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 Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 5, Pages 171–182 (Mi izv404)

The action of an overalgebra on the Plancherel decomposition and shift operators in the imaginary direction

Yu. A. Neretin

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We consider the tensor product of a unitary representation of $G=\mathrm{SL}_2(\mathbb R)$ with a highest weight and the complex-conjugate representation with a lowest weight. The representation space is acted upon by the direct product $G\times G$. We decompose the resulting representation into a direct integral with respect to the diagonal subgroup $G\subset G\times G$. This direct integral is realized as the $L^2$ space on the product of a circle with coordinate $\phi\in[0,2\pi)$ and the semiline $s\geqslant 0$, where $s$ enumerates unitary representations of $G$ of the principal series.
We get explicit formulae for the action of the Lie algebra $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$ on this direct integral. It turns out that the representation operators are second order differential operators with respect to $\phi$ and second order difference operators with respect to $s$, and the difference operators are expressed in terms of the shift $s\mapsto s+i$ in the imaginary direction.

DOI: https://doi.org/10.4213/im404

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English version:
Izvestiya: Mathematics, 2002, 66:5, 1035–1046

Bibliographic databases:

UDC: 519.46
MSC: 22E46, 43A85

Citation: Yu. A. Neretin, “The action of an overalgebra on the Plancherel decomposition and shift operators in the imaginary direction”, Izv. RAN. Ser. Mat., 66:5 (2002), 171–182; Izv. Math., 66:5 (2002), 1035–1046

Citation in format AMSBIB
\Bibitem{Ner02} \by Yu.~A.~Neretin \paper The action of an overalgebra on the Plancherel decomposition and shift operators in the imaginary direction \jour Izv. RAN. Ser. Mat. \yr 2002 \vol 66 \issue 5 \pages 171--182 \mathnet{http://mi.mathnet.ru/izv404} \crossref{https://doi.org/10.4213/im404} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1965938} \zmath{https://zbmath.org/?q=an:1064.22005} \elib{http://elibrary.ru/item.asp?id=14470919} \transl \jour Izv. Math. \yr 2002 \vol 66 \issue 5 \pages 1035--1046 \crossref{https://doi.org/10.1070/IM2002v066n05ABEH000404} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33748507383} 

• http://mi.mathnet.ru/eng/izv404
• https://doi.org/10.4213/im404
• http://mi.mathnet.ru/eng/izv/v66/i5/p171

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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. V. F. Molchanov, “Canonical Representations and Overgroups for Hyperboloids”, Funct. Anal. Appl., 39:4 (2005), 284–295
2. Molchanov V.F., “Canonical representations and overgroups for hyperboloids of one sheet and Lobachevsky spaces”, Acta Appl. Math., 86:1-2 (2005), 115–129
3. J. Math. Sci. (N. Y.), 141:4 (2007), 1452–1478
4. Molchanov V.F., “Canonical representations on lobachevsky spaces: an interaction with an overalgebra”, Acta Appl. Math., 99:3 (2007), 321–337
5. V. F. Molchanov, “Poisson and Fourier Transforms for Tensor Products”, Funct. Anal. Appl., 49:4 (2015), 279–288
6. Molchanov V.F., “Canonical Representations For Hyperboloids: An Interaction With An Overalgebra”, Geometric Methods in Physics, Trends in Mathematics, eds. Kielanowski P., Ali S., Bieliavsky P., Odzijewicz A., Schlichenmaier M., Voronov T., Springer Int Publishing Ag, 2016, 129–138
7. Yu. A. Neretin, “Operational Calculus for the Fourier Transform on the Group $\operatorname{GL}(2,\mathbb{R})$ and the Problem about the Action of an Overalgebra in the Plancherel Decomposition”, Funct. Anal. Appl., 52:3 (2018), 194–202
8. Molchanov V.F., “Polynomial Quantization and Overalgebra”, Algebr. Represent. Theory, 21:5 (2018), 1071–1085
9. Neretin Yu.A., “Restriction of Representations of Gl (N+1, C) to Gl (N, C) and Action of the Lie Overalgebra”, Algebr. Represent. Theory, 21:5 (2018), 1087–1117
10. Neretin Yu.A., “The Fourier Transform on the Group G(l)2(R) and the Action of the Overalgebra Gl(4)”, J. Fourier Anal. Appl., 25:2 (2019), 488–505
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