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Elementary proof of the fundamental lemma for a unitary group

  Published:1998-02-01
 Printed: Feb 1998
  • Yuval Z. Flicker
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Abstract

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).
MSC Classifications: 22E35, 11F70, 11F85, 11S37 show english descriptions Analysis on $p$-adic Lie groups
Representation-theoretic methods; automorphic representations over local and global fields
$p$-adic theory, local fields [See also 14G20, 22E50]
Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]
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]
 

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