  We obtain a $q$-analog of the well known Wallach-Okounkov result on a joint
spectrum of invariant differential operators with polynomial coefficients on a
prehomogeneous vector space of complex $n \times n$-matrices. We are motivated
by applications to the problems of harmonic analysis in the quantum matrix
ball: our main theorem can be used while proving the Plancherel formula (to be
published).
  This paper is dedicated to our friend and colleague Dmitry Shklyarov who
celebrates his 30-th birthday on April 8, 2006.
