Tensor products for the de Sitter group

Martin, Robert P.

doi:10.1090/S0002-9947-1984-0743745-1

Tensor products for the de Sitter group
HTML articles powered by AMS MathViewer

by Robert P. Martin PDF

Trans. Amer. Math. Soc. 284 (1984), 795-814 Request permission

Abstract:

The decomposition of the tensor product of a principal series representation with any other irreducible unitary representation of $G$ is determined for the simply connected double covering, $G = \operatorname {Spin} (4,1)$, of the DeSitter group.

References

Nguyen Huu Anh, Restriction of the principal series of $\textrm {SL}(n,\,\textbf {C})$ to some reductive subgroups, Pacific J. Math. 38 (1971), 295–314. MR 327980, DOI 10.2140/pjm.1971.38.295
Robert Boyer and Robert Martin, The group $C^*$-algebra of the de Sitter group, Proc. Amer. Math. Soc. 65 (1977), no. 1, 177–184. MR 473718, DOI 10.1090/S0002-9939-1977-0473718-X

Uniformly bounded representations of the Lorentz groups

Jacques Dixmier, Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9–41. MR 140614, DOI 10.24033/bsmf.1558

Harmonic analysis on the

dimensional Lorentz group and its application to conformal quantum field theory

Raymond C. Fabec, Localizable representations of the de Sitter group, J. Analyse Math. 35 (1979), 151–208. MR 555303, DOI 10.1007/BF02791065
Adam Kleppner and Ronald L. Lipsman, The Plancherel formula for group extensions. I, II, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459–516; ibid. (4) 6 (1973), 103–132. MR 342641, DOI 10.24033/asens.1235
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
R. A. Kunze and E. M. Stein, Uniformly bounded representations. III. Intertwining operators for the principal series on semisimple groups, Amer. J. Math. 89 (1967), 385–442. MR 231943, DOI 10.2307/2373128
George W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. MR 56611, DOI 10.2307/1969786
Robert Paul Martin, On the decomposition of tensor products of principal series representations for real-rank one semisimple groups, Trans. Amer. Math. Soc. 201 (1975), 177–211. MR 374341, DOI 10.1090/S0002-9947-1975-0374341-0
Robert P. Martin, Tensor products for $\textrm {SL}(2,\,k)$, Trans. Amer. Math. Soc. 239 (1978), 197–211. MR 487045, DOI 10.1090/S0002-9947-1978-0487045-3
Robert P. Martin, Tensor products of principal series for the De Sitter group, Trans. Amer. Math. Soc. 265 (1981), no. 1, 121–135. MR 607111, DOI 10.1090/S0002-9947-1981-0607111-9
Lajos Pukánszky, On the Kronecker products of irreducible representations of the $2\times 2$ real unimodular group. I, Trans. Amer. Math. Soc. 100 (1961), 116–152. MR 172962, DOI 10.1090/S0002-9947-1961-0172962-1
Joe Repka, Tensor products of unitary representations of $\textrm {SL}_{2}(\textbf {R})$, Amer. J. Math. 100 (1978), no. 4, 747–774. MR 509073, DOI 10.2307/2373909
Hugo Rossi and Michèle Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Functional Analysis 13 (1973), 324–389. MR 0407206, DOI 10.1016/0022-1236(73)90056-6
Reiji Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289–433 (French). MR 179296, DOI 10.24033/bsmf.1598
Ernest A. Thieleker, The unitary representations of the generalized Lorentz groups, Trans. Amer. Math. Soc. 199 (1974), 327–367. MR 379754, DOI 10.1090/S0002-9947-1974-0379754-8
N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by V. N. Singh. MR 0229863, DOI 10.1090/mmono/022

Harmonic analysis on semi-simple Lie groups

Floyd L. Williams, Tensor products of principal series representations, Lecture Notes in Mathematics, Vol. 358, Springer-Verlag, Berlin-New York, 1973. Reduction of tensor products of principal series. Representations of complex semisimple Lie groups. MR 0349903, DOI 10.1007/BFb0067484