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

Yu. A. Neretin

Izvestiya. Mathematics , Tome 66 (2002) no. 5, pp. 1035-1046

Résumé

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.