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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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Classification of discrete series by minimal $K$-type
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by Rajagopalan Parthasarathy PDF
Represent. Theory 19 (2015), 167-185 Request permission

Abstract:

Following the proof by Hecht and Schmid of Blattner’s conjecture for $K$ multiplicities of representations belonging to the discrete series it turned out that some results which were earlier known with some hypothesis on the Harish-Chandra parameter of the discrete series representation could be extended removing those hypotheses. For example this was so for the geometric realization problem. Occasionally a few other results followed by first proving them for Harish-Chandra parameters which are sufficiently regular and then using Zuckerman translation functors, wall crossing methods, etc. Recently, Hongyu He raised the question (private communication) of whether the characterization of a discrete series representation by its lowest $K$-type, which was proved by this author and R. Hotta with some hypothesis on the Harish-Chandra parameter of the discrete series representations, can be extended to all discrete series representations excluding none, using a combination of these powerful techniques. In this article we will answer this question using Dirac operator methods and a result of Susana Salamanca-Riba.
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Additional Information
  • Rajagopalan Parthasarathy
  • Affiliation: Raja Ramanna Fellow Bharathiar University Coimbatore
  • Email: sarathy.math.tifr@gmail.com
  • Received by editor(s): July 29, 2014
  • Received by editor(s) in revised form: December 4, 2014, and August 14, 2015
  • Published electronically: October 7, 2015
  • Additional Notes: This research was supported by Raja Ramanna Fellowship from DAE
    The author thanks the referee for suggestions to improve the article by addressing the case of general groups of Harish-Chandra class. His comments on the initial proof of Theorem 1.1 in Section 3 have greatly helped in adding considerable clarity to the original submission.
  • © Copyright 2015 American Mathematical Society
  • Journal: Represent. Theory 19 (2015), 167-185
  • MSC (2010): Primary 22E46; Secondary 22D30
  • DOI: https://doi.org/10.1090/ert/467
  • MathSciNet review: 3405535