Cerejeiras, P.Ferreira, M.Kähler, U.Wirth, J.2023-07-042023-07-042023-05-17Cerejeiras, P., Ferreira, M., Kähler, U. et al. Global Operator Calculus on Spin Groups. J Fourier Anal Appl 29, 32 (2023). https://doi.org/10.1007/s00041-023-10015-51069-5869http://hdl.handle.net/10400.8/8660Acknowledgements The work of P. Cerejeiras, M. Ferreira, and U. Kähler was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT– “Fundação para a Ciência e a Tecnologia”, within project UIDB/04106/2020 and UIDP/04106/2020. The present paper was supported by the project “Global operator calculi on compact and non-compact Lie groups”, Ações Integradas Luso-Alemãs – Acção No. A-42/16. Funding Open access funding provided by FCT|FCCN (b-on).n 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.engSpin groupSpin representationsDifference operatorsPseudo-differential operatorsFourier transformMicrolocal analysisElliptic operatorsGlobal hypoellipticityjournal articlehttps://doi.org/10.1007/s00041-023-10015-51531-5851