Geodesic
Representations of pseudo-orthogonal groups associated with a cone
Sbornik. Mathematics, Tome 10 (1970) no. 3, pp. 333-347 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study representations of the group $SO_0(p,q)$, $p>1$, $q>1$, in the spaces $D_\chi$, $\chi=(\sigma,\varepsilon)$ ($\sigma$ is a complex number; $\varepsilon=0$ or 1), of $C^\infty$-functions $\varphi(x)$ on the cone $-x_1^2-\dots-x_p^2+x_{p+1}^2+\dots+x_{p+q}^2=0$, $x\ne0$, of homogeneous degree $\sigma$ and parity $\varepsilon$: $\varphi(tx)=|t|^\sigma{\operatorname{sign}}^\varepsilon t\cdot\varphi(x)$. We consider the structure of the invariant subspaces, irreducibility, the operators which commute with the group (the intertwining operators), invariant Hermitian forms, and unitarity. Figures: 1. Bibliography: 12 titles. 
@article{SM_1970_10_3_a3,
     author = {V. F. Molchanov},
     title = {Representations of pseudo-orthogonal groups associated with a~cone},
     journal = {Sbornik. Mathematics},
     pages = {333--347},
     year = {1970},
     volume = {10},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_3_a3/}
}
TY  - JOUR
AU  - V. F. Molchanov
TI  - Representations of pseudo-orthogonal groups associated with a cone
JO  - Sbornik. Mathematics
PY  - 1970
SP  - 333
EP  - 347
VL  - 10
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1970_10_3_a3/
LA  - en
ID  - SM_1970_10_3_a3
ER  - 
%0 Journal Article
%A V. F. Molchanov
%T Representations of pseudo-orthogonal groups associated with a cone
%J Sbornik. Mathematics
%D 1970
%P 333-347
%V 10
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1970_10_3_a3/
%G en
%F SM_1970_10_3_a3
V. F. Molchanov. Representations of pseudo-orthogonal groups associated with a cone. Sbornik. Mathematics, Tome 10 (1970) no. 3, pp. 333-347. http://geodesic.mathdoc.fr/item/SM_1970_10_3_a3/

[1] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, Moskva, 1958

[2] I. M. Gelfand, N. Ya. Vilenkin, Nekotorye primeneniya garmonicheskogo analiza. Osnaschennye gilbertovy prostranstva, Fizmatgiz, Moskva, 1961

[3] N. Ya. Vilenkin, “Spetsialnye funktsii, svyazannye s predstavleniyami klassa I grupp dvizhenii prostranstv postoyannoi krivizny”, Trudy Mosk. matem. ob-va, 12 (1963), 185–257 | MR | Zbl

[4] N. Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, Moskva, 1966 | MR

[5] S. Khelgason, Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, Moskva, 1964 | Zbl

[6] R. Takahashi, “Sur les representations unitaires des groupes de Lorentz generalises”, Bull. Soc. Math. France, 91:3 (1963), 289–433 | MR | Zbl

[7] R. Raczka, N. Limic, J. Niederle, “Discrete degenerate representations of noncompact rotation groups”, J. Math. Phys., 7:10 (1967), 1861–1876 | DOI | MR

[8] N. Limic, J. Niederle, R. Raczka, “Continuous degenerate representations of noncompact rotation groups. II”, J. Math. Phys., 7:11 (1967), 2026–2035 | DOI | MR

[9] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ i proizvedenii, Fizmatgiz, Moskva, 1962

[10] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii (gipergeometricheskaya funktsiya, funktsii Lezhandra), Nauka, Moskva, 1965

[11] D. P. Zhelobenko, “O beskonechno differentsiruemykh vektorakh v teorii predstavlenii”, Vestnik MGU, seriya matem., mekh., 1965, no. 1, 3–10

[12] V. F. Molchanov, “Analog formuly Plansherelya dlya giperboloidov”, DAN SSSR, 182:3 (1968)