arXiv subjects

Toshiyuki Kobayashi

Publications and source records attributed to Toshiyuki Kobayashi.

83 records · Page 5Linked to original sources

arXiv2006

Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs

The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh--Gordan--Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua--Kostant--Schmid $K$-type formula, and the canonical representations in the sense of Vershik--Gelfand--Graev. Our method works in a uniform manner for both finite and infinite dimensional cases, for both discrete and continuous spectra, and for both classical and exceptional cases.

math.RT↗

arXiv2006

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $σ$ such that $σ|_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the action of any symmetric subgroup $H$ is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold $S$. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite and infinite dimensional cases, and for both discrete and continuous spectra.

math.DG↗

arXiv2006

A generalized Cartan decomposition for the double coset space U(n_1) x U(n_2) x U(n_3)) U(n) / U(p) x U(q)

Motivated by recent developments on visible actions on complex manifolds, we raise a question whether or not the multiplication of three subgroups $L$, $G'$ and $H$ surjects a Lie group $G$ in the setting that $G/H$ carries a complex structure and contains $G'/G' \cap H$ as a totally real submanifold. Particularly important cases are when $G/L$ and $G/H$ are generalized flag varieties, and we classify pairs of Levi subgroups $(L, H)$ such that $L G' H = G$, or equivalently, the real generalized flag variety $G'/H \cap G'$ meets every $L$-orbit on the complex generalized flag variety $G/H$ in the setting that $(G, G') = (U(n), O(n))$. For such pairs $(L, H)$, we introduce a \textit{herringbone stitch} method to find a generalized Cartan decomposition for the double coset space $L \backslash G/H$, for which there has been no general theory in the non-symmetric case. Our geometric results provides a unified proof of various multiplicity-free theorems in representation theory of general linear groups.

math.RT↗

arXiv2006

On discontinuous group actions on non-Riemannian homogeneous spaces

This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of pseudo-Riemannian manifolds with constant curvatures, and discuss what kind of problems we propose to pursue. For pseudo-Riemannian manifolds, isometric actions of discrete groups are not always properly discontinuous. The fundamental problem is to understand when discrete subgroups of Lie groups $G$ act properly discontinuously on homogeneous spaces $G/H$ for non-compact $H$. For this, we introduce the concepts from a group-theoretic perspective, including the `discontinuous dual' of $G/H$ that recovers $H$ in a sense. We then summarize recent results giving criteria for the existence of properly discontinuous subgroups, and the known results and conjectures on the existence of cocompact ones. The final section discusses the deformation theory and in particular rigidity results for cocompact properly discontinuous groups for pseudo-Riemannian symmetric spaces.

math.DG↗

arXiv2006

Deformation of Properly Discontinuous Actions of Z^k on R^{k+1}

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a `small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concepts from a group theoretic perspective, and focus on `stability' and `local rigidity' of discontinuous groups. As a test case, we give an explicit description of the deformation space of Z^k acting properly discontinuously on R^{k+1} by affine nilpotent transformations. Our method uses an idea of `continuous analogue' and relies on the criterion of proper actions on nilmanifolds.

math.DG↗

arXiv2005

Experimental demonstration of the time reversal Aharonov-Casher effect

We demonstrate the time reversal Aharonov-Casher (AC) effect in small arrays of mesoscopic semiconductor rings. By using an electrostatic gate we can control the spin precession rate and follow the AC phase over several interference periods. We show that we control the precession rate in two different gate voltage ranges; in the lower range the gate voltage dependence is strong and linear and in the higher range the dependence in almost an order of magnitude weaker. We also see the second harmonic of the AC interference, oscillating with half the period. We finally map the AC phase to the spin-orbit interaction parameter alpha and find it is consistent with Shubnikov-de Haas analysis.

cond-mat.mes-hall↗

arXiv2005

Compact Clifford-Klein forms of symmetric spaces -- revisited

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space $G/H$ if $H$ is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for $G/H$, followed by the most update and complete list of symmetric spaces with/without compact quotients. In the second half, as applications of general theory, we prove: (i) there exists a 15 dimensional compact pseudo-Riemannian manifold of signature $(7,8)$ with constant curvature, (ii) there exists a compact quotient of the complex sphere of dimension 1, 3 and 7, and (iii) there exists a compact quotient of the tangential space form of signature $(p,q)$ if and only if $p$ is smaller than the Hurwitz-Radon number of $q$.

math.DG↗

arXiv2003

Branching problems of unitary representations

The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs discretely with finite multiplicity (admissible restrictions). Basic theory and new perspectives of admissible restrictions are presented from both analytic and algebraic view points. We also discuss some applications of admissible restrictions to modular varieties and $L^p$-harmonic analysis.

math.RT↗

arXiv2001

Analysis on the minimal representation of O(p,q) -- I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely, the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov-Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representation, as well as giving new proofs of unitarity and irreducibility. The results in this paper play a basic role in the subsequent papers, where we give explicit branching formulae, and prove unitarization in the various models.

math.RT↗

arXiv2001

Analysis on the minimal representation of O(p,q) -- II. Branching laws

This is a second paper in a series devoted to the minimal unitary representation of O(p,q). By explicit methods from conformal geometry of pseudo-Riemannian manifolds, we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups. In the case of purely discrete spectrum we obtain the full spectrum and give an explicit Parseval-Plancherel formula, and in the general case we construct an infinite discrete spectrum.

math.RT↗

arXiv2001

Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1}

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q = 2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Moebius group of conformal transformations on R^{p-1, q-1}, and preserves a space of solutions of the ultrahyperbolic Laplace equation on R^{p-1, q-1}. We construct in an intrinsic and natural way a Hilbert space of ultrahyperbolic solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L^2(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).

math.RT↗