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Mathematical Communications, Vol.15 No.1 June 2010. -

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 $M\otimes S$.

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

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

Fuente: http://hrcak.srce.hr/


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