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I'm interested in the representation theory of the non-compact real Lie group $\mathrm{SO}^*(2n)$, the subgroup of matrices $M\in\mathrm{SO}(2n,\mathbb{C})$ satisfying $$ M^\dagger\eta \,M=\eta,\qquad \eta = \begin{pmatrix} 0&\mathbb{I}_n \\ -\mathbb{I}_n &0 \end{pmatrix}. $$

Due to its peculiar name, it is extremely difficult to find any information about this group on search engines and library databases. Does anyone know of references about its representation theory?

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    $\begingroup$ are you interested in infinite dimensional unitary representations of this group? If you are only interested in finite dimensional reps, then the Cartan-Weyl theory describes them completely. If you are interested in admissible irreps, then again, the Langlands classification describes them completely. $\endgroup$
    – Venkataramana
    Commented Nov 15, 2015 at 0:17
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    $\begingroup$ @Venkataramana: I'm interested in unitary representations, hence infinite-dimensional. I know that the Laglands classification describes them, but I was hoping for concrete examples of $\mathrm{SO}^*(2n)$ representations. $\endgroup$
    – Giuseppe Sellaroli
    Commented Nov 15, 2015 at 0:35
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    $\begingroup$ These groups do have holomorphic discrete series repns, for example, constructible analogously to those for $Sp(n,\mathbb R)$ and $U(p,q)$. Certainly they have principal series repns, too. What sort of thing more were you wanting? $\endgroup$
    – paul garrett
    Commented Nov 15, 2015 at 2:47
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    $\begingroup$ @sellaroli: actually the Langlands clasification describes only yhe irreducible admissible representations. To pick out the unitary ones among them is (for a general $G$) an open problem, though many special cases are known. As Paul Garrett has already remarked, unitary principal series, discrete series, (and some complementary series) are examples. $\endgroup$
    – Venkataramana
    Commented Nov 15, 2015 at 6:19
  • $\begingroup$ @paulgarrett: I would like to know how the $\mathfrak{so}^*(2n)$ generators act on those representations, and how the basis are contructed. $\endgroup$
    – Giuseppe Sellaroli
    Commented Nov 15, 2015 at 23:49

1 Answer 1

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The Atlas of Lie Groups and Representations software gives copious explicit information about the representation theory of real reductive groups, including $SO^*(2n)$. The software is freely available from http://www.liegroups.org. (It is currently under development and not yet well documented.) There is a web-based version of the software at http://www.liegroups.org/web/atlasInput.html, although this is somewhat out of date.

For example $SO^*(8)$ has $44$ irreducible representations with infinitesimal character $\rho$. These come in three families corresponding to the three conjugacy classes of Cartan subgroups. This information is available from the web version of the software. Also, of these $28$ are unitary. The unitary ones include 8 discrete series (including one holomorphic, one anti-holomorphic) and the trivial representation.

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