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1.
Let G be a linear semisimple Lie group of split rank one with K a maximal compact subgroup. In this paper, we consider the space Cc(G:F) of all functions in Cc(G) whose left and right K-translates span a finite-dimensional space. Using the analytic continuation of the principal series to define the Fourier transform, we give a characterization of the Fourier transform of the space Cc(G:F). This gives an analog of the classical Paley-Wiener theorem which gives a characterization of the Fourier transform of the space Cc(Rn).  相似文献   

2.
The infinitesimal generators of Lévy processes in Euclidean space are pseudodifferential operators with symbols given by the Lévy-Khintchine formula. This classical analysis relies heavily on Fourier analysis which, in the case when the state space is a Lie group, becomes much more subtle. Still the notion of pseudo-differential operators can be extended to connected, simply connected nilpotent Lie groups by employing the Weyl functional calculus. With respect to this definition, the generators of Lévy processes in the simplest step 3 nilpotent Lie group G are pseudodifferential operators which admit C c (G) as its core.  相似文献   

3.
Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by D(G) the space of C functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra UGc of the complexification of the Lie algebra of G for which the linear form ? ψ(π(u 1 ?)) on D(G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra UGc.  相似文献   

4.
Any étale Lie groupoid G is completely determined by its associated convolution algebra Cc(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated Cc(G)-Cc(H)-bimodule Cc(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor Ccgives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.  相似文献   

5.
A super Lie group is a group whose operations are G mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators.In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G, then h is the super Lie algebra of a sub-super Lie group of G. Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G. We also show that if H is a closed sub-super Lie group of a super Lie group G, then GG/H is a principal fiber bundle.We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G category.  相似文献   

6.
In this paper, a better asymptotic order of Fourier transform on SL(2,R) is obtained by using classical analysis and Lie analysis comparing with that of [5]、[6], and the Plancherel theorem on Cc^2(SL(2,R) ) is also obtained as an application.  相似文献   

7.
We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.  相似文献   

8.
We show that the subalgebra of convolution operators with Calderón-Zygmund kernels on a homogeneous group G is inverse-closed in the algebra of all bounded linear operators on the Hilbert space L 2(G). The main tool used is a symbolic calculus, where the convolution of distributions on the group is translated via the abelian Fourier transform into a “twisted product” of symbols on the dual to the Lie algebra g of G.  相似文献   

9.
We show that the kernel of an irreducible unitary representation π of the group algebra L1(G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups.  相似文献   

10.
Let M be a C manifold and G a Lie a group. Let E G be a C principal G-bundle over M. There is a fiber bundle C(E G ) over M whose smooth sections correspond to the connections on E G . The pull back of E G to C(E G ) has a tautological connection. We investigate the curvature of this tautological connection.  相似文献   

11.
We say that a locally compact groupG hasT 1 primitive ideal space if the groupC *-algebra,C *(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC *(G), denoted by PrimG. This means of course that every primitive ideal inC *(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT 1 primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT 1 primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC *(G) are always maximal which implies of course that PrimG isT 1.  相似文献   

12.
Let G be a Lie group which is the union of an ascending sequence G1G2⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.  相似文献   

13.
14.
In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations π:G→GL(V) of an infinite dimensional Lie group G on a locally convex space V. The first class of results concerns the space V of smooth vectors. If G is a Banach-Lie group, we define a topology on the space V of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V is a Fréchet space. This applies in particular to C-dynamical systems (A,G,α), where G is a Banach-Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function 〈π(g)v,v〉 is smooth. The second class of results concerns criteria for Ck-vectors in terms of operators of the derived representation for a Banach-Lie group G acting on a Banach space V. In particular, we provide for each kN examples of continuous unitary representations for which the space of Ck+1-vectors is trivial and the space of Ck-vectors is dense.  相似文献   

15.
An affine symmetric space G/H is said to be exponential if every two points of this space can be joined by a geodesic and weakly exponential if the union of all geodesics issuing from one point is everywhere dense in G/H. For the group space (G × G)/G diag of a Lie group G, these properties are equivalent to the exponentiality and weak exponentiality of G, respectively. We generalize known theorems on the image of the exponential mapping in Lie groups to the case of affine symmetric spaces. We prove the weak exponentiality of the symmetric spaces of solvable Lie groups, and in the semisimple case we obtain criteria for exponentiality and weak exponentiality.  相似文献   

16.
We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers VλN of an irreducible representation Vλ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope.  相似文献   

17.
The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable xRn−1. For the variable x1R+ in the normal direction, we use L2 space or weighted L2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t→∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].  相似文献   

18.
Let β > 1 be a Pisot number andg be a positive Hölder continuous function with period one and g(0) = 1. The multiperiodic functionG(ξ)=Π n=0 g(ξ/βn) is studied and the asymptotic behaviour ofI G(T) = ∫ 0 T G(ξ)dξ investigated. We prove that the limit of logI(T)/ logT exists asT tends to infinity. We also provide a method to calculate this limit for the caseg(ξ) = cos2 2πξ, corresponding to the Fourier transform of the Bernoulli convolution associated to the golden number (or some of its generalizations).  相似文献   

19.
20.
A host algebra of a topological group G is a C *-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C *-algebra C *(S,α), and we describe a setting which permits us to conclude that this C *-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future. To K.H. Hofmann on the occasion of his 75th birthday  相似文献   

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