Narrow Results

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation.

We consider, for which subgroup G′ of G, the restriction π|_G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p₁) × U(p₂) × U(q) for p₁, p₂ ≥ 1 with p₁ + p₂ = p is multiplicity-free, whereas the restriction to U(p₁) × U(p₂) × U(q₁) × U(q₂) for q₁, q₂ ≥ 1 with q₁ + q₂ = q has infinite multiplicities.

In this paper, we study the restrictions of the minimal representation in the analytic continuation of the scalar holomorphic discrete series from Sp(n, ℝ) to GL⁺(n, ℝ), and from SU(n, n) to GL(n, ℂ) respectively. We work with the realizations of the representation spaces as L²-spaces on the boundary orbits of rank one of the corresponding cones, and give explicit integral operators that play the role of the intertwining operators for the decomposition. We prove inversion formulas for dense subspaces and use them to prove the Plancherel theorem for the respective decomposition. The Plancherel measure turns out to be absolutely continuous with respect to the Lebesgue measure in both cases.

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.

For this, we study the Corwin–Greenleaf multiplicity function for Hermitian symmetric spaces G/K. First, we prove that for any G-coadjoint orbit and any K-coadjoint orbit if . Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.

Second, we find a necessary and sufficient condition for by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions.

(1) The restriction π|_H is discretely decomposable in the sense of Kobayashi.

(2) The momentum map is proper.

In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π|_H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π|_H is constant along "continuous parameters". In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π|_H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.

The unitary principal series representations of G = GL(n, ℂ) induced from a character of the maximal parabolic subgroup P = (GL(1, ℂ) × GL(n - 1, ℂ)) ⋉ ℂ^n-1 attain the minimal Gelfand–Kirillov dimension among all infinite-dimensional unitary representations of G. We find the explicit branching laws for the restriction of these representations to all reductive subgroups H of G such that (G, H) forms a symmetric pair.

Zuckerman’s derived functor module of a semisimple Lie group $G$ yields a unitary representation $\pi$ which may be regarded as a geometric quantization of an elliptic orbit ${𝒪}$ in the Kirillov–Kostant–Duflo orbit philosophy. We highlight a certain family of those irreducible unitary representations $\pi$ of the indefinite unitary group $G=U(p,q)$ and a family of subgroups $H$ of $G$ such that the restriction $\pi {|}_H$ is known to be discretely decomposable and multiplicity-free by the general theory of Kobayashi (Discrete decomposibility of the restrictions of $A_q(\lambda )$ with respect to reductive subgroups, II, Ann. of Math.147 (1998) 1–21; Multiplicity-free representations and visible action on complex manifolds, Publ. Res. Inst. Math. Sci.41 (2005) 497–549), where $\pi$ is not necessarily tempered and $H$ is not necessarily compact. We prove that the corresponding moment map $\mu :{𝒪}\to {𝔥}^{\ast }$ is proper, determine the image $\mu ({𝒪})$, and compute the Corwin–Greenleaf multiplicity function explicitly.

Let $G$ be a noncompact connected simple Lie group, and $(G,G^{\mathrm{\Gamma }})$ a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple $(𝔤,K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^{\mathrm{\Gamma }})$, there does not exist a unitarizable simple $(𝔤,K)$-module that is discretely decomposable as a $({𝔤}^{\mathrm{\Gamma }},K^{\mathrm{\Gamma }})$-module. As an application, for $G={\mathrm{\text{E}}}_{6(-14)}$, we obtain a complete classification of Klein four symmetric pairs $(G,G^{\mathrm{\Gamma }})$, with $G^{\mathrm{\Gamma }}$ noncompact, such that there exists at least one nontrivial unitarizable simple $(𝔤,K)$-module that is discretely decomposable as a $({𝔤}^{\mathrm{\Gamma }},K^{\mathrm{\Gamma }})$-module and is also discretely decomposable as a $({𝔤}^{\sigma },K^{\sigma })$-module for some nonidentity element $\sigma \in \mathrm{\Gamma }$.

We study the representation theory of C*-algebras by using semigroup theory and automata theory. The Cuntz algebra is a finitely generated, infinite-dimensional, noncommutative C*-algebra. A certain class of cyclic representations of is characterized by words from the alphabet 1,…,N, which is called a cycle. A class of endomorphisms of is defined by polynomial functions in canonical generators and their conjugates. Such an endomorphism ρ of transforms a cycle π to π ◦ ρ which is a direct sum of cycles π₁,…,π_n unique up to unitary equivalence. The passage from π to π₁,…,π_n is called a branching law for ρ. In this article, we construct a Mealy machine from the endomorphism in order to compute its branching law. We show that the branching law is obtained as outputs from the machine for the input information of a given representation. Furthermore the actual computation of the branching law is executed by using a generalized de Bruijn graph associated with the Mealy machine.

We show that a nontrivial example of universal algebra appears in quantum field theory. For a unital C*-algebra , a sector is a unitary equivalence class of unital *-endomorphisms of . We show that the set of all sectors of is a universal algebra with an N-ary sum which is not reduced to any binary sum when includes the Cuntz algebra as a C*-subalgebra with common unit for N ≥ 3. Next we explain that the set of all unitary equivalence classes of unital *-representations of is a right module of . An essential algebraic formulation of branching laws of representations is given by using submodules of . As an application, we show that the action of on distinguishes elements of .