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

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*Magdy Assem*

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.

#### 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