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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.

22E46 Semisimple Lie groups and their representations
17B25 Exceptional (super)algebras
Full Text: DOI
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