Spherical unitary dual of general linear group over non-Archimidean local field

Marko Tadic

Annales de l'institut Fourier (1986)

  • Volume: 36, Issue: 2, page 47-55
  • ISSN: 0373-0956

Abstract

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Let F be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of G L ( n , F ) is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of G L ( n , F ) by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description of all equivalence classes of irreducible unitary spherical representations of G L ( n , F ) .

How to cite

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Tadic, Marko. "Spherical unitary dual of general linear group over non-Archimidean local field." Annales de l'institut Fourier 36.2 (1986): 47-55. <http://eudml.org/doc/74715>.

@article{Tadic1986,
abstract = {Let $F$ be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of $GL(n,F)$ is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of $GL(n,F)$ by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description of all equivalence classes of irreducible unitary spherical representations of $GL(n,F)$.},
author = {Tadic, Marko},
journal = {Annales de l'institut Fourier},
keywords = {irreducible spherical representations; parabolically induced representations; complementary series},
language = {eng},
number = {2},
pages = {47-55},
publisher = {Association des Annales de l'Institut Fourier},
title = {Spherical unitary dual of general linear group over non-Archimidean local field},
url = {http://eudml.org/doc/74715},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Tadic, Marko
TI - Spherical unitary dual of general linear group over non-Archimidean local field
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 2
SP - 47
EP - 55
AB - Let $F$ be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of $GL(n,F)$ is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of $GL(n,F)$ by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description of all equivalence classes of irreducible unitary spherical representations of $GL(n,F)$.
LA - eng
KW - irreducible spherical representations; parabolically induced representations; complementary series
UR - http://eudml.org/doc/74715
ER -

References

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  1. [1] I. N. BERNSTEIN, P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-archimedean case), in Lie Group Representations II, Proceedings, University of Maryland 1982-1983, Lecture Notes in Math., vol. 1041, Springer-Verlag, Berlin, (1983), 50-102. Zbl0541.22009
  2. [2] P. CARTIER, Representations of p-adic groups: a survey, in Proc. Sympos. Pure Math. Vol. XXXIII, part 1, Amer. Math. Soc., Providence, R.I., 1979, 111-155. Zbl0421.22010MR81e:22029
  3. [3] W. CASSELMAN, The unramified principal series of p-adic groups I, The spherical functions, Comp. Math., vol. 41 (1980), 387-406. Zbl0472.22004MR83a:22018
  4. [4] J. DIEUDONNÉ, Treatise on analysis, vol. VI, Academic Press, New York, 1978. Zbl0435.43001MR58 #29825b
  5. [5] I. M. GELFAND, M. I. GRAEV, Representations of a group of the second order with elements from a locally compact field, Russian Math. Surveys, 18 (1963), 29-100. Zbl0166.40201MR27 #5864
  6. [6] I. G. MACDONALD, Spherical functions on a group of p-adic type, Rammanjan Institute, Univ. of Madras Publ. (1971). Zbl0302.43018
  7. [7] F. I. MAUTNER, Spherical functions over p-adic fields I, II, Amer. J. Math., vol. 80 (1958), 441-457 and vol. 86 (1964), 171-200. Zbl0135.17204
  8. [8] G. I. OLSHANSKY, Intertwining operators and complementary series in the class of representations of the general group of matrices over a locally compact division algebra, induceded from parabolic subgroups, Math. Sb., vol. 93, n°. 2 (1974), 218-253. 
  9. [9] I. SATAKE, Theory of spherical functions on reductive algebraic groups over p-adic fields, Inst. Hautes Études Sci. Publ. Math., 18 (1963), 1-69. Zbl0122.28501MR33 #4059
  10. [10] M. TADIC, Classification of unitary representations in irreducible representations of general linear group (non-archimedean case), to appear in Ann. Scient. École Norm. Sup. Zbl0614.22005
  11. [11] A. V. ZELEVINSKY, Induced representations of reductive p-adic groups II, Ann. Scient. École Norm. Sup., 13 (1980), 165-210. Zbl0441.22014MR83g:22012

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