An analysis of the multiplicity spaces in branching of symplectic groups

Branching of symplectic groups is not multiplicity free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra B{\mathcal{B}}. The algebra B{\mathcal{B}} is a graded algebra whose components encode the multiplicities of irreducible representations of Sp _2n–2 in irreducible representations of Sp _2n . Our first theorem states that the map taking an element of Sp _2n to its principal n × (n + 1) submatrix induces an isomorphism of B{\mathcal{B}} to a different branching algebra B^￠{\mathcal{B}^{\prime}}. The algebra B^￠{\mathcal{B}^{\prime}} encodes multiplicities of irreducible representations of GL _n–1 in certain irreducible representations of GL _n+1. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of Sp _2n to Sp _2n–2 is canonically an irreducible module for the n-fold product of SL ₂. In particular, this induces a canonical decomposition of the multiplicity spaces into one-dimensional spaces, thereby resolving the multiplicities.     

Mixed-symmetry massless gauge fields in AdS5

Using the su(2, 2) spinor language, we describe free mixed-symmetry massless bosonic and fermionic gauge fields of arbitrary spins in the AdS₅ space. We construct manifestly covariant action functionals and derive field equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 47–59, October, 2006.     

The multiplicity function on exponential and completely solvable homogeneous spaces

It has been shown recently that the same multiplicity function appears in direct integral decompositions of both induced and restricted representations on exponential solvable groups — that is, a generalized form of Frobenius Reciprocity is valid. Qualitative results on that multiplicity function in the completely solvable case are proven in this paper — namely, necessary and sufficient conditions for finiteness and boundedness are obtained. The proof uses techniques from pseudo-algebraic geometry.Research supported by DMS 87-00551A02.     

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.

     

Global geometric properties of AdS space and the AdS/CFT correspondence

The Poisson kernels and relations between them for a massive scalar field in a unit ball Bn with Hua's metric and conformal flat metric are obtained by describing the Bn as a submanifold of an (n+1)-dimensional embedding space. Global geometric properties of the AdS space are discussed. We show that the(n+1)-dimensional AdS space AdSn+1 is isomorphic to RP1×Bn and boundary of the AdS is isomorphic to RP1×Sn-1. Bulk-boundary propagator and the AdS/CFT like correspondence are demonstrated based on these global geometric properties of the RP1×Bn.     

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G _n /K _n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N _n nilpotent, in other words pairs (G _n , K _n ) for which G _n /K _n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L ²(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G _n /K _n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L ²(G/K). This allows an interpretation of L ²(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.     

A multiplicity theorem for critical points of functionals on reflexive Banach spaces

In this paper we establish a multiplicity theorem for critical points of functionals on reflexive Banach spaces. Precisely, we deduce the main result using a general variational principle proved by Ricceri. Moreover, we present an application to a Neumann problem which gives a positive answer to some questions formulated by the previous author.Received: 6 February 2003     

Representation theorems for Sobolev spaces on intervals and multiplicity results for nonlinear ODEs

We present some topological isomorphisms between Wm,p((0,1)) and Lp((0,1))×Rm. Combined with some arguments from Fourier analysis, we obtain explicit Schauder bases for Wm,p((0,1)) and some of its subspaces. In addition, we apply such results to treat some boundary value problems involving nonlinear elliptic differential equations.     

Two-interval Sturm-Liouville operators in modified Hilbert spaces

By modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which the Sturm-Liouville equation is defined, we generate self-adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive. This contrasts with the usual theory which requires the coupling matrix to have determinant one.     

APPROXIMATION OF MODIFIED LAGRANGE INTERPOLATION IN ORLICZ SPACES

In this paper, the authors give the Marcinkiewicz-Zygmund inequality based on the zeros of the first kind Chebyshev polynomials in Orlicz norm. As application, the degree of approximation by two kinds of modified Lagrange inter polatory polynomials in Orlicz spaces is studied.     

On a multiplicity one property for the length spectra of even dimensional compact hyperbolic spaces

We prove a multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces, i.e., if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.     

The Paley-Wiener theorem and the local Huygens' principle for compact symmetric spaces: The even multiplicity case

We prove a Paley-Wiener theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new proofs of the strong Huygens' principle for a suitable constant shift of the wave equation on odd-dimensional spaces from our class.     

Minimum distance estimation in normed linear spaces with Donsker-classes

We consider minimum distance estimators where the discrepancy function is defined in terms of a supremum-norm based on a Donsker-class of functions. If the parameter set is contained in a normed linear space we prove a Portmanteau-type theorem. Here, the limit in general is not a probability measure, but an outer measure given by the hitting family of the set of all minimizing points of a certain stochastic process. In case there is exactly one minimizer one obtains traditional weak convergence.     

Tight 4 and 5-designs in projective spaces

Bannai and Hoggar showed that tightt-designs in projective spaces over , , , satisfyt 5. Here we prove that caset = 4 is impossible, and that the two known examples witht = 5 are unique.