Hecke Operators

Let (π, V ) be an irreducible, admissible representation of GSp(4, F) with trivial central character. Assume that π is paramodular. In the previous chapter we proved that, for non-supercuspidal π, the space V (Nπ) is one-dimensional, where Nπ is the minimal paramodular level; we will eventually prove this for all paramodular representations. Thanks to uniqueness, any linear operator on V (Nπ) will act by a scalar, and thus define an invariant. One example will be the Atkin–Lehner eigenvalue επ. In this chapter we introduce the paramodular Hecke algebra and study the action of two of its elements on V (n). When n = Nπ, then the eigenvalues of these two operators will define two more important invariants λπ and µπ. As we will show in the next chapter, Nπ, επ, λπ and µπ will determine the relevant L- and ε-factors of the representation. Besides ultimately defining the invariants λπ and µπ, our two Hecke operators will in fact be an important tool for proving uniqueness at the minimal level and other results.