The Borel-Weil theorem realizes a holomorphic irreducible representation of 
as the dual of a space of sections. Let 
. If 
is the highest weight of a holomorphic irreducible
representation 
of 
(i.e., 
is a dominant integral weight), then the 
-map 
defined by 
, where 
, is an isomorphism.
Thus, 
.
Borel-Weil Theorem
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References
Huang, J.-S. "The Borel-Weil Theorem." §8.7 in Lectures on Representation Theory. Singapore: World Scientific, pp. 105-107, 1999.Referenced on Wolfram|Alpha
Borel-Weil TheoremCite this as:
Weisstein, Eric W. "Borel-Weil Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Borel-WeilTheorem.html