Global pseudo-differential operator calculus over spin groups

In this talk we present a method to construct a global symbol calculus of pseudo-differential operators on spin groups in the sense of Ruzhansky-Turunen-Wirth, focussing on the special case Spin(4). Using representations of Spin(4) we construct a group Fourier transform and establish the calculus of left-invariant differential operators and of difference operators on the group Spin(4). Afterwards we apply this calculus 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.