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Transactions of the Moscow Mathematical Society

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The dual group of a spherical variety


Authors: F. Knop and B. Schalke
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2017, 187-216
MSC (2010): Primary 17B22, 14L30, 11F70
DOI: https://doi.org/10.1090/mosc/270
Published electronically: December 1, 2017
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Abstract: Let $ X$ be a spherical variety for a connected reductive group $ G$. Work of Gaitsgory and Nadler strongly suggests that the Langlands dual group $ G^\vee $ of $ G$ has a subgroup whose Weyl group is the little Weyl group of $ X$. Sakellaridis and Venkatesh defined a refined dual group $ G^\vee _X$ and verified in many cases that there exists an isogeny $ \varphi $ from $ G^\vee _X$ to $ G^\vee $. In this paper, we establish the existence of $ \varphi $ in full generality. Our approach is purely combinatorial and works (despite the title) for arbitrary $ G$-varieties.


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Additional Information

F. Knop
Affiliation: Department of Mathematics, FAU Erlangen-Nürnberg, Germany
Email: friedrich.knop@fau.de

B. Schalke
Affiliation: Department of Mathematics, FAU Erlangen-Nürnberg, Germany
Email: schalke@math.fau.de

DOI: https://doi.org/10.1090/mosc/270
Keywords: Spherical varieties, Langlands dual groups, root systems, algebraic groups, reductive groups
Published electronically: December 1, 2017
Dedicated: To Ernest B.Vinberg on the occasion of his 80th birthday
Article copyright: © Copyright 2017 F.Knop, B.Schalke

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