American Journal of Mathematics
Volume 127, Number 5, October 2005

CONTENTS

Renard, David A.
Trapa, Peter E.

Kazhdan-Lusztig algorithms for nonlinear groups and applications to Kazhdan-Patterson lifting [Access article in PDF] Abstract:

We establish an algorithm to compute characters of irreducible Harish-Chandra modules for a large class of nonalgebraic Lie groups. (Roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in Harish-Chandra's class.) We then apply this theory to study a particular group (the universal cover of the real general linear group), and discover a symmetry of the character computations encoded in a character multiplicity duality. Using this duality theory, we reinterpret a kind of representation-theoretic Shimura correspondence for the general linear group geometrically, and find that it is dual to an analogous lifting for indefinite unitary groups. It seems likely that this example is illustrative of a general framework for studying similar correspondences. (pages 911-971.)
Abstract in Tex

In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. More precisely, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields, by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. Then we define non-abelian L-functions for curves over finite fields using integrations of Eisenstein series associated to L^{2}-automorphic forms over certain generalized moduli spaces. (pages 973-1017.)
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Choi, Suhyoung.
Goldman, William M.

The deformation spaces of convex RP^{2} -structures on 2-orbifolds [Access article in PDF] Abstract:

We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristics is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture notes on hyperbolic 2-orbifolds, the previous work of Goldman on convex real projective structures on surfaces, and some classical geometry. (pages 1019-1102.)
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Martel, Yvan.

Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations [Access article in PDF] Abstract:

We consider the generalized Korteweg-de Vries equations

u_{t} + (u_{xx} + u^{p})_{x} = 0, t,x ∈ ,

in the subcritical and critical cases p=2,3,4 or 5. Let R_{j} (t,x) = Qc_{j} (x - c_{j}t - x_{j}), where j ∈ {1,...,N}, be N soliton solutions of this equation, with corresponding speeds 0 < c_{1} < c_{2} < ∙ ∙ ∙ < c_{N}. In this paper, we construct a solution u(t) of the generalized Korteweg-de Vries equation such that

This solution behaves asymptotically as t → + as the sum of N solitons without loss of mass by dispersion. This is an exceptional behavior, indeed, being given the parameters {c_{j}} _{1≤j≤N}, {x_{j}} _{1≤j≤N}, we prove uniqueness of such a solution.

In the integrable cases p = 2 and 3, such solutions are explicitly known and their properties were extensively studied in the literature (they are called N-soliton solutions). Therefore, the existence result is new only for the nonintegrable cases. The uniqueness result is new for all cases. (pages 1103-1140.)
Abstract in Tex

In this work, we study the distribution of nontrivial zeros of the derivatives of Selberg zeta functions on cocompact hyperbolic surfaces, and obtain an asymptotic formula for the zero density with bounded height, which is an analogue of the Weyl law. We then relate the distribution of the zeros to the multiplicities of Laplacian eigenvalues. (pages 1141-1151.)
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