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By Hung Yean, Loke and Gordan Savin


Abstract. Let G2(R) × Sp 6(R) and G2(R) × F4(R) be split dual pairs in split E7(R) and E8(R), respectively. It is known that the exceptional correspondences for these dual pairs are functorial on the level of infinitesimal characters. In this paper we show that these dual pair correspondences are functorial for the minimal K-types of principal series representations. 1. Split real groups of type En The Cartan decomposition for split real groups of type En can be described by Jordan algebras of rank 4, as it has been shown by Kostant and Brylinski in [4]. To this end, let J = Jn(Q) be a Jordan algebra of n × n-hermitian symmetric matrices over a composition algebra Q. To each Jordan algebra J one can attach a simple Lie algebra k = k(J) with a short Z-filtration k = k−1 ⊕ k ⊕ k1 such that k1 ∼ = J. The algebra k has n strongly orthogonal roots α1,..., αn corresponding to the diagonal entries of J. Let ψ = 1 2 (α1 +... + αn) Of special interest to us is the case n = 4, in which case 〈ψ, ψ 〉 = 2. Let p be the irreducible k-module of highest weight ψ. Then the exceptional lie algebras of type En have Cartan decomposition g = k ⊕ p where p ∼ = Vψ, as a k-module, and k = k(J4(Q)) where Q is a composition algebra over C of dimension 1, 2 and 4 for E6, E7 and E8, respectively. The minimal representation (the corresponding (g, K)-module) has K-types V = ⊕ ∞ i=0Viψ This (g, K)-module corresponds to a representation of the simply connected Chevalley group of type En. This representation is faithful except for E7 when the center µ2 acts trivially

Year: 2011
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