Notes on Stein–Sahi representations and some problems of non-$L^2$-harmonic analysis

Yu. A. Neretin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 125-169

Résumé

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.

Bibliographie

[1] G. S. Andrews, R. Askey, R. Roy, Special Functions, Cambridge Univ. Press, 1999 | MR | Zbl

[2] M. F. Atiyah, “Resolution of singularities and division of distributions”, Comm. Pure Appl. Math., 23 (1970), 145–150 | DOI | MR | Zbl

[3] E. P. van den Ban, H. Schlichtkrull, “Fourier inversion on a reductive symmetric space”, Acta Math., 182:1 (1999), 25–85 | DOI | MR | Zbl

[4] E. P. van den Ban, H. Schlichtkrull, “The most continuous part of the Plancherel decomposition for a reductive symmetric space”, Ann. Math. (2), 145:2 (1997), 267–364 | DOI | MR | Zbl

[5] V. Bargmann, “Irreducible unitary representations of the Lorentz group”, Ann. Math., 48 (1947), 568–640 | DOI | MR | Zbl

[6] F. A. Berezin, “Quantization in complex symmetric spaces”, Izv. Akad. Nauk SSSR, Ser. Math., 39:2 (1975), 363–402 | MR | Zbl

[7] F. A. Berezin, “On relations between covariant and contravariant symbols of operators for complex classical domains”, Dokl. Akad Nauk SSSR, 241:1 (1978), 15–17 | MR | Zbl

[8] M. Berger, “Les espaces symétriques noncompacts”, Ann. Sci. École Norm. Sup., 74 (1957), 85–177 | MR

[9] I. M. Bernshtein, “Analytic continuation of generalized functions with respect to a parameter”, Funkts. Anal. Prilozhen., 6:4 (1972), 26–40 | MR

[10] W. Bertram, The Geometry of Jordan and Lie Structures, Lect. Notes Math., 1754, Springer-Verlag, Berlin, 2000 | MR | Zbl

[11] T. Branson, G. Olafsson, B. Ørsted, “Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup”, J. Funct. Anal., 135:1 (1996), 163–205 | DOI | MR | Zbl

[12] P. Delorme, “Formule de Plancherel pour les espaces symétriques réductifs”, Ann. Math. (2), 147:2 (1998), 417–452 | DOI | MR | Zbl

[13] G. van Dijk, S. C. Hille, “Canonical representations related to hyperbolic spaces”, J. Funct. Anal., 147 (1997), 109–139 | DOI | MR | Zbl

[14] G. van Dijk, V. F. Molchanov, “The Berezin form for rank one para-Hermitian symmetric spaces”, J. Math. Pure. Appl., 78 (1999), 99–119 | DOI | MR | Zbl

[15] G. Dijk, M. Pevzner, “Berezin kernels of tube domains”, J. Funct. Anal., 181:2 (2001), 189–208 | DOI | MR | Zbl

[16] A. Dvorsky, S. Sahi, “Explicit Hilbert spaces for certain unipotent representations, II”, Invent. Math., 138:1 (1999), 203–224 | DOI | MR | Zbl

[17] A. Dvorsky, S. Sahi, “Explicit Hilbert spaces for certain unipotent representations, III”, J. Funct. Anal., 201:2 (2003), 430–456 | DOI | MR | Zbl

[18] J. Faraut, A. Koranyi, Analysis on Symmetric Cones, The Clarendon Press, Oxford, 1994 | MR | Zbl

[19] M. Flensted-Jensen, “Discrete series for semisimple symmetric spaces”, Ann. Math. (2), 111:2 (1980), 253–311 | DOI | MR | Zbl

[20] I. M. Gelfand, M. I. Graev, A. M. Vershik, “Models of representations of current groups”, Representations of Lie Ggroups and Lie Algebras (Budapest, 1971), Akad. Kiado, Budapest, 1985, 121–179 | MR

[21] I. M. Gelfand, M. A. Naimark, “Unitary representations of the Lorentz group”, Izv. Akad. Nauk SSSR., Ser. Mat., 11 (1947), 411–504 | MR

[22] I. M. Gelfand, M. A. Naimark, Unitary Representations of the Classical Groups, Trudy Mat. Inst. Steklov., 36, Izdat. Akad. Nauk SSSR, Moscow–Leningrad, 1950 | MR | Zbl

