# Functoriality properties of the dual group

## TL;DR

This paper establishes the functoriality of the dual group associated with a G-variety, showing that morphisms between varieties induce compatible homomorphisms between their dual groups, extending previous results on their natural homomorphisms.

## Contribution

It proves the functoriality property of the dual group construction for G-varieties, generalizing earlier work on the natural homomorphism to the Langlands dual group.

## Key findings

- Dual groups are functorial with respect to dominant and injective G-morphisms.
- Canonical homomorphisms between dual groups are compatible with original morphisms.
- Extends previous results on the natural homomorphism to the Langlands dual group.

## Abstract

Let $G$ be a connected reductive group. In a previous paper, arxiv:1702.08264, is was shown that the dual group $G^\vee_X$ attached to a $G$-variety $X$ 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 canonical homomorphism $G^\vee_Y\to G^\vee_X$ which is compatible with the homomorphisms to $G^\vee$.