Elementary proof of the fundamental lemma for a unitary group
Printed: Feb 1998
The fundamental lemma in the theory of automorphic forms is proven
for the (quasi-split) unitary group $U(3)$ in three variables
associated with a quadratic extension of $p$-adic fields, and its
endoscopic group $U(2)$, by means of a new, elementary technique.
This lemma is a prerequisite for an application of the trace
formula to classify the automorphic and admissible representations
of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$.
It compares the (unstable) orbital integral of the characteristic
function of the standard maximal compact subgroup $K$ of $U(3)$ at
a regular element (whose centralizer $T$ is a torus), with an
analogous (stable) orbital integral on the endoscopic group $U(2)$.
The technique is based on computing the sum over the double coset
space $T\bs G/K$ which describes the integral, by means of an
intermediate double coset space $H\bs G/K$ for a subgroup $H$ of
$G=U(3)$ containing $T$. Such an argument originates from
Weissauer's work on the symplectic group. The lemma is proven for
both ramified and unramified regular elements, for which endoscopy
occurs (the stable conjugacy class is not a single orbit).
22E35 - Analysis on $p$-adic Lie groups
11F70 - Representation-theoretic methods; automorphic representations over local and global fields
11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]
11S37 - Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]