Example: SL₂ · Part III of III

Growing the example: $\PGL_2$ and the length-zero element

One parameter changes: the group becomes $\PGL_2$, whose cocharacter lattice is twice as large. The tree is the same, but it acquires a symmetry that reverses the base edge. That symmetry is the group $\Om$, and it twists the Hecke algebra by a group ring.

The model, grown by one parameter Definition

(the running example, grown: $\PGL_2(q)$)

Same $F=\Fq((t))$; now $G=\PGL_2=\GL_2/\mathbb G_m$, with $T$, $B$, $K=\PGL_2(\cO)=\GL_2(\cO)/\cO^\times$ and $I$ the images of the diagonal, upper triangular, integral, and Iwahori ($c\in t\cO$) subgroups of $\GL_2$. The new element is

$$\dot\sigma=\sm{0}{1}{t}{0},\qquad \dot\sigma^2=\sm{t}{0}{0}{t}=1\ \text{ in }\PGL_2(F).$$

The cocharacter lattice is $\Xs=\Z\omega^\vee$ with $\omega^\vee:a\mapsto\sm{a}{0}{0}{1}$, and the coroot is $\alpha^\vee=2\omega^\vee$ (since $\sm{a}{0}{0}{a^{-1}}=\sm{a^2}{0}{0}{1}$ modulo scalars). So $\Qv=2\Xs$ and $\Om=\Xs/\Qv=\Z/2$.

What is kept: the tree, $K$, $I$ and $\Waff=\langle s_0,s_1\rangle$ are those of $\SL_2$ (the building depends only on the adjoint group), and $[K:I]=q+1$. What is new: $\Wt=\Waff\rtimes\Om$ is larger than $\Waff$, and the Iwahori–Bruhat decomposition has twice as many double cosets.

How to picture it: $\sigma$ reverses the base edge Proved here

(the length-zero element) Proved here

(i)$\dot\sigma v_0=v_1$ and $\dot\sigma v_1=v_0$: $\sigma$ acts on the tree as the reflection of the edge $e_0$ onto itself with its ends exchanged.

(ii)$\dot\sigma$ normalizes $I$; hence $I\dot\sigma I=\dot\sigma I=I\dot\sigma$ and $\len(\sigma)=0$.

(iii)$\dot\sigma\dot s_1\dot\sigma^{-1}=\dot s_0$, so $\sigma s_1\sigma^{-1}=s_0$ in $\Wt$, and $\sigma=t^{-\omega^\vee}s_1$ in $\Xs\rtimes W$.

(iv)$\Wt=\Waff\sqcup\Waff\sigma$ and $\GF/I$ is the set of oriented edges of the tree; the $I$-orbit of $w\sigma$ consists of the edges of the orbit of $w$ with their orientation reversed, so $|Iw\sigma I/I|=q^{\len(w)}$.

Proof

$\dot\sigma e_1=te_2$ and $\dot\sigma e_2=e_1$, so $\dot\sigma(\cO^2)=\cO e_1\oplus t\cO e_2=L_1$ and $\dot\sigma(L_1)=t\cO e_2\oplus t\cO e_1=t\cO^2\sim\cO^2$.
For $g=\sm abcd\in\GL_2(\cO)$ with $c\in t\cO$: $\dot\sigma g\dot\sigma^{-1}=\sm{0}{1}{t}{0}\sm abcd\sm{0}{t^{-1}}{1}{0}=\sm{d}{t^{-1}c}{tb}{a}$, which lies in $\GL_2(\cO)$ with lower-left entry $tb\in t\cO$. So $\dot\sigma I\dot\sigma^{-1}\subseteq I$, and equality follows by applying the same to $\dot\sigma^{-1}=\dot\sigma$ (modulo scalars). An element normalizing $I$ has $I\dot\sigma I=\dot\sigma I$, a single coset, so $\ind(\sigma)=1=q^{0}$ and $\len(\sigma)=0$ by the index formula (the Concept §4).
$\dot\sigma\dot s_1\dot\sigma^{-1}=\sm{0}{1}{t}{0}\sm{0}{1}{-1}{0}\sm{0}{t^{-1}}{1}{0}=\sm{-1}{0}{0}{t}\sm{0}{t^{-1}}{1}{0}=\sm{0}{-t^{-1}}{t}{0}=\dot s_0$. Also $\dot\sigma=\sm{1}{0}{0}{t}\dot s_1\sm{-1}{0}{0}{1}$, and $\sm{1}{0}{0}{t}=\sm{t^{-1}}{0}{0}{1}$ modulo scalars is $t^{-\omega^\vee}$; the last factor lies in $T(\cO)$. In the coordinate $y$ of Groups §4, $t^{-\omega^\vee}$ translates by $-1$, so $\sigma$ acts by $y\mapsto-y-1$, which maps $A_0=(-1,0)$ to itself.
$\Om=\{e,\sigma\}$ since $\Xs/\Qv=\Z/2$, and $\Wt=\Waff\rtimes\Om$ (Groups, Lemma 5.2). $\PGL_2(F)$ is transitive on vertices, by (i) and the transitivity of $\SL_2(F)$ on each type; the stabilizer of the unoriented edge $e_0$ is $I\sqcup\dot\sigma I$ and the stabilizer of the oriented edge $(v_0\to v_1)$ is $I$. Hence $\GF/I$ is the set of oriented edges, and $\dot w\dot\sigma e_0=\dot w(e_0\text{ reversed})$.

