6.1. Context

Let $G$ be an unramified connected reductive group over $F$. Let $H$ be an unramified endoscopic group of $G$. Let $\gamma \in H(F)$ be a strongly regular semisimple element. Let $T_H = C_H(\gamma )$, and let $T_G$ be a Cartan subgroup of $G$ that is isomorphic to it. More details will be given below about how to choose $T_G$. The choice of $T_G$ matters! Let $\gamma \in T_H(F)$ map to $\gamma _0\in T_G(F)$ under this isomorphism.

By construction, $\gamma _0$ is semisimple. However, as $G$ may have more roots than $H$, it is possible for $\gamma _0$ to be singular, even when $\gamma$ is strongly regular. If $\gamma \in H(F)$ is a strongly regular semisimple element with the property that $\gamma _0$ is also strongly regular, then we will call $\gamma$ a strongly $G$-regular element of $H(F)$.

If $\gamma '$ is stably conjugate to $\gamma _0$ with cocycle $a_\tau$, then $s\in \operatorname {Hom}(X_*,\mathbb {C}^\times )$ gives $\kappa (a_\tau )\in \mathbb {C}^\times$.

Let $K_G$ and $K_H$ be hyperspecial maximal compact subgroups of $G$ and $H$. Let $\chi _{G,K}$ and $\chi _{H,K}$ be the characteristic functions of these hyperspecial subgroups. Set

$$\begin{equation} \Lambda _{G,H}(\gamma ) = \left (\prod _{\alpha \in \Phi _G} |\alpha (\gamma _0)-1|^{1/2}\right ) \left [{\frac {\operatorname {vol}(K_T,dt)}{\operatorname {vol}(K,dg)}}\right ] \sum _{\gamma '\sim \gamma _0} \kappa (a_\tau ) \int _{C_G(\gamma ',F)\backslash G(F)}\chi _{G,K} (g^{-1}\gamma ' g) {\frac {dg}{dt'}}. \tag{6.0.1}\cssId{eqn:kappa}{} \end{equation}$$

The set of roots $\Phi _G$ are taken to be those relative to $T_G$. The sum runs over all stable conjugates $\gamma '$ of $\gamma _0$, up to conjugacy. This is a finite sum. The group $K_T$ is defined to be the maximal compact subgroup of $T_G$. Equation 6.0.1 is a finite linear combination of orbital integrals (that is, integrals over conjugacy classes in the group with respect to an invariant measure). The Haar measures $dt'$ on $C_G(\gamma ',F)$ and $dt$ on $T_G(F)$ are chosen so that stable conjugacy between the two groups is measure preserving. This particular linear combination of integrals is called a $\kappa$-orbital integral because of the term $\kappa (a_\tau )$ that gives the coefficients of the linear combination. Note that the integration takes place in the group $G$, and yet the parameter $\gamma$ is an element of $H(F)$.

The volume terms $\operatorname {vol}(K,dg)$ and $\operatorname {vol}(K_T,dt)$ serve no purpose other than to make the entire expression independent of the choice of Haar measures $dg$ and $dt$, which are only defined up to a scalar multiple.

We can form an analogous linear combination of orbital integrals on the group $H$. Set

$$\begin{equation} \Lambda ^{st}_{H}(\gamma ) = \left (\prod _{\alpha \in \Phi _H} |\alpha (\gamma )-1|^{1/2}\right ) \left [{\frac {\operatorname {vol}(K_T,dt)}{\operatorname {vol}(K_H,dh)}}\right ] \sum _{\gamma '\sim \gamma } \int _{C_H(\gamma ',F)\backslash H(F)}\chi _{H,K} (h^{-1}\gamma ' h) {\frac {dh}{dt'}}. \tag{6.0.2}\cssId{eqn:stable}{} \end{equation}$$

This linear combination of integrals is like $\Lambda _{G,H}(\gamma )$, except that $H$ replaces $G$, $K_H$ replaces $K_G$, $\Phi _H$ (taken relative to $T_H$) replaces $\Phi _G$, and so forth. Also, the factor $\kappa (a_\tau )$ has been dropped. The linear combination of Equation 6.0.2 is called a stable orbital integral, because it extends over all stable conjugates of the element $\gamma$ without the factor $\kappa$. The superscript st in the notation is for `stable.'

6.2. The significance of the fundamental lemma

The Langlands program predicts correspondences $\pi \leftrightarrow \pi '$ between the representation theory of different reductive groups. There is a local program for the representation theory of reductive groups over locally compact fields, and a global program for automorphic representations of reductive groups over the adele rings of global fields.

The Arthur-Selberg trace formula has emerged as a powerful tool in the Langlands program. In crude terms, one side of the trace formula contains terms related to the characters of automorphic representations. The other side contains terms such as orbital integrals. Thanks to the trace formula, identities between orbital integrals on different groups imply identities between the representations of the two groups.

It is possible to work backwards: from an analysis of the terms in the trace formula and a precise conjecture in representation theory, it is possible to make precise conjectures about identities of orbital integrals. The most basic identity that appears in this way is the fundamental lemma, articulated above.

The proofs of many major theorems in automorphic representation theory depend in one way or another on the proof of a fundamental lemma. For example, the proof of Fermat's Last Theorem depends on Base Change for $GL(2)$, which in turn depends on the fundamental lemma for cyclic base change [ 17]. The proof of the local Langlands conjecture for $GL(n)$ depends on automorphic induction, which in turn depends on the fundamental lemma for $SL(n)$ [ 11], [ 12], [ 28]. Properties of the zeta function of Picard modular varieties depend on the fundamental lemma for $U(3)$ [ 26], [ 2]. Normally, the dependence of a major theorem on a particular lemma would not be noteworthy. It is only because the fundamental lemma has not been proved in general, and because the lack of proof has become a serious impediment to progress in the field, that the conjecture has become the subject of increased scrutiny.