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In this article, we clarify the link between the conformal (i.e. Weyl)correspondence from the Minkowski space to the de Sitter space and theconformal (i.e. SO(2,$d$)) invariance of the conformal scalar field on bothspaces. We exhibit the realization on de Sitter space of the massless scalarrepresentation of SO$(2,d)$. It is obtained from the correspondingrepresentation in Minkowski space through an intertwining operator inheritedfrom the Weyl relation between the two spaces. The de Sitter representation iswritten in a form which allows one to take the point of view of a Minkowskianobserver who sees the effect of curvature through additional terms.

Revisiting the conformal invariance of the scalar field: from Minkowski space to de Sitter space