
Journal of Lie Theory 20 (2010), No. 4, 617641 Copyright Heldermann Verlag 2010 Dirac Induction for HarishChandra Modules Pavle Pandzic Dept. of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia pandzic@math.hr David Renard Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau, France renard@math.polytechnique.fr [Abstractpdf] \def\g{{\frak g}} Let $G$ be a connected real reductive Lie group with Cartan involution $\Theta$, such that $K=G^\Theta$ is a maximal compact subgroup of $G$, and such that $G$ and $K$ have equal rank. Let $\g$ be the complexified Lie algebra of $G$. We introduce new notions of Dirac cohomology and homology of a HarishChandra module $X$ for the pair $(\g,K)$. If $X$ is unitary or finitedimensional, then these new notions both coincide with the version of Dirac cohomology introduced by Vogan and further studied by HuangPand\v zi\'c and others. The new notions have certain advantages. Notably, if $X$ is irreducible and has nonzero Dirac cohomology (respectively homology), then $X$ is uniquely determined by its Dirac cohomology (respectively homology). Furthermore, one can define adjoint functors that we call Dirac induction functors. We study basic properties of these functors and we calculate their result explicitly in some examples. Keywords: Lie group, Lie algebra, representation, HarishChandra module, Dirac operator, Dirac cohomology, Dirac induction. MSC: 22E47, 22E46 [ Fulltextpdf (267 KB)] for subscribers only. 