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Let $G$ be a connected reductive algebraic group $G$ over an algebraicallyclosed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In thispaper, we study modular representations of the reductive Lie algebra $\ggg$with $p$-character $\chi$ of standard Levi-form associated with an index subset$I$ of simple roots. With aid of support variety theory we prove a theorem thata $U_\chi(\ggg)$-module is projective if and only if it is a strong "tilting"module, i.e. admitting both $\cz_Q$- and $\cz^{w^I}_Q$-filtrations (to seeTheorem \ref{THMFORINV}). Then by analogy of the arguments in \cite{AK} for$G_1T$-modules, we construct so-called Andersen-Kaneda filtrations associatedwith each projective $\ggg$-module of $p$-character $\chi$, and finally obtainsum formulas from those filtrations.

Filtrations in Modular Representations of Reductive Lie Algebras