Action of overalgebra in Plancherel decomposition and shift operators in imaginary direction

Details Action of overalgebra in Plancherel decomposition and shift operators in imaginary direction Action of overalgebra in Plancherel decomposition and shift operators in imaginary direction

2002-02-02

arxiv.org/abs/math/0202018v1

Mathematics

Representation Theory

12 pages

Scientific paper

Consider the Plancherel decomposition of the tensor product of a highest weight and a lowest weight unitary representations of $SL_2$. We construct explicitly the action of the Lie algebra $sl_2 + sl_2$ in the direct integral of Hilbert spaces. It turns out that a Lie algebra operator is a second order differential operator in one variable and second order difference operator with respect to another variable. The difference operators are defined in terms of the shift in the imaginary direction $f(s)\mapsto f(s+i)$, $i^2=-1$ (the Plancherel measure is supported by real $s$).

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