Entire vectors and holomorphic extension of representations
HTML articles powered by AMS MathViewer
- by Richard Penney PDF
- Trans. Amer. Math. Soc. 198 (1974), 107-121 Request permission
Abstract:
Let $G$ be a connected, simply connected real Lie group and let $U$ be a representation of $G$ in a complete, locally convex, topological vector space $\mathcal {J}$. If $G$ is solvable, it can be canonically embedded in its complexification ${G_c}$. A vector $v \in \mathcal {J}$ is said to be entire for $U$ if the map $g \to {U_g}v$ of $G$ into $\mathcal {J}$ is holomorphically extendible to ${G_c}$. The space of entire vectors is an invariant subspace of the space of analytic vectors. $U$ is said to be holomorphically extendible iff the space of entire vectors is dense. In this paper we consider the question of existence of holomorphic extensions We prove Theorem. A unitary representation $U$ is holomorphically extendible to ${G_C}$ iff $G$ modulo the kernel of $U$ is type $R$ in the sense of Auslander-Moore [1]. In the process of proving the above results, we develop several interesting characterizations of entire vectors which generalize work of Goodman for solvable Lie groups and we prove a conjecture of Nelson concerning the relationship between infinitesimal representations of Lie algebras and representations of the corresponding Lie groups.References
- Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
- Roe W. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc. 143 (1969), 55–76. MR 248285, DOI 10.1090/S0002-9947-1969-0248285-6
- Roe Goodman, Complex Fourier analysis on a nilpotent Lie group, Trans. Amer. Math. Soc. 160 (1971), 373–391. MR 417334, DOI 10.1090/S0002-9947-1971-0417334-3
- Roe W. Goodman, Differential operators of infinite order on a Lie group. I, J. Math. Mech. 19 (1969/1970), 879–894. MR 0255736
- A. Grothendieck, Espaces vectoriels topologiques, Universidade de São Paulo, Instituto de Matemática Pura e Aplicada, São Paulo, 1954 (French). MR 0077884
- G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR 0207883
- Robert T. Moore, Measurable, continuous and smooth vectors for semi-groups and group representations, Memoirs of the American Mathematical Society, No. 78, American Mathematical Society, Providence, R.I., 1968. MR 0229091
- Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615. MR 107176, DOI 10.2307/1970331
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 198 (1974), 107-121
- MSC: Primary 22E45
- DOI: https://doi.org/10.1090/S0002-9947-1974-99938-X
- MathSciNet review: 0364556