  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 $s$).
