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

Abstract

We consider the tensor product of a unitary representation of with a highest weight and the complex-conjugate representation with a lowest weight. The representation space is acted upon by the direct product . We decompose the resulting representation into a direct integral with respect to the diagonal subgroup . This direct integral is realized as the space on the product of a circle with coordinate and the semiline , where enumerates unitary representations of of the principal series.We get explicit formulae for the action of the Lie algebra on this direct integral. It turns out that the representation operators are second order differential operators with respect to and second order difference operators with respect to , and the difference operators are expressed in terms of the shift in the imaginary direction.