  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".
