Double Affine Hecke Algebras and Congruence Groups

Double Affine Hecke Algebras and Congruence Groups
Title Double Affine Hecke Algebras and Congruence Groups PDF eBook
Author Bogdan Ion
Publisher American Mathematical Soc.
Pages 90
Release 2021-06-18
Genre Education
ISBN 1470443260

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The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by auto-morphisms of a finite index subgroup of the Artin group of type A2, which descends to a faithful outer action of a congruence subgroup of SL(2, Z)or PSL(2, Z). This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. It turns out that the structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, even by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products W 􀀁 Q∨ of the Weyl group W with the coroot lattice Q∨. They were defined topologically by van der Lek, and independently, algebraically, by Cherednik. We now describe our results for the adjoint case in greater detail. We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type A2,B2,G2 act by automorphisms on the adjoint DAAG/DAHA associated to affine Lie algebras of twist number r =1, 2, 3, respec-tively. This extends a fundamental result of Cherednik for r =1. We show further that the above rank two Artin group action descends to an outer action of the congruence subgroup Γ1(r). In particular, Γ1(r) acts naturally on the set of isomorphism classes of representations of an adjoint DAAG/DAHA of twist number r, giving rise to a projective representation of Γ1(r)on the spaceof aΓ1(r)-stable representation. We also provide a classification of the involutions of Kazhdan-Lusztig type that appear in the context of these actions.

Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions

Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions
Title Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions PDF eBook
Author Tom H. Koornwinder
Publisher Cambridge University Press
Pages 442
Release 2020-10-15
Genre Mathematics
ISBN 1108916554

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This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.

Eisenstein Series and Automorphic Representations

Eisenstein Series and Automorphic Representations
Title Eisenstein Series and Automorphic Representations PDF eBook
Author Philipp Fleig
Publisher Cambridge Studies in Advanced
Pages 587
Release 2018-07-05
Genre Mathematics
ISBN 1107189926

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Detailed exposition of automorphic representations and their relation to string theory, for mathematicians and theoretical physicists.

Intense Automorphisms of Finite Groups

Intense Automorphisms of Finite Groups
Title Intense Automorphisms of Finite Groups PDF eBook
Author Mima Stanojkovski
Publisher American Mathematical Society
Pages 117
Release 2021-12-09
Genre Mathematics
ISBN 1470450038

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Naturality and Mapping Class Groups in Heegard Floer Homology

Naturality and Mapping Class Groups in Heegard Floer Homology
Title Naturality and Mapping Class Groups in Heegard Floer Homology PDF eBook
Author András Juhász
Publisher American Mathematical Society
Pages 174
Release 2021-12-09
Genre Mathematics
ISBN 1470449722

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Non-Kissing Complexes and Tau-Tilting for Gentle Algebras

Non-Kissing Complexes and Tau-Tilting for Gentle Algebras
Title Non-Kissing Complexes and Tau-Tilting for Gentle Algebras PDF eBook
Author Yann Palu
Publisher American Mathematical Society
Pages 95
Release 2021-12-30
Genre Mathematics
ISBN 1470450046

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Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry

Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry
Title Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry PDF eBook
Author Stuart Margolis
Publisher American Mathematical Society
Pages 135
Release 2021-12-30
Genre Mathematics
ISBN 1470450429

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