A G2-period of a Fourier coefficient of an Eisenstein series on E6

Abstract

We calculate a G2-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group E6, and relate this period to the Ginzburg-Rallis period of cusp forms on GL6. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of GL6.1 2 AARON POLLACK, CHEN WAN, AND MICHA L ZYDOR Let us describe the method in more detail. Let Q be the parabolic subgroup of the split, simply connected E 6 whose Levi subgroup is of D 4 type. In section 1.6 we define a certain generic character ξ of the unipotent radical N of Q and we define a subgroup H of Q to which the character extends. This group turns out to be isomorphic to N ⋊ G 2 , where G 2 is the split exceptional group of type G 2 . We can thus define the periodwhere f is an automorphic form on E 6 (A). Let now P denote the maximal parabolic subgroup of E 6 with Levi factor of type A 5 . Given a cuspidal representation π of GL 6 (A) we can consider the Eisenstein series E(φ, s) which realizes the induction from π ⊗ | det | s to E 6 (A) (see 3.4 for unexplained notation). By the Langlands-Shahidi theory, the non-vanishing of the residue at s = 1/2 of E(φ, s) implies that L( 1 2 , π, Λ 3 ) = 0. The idea is to study P G 2 (Res s=1/2 E(φ, s)) and relate it to the Ginzburg-Rallis period. In fact, we essentially show the two are equal.More specifically, one starts by computing the orbits P (F )\ E 6 (F )/H(F ). This problem is challenging when working with exceptional groups and we do this using the explicit realization of E 6 as determinant preserving linear transformations of the 27-dimensional exceptional Jordan algebra. These computations allow us to compute P G 2 (Λ T E(φ, s)) explicitly. Here Λ T is the Arthur-Langlands truncation operator [Art80]. The truncation operator brings us to the next difficulty in this approach, namely convergence issues. In fact, these can be quite daunting, especially when dealing with non-reductive periods. Indeed, unless we're dealing with compact periods, then even for cusp forms, convergence of periods requires a non-trivial argument, see for example [JS90]. In [IY], Ichino and Yamana handle this using a truncation procedure adapted to their period. However, their case is reductive. Here, we do not introduce any new truncation. Instead, we extend the analysis introduced in the appendix of [BP16] which is based on norms on adelic points of linear algebraic groups and their quotients. In fact, as noticed in loc. cit., the key property is for the quotient to be quasi-affine. We believe this approach should generalize to other periods.The computation of P G 2 (Λ T E(φ, s)) allows us to easily deduce the following theorem.Theorem 0.2. Let π be a cuspidal automorphic representation of GL 6 (A) with trivial central character. Suppose that the Ginzburg-Rallis period P GR (ϕ) is non-zero on the space of π. Then, there exists φ in the induced space of π such that Res s= 1 2 E(φ, s) = 0.Corollary 0.3. Theorem 0.2 implies Theorem 0.1.Proof. By Theorem 0.2 the intertwi...

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