# unitary irreps of O(p,q)

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.

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This link: mathoverflow.net/questions/37021/… doesn't directly answer your question, but may provide some background. My guess is that the unitary irreps. of $O(p,q)$ are not classified in general, but that a lot is known. A key point is that as long as at least one of $p$ or $q$ is even, then the maximal compact $O(p) \times O(q)$ has the same rank as $O(p,q)$, and hence $O(p,q)$ admits discrete series representations. Two possible references are: –  Emerton Dec 31 '11 at 5:39
Knapp's "rep'n theory of semisimple Lie groups: an overview based on examples" (or something to that effect), which is long but gives a lot of detailed background, and Vogan's book "Unitary rep's of reductive Lie groups", which is more of a research monograph. Neither will address $O(p,q)$ specifically, but will rather treat aspects of the general theory, so they won't be exactly what you want by any means. One other thing, just in case you don't know: "unitary dual" is the technical term for the set of unitary irreps. of a semisimple or reductive Lie group, and so googling for this –  Emerton Dec 31 '11 at 5:43
Also, you could try this site: liegroups.org –  Emerton Dec 31 '11 at 6:28
Dear Mark, This paper: liegroups.org/papers/computing.pdf explains the general procedure for computing the unitary dual. It's presumably simpler in your lowish rank cases than in general. The key points are to (a) identify the way that the tempered reps. decompose; (b) find the complementary series. It is the complementary series that become harder to classify as the rank goes up. Regards, –  Emerton Jan 1 '12 at 1:37
Emerton - thank you for pointing me to the Vogan paper. This has opened up a useful search direction. --Mark –  Mark Mueller Jan 1 '12 at 5:21

## 1 Answer

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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