Uniformly bounded representations and exact groups

We characterize groups with Guoliang Yu?s property A (i.e., exact groups) by the existence of a family of uniformly bounded representations which approximate the trivial representation.     

The Cayley transform and uniformly bounded representations

Let G be a simple Lie group of real rank one, with Iwasawa decomposition and Bruhat big cell . Then the space may be (almost) identified with N and with K/M, and these identifications induce the (generalised) Cayley transform . We show that is a conformal map of Carnot-Caratheodory manifolds, and that composition with the Cayley transform, combined with multiplication by appropriate powers of the Jacobian, induces isomorphisms of Sobolev spaces and . We use this to construct uniformly bounded and slowly growing representations of G.     

Polynomial representations of the orthogonal groups

This paper is concerned with realizations of the irreducible representations of the orthogonal group and construction of specific bases for the representation spaces. As is well known, Weyl's branching theorem for the orthogonal group provides a labeling for such bases, called Gelfand-Žetlin labels. However, it is a difficult problem to realize these representations in a way that gives explicit orthogonal bases indexed by these Gelfand-–etlin labels. Thus, in this paper the irreducible representations of the orthogonal group are realized in spaces of polynomial functions over the general linear groups and equipped with an invariant differentiation inner product, and the Gelfand-Žetlin bases in these spaces are constructed explicitly. The algorithm for computing these polynomial bases is illustrated by a number of examples. Partially supported by a grant from the Department of Energy. Partially supported by NSF grant No. MCS81-02345.     

Galois representations of three-dimensional orthogonal motives

 Let M be a motive defined over a number field K and the associated system of -adic representations. Assuming some (mild) conditions on M we determine the image in the case that M is three-dimensional and equipped with a symmetric bilinear form. We can show further that several conjectures on -adic representations hold for M. Received: 4 October 2000 / Revised version: 21 April 2002     

Group representations and matrix spectral problems

We study a connectionvia group representation theory, between the problem of describing the invariant factors of a product of two matrices over a principal ideal domain and the problem of describing the spectrum of a sum of two Hermitian matrices.     

Rational spectral transformations and orthogonal polynomials

We show that generic rational transformations of the Stieltjes function with polynomial coefficients (ST) can be presented as a finite superposition of four fundamental elementary transforms: Christoffel transform (CT), Geronimus transform (GT) and forward and backward associated transformations A⁺T, A⁻T. It is shown that the Laguerre-Hahn polynomials (LHP) on arbitrary nonuniform lattice are covariant with respect to ST (i.e., ST of a LHP yields another LHP), whereas the semi-classical polynomials are covariant with respect to a subclass of linear ST. Some applications of these results to the theory of the semi-classical polynomials are considered.     

Conformal oscillator representations of orthogonal Lie algebras

The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n, C)give rise to a one-parameter(c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n + 2, C). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a ∈ Cn, we prove that the space forms an irreducible o(n + 2, C)-module for any c ∈ C if a is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2, C) on the polynomial algebra C in n variables. Moreover, we prove that C forms an infinite-dimensional irreducible weight o(n + 2, C)-module with finite-dimensional weight subspaces if c ∈ Z/2.     

Motivic orthogonal two-dimensional representations of Gal(ℚ/ℚ)

Given a two-dimensional compatible family of ℓ-adic representations which is motivic and which respects an orthogonal form up to similitudes, we show how to express itsL-function in terms of a Hecke character. We give several examples and in particular we analyze a representation associated to a certainK3 surface which arose in the study of Kloosterman sums.     

Special representations of rank one orthogonal groups

We study the image of the theta correspondence from to a rank one orthogonal group (over a number field). The image consists of cusp forms, the Fourier coefficients of which satisfy a certain invariance property. We show that this property characterizes the image. The proof requires first an analogous local statement (almost everywhere) and then a use of certain Rankin-Selberg integrals. Partially supported by the Bat-Sheva de Rothschild Fund for the Advancement of Science and Technolog.     

Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d) ^log ₂ ^d. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119. Translated by S. A. Evdokimov.     

Orthogonal F-rectangles,orthogonal arrays,and codes

The relationships between a set of orthogonal F-squares or F-rectangles and orthogonal arrays are described. The relationship between orthogonal arrays and error-correcting codes is demonstrated. The development of complete sets of orthogonal F-rectangles allows construction of codes of any word length and for any number of words. Likewise, the development of F-rectangle theory makes code construction much more flexible in terms of a variable number of symbols. The relationship among sets of orthogonal hyperrectangles, orthogonal arrays, and codes is also described.     

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 .     

Orthogonal, symplectic and unitary representations of finite groups

Let be a field, a finite group, and a linear representation on the finite dimensional -space . The principal problems considered are:

I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms which are -invariant.

II. If is such a form, enumerate the equivalence classes of representations of into the corresponding group (orthogonal, symplectic or unitary group).

III. Determine conditions on or under which two orthogonal, symplectic or unitary representations of are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically' equivalent.

This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) -module components of their spaces are equivalent.

We assume throughout that the characteristic of does not divide .

Solutions to I and II are given when is a finite or local field, or when is a global field and the representation is ``split'. The results for III are strongest when the degrees of the absolutely irreducible representations of are odd - for example if has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index - and, in the case that is a local or global field, when the representations are split.

     

Minimal permutation representations of finite simple orthogonal groups

In the paper, nontrivial permutation representations of minimal degree are studied for finite simple orthogonal groups. For them, we find degrees, ranks, subdegrees, point stabilizers and their pairwise intersections.Translated fromAlgebra i Logika, Vol. 33, No. 6, pp. 603–627, November–December, 1994.     

On spectral variations under relatively bounded perturbations

Linear operators in Banach and Hilbert spaces are considered. Bounds for the spectrum are established under relatively bounded perturbations. An application to nonselfadjoint differential operators is discussed.