The central result of this chapter is the celebrated Weyl character formula for irreducible representations of a connected semisimple algebraic group G. We give two (logically independent) proofs of this formula. The first is algebraic and uses the theorem of the highest weight, some arguments involving invariant regular functions, and the Casimir operator. The second is Weyl’s original analytic proof based on his integral formula for the compact real form of G.
KeywordsAlgebraic Group Casimir Operator Dominant Weight Weyl Function Character Formula
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