Dirac Operators in the Algebraic Setting

Part of the Mathematics: Theory & Applications book series (MTA)


Dirac operators were introduced into representation theory by Parthasarathy [Par] as a tool to construct the discrete series representations. The final results, which applied to all discrete series, were obtained by Atiyah and Schmid in [AS]. In this chapter we study an algebraic version of Parthasarathy’s Dirac operator, due to Vogan. In particular, we explain the notion of Dirac cohomology of Harish-Chandra modules, and Vogan’s conjecture which predicts the infinitesimal character of modules with nonzero Dirac cohomology [V3]. We present a proof of this conjecture following [HP1].


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© Birkhäuser Boston 2006

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