Higher Souslin trees and the GCH, revisited

It is proved that for every uncountable cardinal λ, GCH+□(λ+) entails the existence of a cf(λ)-complete λ+-Souslin tree. In particular, if GCH holds and there are no ℵ2-Souslin trees, then ℵ2 is weakly compact in Gödel's constructible universe, improving Gregory's 1976 lower bound. Furthermore, it follows that if GCH holds and there are no ℵ2 and ℵ3 Souslin trees, then the Axiom of Determinacy holds in L(R).