[23] S. G. Gindikin, F. I. Karpelevich, “On an integral connected with symmetric Riemannian space of nonpositive curvature”, Izv. Akad. Nauk SSSR, Ser. Mat., 30 (1966), 1147–1156 | MR | Zbl

[24] W. Groenevelt, Tensor product representations and special functions, Ph. D. Thesis, Deift University, 2004

[25] P. Harinck, “Plancherel formula pour $\mathrm{GL}(n,\mathbb{C})/\mathrm{U}(p,q)$”, J. Reine Angew. Math., 428 (1992), 45–95 | DOI | MR | Zbl

[26] P. Harinck, “Fonctions orbitales sur $G_\mathbb{C}/G_\mathbb{R}$. Formule d'inversion des intégrales orbitales et formule de Plancherel”, J. Funct. Anal., 153 (1998), 52–107 | DOI | MR | Zbl

[27] Harish-Chandra, “Harmonic analysis on real semisimple groups. III: The Maas–Selberg relations and the Plancherel formula”, Ann. Math., 104 (1976), 117–201 | DOI | MR | Zbl

[28] G. Heckman, H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Academic Press, San Diego, 1994 | MR | Zbl

[29] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York–London, 1962 | MR | Zbl

[30] S. Helgason, Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, Orlando, 1984 | MR | Zbl

[31] S. C. Hille, Canonical representations, Ph. D. Thesis, Leiden University, 1999

[32] Hua Loo-Keng, Harmonic Analysis of Functions of Several Complex Variables in Classical Domains, Beijing, 1958 | MR

[33] H. P. Jakobsen, M. Vergne, “Restrictions and expansions of holomorphic representations”, J. Funct. Anal., 34:1 (1979), 29–53 | DOI | MR | Zbl

[34] K. W. J. Kadell, “The Selberg–Jack symmetric functions”, Adv. Math., 130:1 (1997), 33–102 | DOI | MR | Zbl

[35] A. A. Kirillov, Elements of Representation Theory, Nauka, Moscow, 1972 | MR

[36] A. W. Knapp, E. M. Stein, “Intertwining operators for semisimple groups”, Ann. Math. (2), 93 (1971), 489–578 | DOI | MR | Zbl

[37] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition. With contributions by A. Zelevinsky, Oxford Univ. Press, New York, 1995 | MR | Zbl

[38] B. O. Makarevich, “Open symmetric orbits of reductive groups in symmetric $R$-spaces”, Mat. Sb., 91(133) (1973), 390–401, 472 | MR | Zbl

[39] V. F. Molchanov, “Decomposition of the tensor square of a representation of the complementary series of a group”, Funkts. Anal. Prilozhen, 9:4 (1975), 79–80 | MR | Zbl

[40] V. F. Molchanov, “Tensor products of unitary representations of the three-dimensional Lorentz group”, Izv. Akad. Nauk SSSR, Ser. Mat., 43:4 (1979), 860–891 | MR | Zbl

[41] V. F. Molchanov, “Plancherel's formula for the pseudo-Riemannian space SL $(3,R)/$GL $(2,R)$”, Sib. Mat. Zh., 23:5 (1982), 142–151, 224 | MR | Zbl

[42] V. F. Molchanov, “The Plancherel formula for pseudo-Riemannian symmetric spaces of rank 1”, Dokl. Akad. Nauk SSSR, 290:3 (1986), 545–549 | MR

[43] V. G. Molchanov, “Harmonic analysis on homogeneous spaces”, Representation Theory and Noncommutative Harmonic Analysis, II, Encyclopaedia Math. Sci., 59, Springer, Berlin, 1995, 1–135 | MR

[44] N. Mukunda, “Unitary representations of the Lorentz groups: Reduction of the supplementary series under a noncompact subgroup”, J. Math. Phys., 9 (1968), 417–431 | DOI | MR | Zbl

[45] M. Muro, “Singular invariant hyperfunctions on the space of complex and quaternion Hermitian matrices”, J. Math. Soc. Japan, 53:3 (2001), 589–602 | DOI | MR | Zbl

[46] T. Nagano, “Transformation groups on compact symmetric spaces”, Trans. Amer. Math. Soc., 118 (1965), 428–453 | DOI | MR | Zbl

