Low Level Definability Above Large Cardinals

We study connections between definability in generalized descriptive set theory and large cardinals under ZFC. We show that if κ is a limit of measurables, then no Σ1(Vκ∪OR) wellorder of a subset of P(κ) of length ≥κ+ exists, answering a question of Lücke and Müller. However, consistently, a Woodin cardinal exists and for every uncountable cardinal κ which is not a limit of measurables, a Σ1(Hκ∪{κ})-good wellorder of Hκ+ exists. If κ is a limit of measurables and cof(κ)>ω, then no Σ1(Vκ∪OR) almost disjoint family F⊆P(κ) of cardinality >κ exists. Consistently, Π1({κ}) mad families and maximal independent families F⊆P(κ) exist, κ is a limit of measurables, and more. If κ is weakly compact and every Σ1(Vκ∪{κ}) subset of P(κ) of cardinality >κ contains a perfect subset of the right kind, then there is an inner model with a weakly compact limit of measurables. We prove some related facts regarding Σ1(Vλ∪{Vλ}∪OR) when I2(λ) holds. These depend on an analysis of fixed points of linear iterations involving I2(λ)-extenders.