##
**On the Langlands correspondence for symplectic motives.**
*(English.
Russian original)*
Zbl 1391.11075

Izv. Math. 80, No. 4, 678-692 (2016); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 4, 49-64 (2016).

Given a discrete symplectic motive \(M\) of rank 2n over \(\mathbb Q\), Langlands conjectured that it associates to a generic, automorphic representation \(\pi\) of the split orthogonal group \(G = \mathrm{SO}(2n+1,\mathbb Q)\), which appears with multiplicity one in the cuspidal spectrum. This paper proposes a refinement of the above conjecture using the results on local test vectors for the Whittaker functionals [A. A. Popa, J. Number Theory 128, No. 6, 1637–1645 (2008; Zbl 1146.11030)]. We will use freely the references as listed in the reference section in the paper.

Let \(M\) be a pure motive of weight \(-1\) and rank \(2n\) over \(\mathbb Q\) with a non-degenerate symplectic polarization \(\psi:\wedge^2 M \rightarrow \mathbb Q(1)\). An example is the primitive odd cohomology groups of geometrically irreducible, non-singular, projective varieties \(X\) over \(\mathbb Q\) of dimension \(r\). For an integer \(m\) with \(1 \leq (2m - 1) \leq r\), we set \(M = {\mathrm{H}}^{2m-1}(X)_{\mathrm{prim}} (m)\).

The symplectic motive \((M, \psi)\) is call discrete if its automorphism group \(C = \mathrm{Aut}(M, \psi)\) is a finite group scheme over \(\mathbb Q\). The \(l\)-adic realization \(M_l\) of \(M\) gives a continuous Galois representation \(\rho:\mathrm{Gal}(\mathbb Q/\mathbb Q) \rightarrow\mathrm{CSp}(2n, \mathbb Q_l)\). Here the similitude character gives the \(l\)-adic cyclotomic character. The Galois representation in turn gives a complex representation of the Weil-Deligne group \(\phi_{l,p}:{\mathrm{WD}}(\mathbb Q_p) \rightarrow\mathrm{CSp}(2n,\mathbb C)\). By twisting it with an unramified character we get a local Langlands parameter \[ \phi_p:{\mathrm{WD}}(\mathbb Q_p) \rightarrow\mathrm{rSp}(2n,\mathbb C). \] It is assume that this parameter is tempered.

Let \(\pi_p\) be the irreducible tempered generic representation of \(G(\mathbb Q_p)\) attached to \(\phi_p\). In Section 5 of the paper, the author defines a compact subgroup \(K_0(p^m)\) of \(G(\mathbb Q_l)\) such that \(\dim \pi_p^{K_0(p^m)} = 1\). A nonzero vector \(z_p\) in \(\pi_p^{K_0(p^m)}\), is called a new vector. Let \(U\) be the unipotent radical of a Borel subgroup of \(G\). Following [N. R. Wallach, in: Operator algebras and group representations, Proc. int. Conf., Neptun/Rom. 1980, Vol. II, monogr. Stud. Math. 18, 227–237 (1984; Zbl 0554.22004); J. P. Labesse and R. P. Langlands, Can. J. Math. 31, 726–785 (1979; Zbl 0421.12014)] and [H. Jacquet and R. P. Langlands, Automorphic forms on GL (2). Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0236.12010)], a (nonzero) Whittaker functional in \(\mathrm{Hom}_{U(\mathbb Q_p)}(\pi_p,\theta_p)\) is nonzero when evaluated at a new vector. Here \(\theta_p\) is a generic character of \(U(\mathbb Q_p)\). In Section 4, the author defines a (nonzero) new vector \(v_\infty\) in \(\pi_\infty\).

We recall and note that \(M\) is discrete and the local representations \(\pi_p\) are all tempered. We consider \(\pi = \pi(M) = \otimes_v' \pi_v\) which is an irreducible complex representation of \(G(\mathbb A)\). The Langlands Conjecture states that \(\pi\) is cuspidal and appears with multiplicity one in the discrete spectrum of \(G\). Assuming this conjecture, we fix an embedding of \(\pi\) into the space \(\mathcal A_0\) of cusp forms on the group \(G\). It is conjectured that the linear form \(\pi \rightarrow \mathcal A_0 \rightarrow \mathbb C\) taking a cusp form \(f\) in the image of \(\pi\) to the integral \[ \int_{U(\mathbb Q) \backslash U(\mathbb A)} f(u) \overline{\theta(u)} du \] is non-zero, and gives a basis of the one dimensional vector space \(\mathrm{Hom}_{U(\mathbb A)}(\pi, \theta)\). Here \(\theta\) is a character of \(U(\mathbb A)\). Let \(F \in \mathcal A_0\) be the image of the tensor product of test vectors \(\otimes_v z_v\). The proposed refinement of the Langlands conjecture is that the integral \[ a_1(F) = \int_{U(\mathbb Q) \backslash U(\mathbb A)} F(u) \overline{\theta(u)} du= \int_{U(\mathbb R) \backslash U(\mathbb R)} F_\infty(u) \overline{\theta_\infty(u)} du \] is non-zero.

In Section 8, the author generalizes \(a_1(F)\) and defines the \(t\)-th Fourier coefficient of \(F\) for \(t \in T(\mathbb Q)\). He then explains how to construct \(F\) following the ideas of D. Jiang and D. Soudry [in: Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002. Baltimore, MD: Johns Hopkins University Press. 457–519 (2004; Zbl 1062.11077)]. In Section 9, he discusses the case when \((A,\psi)\) is a polarized abelian variety of dimensional \(n\) over \(\mathbb Q\) and \(M = (H_1(A), \psi)\).

