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Support Varieties of Non‐Restricted Modules over Lie Algebras of Reductive Groups
Author(s) -
Premet Alexander
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797004900
Subject(s) - algebraically closed field , mathematics , variety (cybernetics) , conjecture , component (thermodynamics) , reductive group , field (mathematics) , type (biology) , lie algebra , combinatorics , character (mathematics) , group (periodic table) , set (abstract data type) , pure mathematics , physics , group theory , computer science , geometry , ecology , statistics , quantum mechanics , biology , thermodynamics , programming language
Let G be a connected semisimple group over an algebraically closed field K of characteristic p >0, and = Lie ( G ). Fix a linear function χ∈g* and let Z g (χ) denote the stabilizer of χ in g. Set N p (g)={ x ∈g∣ x [ p ] =0}. Let C χ (g) denote the category of finite‐dimensional g‐modules with p ‐character χ. In [ 7 ], Friedlander and Parshall attached to each M ∈Ob(C χ (g)) a Zariski closed, conical subset V g ( M )⊂N p (g) called the support variety of M . Suppose that G is simply connected and p is not special for G , that is, p ≠2 if G has a component of type B n , C n or F 4 , and p ≠3 if G has a component of type G 2 . It is proved in this paper that, for any nonzero M ∈Ob(C χ (g)), the support variety V g ( M ) is contained in N p (g)∩Z g (χ). This allows one to simplify the proof of the Kac–Weisfeiler conjecture given in [ 18 ].