DOMAINS OF HOLOMORPHY FOR IRREDUCIBLE ADMISSIBLE UNIFORMLY BOUNDED BANACH REPRESENTATIONS OF SIMPLE LIE GROUPS
- 43 Downloads
In this note, we address a question raised by B. Krötz on the classification of G-invariant domains of holomorphy for irreducible admissible Banach representations of connected non-compact simple real linear Lie groups G. When G is not of Hermitian type, we give a complete description of such G-invariant domains for irreducible admissible uniformly bounded representations on reflexive Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine such G-invariant domains only when the representations considered are highest or lowest weight representations.
Unable to display preview. Download preview PDF.
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.Google Scholar
- A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. of Math. Stud., Vol. 94, Princeton, NJ, 1980.Google Scholar
- W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G, Can. J. Math. XLI (1989), no. 3, 385–438.Google Scholar
- M. Cowling, Applications of representation theory to harmonic analysis of Lie groups (and vice versa), in: Representation Theory and Complex Analysis, Lecture Notes in Mathematics, Vol. 1931, Springer, Berlin, 2008, pp. 1–50.Google Scholar
- A. W. Knapp, Lie Groups Beyond An Introduction, 2nd ed., Progress in Mathematics, Vol. 140, Birkhäuser Boston, Boston, MA, 2002.Google Scholar
- B. Kostant, \A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem, in: Noncommutative Harmonic Analysis, Progress in Mathematics, Vol. 220, Birkhäuser Boston, Boston, MA, 2004, pp. 291–353.Google Scholar
- B. Krötz, R. J. Stanton, Holomorphic extensions of representations. (II). Geometry and harmonic analysis, Geom. Funct. Anal. 15 (2005), 190–245.Google Scholar
- N. R. Wallach, Real Reductive Groups. II, Pure and Applied Mathematics, Vol. 132-II, Academic Press, Boston, MA, 1992.Google Scholar
- G. Warner, Harmonic Analysis On Semi-simple Lie Groups, Vol. I, Springer-Verlag, Berlin, 1972.Google Scholar