5. Stable Conjugacy

Definition 5.1

Let $\delta$ and $\delta '$ be strongly regular semisimple elements in $G(F)$ . They are conjugate if $g^{-1}\delta g = \delta '$ for some $g\in G(F)$ . They are stably conjugate if $g^{-1}\delta g = \delta '$ for some $g\in G(\bar F)$ .

Example 5.2

Let $G= SL(2)$ and $F=\mathbb {Q}_p$ . Assume that $p\ne 2$ and that $u$ is a unit that is not a square in $\mathbb {Q}_p$ . Let $\epsilon = \sqrt {u}$ in an unramified quadratic extension of $\mathbb {Q}_p$ . We have the matrix calculation

\begin{pmatrix} 1+p&1\\ 2p+p^2&1+p \end{pmatrix} \begin{pmatrix} \epsilon &0\\ 0&\epsilon ^{-1} \end{pmatrix}= \begin{pmatrix} \epsilon &0\\ 0&\epsilon ^{-1} \end{pmatrix} \begin{pmatrix} 1+p&u^{-1}\\ (2p+p^2)u&1+p \end{pmatrix}.

This matrix calculation shows that the matrices

\begin{equation} \begin{pmatrix} 1+p&1\\ 2p+p^2&1+p \end{pmatrix}\text{ and } \begin{pmatrix} 1+p&u^{-1}\\ (2p+p^2)u&1+p \end{pmatrix} \tag{5.2.1}\cssId{eqn:2matrix}{} \end{equation}

of $SL(2,\mathbb {Q}_p)$ are stably conjugate. The diagonal matrix that conjugates one to the other has coefficients that lie in a quadratic extension. A short calculation shows that the matrices 5.2.1 are not conjugate by a matrix of $SL(2,\mathbb {Q}_p)$ .

5.1. Cocycles

Let $\gamma _0$ and $\gamma '$ be stably conjugate strongly regular semisimple elements of $G(F)$ . We view $\gamma _0$ as a fixed base point and $\gamma '$ as variable. If $\tau \in \operatorname {Gal}(\bar F/F)$ , then

\begin{equation} \begin{array}{lll} g^{-1}\gamma _0 g &= \gamma ', (\text{with } g\in G(\bar F), \gamma _0,\gamma '\in G(F))\\\tau (g)^{-1}\tau (\gamma _0) \tau (g) &= \tau (\gamma '),\\\tau (g)^{-1}\gamma _0\tau (g) &= g^{-1}\gamma _0 g,\\\gamma _0 \left ( \tau (g)g^{-1}\right ) &= \left (\tau (g)g^{-1}\right ) \gamma _0,\\\gamma _0 a_\tau = a_\tau \gamma _0, \text{ with } a_\tau = \tau {(g)}g^{-1}. \end{array} \tag{5.2.2} \end{equation}

The element $a_\tau$ centralizes $\gamma _0$ and hence gives an element of the centralizer $T$ . Viewed as a function of $\tau \in \operatorname {Gal}(\bar F/F)$ , $a_\tau$ satisfies the cocycle relation

\tau _1(a_{\tau _2})a_{\tau _1} = a_{\tau _1\tau _2}.

It is continuous in the sense that there exists a field extension $F_1/F$ for which $a_\tau =1$ , for all $\tau \in \operatorname {Gal}(\bar F/F_1)$ . Thus, $a_\tau$ gives a class in

H^1(\operatorname {Gal}(\bar F/F),T(\bar F)),

which is defined to be the group of all continuous cocycles with values in $T$ , modulo the subgroup of all continuous cocycles of the form

b_\tau = \tau (t)t^{-1},

for some $t\in T(\bar F)$ .

A general calculation of the group $H^1(\operatorname {Gal}(\bar F/F),T)$ is achieved by the Tate-Nakayama isomorphism. Let $F_1/F$ be a Galois extension that splits the Cartan subgroup $T$ .

Theorem 5.3

(Tate-Nakayama isomorphism [ 27]) The group $H^1(\operatorname {Gal}(\bar F/F),T)$ is isomorphic to the quotient of the group

\{u \in X_*{\ \vert \ }\sum _{\tau \in \operatorname {Gal}(F_1/F)} \tau u = 0\}

by the subgroup generated by the set

\{u \in X_* {\ \vert \ }\exists \tau \in \operatorname {Gal}(F_1/F)\ \ \exists v \in X_*.\ \ u = \tau v - v\}.

Example 5.4

Let $T = U_E(1)$ (the torus that made an appearance earlier as an endoscopic group of $SL(2)$ ). As was shown above, the group of cocharacters can be identified with $\mathbb {Z}$ . The splitting field of $T$ is the quadratic extension field $E$ . The nontrivial element $\tau \in \operatorname {Gal}(E/F)$ acts by reflection on $X_*\cong \mathbb {Z}$ : $\tau (u) = -u$ . By the Tate-Nakayama isomorphism, the group $H^1(\operatorname {Gal}(\bar F/F),U_E(1))$ is isomorphic to

\{u\in \mathbb {Z} {\ \vert \ }u+\tau u = 0\}/\{u\in \mathbb {Z}{\ \vert \ }\exists v.\ u = \tau v - v\}=\mathbb {Z}/2\mathbb {Z}.

Let $H$ be an unramified endoscopic group of $G$ . Suppose that $T_H$ is a Cartan subgroup of $H$ . Let $T_G$ be an isomorphic Cartan subgroup in $G$ . The data defining $H$ includes the existence of an element $s\in \operatorname {Hom}(X_*,\mathbb {C}^\times )$ ; that is, a character of the abelian group $X_*$ . Fix one such character $s$ . We can restrict this character to get a character of

\{u \in X_*{\ \vert \ }\sum _{\tau \in \operatorname {Gal}(F_1/F)} \tau u = 0\}.

It can be shown that the character $s$ is trivial on

\{u \in X_* {\ \vert \ }\exists \tau \in \operatorname {Gal}(F_1/F)\ \ \exists v \in X_*.\ \ u = \tau v - v\}.

Thus, by the Tate-Nakayama isomorphism, the character $s$ determines a character $\kappa$ of the cohomology group

H^1(\operatorname {Gal}(\bar F/F),T).

In this way, each cocycle $a_\tau$ gives a complex constant $\kappa (a_\tau )\in \mathbb {C}^\times$ .

Example 5.5

The element $s\in \mathbb {C}^\times$ giving the endoscopic group $H=U_E(1)$ of $SL(2)$ is $s=-1$ , which may be identified with the character $n\mapsto (-1)^n$ of $\mathbb {Z}$ . This gives the nontrivial character $\kappa$ of

H^1(\operatorname {Gal}(\bar F/F),U_E(1)) \cong \mathbb {Z}/2\mathbb {Z}.