Motives with Galois group of type \(G_2\): An exceptional theta-correspondence.

*(English)*Zbl 0931.11015This substantial paper concerns arithmetic automorphic forms and an exceptional theta correspondence. It is the first step in a project to construct a motive \(M\) of rank 7 and weight 0 over the base field \({\mathbb Q}\) with motivic Galois group of type \(G_2\).

The authors begin by developing the arithmetic theory of modular forms for semi-simple algebraic groups \(G\) over \({\mathbb Q}\) with \(G({\mathbb R})\) compact. By giving a rational structure on the space of such forms, they show that the coefficients of the corresponding \(L\)-polynomials lie in a totally real number field \(E\subset {\mathbb C}\). They show that there are many automorphic representations with local component given by the Steinberg representation. Then specializing to the case when \(G\) is the automorphism group of Cayley’s octonion algebra (\(G\) is the anisotropic form of \(G_2\)), the authors give explicit examples of such modular forms. For example, they show that there is a unique automorphic representation \(\pi=\otimes \pi_v\) of \(G(A_{\mathbb Q})\) with \(\pi_\infty\cong{\mathbb C}\), \(\pi_5\) isomorphic to the Steinberg representation of \(G({\mathbb Q}_5)\), and \(\pi_p\) unramified for \(p\neq 5\).

The authors next turn to motives of rank 7 over \({\mathbb Q}\) associated to automorphic forms on the anisotropic form of \(G_2\). They formulate a conjecture concerning the existence of these motives, which should be attached to certain automorphic representations \(\pi\) of \(G(A_{\mathbb Q})\). They expect to obtain such motives by first lifting \(\pi\) to an automorphic representation \(\pi'\) of \(G'=PGSp_6\) (the lifting is compatible with Langlands functoriality), then using \(\pi'\) to define a motive \(M'\) in the cohomology of a 6-dimensional Siegel modular variety over \({\mathbb Q}\). As \(\pi'\) is a lift, \(M'\) should decompose as a sum of the desired motive \(M\) for \(\pi\) and a Hodge class. With this as motivation, they give local criteria which allow one to show that a subgroup \(\Gamma\) of \(SO_8\) is contained in either \(\text{Spin}_7\) or \(G_2=\text{Spin}_7\cap SO_7\). They also begin their consideration of the unique form \(H\) of the split adjoint group of type \(E_7\), which has rank 3 over \({\mathbb Q}\), and of the dual pair \(G\times G'\) in \(H\).

The next three sections of the paper take up this dual pair in earnest. The authors study the theta correspondence obtained by restricting the minimal representation \(\Pi\) of \(H\) to \(G\times G'\). First they work out the real correspondence. In the \(p\)-adic case they formulate a conjectural description of the irreducible representations \(\pi\times\pi'\) of \(G({\mathbb Q}_p)\times G'({\mathbb Q}_p)\) whch occur as quotients of \(\Pi\) in terms of the Langlands-Deligne-Lusztig-Vogan parametrization of irreducible representations. They derive some implications of this description which they are able to check. In particular, they show that if \(\pi'\) is a unitarizable theta-lift of the Steinberg representation of \(G({\mathbb Q}_p)\), then it is isomorphic to Steinberg for \(G'({\mathbb Q}_p)\). They then turn to the global theta correspondence. This is constructed using the automorphic realization of the adelic minimal representation, which was obtained by H. H. Kim [Rev. Mat. Iberoam. 9, 139-200 (1993; Zbl 0777.11015)]. They study the nonvanishing of the theta lift, giving a criterion in terms of period integrals. They also give a criterion for cuspidality. They apply these results to show that the two automorphic forms for \(G(A_{\mathbb Q})\) constructed earlier have cuspidal theta lifts. Using this and their local work, they show, for example, that the level 5 form described above lifts to an automorphic representation corresponding to a classical holomorphic Siegel modular form of weight 4 and level 5.

In the final section of the paper, the authors present a discussion of lifting and local periods. Recall that the complement of the motive \(M\) in \(M'\) should be given by the classes of Hilbert modular 3-folds. Hence the forms on \(G'\) coming from \(G\) should be characterized as having non-zero periods over the cycles given by the Hilbert modular 3-folds. This is established locally.

