Geometric theta-lifting for the dual pair ${\mathrm{𝕊𝕆}}_{2m},𝕊{\mathrm{p}}_{2n}$
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 3, pp. 427-493.

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $>2$. Consider the dual pair $H={\mathrm{SO}}_{2m},G={\mathrm{Sp}}_{2n}$ over $X$ with $H$ split. Write ${\mathrm{Bun}}_{G}$ and ${\mathrm{Bun}}_{H}$ for the stacks of $G$-torsors and $H$-torsors on $X$. The theta-kernel ${\mathrm{Aut}}_{G,H}$ on ${\mathrm{Bun}}_{G}×{\mathrm{Bun}}_{H}$ yields theta-lifting functors ${F}_{G}:\mathrm{D}\left({\mathrm{Bun}}_{H}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{G}\right)$ and ${F}_{H}:\mathrm{D}\left({\mathrm{Bun}}_{G}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{H}\right)$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non ramified case). Namely, we show that for $n=m$ the functor ${F}_{G}:\mathrm{D}\left({\mathrm{Bun}}_{H}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{G}\right)$ commutes with Hecke operators with respect to the inclusion of the Langlands dual groups $\stackrel{ˇ}{H}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}{\mathrm{SO}}_{2n}\stackrel{}{↪}{\mathrm{SO}}_{2n+1}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}\stackrel{ˇ}{G}$. For $m=n+1$ we show that the functor ${F}_{H}:\mathrm{D}\left({\mathrm{Bun}}_{G}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{H}\right)$ commutes with Hecke operators with respect to the inclusion of the Langlands dual groups $\stackrel{ˇ}{G}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}{\mathrm{SO}}_{2n+1}\stackrel{}{↪}{\mathrm{SO}}_{2n+2}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}\stackrel{ˇ}{H}$. In other cases the relation is more complicated and involves the ${\mathrm{SL}}_{2}$ of Arthur. As a step of the proof, we establish the geometric theta-lifting for the dual pair ${\mathrm{GL}}_{m},{\mathrm{GL}}_{n}$. Our global results are derived from the corresponding local ones, which provide a geometric analog of a theorem of Rallis.

Soit $X$ une courbe projective lisse sur un corps algébriquement clos de caractéristique $>2$. On considère la paire duale $H={\mathrm{SO}}_{2m}$, $G={\mathrm{Sp}}_{2n}$ sur $X$$H$ est déployé. Notons ${\mathrm{Bun}}_{G}$ et ${\mathrm{Bun}}_{H}$ les champs de modules des $G$-torseurs et des $H$-torseurs sur $X$. Le faisceau thêta ${\mathrm{Aut}}_{G,H}$ sur ${\mathrm{Bun}}_{G}×{\mathrm{Bun}}_{H}$ donne lieu aux foncteurs de thêta-lifting ${F}_{G}:\mathrm{D}\left({\mathrm{Bun}}_{H}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{G}\right)$ et ${F}_{H}:\mathrm{D}\left({\mathrm{Bun}}_{G}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{H}\right)$ entre les catégories dérivées correspondantes. On décrit la relation entre ces foncteurs et les opérateurs de Hecke. Dans deux cas particuliers cela devient la fonctorialité de Langlands géométrique pour cette paire (cas partout non ramifié). À savoir, on montre que pour $n=m$ le foncteur ${F}_{G}:\mathrm{D}\left({\mathrm{Bun}}_{H}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{G}\right)$ commute avec les opérateurs de Hecke par rapport à l’inclusion des groupes duaux de Langlands $\stackrel{ˇ}{H}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}{\mathrm{SO}}_{2n}\stackrel{}{↪}{\mathrm{SO}}_{2n+1}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}\stackrel{ˇ}{G}$. Pour $m=n+1$ on montre que le foncteur ${F}_{H}:\mathrm{D}\left({\mathrm{Bun}}_{G}\right)\to \mathrm{D}\left({\mathrm{Bun}}_{H}\right)$ commute avec les opérateurs de Hecke par rapport à l’inclusion des groupes duaux de Langlands $\stackrel{ˇ}{G}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}{\mathrm{SO}}_{2n+1}\stackrel{}{↪}{\mathrm{SO}}_{2n+2}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\to }\phantom{\rule{0.166667em}{0ex}}\stackrel{ˇ}{H}$. Dans d’autres cas la relation est plus complexe et fait intervenir le ${\mathrm{SL}}_{2}$ d’Arthur. Comme une étape de la preuve, on établit le thêta-lifting géométrique pour la paire duale ${\mathrm{GL}}_{m},{\mathrm{GL}}_{n}$. Nos résultats globaux sont déduits de résultats locaux correspondants, qui géométrisent un théorème de Rallis.

DOI: 10.24033/asens.2147
Classification: 11R39,  14H60
Keywords: theta-lifting, geometric Langlands, Langlands functoriality, theta-sheaf
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title = {Geometric theta-lifting for the dual pair $\mathbb {SO}_{2m}, \mathbb {S}\mathrm {p}_{2n}$},
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Lysenko, Sergey. Geometric theta-lifting for the dual pair $\mathbb {SO}_{2m}, \mathbb {S}\mathrm {p}_{2n}$. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 3, pp. 427-493. doi : 10.24033/asens.2147. http://archive.numdam.org/articles/10.24033/asens.2147/

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