Inversion quasi-invariant Gaussian measures on the group of upper-triangular matrices of infinite order
In [Measures on infinite-dimensional groups, quasi-invariant with respect to inverse mapping and the commutant theorem (Singapore 1998; Zbl 0935.43001)], the author presented a scheme for generalising, to non-locally compact groups, the commutation theorem for locally compact groups with (left) Haar measure, viz. that the right regular representation is generated by the operators of the left regular representation. He showed that, for \({\tilde{G}}\) the group of upper triangular infinite matrices, which he provides with a Gaussian measure with variance determined by a double sequence \(b = (b_{k,n})_{k<n}\) of positive numbers, if \(E(b)\), as defined there, is finite then the measure is quasi-invariant with respect to the inverse mapping and an analogue of the commutation theorem holds. The article under review provides a slightly modified version of the proof.
left and right regular representations of a group
scientific article; zbMATH DE number 1497245