Local and multilinear noncommutative de Leeuw theorems

LetGbe a locally compact unimodular group, and let$\phi $be some function ofnvariables onG. To such a$\phi $, one can associate a multilinear Fourier multiplier, which acts on somen-fold product of the noncommutative$L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on ann-fold product of Schatten classes$S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called “multiplicatively bounded$(p_1,\ldots ,p_n)$-norm” of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map$L_{p_1}(\mathbb {R}, S_{p_1}) \times L_{p_2}(\mathbb {R}, S_{p_2}) \rightarrow L_{1}(\mathbb {R}, S_{1})$, whenever$p_1$and$p_2$are such that$\frac {1}{p_1} + \frac {1}{p_2} = 1$. A similar result holds for certain Calderón–Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.

This is a semi–expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so many years, much of that Memoir is up to date, but of course there have been a number of refinements, advances and new developments, most of which have applied to smaller classes of real reductive Lie groups. Here we rewrite that AMS Memoir in in view of these advances and indicate the ties with some of the more recent (or at least less classical) approaches to geometric realization of unitary representations.

Discrete Series Representations for the Orbit Spaces Arising from Two Involutions of Real Reductive Lie Groups

Stochastic calculus of variation on a Lie group: Reduced variation and adjoint representation — Path groups: left infinitesimal quasi-invariance of Wiener measure — Path group on a compact Lie group — Orthonormal frame bundles over a Riemannian manifold: Levi-Civita parallelism, structure equations — Lifting to the frame bundle of the Riemannian diffusion: the stochastic parallel displacement — The horizontal stochastic flow, determination of its Jacobian on a Riemannian diffusion — Invariance of Brownian motion by orthogonal transformations — Tangent processes — Stochastic analysis on the path space of a Riemannian manifold: Twisted differential, Clark-Bismut-Ocone representation — Harnack estimates via the reduced variation — Loop spaces.KeywordsVector FieldRiemannian ManifoldTangent SpaceLoop SpaceHorizontal FlowThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A new concept of “generalized enveloping algebra” is introduced by means of the generalized Heisenberg commutation relations of non-Abelian quantum kinematics. This concept is examined within the quantum-kinematic formalism of some noncompact Lie groups of a special kind. The well known Gel'fand theorem (which relates the center of the traditional enveloping algebra with the adjoint representation) is then extended to the generalized enveloping algebra of the group. In this way, the isomorphism of the “generalized left-center” and the “traditional right-center” of the corresponding enveloping algebras is proved within the left regular representation of noncompact Lie groups of the chosen kind. As an interesting application of generalized enveloping algebras, this paper contains a brief discussion of quantum-kinematic (boson) ladder operators for non-Abelian noncompact finite Lie groups and of their corresponding coherent states.

Let H⊂G be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module , and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| H of discrete series π for a symmetric space G/H is also given.

Approximating Fourier transformation of orbital integrals

In recent years there has developed a satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters. These polynomials include as special cases: symmetric functions; zonal spherical functions on real and p-adic reductive Lie groups; the Jacobi polynomials of Heckman and Opdam; and the Askey–Wilson polynomials, which themselves include as special or limiting cases all the classical families of orthogonal polynomials in one variable. This book, first published in 2003, is a comprehensive and organised account of the subject aims to provide a unified foundation for this theory, to which the author has been a principal contributor. It is an essentially self-contained treatment, accessible to graduate students familiar with root systems and Weyl groups. The first four chapters are preparatory to Chapter V, which is the heart of the book and contains all the main results in full generality.

We establish regularity conditions for ‐boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander–Mikhlin (HM) criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard, and de la Salle for . Our approach is partly based on a sharp local HM theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular non‐Toeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle‐based approach to Fourier multipliers in group algebras by Junge, Mei, and Parcet. A few related open problems are also discussed.

