G. van Dijk

TENSOR PRODUCTS OF MAXIMAL DEGENERATE SERIES REPRESENTATIONS OF THE GROUP SL(n, R) (*)

Abstract This paper has two goals: firstly to study the maximal degenerate series representations π±μ, gn, μ Π±u,v, u∈C ν = 0, 1 of SL(n, ℝ) and, secondly, to decompose the tensor products π+μ, gn ⊗ π−μ, gn μ ℝ, ν =0, 1 (or canonical representations) into irreducible representations. These tensor products have an invariant Hermitian form, which is closely connected with the Berezin form, considered in an earlier paper. We determine the decomposition of the tensor products with respect to this Hermitian form. Generally speaking, this form is not positive-definite. Though one can consider all concepts in the framework of para-Hermitian symmetric spaces, explicit results are only known so far for the rank one spaces, considered in this paper.

The Berezin form for rank one para-Hermitian symmetric spaces☆

Abstract Recently one of the authors presented a general scheme of quantization (in the spirit of Berezin) for para-Hermitian symmetric spaces. One of the main notions of this theory is the Berezin form. It is an invariant Hermitian form, which, generally speaking, is not positive-definite. The rank one para-Hermitian spaces are exhausted by the spaces X = SL(n, R )/GL(n − 1, R ), up to coverings. In this paper we decompose the Berezin form B μ, ν, μ ϵ R , ν = 0, 1 for these spaces X into irreducible components. For values μ −n+1 2 the decomposition contains the same irreducible components as L2(X), but the Plancherel measure is not necessarily positive. For μ > −n+1 2 a finite number of irreducible components, belonging to the (non-unitary) complementary series, is added. We also pay attention to the behaviour of B μ, ν as μ → −∞, which has a meaning in the context of quantization. It turns out that the correspondence principle holds on the discrete spectrum for n even.

A new class of Gelfand pairs

The well-known result for rank 1 real symmetric spacesX=G/H, stating the multiplicity-free decomposition of the natural action ofG onL2(X), is extended to the classical rank 1p-adic symmetric spaces. The absence of ap-adic analogue of the Laplace-Beltrami operator causes additional complications and necessitates a global study of the invariant distributions on the quotient space.

Maximal Degenerate Representations, Berezin Kernels and Canonical Representations

In this work we present a new context of the canonical representations which have been introduced by Berezin, Gel’fand, Graev and Vershik for simple Lie groups G of Hermitian type. We discuss maximal-degenerate representations of the complexification of G and the decomposition of the canonical representations into irreducible parts.

Spherical distributions on the pseudo-Riemannian space SL(n, R)/GLSL(n−R)

In this paper we determine the spherical distributions on the pseudo-Riemannian symmetric space SL(n, R)/GLSL(n−R and study their asymptotics, thus providing the necessary ingredients for the Plancherel formula for this space.

THE PLANCHEREL FORMULA FOR LINE BUNDLES ON COMPLEX HYPERBOLIC SPACES

In this paper we obtain the Plancherel formula for the spaces of L 2 -sections of line bundles over the complex projective hyperboloids G=H withGD U.p;qIC/ andHD U.1IC/ U.p 1;qIC/. The Plancherel formula is given in an explicit form by means of spherical distributions associated with a character of the subgroupH . We obtain the Plancherel formula by a special method which is also suitable for other problems, for example, for quantization in the spirit of Berezin.

A New Approach to Berezin Kernels and Canonical Representations

We propose, following ideas of S. C. Hille, another approach to the theory of Berezin kernels than the ususal one in the context of Hermitian symmetric spaces and spaces of Hermitian type. Our context is much more general and circles around so-called σθ-stable parabolic subgroups and intertwining operators. We present new examples of Berezin kernels and also highlight the new approach in the context of interpolation of representations between (L 2(SU(1, n, F)/S(U(1, F) × U(n, F))) and its compact analogue L 2(SU(n + 1, F)/S(U(1, F) × (U(n, F))) for F = ℝ, ℂ, ℍ, in the spirit of Neretin [27].

Harmonic analysis on the finite symmetric space GL(n, K)GL(1, K) × GL(n − 1, K)

Abstract Let Gn = GL(n, K) and Hn = GL(1, K) × GL(n − 1, K) with K a finite field of odd characteristic. We determine all irreducible Hn-spherical representations of Gn and compute the associated spherical functions. Main by-products are: - (Gn, Hn) is not a Gelfand pair: there is one special representation which occurs twice in the regular representation of Gn on X = G n H n ; - there are no irreducible spherical cuspidal representations for n>2.

Canonical representations associated to hyperbolic spaces II

Abstract In this paper we determine explicit models for the unitary representations which occur discretely in the decomposition of canonical representations for hyperbolic spaces. This generalizes work of Gelfand, Graev and Vershik for the case SU(1,1). In a previous paper [1] we have introduced canonical representations π λ for hyperbolic spaces of Riemannian type, generalizing Berezins definition for the case of Hermitian symmetric spaces. These unitary representations have a rich internal structure and prove that not only quasi-regular representations are important in harmonic analysis. They have an interesting decomposition into irreducible constituents: for large λ only unitary principal series representations occur, for small λ however a finite number of complementary series representations has to be added. A detailed description of the latter, as subrepresentations of π λ , was started in [1]. The present paper contains much more precise results. The classical notions of Fourier and Poisson transform play a dominant role and are treated in Section 4. In Section 5 the actual projection of a given function onto the complementary series is then given. Since the proof makes use of some results on maximal degenerate series representations of SL( n + 1), these representations are studied in Section 1 in some detail. As a by-product we obtain an alternative introduction of the canonical representations, see Sections 2 and 5. This paper has a considerable, but unevitable optical overlap with our joint paper [2] with V.F. Molchanov. The method described here has been introduced in a different setting by Molchanov, but the observation that it works also for hyperbolic spaces is new and surprising. To leave out parts of the proof would make this paper almost unreadable. We thank S.C. Hille and V.F. Molchanov for several discussions on these matters and for their encouragement to publish the results obtained in this paper.

On the regular embedding of representations of motion groups

Abstract In this short note we extend a result proved by Klamer in 1979 on the regular realization of representations of Poincare groups, to a much larger class of motion groups.