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

• Introduction
• Unipotent orbits and prehomogeneous spaces
• The Hecke algebra and some Igusa local orbital zeta functions
• The evaluation of \(f^H\) at the identity
• Matching of unipotent orbital integrals
• Remarks on stability and endoscopic transfer
• Appendix I
• Appendix II
• References