Finite-Dimensional Irreducible Representations of Semisimple Lie Groups

AbstractPolynomial identities satisfied by the generators of the Lie groups O(n) and U(n) are rederived. Using these identities the reduced matrix elements of the Lie groups U(n) and O(n) are evaluated as rational functions of the IR labels occurring in the canonical chainsThis method does not require an explicit realization of the Lie algebras and their representations using bosons. Finally, trace formulae encountered previously by several authors for finite dimensional irreducible representations are shown to hold on arbitrary representations admitting an infinitesimal character.

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Characters of irreducible representations of real semisimple Lie groups

Functional Analysis and Its Applications ◽

10.1007/bf01075540 ◽

1977 ◽

Vol 10 (3) ◽

pp. 246-248 ◽

Cited By ~ 1

Author(s):

A. I. Fomin

Keyword(s):

Lie Groups ◽

Irreducible Representations ◽

Semisimple Lie Groups

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Mumford-Tate Groups and Domains

10.23943/princeton/9780691154244.001.0001 ◽

2012 ◽

Cited By ~ 3

Author(s):

Mark Green ◽

Phillip A. Griffiths ◽

Matt Kerr

Keyword(s):

Representation Theory ◽

Lie Groups ◽

Complex Geometry ◽

Discrete Series ◽

Classical Case ◽

Complete Classification ◽

Semisimple Lie Groups ◽

Dimensional Representation ◽

Finite Dimensional Representation ◽

Finite Dimensional

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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