A Walk Through $Spin(1,d+1)$

Abstract: We construct unitary irreducible representation of the de Sitter group, that forms the basis for the study of $dS_{d+1}$ QFT. Using the intertwining kernel analysis, we discuss bosonic symmetric tensor, and fermionic higher-spinors. Particular care is given to the structure and construction of exceptional series and fermionic operators. We discuss the discrete series, and explain how and why the exceptional series give rise to seemingly non-invariant correlators in de Sitter. Using tools from Clifford analysis, we show that for $d>3$, there are no exceptional fermionic representations, and so no unitary (higher)-gravitino fields. We also consider the structure of representations for $d=3$ and $d=2$, as relevant for the study of $dS_4$, the only space allowing for unitary gravitino and its generalisation, and of $dS_3$.