Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups

We develop a pseudo-differential Weyl calculus on nilpotent Lie groups, which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose, we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie group and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.     

Paley-Wiener-type theorem for nilpotent Lie groups

A Paley-Wiener-type theorem is proved for connected and simply connected Lie groups. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1564–1566, October, 1998.     

Matrix coefficients and coadjoint orbits of compact Lie groups

Let be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the -function and the pull-back of an arbitrary unitary irreducible representation of to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups.

     

Anosov actions of nilpotent Lie groups

We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70's. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation.     

Holomorphic representations of nilpotent Lie groups

Let G be a connected, simply-connected complex nilpotent Lie group, and G_r ?( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of G_R to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of G_R, we study the convex cone of entire functions on G whose restrictions to G_R are positive-definite, and determine the minimal order of growth at infinity of such functions.     

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if ), we define the twisted Weyl group W = W(G,A,T), which acts on T and H ¹(A,T), where T is a maximal compact torus of , the identity component of the group of invariants G ^A . We then prove that the natural map is a bijection, reducing the calculation of H ¹(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.      

Stratonovich-Weyl correspondence for compact semisimple Lie groups

Let G be a compact Lie group, M a G-homogeneous space and π a unitary representation of G realized on a Hilbert space of functions on M. We give a general presentation of the Stratonovich-Weyl correspondence associated with π. In the case when G is a compact semisimple Lie group and π _λ an irreducible representation of G with highest weight λ, we study the Stratonovich-Weyl symbol of the derived operator d π _λ (X) for X in the Lie algebra of G and its behavior as λ goes to infinity.     

Symbols and orbits for 3-step nilpotent Lie groups

Sufficient conditions are derived for L²-boundedness of a convolution operator on a 3-step nilpotent Lie group. This is achieved by producing estimates on the Kirillov symbol of the operator, and is closely linked to the co-adjoint orbit structure of the group. A structure theorem for 3-step nilpotent Lie groups with 1-dimensional center is proved.     

The behavior of Fourier transforms for nilpotent Lie groups

We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on extends to be ``holomorphic' on an appropriate complexification of (a part of) . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of when is two-step nilpotent.

     

Radon transforms on Siegel-type nilpotent Lie groups

Let \mathfrak{N}:={{\displaystyle H}}_n\times {ℂ}^n be the Siegel-type nilpotent group, which can be identified as the Shilov boundary of Siegel domain of type II, where {{\displaystyle H}}_n denotes the set of all n\times n Hermitian matrices. In this article, we use singular convolution operators to define Radon transform on \mathfrak{N} and obtain the inversion formulas of Radon transforms. Moveover, we show that Radon transform on \mathfrak{N} is a unitary operator from Sobolev space W^n;2 into L²(\mathfrak{N}):     

The existence of soliton metrics for nilpotent Lie groups

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation U v = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.     

Spectral synthesis for coadjoint orbits of nilpotent Lie groups

We determine the space of primary ideals in the group algebra \(L^{1}(G) \) of a connected nilpotent Lie group by identifying for every \(\pi \in \widehat{G} \) the family \(\mathcal I^\pi \) of primary ideals with hull \(\{\pi \} \) with the family of invariant subspaces of a certain finite dimensional sub-space \(\mathcal P_Q^\pi \) of the space of polynomials \(\mathcal P(G) \) on G.     

Convexity and unitary representations of nilpotent Lie groups

Summary The moment map of symplectic geometry is extended to associate to any unitary representation of a nilpotent Lie group aG-invariant subset of the dual of the Lie algebra. We prove that this subset is the closed conex hull of the Kirillov orbit of the representation.Supported by NSERC research grant no. A7918     

Littlewood-Paley and Lusin functions on nilpotent Lie groups

In this paper, we define the Littlewood-Paley and Lusin functions associated to the sub-Laplacian operator on nilpotent Lie groups. Then we prove the Lp (1<p<∞) boundedness of Littlewood-Paley and Lusin functions.