Electronic ISBN:  9781470409029 
Product Code:  MEMO/99/476.E 
List Price:  $34.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 99; 1992; 142 ppMSC: Primary 20; Secondary 12; 22;
Langlands theory predicts deep relationships between representations of different reductive groups over a local or global field. The trace formula attempts to reduce many such relationships to problems concerning conjugacy classes and integrals over conjugacy classes (orbital integrals) on \(p\)adic groups. It is possible to reformulate these problems as ones in algebraic geometry by associating a variety \(Y\) to each reductive group. Using methods of Igusa, the geometrical properties of the variety give detailed information about the asymptotic behavior of integrals over conjugacy classes. This monograph constructs the variety \(Y\) and describes its geometry. As an application, the author uses the variety to give formulas for the leading terms (regular and subregular germs) in the asymptotic expansion of orbital integrals over \(p\)adic fields. The final chapter shows how the properties of the variety may be used to confirm some predictions of Langlands theory on orbital integrals, Shalika germs, and endoscopy.
ReadershipResearchers in the fields of automorphic forms, representation theory and \(p\)adic groups.

Table of Contents

Chapters

I. Basic constructions

II. Coordinates and coordinate relations

III. Groups of rank two

IV. The subregular spurious divisor

V. The subregular fundamental divisor

VI. Rationality and characters

VII. Applications to endoscopic groups


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Langlands theory predicts deep relationships between representations of different reductive groups over a local or global field. The trace formula attempts to reduce many such relationships to problems concerning conjugacy classes and integrals over conjugacy classes (orbital integrals) on \(p\)adic groups. It is possible to reformulate these problems as ones in algebraic geometry by associating a variety \(Y\) to each reductive group. Using methods of Igusa, the geometrical properties of the variety give detailed information about the asymptotic behavior of integrals over conjugacy classes. This monograph constructs the variety \(Y\) and describes its geometry. As an application, the author uses the variety to give formulas for the leading terms (regular and subregular germs) in the asymptotic expansion of orbital integrals over \(p\)adic fields. The final chapter shows how the properties of the variety may be used to confirm some predictions of Langlands theory on orbital integrals, Shalika germs, and endoscopy.
Researchers in the fields of automorphic forms, representation theory and \(p\)adic groups.

Chapters

I. Basic constructions

II. Coordinates and coordinate relations

III. Groups of rank two

IV. The subregular spurious divisor

V. The subregular fundamental divisor

VI. Rationality and characters

VII. Applications to endoscopic groups