Geometrizing the minimal representations of even orthogonal groups
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- by Vincent Lafforgue and Sergey Lysenko
- Represent. Theory 17 (2013), 263-325
- DOI: https://doi.org/10.1090/S1088-4165-2013-00431-4
- Published electronically: May 28, 2013
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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.References
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Bibliographic 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