Dirac cohomology and $\Theta$-correspondence for complex dual pairs

Spyridon Afentoulidis-Almpanis; Gang Liu; Salah Mehdi

Dirac cohomology and $\Theta$-correspondence for complex dual pairs

Representation Theory 2023-07-27 v1

Authors: Spyridon Afentoulidis-Almpanis , Gang Liu , Salah Mehdi

Abstract

We study the behavior of Dirac cohomology under Howe's $\Theta$ -correspondence in the case of complex reductive dual pairs. More precisely, if $(G_1,G_2)$ is a complex reductive dual pair with $G_1$ and $G_2$ viewed as real groups, we describe those Harish-Chandra modules $\pi_1$ of $G_1$ with nonzero Dirac cohomology whose $\Theta$ -liftings $\Theta(\pi_1)$ still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of $\Theta(\pi_1)$ .

Keywords

dirac cohomology theta-correspondence complex reductive dual pairs harish-chandra modules howe correspondence

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@article{arxiv.2307.13742,
  title  = {Dirac cohomology and $\Theta$-correspondence for complex dual pairs},
  author = {Spyridon Afentoulidis-Almpanis and Gang Liu and Salah Mehdi},
  journal= {arXiv preprint arXiv:2307.13742},
  year   = {2023}
}

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19 pages

Dirac cohomology and $\Theta$-correspondence for complex dual pairs

Abstract

Keywords

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