Siegel p² Vectors for Representations of $GSp(4)$

Let F be a p-adic field and (π, V) an irreducible complex representation of $G=GSp(4, F)$ with trivial central character. Let Si(p²)⊂ G denote the Siegel congruence subgroup of level p² and u∈ NG( Si(p²)) the Atkin-Lehner element. We compute the dimension of the space of Si(p²)-fixed vectors in V as well as the signatures of the involutions π(u) acting on these spaces.