Let \(G\) be a connected reductive group. Previously, it was shown that for any \(G\)-variety \(X\) one can define the dual group \(G^\vee_X\) which admits a natural homomorphism with finite kernel to the Langlands dual group \(G^\vee\) of \(G\). Here, we prove that the dual group is functorial in the following sense: if there is a dominant \(G\)-morphism \(X\to Y\) or an injective \(G\)-morphism \(Y\to X\) then there is a unique homomorphism with finite kernel \(G^\vee_Y\to G^\vee_X\) which is compatible with the homomorphisms to \(G^\vee\).
1. Ahiezer, Dmitry, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom., 1, 49-78, (1983); DOI 10.1007/BF02329739; zbl 0537.14033; MR0739893.
2. Bravi, Paolo; Luna, Domingo, An introduction to wonderful varieties with many examples of type $\rm F_4$, J. Algebra, 329, 4-51, (2011); DOI 10.1016/j.jalgebra.2010.01.025; zbl 1231.14040; MR2769314; arxiv 0812.2340.
4. Brion, Michel, On spherical varieties of rank one (after D. Ahiezer, A. Huckleberry, D. Snow). In: CMS Conf. Proc., 10, Amer. Math. Soc., Providence, RI, Group actions and invariant theory, (1988); zbl 0702.20029; MR1021273.
5. Brion, Michel, Vers une généralisation des espaces symétriques, J. Algebra, 134, 115-143, (1990); DOI 10.1016/0021-8693(90)90214-9; zbl 0729.14038; MR1068418.