Abstract
A method for constructing special nonunitary representations of semisimple Lie groups by using representations of Iwasawa subgroups is suggested. As a typical example, the group U(2, 2) is studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. M. Vershik and S. I. Karpushev, “Cohomology of groups in unitary representations, the neighborhood of the identity, and conditionally positive definite functions,” Mat. Sb., 119:4 (1982), 521–533; English transl.: Math. USSR-Sbornik, 47 (1984), 513–526.
Y. Shalom, “Rigidity of commensurators and irreducible lattices,” Invent. Math., 141:1 (2000), 1–54.
B. Bekka, P. de la Harpe, and A. Valette, Kazdan’s Property (T), Cambridge University Press, Cambridge, 2007.
U. Haagrup, “An example of nonnuclear C*-algebra, which has the metric approximation property,” Invent. Math., 50:3 (1978), 279–293.
P.-A. Cherix et al., Groups with the Haagerup Property: Gromov’s A-T-Menability, Birkhäuser Verlag, Basel, 2001.
B. Kostant, “On existence and irreducibility of certain series of representations,” Bull. Amer. Math Soc., 75 (1969), 627–642.
A. M. Vershik, I. M. Gelfand, and M. I. Graev, “Representations of the group SL(2,R), where R is a ring of functions,” Uspekhi Mat. Nauk, 28:5(173) (1973), 83–128; English transl.: Russian Math. Surveys, 28: 5 (1973), 87–132.
A. M. Vershik, I. M. Gelfand, and M. I. Graev, “Irreducible representations of the group G X and cohomologies,” Funkts. Anal. Prilozhen., 8:2 (1974), 67–69; English transl.: Functional Anal. Appl., 8:2 (1974), 151–153.
I. M. Gelfand, M. I. Graev, and A. M. Vershik, “Models of representations of current groups,” in: Representations of Lie groups and Lie algebras (Budapest, 1971), Akad. Kiadó, Budapest, 1985, 121–179.
R. S. Ismagilov, Representations of infinite-dimensional groups, Transl. Math. Monographs, vol. 152, Amer. Math. Soc., Providence, RI, 1996.
A. Guichardet, “Sur la cohomologie des groupe topologiques II,” Bull. Soc. Math. France, 96 (1972), 305–332.
P. Delorm, “1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubes. Produits tensoriels continus de représentations,” Bull. Soc. Math. France, 105 (1977), 281–336.
F. A. Berezin, “Representations of the continuous direct product of universal coverings of the group of motions of a complex ball.,” Trudy Mosk. Matem. Obshch., 36 (1978), 275–293; English transl.: Trans. Mosc. Math. Soc., 36 (1979), 281–298.
H. Araki, “Factorisable representations of current algebra,” Publ. Res. Inst. Math. Sci., Kyoto Univ., Ser. A, 5 (1969/1970), 361–422.
K. Parthasarathi and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory, Springer-Verlag, Berlin-New York, 1972.
A. M. Vershik and M. I. Graev, “The basic representation of the current group O(n, 1)X in the L 2 space over the generalized Lebesgue measure,” Indag. Math., 16:3/4 (2005), 499–529.
A. M. Vershik and M. I. Graev, “Structure of the complementary series and special representations of the groups O(n, 1) and U(n, 1),” Uspekhi Matem. Nauk, 61:5 (2006), 3–88; English transl.: Russian Math. Surv., 61:5 (2006), 799–884.
A. M. Vershik and M. I. Graev, “Integral models of representations of current groups,” Funkts. Anal. Prilozhen., 42:1 (2008), 22–32; English transl.: Functional Anal. Appl., 42:1 (2008), 19–27.
A. M. Vershik and M. I. Graev, “Integral models of representations of the current groups of simple Lie groups,” Uspekhi Mat. Nauk, 64:2(386) (2009), 5–72; English transl.: Russian Math. Surveys, 64: 2 (2009), 205–271.
A. M. Vershik and M. I. Graev, “The Poisson model of the Fock space and representations of current groups,” Algebra i Analiz, 23:3 (2011), 63–136; English transl.: St. Petersburg Math. J., 23:3 (2012), 459–510.
A. M. Vershik and M. I. Graev, “Special representations of the groups U(∞, 1) and O(∞, 1) and the associated representations of the current groups U(∞, 1)X and O(∞, 1)X in quasi-Poisson spaces,” Funkts. Anal. Prilozhen., 46:1 (2012), 1–12; English transl.: Functional Anal. Appl., 46:1 (2012), 1–10.
A. M. Vershik, “Does there exist a Lebesgue measure in the infinite-dimensional space?” Trudy Mat. Inst. Steklov., 259 (2007), 256–281; English transl.: Proc. Steklov Institute of Mathematics, 259 (2007), 248–272.
A. M. Vershik, “Invariant measures for the continual Cartan subgroup,” J. Funct. Anal., 255:9 (2008), 2661–2682.
A. M. Vershik, “The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension,” J. Fixed Point Theory Appl., 3:2 (2008), 317–329.
A. L. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, 1990.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 48, No. 3, pp. 1–13, 2014
Original Russian Text Copyright © by A. M. Vershik and M. I. Graev
To the 100th birthday of Israel Moiseevich Gelfand
A. M. Vershik acknowledges the support of RFBR grants 14-01-00373 and 13-01-12422-ofi_m. M. I. Graev acknowledges the support of RFBR grant 13-01-00190-a.
Rights and permissions
About this article
Cite this article
Vershik, A.M., Graev, M.I. Cohomology in nonunitary representations of semisimple Lie groups (the group U(2, 2)). Funct Anal Its Appl 48, 155–165 (2014). https://doi.org/10.1007/s10688-014-0057-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-014-0057-9