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- W1964841036 abstract "Let $cA$ be a commutative unital Banach algebra, $g$ be a semisimple complex Lie algebra and $G(cA)$ be the 1-connected Banach--Lie group with Lie algebra $g otimes cA$. Then there is a natural concept of a parabolic subgroup $P(cA)$ of $G(cA)$ and we obtain generalizations $X(cA) := G(cA)/P(cA)$ of the generalized flag manifolds. In this note we provide an explicit description of all homogeneous holomorphic line bundles over $X(cA)$ with non-zero holomorphic sections. In particular, we show that all these line bundles are tensor products of pullbacks of line bundles over $X(C)$ by evaluation maps. For the special case where $cA$ is a $C^*$-algebra, our results lead to a complete classification of all irreducible involutive holomorphic representations of $G(cA)$ on Hilbert spaces." @default.
- W1964841036 created "2016-06-24" @default.
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- W1964841036 date "2009-09-09" @default.
- W1964841036 modified "2023-09-27" @default.
- W1964841036 title "Borel--Weil Theory for Groups over Commutative Banach Algebras" @default.
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