Contents 0. Introduction 1. Unipotent orbits and prehomogeneous spaces 1. Ranga Rao data 2. Some classes of examples 3. Induction of G(F)-unipotent classes 2. The Hecke algebra and some Igusa local orbital zeta functions 1. Unipotent orbital integrals as special values of orbital Igusa zeta functions 2. GL(n, 0/r)-orbit decomposition of Sym(n) and local densitites 3. The ^-decomposition of supp (X *-• / m ( l + X)) n g(2) 3. The evaluation of fH at the identity 1. The integral of $fc, 2 k n, Part A 2. The integral of $fc, 2 k n, Part B 3. The integral of $i 4. The value / H ( l ) for H = SOE(4) x SL(2) 5. Matching results 4. Matching of unipotent orbital integrals 1. Unramified endoscopic data 2. The map f *- fH 3. Endoscopic induction of unipotent orbits 4. Matching of regular unipotent orbital integrals 5. Matching of unipotent orbital integrals for G = Sp(6) and its unramified en 6. Matching of subregular orbital integrals 7. Matching of the orbits 2 r l 2 n - 2 r , for r = 2,3 8. Matching results for Sp(8) 9. Endoscopic transfer of the trivial orbital integral 10. Endoscopic transfer of other orbital integrals 11. Some remarks on the transfer factors 5. Remarks on stability and endoscopic transfer 1. Stable distributions 2. Formal properties of endoscopic induction and stability 3. Remarks on Shalika germs 4. Conjecture (B) implies Conjecture (A) 5. Stability and subregular packets 6. Heuristics Appendix I Appendix II References Vl l

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