New graded methods in the homological algebra of semisimple algebraic groups

Abstract

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of $\Ext^\bullet_{G_1}(V,W)$ when $V,W$ belong to several important classes of rational $G$-modules, and $G_1$ denotes the first Frobenius kernel of $G$. For example, it is proved that, if $L,L^{\prime }$ are ($p$-regular) irreducible $G_1$-modules, then $\Ext^n_{G_1}(L,L')^{[-1]}$ has a good filtration with computable multiplicities. This and many other results depend on the entirely new technique of using methods of what we call forced gradings in the representation theory of $G$, as developed by the authors in recent papers, and extended here. In addition to providing proofs, these methods lead effectively to a new conceptual framework for the study of rational $G$-modules, and, in this context, to the introduction of a new class of graded finite dimensional algebras, which we call Q-Koszul algebras. These algebras are similar to Koszul algebras, but are quasi-hereditary, rather than semisimple, in grade 0.

Numerous typos and other corrections have been made

References in corpus (3)

• Schubert calculus and torsion explosion
• On $p$-filtrations of Weyl modules
• On extensions for Ree groups of type F_4

Cited by in corpus (3)

• Q-Koszul Algebras and Three Conjectures
• Stabilisation of the LHS spectral sequence for algebraic groups
• On modules arising from quantum groups at $p^r$-th roots of unity