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The action of an overalgebra on the Plancherel decomposition and shift operators in the imaginary direction , 66:5 (2002), 1035–1046статья
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Дата последнего поиска статьи во внешних источниках: 13 мая 2020 г.
- Автор: Neretin Yu A.
- Журнал: Izvestiya. Mathematics
- Том: 66
- Номер: 5
- Год издания: 2002
- Издательство: American Mathematical Society
- Местоположение издательства: United States
- Первая страница: 1035
- Последняя страница: 1046
- DOI: 10.1070/IM2002v066n05ABEH000404
- Аннотация: We consider the tensor product of a unitary representation of $G=SL_2(R)$ 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 GG. 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 $\phi∈[0,2\pi)$ and the semiline $s\ge 0$, where $s$ enumerates unitary representations of $G$ of the principal series. We get explicit formulae for the action of the Lie algebra $sl_2\oplus sl_2$ 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.
- Добавил в систему: Неретин Юрий Александрович