Galois representations for even general special orthogonal groups (Q6637218)

``Inspired by conjectures of Langlands and Clozel's work [\textit{L. Clozel}, Perspect. Math. 10, 77--159 (1990; Zbl 0705.11029)] for the group \(G = \mathrm{GL}_n\), Buzzard-Gee [\textit{K. Buzzard} and \textit{T. Gee}, Lond. Math. Soc. Lect. Note Ser. 414, 135--187 (2014; Zbl 1377.11067), Conj. 5.16] formulate a version of the Langlands correspondence for an arbitrary connected reductive group \(G\) over a number field \(F\) (Conjecture 1).''\N\NWrite \(\hat{G}\) (resp. \({}^{L}G\)) for the Langlands dual group (resp. \(L\)-group) of \(G\) over \(\overline{\mathbb Q}_l\). When \(g \in {}^{L}G(\overline{\mathbb Q}_l)\), let \(g_{ss}\) denote its semisimple part.\N\NConjecture 1. Let \(l\) be a prime number and fix an isomorphism \(\iota: \mathbb C \xrightarrow{\sim} \overline{\mathbb Q}_l\). Let \(\pi\) be a cuspidal \(L\)-algebraic automorphic representation of \(G(\mathbb A_F)\). Then there exists a Galois representation \N\[ \rho_{\pi} = \rho_{\pi,\iota}: \mathrm{Gal}(\overline{F}/F) \to {}^{L}G(\overline{\mathbb Q}_l), \] \Nsuch that for all but finitely many primes \(\mathfrak q\) of \(F\) (excluding \(\mathfrak q | l\) and those such that \(\pi_{\mathfrak q}\) are ramified), the \(\hat{G}\)-conjugacy class of \(\rho_{\pi}(\mathrm{Frob}_{\mathfrak q})_{ss} \in {}^{L}G(\mathbb {Q}_l)\) is the Satake parameter of \(\pi_{\mathfrak q}\) via \(\iota\).\N\N``Our goal is to prove Conjecture 1 for a quasi-split form \(G^{*}\) of \(\mathrm{GSO}_{2n}\) over a totally real field under certain local hypotheses, as a sequel to our work [\textit{A. Kret} and \textit{S. W. Shin}, J. Eur. Math. Soc. (JEMS) 25, No. 1, 75--152 (2023; Zbl 1519.11064)] where we proved the conjecture for \(\mathrm{GSp}_{2n}\) under similar local hypotheses. The group \(\mathrm{GSO}_{2n}\) is closely related to the classical group \(\mathrm{SO}_{2n}\), just like \(\mathrm{GSp}_{2n}\) is to \(\mathrm{Sp}_{2n}\), but the similitude groups may well be regarded as non-classical groups. An important reason is that the Langlands dual groups of \(\mathrm{GSO}_{2n}\) and \(\mathrm{GSp}_{2n}\), namely, the general spin groups \(\mathrm{GSpin}_{2n}\) and \(\mathrm{GSpin}_{2n+1}\), do not admit standard embeddings (into general linear groups of proportional rank). This makes the problem both nontrivial and interesting.''\N\NMain result of this paper is formulated in Theorem A. Proof of this result (given in section 12) relies crucially on the main results of Arthur's book [\textit{J. Arthur}, The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)], and uses deep results/constructions concerning the cohomology of some Shimura varieties. The last three sections present applications of Theorem A to the construction of Galois representations for \(\mathrm{SO}_{2n}^{E/F}\), automorphic multiplicity and meromorphic continuation of (half)-spin \(L\)-functions.