On the type-definability of the binding group in simple theories

Let T be simple, work in Ceq over a boundedly closed set. Let p ∈ S(∅) be internal in a quasi-stably-embedded type-definable set Q (e.g. Q is definable or stably-embedded) and suppose (p,Q) is ACLembedded in Q (see definitions below). Then Aut(p/Q) = {σ|pC | σ ∈ Aut(C/Q)} with its action on pC is type-definable in Ceq over ∅. In particular, if p ∈ S(∅) is internal in a stably-embedded type-definable set Q, and pC∪Q is stably-embedded, then Aut(p/Q) is type-definable with its action on pC .