as a homomorphism $$ k_F^\times \to X_\ast(\hat{T}) \otimes \mathbb{C} = \hat{T}(\mathbb{C}). $$ So choosing a generator for $k_F^\times$, we may regard $\chi$ has an element $\chi \in \hat{T}[q-1]$.
The endoscopic group
In general for $s \in \hat{T}$ we have $$ 1 \to \hat{H}_ s = Z_{\hat{G}}(s)^0 \to Z_{\hat{G}}(s) \to \pi_0(s) = \pi_0(Z_{\hat{G}}(s)) \to 1. $$ Then we can define $$ \Phi_s^\vee = \lbrace a^\vee \in \Phi^\vee(G, T) = \Phi(\hat{G}, \hat{T}) : a^\vee(s) = 1 \rbrace. $$ When $s = \chi$, this exactly recovers $\Phi_s^\vee = \Phi_\chi^\vee$. We also have $$ W_s = \lbrace w \in W(\hat{G}, \hat{T}) : w(s) = s \rbrace \supseteq W_s^0 = \langle s_{a^\vee} : a^\vee \in \Phi_s^\vee \rangle. $$
Lemma 1. The subgroup $\hat{T} \subseteq \hat{H}$ is a maximal torus of $\hat{H}$, and moreover $\Phi(\hat{H}, \hat{T}) = \Phi_s^\vee$ and $W(\hat{H}, \hat{T}) = W_s^0$. There is also a corresponding Borel $B_{\hat{H}} = \hat{T} \prod_{a^\vee \in \Phi_s^\vee \cap \Phi_+^\vee} U_{a^\vee}$.
Remark 2. But simple roots in $\Phi_s^\vee$ need not be simple in $\Phi^\vee$.
Lemma 3. We have $\pi_0(s) \cong W_s / W_s^0$.
Proof.
If we look at $N_{Z_{\hat{G}}(s)}(\hat{T}) \hookrightarrow N_{\hat{G}}(\hat{T})$, this means every component of $Z_{\hat{G}}(s)$. This is because if conjugate $\hat{T}$ by some element of $Z_{\hat{G}}(s)$, this is still a maximal torus in $Z_{\hat{G}}(s)^0$ and so we can conjugate by an element of $Z_{\hat{G}}(s)^0$ to get it back to $\hat{T}$.
By a similar argument, we see that $$ 1 \to \hat{T} \to N_{Z_{\hat{G}}(s)}(B_{\hat{H}}, \hat{T}) \to \pi_0(s) \to 1. $$ So we have $$ \pi_0(s) = \frac{N_{Z_{\hat{G}}(s)}(B_{\hat{H}}, \hat{T})}{\hat{T}} \hookrightarrow \frac{N_{Z_{\hat{G}}(s)}(\hat{T})}{\hat{T}} = W_s. $$ In particular, the short exact sequence $$ 1 \to W_s^0 \to W_s \to \pi_0(s) \to 1 $$ canonically splits.
Remark 4. The sequence $$ 1 \to Z(\hat{H}) \to N_{Z_{\hat{G}}(s)}(B_{\hat{H}}, \hat{T}, e_{\hat{H}}) \to \pi_0(s) \to 1 $$ is usually not split. This causes a lot of problems in works of Langlands.
Remark 5. If $\hat{G}_ \mathrm{der}$ is simply connected, then $Z_{\hat{G}}(s)$ is automatically connected. But for $\hat{G} = \mathrm{PGL}_ 2$, for $s = \operatorname{diag}(1, -1)$ we have $Z_{\hat{G}}(s) = N_{\hat{G}}(\hat{T})$ not connected.
Lemma 6. The group $\pi_0(s)$ is finite abelian.
Proof.
We reduce to the case when $\hat{G}$ is semisimple, and lift $s \in \hat{G}$ to $\tilde{s} \in \hat{G}_ \mathrm{sc}$. Writing $\Gamma = \ker(\hat{G}_ \mathrm{sc} \to \hat{G}0$, we see that $$ \pi_0(s) \hookrightarrow \Gamma; \quad w \mapsto w(\tilde{s}) / \tilde{s} $$ is an injective group homomorphism.
