Original scientific paper

Dirac operators and unitarizability of Harish-Chandra modules

Pavle Pandžić orcid.org/0000-0002-7405-4381 ; Department of Mathematics, University of Zagreb, Zagreb, Croatia

Abstract

Let $G$ be a simple noncompact Lie group. Let $K$ be a maximal compact subgroup of $G$, and let
$\frg=\frk\oplus\frp$ be the corresponding Cartan decomposition of the complexified Lie algebra $\frg$ of $G$.
We give a criterion for a $(\frg,K)$-module $M$ to be unitary in terms of the action of the Dirac
operator $D$ on $M\otimes S$, where $S$ is a spin module for the Clifford algebra $C(\frp)$. More precisely, we show that an arbitrary
Hermitian inner product on $M$ will be invariant if and only if $D$ is symmetric with respect to the corresponding
inner product on $M\otimes S$.

Keywords

reductive Lie group; unitary representation; Harish-Chandra module; Dirac operator

Hrčak ID:

53236

URI

https://hrcak.srce.hr/53236

Publication date:

10.6.2010.

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