Affine transitions for involution Stanley symmetric functions

We study a family of symmetric functions $\hat F_z$ indexed by involutions $z$ in the affine group. These power series are analogues Lam's Stanley and generalizations involution introduced Hamaker, Pawlowski, first author. Our main result is to prove transition formula for which can be used define an analogue Lascoux-Sch\"utzenberger tree. proof this relies on Lam Shimozono's some new technical properties strong Bruhat order permutations.