Minimal Idempotents on Solvable Groups

TL;DR

This paper develops a theory of character sheaves on affine algebraic groups over fields of positive characteristic, focusing on minimal idempotents in the derived category and proving conjectures for solvable groups.

Contribution

It proves Drinfeld's conjecture on minimal idempotents for solvable groups and reduces the general problem to the Heisenberg case.

Findings

Proved that admissible pairs produce minimal idempotents in solvable groups.

Showed the equivalence of conjectures for general groups to a weaker form.

Reduced the problem of character sheaves to the Heisenberg case.

Abstract

In this paper, we begin to develop a theory of character sheaves on an affine algebraic group $G$ defined over an algebraically closed field $k$ of characteristic $p>0$ using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let $l$ be a prime different from $p$. Following Boyarchenko and Drinfeld, we define the notion of an admissible pair on $G$ and the corresponding idempotent in the $\overline{\mathbb{Q}_l}$-linear triangulated braided monoidal category $\mathscr{D}_G(G)$ of conjugation equivariant $\overline{\mathbb{Q}_l}$-complexes (under convolution with compact support) and study their properties. We aim to break up the braided monoidal category $\mathscr{D}_G(G)$ into smaller and more manageable pieces corresponding to these idempotents in $\mathscr{D}_G(G)$. Drinfeld has conjectured that the idempotent in $\mathscr{D}_G(G)$ obtained from an admissible pair is in fact a minimal idempotent and that any minimal idempotent in $\mathscr{D}_G(G)$ can be obtained from some admissible pair on $G$. We will prove this conjecture in the case when the neutral connected component $G^\circ \subset G$ is a solvable group. For general groups, we prove that this conjecture is in fact equivalent to an a priori weaker conjecture. Using these results, we reduce the problem of defining character sheaves on general algebraic groups to a special case which we call the "Heisenberg case".