Publication Date:
2015-01-20
Description:
This paper considers Weyl modules for a semisimple, simply connected algebraic group $G$ over an algebraically closed field $k$ of positive characteristic $p\neq 2$ . The main result proves, if $p\geq 2h-2$ (where $h$ is the Coxeter number) and if the Lusztig character formula holds for all (irreducible modules with) $p$ -regular $p$ -restricted highest weights, then any Weyl module $\Delta (\lambda )$ has a $\Delta ^p$ -filtration, namely a filtration with sections of the form $\Delta ^p(\mu _0+p\mu _1):=L(\mu _0)\otimes \Delta (\mu _1)^{[1]}$ , where $\mu _0$ is $p$ -restricted and $\mu _1$ is arbitrary dominant. In case the highest weight $\lambda$ of the Weyl module $\Delta (\lambda )$ is $p$ -regular, the $p$ -filtration is compatible with the $G_1$ -radical series of the module. The problem of showing that Weyl modules have $\Delta ^p$ -filtrations was first proposed by Jantzen in 1980. The proof in this paper is based on new methods involving ‘forced gradings’ arising from orders associated to quantum enveloping algebras. A new Ext $^{1}$ -criterion is proved for $\Delta ^p$ -filtrations, but only in the context of such forced gradings. Finally, in subsequent work, these results have already had applications to the $G$ -module structure of Ext-groups for the restricted enveloping algebra of $G$ .
Print ISSN:
0024-6107
Electronic ISSN:
1469-7750
Topics:
Mathematics
