Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τ_λ of U(n) has nonnegative highest weight λ of depth n/2. It says that the multiplicity in τ_λ|_SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τ_λ in the U(n) tensor product τ_μτ_ν, the allowable μ's being all even nonnegative highest weights for U(n). Littlewood's proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τ_μτ_ν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.     

The Clebsch-Gordan coefficients for the quantum group SμU(2) and q-Hahn polynomials

The tensor product of two unitary irreducible representations of the quantum group S_μU(2) is decomposed in a direct sum of unitary irreducible representations with explicit realizations. The Clebsch-Gordan coefficients yield the orthogonality relations for q-Hahn and dual q-Hahn polynomials.     

Representations in L-Spaces on Infinite-Dimensional Symmetric Cones

In this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L²-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert-Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.”     

Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group

We prove the pairwise disjointness of representations T _z,w of the infinite-dimensional unitary group. These representations are a natural generalization of the regular representation to the “big” group U(∞). They were introduced and studied by G. Olshanski and A. Borodin. The disjointness of these representations reduces to that of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.     

A contraction ofS U (2) to the Heisenberg group

A contraction ofS U (2) to the three-dimensional Heisenberg group is defined by considering the action of the two groups on appropriate manifolds in ². The infinite-dimensional irreducible unitary representations of the Heisenberg group are then shown to be limits of (finite-dimensional) irreducible representations ofS U (2).     

Representations of rational Cherednik algebras of rank one in positive characteristic

In this paper, we classify the irreducible representations of the rational Cherednik algebras of rank 1 in characteristic p>0. There are two cases. One is the “quantum” case, where “Planck's constant” is nonzero and generic irreducible representations have dimension pr, where r is the order of the cyclic group contained in the algebra. The other is the “classical” case, where “Planck's constant” is zero and generic irreducible representations have dimension r.     

Grauert- and Lax-Halmos-type theorems and extension of matrices with entries in H

In the paper we prove an extension theorem for matrices with entries in H^∞(U) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on maximal ideal spaces of certain Banach algebras.     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Representations of trigonometric Cherednik algebras of rank one in positive characteristic

In this paper, we classify the irreducible representations of the trigonometric Cherednik algebras of rank 1 in characteristic p>0. There are two cases. One is the “quantum” case, where “Planck’s constant” is nonzero and generic irreducible representations have dimension 2p. In this case, smaller representations exist if and only if the “coupling constant” k is in ; namely, if k is an even integer such that 0≤k≤p−1, then there exist irreducible representations of dimensions p−k and p+k, and if k is an odd integer such that 1≤k≤p−2, then there exist irreducible representations of dimensions k and 2p−k. The other case is the “classical” case, where “Planck’s constant” is zero and generic irreducible representations have dimension 2. In that case, one-dimensional representations exist if and only if the “coupling constant” k is zero.     

Irreducible decompositions of unitary representations of infinite- dimensional groups

This paper concerns the problem of irreducible decompositions of unitary representations of topological groups G, including the group Diff₀(M) of diffeomorphisms with compact support on smooth manifolds M. It is well known that the problem is affirmative, when G is a locally compact, separable group (cf. [3, 4]). We extend this result to infinite-dimensional groups with appropriate quasi-invariant measures, and, in particular, we show that every continuous unitary representation of Diff₀(M) has an irreducible decomposition under a fairly mild condition. This research was partially supported by a Grant-in-Aid for Scientific Research (No.14540167), Japan Socieity of the Promotion of Science.     

Irreducible unitary representations of the groups U(p,q) sustaining passage to the limit aS q → ∞

The results of the note are inspired by the theory of representations of the infinite-dimensional classical groups. A new series of irreducible unitary representations of the group U(p,q) is described. These representations are constructed in the Gel'fand-Tsetlin basis and also as induced by nonunitary finite-dimensional representations of a maximal parabolic subgroup. As q they approximate irreducible unitary representations of the group U(p,).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 114–120, 1989.     

The universality of the quantum Fourier transform in forming the basis of quantum computing algorithms

The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by the Walsh-Hadamard transform H and phase shifts P(·), both of which are 2×2 unitary matrices as operators on the two-dimensional 1-qubit space. In this paper, we show that H and P(·) suffice to generate the unitary group U(2) and, consequently, through controlled-U operations and their concatenations, the entire unitary group U(2ⁿ) on n qubits can be generated. Since any quantum computing algorithm in an n-qubit quantum computer is based on operations by matrices in U(2ⁿ), in this sense we have the universality of the QFT.     

Two Constructions of Markov Chains on the Dual of <Emphasis Type="Italic">U</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)

We provide two new constructions of Markov chains which had previously arisen from the representation theory of \(U(\infty )\). The first construction uses the combinatorial rule for the Littlewood–Richardson coefficients, which arise from tensor products of irreducible representations of the unitary group. The second arises from a quantum random walk on the von Neumann algebra of U(n), which is then restricted to the center. Additionally, the restriction to a maximal torus can be expressed in terms of weight multiplicities, explaining the presence of tensor products.     

Free group representations from vector-valued multiplicative functions,I

Let Γ denote a noncommutative free group and let Ω stand for its boundary. We construct a large class of unitary representations of Γ. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of Γ ⋉_λ C(Ω). We prove here that each of them is irreducible as a representation of Γ ⋉_λ C(Ω). Actually, as will be shown in further work, each of them is irreducible as a representation of Γ, or is the direct sum of exactly two irreducible, inequivalent Γ-representations. This research was supported by the Italian CNR.     

On Bessel integrals for reducible degenerate principal series representations

We discuss generalized Bessel integrals with nondegenerate characters, which are assigned to irreducible submodules of a reducible degenerate principal series representation of Sp(n,R). Then we give sufficient conditions for their vanishings which are based on the signatures of the nondegenerate characters. This consequently suggests a reasonable correspondence between open GLn(R)-orbits in the set of real symmetric matrices of size n and irreducible submodules of the reducible principal series representations.     

Functors for unitary representations of classical real groups and affine Hecke algebras

We define exact functors from categories of Harish–Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with “real infinitesimal character”) into the corresponding p-adic spherical unitary dual.     

Harmoniques Sphériques et Représentations Irréductibles de O ( ∞ )

We build an irreducible unitary representation of SO( ) from the usual one of SO( n ) in the space of harmonic homogeneous polynomials of degree m of ⁿ . We give a characterization of these new representations which extends in a natural way the finite dimensional characterization. In the particular case of SO( ), we thus get some results of Olshanskii (cf. [12]). This leads to a new proof of McKean conjecture about irreducible representations of ( infin ) (cf. [10]).