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?



reference on representation theory of SO*(2n)
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, although this is somewhat out of date.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. 


