Geodesic
A Commutative Model of a Representation of the Group $O(n,1)^X$ and a Generalized Lebesgue Measure in the Space of Distributions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 1-12.

Voir la notice de l'article provenant de la source Math-Net.Ru

For an irreducible unitary representation of an $O(n,1)$ current group, we consider a commutative model obtained by diagonalization with respect to a maximal unipotent subgroup. This model leads to a new measure on the space of distributions. The measure is invariant with respect to an infinite-dimensional linear symmetry group.
Keywords: current group, basic representation, commutative model.
Mots-clés : orthogonal group 
@article{FAA_2005_39_2_a0,
     author = {A. M. Vershik and M. I. Graev},
     title = {A {Commutative} {Model} of a {Representation} of the {Group} $O(n,1)^X$ and a {Generalized} {Lebesgue} {Measure} in the {Space} of {Distributions}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {1--12},
     publisher = {mathdoc},
     volume = {39},
     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a0/}
}
TY  - JOUR
AU  - A. M. Vershik
AU  - M. I. Graev
TI  - A Commutative Model of a Representation of the Group $O(n,1)^X$ and a Generalized Lebesgue Measure in the Space of Distributions
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2005
SP  - 1
EP  - 12
VL  - 39
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a0/
LA  - ru
ID  - FAA_2005_39_2_a0
ER  - 
%0 Journal Article
%A A. M. Vershik
%A M. I. Graev
%T A Commutative Model of a Representation of the Group $O(n,1)^X$ and a Generalized Lebesgue Measure in the Space of Distributions
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2005
%P 1-12
%V 39
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a0/
%G ru
%F FAA_2005_39_2_a0
A. M. Vershik; M. I. Graev. A Commutative Model of a Representation of the Group $O(n,1)^X$ and a Generalized Lebesgue Measure in the Space of Distributions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 1-12. http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a0/

[1] Gelfand I. M., Graev M. I., Pyatetskii-Shapiro I. I., Teoriya predstavlenii i avtomorfnye funktsii, Nauka, M., 1966 | MR

[2] Vershik A. M., Gelfand I. M., Graev M. I., “Neprivodimye predstavleniya gruppy $G^X$ i kogomologii”, Funkts. analiz i ego pril., 8:2 (1974), 67–69 | MR | Zbl

[3] Vershik A. M., Gelfand I. M., Graev M. I., “Predstavleniya gruppy $SL(2,R)$, gde $R$ — koltso funktsii”, UMN, 28:5 (1973), 83–128 | MR

[4] Vershik A. M., Gelfand I. M., Graev M. I., “Kommutativnaya model osnovnogo predstavleniya gruppy $GL(2,\mathbb R)^X$ otnositelno unipotentnoi podgruppy”, Teoretiko-gruppovye metody v fizike, t. 2, Nauka, M., 1983, 472–487

[5] Vershik A. M., Gelfand I. M., Graev M. I., “Kommutativnaya model predstavleniya gruppy tokov $SL(2,\mathbb R)^X$, svyazannaya s unipotentnoi podgruppoi”, Funkts. analiz i ego pril., 17:2 (1983), 70–72 | MR | Zbl

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

[7] Gelfand I. M., Vilenkin N. Ya., Nekotorye primeneniya garmonicheskogo analiza. Obobschennye gilbertovy prostranstva, Fizmatgiz, M., 1951

[8] Tsilevich N., Vershik A., Yor M., “An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process”, J. Funct. Anal., 185:1 (2001), 274–296 | DOI | MR | Zbl

[9] Vershik A. M., Karpushev S. I., “Kogomologii grupp v unitarnykh predstavleniyakh, okrestnost edinitsy i uslovno polozhitelno opredelennye funktsii”, Matem. sb., 119:4 (1982), 521–533 | MR | Zbl

[10] Shalom Y., “Rigidity, unitary representation of semisimple groups and fundamental groups of manifold with rank one transformation group”, Ann. of Math. (2), 152 (2000), 113–182 | DOI | MR | Zbl

[11] Kuhn G., Vershik A., “Canonical semigroups of states and cocycles for the group of automorphisms of a homogeneous tree”, Algebr. Represent. Theory, 6:3 (2003), 333–352 | DOI | MR | Zbl

[12] Olshanskii G. I., “Sfericheskie funktsii i kharaktery na gruppe $U(\infty)$”, UMN, 37:2 (1982), 217–218 | MR

[13] Beitmen G., Erdeii L., Vysshie transtsendentnye funktsii, t. 2, Fizmatgiz, M., 1974

[14] Gikhman I. I., Skorokhod A. V., Teoriya sluchainykh protsessov, t. 2, Nauka, M., 1973 | MR