Weingarten integration over noncommutative homogeneous spaces (Intégration de Weingarten sur les espaces homogènes non commutatifs)

摘要

We discuss an extension of the Weingarten formula, to the case of noncommutative homogeneous spaces, under suitable “easiness” assumptions. The spaces that we consider are noncommutative algebraic manifolds, generalizing the spaces of type $X=G/Hsubset mathbb{C}^N$, with $Hsubset Gsubset U_N$ being subgroups of the unitary group, subject to certain uniformity conditions. We discuss various axiomatization issues, then we establish the Weingarten formula, and we derive some probabilistic consequences.