scholarly journals One-dimensional $K$-types in finite dimensional representations of semisimple Lie groups: A generalization of Helgason's theorem.

1984 ◽  
Vol 54 ◽  
pp. 279 ◽  
Author(s):  
Henrik Schlichtkrull
Keyword(s):  
Lie Groups ◽  
Semisimple Lie Groups ◽  
One Dimensional ◽  
Finite Dimensional

Mumford-Tate Groups and Domains

2012 ◽  
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.

scholarly journals Harmonic morphisms from the classical compact semisimple Lie groups

2007 ◽  
Vol 33 (4) ◽  
pp. 343-356 ◽  
Author(s):  
Sigmundur Gudmundsson ◽  
Anna Sakovich
Keyword(s):  
Lie Groups ◽  
Harmonic Morphisms ◽  
Semisimple Lie Groups

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

scholarly journals Representations of complex semisimple Lie groups and Lie algebras

1966 ◽  
Vol 72 (3) ◽  
pp. 522-526 ◽  
Author(s):  
K. R. Parthasarathy ◽  
R. Ranga Rao ◽  
V. S. Varadarajan
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Semisimple Lie Groups ◽  
Complex Semisimple

scholarly journals A geometric construction of the discrete series for semisimple Lie groups

1979 ◽  
Vol 54 (2) ◽  
pp. 189-192 ◽  
Author(s):  
Michael Atiyah ◽  
Wilfried Schmid
Keyword(s):  
Lie Groups ◽  
Discrete Series ◽  
Geometric Construction ◽  
Semisimple Lie Groups
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