Support Varieties of Non‐Restricted Modules over Lie Algebras of Reductive Groups

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 ].