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The Subregular Germ of Orbital Integrals

Available Formats:
Electronic ISBN: 978-1-4704-0902-9
Product Code: MEMO/99/476.E
List Price: $34.00 MAA Member Price:$30.60
AMS Member Price: $20.40 Click above image for expanded view The Subregular Germ of Orbital Integrals Available Formats:  Electronic ISBN: 978-1-4704-0902-9 Product Code: MEMO/99/476.E  List Price:$34.00 MAA Member Price: $30.60 AMS Member Price:$20.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 991992; 142 pp
MSC: 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.

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
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Volume: 991992; 142 pp
MSC: 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.

Researchers in the fields of automorphic forms, representation theory and $p$-adic groups.