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Minimal Idempotents on Solvable Groups

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".