Cartan Motion Groups Research Articles

Limits of Bessel functions for root systems as the rank tends to infinity

We study the asymptotic behaviour of Bessel functions associated to root systems of type An−1 and type Bn with positive multiplicities as the rank n tends to infinity. In both cases, we characterize the possible limit functions and the Vershik–Kerov type sequences of spectral parameters for which such limits exist. In the type A case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of β-ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over F=R,ℂ,H (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.

On the analogy between real reductive groups and Cartan motion groups: the Mackey–Higson bijection

On the analogy between real reductive groups and Cartan motion groups: the Mackey–Higson bijection

On the analogy between real reductive groups and Cartan motion groups: Contraction of irreducible tempered representations

Attached to any reductive Lie group $G$ is a "Cartan motion group" $G_0$ $-$ a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_0$, whose existence had been suggested by Mackey in the 1970s, has recently been described by the author. In the present notes, we use the existence of a family of groups interpolating between $G$ and $G_0$ to realize the bijection as a deformation: for every irreducible tempered representation $\pi$ of G, we build, in an appropriate Fr\'echet space, a family of subspaces and evolution operators that contract $\pi$ onto the corresponding representation of $G_0$.

On the analogy between real reductive groups and Cartan motion groups: A proof of the Connes-Kasparov isomorphism

Alain Connes and Nigel Higson pointed out in the 1990s that the Connes-Kasparov “conjecture” for the K-theory of reduced group C⁎-algebras seemed, in the case of reductive Lie groups, to be a cohomological echo of a conjecture of George Mackey concerning the rigidity of representation theory along the deformation from a real reductive group to its Cartan motion group. For complex semisimple groups, Nigel Higson established in 2008 that Mackey's analogy is a real phenomenon, and does lead to a simple proof of the Connes-Kasparov isomorphism. We here turn to more general reductive groups and use our recent work on Mackey's proposal, together with Higson's work, to obtain a new proof of the Connes-Kasparov isomorphism.

Synchronization over Cartan Motion Groups via Contraction

Group contraction is an algebraic map that relates two classes of Lie groups by a limiting process. We utilize this notion for the compactification of the class of Cartan motion groups, which includes the important special case of rigid motions. The compactification process is then applied to reduce a noncompact synchronization problem to a problem where the solution can be obtained by means of a unitary, faithful representation. We describe this method of synchronization via contraction in detail and analyze several important aspects of this application. We then show numerically the advantages of our approach compared to some current state-of-the-art synchronization methods on both synthetic and real data.

Bessel convolutions on matrix cones

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras ${\mathbb F} = {\mathbb R}, {\mathbb C}$ or ${\mathbb H}$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}({\mathbb F})$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p({\mathbb F})$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping $r\mapsto \text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\times U_q)$ over ${\mathbb F}$.

CRITERION OF PROPER ACTIONS ON HOMOGENEOUS SPACES OF CARTAN MOTION GROUPS

The Cartan motion group associated to a Riemannian symmetric space X is a semidirect product group acting isometrically on its tangent space. For two subsets in a locally compact group G, Kobayashi introduced the concept of "properness" as a generalization of properly discontinuous actions of discrete subgroups on homogeneous spaces of G. In this paper, we give a criterion of properness for homogeneous spaces of Cartan motion groups. Our criterion has a similar feature to the case where G is a reductive Lie group.

Transformation de Poisson et irréductibilité de certaines représentations induites non-unitaires

Let G × N be a semi-direct product of Lie groups; we define a family of non-unitarily induced representations of it and then give two sufficient conditions for their irreducibility. Under certain hypotheses ( G is semi-algebraic and acts regularly on the vector space N), plus conditions on some orbit of G in N ∗ , we prove that the first criterion is also necessary. The proof of this result relies on a characterization of the Schwartz space of a semi-algebraic smooth variety that we give. We particularly study the case of Cartan motion groups related to pseudo-Riemannian rank one symmetric spaces.

A Nonabelian Version of the Shannon Sampling Theorem

Using techniques from the theory of contraction of Lie groups, a version of the Shannon sampling theorem appropriate to Cartan motion groups is proved. This work has possible applications in digital signal analysis.

Spherical Functions on Cartan Motion Groups

Spherical Functions on Cartan Motion Groups

Spherical functions on Cartan motion groups

This paper gives a reasonably complete treatment of harmonic analysis on Cartan motion groups. Included is an explicit parameterization of irreducible spherical functions of general K K -type, and of the nonunitary dual (and its topology). Also included is the explicit Plancherel measure, the Paley Wiener theorem, and an asymptotic expansion of general matrix entries. (These are generalized Bessel functions.) However the main result is Theorem 19, a technical result which measures the size of the centralizer of K K in the universal enveloping algebra of the corresponding reductive group.

Motion Groups and Absolutely Convergent Fourier Transforms

According to an extension of a classical theorem of Bernstein, due to C. Herz, a function on Rn belonging to a Besov space of appropriate order has an absolutely convergent Fourier transform. We establish extensions of this result to Cartan motion groups, for Besov spaces defined with respect to both isotropic and non-isotropic differences.

Le comportement à l'infini des fonctions de Bessel généralisées, II

Sharp uniform estimates are given for the zonal functions (with regular parameter) associated to Cartan motion groups; they generalize the classical O(|X| − 1 2 ) estimate for Bessel functions. The proof uses mainly the differential system satisfied by the zonal functions.

A Hardy-Littlewood theorem for spherical Fourier transforms on symmetric spaces

Let q ⩾ 2. If f is a measurable function on R n such that f(x) ¦x¦ n(1 − 2 q ) ϵ L q(R n) , then its Fourier transform f ɞ can be defined and there exists a constant A q such that the inequality ∥ f ɞ ∥ q ∥ f ¦ · ¦ n(1 − 2 q ∥ q holds. This result is called the Hardy-Littlewood theorem. This paper studies what the corresponding function to ¦x¦ n is for the spherical Fourier transform on Riemannian symmetric spaces and gives an analogue of this theorem. Also it is shown that the spherical Fourier transform of function f having the corresponding property on G K extends holomorphically to a tube domain and that an analogue of Riemann-Lebesgue lemma holds there. The following three cases are treated: the Riemannian symmetric spaces of the noncompact type, those of the compact type and the symmetric spaces on which the Cartan motion groups are acting.

On Lp multipliers of Cartan motion groups

Let ( G, K) be a Riemannian symmetric pair of the compact type and let V × K be the associated Cartan motion group. We establish a body of approximation theorems that permit the transfer of Fourier analysis from the group G to the Cartan motion group. As a consequence, we prove an analogue of a theorem of de Leeuw concerning restriction of Fourier multipliers of the group R to the group Z . The latter theorem extends an earlier result of R. L. Rubin for the case where G = SO(3), K = SO(2), and V × K is the Euclidean motion group.

On the injectivity of the Poisson transform

The injectivity of the Poisson transform associated to the Cartan motion groups is proved with the aid of a Fatou-type theorem.

Cartan motion groups and dual topology

Cartan motion groups and dual topology