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I am trying to understand cohomology of $G := GL(N)$. For this I need to understand representations of $G(\mathbb{R})$ with nontrivial $(\mathfrak{g},K_\infty)$-cohomology, where $\mathfrak{g}$ is the Lie algebra of $G(\mathbb{R})$ and $K_\infty$ is $O(N,\mathbb{R})$. Is there a classification of such representations?

Thank you.

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    $\begingroup$ In case you're interested, here are the numbers of representations of GL(n,R) with cohomology for n=2...10: [3,2,6,4,12,8,24,16,48] (computed by the Atlas of Lie groups and representations software). $\endgroup$ Jul 28, 2017 at 15:10

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Yes. (That's assuming your definition of “representation” gives at least a $(\mathfrak g, K_\infty)$-module.)

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    $\begingroup$ The linked reference is to: Vogan, David A., Jr.; Zuckerman, Gregg J. Unitary representations with nonzero cohomology. Compositio Math. 53 (1984), no. 1, 51-90. $\endgroup$
    – YCor
    Jul 28, 2017 at 9:37
  • $\begingroup$ Right; I believe the AMS site will show title and numdam link even without subscription. The review explains that they describe all Harish-Chandra modules (unitary or not) with nontrivial $(\mathfrak g, K_\infty)$-cohomology, and the paper adds that the OP's $GL(N)$ case is already in Speh (1983). $\endgroup$ Jul 28, 2017 at 12:33
  • $\begingroup$ @Ziegler Thank you; the paper is useful. $\endgroup$
    – Vanya
    Jul 29, 2017 at 17:26

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