A distinguished family of unitary representations for the exceptional groups of real rank \(= 4\). (English) Zbl 0839.22006

Gross, Benedict H.; Wallach, Nolan R.

Brylinski, Jean-Luc (ed.) et al., Lie theory and geometry: in honor of Bertram Kostant on the occasion of his 65th birthday. Invited papers, some originated at a symposium held at MIT, Cambridge, MA, USA in May 1993. Boston, MA: Birkhäuser. Prog. Math. 123, 289-304 (1994).

In this note, we will construct three small unitary representations for each of the four simply-connected exceptional Lie groups \(G\) of real rank = 4. We will describe the restrictions of these representations to a maximal compact subgroup \(K\) of \(G\), and will show they are multiplicity-free. The method of construction is by a continuation of the “quaternionic discrete series” for \(G\). This works in more generality, and we will treat it fully in another paper, so we have only sketched the proofs here.
The four exceptional groups considered here form a family, indexed by the real division algebras \(\mathbb{R}, \mathbb{C}, \mathbb{H}\), and \(\mathbb{O}\) of dimensions \(f = 1,2,4\), and 8. The disconnected “exceptional” group \(\widetilde {\text{Spin}} (4,4) \rtimes S_3\) of type \(D_4\) forms the 0-th member of this family, corresponding to the \(\mathbb{R}\)-algebra of dimension \(f = 0\). The structural results we obtain on the quaternionic discrete series and their unitary continuations are all uniform in the family, and extend to the \(D_4\) case. In particular, we obtain a new construction of the ladder representation of \(SO (4,4)^+\), which was studied by B. Kostant [Differential geometrical methods in theoretical physics, NATO ASI Ser., Ser. C 250, 65-108 (1988; Zbl 0663.22009)].
We end with some remarks on the restriction of the ladder representation of \(G = E_{8,4}\) to its symmetric subgroups \(H\), and give a new construction of the ladder representation of the split group \(G' = E_{8,8}\).
For the entire collection see [Zbl 0807.00014].

MSC:

22E15	General properties and structure of real Lie groups
22E47	Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)

Keywords:

quaternionic discrete series; unitary representations; exceptional Lie groups; ladder representation

Citations:

Zbl 0663.22009