Explicit Hilbert spaces for certain unipotent representations III

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L²-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L²-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of these representations and describe their decomposition.     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces

We provide an alternate approach to an intertwining lifting theorem obtained by Ball, Trent and Vinnikov. The results are an exact analogue of the classical Sz-Nagy-Foias theorem in the case of multipliers on a class of reproducing kernel spaces, which satisfy the Nevanlinna-Pick property.     

A general multiplicity theorem for certain nonlinear equations in Hilbert spaces

In this paper, we prove the following general result. Let be a real Hilbert space and a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that

Then, for each for which the set is not convex and for each convex set dense in , there exist and 0$"> such that the equation

has at least three solutions.

     

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the $L^2$ norm, and study their approximation properties over Hilbert subspaces of $L^2$. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysis of $M$-estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.     

Orthoscalar representations of quivers in the category of Hilbert spaces

As is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers correspond to extended Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article “Locally scalar representations of graphs in the category of Hilberts spaces” (Func. Anal. Apps., 2005), the authors showed a way for carrying over these results to Hilbert spaces, constructed Coxeter functors, and proved an analog of the Gabriel theorem for locally scalar representations (up to unitary equivalence). The category of locally scalar representations of a quiver can be regarded as a subcategory in the category of all representations (over the field ℂ). In the present paper, we study the relationship between the indecomposability of locally scalar representations in the subcategory and in the category of all representations (it is proved that for a class of quivers wide enough indecomposability in the subcategory implies indecomposability in the category). For a quiver corresponding to the extended Dynkin graph , locally scalar representations that cannot be obtained from the simplest ones by Coxeter functors (regular representations) are classified. Bibliography: 21 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 180–201.     

Irreducibility of unitary group representations and reproducing kernels Hilbert spaces

We discuss unitary representations of groups in Hilbert spaces of functions given together with reproducing kernels, and in particular irreducibility. Our main focus is on examples, including spherical representations of orthogonal groups, distance-transitive finite graphs, irreducibility of induced representations, the discrete series of SL(2, ), and some representations of PGL(2, _p).The appendix contains examples involving groups acting on rooted trees.     

Explicit determination of images of Galois representations attached to Hilbert modular forms

In a previous paper the second author proved that the image of the Galois representation modulo λ attached to a Hilbert modular newform is “large” for all but finitely many primes λ, if the newform is not a theta series. In this brief note, we give an explicit bound for this exceptional finite set of primes and determine the images in three different examples. Our examples are of Hilbert newforms on real quadratic fields, of parallel or non-parallel weight and of different levels.     

Singular locally scalar representations of quivers in Hilbert spaces and separating functions

We consider locally scalar representations of extended Dynkin graphs in Hilbert spaces. The relation between these representations and the function (n) = 1 + (n – 1)/(n + 1) is established. We construct a family of separating functions that generalize the function and play a similar role in a broader class of graphs.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 796–809, June, 2004.     

Some small unipotent representations of indefinite orthogonal groups

We study a family of small unitary representations of indefinite orthogonal groups. These representations arise as analytic continuations of the discrete series and were studied extensively by Knapp in [K3]. We complete Knapp's analysis by proving that they are irreducible. In order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one. Moreover, we prove that the K-type structure of each representation matches (up to a shift) the K-type structure of the ring of functions on the closure a nilpotent orbit on .     

Rigged Hilbert spaces and contractive families of Hilbert spaces

The existence of a rigged Hilbert space whose extreme spaces are, respectively, the projective and the inductive limit of a directed contractive family of Hilbert spaces is investigated. It is proved that, when it exists, this rigged Hilbert space is the same as the canonical rigged Hilbert space associated to a family of closable operators in the central Hilbert space.     

Certain conjectures relating unipotent orbits to automorphic representations

In this paper I formulate certain conjectures relating the structure of unipotent orbits to automorphic representations. We consider a few examples and prove some of these conjectures.     

Integral representations in Hardy classes and best approximations in certain functional spaces

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H¹ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 342–347, March, 1991.     

Integral representations in hardy classes and best approximations in certain functional spaces

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H¹ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 342–347, March, 1991.