  It was observed recently that relations between matrix elements of certain
operators in the ${\rm SL}(2,\mathbb R)$ spin chain models take the form of
multidimensional integrals derived by R.A. Gustafson. The spin magnets with
${\rm SL}(2,\mathbb C)$ symmetry group and ${\rm L}_2(\mathbb C)$ as a local
Hilbert space give rise to a new type of $\Gamma$-function integrals. In this
work we present a direct calculation of two such integrals. We also analyse
properties of these integrals and show that they comprise the star-triangle
relations recently discussed in the literature. It is also shown that in the
quasi-classical limit these integral identities are reduced to the duality
relations for Dotsenko-Fateev integrals.
