Tensor products of maximal degenerate series representations of the group SL(n, R)

Journal des Mathematiques Pures et Appliquees

Volumn 78, Issue 1, 1999, Pages 99-119

  Van Dijk, G ^a   Molchanov, V F ^b  

^a LEIDEN UNIVERSITY   (Netherlands)

^b DERZHAVIN TAMBOV STATE UNIVERSITY   (Russian Federation)

Author keywords

Berezin form; Decomposition into irreducible components; Maximal degenerate representations; Tensor products of representations

Indexed keywords

EID: 0032622571     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-7824(99)80011-7     Document Type: Article

References (34)

1. N. Bourbaki, Eléments de mathématique, VI Intégration, Ch. VI-VIII. Hermann, Paris, 1965.
2. F.A. Berezin, Quantization in complex bounded domains, Dokl. Akad. Nauk SSSR, 211, 1973, pp. 1263-1266, Engl. transl., Sov. Math. Dokl. 14, 1973, pp. 1209-1213.
3. F.A. Berezin, Quantization in complex bounded domains, Dokl. Akad. Nauk SSSR, 211, 1973, pp. 1263-1266, Engl. transl., Sov. Math. Dokl. 14, 1973, pp. 1209-1213.
4. F.A. Berezin, Quantization, Izv. Akad. Nauk SSSR, Ser. Mat. 38, No. 5, 1116-1175, Engl. transl.: Math. USSR-Izv., 8, 1974, pp. 1109-1165, 1974.
5. F.A. Berezin, Quantization, Izv. Akad. Nauk SSSR, Ser. Mat. 38, No. 5, 1116-1175, Engl. transl.: Math. USSR-Izv., 8, 1974, pp. 1109-1165, 1974.
6. F.A. Berezin, Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR, Ser. Mat., 39, 1975, pp. 363-402, Engl. transl.: Math. USSR, Izv., 9, 1975, pp. 341-379.
7. F.A. Berezin, Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR, Ser. Mat., 39, 1975, pp. 363-402, Engl. transl.: Math. USSR, Izv., 9, 1975, pp. 341-379.
8. G. Van Dijk and S.C. Hille, Canonical representations related to hyperbolic spaces, J. Funct. Anal., 147, 1997, pp. 109-139.
9. G. Van Dijk and S.C. Hille, Maximal degenerate representations, Berezin kernels and canonical representations. in "Lie Groups and Lie Algebra, Their Representations, Generalizations, and Applications" (B. Komrakov, J. Krasil'shchik, G. Litvinov and A. Sossinsky, Eds.), Kluwer Academic, Dordrecht, 1997.
10. G. Van Dijk and V.F. Molchanov, The Berezin Form for Rank One Para-Hermitian Symmetric Spaces. Preprint Leiden University, 1997.
11. A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions I, II. McGraw-Hill, New York, 1953, 1955.
12. I.S. Gradshtein and I.M. Ryzhik, Table of Integrals, Series and Products. Academic Press, New York etc, 1980.
13. S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math., 8, 1985, pp. 81-98.
14. V.F. Molchanov, Representations of the pseudo-orthogonal group associated with a cone, Mat. Sb., 81, 1970, pp. 358-375, Engl. transl. Math. USSR, Sb. 10, pp. 333-347.
15. V.F. Molchanov, Representations of the pseudo-orthogonal group associated with a cone, Mat. Sb., 81, 1970, pp. 358-375, Engl. transl. Math. USSR, Sb. 10, pp. 333-347.
16. V.F. Molchanov, Decomposition of the tensor square of a representation of the complementary series of the unimodular group of real second order matrices, Sib. Mat. Zh., 18, 1977, pp. 174-188, Engl. transl., Sib. Math. J., 18, 1977, pp. 128-138.
17. V.F. Molchanov, Decomposition of the tensor square of a representation of the complementary series of the unimodular group of real second order matrices, Sib. Mat. Zh., 18, 1977, pp. 174-188, Engl. transl., Sib. Math. J., 18, 1977, pp. 128-138.
