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
Publication date:
10.6.2010.
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