Algebraic 𝒟-modules and representation theory of semisimple Lie groups

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it is an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The book gives the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. It also indicates that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

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Multiplicity One Theorems for Generalized Gelfand–Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

Representations of Lie Groups, Kyoto, Hiroshima, 1986 ◽

10.2969/aspm/01410031 ◽

2018 ◽

Cited By ~ 2

Author(s):

Hiroshi Yamashita

Keyword(s):

Lie Groups ◽

Discrete Series ◽

Semisimple Lie Groups ◽

Multiplicity One ◽

Whittaker Models

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Harmonic morphisms from the classical compact semisimple Lie groups

Annals of Global Analysis and Geometry ◽

10.1007/s10455-007-9090-8 ◽

2007 ◽

Vol 33 (4) ◽

pp. 343-356 ◽

Cited By ~ 6

Author(s):

Sigmundur Gudmundsson ◽

Anna Sakovich

Keyword(s):

Lie Groups ◽

Harmonic Morphisms ◽

Semisimple Lie Groups

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NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

International Journal of Modern Physics A ◽

10.1142/s0217751x14300014 ◽

2014 ◽

Vol 29 (03n04) ◽

pp. 1430001 ◽

Cited By ~ 10

Author(s):

V. K. DOBREV

Keyword(s):

Representation Theory ◽

Theoretical Perspective ◽

Explicit Construction ◽

Intertwining Operators ◽

Difference Operators ◽

Semisimple Lie Groups ◽

Boundary Operators ◽

Boundary Fields ◽

Theoretical Results ◽

Schrödinger Algebra

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

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