7.1. Topological Jordan decomposition

A somewhat less trivial reduction of the problem is provided by the topological Jordan decomposition. Suppose that $\gamma$ lies in a compact subgroup. It can be written uniquely as a product

$$\gamma = \gamma _s \gamma _u = \gamma _u\gamma _s,$$

where $\gamma _s$ has finite order, of order prime to the residue field characteristic $p$, and $\gamma _u$ is topologically unipotent. That is,

$$\lim _{n\to \infty } \gamma _u^{p^n} = 1.$$

The limit is with respect to the $p$-adic topology. A special case of the topological Jordan decomposition $\gamma \in O_F^\times \subset \mathbb {G}_m(F)$ is treated in [ 13, p20]. In that case, $\gamma _s$ is defined by the formula

$$\gamma _s = \lim _{n\to \infty } \gamma ^{q^n}.$$

Let $\gamma$, $\gamma _0$, and $\gamma '$ be chosen as in Section 6.1. Each of these elements has a topological Jordan decomposition. Let $G_s = C_G(\gamma _{0 s})$ and $H_s = C_H(\gamma _s)$. It turns out that $G_s$ is an unramified reductive group with unramified endoscopic group $H_s$. Descent for orbital integrals gives the formulas [ 20] [ 8]

$$\begin{array}{lll} \Lambda _{G,H}(\gamma ) &= \Lambda _{G_s,H_s}(\gamma _u)\\&\\\Lambda _{H}^{st}(\gamma ) &=\Lambda _{H_s}^{st}(\gamma _u). \end{array}$$

This reduces the fundamental lemma to the case that $\gamma$ is a topologically unipotent elements.

7.2. Lie algebras

It is known (at least when the $p$-adic field $F$ has characteristic zero), that the fundamental lemma holds for fields of arbitrary residual characteristic provided that it holds when the $p$-adic field has sufficiently large residual characteristic [ 9]. Thus, if we are willing to restrict our attention to fields of characteristic zero, we may assume that the residual characteristic of $F$ is large. In fact, in our discussion of a reduction to Lie algebras in this section, we simply assume that the characteristic of $F$ is zero.

A second reduction is based on Waldspurger's homogeneity results for classical groups. (Homogeneity results have since been reworked and extended to arbitrary reductive groups by DeBacker, again assuming mild restrictions on $G$ and $F$.)

When the residual characteristic is sufficiently large, there is an exponential map from the Lie algebra to the group that has every topologically unipotent element in its image. Write

$$\gamma _u = \exp ( X),$$

for some element $X$ in the Lie algebra. We may then consider the behavior of orbital integrals along the curve $\exp (\lambda ^2 X)$. A difficult result of Waldspurger for classical groups states that if $|\lambda |\le 1$, then

$$\begin{array}{lll} \Lambda _{G,H}(\exp (\lambda ^2 X)) &= \sum a_i |\lambda |^i\\\Lambda _{H}^{st}(\exp (\lambda ^2 X))&=\sum b_i |\lambda |^i; \end{array}$$

that is, both sides of the fundamental lemma identity are polynomials in $|\lambda |$. If a polynomial identity holds when $|\lambda |<\epsilon$ for some $\epsilon >0$, then it holds for all $|\lambda |\le 1$. In particular, it holds at $\gamma _u$ for $\lambda =1$. The polynomial growth of orbital integrals makes it possible to prove the fundamental lemma in a small neighborhood of the identity element, and then conclude that it holds in general. In this manner, the fundamental lemma can be reduced to a conjectural identity in the Lie algebra.