**Memoirs of the American Mathematical Society**

1998;
101 pp;
Softcover

MSC: Primary 22;

Print ISBN: 978-0-8218-0765-1

Product Code: MEMO/134/635

List Price: $49.00

AMS Member Price: $29.40

MAA Member Price: $44.10

**Electronic ISBN: 978-1-4704-0224-2
Product Code: MEMO/134/635.E**

List Price: $49.00

AMS Member Price: $29.40

MAA Member Price: $44.10

# On Stability and Endoscopic Transfer of Unipotent Orbital Integrals on \(p\)-adic Symplectic Groups

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*Magdy Assem*

The theory of endoscopy is an intriguing part of the Langlands
program, as it provides a way to attack the functoriality principle of
Langlands for certain pairs of reductive groups \((G,H)\), in which
\(H\)
is what is known as an endoscopic group for \(G\). The starting point
for this method is a close study of the relationship of orbital
integrals on \(G\) with stable orbital integrals on \(H\).

This volume investigates unipotent orbital integrals of spherical
functions on \(p\)-adic symplectic groups. The results are then put
into a conjectural framework, that predicts (for split classical
groups) which linear combinations of unipotent orbital integrals are
stable distributions.

#### Readership

Research mathematicians interested in analysis on \(p\)-adic Lie groups.

#### Table of Contents

# Table of Contents

## On Stability and Endoscopic Transfer of Unipotent Orbital Integrals on $p$-adic Symplectic Groups

- Contents vii8 free
- 0. Introduction 112 free
- 1. Unipotent orbits and prehomogeneous spaces 617 free
- 2. The Hecke algebra and some Igusa local orbital zeta functions 1324
- 3. The evaluation of f[sup(H)] at the identity 2435
- 4. Matching of unipotent orbital integrals 5869
- 1. Unramified endoscopic data 5869
- 2. The map f [omitted] f[sup(H) 5970
- 3. Endoscopic induction of unipotent orbits 5970
- 4. Matching of regular unipotent orbital integrals 6172
- 5. Matching of unipotent orbital integrals for G = Sp(6) and its unramified endoscopic groups 6374
- 6. Matching of subregular orbital integrals 6879
- 7. Matching of the orbits 2[sup(r)] 1[sup(2n-2r)], for r = 2,3 6980
- 8. Matching results for Sp(8) 7081
- 9. Endoscopic transfer of the trivial orbital integral 7687
- 10. Endoscopic transfer of other orbital integrals 7788
- 11. Some remarks on the transfer factors 7889

- 5. Remarks on stability and endoscopic transfer 8091
- Appendix I 89100
- Appendix II 97108
- References 100111