  We discuss the spectral decomposition of the hypergeometric differential
operators on the line $\mathrm{Re}\, z=1/2$. Such operators arise in the
problem of decomposition of tensor products of unitary representations of the
universal covering of the group $SL(2\,{\mathbb R}$. Our main purpose is a
search of natural bases in generalized eigenspaces and variants of the
inversion formula.