18. V.F. Molchanov, The restriction of a representation of the complementary series of a pseudo-orthogonal group to a pseudo-orthogonal group of lower dimension, Dokl. Akad. Nauk SSSR, 237, 1977, pp. 782-785, Engl. transl., Sov. Math., Dokl., 18, pp. 1493-1497, 1977.
19. V.F. Molchanov, The restriction of a representation of the complementary series of a pseudo-orthogonal group to a pseudo-orthogonal group of lower dimension, Dokl. Akad. Nauk SSSR, 237, 1977, pp. 782-785, Engl. transl., Sov. Math., Dokl., 18, pp. 1493-1497, 1977.
20. V.F. Molchanov, Tensor products of unitary representations of the three-dimensional Lorentz group, Izv. Akad. Nauk SSSR, Ser. Mat., 43, 1979, pp. 860-891, Engl. transl. Math. USSR, Izv., 15, 1980, pp. 113-143.
21. V.F. Molchanov, Tensor products of unitary representations of the three-dimensional Lorentz group, Izv. Akad. Nauk SSSR, Ser. Mat., 43, 1979, pp. 860-891, Engl. transl. Math. USSR, Izv., 15, 1980, pp. 113-143.
22. V.F. Molchanov, Quantization on the imaginary Lobachevsky plane, Funkts. Anal. Prilozh., 14, No. 2, 1980, pp. 73-74, Engl. transl. Funct. Anal. Appl., 14, 1980, pp. 142-144.
23. V.F. Molchanov, Quantization on the imaginary lobachevsky plane, Funkts. Anal. Prilozh., 14, No. 2, 1980, pp. 73-74, Engl. transl. Funct. Anal. Appl., 14, 1980, pp. 142-144.
24. V.F. Molchanov, The plancheral formula for the pseudo-Riemannian space SL(3, ℝ)/Gl(2, ℝ), Sib. Mat. Zh., 23, No. 5, 1982, 142-151, Engl. transl. Sib. Math. J., 23, 1982, pp. 703-711.
25. V.F. Molchanov, The plancheral formula for the pseudo-Riemannian space SL(3, ℝ)/Gl(2, ℝ), Sib. Mat. Zh., 23, No. 5, 1982, 142-151, Engl. transl. Sib. Math. J., 23, 1982, pp. 703-711.
26. V.F. Molchanov, Quantization on para-Hermitian symmetric spaces, Adv. in Math. Sci., AMS Transl., Ser. 2, 175, 1996, pp. 81-95.
27. Yu. A. Neretin, Representations of complementary series entering discretely in tensor products of unitary representations, Funkts. Anal. Prilozh., 20, No. 1, 1986, pp. 79-80, Engl. transl. Funct. Anal. Appl., 20, No. 1, 1986, pp. 68-70.
28. Yu. A. Neretin, Representations of complementary series entering discretely in tensor products of unitary representations, Funkts. Anal. Prilozh., 20, No. 1, 1986, pp. 79-80, Engl. transl. Funct. Anal. Appl., 20, No. 1, 1986, pp. 68-70.
29. Yu. A. Neretin and G.I. Olshanskii, Boundary values of holomorphic functions, singular unitary representations of the groups O(p, q) and their limits when q → ∞, Zapiski Nauchn. Sem. POMI RAN, No. 223, 1995, pp. 9-91 (in Russian).
30. G. Olafsson, Causal symmetric spaces, Math. Goetting., 15, 1990.
31. A.M. Vershik, I.M. Gel'fand and M.I. Graev, Representations of the group SL(2, R) where R is a ring of functions, Usp. Mat. Nauk 28, No. 5, 1973, pp. 82-128, Engl. transl., Russ. Math. Surv., 28, No. 5, 1973, pp. 87-132.
32. A.M. Vershik, I.M. Gel'fand and M.I. Graev, Representations of the group SL(2, R) where R is a ring of functions, Usp. Mat. Nauk 28, No. 5, 1973, pp. 82-128, Engl. transl., Russ. Math. Surv., 28, No. 5, 1973, pp. 87-132.
33. N. Ya. Vilenkin, Special functions and the theory of group representations, Nauka, Moscow. Engl. transl. Transl. Math. Monographs, 22, (1965), Amer. Math. Soc., Providence RI 1968.
34. N. Ya. Vilenkin, Special functions and the theory of group representations, Nauka, Moscow. Engl. transl. Transl. Math. Monographs, 22, (1965), Amer. Math. Soc., Providence RI 1968.