Weak containment and weak Frobenius reciprocity

Gootman, Elliot C.

doi:10.1090/S0002-9939-1976-0435286-7

Weak containment and weak Frobenius reciprocity
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Proc. Amer. Math. Soc. 54 (1976), 417-422 Request permission

Abstract:

We study weak containment relations between unitary representations of a group $G$ and a closed normal subgroup $K$ by exploiting a property of $G$-ergodic quasi-invariant measures on the primitive ideal space of $K$. By this means, we prove that every irreducible representation of $G$ is weakly contained in a representation induced from an irreducible representation of $K$ if the quotient group $G/K$ is amenable; and that the pair $(G,K)$ satisfies a weak Frobenius reciprocity property if and only if $G/K$ is amenable and $G$ acts minimally on the primitive ideal space of $K$. If $G/K$ is compact, $G$ acts minimally if and only if the primitive ideal space of $K$ is ${T_1}$.

References

Lawrence Baggett, A weak containment theorem for groups with a quotient $R$-group, Trans. Amer. Math. Soc. 128 (1967), 277–290. MR 220869, DOI 10.1090/S0002-9947-1967-0220869-9
Robert J. Blattner, Group extension representations and the structure space, Pacific J. Math. 15 (1965), 1101–1113. MR 188346
Jacques Dixmier, Sur les $C^{\ast }$-algèbres, Bull. Soc. Math. France 88 (1960), 95–112 (French). MR 121674
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
Edward G. Effros and Frank Hahn, Locally compact transformation groups and $C^{\ast }$- algebras, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, R.I., 1967. MR 0227310
J. M. G. Fell, Weak containment and induced representations of groups, Canadian J. Math. 14 (1962), 237–268. MR 150241, DOI 10.4153/CJM-1962-016-6
J. M. G. Fell, Weak containment and induced representations of groups. II, Trans. Amer. Math. Soc. 110 (1964), 424–447. MR 159898, DOI 10.1090/S0002-9947-1964-0159898-X
James Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885–911. MR 146297
Elliot C. Gootman and Robert R. Kallman, The left regular representation of a $p$-adic algebraic group is type $\textrm {I}$, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 273–284. MR 535768
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
F. P. Greenleaf, Amenable actions of locally compact groups, J. Functional Analysis 4 (1969), 295–315. MR 0246999, DOI 10.1016/0022-1236(69)90016-0
Roger E. Howe, On representations of discrete, finitely generated, torsion-free, nilpotent groups, Pacific J. Math. 73 (1977), no. 2, 281–305. MR 499004
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
L. Pukanszky, The primitive ideal space of solvable Lie groups, Invent. Math. 22 (1973), 75–118. MR 333078, DOI 10.1007/BF01392298