What the tree gains Nothing is added to the picture of Part I except an arrow on each edge. The orbit of $w$ is a set of arrows pointing away from $e_0$ along the paths of type $w$; the orbit of $w\sigma$ is the same set of edges with the arrows reversed.

The Hecke algebra of $\PGL_2$: a twisted tensor product Proved here

($\cH$ for $\PGL_2$) [IM65, Prop 3.8 and p. 279, type $A_1$] Proved here

$$T_\sigma T_w=T_{\sigma w},\qquad T_wT_\sigma=T_{w\sigma},\qquad T_\sigma^2=T_e,\qquad T_\sigma T_{s_1}T_\sigma=T_{s_0},$$

so $\cH=\Haff\oplus\Haff T_\sigma$ with $\Haff=\langle T_{s_0},T_{s_1}\rangle$ the affine Hecke algebra of $\SL_2$, and $T_\sigma$ acts on $\Haff$ by the automorphism exchanging $T_{s_0}$ and $T_{s_1}$: $\cH=\C[\Z/2]\,\widetilde\otimes\,\Haff$.

Proof

$(T_\sigma\ast T_w)(z)=\#\{uI\subseteq\dot\sigma I:\ u^{-1}z\in IwI\}$ by Hecke algebras, Lemma 2.2; the only coset is $\dot\sigma I$, so this is $\one_{\dot\sigma IwI}(z)=\one_{I\sigma wI}(z)$ because $\dot\sigma I=I\dot\sigma$. Similarly on the right. $T_\sigma^2=T_{\sigma^2}=T_e$, and $T_\sigma T_{s_1}T_\sigma=T_{\sigma s_1\sigma}=T_{s_0}$ by (iii) of . The two $\Haff$-summands are spanned by the $T_w$ and the $T_{w\sigma}$, $w\in\Waff$, which together are the basis of $\cH$.

The characters of Part II that survive are those with $\lambda_0=\lambda_1$: $\ind$ (with $\ind(T_\sigma)=1$) and $\eps$ (with $\eps(T_\sigma)=1$), each in two versions $T_\sigma\mapsto\pm1$. The mixed characters of $\SL_2$ are exchanged by $\sigma$, so they do not extend.

The next size up: $\GL_2$ For $\GL_2$, $\Xs=\Z^2$, $\Qv=\Z(1,-1)$ and $\Om\simeq\Z$, generated by the same matrix $\dot\sigma$, now of infinite order since $\dot\sigma^2=t\cdot1$ is central but not trivial. Then $\cH=\C[\Z]\,\widetilde\otimes\,\Haff$, and $T_\sigma$ is an invertible element of infinite order: $\Om$ need not be finite for a reductive group, only for a semisimple one.

What this adds to the Concept Context

The Concept's third relation, $T_\sigma T_w=T_{\sigma w}$, and its description of $\cH$ as $\C[\Om]\,\widetilde\otimes\,\Haff$ are exactly what this page checks for the smallest $\Om\neq1$. The Bernstein presentation also changes: $\C[\Xs]$ now contains $\Th_{\omega^\vee}$, a square root of $\Th_{\alpha^\vee}$ (source page §3).