Guantum matrix ball: the Cauchy--Szeg\"o kernel and the Shilov boundary

L. Vaksman

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2001), pp. 366-384.

Résumé

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.