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 Mosc. Math. J., 2002, Volume 2, Number 4, Pages 635–645 (Mi mmj66)

An analytic separation of series of representations for $SL(2;\mathbb R)$

S. G. Gindikin

Rutgers, The State University of New Jersey, Department of Mathematics

Abstract: For the group $SL(2;\mathbb R)$, holomorphic wave fronts of the projections on different series of representations are contained in some disjoint cones. These cones are convex for holomorphic and antiholomorphic series, which corresponds to the well-known fact that these projections can be extended holomorphically to some Stein tubes in $SL(2;\mathbb C)$. For the continuous series, the cone is not convex, and the projections are boundary values of 1-dimensional $\bar\partial$-cohomology in a non-Stein tube.

Key words and phrases: ntegral geometry, horospherical transform, series of representations, $\bar\partial$-cohomology, holomorphic wave front.

DOI: https://doi.org/10.17323/1609-4514-2002-2-4-635-645

Full text: http://www.ams.org/.../abst2-4-2002.html
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MSC: 22E46, 32A45, 44A12
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Citation: S. G. Gindikin, “An analytic separation of series of representations for $SL(2;\mathbb R)$”, Mosc. Math. J., 2:4 (2002), 635–645

Citation in format AMSBIB
\Bibitem{Gin02} \by S.~G.~Gindikin \paper An analytic separation of series of representations for ${\rm SL}(2;\mathbb R)$ \jour Mosc. Math.~J. \yr 2002 \vol 2 \issue 4 \pages 635--645 \mathnet{http://mi.mathnet.ru/mmj66} \crossref{https://doi.org/10.17323/1609-4514-2002-2-4-635-645} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1986084} \zmath{https://zbmath.org/?q=an:1026.22014} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208593600001} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Gindikin S., “Holomorphic horospherical duality “sphere-cone””, Indag Math (N S ), 16:3–4 (2005), 487–497
2. S. G. Gindikin, “The horospherical Cauchy–Radon transform on compact symmetric spaces”, Mosc. Math. J., 6:2 (2006), 299–305
3. Gindikin S., Kroetz B., Olafsson G., “Horospherical model for holomorphic discrete series and horospherical Cauchy transform”, Compositio Mathematica, 142:4 (2006), 983–1008
4. Gindikin S., “Harmonic analysis on symmetric Stein manifolds from the point of view of complex analysis”, Jpn J Math, 1:1 (2006), 87–105
5. Gindikin S., “The horospherical duality”, Science in China Series A-Mathematics, 51:4 (2008), 562–567
6. Frenkel I., Libine M., “Split quaternionic analysis and separation of the series for SL(2, R) and SL(2, C)/SL(2, R)”, Adv Math, 228:2 (2011), 678–763