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Classification of discrete series by minimal $ K$-type

Author: Rajagopalan Parthasarathy
Journal: Represent. Theory 19 (2015), 167-185
MSC (2010): Primary 22E46; Secondary 22D30
Published electronically: October 7, 2015
MathSciNet review: 3405535
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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

Keywords: Dirac operator inequality, Dirac cohomology, Harish-Chandra class group, Mackey's criterion, $A_{\mathfrak q, \lambda}$
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.
Article copyright: © Copyright 2015 American Mathematical Society

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