123 research outputs found
Two geometric character formulas for reductive Lie groups
In this paper we prove two formulas for the characters of representations of
reductive groups. Both express the character of a representation in terms of
the same geometric data attached to it. When specialized to the case of a
compact Lie group, one of them reduces to Kirillov's character formula in the
compact case, and the other, to an application of the Atiyah-Bott fixed point
formula to the Borel-Weil realization of the representation
Distributions and Analytic Continuation of Dirichlet Series
This paper is second in a series of three papers; the first of which is
"Summation Formulas, from Poisson and Voronoi to the Present"
(math.NT/0304187), and the third of which is "Automorphic Distributions,
L-functions, and Voronoi Summation for GL(3)". The first paper is primarily an
expository paper, while the third proves a Voronoi-style summation formula for
the coefficients of a cusp form on GL(3,Z)\GL(3,R). This present paper contains
the distributional machinery used in the third paper for rigorously deriving
the summation formula, and also for the proof of the GL(3)xGL(1) converse
theorem given in the third paper. The primary concept studied is a notion of
the order of vanishing of a distribution along a closed submanifold.
Applications are given to the analytic continuation of Riemann's zeta function;
degree 1 and degree 2 L-functions; the converse theorem for GL(2); and a
characterization of the classical Mellin transform/inversion relations on
functions with specified singularities.Comment: 72 page
Pairings of automorphic distributions
We present a pairing of automorphic distributions that applies in situations
where a Lie group acts with an open orbit on a product of generalized flag
varieties. The pairing gives meaning to an integral of products of automorphic
distributions on these varieties. This generalizes classical integral
representations or "Rankin-Selberg integrals" of L-functions, and gives new
constructions and analytic continuations of automorphic L-functions.
Keywords: Automorphic forms, invariant pairings, automorphic distributions,
L-functions, analytic continuation, rapid decay.Comment: 19 pages, to appear in Mathematische Annale
Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)
This paper is third in a series of three, following "Summation Formulas, from
Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and
Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is
primarily an expository paper explaining the present one, whereas the second
contains some distributional machinery used here as well. These papers concern
the boundary distributions of automorphic forms, and how they can be applied to
study questions about cusp forms on semisimple Lie groups. The main result of
this paper is a Voronoi-style summation formula for the Fourier coefficients of
a cusp form on GL(3,Z)\GL(3,R). We also give a treatment of the standard
L-function on GL(3), focusing on the archimedean analysis as performed using
distributions. Finally a new proof is given of the GL(3)xGL(1) converse theorem
of Jacquet, Piatetski-Shapiro, and Shalika. This paper is also related to the
later papers math.NT/0402382 and math.NT/0404521.Comment: 66 pages, published versio
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