Geodesic
The dual group of a~spherical variety
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 227-260

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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. 
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     author = {F. Knop and B. Schalke},
     title = {The dual group of a~spherical variety},
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F. Knop; B. Schalke. The dual group of a~spherical variety. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 227-260. http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a2/