I am interested in the irreducible unitary representations of the orthogonal groups $O(p,q)$. By $O(p,q)$ I mean the real Lie groups which preserve the quadratic form of signature $(p,q)$ in $\mathbb{R}^n$, $n = p+q$ dimensions. Special cases of interest in physics are the conformal group O(4,2), the deSitter group O(4,1) and the anti-deSitter group O(3,2) in dimension 4 = 3+1 (i.e. Minkowski space). I am only interested in the non-compact groups, the compact cases being well-understood. (So I expect that the irreducible unitary representations are all infinite dimensional.) I am not exclusively interested in Lorentzian signature $n = (n-1) + 1$, nor am I exclusively interested in $n=4$. As a theoretical physicist, I am not familiar with the undoubtedly vast literature on representations of non-compact Lie groups, and I would appreciate a few pointers to the most relevant references; those that review the general setting, but especially which address these groups specifically.

The unitary dual of Spin(n,1) is known for all n (Hirai, 1962, see Math Reviews MR0696689). This gives the unitary dual of the identity component of SO(n,1) (which is a quotient of Spin(n,1)). The unitary dual of any group can be deduced readily from that of its identity component. Also SL(2,C)=Spin(3,1), and SL(2,R)=Spin(2,1).

The unitary dual of Sp(4,R) is known (Nzoukoudi, 1983, MR0736241), and Sp(4,R)=Spin(3,2). The unitary dual of SU(2,2) is known (Knapp/Speh, 1982, MR0645645), and SU(2,2)=Spin(4,2). These give the unitary duals of SO(3,2) and SO(4,2).

Besides this (and the compact groups) I think the entire unitary dual is not known for any O(p,q), although for any fixed group a large part of its unitary dual is known. You might look at some papers by Welleda Baldoni and Tony Knapp from the 1980s or so.

Note: much of the literature applies to groups of "Harish-Chandra's class". This includes all SO(p,q), but O(p,q) only if p+q is odd. So if p+q is even you have to do a little extra work to get from the unitary dual of SO(p,q) to that of O(p,q) (for each irreducible unitary representation of SO(p,q) you have to decide if its induction to O(p,q) has 1 or 2 irreducible summands).

As for www.liegroups.org, we hope to have the complete answer for any group, but that day has not yet arrived.

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