Perfect subsets of generalized Baire spaces and long games

We extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire space ${}^{\lambda }\lambda$, where $\lambda$ is an uncountable cardinal with ${\lambda }^{<\lambda }=\lambda$. In the first main theorem, we show that that the perfect set property for all subsets of ${}^{\lambda }\lambda$ that are definable from elements of ${}^{\lambda }\mathrm{O}\mathrm{r}\mathrm{d}$ is consistent relative to the existence of an inaccessible cardinal above $\lambda$. In the second main theorem, we introduce a Banach-Mazur type game of length $\lambda$ and show that the determinacy of this game, for all subsets of ${}^{\lambda }\lambda$ that are definable from elements of ${}^{\lambda }\mathrm{O}\mathrm{r}\mathrm{d}$ as winning conditions, is consistent relative to the existence of an inaccessible cardinal above $\lambda$. We further obtain some related results about definable functions on ${}^{\lambda }\lambda$ and consequences of resurrection axioms for definable subsets of ${}^{\lambda }\lambda$.