## Higher Nullcones and Commuting Varieties |

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### Contents

The Higher Nullcones | 8 |

The Case r 1 | 18 |

The Nullcone and the Commuting Variety | 25 |

3 other sections not shown

### Common terms and phrases

a r r action of G(k affine algebraic k-group affine k-group algebra f algebraic group defined augmentation ideal bijective Borel subgroup clear closed irreducible subsets closed subvariety coefficient commutative k-algebra commuting elements commuting variety comorphism comultiplication connected reductive group coordinate ring corollary Coxeter number defined over F dimension equal divided powers elements in Lie(G finite dimensional following string G denotes G(k)-equivariant GLn(k group G HIGHER NULLCONES Hopf-algebra map hy(G hy(Gr i,i+k induced injective irreducible component irreducible variety isomorphic Jordan canonical form k-Alg k-group homomorphisms k-vector Let G Lie algebra homomorphism Lie(U linear algebraic group maximal torus morphism of varieties normal subgroup Nr(U partition Proof Proposition pth-power representing algebra restricted Lie algebra restricted structure result is proved Section semisimple sp+t string of equalities surjective Theorem unipotent radical upper triangular matrices variety of commuting vector space