Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

Principal nilpotent orbits and reducible principal series


Author: Wentang Kuo
Journal: Represent. Theory 6 (2002), 127-159
MSC (2000): Primary 22E50; Secondary 22E35
DOI: https://doi.org/10.1090/S1088-4165-02-00132-2
Published electronically: July 25, 2002
MathSciNet review: 1915089
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a split reductive $p$-adic group. In this paper, we establish an explicit link between principal nilpotent orbits of $G$ and the irreducible constituents of principal series of $G$. A geometric characterization of certain irreducible constituents is also provided.


References [Enhancements On Off] (What's this?)

  • 1. Barbasch, D. and Vogan, D., ``The local structure of characters,'' J. Funct. Anal., 37, 1980, 27-55. MR 82e:22024
  • 2. Borel, A., ``Automorphic $L$-functions,'' Proc. Sympos. Pure Math., Amer. Math. Soc., Vol. 33, II, 1979, 27-61. MR 81m:10056
  • 3. Bourbaki, N., ``Groupes et algèbres de Lie,'' Chapters 7 and 8. Fasc. XXXVIII, Paris, Hermann 1975.
  • 4. Bruhat, F. and Tits, J., ``Groupes reductifs sur un corps local,'' Paris, Institut des Hautes Etudes Scientifiques, Publ. Math., Vol. 41, 1972, 5-252. MR 48:6265
  • 5. Casselman, W. and Shalika, J. A., ``The unramified principal series of $p$-adic groups II: The Whittaker function,'' Comp. Math., Vol. 1980, 207-231. MR 83i:22027
  • 6. Harish-Chandra, ``Admissible invariant distributions on reductive $p$-adic groups,'' Lie Theories and their applications, Queen's Papers on Pure and Applied Mathematics, Queen's University, Kingston, Ontario, 1987, 281-347. MR 58:28313
  • 7. Gelbart, S.S. and Knapp, A.W., ``Irreducible Constituents of Principal Series of $SL_n(k)$,'' Duke Mathematical Journal, Vol. 48, No. 2, 1981, 313-326. MR 82j:22018
  • 8. Gelbart, S.S. and Knapp, A.W., ``$L$-Indistinguishability and $R$ Groups for the Special Linear Group,'' Advances in Mathematics 43, 1982, 101-121. MR 83j:22009
  • 9. Gel'fand, I.M., Graev, M.I., and Pyatetskii-Shapiro, I.I., ``Representation Theory and Automorphic Functions,'' W.B. Saunders Company, 1969. MR 38:2093
  • 10. Keys, Charles D., ``On the decomposition of reducible principal series representations of $p$-adic Chevalley groups,'' Pacific Journal of Mathematics, Vol. 101, No. 2, 1982, 351-388. MR 84d:22032
  • 11. Keys, Charles D., ``$L$-indistinguishability and $R$-groups for quasisplit groups: unitary groups in even dimension,'' Ann. Sci. École Norm. Sup. (4) 20, 1987, 31-64. MR 88m:22042
  • 12. Keys, Charles D. and Shahidi, F., ``Artin $L$-function and normalization of intertwining operators,'' Ann. Sci. École Norm. Sup., 4 série, t. 21, 1988, 67-89. MR 89k:22034
  • 13. Knapp, A.W., ``Commutativity of intertwining operators,'' Bull. Amer. Math. Soc. 79, 1973, 1016-1018. MR 48:11399
  • 14. Knapp, A.W., ``Commutativity of intertwining operators II,'' Bull. Amer. Math. Soc. 82, 1976, no. 2, 271-273. MR 53:10986
  • 15. Knapp, A.W. and Stein, E., ``Irreducibility theorems for the principal series,'' in Conference on Harmonic Analysis, Lecture Notes in Mathematics, 266, Springer-Verlag, New York, 1972, 197-214. MR 54:10499
  • 16. Kostant, B., ``On Whittaker vectors and representation theory,'' Invent. Math, (1978), no. 2, 101-184. MR 80b:22020
  • 17. Matumoto. H., `` $C^{-\infty}$-Whittaker vectors corresponding to a principal nilpotent orbit of a real reductive linear Lie group, and wave front sets,'' Compositio Math, 82, 1992, 189-244. MR 93c:22026
  • 18. M\oeglin, C. and Waldspurger, J.L., ``Modèles de Whittaker dégénérés pour des groupes $p$-adiques,'' Math. Z. 196, 1987, 427-452. MR 89f:22024
  • 19. Rodier, F., ``Modèle de Whittaker et caractères de représentations, in Noncommutative harmonic analysis'', edited by H. Carmona and M. Vergne, Lect. Notes Math. 466, 151-171, Springer, Berlin 1975. MR 52:14165
  • 20. Sally, P.J. and Taibleson, M.H., ``Special functions on locally compact fields,'' Acta, Math. 116 (1966), 279-309. MR 34:6168
  • 21. Shahidi, F., ``Functional equation satisfied by certain $L$-functions,'' Comp. Math., Vol. 37, 1978, 171-208. MR 58:16603
  • 22. Shahidi, F., ``On Certain $L$-Functions,'' Amer. J. Math., Vol. 103, 1981, 297-356. MR 82i:10030
  • 23. Shahidi, F., ``A proof of Langlands' conjecture on Plancherel measures; Complementary series for $p$-adic groups,'' Ann. of Math., 132, 1990, 273-330. MR 91m:11095
  • 24. Steinberg, R., ``Lectures on Chevalley Groups,'' Yale University Lecture Notes, New Haven, 1968. MR 57:6215
  • 25. Vogan, D., ``Gel'fand-Kirillov dimensions for Harish-Chandra modules,'' Invent. Math, 48, 1978, 75-98. MR 58:22205

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 22E50, 22E35

Retrieve articles in all journals with MSC (2000): 22E50, 22E35


Additional Information

Wentang Kuo
Affiliation: Department of Mathematics, Purdue University, MATH 726, West Lafayette, Indiana 47906
Address at time of publication: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6
Email: wtkuo@mast.queensu.ca

DOI: https://doi.org/10.1090/S1088-4165-02-00132-2
Keywords: Nilpotent orbits, reducible principal series, $p$-adic groups
Received by editor(s): July 15, 2001
Received by editor(s) in revised form: April 11, 2002
Published electronically: July 25, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society