Guantum matrix ball: the Cauchy–Szegö kernel and the Shilov boundary

This work produces a q-analogue of the Cauchy–Szegö integral representation that retrieves a holomorphic function in the matrix ball from its values on the Shilov boundary. Besides that, the Shilov boundary of the quantum matrix ball is described and the $U_q\mathfrak{su}_{m,n}$-covariance of the $U_q\mathfrak{s}(\mathfrak{u}_m \times \mathfrak{u}_n)$-invariant integral on this boundary is established. The latter result allows one to obtain a q-analogue for the principal degenerate series of unitary representations related to the Shilov boundary of the matrix ball.