The irreducible unitary GL ( n - 1 , ) -spherical representations of SL ( n , )

G. Van Dijk; M. Poel

Compositio Mathematica (1990)

  • Volume: 73, Issue: 1, page 1-30
  • ISSN: 0010-437X

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Van Dijk, G., and Poel, M.. "The irreducible unitary $\mathrm {GL} (n-1,\mathbb {R})$-spherical representations of $\mathrm {SL} (n, \mathbb {R})$." Compositio Mathematica 73.1 (1990): 1-30. <http://eudml.org/doc/89995>.

@article{VanDijk1990,
author = {Van Dijk, G., Poel, M.},
journal = {Compositio Mathematica},
keywords = {involution; -fixed points; unitary irreducible representations; invariant distribution vector; symmetric space; spherical distributions},
language = {eng},
number = {1},
pages = {1-30},
publisher = {Kluwer Academic Publishers},
title = {The irreducible unitary $\mathrm \{GL\} (n-1,\mathbb \{R\})$-spherical representations of $\mathrm \{SL\} (n, \mathbb \{R\})$},
url = {http://eudml.org/doc/89995},
volume = {73},
year = {1990},
}

TY - JOUR
AU - Van Dijk, G.
AU - Poel, M.
TI - The irreducible unitary $\mathrm {GL} (n-1,\mathbb {R})$-spherical representations of $\mathrm {SL} (n, \mathbb {R})$
JO - Compositio Mathematica
PY - 1990
PB - Kluwer Academic Publishers
VL - 73
IS - 1
SP - 1
EP - 30
LA - eng
KW - involution; -fixed points; unitary irreducible representations; invariant distribution vector; symmetric space; spherical distributions
UR - http://eudml.org/doc/89995
ER -

References

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  2. [B-G] J.N. Bernstein and S.I. Gelfand: Tensor products of finite and infinite dimensional representations of semisimple Lie algebras. Comp. Math.41 (1980), p. 245-285. Zbl0445.17006MR581584
  3. [D-P] G van Dijk & M. Poel: The Plancherel formula for the pseudo-Riemannian space SL(n, R)/GL(n-1, R). Comp. Math.58 (1986) p. 371-397. Zbl0593.43009MR846911
  4. [D] J. Dixmier: Enveloping Algebras. North-Holland Publishing Company, Amsterdam/ New York/Oxford, 1977. Zbl0339.17007MR498740
  5. [Fa] J. Faraut: Distributions sphériques sur les espaces hyperboliques. J. Math. Pures et Appl.58 (1979), p. 369-444. Zbl0436.43011MR566654
  6. [F-K] M. Flensted-Jensen & T.H. Koornwinder: Positive definite spherical functions on a non-compact rank one symmetric space. In: Lect. Notes in Math. 739, Springer VerlagBerlin etc., 1979, p. 249-282. Zbl0433.43014MR560841
  7. [H] S. Helgason: A Duality for Symmetric Spaces with Applications to Group Representations I. Adv. Math.5 (1970), p. 1-154. Zbl0209.25403MR263988
  8. [Kn] A.W. Knapp: Representation Theory of Semisimple Groups. An Overview Based on Examples. Princeton University Press, Princeton N.J., 1986. Zbl0604.22001MR855239
  9. [Ko] B. Kostant: On the existence and irreducibility of certain series of representations. In: I.M. Gelfand (ed.) Lie groups and their representations, Halsted Press, New-York, 1975, p. 231-329.(See also Bull. A.M.S. 75 (1969), p. 627-642.) Zbl0229.22026MR245725
  10. [MKo] M.T. Kosters: Spherical distributions on rank one symmetric spaces. Thesis University of Leiden, 1983. 
  11. [MKo-D] M.T. Kosters & G. van Dijk: Spherical Distributions on the Pseudo-Riemannian space SL(n, R)/GL(n - 1, R). J. Funct. Anal.68 (1986), p. 168-213. Zbl0607.43008MR852659
  12. [WKo] W.A. Kosters: Eigenspaces of the Laplace-Beltrami operator on SL(n, R)/S(GL(1) x GL(n - 1)), I and II. Ind. Math.47 (1985), p. 99-145. Zbl0576.43006MR783010
  13. [M] V.F. Molčanov: The Plancherel formula for the pseudo-Riemannian space SL(3, R)/ GL(2, R). Sibirsk Math. J.23 (1982), p. 142-151. Zbl0515.22012MR673546
  14. [P] M. Poel: Harmonic Analysis on SL(n, R)/GL(n - 1, R). Thesis University of Utrecht, 1986. 
  15. [T] E.G.F. Thomas: The theorem of Bochner-Schwartz-Godement for generalized Gelfand pairs. In: K.D. Bierstedt and B. Fuchsteiner (eds.), Functional Analysis: Surveys and recent results III, Elseviers Science Publishers B.V. (North Holland) (1984). Zbl0564.43008MR761388
  16. [Va] V.S. Varadarajan: Harmonic Analysis on Real Reductive Groups. Lecture Notes in Mathematics, No. 576, Springer-Verlag, Berlin/ Heidelberg/New York, 1977. Zbl0354.43001MR473111
  17. [Vo1] D.A. Vogan: Representations of Real Reductive Lie Groups. Birkhäuser, Boston/ Basel/ Stuttgart, 1981. Zbl0469.22012MR632407
  18. [Vo2] D.A. Vogan: The unitary dual of GL(n) over an archimedian field. Invent. Math.83 (1985), p. 449-505. Zbl0598.22008MR827363
  19. [W] N.R. Wallach: Harmonic Analysis on Homogeneous Spaces. Dekker, New-York, 1977. Zbl0265.22022MR498996
  20. [Z] G. Zuckerman: Tensor products of finite and infinite dimensional representations of semisimple Lie groups. Ann. Math.106 (1977), p. 295-308. Zbl0384.22004MR457636

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