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Irreducible Components of Fixed Point Subvarieties of Flag Varieties
Author(s) -
Douglass J. Matthew
Publication year - 1998
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19981890107
Subject(s) - mathematics , borel subgroup , maximal torus , algebraic group , reductive group , weyl group , flag (linear algebra) , algebraically closed field , nilpotent , type (biology) , combinatorics , (g,k) module , orbit (dynamics) , pure mathematics , group (periodic table) , lie algebra , algebraic number , algebra over a field , group theory , mathematical analysis , lie conformal algebra , fundamental representation , aerospace engineering , ecology , chemistry , engineering , biology , weight , organic chemistry , adjoint representation of a lie algebra
Suppose G is a connected reductive algebraic group, P is a parabolic subgroup of G, L is a Levi factor of P , and e is a regular nilpotent element in Lie L. We assume that the characteristic of the underlying field is good for G. Choose a maximal torus, T , and a Borel subgroup, B , of G, so that T⊆ B∩L , B ⊆ P and e ∈ Lie B. Let β be the variety of Borel subgroups of G and let e be the subset of consisting of Borel subgroups whose Lie algebras contain e. Finally, let W be the Weyl group of G with respect to T. For ω ∈ W let ω be the B‐orbit in containing ω B. We consider the intersections ω ∩ e . The main result is that if dim ω ∩ e = dim e , then ω ∩ e is an affine space. Thus, the irreducible components of e are indexed by Weyl group elements. It is also shown that if G is of type A , then this set of Weyl group elements is a right cell in W.