Beilinson–Bernstein Localization Over ℚ and Periods of Automorphic Forms

By Michael Harris. 2013

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The present paper is the first in a projected series of articles whose purpose is to draw conclusions from the comparison between an arithmetic conjecture of Deligne and an analytic conjecture of Ichino and Ikeda [D2,II]. Let F be a totally real field. The Ichino-Ikeda conjecture for unitary groups over F [II,H*] relates certain expressions in special values of automorphic L-functions with what we call GrossPrasad period integrals of automorphic representations. When these automorphic representations are of discrete series type at archimedean places, the L-functions are attached to motives over F , with coefficients in some number field E, and the special values are critical in Deligne’s sense [D2]. Deligne’s conjecture then implies that the normalized quotients of the special values by the motivically defined Deligne periods are algebraic numbers whose Gal(Q/Q)-conjugates are again normalized special values of L-functions of the same type. In order to compare the Deligne and Ichino-Ikeda conjectures, one needs to consider the Gross-Prasad period integrals attached to all real embeddings of F simultaneously. This leads inevitably to the problem of constructing models over number fields of discrete series (Lie(G), K)modules that are compatible under conjugation by Gal(Q/Q). Although the discrete series representations of a reductive Lie group G are initially defined in terms of the decomposition of the regular representation of G×G on the Hilbert space L2(G), many of their principal features can be studied by purely algebraic means. The Beilinson-Bernstein localization theory, which realizes these representations in terms of equivariant perverse sheaves or D-modules on the flag variety, provides an efficient way to encode these features in terms of algebraic geometry. In particular, when G is the group of real points of an algebraic group given with a rational structure over a number field, one sees easily that the (Lie(G), K)-module attached to a given discrete series representation π also can be defined over Q, provided the central character of π is algebraic and the maximal compact subgroup K of G is rational over some number field.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d272e07366aa5393fd31e703ecee43dc297e24ee