Duality for Spherical Representations in Exceptional Theta Correspondences
Duality for Spherical Representations in Exceptional Theta Correspondences
Work of Professor LOKE Hung Yean, NUS and Professor Gordan SAVIN, University of Utah  Let H be a real split Lie group of type En where n = 6,7,8. If n = 6, we set H = H0 o Z/2Z where H0 is the real points of a split, simply connected algebraic group, simply connected group of type E6. It contains a split dual pair G×G0 is such that G ∼ = SL3(R)oZ/2Z and G is of the type G2. If n = 7 or 8, we set H to be the group of real points of a split, simply connected algebraic group of the type En. The group H contains a split dual pair G×G0 where G is of the type G2, while G0 is a simply connected group of the type C3 and F4 respectively. Let g and g0 be the Lie algebras of G and G0’ respectively, and let K and K0 be the maximal compact subgroups of G and G0, respectively. Let V be the Harish-Chandra module of the minimal representation of H. Let V be an irreducible (g, K)-module and V 0 be an irreducible (g0, K0)-module. We say that V and V 0 correspond if V ⊗ V 0 is a quotient of V. Let V be an irreducible (g, K)-module. There is a (g0, K0)-module Θ(V ) such that V/ ∩φ ker φ ‘ Θ(V ) ⊗ Vwhere the intersection is taken over all (g, K)-module homomorphisms φ: V → V . Motivated by the classical dual pair correspondences, it is conjectured that Θ(V ) is a finite length (g0, K0)-module with a unique irreducible quotient V 0, and then conversely, that Θ(V 0) is a finite length (g, K)-module with V as a unique irreducible quotient. We call this a strong duality. Let λ and λ0 denote the infinitesimal characters of V and V 0 respectively. It is known previously that there is an explicit correspondence of the infinitesimal characters. A spherical representation V of G is an irreducible representation in which V K◦ is nonzero. It is a fact that a spherical representation is uniquely determined by its infinitesimal character. We let Sλ denote the spherical representation of G with infinitesimal char-acter λ. Likewise we let Sλ0 to be the spherical representation of G0 with infinitesimal character λ 0. Our first main result is that if Θ(Sλ0 ) 6= 0 then it is a finite length (g, K)-module with the unique irreducible quotient isomorphic to Sλ. Here λ is the infinitesimal character corresponding to λ0. Next suppose H is of the type E6 or E7. As before let λ be the infinitesimal character corresponding to λ0. Then Θ(Sλ0 ) 6= 0. In addition Θ(Sλ) is a finite length (g0, K0)-module with the unique irreducible quotient isomorphic to Sλ0 . In summary we establish the strong duality for spherical representations in the split E6 and E7 cases, but only one for the dual pair in the split E8 case. Reference: H.Y. Loke, G. Savin, “Duality for spherical representations in exceptional theta correspondences”. Transactions of the American Mathematical Society, 371, No. 9 (2019): 6359-6375.