Geometric theta-lifting for the dual pair 𝕊𝕆 2 m , 𝕊 p 2 n

Sergey Lysenko

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 3, page 427-493
  • ISSN: 0012-9593

Abstract

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Let X be a smooth projective curve over an algebraically closed field of characteristic  > 2 . Consider the dual pair H = SO 2 m , G = Sp 2 n over X with H split. Write Bun G and Bun H for the stacks of G -torsors and H -torsors on X . The theta-kernel Aut G , H on Bun G × Bun H yields theta-lifting functors F G : D ( Bun H ) → D ( Bun G ) and F H : D ( Bun G ) → D ( Bun H ) 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 : D ( Bun H ) → D ( Bun G ) commutes with Hecke operators with respect to the inclusion of the Langlands dual groups H ˇ → ˜ SO 2 n ↪ SO 2 n + 1 → ˜ G ˇ . For m = n + 1 we show that the functor F H : D ( Bun G ) → D ( Bun H ) commutes with Hecke operators with respect to the inclusion of the Langlands dual groups G ˇ → ˜ SO 2 n + 1 ↪ SO 2 n + 2 → ˜ H ˇ . In other cases the relation is more complicated and involves the SL 2 of Arthur. As a step of the proof, we establish the geometric theta-lifting for the dual pair GL m , GL n . Our global results are derived from the corresponding local ones, which provide a geometric analog of a theorem of Rallis.

How to cite

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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 44.3 (2011): 427-493. <http://eudml.org/doc/272238>.

@article{Lysenko2011,
abstract = {Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $&gt;2$. 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 $\mathrm \{Bun\}_G\times \mathrm \{Bun\}_H$ yields theta-lifting functors $F_G: \mathrm \{D\}(\mathrm \{Bun\}_H)\rightarrow \mathrm \{D\}(\mathrm \{Bun\}_G)$ and $F_H: \mathrm \{D\}(\mathrm \{Bun\}_G)\rightarrow \mathrm \{D\}(\mathrm \{Bun\}_H)$ 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 $F_G: \mathrm \{D\}(\mathrm \{Bun\}_H)\rightarrow \mathrm \{D\}(\mathrm \{Bun\}_G)$ commutes with Hecke operators with respect to the inclusion of the Langlands dual groups $\check\{H\}\,\widetilde\{\rightarrow \}\, \mathrm \{SO\}_\{2n\}\stackrel\{\}\{\hookrightarrow \} \mathrm \{SO\}_\{2n+1\}\,\widetilde\{\rightarrow \}\,\check\{G\}$. For $m=n+1$ we show that the functor $F_H: \mathrm \{D\}(\mathrm \{Bun\}_G)\rightarrow \mathrm \{D\}(\mathrm \{Bun\}_H)$ commutes with Hecke operators with respect to the inclusion of the Langlands dual groups $\check\{G\}\,\widetilde\{\rightarrow \}\, \mathrm \{SO\}_\{2n+1\}\stackrel\{\}\{\hookrightarrow \} \mathrm \{SO\}_\{2n+2\}\,\widetilde\{\rightarrow \}\, \check\{H\}$. 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.},
author = {Lysenko, Sergey},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {theta-lifting; geometric Langlands; Langlands functoriality; theta-sheaf},
language = {eng},
number = {3},
pages = {427-493},
publisher = {Société mathématique de France},
title = {Geometric theta-lifting for the dual pair $\mathbb \{SO\}_\{2m\}, \mathbb \{S\}\mathrm \{p\}_\{2n\}$},
url = {http://eudml.org/doc/272238},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Lysenko, Sergey
TI - Geometric theta-lifting for the dual pair $\mathbb {SO}_{2m}, \mathbb {S}\mathrm {p}_{2n}$
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 3
SP - 427
EP - 493
AB - Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $&gt;2$. 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 $\mathrm {Bun}_G\times \mathrm {Bun}_H$ yields theta-lifting functors $F_G: \mathrm {D}(\mathrm {Bun}_H)\rightarrow \mathrm {D}(\mathrm {Bun}_G)$ and $F_H: \mathrm {D}(\mathrm {Bun}_G)\rightarrow \mathrm {D}(\mathrm {Bun}_H)$ 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 $F_G: \mathrm {D}(\mathrm {Bun}_H)\rightarrow \mathrm {D}(\mathrm {Bun}_G)$ commutes with Hecke operators with respect to the inclusion of the Langlands dual groups $\check{H}\,\widetilde{\rightarrow }\, \mathrm {SO}_{2n}\stackrel{}{\hookrightarrow } \mathrm {SO}_{2n+1}\,\widetilde{\rightarrow }\,\check{G}$. For $m=n+1$ we show that the functor $F_H: \mathrm {D}(\mathrm {Bun}_G)\rightarrow \mathrm {D}(\mathrm {Bun}_H)$ commutes with Hecke operators with respect to the inclusion of the Langlands dual groups $\check{G}\,\widetilde{\rightarrow }\, \mathrm {SO}_{2n+1}\stackrel{}{\hookrightarrow } \mathrm {SO}_{2n+2}\,\widetilde{\rightarrow }\, \check{H}$. 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.
LA - eng
KW - theta-lifting; geometric Langlands; Langlands functoriality; theta-sheaf
UR - http://eudml.org/doc/272238
ER -

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