math

(what is this?)

# Title: Geometric theta-lifting for the dual pair SO_{2m}, Sp_{2n}

Abstract: Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=SO_{2m}, G=Sp_{2n} 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 on Bun_G\times Bun_H yields the theta-lifting functors between the derived categories of l-adic sheaves on Bun_G and Bun_H. We describe the relation of these functors with Hecke operators.
In two particular cases it becomes the geometric Langlands functoriality for this pair (in the nonramified case). Namely, for n=m the functor from the derived category on Bun_H to that on Bun_G commutes with Hecke functors with respect to the inclusion of the Langlands dual groups SO_{2n}\to SO_{2n+1}. For m=n+1 the functor from the derived category on Bun_G to that on Bun_H commutes with Hecke functors with respect to the inclusion of the Langlands dual groups SO_{2n+1}\to \SO_{2n+2}.
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.
 Comments: LaTeX2e, 70 pages, final version to appear in Annales ENS Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG) MSC classes: 11R39, 14H60 Cite as: arXiv:math/0701170 [math.RT] (or arXiv:math/0701170v6 [math.RT] for this version)

## Submission history

From: Sergey Lysenko [view email]
[v1] Fri, 5 Jan 2007 16:30:00 GMT (51kb)
[v2] Tue, 26 Jun 2007 12:10:43 GMT (56kb)
[v3] Wed, 26 Dec 2007 09:33:33 GMT (58kb)
[v4] Sun, 3 Aug 2008 18:32:24 GMT (57kb)
[v5] Wed, 11 Feb 2009 21:53:13 GMT (58kb)
[v6] Tue, 29 Jun 2010 13:30:08 GMT (58kb)
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9340149164199829, "perplexity": 4575.102353483529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814801.45/warc/CC-MAIN-20180223154626-20180223174626-00397.warc.gz"}