[47] M. A. Naimark, “Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations. I: The case of a tensor product of representations of the fundamental series”, Trudy Moskov. Mat. Obšč, 8, 1959, 121–153 | MR

[48] M. A. Naimark, “Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations. II: The case of a tensor product of representations of the fundamental and complementary series”, Trudy Moskov. Mat. Obšč, 9, 1960, 237–282 | MR

[49] M. A. Naimark, “Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations, III”, Trudy Moskov. Mat. Obšč, 10, 1961, 181–216 | MR

[50] Yu. A. Neretin, “Discrete occurrences of representations of the complementary series in tensor products of unitary representations”, Funkts. Anal. Prilozhen, 20:1 (1986), 79–80 | MR | Zbl

[51] Yu. A. Neretin, “Conformal geometry of symmetric spaces, and generalized linear fractional Krein–Smul'yan mappings”, Mat. Sb., 190:2 (1999), 93–122 | MR | Zbl

[52] Yu. A. Neretin, Categories of Symmetries and Infinite-Dimensional Groups, Oxford Univ. Press, New York, 1996 | MR | Zbl

[53] Yu. A. Neretin, “Pseudoriemannian symmetric spaces: one type realizations and open embeddings to Grassmannians”, Zap. Nauchn. Semin. POMI, 256, 1999, 145–167 ; arXiv: /math/9905014 | MR | Zbl

[54] Yu. A. Neretin, “Matrix analogs of $B$-function and Plancherel formula for Berezin kernel representations”, Mat. Sb., 191:5 (2000), 67–100 ; arXiv: /math.RT/9905045 | MR | Zbl

[55] Yu. A. Neretin, “On the separation of spectra in the analysis of Berezin kernels”, Funkts. Anal. Prilozhen., 34:3 (2000), 49–62 | MR | Zbl

[56] Yu. A. Neretin, “Plancherel formula for Berezin deformation of $L^2$ on Riemannian symmetric space”, J. Funct. Anal., 189 (2002), 336–408 ; arXiv: /math.RT/9911020 | DOI | MR | Zbl

[57] Yu. A. Neretin, “Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants”, Moscow Math. J., 1:2 (2001), 157–220 | MR | Zbl

[58] Yu. A. Neretin, “The index hypergeometric transform and an imitation of the analysis of Berezin kernels on hyperbolic spaces”, Mat. Sb., 192:3 (2001), 83–114 | MR | Zbl

[59] Yu. A. Neretin, “he action of an overalgebra in the Plancherel decomposition and shift operators in an imaginary direction”, Izv. Ross. Akad. Nauk Ser. Mat., 66:5 (2002), 171–182 | MR | Zbl

[60] Yu. A. Neretin, “The beta function of the Bruhat–Tits building and the deformation of the space $l\sp 2$ on the set of $p$-adic lattices”, Mat. Sb., 194:12 (2003), 31–62 | MR | Zbl

[61] Yu. A. Neretin, Some continuous analogs of expansion in Jacobi polynomials and vector valued hypergeometric orthogonal bases, arXiv: /math.CA/0309445 | MR

[62] Yu. A. Neretin, Notes on Sobolev spaces on compact classical groups and Stein–Sahi representations, arXiv: /math.RT/0411420

[63] Yu. A. Neretin, A class of kernels on pseudo-Riemannian symmetric spaces, in preparation

[64] Yu. A. Neretin, G. I. Olshanskii, “Boundary values of holomorphic functions, singular unitary representations of groups $O(p,q)$ and their limits as $q\to\infty$”, Zap. Nauchn. Semin. POMI, 223, 1995, 9–91 ; http://wwwth.itep.ru/~neretin | MR | Zbl

[65] G. I Olshansky, “Irreducible unitary representations of the groups $\mathrm U(p,q)$ that allow passage to a limit as $q\to\infty$”, Zap. Nauchn. Semin. LOMI, 172, 1989, 114–120 | MR

[66] T. Oshima, “A method of harmonic analysis on semisimple symmetric spaces”, Algebraic Analysis, II, Academic Press, Boston, 1988, 667–680 | MR

