Electronic ISBN:  9781470402440 
Product Code:  MEMO/137/655.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 137; 1999; 112 ppMSC: Primary 11;
The trace formula is the most powerful tool currently available to establish liftings of automorphic forms, as predicted by Langlands principle of functionality. The geometric part of the trace formula consists of orbital integrals, and the lifting is based on the fundamental lemma. The latter is an identity of the relevant orbital integrals for the unit elements of the Hecke algebras.
This volume concerns a proof of the fundamental lemma in the classically most interesting case of Siegel modular forms, namely the symplectic group \(Sp(2)\). These orbital integrals are compared with those on \(GL(4)\), twisted by the transpose inverse involution. The technique of proof is elementary. Compact elements are decomposed into their absolutely semisimple and topologically unipotent parts also in the twisted case; a double coset decomposition of the form \(H\backslash G/K\)—where H is a subgroup containing the centralizer—plays a key role.ReadershipGraduate students and research mathematicians working in automorphic forms, trace formula, orbital integrals, conjugacy classes of rational elements in a classical group and in stable conjugacy.

Table of Contents

Chapters

Introduction

I. Preparations

II. Main comparison

III. Semi simple reduction


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The trace formula is the most powerful tool currently available to establish liftings of automorphic forms, as predicted by Langlands principle of functionality. The geometric part of the trace formula consists of orbital integrals, and the lifting is based on the fundamental lemma. The latter is an identity of the relevant orbital integrals for the unit elements of the Hecke algebras.
This volume concerns a proof of the fundamental lemma in the classically most interesting case of Siegel modular forms, namely the symplectic group \(Sp(2)\). These orbital integrals are compared with those on \(GL(4)\), twisted by the transpose inverse involution. The technique of proof is elementary. Compact elements are decomposed into their absolutely semisimple and topologically unipotent parts also in the twisted case; a double coset decomposition of the form \(H\backslash G/K\)—where H is a subgroup containing the centralizer—plays a key role.
Graduate students and research mathematicians working in automorphic forms, trace formula, orbital integrals, conjugacy classes of rational elements in a classical group and in stable conjugacy.

Chapters

Introduction

I. Preparations

II. Main comparison

III. Semi simple reduction