# The Cauchy Harish-Chandra Integral, for the pair ${\U0001d532}_{p,q},{\U0001d532}_{1}$

Andrzej Daszkiewicz; Tomasz Przebinda

Open Mathematics (2007)

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

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topAndrzej 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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- [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
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- [4] L. Hörmander: The analysis of linear partial differential operators, I, Springer Verlag, Berlin, 1983.
- [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] 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] V.S. Varadarajan: Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, Springer Verlag, Berlin-New York, 1977. Zbl0354.43001
- [8] N. Wallach: Real Reductive Groups, I, Pure and Applied Mathematics, 132, Academic Press, Inc., Boston, MA, 1988.

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