Shift Symmetries in (A)dS Space

I will discuss the generalizations of shift symmetries, galileon symmetries, and extended galileon symmetries to (A)dS space and to higher spin.   Unlike flat space, these symmetries are present only for particles with particular masses, and are related to partially massless symmetries.  For the case of scalars, I will discuss non-linear extensions of the symmetries and invariant interactions. This leads to a unique ghost-free theory in (A)dS space that is an (A)dS extension of the special Galileon. This theory has interactions that are completely fixed by the nonlinear symmetry, including a nontrivial potential.  I will speculate on possible generalizations to interacting massive higher-spin particles.