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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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