A note on Borel’s density theorem
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- by Harry Furstenberg PDF
- Proc. Amer. Math. Soc. 55 (1976), 209-212 Request permission
Abstract:
We prove the following theorem of Borel: If $G$ is a semisimple Lie group, $H$ a closed subgroup such that the quotient space $G/H$ carries finite measure, then for any finite-dimensional representation of $G$, each $H$-invariant subspace is $G$-invariant. The proof depends on a consideration of measures on projective spaces.References
- Armand Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179–188. MR 123639, DOI 10.2307/1970150
- Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR 146298, DOI 10.2307/1970220
- J. v. Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc. 36 (1934), no. 3, 445–492. MR 1501752, DOI 10.1090/S0002-9947-1934-1501752-3
- J. v. Neumann and E. P. Wigner, Minimally almost periodic groups, Ann. of Math. (2) 41 (1940), 746–750. MR 2891, DOI 10.2307/1968853
- M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 209-212
- MSC: Primary 22E40; Secondary 28A65
- DOI: https://doi.org/10.1090/S0002-9939-1976-0422497-X
- MathSciNet review: 0422497