Geometrizing the minimal representations of even orthogonal groups (Q2836486)

This is one of a series of papers of the authors extending the theory of minimal representations to the setting of the geometric Langlands program. The motivation is to construct automorphic sheaves corresponding to minimal representations which will yield new cases of the geometric Langlands functoriality. In previous papers of both authors and the second author, the geometric Weil representation was constructed [Compos. Math. 145, No. 1, 56--88 (2009; Zbl 1220.22015)] and geometric theta-lifting is studied [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 3, 427--493 (2011; Zbl 1229.22015)]. This paper is concerned with minimal representations of even orthogonal groups.NEWLINENEWLINELet \(k\) be a finite field and \(X\) be a smooth projective geometrically connected curve over \(k\). Let \(H=\mathbb{SO}_{2n}\) be a split orthogonal group of type \(D_{n}\) with \(n\geq 4\). Let \(\text{Bun}_{H}\) be the stack of \(H\)-torsors on \(X\) and \(\text{D}(\text{Bun}_{H})\) be the derived category of \(\mathbb{\bar Q}_\ell\)-sheaves on \(\text{Bun}_H\). In Appendix A, the authors introduce almost constant sheaves on \(\text{Bun}_{H}\). Denote by \(\text{D}(\text{Bun}_{H})_{ls}\) the full triangulated subcaetegory generated by them. The main results of this paper, Theorems 2.1 and 2.2, provide a perverse sheaf \({\mathcal{K}} _H\in \text{D}(\text{Bun}_{H})\) such that its image in \(\text{D}(\text{Bun}_{H})/\text{D}(\text{Bun}_{H})_{ls}\) satisfies the Hecke property for the Arthur parameter \(\phi\) of the minimal representation of \(H\). Hence, a conjecturally existed automorphic sheaf \(K_{\phi}\in \text{D}(\text{Bun}_{H})\) should have image \({\mathcal{K}}_H\) in \(\text{D}(\text{Bun}_{H})/\text{D}(\text{Bun}_{H})_{ls}\). The construction of \({\mathcal{K}}_H\) is based on the theta-lifting via the \(P\)-model, where \(P\) is specific parabolic subgroup of \(H\). The authors also discuss \(Q\)- and \(R\)-models, as well as a conjectural construction via residues of Eisenstein series.NEWLINENEWLINEThe reader could consult \textit{W. T.~Gan} and \textit{G.~Savin} [Represent. Theory 9, 46--93 (2005; Zbl 1092.22012)] for a survey of minimal representations and \textit{E.~Frenkel} [Bull. Am. Math. Soc., New Ser. 41, No. 2, 151--184 (2004; Zbl 1070.11051)] for an introduction to the geometric Langlands program.