[67] T. Oshima, “A calculation of $c$-functions for semisimple symmetric spaces”, Lie Groups and Symmetric Spaces, Amer. Math. Soc. Transl., Ser. 2, 210, Amer. Math. Soc., Providence, RI, 2003, 307–330 | MR | Zbl

[68] T. Oshima, T. Matsuki, “A description of discrete series for semisimple symmetric spaces”, Adv. Stud. Pure Math., 4 (1984), 331–390 | MR | Zbl

[69] T. Oshima, J. Sekiguchi, “Eigenspaces of invariant differential operators on a semisimple symmetric space”, Inv. Math., 57 (1980), 1–81 | DOI | MR | Zbl

[70] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. More Special Functions, V. 3, Gordon and Breach Science Publishers, New York, 1990 | MR | Zbl

[71] I. I. Pyateskii-Shapiro, Automorphic Functions and the Geometry of Classical Domains, Fizmatlit, Moscow, 1961

[72] L. Pukanszky, “On the Kronecker products of irreducible unitary representations of the $2\times2$ real unimodular group”, Trans. Amer. Math. Soc., 100 (1961), 116–152 | DOI | MR | Zbl

[73] M. Rais, Distributions Homogènes sur des Espaces de Matrices, Bull. Soc. Math. France Mém., 30, 1972, Supplément | MR | Zbl

[74] F. Ricci, E. M. Stein, “Homogeneous distributions on spaces of Hermitian matrices”, J. Reine Angew. Math., 368 (1986), 142–164 | MR | Zbl

[75] S. Sahi, “A simple construction of Stein's complementary series representations”, Proc. Amer. Math. Soc., 108:1 (1990), 257–266 | DOI | MR | Zbl

[76] S. Sahi, “Unitary representations on the Shilov boundary of a symmetric tube domain”, Representation theory of groups and algebras, Contemp. Math., 145, American Mathematical Society, Providence, RI, 1993, 275–286 | MR | Zbl

[77] S. Sahi, “Jordan algebras and degenerate principal series”, J. Reine Angew. Math., 462 (1995), 1–18 | DOI | MR | Zbl

[78] S. Sahi, “Explicit Hilbert spaces for certain unipotent representations”, Inv. Math., 110:2 (1992), 409–418 | DOI | MR | Zbl

[79] S. Sahi, E. M. Stein, “Analysis in matrix space and Speh's representations”, Inv. Math., 101:2 (1990), 379–393 | DOI | MR | Zbl

[80] M. Sato, T. Shintani, “On zeta functions associated with prehomogeneous vector spaces”, Ann. Math. (2), 100 (1974), 131–170 | DOI | MR | Zbl

[81] E. M. Stein, “Analysis in matrix spaces and some new representations of $\mathrm{SL}\,(N,C)$”, Ann. Math. (2), 86 (1967), 461–490 | DOI | MR | Zbl

[82] A. Unterberger, H. Upmeier, “The Berezin transform and invariant differential operators”, Comm. Math. Phys., 164 (1994), 563–597 | DOI | MR | Zbl

[83] M. Vergne, H. Rossi, “Analytic continuation of the holomorphic discrete series of a semi-simple Lie group”, Acta Math., 136:1–2 (1976), 1–59 | DOI | MR | Zbl

[84] A. M. Vershik, I. M. Gelfand, M. I. Graev, “Representations of the group $\mathrm{SL}\,(2,R)$, where $R$ is a ring of functions”, Uspekhi Mat. Nauk, 28:5(173) (1973), 83–128 | MR

[85] N. Yu. Vilenkin, Special Functions and Theory of Group Representations, Nauka, Moscow, 1965 | MR

[86] N. Ya. Vilenkin, A. U. Klimyk, Representations of Lie Groups and Special Functions, Vol. 1–2, Kluwer, 1991 | MR | Zbl

[87] D. A. Vogan, “The unitary dual of $\mathrm{GL}(n)$ over an Archimedean field”, Inv. Math., 83:3 (1986), 449–505 | DOI | MR | Zbl

[88] G. Zhang, “Berezin transform on line bundles over bounded symmetric domains”, J. Lie Theory, 10:1 (2000), 111–126 | MR | Zbl

[89] G. Zhang, “Branching coefficients of holomorphic representations and Segal–Bargmann transform”, J. Funct. Anal., 195:2 (2002), 306–349 | DOI | MR | Zbl