Overconvergent Algebraic Automorphic Forms

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions for forms of GLn due to Buzzard, Chenevier and Yamagami. This leads to some new phenomena, including the appearance of intermediate spaces of “semiclassical” automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p). The construction of these spaces relies on methods of locally analytic representation theory, combined with the theory of compact operators on Banach modules. Using a result of Owen Jones relating the representation at a locally algebraic weight to a Verma module, I also prove an analogue in this situation of Coleman’s theorem that forms of small slope are classical, implying that the eigenvarieties I construct contain a dense set of points corresponding to classical eigenforms.