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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Geometrizing the minimal representations of even orthogonal groups
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by Vincent Lafforgue and Sergey Lysenko PDF
Represent. Theory 17 (2013), 263-325 Request permission

Abstract:

Let $X$ be a smooth projective curve. Write $\mathrm {Bun}_{\mathrm {S}\mathbb {O}_{2n}}$ for the moduli stack of $\mathrm {S}\mathbb {O}_{2n}$-torsors on $X$. We give a geometric interpretation of the automorphic function $f$ on $\mathrm {Bun}_{\mathrm {S}\mathbb {O}_{2n}}$ corresponding to the minimal representation. Namely, we construct a perverse sheaf $\mathcal {K}_H$ on $\mathrm {Bun}_{\mathrm {S}\mathbb {O}_{2n}}$ such that $f$ should be equal to the trace of the Frobenius of $\mathcal {K}_H$ plus some constant function. The construction is based on some explicit geometric formulas for the Fourier coefficients of $f$ on one hand, and on the geometric theta-lifting on the other hand. Our construction makes sense for more general simple algebraic groups, we formulate the corresponding conjectures. They could provide a geometric interpretation of some unipotent automorphic representations in the framework of the geometric Langlands program.
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Additional Information
  • Vincent Lafforgue
  • Affiliation: CNRS et MAPMO, UMR 7349, Université d’Orléans, Rue de Chartres, B.P. 6759 - 45067 Orléans cedex 2, France
  • Email: vlafforg@math.jussieu.fr
  • Sergey Lysenko
  • Affiliation: Institut Elie Cartan Nancy, Université de Lorraine, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France
  • MR Author ID: 614865
  • Email: Sergey.Lysenko@univ-lorraine.fr
  • Received by editor(s): April 22, 2011
  • Received by editor(s) in revised form: February 7, 2012, and November 26, 2012
  • Published electronically: May 28, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 17 (2013), 263-325
  • MSC (2010): Primary 14D24; Secondary 22E57, 11R39
  • DOI: https://doi.org/10.1090/S1088-4165-2013-00431-4
  • MathSciNet review: 3057297