8. The problem of base points

The fundamental lemma was formulated above with one omission: we never made precise how to fix an isomorphism $T_H\leftrightarrow T_G$ between Cartan subgroups in $H$ and $G$. Such isomorphisms exist, because the two Cartan subgroups have the same root data. But the statement of the fundamental lemma is sensitive to how an isomorphism is selected between $T_H$ and a Cartan subgroup of $G$. If we change the isomorphism, we change the $\kappa$-orbital integral by a root of unity $\zeta \in \mathbb {C}^\times$. The correctly chosen isomorphism will depend on the element $\gamma \in H(F)$.

The ambiguity of isomorphism was removed by Langlands and Shelstad in [ 19]. They define a transfer factor $\Delta (\gamma _H,\gamma _G)$, which is a complex valued function on $H(F)\times G(F)$. The transfer factor can be defined to have the property that it is zero unless $\gamma _H\in H(F)$ is strongly regular semisimple, $\gamma _G\in G(F)$ is strongly regular semisimple, and there exists an isomorphism (preserving character groups) from the centralizer of $\gamma _H$ to the centralizer of $\gamma _G$. There exists $\gamma _0\in G(F)$ such that

$$\begin{equation} \Delta (\gamma _H,\gamma _0)=1. \tag{8.0.1}\cssId{eqn:delta}{} \end{equation}$$

The correct formulation of the fundamental lemma is to pick the base point $\gamma _0\in G(F)$ so that Condition 8.0.1 holds.

For classical groups, Waldspurger gives a simplified formula for the transfer factor $\Delta$ in [ 31]. Furthermore, because of the reduction of the fundamental lemma to the Lie algebra (Section 7.2), the transfer factor may be expressed as a function on the Lie algebras of $G$ and $H$, rather than as a function on the group.

8.1. Base points for unitary groups

More recently, Laumon (while working on the fundamental lemma for unitary groups) observed a similarity between Waldspurger's simplified formula for the transfer factor and the explicit formula for differents that is found in [ 27]. In this way, Laumon found a simple description of the matching condition $\gamma \leftrightarrow \gamma _0$ implicit in the statement of the fundamental lemma.