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Preservation of depth in the local geometric Langlands correspondence


Authors: Tsao-Hsien Chen and Masoud Kamgarpour
Journal: Trans. Amer. Math. Soc. 369 (2017), 1345-1364
MSC (2010): Primary 17B67, 17B69, 22E50, 20G25
DOI: https://doi.org/10.1090/tran/6794
Published electronically: July 20, 2016
MathSciNet review: 3572276
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Abstract: It is expected that, under mild conditions, the local Langlands correspondence preserves depths of representations. In this article, we formulate a conjectural geometrisation of this expectation. We prove half of this conjecture by showing that the depth of a categorical representation of the loop group is greater than or equal to the depth of its underlying geometric Langlands parameter. A key ingredient of our proof is a new definition of the slope of a meromorphic connection, a definition which uses opers.


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Additional Information

Tsao-Hsien Chen
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: chenth@math.northwestern.edu

Masoud Kamgarpour
Affiliation: School of Mathematics and Physics, The University of Queensland, St. Lucia, Queensland 4072, Australia
Email: masoud@uq.edu.au

DOI: https://doi.org/10.1090/tran/6794
Keywords: Local geometric Langlands, Moy-Prasad Theory, slope of connections, opers, affine vertex algebras, Segal-Sugwara operators
Received by editor(s): November 24, 2014
Received by editor(s) in revised form: January 8, 2015, and July 22, 2015
Published electronically: July 20, 2016
Dedicated: To Volodya Drinfeld, on the occasion of his sixtieth birthday
Article copyright: © Copyright 2016 American Mathematical Society