Actions of the Derived Group of a Maximal Unipotent Subgroup on G-varieties
نویسندگان
چکیده
The ground field k is algebraically closed and of characteristic zero. Let G be a semisimple simply-connected algebraic group over k and U a maximal unipotent subgroup of G. One of the fundamental invariant-theoretic facts, which goes back to Hadžiev [9], is that k[G/U ] is a finitely generated k-algebra and regarded as G-module it contains every finite-dimensional simple G-module exactly once. From this, one readily deduces that the algebra of U-invariants, k[G/U ] , is polynomial. More precisely, choose a maximal torus T ⊂ NormG(U). Let r be the rank of G, ̟1, . . . , ̟r the fundamental weights of T corresponding to U , and α1, . . . , αr the respective simple roots. Set X+ = ∑r i=1 N̟i, and let R(λ) denote the simple G-module with highest weight λ ∈ X+. Then
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