arXiv subjects
Publications and source records attributed to Ye. Kolisnyk.
Plancherel formula is one of the celebrated result of harmonic analysis on semisimple Lie groups and their homogeneous spaces. The main goal of this work is to find a q-analog of the Plancherel formula for spherical transform the unit matrix ball. Here we present an explicit formula for the radial part of the Plancherel measure. q-Jacobi polynomials as spherical functions naturally arise on the way.
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.