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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2002 Volume 66, Issue 5, Pages 171–182 (Mi im404)

This article is cited in 12 papers

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.

UDC: 519.46

MSC: 22E46, 43A85

Received: 06.04.2001

DOI: 10.4213/im404


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

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