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On Twisted Orbital Integral Identities for PGL(3) Over A p-adic Field

Published online by Cambridge University Press:  20 November 2018

David Joyner
David Joyner*
Affiliation:
Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402
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Abstract

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The object of this paper is to prove certain p-adic orbital integral identities needed in order to accomplish the symmetric square transfer via the twisted Arthur trace formula. Only §5 of this article contains original material, the rest of it is due to R. Langlands. Very briefly, we reduce the problem of proving certain orbital integral identities for “matching” functions in the respective Hecke algebras to two counting problems on the buildings. We give Langlands’ solution of one of these problems in the case of the unit elements of the respective Hecke algebras and §5 provides the solution to the other one, again, in the unit element case. The main results assume p ≠ 2.

Keywords

11F7011R3922E50
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 6 , 01 December 1990 , pp. 1098 - 1130
Copyright
Copyright © Canadian Mathematical Society 1990

Footnotes

Partially supported by an NSF fellowship.

References

1. Blondel, C., Les représentations supercuspidales des groupes métaplectiques sur GL(2) et leurs caractères, Bull. Soc. Math, de France, (Mémoire 18) 113 (1985).Google Scholar
2. Borel, A., Automophic L-functions, in Proc. Symp. Pure Math., vol 33, part 2, AMS, Providence, RI, 1979.Google Scholar
3. Cartier, P., Representations of p-adic groups, in Proc. Symp. Pure Math., vol 33, part 2, AMS, Providence, RI, 1979.10.1090/pspum/033.1/546593CrossRefGoogle Scholar
4. Flicker, Y., On the symmetric square: orbital integrals, Math. Ann. 279 (1987), 173–191.Google Scholar
5. Gelbart, S. and Jacquet, H., A relation between automorphie forms on GL(2) and GL(3), Ann. Sci. École Norm. Sup. 11 (1978), 471–552.Google Scholar
6. Jacquet, H., Piatetski-Shapiro, I., and Shalika, J., Automorphic forms on GL(3),I,II, Ann. Math. 109 (1979), 169–258.Google Scholar
7. Kottwitz, R. and Shelstad, D., in preparation.Google Scholar
8. Kottwitz, R., Orbital integrals on GL(3), Amer. J. Math. 102 (1980), 327–384.Google Scholar
9. Kottwitz, R., Unstable orbital integrals on SL(3), Duke Math. J. 48 (1981), 649–664 Google Scholar
10. Labesse, J.-P. and Langlands, R., L-indistinguishability for SL(2), Can. J. Math., 31 (1979), 726–785.Google Scholar
11. Lang, S., Algebra, 2nd ed., Addison-Wesley, 1984.Google Scholar
12. Langlands, R., Some identities for orbital integrals attached to GL(3), manuscript.Google Scholar
13. Langlands, R., Les Debut d'une Formule des Traces Stable, Publ. Math. Univ. Paris VII, 1980.Google Scholar
14. Langlands, R., Base Change for GL(2), Ann. Math. Studies, Princeton Univ. Press, 1980.Google Scholar
15. Langlands, R. and Shelstad, D., On the definition of transfer factors, Math. Ann (1987).10.1007/BF01458070CrossRefGoogle Scholar
16. Macdonald, I., Spherical Functions on a Group of p-adic Type, Publ. Ramanujan Inst., no. 2, Madras, 1971.Google Scholar
17. Shimura, G., On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975) 79–98.Google Scholar
18. Tits, J., Reductive groups over local fields, in Proc. Symp. Pure Math., vol 33, part 1, AMS, Providence, RI, 1979.Google Scholar