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Representation Theory

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Geometrizing the minimal representations of even orthogonal groups

Authors: Vincent Lafforgue and Sergey Lysenko
Journal: Represent. Theory 17 (2013), 263-325
MSC (2010): Primary 14D24; Secondary 22E57, 11R39
Published electronically: May 28, 2013
MathSciNet review: 3057297
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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.

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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

Sergey Lysenko
Affiliation: Institut Elie Cartan Nancy, Université de Lorraine, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France
MR Author ID: 614865

Keywords: Geometric Langlands, minimal representation, theta lifting
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.