Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Suborbits of a point-stabilizer in the unitary group on the last subconstituent of Hermitean dual polar graphs

Let Γ be a dual polar graph in a unitary space. It is well-known that a point-stabilizer in the unitary group is transitive on the last subconstituent Λ of Γ. In this paper, we determine all the suborbits of this action, calculate its rank and the length of each suborbit. Note that the induced subgraph on Λ is quasi-strongly regular. As an application of our results, all its parameters are computed.     

Distinction by the quasi-split unitary group

In earlier work, we proved that any quadratic base change automorphic cuspidal representation of GL(n) is distinguished by a unitary group. Here we prove that we can take the unitary group to be quasi-split     

Generalized isometries of the special unitary group

In this paper we determine the structure of all so-called generalized isometries of the special unitary group which are transformations that respect any member of a large collection of generalized distance measures.     

Harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on the infinite-dimensional unitary group is to decompose a certain family of unitary representations of this group, which is a substitute for the nonexisting regular representations and depends on two complex parameters (Olshanski, 2003). In the case of noninteger parameters, the decomposing measure is described in terms of determinantal point processes (Borondin and Olshanski, 2005). The aim of the present paper is to describe the decomposition for integer parameters; in this case, the spectrum of the decompositions changes drastically. A similar result was earlier obtained for the infinite symmetric group (Kerov, Olshanski, and Vershik, 2004), but the case of the unitary group turned out to be much more complicated. In the proof we use Gustafson’s multilateral summation formula for hypergeometric series. Bibliography: 6 titles.     

Isomorphisms of the unitary group over a ring

We describe isomorphisms of unitary groups over associative rings with 1/2 that contain a compact system of idempotents. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 55–70, 2006.     

The fundamental group of a unitary orbit

The unitary orbit of a complex n × n matrix A is simply connected if and only if the portion of the commutant {A} which resides in the special unitary group is path connected.     

Asymptotics of Plancherel measures for the infinite-dimensional unitary group

We study a two-dimensional family of probability measures on infinite Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters of the infinite-dimensional unitary group. These measures are unitary group analogs of the well-known Plancherel measures for symmetric groups.We show that any measure from our family defines a determinantal point process on Z₊×Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes.     

Maximizing concave piecewise affine functions on the unitary group

We show that a convex relaxation, introduced by Sridharan, McEneaney, Gu and James to approximate the value function of an optimal control problem arising from quantum gate synthesis, is exact. This relaxation applies to the maximization of a class of concave piecewise affine functions over the unitary group.