On the action of the U_p operator on the local (at p) representation attached to congruence level Siegel modular forms

In this article, we study the action of the \(U_p\) Hecke operator on the normalized spherical vector \(\phi \) in the representation of \({{\mathrm{GSp}}}_4(\mathbf {Q}_p)\) induced from a character on the Borel subgroup. We compute the Petersson norm of \(U_p \phi \) in terms of certain local L-values associated with \(\phi \).