Geodesic
Special Representations of the Groups $U(\infty,1)$ and $O(\infty,1)$ and the Associated Representations of the Current Groups $U(\infty,1)^X$ and $O(\infty,1)^X$ in Quasi-Poisson Spaces
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 1, pp. 1-12 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A method for constructing representations of the current groups $O(n,1)^X$ and $U(n,1)^X$, $n\in\mathbb N$, developed in the previous papers by the authors is generalized to the case of infinite $n$. This leads to an interesting difference in construction (absent for finite $n$) between the cases of the orthogonal and unitary groups, which is due to the different character of special representations of the groups of coefficients.
Keywords: current group, integral model, Fock representation, canonical representation, special representation, infinite-dimensional Lebesgue measure. 
@article{FAA_2012_46_1_a0,
     author = {A. M. Vershik and M. I. Graev},
     title = {Special {Representations} of the {Groups} $U(\infty,1)$ and $O(\infty,1)$ and the {Associated} {Representations} of the {Current} {Groups} $U(\infty,1)^X$ and $O(\infty,1)^X$ in {Quasi-Poisson} {Spaces}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {1--12},
     year = {2012},
     volume = {46},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2012_46_1_a0/}
}
TY  - JOUR
AU  - A. M. Vershik
AU  - M. I. Graev
TI  - Special Representations of the Groups $U(\infty,1)$ and $O(\infty,1)$ and the Associated Representations of the Current Groups $U(\infty,1)^X$ and $O(\infty,1)^X$ in Quasi-Poisson Spaces
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2012
SP  - 1
EP  - 12
VL  - 46
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/FAA_2012_46_1_a0/
LA  - ru
ID  - FAA_2012_46_1_a0
ER  - 
%0 Journal Article
%A A. M. Vershik
%A M. I. Graev
%T Special Representations of the Groups $U(\infty,1)$ and $O(\infty,1)$ and the Associated Representations of the Current Groups $U(\infty,1)^X$ and $O(\infty,1)^X$ in Quasi-Poisson Spaces
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2012
%P 1-12
%V 46
%N 1
%U http://geodesic.mathdoc.fr/item/FAA_2012_46_1_a0/
%G ru
%F FAA_2012_46_1_a0
A. M. Vershik; M. I. Graev. Special Representations of the Groups $U(\infty,1)$ and $O(\infty,1)$ and the Associated Representations of the Current Groups $U(\infty,1)^X$ and $O(\infty,1)^X$ in Quasi-Poisson Spaces. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 1, pp. 1-12. http://geodesic.mathdoc.fr/item/FAA_2012_46_1_a0/

[1] A. M. Vershik, M. I. Graev, “Puassonova model fokovskogo prostranstva i predstavleniya grupp tokov”, Algebra i analiz, 23:3 (2011), 63–136 | MR

[2] A. M. Vershik, M. I. Graev, “Kommutativnaya model predstavleniya gruppy $O(n,1)^X $ i obobschennaya lebegova mera v prostranstve raspredelenii”, Funkts. analiz i ego pril., 39:2 (2005), 1–12 | DOI | MR | Zbl

[3] A. M. Vershik, M. I. Graev, “Struktura dopolnitelnykh serii i osobykh predstavlenii grupp $O(n,1)$ i $U(n,1)$”, UMN, 61:5 (2006), 3–88 | DOI | MR | Zbl

[4] I. M. Gelfand, M. I. Graev, A. M. Vershik, “Models of representations of current groups”, Representations of Lie groups and Lie algebras, Akad. Kiado, Budapest, 1985, 121–179 | MR

[5] R. S. Ismagilov, Representations of Infinite-Dimensional Groups, Transl. Math. Monographs, 152, Amer. Math. Soc., Providence, RI, 1996 | DOI | MR | Zbl

[6] Dzh. Kingman, Puassonovskie protsessy, MTsNMO, M., 2007

[7] Yu. A. Neretin, Kategorii simmetrii i beskonechnomernye gruppy, Editorial URSS, M., 1998