Strong functorial lift ($L$- and $\varepsilon$-factor compatibility) and full Ramanujan conjecture for the quasi-split non-split even orthogonal group $\mathbf{SO}_{2n}^{*}$ over function fields

For a global function field $F$, del Castillo (2022) proved the Langlands functoriality conjecture for globally generic cuspidal automorphic representations $\pi$ of the quasi-split non-split even special orthogonal group $\mathbf{SO}_{2n}^{*}(\mathbb{A}_F)$ by transferring them to automorphic representations $\Pi$ of $\mathbf{GL}_{2n}(\mathbb{A}_F)$, and established equality of local $\gamma$-factors of $\pi$ and $\Pi$ at every place, from which unramified Ramanujan follows. The paper explicitly leaves open the upgrade of this transfer to a strong lift — equality of local $L$-functions and $\varepsilon$-factors against all generic twists of general linear groups at every place — and the resulting generalized Ramanujan conjecture (temperedness at ramified places). The problem is open: the analogous strong lift is known for split classical groups and for quasi-split unitary groups over function fields (where full Ramanujan is proved via stable base change), while for $\mathbf{SO}_{2n}^{*}$ the later literature provides only local machinery (standard module and tempered $L$-function conjectures in positive characteristic), not the strong lift.

Background

Let $F$ be a global function field, i.e. a finite extension of $\mathbb{F}_q(t)$, with ring of adèles $\mathbb{A}_F$ (the restricted product of the completions $F_x$ over all places $x$). An automorphic representation of a reductive group $G(\mathbb{A}_F)$ is an irreducible constituent of the space of automorphic forms on $G(F)\backslash G(\mathbb{A}_F)$; it is cuspidal when its constant terms vanish, and globally generic when it admits a nontrivial Whittaker functional, i.e. realizes an embedding into a space of generic (Whittaker) functions. Langlands's functoriality conjecture predicts that for an $L$-homomorphism $\rho:{}^{L}G\to{}^{L}H$ between the Langlands dual groups (semi-direct products of the complex dual group with the Galois group of $F$), every cuspidal automorphic representation $\pi=\otimes'_x\pi_x$ of $G(\mathbb{A}_F)$ transfers to an automorphic representation $\Pi=\otimes'_x\Pi_x$ of $H(\mathbb{A}_F)$ whose Satake parameters (the conjugacy classes encoding unramified local representations) at almost all places are the images under $\rho$ of those of $\pi$; the transfer should moreover respect the arithmetic invariants of $\pi$, namely its $L$-functions (Euler products attached to representations), $\varepsilon$-factors (root numbers appearing in functional equations), and $\gamma$-factors (the ratios $\varepsilon(s,\cdot)\,L(1-s,\tilde{\cdot})/L(s,\cdot)$).

For a classical group $G$, the dual group embeds into ${}^{L}\mathbf{GL}_N$, and the target of the transfer is a general linear group. Over number fields, Cogdell–Kim–Piatetski-Shapiro–Shahidi proved such functoriality for globally generic cuspidal representations of quasi-split symplectic, unitary, and special orthogonal groups via the converse theorem (a criterion recognizing automorphic representations of $\mathbf{GL}_N$ from the analytic behavior of twisted $L$-functions), combined with the Langlands–Shahidi method, which defines local factors for generic representations through analytic continuation of Eisenstein series and intertwining operators. Lomelí extended this approach to positive characteristic, proving functoriality for the split classical groups and unitary groups over global function fields; for the split classical groups the $L$-function and $\varepsilon$-factor relations between $\pi$ and its transfer are available, and for quasi-split unitary groups Lomelí later proved the full local-global theory via stable base change (a functorial transfer to $\mathbf{GL}_N$ over a quadratic extension preserving local $L$-functions, $\gamma$-factors, and root numbers), deducing the complete Ramanujan conjecture: every local component of a globally generic cuspidal representation is tempered, i.e. a unitary representation whose matrix coefficients decay like those of a representation induced from a discrete series (equivalently, in the unramified case, Satake parameters of absolute value $1$). For $\mathbf{GL}_N$ itself over function fields, Ramanujan is a theorem of L. Lafforgue, obtained from the Langlands correspondence for $\mathbf{GL}_N$ via Drinfeld shtukas (certain infinite-dimensional algebraic varieties equipped with a Frobenius action, the geometric objects underlying the proof).

