The PRV-formula for tensor product decompositions and its applications

Let G be a semisimple algebraic group, V a simple finite-dimensional self-dual G-module, and W an arbitrary simple finite-dimensional G-module. Using the triple multiplicity formula due to Parthasarathy, Ranga Rao, and Varadarajan, we describe the multiplicities of W in the symmetric and exterior squares of V in terms of the action of a maximum-length element of the Weyl group on some subspace in V ^T , where T ? G is a maximal torus. By way of application, we consider the cases in which V is the adjoint, little adjoint, or, more generally, a small G-module. We also obtain a general upper bound for triple multiplicities in terms of Kostant’s partition function.     

Degenerate principal series representations and their holomorphic extensions

Let X=H/L be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain D=G/K. The intersection S of the Shilov boundary of D with X defines a distinguished subset of the topological boundary of X and is invariant under H. It can be realized as S=H/P for certain parabolic subgroup P of H. We study the spherical representations of H induced from P. We find formulas for the spherical functions in terms of the Macdonald hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces D. We consider a class of H-invariant integral intertwining operators from the representations on L²(S) to the holomorphic representations of G restricted to H. We construct a new class of complementary series for the groups H=SO(n,m), SU(n,m) (with n−m>2) and Sp(n,m) (with n−m>1). We realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of G=SU(n,m), SU(n,m)×SU(n,m) and SU(2n,2m) respectively.     

B型和C型Weyl模的一个新重数公式

A monomial basis and a filtration of subalgebras for the universal enveloping algebra U(gl) of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V (λ) as a U(gl)-module into a direct sum of Weyl modules V (μ)’s as U(gl-1)modules is described. In particular, a new multiplicity formula for the Weyl module V (λ) is obtained in this note.     

Whittaker functions for generalized principal series representations of GSp(2,R)

We study Whittaker functions for generalized principal series representations of $\mathrm{GSp}(2,\mathbf{R})$ induced from Siegel parabolic subgroup. We give Mellin–Barnes type integral representations of Whittaker functions belonging to certain K-types. As an application of our explicit formulas, we compute the archimedean parts of Novodvorsky's zeta integrals. As a consequence we can show the entireness of the spinor L-functions for generic cusp forms on $\mathrm{GSp}(2)$.     

On Bessel integrals for reducible degenerate principal series representations

We discuss generalized Bessel integrals with nondegenerate characters, which are assigned to irreducible submodules of a reducible degenerate principal series representation of Sp(n,R). Then we give sufficient conditions for their vanishings which are based on the signatures of the nondegenerate characters. This consequently suggests a reasonable correspondence between open GLn(R)-orbits in the set of real symmetric matrices of size n and irreducible submodules of the reducible principal series representations.     

Recurrence relations and convergence theory of the generalized polar decomposition on Lie groups

The subject matter of this paper is the analysis of some issues related to generalized polar decompositions on Lie groups. This decomposition, depending on an involutive automorphism , is equivalent to a factorization of , being a Lie group, as with and , and was recently discussed by Munthe-Kaas, Quispel and Zanna together with its many applications to numerical analysis. It turns out that, contrary to , an analysis of is a very complicated task. In this paper we derive the series expansion for , obtaining an explicit recurrence relation that completely defines the function in terms of projections on a Lie triple system and a subalgebra of the Lie algebra , and obtain bounds on its region of analyticity. The results presented in this paper have direct application, among others, to linear algebra, integration of differential equations and approximation of the exponential.

     

Qualitative uncertainty principles for groups with finite dimensional irreducible representations

Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations.     

Tensor products of certain modules for the generalized cartan matrix lie algebra

The outer multiplicites of tensor product decompositions are computed for the smallest Euclidean Kac-Moody Lie algebra when one of the tensor factors is a fundamental module or the p-module. Some combinatorial identities are obtained.     

单变量自守形式导引

本讲义是关于自守形式解析理论的导论，这些自守形式定义在上半平面上，对应的群是Ｇ＝ＳＬ２（Ｒ）或是Ｇ的具有有限余维数的离散子群Г。     

Chebyshev series method for computing weighted quadrature formulas

In this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions. The main goal is to evaluate accurately a definite integral, whose mass is highly concentrated near some points. The numerical implementation of this approach is based on the calculation of Chebyshev series and some integration formulas which are exact for polynomials. In terms of accuracy, the proposed method can be compared with rational Gauss quadrature formula.     

Family algebras and generalized exponents for polyvector representations of simple Lie algebras of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We give an explicit formula for the exterior powers ∧ ^k π ₁ of the defining representation π ₁ of the simple Lie algebra ?ο(2n + 1, ?). We use the technique of family algebras. All representations in question are children of the spinor representation σ of g2ο(2n + 1, ?). We also give a survey of main results on family algebras.     

Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Newton polytopes,increments, and roots of systems of matrix functions for finite-dimensional representations

The asymptotic root distribution is computed for systems of matrix functions associated with finite-dimensional holomorphic representations of a Lie group. This distribution can be expressed via the increments of the representations involved. If the group is reductive, then the number of equations in the system can be arbitrary, from 1 to the dimension of the group. In this case, the computation results are stated in the language of convex geometry. These computations imply the previously known formulas for the density of the solution variety of a system of exponential equations as well as for the number of solutions of a polynomial system and, more generally, of a system formed by matrix functions of representations of a complex reductive Lie group.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 22–35, 2004Original Russian Text Copyright © by B. Ya. KazarnovskiiSupported by INTAS grant 00-259 and NWO grant 047.008.005.     

On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem on ; on , where is a bounded region in , is an indefinite weight function and may be positive, negative or zero.

     

Taylor series for the Askey-Wilson operator and classical summation formulas

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As an application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context.

     

Isomorphism problems and groups of automorphisms for generalized Weyl algebras

We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of by finding sets of generators for the group of automorphisms.