Bundle-connection pairs and loop group representations

LetM be a connected differentiable manifold. Denote by Ω _m (M) the space ofH ¹-loops based at a fixed pointm∈M. Associated to Ω _m (M) one has , the group of unparameterized loops. Given a bundle-connection pair (E,∇) overM with fiber the finite-dimensional vector spaceV and structure groupG⊂GL(V) we get (up to equivalence) a smooth representation of inG given by the parallel transport operatorP ^∇. It is possible to find in the literature several versions of the converse theorem, namely: all (smooth) representations of arise in the above described way from a bundle-connection pair. It is shown in the present paper that the correct setting for this theorem is the theory of induced representations for groupoids. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 503–518, April, 1997.     

The space of surface group representations

In this note we prove that the number of irreducible components of Hom (π,G) is the same as π₁(G), where π is a surface group andG is complex semisimple. This is established by studying the flat bundles on Riemann surfaces. The present work is partially supported by NSF grant DMS89-04922     

Isotopic pairs and their representations

The paper presents representations of isotopic pairs, the algebraic objects which can, probably, be convenient for describing some forms of the non-Hamiltonian interaction of Hamiltonian systems on the quantum level. General constructions are illustrated by simple examples.This work was given partial support from C.N.R.S. (France), M.P.I.M. (Germany), and E.S.I. (Austria). The preliminary electronic version of the paper in English is in the Texas Electronic Archive on Mathematical Physics under number mp arc/94-267.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 1, pp. 18–28, October, 1995.     

Bailey pairs and indefinite quadratic forms

We construct classes of Bailey pairs where the exponent of q   in _αn${\alpha }_n$ is an indefinite quadratic form. As an application we obtain families of q-hypergeometric mock theta multisums.     

Rank restrictions on the quadratic form in indefinite quadratic programming

Summary A quadratic programming problem, whereq(x) =a ^T x +x ^T Qx is an indefinite objective function, can be solved withSwarup's approach of optimizing (c ^T x + )(d ^T x + ) only if the rank ofQ is two; ifQ is definite, the rank ofQ must be one.

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Zusammenfassung Ein quadratisches Optimierungsproblem, in demq(x) =a ^T x +x ^T Qx eine indefinite Zielfunktion ist, kann mitSwarups Optimierungsansatz (c ^T x + )(d ^T x + ) nur gelöst werden, wenn der Rang vonQ gleich zwei ist; wennQ definit ist, muß der Rang vonQ gleich eins sein.

     

Small quadratic elements in representations of the special linear group with large highest weights

For almost all p-restricted irreducible representations of the group A_n(K) in characteristic p > 0 with highest weights large enough with respect to p, the Jordan block structure of the images of small quadratic unipotent elements in these representations is determined. It is proved that if φ is an irreducible p-restricted representation of A_n(K) with highest weight

Boundary pairs associated with quadratic forms

We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet‐to‐Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non‐smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.     

Separation of representations with quadratic overgroups

Any unitary irreducible representation π of a Lie group G defines a moment set Iπ, subset of the dual g^? of the Lie algebra of G. Unfortunately, Iπ does not characterize π. If G is exponential, there exists an overgroup G⁺ of G, built using real-analytic functions on g^?, and extensions π⁺ of any generic representation π to G⁺ such that Iπ⁺ characterizes π.In this paper, we prove that, for many different classes of group G, G admits a quadratic overgroup: such an overgroup is built with the only use of linear and quadratic functions.     

Modular representations of simple Jordan pairs

Omsk. Translated fromSibirskii Matematicheskii Zhurnal, Vol. 32, No. 2, pp. 200–203, March–April, 1991.     

Local group representations

As a generalization of the usual notion of a group representation, a local group representation of connected and locally connected groups G is introduced as a certain sheaf A with base G, such that all fibres A _x may be identified with a fixed linear subspace D of the representation space. The sheaf provides a means of expressing a group symmetry which is known to hold only locally (in a sense which will be made precise). The problem of extending a local representation to a global one may be considered as a noncommutative counterpart of the dilation theory of systems of doubly commuting contractions. If G is a Lie group, one may associate a derived representation of the enveloping algebra (g) of the Lie algebra g of G. In case of isometric local representations in a Hilbert space, these are *-representations and determine the appropriate local representation uniquely. As an application we obtain that the n-tuples of antisymmetric operators of partial derivatives on L ²(), where is a connected open subset of ⁿ, determine up to translation.     

Local-global principles for representations of quadratic forms

We prove a local-global principle for the problem of representations of quadratic forms by quadratic forms over ℤ, in codimension ≥5. The proof uses the ergodic theory of p-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.     

Equivalence of two series of spherical representations of a free group

Summary The spherical principal series of a non-commutative free group may be analytically continued to yield a series of uniformly bounded representations, much as the spherical representations (in1/2) + it of SL (2,R) may be analytically continued in the strip 0 < Rez < 1. This series of uniformly bounded representations was constructed and studied by A. M.Mantero and A.Zappa. Independently T.Pytlik and R.Szwarc introduced and studied representations of the free group which contain a series of subrepresentations indexed by spherical functions. Both series consist of irreducible representations and include the spherical complementary series. The aim of this paper is to prove that the non-unitary uniformly bounded representations of the two series are also equivalent.