We discuss a Plancherel formula for countable groups, which provides canonical decomposition of the regular representation such group Γ into direct integral factor representations. Our main result gives precise description this in terms FC-center Γfc (that is, normal subgroup consisting elements with finite conjugacy class); involves action an appropriate totally disconnected compact automorphi...

We construct the Plancherel measure corresponding to the Bessel model on the split special orthogonal group of odd degree ≥ 5 and the Whittaker model on a connected split reductive group in general. As an application we prove the inversion formula which expresses the related integral transform in terms of the Satake transform.

We show that the map from minimal-parabolic pseudo-Eisenstein series to their decomposition coefficients against minimal-parabolic Eisenstein series is an isometry to its image. The argument characterizing the image is sketched, but is incomplete without characterizing the other, less-continuous parts of the spectral decomposition. Thus, we do not quite prove Plancherel for this part of the spe...

We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n steps are necessary and suffice for convergence to the stationary distribution.