Menu: Home :: go to Journal :: switch to Russian :: switch to English
You are here: all Journals and Issues→ Journal→ Issue→ Article

Canonical representation

Annotation

These lecture notes provide an introduction to the theory of so-called canonical representations, a special type of (reducible) unitary representations. The simplest way to define them and to see their importance is done in the context of the group Sl(2, R), the group of 2x2 matrices of determinant one, see [16]. We have chosen a class of groups, namely G=SU(1, n), n>/1. Notice that SL(2, R) is isomorphic to SU (1, 1). Canonical representations can be seen, generally speaking, as the completion of L2 (G/K) with respect to a new G-invariant inner product, in the same spirit as the “complementary series” is obtained from the “principal series” for G. Here K=SU(n). But this is only one (but important) point of view, see section 5. Canonical representations occur also when studying tensor products of holomorphic and anti-holomorphic discrete series representations. This is explained in section 4. The connection with quantization in the sense of Berezin is not treated in these notes because we will emphasize the representation theory. On the other hand, Berezin has made a large contribution to the understanding of canonical representations. The main problem is to decompose the canonical representations into irreducible constituents. This is not an easy task. It has been done by Berezin [1] and, later, by Upmeier and Unterberger [15]. There are however, in both treatments, conditions on the set of parameters of the representations: only large parameters are allowed. For small parameters (see [3]) an interesting new phenomenon occurs: finitely many complementary series representations take part in the decomposition. We shall treat the case G=SU(1, n) in detail in these notes and try to illustrate all aspects of the theory of canonical representations we have mentioned.

Full-text in one file

Download

Pages

350-366

References

[1] Berezin, F.A. (1973): Quantization in complex bounded domains. Dokl. Akad. Nauk SSSR 211, 1263–1266. Engl. transl.: Sov. Math., Dokl. 14, 1209–1213 (1973). [2] Dijk, G. van (1994): Group representations on spaces of distributions. Russian J. Math. Physics 2, No. 1, 57-68. [3] Dijk G. van, Hille, S.C. (1995): Canonical representations related to hyperbolic spaces. Report no. 3, Institut Mittag Leffler 1995/96. [4] Dijk, G. van, Hille S.C. (1996): Maximal degenerate representations, Berezin kernels and canonical representations. Report W 96-04, Leiden University. [5] Dijk, G. van, Molchanov V.F. (1997): The Berezin form for the space SL(n,R)/GL(n–1,R). Preprint Leiden University. [6] Dijk, G. van, Molchanov, V.F. (1997): Tensor products of maximal degenerate series representations of the group SL(n,R). Preprint Leiden University. [7] Erdelyi, A. et al. (1954): Tables of Integral Transforms, volume II. McGraw-Hill, New York. [8] Erdelyi, A. et al. (1953): Higher Transcental Functions, volume I. McGraw-Hill, New York. [9] Faraut, J. (1982): Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques. In: “Analyse Harmonique”, Les Cours de CIMPA, 315–446. [10] Faraut, J. (1979): Distributions spheriques sur les espaces hyperboliques. J. Math. Pures et Appl. 58, 369–444. [11] Flensted-Jensen, M., Koornwinder, T.H. (1979): Positive-definite spherical functions on a non-compact rank one symmetric space. Springer Lecture Notes in Math. 739, 249–282. [12] Hille S.C. (1996): Canonical representations associated to a character on a Hermitian symmetric space of rank one. Preprint Leiden University. [13] Molchanov, V.F. (1996): Quantization on para-Hermitian symmetric spaces. Adv. in Math. Sci., AMS Transl., Ser. 2, 175, 81–96. [14] Repka, J. (1979): Tensor products of holomorphic discrete series presentations. Can. J. Math. 31, 836–844. [15] Unterberger A., Upmeier, H. (1994): The Berezin transform and invariant differential operators. Comm. Math. Physics, 164, No 3, 563–597. [16] Vershik, A.M., Gel`fand, I.M., Graev, M.I. (1973): Representations of the group SL(2,R) where R is a ring of functions. Usp. MMMat. Nauk 28, No. 5, 82–128. Engl. transl.: Russ. Math. Surv. 28, No. 5, 87–132(1973).

Section of issue

Articles

Для корректной работы сайта используйте один из современных браузеров. Например, Firefox 55, Chrome 60 или более новые.