The Cauchy Harish-Chandra Integral, for the pair 𝔲 p , q , 𝔲 1

Andrzej Daszkiewicz; Tomasz Przebinda

Open Mathematics (2007)

  • Volume: 5, Issue: 4, page 654-664
  • ISSN: 2391-5455

Abstract

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For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals.

How to cite

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Andrzej Daszkiewicz, and Tomasz Przebinda. "The Cauchy Harish-Chandra Integral, for the pair \[\mathfrak {u}_{p,q} ,\mathfrak {u}_1 \]." Open Mathematics 5.4 (2007): 654-664. <http://eudml.org/doc/269730>.

@article{AndrzejDaszkiewicz2007,
abstract = {For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals.},
author = {Andrzej Daszkiewicz, Tomasz Przebinda},
journal = {Open Mathematics},
keywords = {Orbital integrals; dual pairs},
language = {eng},
number = {4},
pages = {654-664},
title = {The Cauchy Harish-Chandra Integral, for the pair \[\mathfrak \{u\}\_\{p,q\} ,\mathfrak \{u\}\_1 \]},
url = {http://eudml.org/doc/269730},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Andrzej Daszkiewicz
AU - Tomasz Przebinda
TI - The Cauchy Harish-Chandra Integral, for the pair \[\mathfrak {u}_{p,q} ,\mathfrak {u}_1 \]
JO - Open Mathematics
PY - 2007
VL - 5
IS - 4
SP - 654
EP - 664
AB - For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals.
LA - eng
KW - Orbital integrals; dual pairs
UR - http://eudml.org/doc/269730
ER -

References

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  1. [1] A. Bouaziz: “Intégrales orbitales sur les algèbres de Lie réductives”, Invent. Math., Vol. 115, (1994), pp. 163–207. http://dx.doi.org/10.1007/BF01231757 Zbl0814.22005
  2. [2] A. Daszkiewicz and T. Przebinda: “The oscillator character formula, for isometry groups of split forms in deep stable range”, Invent. Math., Vol. 123, (1996), pp. 349–376. Zbl0845.22007
  3. [3] R. Howe: “Transcending classical invariant theory”, J. Amer. Math. Soc., Vol. 2, (1989), pp. 535–552. http://dx.doi.org/10.2307/1990942 Zbl0716.22006
  4. [4] L. Hörmander: The analysis of linear partial differential operators, I, Springer Verlag, Berlin, 1983. 
  5. [5] T. Przebinda: “A Cauchy Harish-Chandra integral, for a real reductive dual pair”, Invent. Math., Vol. 141, (2000), pp. 299–363. http://dx.doi.org/10.1007/s002220000070 Zbl0953.22014
  6. [6] W. Schmid: “On the characters of the discrete series. The Hermitian symmetric case”, Invent. Math., Vol. 30, (1975), pp. 47–144. http://dx.doi.org/10.1007/BF01389847 Zbl0324.22007
  7. [7] V.S. Varadarajan: Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, Springer Verlag, Berlin-New York, 1977. Zbl0354.43001
  8. [8] N. Wallach: Real Reductive Groups, I, Pure and Applied Mathematics, 132, Academic Press, Inc., Boston, MA, 1988. 

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