Cartan Motion Group Research Articles

Mackey analogy as deformation of $${\mathcal {D}}$$-modules

Given a real reductive linear Lie group \(G_{\mathbb {R}}\), the Mackey analogy is a bijection between the set of irreducible tempered representations of \(G_{\mathbb {R}}\) and the set of irreducible unitary representations of its Cartan motion group, established by Higson and Afgoustidis. We show that this bijection arises naturally from families of twisted \({\mathcal {D}}\)-modules over the ag variety of \(G_{\mathbb {R}}\).

Continuity of the Mackey–Higson bijection

When $G$ is a real reductive group and $G_0$ is its Cartan motion group, the Mackey-Higson bijection is a natural one-to-one correspondence between all irreducible tempered representations of $G$ and all irreducible unitary representations of $G_0$. In this short note, we collect some known facts about the topology of the tempered dual $\widetilde{G}$ and that of the unitary dual $\widehat{G_0}$, then verify that the Mackey-Higson bijection $\widetilde{G} \to \widehat{G_0}$ is continuous.

The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebras

The purpose of this paper is to make a further contribution to the Mackey bijection for a complex reductive groupGG, between the tempered dual ofGGand the unitary dual of the associated Cartan motion group. We shall construct an embedding of theC∗C^*-algebra of the motion group into the reducedC∗C^*-algebra ofGG, and use it to characterize the continuous field of reduced groupC∗C^*-algebras that is associated to the Mackey bijection. We shall also obtain a new characterization of the Mackey bijection using the same embedding.

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.

Complex quantum groups and a deformation of the Baum–Connes assembly map

We define and study an analogue of the Baum–Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of qdeformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum–Connes assembly map for a complex semisimple Lie group G, which allows one to express the K-theory of the reduced group C∗-algebra of G in terms of the K-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup K acting on k∗ via the coadjoint action. In the quantum case the role of the Cartan motion group is played by the Drinfeld double of the classical group K, whose associated group C∗-algebra is the crossed product of C(K) with respect to the adjoint action of K. Our quantum assembly map is obtained by varying the deformation parameter in the Drinfeld double construction applied to the standard deformation Kq of K. We prove that the quantum assembly map is an isomorphism, thus providing a description of the K-theory of complex quantum groups in terms of classical topology. Moreover, we show that there is a continuous field of C∗-algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them. It follows in particular that the quantum assembly map contains the classical Baum–Connes assembly map as a direct summand.

Mackey analogy via 𝒟-modules for SL(2, ℝ)

A conjecture by Mackey and Higson claims that there is close relationship between irreducible representations of a real reductive group and those of its Cartan motion group. The case of irreducible tempered unitary representations has been verified recently by Afgoustidis. We study the admissible representations of [Formula: see text] by considering families of [Formula: see text]-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makecy–Higson bijection in the general case.

A contraction of the principal series by Berezin-Weyl quantization

We study a contraction of the principal series representations of a noncompact semisimple Lie group to the unitary irreducible representations of its Cartan motion group by means of the Berezin-Weyl quantization on the coadjoint orbits associated with these representations.

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.

Paley–Wiener theorems for the Dunkl transform

We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Paley–Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrary smooth functions. Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coincide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl’s intertwining operator to the invariants can be interpreted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.

An analogue of the Hardy theorem for the Cartan motion group

An analogue of the Hardy theorem for the Cartan motion group

The Radon transform on tangent space of bounded symmetric domainSO * (8)/U(4)

We study the Radon transform on the Cartan motion group associated with the bounded symmetric domainSO* (8)/U(4), and prove that certain orbit of the regular elements is not the spectral synthesis set for the Fourier algebra on this Cartan motion group.

Scalar irreducibility of eigenspaces on the tangent space of a reductive symmetric space

Let X 0 be the tangent space at eH of the reductive symmetric space G H , and let G 0 denote the group of affine transformations of X 0 generated by the translations and the natural action of H. We show that any joint eigenspace of the G 0-invariant differential operators on X 0 is scalarly irreducible under the action of G 0. This holds in particular for a Riemannian symmetric space of the non-compact type, where G 0 is the Cartan motion group.

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

We give asymptotic expansions for zonal functions of the Cartan motion group; they generalize the classical expansion for Bessel functions.

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.

Scalar irreducibility of certain eigenspace representations

A sufficient condition for scalar irreducibility of a representation of a group on a topological vector space of the form {ƒ ∈ C ∞(R ∞)∣p ∞(∇)ƒ = 0 for j=1,…,r} where p 1,…, p r are polynomials is given. Applications include the differential operators that are invariant under the Cartan motion group of a symmetric space and the Laplace operator.

On contractions of semisimple Lie groups

A limiting formula is given for the representation theory of the Cartan motion group associated to a Riemannian symmetric pair ( G , K ) (G,K) in terms of the representation theory of G G .

A duality for symmetric spaces with applications to group representations. III. Tangent space analysis

1. IN~ODUCTI~N Let X = G/K be a symmetric space of the noncompact type, G being a connected semisimple Lie group with finite center and K a maximal compact subgroup. Let X0 be the tangent space to X at the fixed pont o of K and let G, denote the group of af%ne transformations of X,, generated by the translations and the natural action of K on X,, . The group GO which is the semi-direct product K x S X,, is often called the Car-tan motion group. While the irreducible unitary representations of GO are described by Mackey’s general semi-direct product theorem [13, p. 1311, and their characters given by Gindikm [4], our purpose here is to develop GO-invariant analysis on the space X0 . More specifically, let D(G,,/K) denote the algebra of G,,-invariant differential operators on Go/K = X,,. We are interested in the joint eigenfunctions of D(G,,/K) and in the representations of GO on the joint eigenspaces, ([8(d)] Ch. IV). As tools for this study we prove Paley-Wiener type theorems for the spherical transform and for the generalized Bessel transform on X0. The first of these comes from Theorem 2.1 which is an analog for entire functions of exponential type of Chevalley’s restriction theorem for polynomials. An important property of the generalized Bessel transform comes from Theorem 4.6 which shows that the generalized Bessel function is divisible by the Jg-matrix. Another useful tool is Theorem 4.8 which represents the generalized Bessel function as a certain derivative of the zonal spherical function. These results are used in Section 6 to derive an integral formula for the K-finite joint eigenfunctions of D(G,/K). This in turn is used to give (in Theorem 6.6) an explicit irreducibility criterion for the associated eigenspace representation of G,, . This is the principal application of our results. The results for the flat space X, = GO/K are to a large extent analogous to those proved in [8(g)] for the curved space X = G/K, most of the proofs are

Fourier transforms on the Cartan motion group

Fourier transforms on the Cartan motion group