Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

The Poisson model of the Fock space and representations of current groups


Authors: A. M. Vershik and M. I. Graev
Translated by: N. V. Tsilevich
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 459-510
MSC (2010): Primary 81R10
DOI: https://doi.org/10.1090/S1061-0022-2012-01204-5
Published electronically: March 2, 2012
MathSciNet review: 2896165
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The quasi-Poisson measures are considered, i.e., the $ \sigma $-finite measures given by a density with respect to a Poisson measure. Representations of current groups are constructed in Hilbert spaces of functionals integrable with respect to a quasi-Poisson measure. For the groups $ O(n,1)$ and $ U(n,1)$, these models give new, more convenient, realizations of the representations in Fock spaces constructed in the previous papers by the authors. A crucial role in considerations is played by spaces of configurations and an analogy between quasi-Poisson and stable measures.


References [Enhancements On Off] (What's this?)

  • 1. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vols. I, II, Robert E. Krieger Publ. Co., Inc., Melbourne, FL, 1981. MR 0698779 (84h:33001a); MR 0698780 (84h:33001b)
  • 2. F. A. Berezin, Representations of a continuous direct product of universal coverings of the group of motions of the complex ball, Trudy Moskov. Mat. Obshch. 36 (1978), 275-293; English transl. in Trans. Moscow Math. Soc. 1979, no. 2. MR 0507574 (80e:22009)
  • 3. A. M. Vershik, Does a Lebesgue measure in an infinite-dimensional space exist? Trudy Mat. Inst. Steklova 259 (2007), 256-281; English transl. in Proc. Steklov Inst. Math. 259 (2007), no. 1. MR 2433687 (2011i:28014)
  • 4. 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, Uspekhi Mat. Nauk 28 (1973), no. 5, 83-128; English transl., Russian Math. Surveys 28 (1973), no. 5, 87-132. MR 0393337 (52:14147)
  • 5. -, Irreducible representations of the group $ G^X$ and cohomology, Funktsional. Anal. i Prilozhen. 8 (1974), no. 2, 67-69; English transl., Funct. Anal. Appl. 8 (1974), 151-153. MR 0348032 (50:530)
  • 6. -, Representations of the group of diffeomorphisms, Uspekhi Mat. Nauk 30 (1975), no. 6, 3-50; English transl. in Russian Math. Surveys 30 (1975), no. 6. MR 0399343 (53:3188)
  • 7. -, Commutative model of the representation of the group of flows $ SL (2,\, {\mathbb{R}\,})^X $ connected with a unipotent subgroup, Funktsional. Anal. i Prilozhen. 17 (1983), no. 2, 70-72; English transl., Funct. Anal. Appl. 17 (1983), no. 2, 137-139. MR 0705049 (85j:22037b)
  • 8. A. M. Vershik and M. I. Graev, A commutative model of a representation of the group $ O(n,1)^X $ and a generalized Lebesgue measure in a distribution space, Funktsional. Anal. i Prilozhen. 39 (2005), no. 2, 1-12; English transl., Funct. Anal. Appl. 39 (2005), no. 2, 81-90. MR 2161512 (2006h:22013)
  • 9. -, The structure of complementary series and special representations of the groups $ O(n,1)$ and $ U(n,1)$, Uspekhi Mat. Nauk 61 (2006), no. 5, 3-88; English transl., Russian Math. Surveys 61 (2006), no. 5, 799-884. MR 2328257 (2009d:22020)
  • 10. -, Integral models of representations of current groups, Funktsional. Anal. i Prilozhen. 42 (2008), no. 1, 22-32; English transl., Funct. Anal. Appl. 42 (2008), no. 1, 19-27. MR 2423975 (2009e:22029)
  • 11. -, Integral models of unitary representations of current groups with values in semidirect products, Funktsional. Anal. i Prilozhen. 42 (2008), no. 4, 37-49; English transl., Funct. Anal. Appl. 42 (2008), no. 4, 279-289. MR 2492425 (2010m:22009)
  • 12. -, Integral models of representations of current subalgebras of simple Lie groups, Uspekhi Mat. Nauk 64 (2009), no. 2, 5-72; English transl., Russian Math. Surveys 64 (2009), no. 2, 205-271. MR 2530917 (2011c:22035)
  • 13. A. M. Vershik and S. I. Karpushev, Cohomology of groups in unitary representations, neighborhood of the identity and conditionally positive definite functions, Mat. Sb. (N. S.) 119 (1982), no. 4, 521-533; English transl. in Math. USSR-Sb. 47 (1984), no 2. MR 0682497 (84d:22010)
  • 14. A. M. Vershik and N. V. Tsilevich, Fock factorizations and decompositions of the $ L^2$ spaces over general Lévy processes, Uspekhi Mat. Nauk 58 (2003), no. 3, 3-50; English transl., Russian Math. Surveys 58 (2003), no. 3, 427-472. MR 1998773 (2004e:46087)
  • 15. I. M. Gel'fand and M. I. Graev, Special representations of the group $ SU(n,1)$ and projective unitary representations of the current group $ SU(n,1)^X$, Dokl. Akad. Nauk 332 (1993), no. 3, 280-282; English transl., Russian Acad. Sci. Dokl. Math. 48 (1994), no. 2, 291-295. MR 1257017 (95b:22013)
  • 16. A. Guichardet, Cohomologie des groupes topologiques et des algèbres de Lie, Textes Math., vol. 2, CEDIC, Paris, 1980. MR 0644979 (83f:22004)
