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

top
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

top

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 -

References

top
  1. [1] J. Adams, L -functoriality for dual pairs, Astérisque 171-172 (1989), 85–129. Zbl0715.22016
  2. [2] A. A. Beilinson, J. Bernstein & P. Deligne, Faisceaux pervers, Astérisque100 (1982), 5–171. Zbl0536.14011MR751966
  3. [3] A. A. Beilinson & V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigen-sheaves, preprint http://www.math.uchicago.edu/~mitya/langlands.html. Zbl0864.14007MR1385674
  4. [4] T. Braden, Hyperbolic localization of intersection cohomology, Transform. Groups8 (2003), 209–216. Zbl1026.14005MR1996415
  5. [5] A. Braverman & D. Gaitsgory, Geometric Eisenstein series, Invent. Math.150 (2002), 287–384. Zbl1046.11048MR1933587
  6. [6] E. Frenkel, D. Gaitsgory & K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. of Math.153 (2001), 699–748. Zbl1070.11050MR1836286
  7. [7] E. Frenkel, D. Gaitsgory & K. Vilonen, On the geometric Langlands conjecture, J. Amer. Math. Soc.15 (2002), 367–417. Zbl1071.11039MR1887638
  8. [8] D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math.144 (2001), 253–280. Zbl1072.14055
  9. [9] D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, Ann. of Math.160 (2004), 617–682. Zbl1129.11050MR2123934
  10. [10] D. Gaitsgory, On de Jong’s conjecture, Israel J. Math.157 (2007), 155–191. Zbl1123.11020MR2342444
  11. [11] D. Gaitsgory & D. Nadler, Spherical varieties and Langlands duality, Mosc. Math. J.10 (2010), 65–137. Zbl1207.22013MR2668830
  12. [12] R. Howe, Another look at the local θ -correspondence for an unramified dual pair, in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc. 2, Weizmann, 1990, 93–124. Zbl0722.22010MR1159101
  13. [13] M. Kapranov & E. Vasserot, Vertex algebras and the formal loop space, Publ. Math. I.H.É.S. 100 (2004), 209–269. Zbl1106.17038MR2102701
  14. [14] S. S. Kudla, On the local theta-correspondence, Invent. Math.83 (1986), 229–255. Zbl0583.22010MR818351
  15. [15] S. S. Kudla, Notes on the local theta correspondence, lecture notes http://www.math.toronto.edu/~skudla/castle.pdf, 1996. Zbl0583.22010
  16. [16] V. Lafforgue & S. Lysenko, Geometric Weil representation: local field case, Compos. Math.145 (2009), 56–88. Zbl1220.22015
  17. [17] Y. Laszlo & M. Olsson, The six operations for sheaves on Artin stacks. II. Adic coefficients, Publ. Math. I.H.É.S. 107 (2008), 169–210. Zbl1191.14003
  18. [18] G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Publ. Math. I.H.É.S. 65 (1987), 131–210. Zbl0641.14009
  19. [19] S. Lysenko, Moduli of metaplectic bundles on curves and theta-sheaves, Ann. Sci. École Norm. Sup.39 (2006), 415–466. Zbl1111.14029
  20. [20] S. Lysenko, Geometric Waldspurger periods, Compos. Math.144 (2008), 377–438. Zbl1209.14010
  21. [21] I. Mirković & K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math.166 (2007), 95–143. Zbl1138.22013
  22. [22] C. Mœglin, M.-F. Vignéras & J.-L. Waldspurger, Correspondances de Howe sur un corps p -adique, Lecture Notes in Math. 1291, Springer, 1987. Zbl0642.22002
  23. [23] S. Rallis, Langlands’ functoriality and the Weil representation, Amer. J. Math.104 (1982), 469–515. Zbl0532.22016

NotesEmbed ?

top

You must be logged in to post comments.