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New dual pair correspondences. (English) Zbl 0865.22009
The authors construct new dual pair correspondences for subgroups of the four exceptional (real) groups $$F_4$$, $$E_6$$, $$E_7$$, $$E_8$$ with real rank four. Each dual pair is of the form $$\widetilde G_2 \times H$$ where $$\widetilde G_2$$ is the split $$G_2$$ and $$H$$ is (the connected component of) the automorphism group of the Jordan algebra of Hermitian three-by-three matrices with entries in one of the alternative division algebras $$R,C,H$$ or $$O$$. In each instance, $$H$$ is compact, which is a boon to Howe’s general method of reductive dual pairs. Specifically, for $$F_4$$, $$H\cong SO(3)$$. For $$E_6$$, $$H\cong U(3)$$. For $$E_7$$, $$H\cong Sp(3)$$. For $$E_8$$, $$H\cong F^c_4$$, the compact form of $$F_4$$. In contrast to the classical case of a reductive dual pair $$(G,G')$$ inside a symplectic group, the host group here has no metaplectic representation, but does have a reasonably well-understood minimal (unitary, irreducible) representation, $$\widetilde V$$. [Cf. B. H. Gross and N. R. Wallach in: J.-L. Brylinski (ed.) et al., Lie theory and geometry, Boston: Birkhäuser, Prog. Math. 123, 289-304 (1994; Zbl 0839.22006) or R. Brylinski and B. Kostant, Proc. Natl. Acad. Sci. USA 91, 6026-6029 (1994; Zbl 0803.58023) and the same authors in: S. Glindikin (ed.) et al., Functional analysis on the eve of the 21st century, Boston: Birkhäuser, Progr. Math. 131, 13-63 (1995; Zbl 0851.22017).] The main results of the paper are decompositions of the minimal representation $$\widetilde V$$ as restricted to the subgroup $$\widetilde G_2\times H$$ for the four choices of the host group. Just as one would hope, there is a decomposition of the form $$\widetilde V|_{\widetilde G_2 \times H} \cong \bigoplus_E(\Theta(E) \otimes E)$$, where each $$E$$ is a finite-dimensional irreducible representation of $$H$$ and $$\Theta(E)$$ is an infinite-dimensional irreducible representation of $$\widetilde G_2$$ (which is necessarily unitary). The sum is taken over a discrete set of irreducible representations of $$H$$, which is given explicitly in each case. Moreover, in three of the four cases, the correspondence $$E\leftrightarrow \Theta(E)$$ is one-to-one. The exception comes for $$E_6$$, where $$\Theta(E) \cong \Theta(E^*)$$. The representations of $$\widetilde G_2$$ that occur are described according to D. A. Vogan’s classification [Invent. Math. 116, 677-791 (1994; Zbl 0808.22003)]. In most cases, they are discrete series representations, which are described by their lowest $$K$$-types. The exceptions are for $$E_6$$, where $$\Theta (E)$$ is sometimes a limit of the discrete series and for $$F_4$$, where $$\Theta (E)$$ is sometimes the subquotient of a generalized principal series described in D. A. Vogan [op. cit., Theorem 10.9(h)]. For the groups $$E_6$$, $$E_7$$ and $$E_8$$, the authors also describe their results in the language of Langlands correspondences. In particular, for $$E_6$$ and $$E_7$$ they describe the correspondences in terms of $$L$$-packets. For $$E_8$$, they formulate the correspondence in terms of Arthur’s packets. The authors also include a section on branching rules for restrictions of representations of the complex groups $$E_6$$ and $$SL(6,\mathbb{C})$$. The latter case was already known. However, the methods used here for both cases are natural variations on the methods of the rest of the paper.

##### MSC:
 22E46 Semisimple Lie groups and their representations 17B25 Exceptional (super)algebras
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