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1.
LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call admissible but we call them of Harish-Chandra type. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, ) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations UV, with imaginary and withV from the quasi-discrete series of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.We show that irreducible continuous unitary representations of really reductive groups are of Harish-Chandra type. Then the results above yield the canonical decomposition of the unitary spectrum>G for any really reductiveG. In particular, this holds ifG/G 0 is finite, so the center of the connected semi-simple subgroup with Lie algebra [g, g] may be infinite!Research supported, in part, by the Hungarian National Fund for Scientific Research (grant Nos. 1900 and 2648).  相似文献   

2.
We relate the distribution characters and the wave front sets of unitary representation for real reductive dual pairs of type I in the stable range.  相似文献   

3.
    
We extend, in the context of connected noncompact semisimple Lie group, two results of Antezana, Massey, and Stojanoff: Given 0<λ<1, (a) the limit points of the sequence are normal, and (b) , where ‖X‖ is the spectral norm and r(X) is the spectral radius of XCn×n and Δλ(X) is the λ-Aluthge transform of X.  相似文献   

4.
We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman. The latter provides an extension of the works of Rocha-Caridi and Wallach [A. Rocha-Caridi, N.R. Wallach, Projective modules over infinite dimensional graded Lie algebras, Math. Z. 180 (1982) 151-177] and Deodhar, Gabber and Kac [V. Deodhar, O. Gabber, V. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. Math. 45 (1982) 92-116] on block decompositions of BGG categories for Kac-Moody algebras. We also derive a compatibility relation between the affine Jacquet functor and the Kazhdan-Lusztig tensor product and apply it to prove that the affine Harish-Chandra category is stable under fusion tensoring with the Kazhdan-Lusztig category. This compatibility will be further applied in studying translation functors for the affine Harish-Chandra category, based on the fusion tensor product.  相似文献   

5.
Let G and H be Lie groups with Lie algebras and . Let G be connected. We prove that a Lie algebra homomorphism is exact if and only if it is completely positive. The main resource is a corresponding theorem about representations on Hilbert spaces. This article summarizes the main results of [1]. Received: 6 December 2005  相似文献   

6.
Let II be the upper half-plane in, consider the Bergman space , the subspace of all analytic functions fromL 2(II). The complete decomposition ofL 2(II) onto Bergman and Bergman type spaces of poly-analytic and poly-anti-analytic functions is obtained. The orthogonal Bergman type projections onto each of these subspaces are described. Connections with the Hardy spaces and the Szegö projections are established.This work was partially supported by CONACYT Project 3114P-E9607, México.  相似文献   

7.
Using a new definition of rank for representations of semisimple groups sharp results are proved for the decay of matrix coefficients of unitary representations of two types of non-split p-adic simple algebraic groups of exceptional type. These sharp bounds are achieved by minimal representations. It is also shown that in one of the cases considered, the minimal representation is isolated in the unitary dual.  相似文献   

8.
9.
We study reproducing kernels for harmonic Bergman spaces of the unit ball inR n . We establish some new properties for the reproducing kernels and give some applications of these properties.  相似文献   

10.
Let G be a simply connected Chevalley group of type D n , E n or G2. In this paper, we show that the minimal representation of G is unique for types D n and E n and it does not exist for the type G2.  相似文献   

11.
Let E be a finite dimensional symplectic space over a local field of characteristic zero. We show that for every element in the metaplectic double cover of the symplectic group Sp(E), and are conjugate by an element of GSp(E) with similitude −1.  相似文献   

12.
13.
We use Kirillov's orbit method to construct all unitarizable highest weight representations with scalar lowest K-type.  相似文献   

14.
For a general Carnot group G with homogeneous dimension Q we prove the existence of a fundamental solution of the Q-Laplacian uQ and a constant aQ>0 such that exp(−aQuQ) is a homogeneous norm on G. This implies a representation formula for smooth functions on G which is used to prove the sharp Carnot group version of the celebrated Moser-Trudinger inequality.  相似文献   

15.
《Advances in Mathematics》2007,208(1):299-317
Geometric realizations for the restrictions of GNS representations to unitary groups of C-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.  相似文献   

16.
We give a new proof of the Bernstein-Lunts equivalence of ordinary and equivariant derived categories of Harish-Chandra modules. This proof requires no boundedness assumptions. It uses K-projective resolutions of equivariant complexes, which are shown to exist and constructed explicitly when the group under consideration is reductive. The general case can be obtained from the reductive case by techniques of [MP].  相似文献   

17.
Let F be a non-Archimedean local field and D a central F-division algebra of dimension n2, n?1. We consider first the irreducible smooth representations of D× trivial on 1-units, and second the indecomposable, n-dimensional, semisimple, Weil-Deligne representations of F which are trivial on wild inertia. The sets of equivalence classes of these two sorts of representations are in canonical (functorial) bijection via the composition of the Jacquet-Langlands correspondence and the Langlands correspondence. They are also in canonical bijection via explicit parametrizations in terms of tame admissible pairs. This paper gives the relation between these two bijections. It is based on analysis of the discrete series of the general linear group GLn(F) in terms of a classification by extended simple types.  相似文献   

18.
《Quaestiones Mathematicae》2013,36(2):185-214
Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.  相似文献   

19.
20.
It is known that for particular classes of operators on certain reproducing kernel Hilbert spaces, key properties of the operators (such as boundedness or compactness) may be determined by the behaviour of the operators on the reproducing kernels. We prove such results for Toeplitz operators on the Paley-Wiener space, a reproducing kernel Hilbert space over . Namely, we show that the norm of such an operator is equivalent to the supremum of the norms of the images of the normalised reproducing kernels of the space. In particular, therefore, the operator is bounded exactly when this supremum is finite. In addition, a counterexample is provided which shows that the operator norm is not equivalent to the supremum of the norms of the images of the real normalised reproducing kernels. We also give a necessary and sufficient condition for compactness of the operators, in terms of their limiting behaviour on the reproducing kernels.  相似文献   

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