Global Operator Calculus on Spin Groups

In this paper, we use the representation theory of the group Spin ( m ) to develop aspects of the global symbolic calculus of pseudo-differential operators on Spin ( 3 ) and Spin ( 4 ) in the sense of Ruzhansky–Turunen–Wirth. A detailed study of Spin ( 3 ) and Spin ( 4 ) -representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group Spin ( 4 ) and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.