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Some results on unipotent orbital integrals. (English) Zbl 0737.22005
This paper deals with some questions concerning orbital integrals important in applications of the Selberg trace formula. Let $$F$$ be a local field and $$G$$ be a reductive algebraic group defined over $$F$$. Let $$H$$ be an endoscopic group of $$G$$. Then one should be able to transfer functions on $$G(F)$$ to functions on $$H(F)$$ so that certain corresponding orbital integrals are equal. Moreover this transfer should extend the natural map between Hecke algebras. At present this is not known in general for non-archimedean fields and the purpose of this paper is to produce evidence for it. One approach to the problem requires the analysis of certain unipotent orbital integrals for functions in the Hecke algebra of $$G$$ and of corresponding functions for $$H$$ with the intention of proving linear relations between them. The author does this by direct and difficult computations in the cases $$G=SO(2k+1)$$, $$H=SO(2k- 1)\times PGL(2)$$ and $$G=Sp(2k)$$, $$H=SO(2k)$$.

MSC:
 22E35 Analysis on $$p$$-adic Lie groups 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 22E50 Representations of Lie and linear algebraic groups over local fields
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References:
 [1] M. Assem , Marching of certain unipotent orbital integrals on P-adic orthogonal groups , Ph.D. Thesis, University of Washington (1988). [2] D. Barbasch and D. Vogan , Primitive ideals and orbital integrals in complex classical groups , Math. Ann., 259 (1982), pp. 153-199. · Zbl 0489.22010 · doi:10.1007/BF01457308 · eudml:163626 [3] R. Carter , Finite Groups of Lie Type: Conjugacy Classes and Complex Characters , John Wiley, 1985. · Zbl 0567.20023 [4] L. Clozel and P. Delorme , Le Théorème de Paley-Wiener invariant pour les groupes die Lie reductifs (I) , Inv. Math., 77 (1984), pp. 427-453. · Zbl 0584.22005 · doi:10.1007/BF01388832 · eudml:143154 [5] V. Ginsburg , Integrales sur le orbites nilpotentes et representations de groupes de Weyl , C. R. Acad. Sc., Paris (1983). · Zbl 0544.22009 [6] T. Hales , Unipotent classes induced from endoscopic groups , Mathematical Sciences Research Institute (September 1987). [7] S. Helgason , Groups and Geometric Analysis , Academic Press, 1984. · Zbl 0543.58001 [8] J. Igusa , Forms of Higher Degree , Tata Institute of Fundamental Research, Bombay, 1978. · Zbl 0417.10015 [9] K. Ireland and M. Rosen , A classical Introduction to Modern Number Theory , Graduate Texts in Mathematics, 58, Springer Verlag, 1982. · Zbl 0482.10001 [10] R. Kottwitz , Tamagawa numbers , Ann. of Math., 127 (1986), pp. 629-646. · Zbl 0678.22012 · doi:10.2307/2007007 [11] R. Langlands , Les Débuts d’une Formule de Traces Stables , Publ. Math. de l’Univ. de Paris VII (13), 1983. · Zbl 0532.22017 [12] G. Lusztig and J. Spaltenstein , Induced unipotent classes , J. London Math. Soc., 19 (1979), pp. 41-52. · Zbl 0407.20035 · doi:10.1112/jlms/s2-19.1.41 [13] I. Macdonald , Some irreducible representations of Weyl groups , Bull. London Math. Soc., 4 (1972), pp. 148-150. · Zbl 0251.20043 · doi:10.1112/blms/4.2.148 [14] I. Macdonald , Spherical Functions on a Group of p-adic Type , Ramanujan Institute Publications, Madras, 1971. · Zbl 0302.43018 [15] I. Macdonald , The Poincaré series of a Coxeter group , Math. Ann., 199 (1972), pp. 161-174. · Zbl 0286.20062 · doi:10.1007/BF01431421 · eudml:162322 [16] R. Rao , Orbital integrals in reductive groups , Ann. of Math., 96:3 (1972), pp. 503-510. · Zbl 0302.43002 · doi:10.2307/1970822 [17] D. Shelstad , L -indistinguishability for real groups , Math. Ann., 259 (1982), pp. 385-430. · Zbl 0506.22014 · doi:10.1007/BF01456950 · eudml:163641 [18] J. Spaltenstein , Classes Unipotents et Sous-Groupes de Borel , Springer Lecture Notes in Math., 1982. · Zbl 0486.20025 [19] T. Springer , A construction of representations of Weyl groups , Inv. Math., 44 (1978), pp. 279-293. · Zbl 0376.17002 · doi:10.1007/BF01403165 · eudml:142532 [20] T. Springer and R. Steinberg , Conjugacy classes, seminar on algebraic groups and related finite groups , Lecture Notes in Math, 131. · Zbl 0249.20024
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