Example 7. We consider $G = \mathrm{Sp}(2n)$ and $\hat{G} = \mathrm{SO}(2n+1)$. Then we have $X^\ast(T) = \bigoplus \mathbb{Z}\epsilon_i$, $$ \Phi = \lbrace \pm \epsilon_i \pm \epsilon_j, \pm 2\epsilon_j \rbrace $$ where simple roots are $\epsilon_i - \epsilon_{i+1}$ and $2\epsilon_n$. On the dual side, we have $$ \Phi^\vee = \lbrace \pm \epsilon_i^\vee \pm \epsilon_j^\vee, \pm \epsilon_j^\vee \rbrace, $$ with simple roots $\epsilon_i^\vee - \epsilon_{i+1}^\vee$ and $\epsilon_n^\vee$. If we take $s \in \hat{T}$ such that $\epsilon_i^\vee(s) = -1$ then we have $$ \Phi_s^\vee = \lbrace \pm \epsilon_i^\vee \pm \epsilon_j^\vee \rbrace $$ which is of type D. So we have $$ 1 \to \hat{H} = \mathrm{SO}(2n) \to Z_{\hat{G}}(s) = Z_{\hat{G}}(s) = \mathrm{O}(2n) \to \mathbb{Z}/2\mathbb{Z} \to 1. $$ Here, the simple roots are $\epsilon_i^\vee - \epsilon_{i+1}^\vee$ and $2\epsilon_n^\vee$. We also have $$ \Phi_s = \lbrace \pm \epsilon_i \pm \epsilon_j \rbrace $$ with simple roots $\epsilon_i - \epsilon_{i+1}$ and $\epsilon_{n-1} + \epsilon_n$. This is not a subroot system.
Now given $s \in \hat{T} \subseteq \hat{G}$ we get $(\hat{H}, B_{\hat{H}}, \hat{H}, e_\hat{H})$ and dualize it to $$ (H, B_H, T_H, e_H). $$ But we note that $H$ is not a subgroup of $G$. This $H$ is usually called the endoscopic group attached to $s \in \hat{T}$.
Back to endoscopic Weyl groups
We now come back to $$ \tilde{W}_ \chi \supseteq \tilde{W}_ \chi^1 \supseteq \tilde{W}_ \chi^0. $$ This $\tilde{W}_ \chi^0$ is the affine Weyl group of $H$, and $\tilde{W}_ \chi^1$ is the Iwahori–Weyl group of $H$. We have seen that $$ \pi_0(\chi) = W_\chi / W_\chi^0 = \tilde{W}_ \chi / \tilde{W}_ \chi^1 $$ and this is always trivial when $Z_G$ is connected, or equivalently when $\hat{G}_ \mathrm{der}$ is simply connected.
From before, we had a splitting of $$ 1 \to \tilde{W}_ \chi^0 \to \tilde{W}_ \chi^1 \to \Omega_\chi^1 = \pi_1(H) \to 1. $$ Now $\tilde{W}_ \chi$ is corresponds to some coarser thin chamber complex structure on the same apartment. So we similarly get a splitting of $$ 1 \to \tilde{W}_ \chi^0 \to \tilde{W}_ \chi \to \Omega_\chi \to 1. $$ Here, this $\Omega_\chi$ fits in a short exact sequence $$ 1 \to \Omega_\chi^1 \to \Omega_\chi \to \pi_0(\chi) \to 1. $$
Lemma 8. 1. For every $w \in \Omega_\chi$ we have $\ell(wv) \ge \ell(w)$ for all $v \in \tilde{W}_ \chi^0$.
- If $v_1 \le_\chi v_2$ then $wv_1 \le wv_2$ in $\tilde{W}$ for all $w \in \Omega_\chi$.
- There is a length function $\ell_\chi \colon \tilde{W}_ \chi \to \mathbb{Z}_ {\ge 0}$ obtained by looking at the alcove for $H$. Then we have $$ \ell(v) - \ell(v^\prime) \ge \ell_\chi(v) - \ell_\chi(v^\prime) $$ if $v^\prime \le_\chi v$.
What happens for $\chi \neq \chi^\prime$? We have simply transitive actions of $$ \tilde{W}_ \chi \curvearrowright {}_ \chi \tilde{W}_ {\chi^\prime} \curvearrowleft \tilde{W}_ {\chi^\prime} $$ We can then show that $$ \tilde{W}_ \chi^0 \backslash {}_ \chi \tilde{W}_ {\chi^\prime} \cong {}_ \chi \tilde{W}_ {\chi^\prime} / \tilde{W}_ {\chi^\prime}^0 $$ and there is a minimal length lifting $\tilde{W}_ \chi^0 \backslash {}_ \chi \tilde{W}_ {\chi^\prime} \hookrightarrow {}_ \chi \tilde{W}_ {\chi^\prime}$ and so forth.