This paper is concise and informative but assumes quite a bit of background knowledge in Langlands conjectures and motives. It is carefully and clearly written and it has been a joy to read.

Let \(M\) be a pure motive of weight \(-1\) and rank \(2n\) over \(\mathbb Q\) with a non-degenerate symplectic polarization \(\psi:\wedge^2 M \rightarrow \mathbb Q(1)\). An example is the primitive odd cohomology groups of geometrically irreducible, non-singular, projective varieties \(X\) over \(\mathbb Q\) of dimension \(r\). For an integer \(m\) with \(1 \leq (2m - 1) \leq r\), we set \(M = {\mathrm{H}}^{2m-1}(X)_{\mathrm{prim}} (m)\).

The symplectic motive \((M, \psi)\) is call discrete if its automorphism group \(C = \mathrm{Aut}(M, \psi)\) is a finite group scheme over \(\mathbb Q\). The \(l\)-adic realization \(M_l\) of \(M\) gives a continuous Galois representation \(\rho:\mathrm{Gal}(\mathbb Q/\mathbb Q) \rightarrow\mathrm{CSp}(2n, \mathbb Q_l)\). Here the similitude character gives the \(l\)-adic cyclotomic character. The Galois representation in turn gives a complex representation of the Weil-Deligne group \(\phi_{l,p}:{\mathrm{WD}}(\mathbb Q_p) \rightarrow\mathrm{CSp}(2n,\mathbb C)\). By twisting it with an unramified character we get a local Langlands parameter \[ \phi_p:{\mathrm{WD}}(\mathbb Q_p) \rightarrow\mathrm{rSp}(2n,\mathbb C). \] It is assume that this parameter is tempered.

Let \(\pi_p\) be the irreducible tempered generic representation of \(G(\mathbb Q_p)\) attached to \(\phi_p\). In Section 5 of the paper, the author defines a compact subgroup \(K_0(p^m)\) of \(G(\mathbb Q_l)\) such that \(\dim \pi_p^{K_0(p^m)} = 1\). A nonzero vector \(z_p\) in \(\pi_p^{K_0(p^m)}\), is called a new vector. Let \(U\) be the unipotent radical of a Borel subgroup of \(G\). Following [N. R. Wallach, in: Operator algebras and group representations, Proc. int. Conf., Neptun/Rom. 1980, Vol. II, monogr. Stud. Math. 18, 227–237 (1984; Zbl 0554.22004); J. P. Labesse and R. P. Langlands, Can. J. Math. 31, 726–785 (1979; Zbl 0421.12014)] and [H. Jacquet and R. P. Langlands, Automorphic forms on GL (2). Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0236.12010)], a (nonzero) Whittaker functional in \(\mathrm{Hom}_{U(\mathbb Q_p)}(\pi_p,\theta_p)\) is nonzero when evaluated at a new vector. Here \(\theta_p\) is a generic character of \(U(\mathbb Q_p)\). In Section 4, the author defines a (nonzero) new vector \(v_\infty\) in \(\pi_\infty\).

We recall and note that \(M\) is discrete and the local representations \(\pi_p\) are all tempered. We consider \(\pi = \pi(M) = \otimes_v' \pi_v\) which is an irreducible complex representation of \(G(\mathbb A)\). The Langlands Conjecture states that \(\pi\) is cuspidal and appears with multiplicity one in the discrete spectrum of \(G\). Assuming this conjecture, we fix an embedding of \(\pi\) into the space \(\mathcal A_0\) of cusp forms on the group \(G\). It is conjectured that the linear form \(\pi \rightarrow \mathcal A_0 \rightarrow \mathbb C\) taking a cusp form \(f\) in the image of \(\pi\) to the integral \[ \int_{U(\mathbb Q) \backslash U(\mathbb A)} f(u) \overline{\theta(u)} du \] is non-zero, and gives a basis of the one dimensional vector space \(\mathrm{Hom}_{U(\mathbb A)}(\pi, \theta)\). Here \(\theta\) is a character of \(U(\mathbb A)\). Let \(F \in \mathcal A_0\) be the image of the tensor product of test vectors \(\otimes_v z_v\). The proposed refinement of the Langlands conjecture is that the integral \[ a_1(F) = \int_{U(\mathbb Q) \backslash U(\mathbb A)} F(u) \overline{\theta(u)} du= \int_{U(\mathbb R) \backslash U(\mathbb R)} F_\infty(u) \overline{\theta_\infty(u)} du \] is non-zero.

In Section 8, the author generalizes \(a_1(F)\) and defines the \(t\)-th Fourier coefficient of \(F\) for \(t \in T(\mathbb Q)\). He then explains how to construct \(F\) following the ideas of D. Jiang and D. Soudry [in: Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002. Baltimore, MD: Johns Hopkins University Press. 457–519 (2004; Zbl 1062.11077)]. In Section 9, he discusses the case when \((A,\psi)\) is a polarized abelian variety of dimensional \(n\) over \(\mathbb Q\) and \(M = (H_1(A), \psi)\).

This paper is concise and informative but assumes quite a bit of background knowledge in Langlands conjectures and motives. It is carefully and clearly written and it has been a joy to read.

Reviewer: Hung Yean Loke (Singapore)

### MSC:

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

14D24 | Geometric Langlands program (algebro-geometric aspects) |

20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |

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