Gamma-functions of representations and lifting

Details Gamma-functions of representations and lifting Gamma-functions of representations and lifting

1999-12-27

arxiv.org/abs/math/9912208v4

Mathematics

Algebraic Geometry

Main text by Alexander Braverman and David Kazhdan with an appendix by V. Vologodsky. Several minor mistakes corrected

Scientific paper

Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual group of F one can associate certain meromorphic function on the set of isomorphism classed of irreducible representations of G (which we call gamma-functions). In this paper we describe a general framework for an explicit construction of these gamma-functions. We make this idea precise in certain special cases. As a byproduct we give explicit conjectural formulas for the representation $l_E(\theta)$ of the group GL(n,F) associated to a character $\theta$ of the multiplicative group $E^*$ of a separable extension E of F of degree n. We also describe a conjectural analogue of the Poisson summation formula associated to a representation of the Langlands dual group, which implies the existence of the meromorphic continuation and functional equation for the corresponding automorphic L-functions.

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