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.