Filtrations in Modular Representations of Reductive Lie Algebras

Abstract

Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $⋙$ 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 that a $U_{\chi }(⋙)$-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 see Theorem \ref{THMFORINV}). Then by analogy of the arguments in \cite{AK} for $G_{1}T$-modules, we construct so-called Andersen-Kaneda filtrations associated with each projective $⋙$-module of $p$-character $\chi$, and finally obtain sum formulas from those filtrations.

The current version of this paper will appear in Algebra Colloquium