  • 17. D. A. Kazhdan, On the connection of the dual space of a group with the structure of its closed subgroups, Funktsional. Anal. i Prilozhen. 1 (1967), no. 1, 71-74. (Russian) MR 0209390 (35:288)
  • 18. J. F. C. Kingman, Poisson processes, Oxford Stud. in Probab., vol. 3, The Clarendon Press, Oxford Univ. Press, New York, 1993. MR 1207584 (94a:60052)
  • 19. A. A. Kirillov, Lectures on the orbit method, Novosibirsk, 2002; English transl., Grad. Stud. in Math., vol. 64, Amer. Math. Soc., Providence, RI, 2004. MR 2069175 (2005c:22001)
  • 20. V. F. Molchanov, Canonical representations and overgroups for hyperboloids. Funktsional. Anal. i Prilozhen. 39 (2005), no. 4, 48-61; English transl., Funct. Anal. Appl. 39 (2005), no. 4, 284-295. MR 2197513 (2007d:22018)
  • 21. Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups, Editorial URSS, Moscow, 1998; English transl., London Math. Soc. Monogr. (N. S.), vol. 16, Clarendon Press, Oxford Univ. Press, New York, 1996. MR 1418863 (98b:22003)
  • 22. A. M. Perelomov, Generalized coherent states and their applications, Nauka, Moscow, 1987; English transl., Springer-Verlag, Berlin, 1986. MR 0899736 (88i:22029); MR 0858831 (87m:22035)
  • 23. N. E. Hart, Geometric quantization in action, Math. Appl. (East European Ser.), vol. 8, D. Reidel Publ. Co., Dordrecht-Boston, MA, 1983. MR 0689710 (84f:58053)
  • 24. H. Araki, Factorizable representation of current algebra. Non commutative extension of the Lévy-Kinchin formula and cohomology of a solvable group with values in a Hilbert space, Publ. Res. Inst. Math. Sci. 5 (1969/1970), 361-422. MR 0263326 (41:7931)
  • 25. P. Delorme, 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull Soc. Math. France 105 (1977), 281-336. MR 0578893 (58:28272)
  • 26. -, Irreductibilité de certaines représentations de $ G^X$, J. Funct. Anal. 30 (1978), 36-47. MR 0513476 (80a:22022)
  • 27. I. M. Gel'fand, M. I. Graev, and A. M. Vershik, Representations of the group of smooth mappings of a manifold $ X$ into a compact Lie group, Compositio Math. 35 (1977), 299-334. MR 0578652 (58:28257)
  • 28. -, Representations of the group of functions taking values in a compact Lie group, Compositio Math. 42 (1981), 217-243. MR 0596877 (83g:22002)
  • 29. -, Models of representations of current groups, Representations of Lie Groups and Lie Algebras (Budapest, 1971), Akad. Kiadó, Budapest, 1985, pp. 121-179. MR 0829048 (88a:22032)
  • 30. M. I. Graev and A. M. Vershik, The basic representation of the current group $ O(n,1)^X $ in the $ L^2$ space over the generalized Lebesgue measure, Indag. Math. 16 (2005), 499-529. MR 2313636 (2008g:22028)
  • 31. A. Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes in Math., vol. 261, Springer-Verlag, Berlin-New York, 1972. MR 0493402 (58:12416)
  • 32. R. S. Ismagilov, Representations of infinite-dimensional groups, Transl. Math. Monogr., vol. 152, Amer. Math. Soc., Providence, RI, 1996. MR 1393939 (97g:22014)
  • 33. K. R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Lecture Notes in Math., vol. 272, Springer-Verlag, Berlin-New York, 1972. MR 0622034 (58:29849)
  • 34. N. Tsilevich, A. Vershik, and M. Yor, An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process, J. Funct. Anal. 185 (2001), 274-296. MR 1853759 (2002g:46071)
  • 35. A. Vershik, Invariant measures for the continual Cartan subgroup, J. Funct. Anal. 255 (2008), 2661-2682. MR 2473272 (2010f:37002)
  • 36. -, The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension, J. Fixed Point Theory Appl. 3 (2008), 317-329. MR 2434451 (2009g:58016)
  • 37. A. Vershik and B. Tsirelson, Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations, Rev. Math. Phys. 10 (1998), 81-145. MR 1606855 (99c:60085)
  • 38. J. A. Wolf, Unitary representations of maximal parabolic subgroups of the classical groups, Mem. Amer. Math. Soc. 8 (1976), no. 180, 193 pp. MR 0444847 (56:3194)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 81R10

Retrieve articles in all journals with MSC (2010): 81R10


Additional Information

A. M. Vershik
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: vershik@pdmi.ras.ru

M. I. Graev
Affiliation: Scientific-Research Institute for System Studies, Russian Academy of Sciences, Nakhimovskii Prospekt 36–1, Moscow 117218, Russia
Email: graev_36@mtu-net.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01204-5
Keywords: Current group, integral model, Fock representation, canonical representation, special representation, infinite-dimensional Lebesgue measure
Received by editor(s): June 29, 2010
Published electronically: March 2, 2012
Additional Notes: Supported by the RFBR grants 08-01-00379a, 09-01-12175-ofi-m (the first author) and 10-01-00041a (the second author).
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society