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 $$\mathtt {k}$$k of characteristic $$p>0$$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 (Sel Math, 2008. doi:10.1007/s00029-013-0133-7, arXiv:0810.0794v1), we define the notion of an admissible pair on G and the corresponding idempotent in the $$\overline{\mathbb {Q}}_l$$Q¯l-linear triangulated braided monoidal category $$\mathscr {D}_G(G)$$DG(G) of conjugation equivariant $$\overline{\mathbb {Q}}_l$$Q¯l-complexes (under convolution with compact support) and study their properties. In the spirit of Boyarchenko and Drinfeld (2008), we aim to break up the braided monoidal category $$\mathscr {D}_G(G)$$DG(G) into smaller and more manageable pieces corresponding to these idempotents in $$\mathscr {D}_G(G)$$DG(G). Drinfeld has conjectured that the idempotent in $$\mathscr {D}_G(G)$$DG(G) obtained from an admissible pair is in fact a minimal idempotent and that any minimal idempotent in $$\mathscr {D}_G(G)$$DG(G) can be obtained from some admissible pair on G. In this paper, we prove this conjecture in the case when the neutral connected component $$G^\circ \subset G$$G∘⊂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.” Moreover, as we will see in this paper, the study of character sheaves in the Heisenberg case may be considered, in a certain sense, as a twisted version of the theory of character sheaves on reductive groups as developed by Lusztig (Adv Math 56, 57, 59, 61, 1985, 1986).