The group $\mathbf{SO}_{2n}^{*}$ denotes the quasi-split but non-split even special orthogonal group over $F$ (quasi-split meaning that the group contains a Borel subgroup, i.e. a maximal connected solvable subgroup, defined over $F$): the special orthogonal group of the unique quasi-split quadratic form in $2n$ variables whose discriminant class is nontrivial (in the split case the discriminant is trivial). It is the one quasi-split classical group family over function fields not covered by Lomelí's results, because the Langlands–Shahidi machine had not previously been developed for the pair $(\mathbf{GL}_m,\mathbf{SO}_{2n}^{*})$. del Castillo's preprint fills the functoriality gap: every globally generic cuspidal $\pi$ of $\mathbf{SO}_{2n}^{*}(\mathbb{A}_F)$ lifts to an isobaric sum $\Pi=\Pi_1\boxplus\cdots\boxplus\Pi_d$ of distinct unitary self-dual (isomorphic to their contragredient) cuspidal representations of $\mathbf{GL}_{N_i}(\mathbb{A}_F)$ with $N_1+\cdots+N_d=2n$ — the isobaric sum being the automorphic representation of $\mathbf{GL}_{2n}(\mathbb{A}_F)$ assembled from the cuspidal summands $\Pi_i$ by automorphic induction — and for every place $x$ and every irreducible generic representation $\tau_x$ of $\mathbf{GL}_m(F_x)$ the local factors satisfy

$$\gamma(s,\pi_x\times\tau_x,\psi_x)=\gamma(s,\Pi_x\times\tau_x,\psi_x),$$

where $\psi_x$ is a nontrivial additive character of the local field $F_x$, the left-hand factor is defined by the Langlands–Shahidi method (as adapted to $\mathbf{SO}_{2n}^{*}$ in the preprint), and the right-hand factor is the Rankin–Selberg factor for $\mathbf{GL}_{2n}\times\mathbf{GL}_m$, defined by Rankin–Selberg zeta integrals for pairs of general linear groups rather than by Eisenstein series. From this and L. Lafforgue's theorem the preprint deduces unramified Ramanujan: at unramified places the Satake parameters of $\pi_x$ have absolute value $1$. The preprint calls a lift satisfying the analogous equalities of local $L$-functions and $\varepsilon$-factors a strong lift and states that a strong lift would imply the Ramanujan conjecture at all places; its Remark 6.2.4 records that these $L$- and $\varepsilon$-relations are expected but unproved for $\mathbf{SO}_{2n}^{*}$, while being already established for the split classical groups over function fields.

Problem Statement

Let $F$ be a global function field and let $\mathbf{SO}_{2n}^{*}$ be the quasi-split non-split even special orthogonal group over $F$. It is known (del Castillo, 2022) that every globally generic cuspidal automorphic representation $\pi=\otimes'_x\pi_x$ of $\mathbf{SO}_{2n}^{*}(\mathbb{A}_F)$ admits a transfer: an automorphic representation $\Pi=\otimes'_x\Pi_x$ of $\mathbf{GL}_{2n}(\mathbb{A}_F)$ that is an isobaric sum of distinct unitary self-dual cuspidal representations and satisfies, at every place $x$ of $F$ and for every irreducible generic representation $\tau_x$ of $\mathbf{GL}_m(F_x)$, the $\gamma$-factor equality $\gamma(s,\pi_x\times\tau_x,\psi_x)=\gamma(s,\Pi_x\times\tau_x,\psi_x)$, with Langlands–Shahidi factors on the left and Rankin–Selberg factors on the right.

The open problem, stated by the source as its expected next step, is to upgrade this transfer to a strong lift: prove that for every place $x$ of $F$ and every irreducible generic representation $\tau_x$ of $\mathbf{GL}_m(F_x)$,

$$L(s,\pi_x\times\tau_x)=L(s,\Pi_x\times\tau_x) \quad\text{and}\quad \varepsilon(s,\pi_x\times\tau_x,\psi_x)=\varepsilon(s,\Pi_x\times\tau_x,\psi_x),$$

with the factors on the left defined by the Langlands–Shahidi method for the pair $(\mathbf{GL}_m,\mathbf{SO}_{2n}^{*})$ and those on the right by the Rankin–Selberg method for $\mathbf{GL}_{2n}\times\mathbf{GL}_m$; and, as the corresponding application, deduce from these relations the generalized Ramanujan conjecture for these representations: every local component $\pi_x$ of a globally generic cuspidal $\pi$ of $\mathbf{SO}_{2n}^{*}(\mathbb{A}_F)$ is tempered, at ramified as well as unramified places. The common objective uniting the two deliverables is the strong-lift equality of local $L$- and $\varepsilon$-factors; the temperedness statement is its explicitly intended consequence, exactly as in the completed analogs for split classical groups and quasi-split unitary groups over function fields.

References

1. Héctor del Castillo, Langlands functoriality conjecture for ${\bf SO}_{2n}^*$ in positive characteristic, arXiv:2201.03119 (2022), arXiv DOI 10.48550/arXiv.2201.03119, https://arxiv.org/abs/2201.03119
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