The authors begin by developing the arithmetic theory of modular forms for semi-simple algebraic groups \(G\) over \({\mathbb Q}\) with \(G({\mathbb R})\) compact. By giving a rational structure on the space of such forms, they show that the coefficients of the corresponding \(L\)-polynomials lie in a totally real number field \(E\subset {\mathbb C}\). They show that there are many automorphic representations with local component given by the Steinberg representation. Then specializing to the case when \(G\) is the automorphism group of Cayley’s octonion algebra (\(G\) is the anisotropic form of \(G_2\)), the authors give explicit examples of such modular forms. For example, they show that there is a unique automorphic representation \(\pi=\otimes \pi_v\) of \(G(A_{\mathbb Q})\) with \(\pi_\infty\cong{\mathbb C}\), \(\pi_5\) isomorphic to the Steinberg representation of \(G({\mathbb Q}_5)\), and \(\pi_p\) unramified for \(p\neq 5\).

The authors next turn to motives of rank 7 over \({\mathbb Q}\) associated to automorphic forms on the anisotropic form of \(G_2\). They formulate a conjecture concerning the existence of these motives, which should be attached to certain automorphic representations \(\pi\) of \(G(A_{\mathbb Q})\). They expect to obtain such motives by first lifting \(\pi\) to an automorphic representation \(\pi'\) of \(G'=PGSp_6\) (the lifting is compatible with Langlands functoriality), then using \(\pi'\) to define a motive \(M'\) in the cohomology of a 6-dimensional Siegel modular variety over \({\mathbb Q}\). As \(\pi'\) is a lift, \(M'\) should decompose as a sum of the desired motive \(M\) for \(\pi\) and a Hodge class. With this as motivation, they give local criteria which allow one to show that a subgroup \(\Gamma\) of \(SO_8\) is contained in either \(\text{Spin}_7\) or \(G_2=\text{Spin}_7\cap SO_7\). They also begin their consideration of the unique form \(H\) of the split adjoint group of type \(E_7\), which has rank 3 over \({\mathbb Q}\), and of the dual pair \(G\times G'\) in \(H\).

The next three sections of the paper take up this dual pair in earnest. The authors study the theta correspondence obtained by restricting the minimal representation \(\Pi\) of \(H\) to \(G\times G'\). First they work out the real correspondence. In the \(p\)-adic case they formulate a conjectural description of the irreducible representations \(\pi\times\pi'\) of \(G({\mathbb Q}_p)\times G'({\mathbb Q}_p)\) whch occur as quotients of \(\Pi\) in terms of the Langlands-Deligne-Lusztig-Vogan parametrization of irreducible representations. They derive some implications of this description which they are able to check. In particular, they show that if \(\pi'\) is a unitarizable theta-lift of the Steinberg representation of \(G({\mathbb Q}_p)\), then it is isomorphic to Steinberg for \(G'({\mathbb Q}_p)\). They then turn to the global theta correspondence. This is constructed using the automorphic realization of the adelic minimal representation, which was obtained by H. H. Kim [Rev. Mat. Iberoam. 9, 139-200 (1993; Zbl 0777.11015)]. They study the nonvanishing of the theta lift, giving a criterion in terms of period integrals. They also give a criterion for cuspidality. They apply these results to show that the two automorphic forms for \(G(A_{\mathbb Q})\) constructed earlier have cuspidal theta lifts. Using this and their local work, they show, for example, that the level 5 form described above lifts to an automorphic representation corresponding to a classical holomorphic Siegel modular form of weight 4 and level 5.

In the final section of the paper, the authors present a discussion of lifting and local periods. Recall that the complement of the motive \(M\) in \(M'\) should be given by the classes of Hilbert modular 3-folds. Hence the forms on \(G'\) coming from \(G\) should be characterized as having non-zero periods over the cycles given by the Hilbert modular 3-folds. This is established locally.

Reviewer: S.Friedberg (Chestnut Hill)

##### MSC:

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F27 | Theta series; Weil representation; theta correspondences |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

11G18 | Arithmetic aspects of modular and Shimura varieties |

22E50 | Representations of Lie and linear algebraic groups over local fields |