Subelliptic Peter–Weyl and Plancherel theorems on compact, connected, semisimple Lie groups

We will study the connections between the elliptic and subelliptic versions of the Peter–Weyl and Plancherel theorems, in the case when the sub-Riemannian structure is generated naturally by the choice of a Cartan subalgebra. Along the way we will introduce and study the subelliptic Casimir operator associated to the subelliptic Laplacian.