Discrete Series Representation - Constructions of The Discrete Series

Constructions of The Discrete Series

Harish-Chandra's original construction of the discrete series was not very explicit. Several authors later found more explicit realizations of the discrete series.

  • Narasimhan & Okamoto (1970) constructed most of the discrete series representations in the case when the symmetric space of G is hermitean.
  • Parthasarathy (1972) constructed many of the discrete series representations for arbitrary G.
  • Langlands (1966) conjectured, and Schmid (1976) proved, a geometric analogue of the Borel–Bott–Weil theorem, for the discrete series, using L2 cohomology instead of the coherent sheaf cohomology used in the compact case.
  • An application of the index theorem, Atiyah & Schmid (1977) constructed all the discrete series representations in spaces of harmonic spinors. Unlike most of the previous constructions of representations, the work of Atiyah and Schmid did not use Harish-Chandra's existence results in their proofs.
  • Discrete series representations can also be constructed by cohomological parabolic induction using Zuckerman functors.

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