Conjugacy Classes in Algebraic Groups |
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abelian affine algebraic group affine algebraic variety affine variety assume automorphism B₁ Borel subgroup Bruhat lemma Cartan subgroup char G Choose Clearly closed subgroup codimension commute complete conjugacy classes conjugate Consider contains corollary cosets defined denote diagonalizable diagonalizable group dim G dimension elements of G endomorphism exists fibre finite dimensional fixed point follows immediately functions G acts G₁ G₂ given GL(V group G Hence dim Hence G invariant irreducible components irregular unipotent elements Jordan decomposition k-algebra k-algebra homomorphism Let f Let G Lie algebra line of type linear maximal torus morphism nilpotent open set orbit P₁ P₂(S polynomial projective variety Proof proves the proposition quotient reductive group regular elements Remark root system S₁ semisimple elements simple root simply connected solvable subgroup of G subspace subvariety surjective Symm U₁ unipotent elements unique V₁ V₂ zero ZG(t Zg(x