Resumen de 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 kk of characteristic p>0p>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 Q¯¯¯¯lQ¯l -linear triangulated braided monoidal category DG(G)DG(G) of conjugation equivariant Q¯¯¯¯lQ¯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 DG(G)DG(G) into smaller and more manageable pieces corresponding to these idempotents in DG(G)DG(G) . Drinfeld has conjectured that the idempotent in DG(G)DG(G) obtained from an admissible pair is in fact a minimal idempotent and that any minimal idempotent in DG(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∘⊂GG∘⊂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).
