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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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The Dynkin diagram $R$-group
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by Dana Pascovici PDF
Represent. Theory 5 (2001), 1-16 Request permission

Abstract:

We define an abelian group from the Dynkin diagram of a split real linear Lie group with abelian Cartan subgroups, $G$, and show that the $R_{\delta , 0}$-groups defined by Knapp and Stein are subgroups of it. The proof relies on Vogan’s approach to the $R$-groups. The $R$-group of a Dynkin diagram is easily computed just by looking at the diagram, and so it gives, for instance, quick proofs of the fact that the principal series with zero infinitesimal character of the split groups $E_6$, $E_8$, $G_2$ or $F_4$ are irreducible. The Dynkin diagram subgroup also implicitly describes a small Levi subgroup, which we hope might play a role in computing regular functions on principal nilpotent orbits. We present in the end a conjecture and some evidence in this direction.
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Additional Information
  • Dana Pascovici
  • Affiliation: Department of Mathematics, Purdue University, MATH 602, West Lafayette, Indiana 47906
  • Email: pascovic@math.purdue.edu
  • Received by editor(s): May 31, 2000
  • Received by editor(s) in revised form: October 10, 2000
  • Published electronically: January 18, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Represent. Theory 5 (2001), 1-16
  • MSC (2000): Primary 22E46; Secondary 22E45
  • DOI: https://doi.org/10.1090/S1088-4165-01-00104-2
  • MathSciNet review: 1826426