The dual group of a spherical variety

TL;DR

This paper proves the existence of a refined dual group associated with a spherical variety for any reductive group, confirming a conjecture and extending previous partial results through a combinatorial approach.

Contribution

It establishes the existence of the isogeny from the refined dual group to the Langlands dual group for all spherical varieties, generalizing prior partial cases.

Findings

Confirmed the existence of the isogeny in full generality.

Applied a purely combinatorial method.

Extended results to arbitrary G-varieties.

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 $\phi$ from $G^\vee_X$ to $G^\vee$. In this paper, we establish the existence of $\phi$ in full generality. Our approach is purely combinatorial and works (despite the title) for arbitrary $G$-varieties.