Graded characters of induced representations for real reductive Lie groups I

It is shown that every compact group G is a -simplicial space where Q is any field of characteristic zero.As a consequence it follows that G satisfies a variation of the Lefschetz fixed point theorem.It has been known for some time that the Lefschetz fixed point theorem applies to a few spaces other than just ANR spaces, especially if some care is taken to use coefficients in certain fields [2].The case of all compact groups provides a broad class of spaces which may not have local connectivity of any order.It is shown that every compact group G satisfies the Lefschetz fixed point theorem when coefficients for the homology groups are taken in a field of characteristic zero.1. Theorem.Let G be a compact group.Then G is an inverse limit of Lie groups G = proj lim GM and if f : G" -> G is a homomorphism belonging to this inverse system, then there is a commutative diagram of covering compact Lie groups Presented to the Society, January 18,

Representation theory in complex rank, II

A theorem of Drinfel'd (Drinfel'd (1993)) classifies the Poisson homogeneous spaces of a Poisson Lie group (G,πG) via a special class of Lagrangian subalgebras of the Drinfel'd double of its Lie bialgebra. This result is extended in Liu et al. (1998) to a classification of the Poisson homogeneous spaces of a Poisson groupoid (G⇉P, πG) via a special class of Dirac structures in the Courant algebroid defined by the corresponding Lie bialgebroid. It is hence natural to ask what corresponds via these classifications, or extensions of them, to arbitrary Dirac structures in the Drinfel'd double or the Courant algebroid associated to a Poisson group(oid). We show in this thesis that there is a bigger class of Dirac structures that fits in this correspondence. They correspond naturally to Dirac homogeneous spaces of the Poisson group(oid). Dirac Lie groups and Dirac groupoids have been defined by Ortiz (2009) as a generalization of Poisson Lie groups, Poisson groupoids and presymplectic groupoids. We prove in an alternative manner the existence of a natural Lie bialgebra associated to a Dirac Lie group, and we find good candidates for the infinitesimal data of a Dirac groupoid; a square of morphisms of Lie algebroids associated to the multiplicative Dirac structure. These objects generalize simultaneously the Lie bialgebroid of Poisson groupoids and the IM-2-forms associated to presymplectic groupoids (Bursztyn et al. (2004)). We show also the existence of a Courant algebroid associated to these algebroids. This Courant algebroid structure is induced in a natural way by the ambient standard Courant algebroid TG ×G T*G and is isomorphic to the Courant algebroid AG ×P A*G in the Poisson case. We show that Dirac homogeneous spaces of Dirac Lie groups (respectively Dirac groupoids) correspond to certain classes of Dirac structures in the Drinfel'd double of the Lie bialgebra (respectively in the Courant algebroid). These results generalize the classification theorems in Drinfel'd (1993) and Liu et al. (1998). Along the way, we study involutive multiplicative foliations on Lie groupoids and show when and how there is a natural groupoid structure defined on the leaf spaces in the set of objects and the set of units of the Lie groupoid. This is a fact that is always true in the Lie group case, where a multiplicative foliation is automatically the left and right invariant image of an ideal in the Lie algebra, i.e., the leaves are the cosets of a normal subgroup of the Lie group.

For the characterization of multipliers of Lp(Rd) or more generally, of Lp(G) for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra Ap(G) plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of Ap(G). In fact, we demonstrate that it coincides with the space of multipliers of Lp(G),∥·∥p. Given a multiplier T of (Ap(G),∥·∥Ap(G)) and using the embedding (Ap(G),∥·∥Ap(G))↪C0(G),∥·∥∞, the linear functional f↦[T(f)(0)] is bounded, and T can be written as a moving average for some element in the dual PMp(G) of (Ap(G),∥·∥Ap(G)). A key step for this identification is another elementary fact: showing that the multipliers from Lp(G),∥·∥p to C0(G),∥·∥∞ are exactly the convolution operators with kernels in Lq(G),∥·∥q for 1<p<∞ and 1/p+1/q=1. The proofs make use of the space of mild distributions, which is the dual of the Segal algebra S0(G),∥·∥S0, and the fact that multipliers T from S0(G) to S0′(G) are convolution operators of the form T:f↦σ∗f for some uniquely determined σ∈S0′. This setting also allows us to switch from the description of these multipliers as convolution operators (by suitable pseudomeasures) to their description as Fourier multipliers, using the extended Fourier transform in the setting of S0′(G),∥·∥S0′. The approach presented here extends to other function spaces, but a more detailed discussion is left to future publications.

We show that the analyticity of semigroups $$(T_t)_{t \geqslant 0}$$ of selfadjoint contractive Fourier multipliers on $$L^p$$ -spaces of compact abelian groups is preserved by the tensorisation of the identity operator of a Banach space for a large class of K-convex Banach spaces, answering partially a conjecture of Pisier. We also give versions of this result for some semigroups of Schur multipliers and Fourier multipliers on noncommutative $$L^p$$ -spaces. Finally, we give a precise description of semigroups of Schur multipliers to which the result of this paper can be applied.