SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS

Abstract

Let G be a connected semisimple group over an algebraically closed field K of characteristic p>0, and [gfr ]=Lie (G). Fix a linear function χ∈[gfr ]* and let [zfr ]_{[gfr ]}(χ) denote the stabilizer of χ in [gfr ]. Set [Nscr ]_p([gfr ]) ={x∈[gfr ][mid ]x^[p]=0}. Let [Cscr ]_χ([gfr ]) denote the category of finite-dimensional [gfr ]-modules with p-character χ. In [7], Friedlander and Parshall attached to each M∈Ob([Cscr ]_χ([gfr ])) a Zariski closed, conical subset [Vscr ]_{[gfr ]}(M)⊂[Nscr ]_p([gfr ]) 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 F₄, and p≠3 if G has a component of type G₂. It is proved in this paper that, for any nonzero M∈Ob([Cscr ]_χ([gfr ])), the support variety [Vscr ]_{[gfr ]}(M) is contained in [Nscr ]_p([gfr ])∩[zfr ]_{[gfr ]}(χ). This allows one to simplify the proof of the Kac–Weisfeiler conjecture given in [18].