Duality for spherical representations in exceptional theta correspondences

Abstract

Abstract. We study the exceptional theta correspondence for real groups obtained by restricting the minimal representation of the split exceptional group of the type E n , to a split dual pair where one member is the exceptional group of the type G 2 . We prove that the correspondence gives a bijection between spherical representations if n = 6, 7, and a slightly weaker statement if n = 8.

An exceptional Siegel–Weil formula and poles of the Spin L-function of

“…The Siegel-Weil identity (i.e. Theorem 1.2) then implies that π appears in the exceptional theta correspondence for the dual pair Aut(O) × PGSp 6 , for some O containing D. Since this exceptional theta correspondence is known to be functorial for spherical representations (see [LS19] and [SW15]), this completes the proof that π is a weak lift from a group of absolute type G 2 .…”

“…However, if G is split, G 1 is the smaller member of the dual pair (also split) and Π the minimal representation, then the spherical matrix coefficient Φ of Π, when restricted to G 1 , is typically contained in L 2− (G 1 ) for some > 0. (This is easy to check in any given situation; see [LS19]). Thus, in such situations, it makes sense to integrate Φ against spherical tempered functions of G 1 , that is, to consider the spherical transform of Φ on G 1 (k v ).…”

“…Here G 1 = SL 2 corresponds to the highest root and the highest weight is also a fundamental weight for E 7 (the ambient group G). • the split exceptional dual pairs in G of type E n where one member of the dual pair is the type G 2 ; see [LS19]. In particular, this includes the case PGL 3 × G 2 treated in [Gan11].…”

“…Proof. Adopting the notation from [LS19], let G = G 2 (R), g the Lie algebra of G , K a maximal compact subgroup of G , and g = k ⊕ p the corresponding Cartan decomposition. Let…”

“…as explained in the introduction of [LS19], where the case r = 0 is discussed. After taking det 2r -invariants in Π → Θ(σ) σ, it follows that Θ(σ) is a quotient of U (g ) ⊗ U (k ) C. Thus any irreducible quotient σ of Θ(σ) is spherical.…”