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Tr. Mosk. Mat. Obs., 2017, Volume 78, Issue 2, Pages 227–260 (Mi mmo599)  

The dual group of a spherical variety

F. Knop, B. Schalke

Dept. Mathematik, FAU Erlangen-Nürnberg, Germany

Abstract: Let $X$ be a spherical variety for a connected reductive group $G$. Work of Gaitsgory–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–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.

Key words and phrases: spherical varieties, Langlands dual groups, root systems, algebraic groups, reductive groups

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English version:
Transactions of the Moscow Mathematical Society, 2017, 78, 187–216

Document Type: Article
UDC: 512.745, 512.813.4, 512.743.5
MSC: 17B22, 14L30, 11F70
Received: 27.03.2017
Revised: 14.05.2017
Language: English

Citation: F. Knop, B. Schalke, “The dual group of a spherical variety”, Tr. Mosk. Mat. Obs., 78, no. 2, MCCME, M., 2017, 227–260; Trans. Moscow Math. Soc., 78 (2017), 187–216

Citation in format AMSBIB
\Bibitem{KnoSch17}
\by F.~Knop, B.~Schalke
\paper The dual group of a~spherical variety
\serial Tr. Mosk. Mat. Obs.
\yr 2017
\vol 78
\issue 2
\pages 227--260
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo599}
\elib{http://elibrary.ru/item.asp?id=37045065}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2017
\vol 78
\pages 187--216
\crossref{https://doi.org/10.1090/mosc/270}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85037668141}


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