Memoirs of the American Mathematical Society 1998; 101 pp; softcover Volume: 134 ISBN10: 082180765X ISBN13: 9780821807651 List Price: US$49 Individual Members: US$29.40 Institutional Members: US$39.20 Order Code: MEMO/134/635
 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
