The Fundamental Lemma for the Shalika Subgroup of $GL(4)$
The Fundamental Lemma for the Shalika Subgroup of $GL(4)$

Author: Solomon Friedberg

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 167

ISBN-13: 0821805401

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The authors establish the fundamental lemma for a relative trace formula. The trace formula compares generic automorphic representations of [italic capitals]GS[italic]p(4) with automorphic representations of [italic capitals]GS(4) which are distinguished with respect to a character of the Shalika subgroup, the subgroup of matrices of 2 x 2 block form ([superscript italic]g [over] [subscript capital italic]X [and] 0 [over] [superscript italic]g). The fundamental lemma, giving the equality of the orbital integrals of the unit elements of the respective Hecke algebras, amounts to a comparison of certain exponential sums arising from these two different groups.

Conjugacy of $\mathrm {Alt}_5$ and $\mathrm {SL}(2, 5)$ Subgroups of $E_8(\mathbb C)$
Conjugacy of $\mathrm {Alt}_5$ and $\mathrm {SL}(2, 5)$ Subgroups of $E_8(\mathbb C)$

Author: Darrin D. Frey

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 177

ISBN-13: 0821807781

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Exceptional complex Lie groups have become increasingly important in various fields of mathematics and physics. As a result, there has been interest in expanding the representation theory of finite groups to include embeddings into the exceptional Lie groups. Cohen, Griess, Lisser, Ryba, Serre and Wales have pioneered this area, classifying the finite simple and quasisimple subgroups that embed in the exceptional complex Lie groups. This work contains the first major results concerning conjugacy classes of embeddings of finite subgroups of an exceptional complex Lie group in which there are large numbers of classes. The approach developed in this work is character theoretic, taking advantage of the classical subgroups of Eg(C). The machinery used is relatively elementary and has been used by the author and others to solve other conjugacy problems. The results presented here are very explicity. Each known conjugacy class if listed by its fusion pattern with an explicit character afforded by an embedding in that class.

Collected Works of Herve Jacquet
Collected Works of Herve Jacquet

Author: Hervé Jacquet

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 618

ISBN-13: 0821853562

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Herve Jacquet is one of the founders of the modern theory of automorphic representations and their associated $L$-functions. This volume represents a selection of his most influential papers not already available in book form. The volume contains papers on the $L$-function attached to a pair of representations of the general linear group. Thus, it completes Jacquet's papers on the subject (joint with Shalika and Piatetski-Shapiro) that can be found in the volume of selected works of Piatetski-Shapiro. In particular, two often quoted papers of Jacquet and Shalika on the classification of automorphic representations and a historically important paper of Gelbart and Jacquet on the functorial transfer from $GL(2)$ to $GL(3)$ are included. Another series of papers pertains to the relative trace formula introduced by Jacquet. This is a variant of the standard trace formula which is used to study the period integrals of automorphic forms. Nearly complete results are obtained for the period of an automorphic form over a unitary group.

Compact Connected Lie Transformation Groups on Spheres with Low Cohomogeneity. II
Compact Connected Lie Transformation Groups on Spheres with Low Cohomogeneity. II

Author: Eldar Straume

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 90

ISBN-13: 0821804839

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The cohomogeneity of a transformation group ([italic capitals]G, X) is, by definition, the dimension of its orbit space, [italic]c = dim [italic capitals]X, G. We are concerned with the classification of differentiable compact connected Lie transformation groups on (homology) spheres, with [italic]c [less than or equal to symbol] 2, and the main results are summarized in five theorems, A, B, C, D, and E in part I. This paper is part II of the project, and addresses theorems D and E. D examines the orthogonal model from theorem A and orbit structures, while theorem E addresses the existence of "exotic" [italic capital]G-spheres.

Abelian Galois Cohomology of Reductive Groups
Abelian Galois Cohomology of Reductive Groups

Author: Mikhail Borovoi

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 65

ISBN-13: 0821806505

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In this volume, a new function H 2/ab (K, G) of abelian Galois cohomology is introduced from the category of connected reductive groups G over a field K of characteristic 0 to the category of abelian groups. The abelian Galois cohomology and the abelianization map ab1: H1 (K, G) -- H 2/ab (K, G) are used to give a functorial, almost explicit description of the usual Galois cohomology set H1 (K, G) when K is a number field

Cyclic Phenomena for Composition Operators
Cyclic Phenomena for Composition Operators

Author: Paul Bourdon

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 122

ISBN-13: 0821806300

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We undertake a systematic study of cyclic phenomena for composition operators. Our work shows that composition operators exhibit strikingly diverse types of cyclic behavior, and it connects this behavior with classical problems involving complex polynomial approximation and analytic functional equations.

A Continuum Limit of the Toda Lattice
A Continuum Limit of the Toda Lattice

Author: Percy Deift

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 233

ISBN-13: 0821806912

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In this book, the authors describe a continuum limit of the Toda ODE system, obtained by taking as initial data for the finite lattice successively finer discretizations of two smooth functions. Using the integrability of the finite Toda lattice, the authors adapt the method introduced by Lax and Levermore for the study of the small dispersion limit of the Korteweg de Vries equations to the case of the Toda lattice. A general class of initial data is considered which permits, in particular, the formation of shocks. A feature of the analysis in this book is an extensive use of techniques from the theory of Riemann-Hilbert problems.

CR-Geometry and Deformations of Isolated Singularities
CR-Geometry and Deformations of Isolated Singularities

Author: Ragnar-Olaf Buchweitz

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 111

ISBN-13: 082180541X

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In this power we show how to compute the parameter space [italic capital]X for the versal deformation of an isolated singularity ([italic capital]V, 0) under the assumptions [italic]dim [italic capital]V [greater than or equal to symbol] 4, depth {0} [italic capital]V [greater than or equal to symbol] 3, from the CR-structure on a link [italic capital]M of the singularity. We do this by showing that the space [italic capital]X is isomorphic to the space (denoted here by [script capital]K[subscript italic capital]M) associated to [italic capital]M by Kuranishi in 1977. In fact we produce isomorphisms of the associated complete local rings by producing quasi-isomorphisms of the controlling differential graded Lie algebras for the corresponding formal deformation theories.

Relations Related to Betweenness: Their Structure and Automorphisms
Relations Related to Betweenness: Their Structure and Automorphisms

Author: Samson Adepoju Adeleke

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 141

ISBN-13: 0821806238

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This volume is about tree-like structures, namely semilinear ordering, general betweenness relations, C-relations and D-relations. It contains a systematic study of betweenness and introduces C- and D- relations to describe the behaviour of points at infinity (leaves or ends or directions of trees). The focus is on structure theorems and on automorphism groups, with applications to the theory of infinite permutation groups.

The Study of Minimax Inequalities and Applications to Economies and Variational Inequalities
The Study of Minimax Inequalities and Applications to Economies and Variational Inequalities

Author: George Xian-Zhi Yuan

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 157

ISBN-13: 0821807471

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This book provides a unified treatment for the study of the existence of equilibria of abstract economics in topological vector spaces from the viewpoint of Ky Fan minimax inequalities, which strongly depend on his infinite dimensional version of the classical Knaster, Kuratowski and Mazurkiewicz Lemma (KKM Lemma) in 1961. Studied are applications of general system versions of minimax inequalities and generalized quasi-variational inequalities, and random abstract economies and its applications to the system of random quasi-variational inequalities are given.