Local Newforms for GSp(4) pp 187-237 | Cite as

# 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.

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