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Local newforms for $$\text{GSp}(4)$$. (English) Zbl 1126.11027
Lecture Notes in Mathematics 1918. Berlin: Springer (ISBN 978-3-540-73323-2/pbk). viii, 307 p. (2007).
Let $$F$$ be a non-archimedean local field of characteristic 0. The present work develops a local theory of newforms for irreducible admissible representations of $$\text{PGSp}(4,F)$$. Let $$(\pi,V)$$ be such a representation. Let $$V(n)$$ denote the space of all vectors in $$V$$ fixed by the paramodular group of level $${\mathfrak p}^n$$ (this is a certain compact open subgroup of $$\text{GSp}(4,F)$$ containing the Klingen congruence subgroup of level $${\mathfrak p}^n$$; these groups do not form a descending sequence of groups).
One of the main results is the following. Assume $$\pi$$ is such that $$V(n)\neq 0$$ for some $$n$$. Let $$N_{\pi}=\min\{n\geq 0\mid V(n)\neq 0\}$$. Then $$\dim V(N_\pi)=1$$. An element $$\neq 0$$ of $$V(N_{\pi})$$ is called a newform and $$N_\pi$$ is called the level of $$\pi$$. Newforms exist in particular for all generic representations. The spaces $$V(n)$$ with $$n>N_\pi$$ can be obtained from $$V(N_{\pi})$$ by applying certain level raising operators. The $$L$$- and $$\varepsilon$$-factors attached to the representation $$\pi$$ can be expressed in terms of invariants whose value is determined by a newform. Let us explain this more precisely.
There is a classification, by Sally and Tadić, of all non-supercuspidal irreducible representations of $$\text{GSp}(4,F)$$ as constituents of parabolically induced representations. The authors assign to each of these representations an $$L$$-parameter (which is a homomorphism from the Weil-Deligne group of $$F$$ to $$\text{GSp}(4,{\mathbb C})$$), using the fact that the local Langlands correspondence is known for the proper Levi subgroups of $$\text{GSp}(4)$$. Thus, for non-supercuspidal $$\pi$$ we have the correct $$L$$-parameter $$\varphi_{\pi}$$ and $$L(s,\varphi_{\pi})$$ and $$\varepsilon(s,\varphi_{\pi},\psi)$$ can be computed. On the other hand, for a generic irreducible representation there is the theory of Novodvorsky’s zeta integral, which provides factors $$L(s,\pi)$$ and $$\varepsilon(s,\pi,\psi)$$. These factors $$L(s,\pi)$$ have been computed by Takloo-Bighash. If $$\pi$$ is both non-supercuspidal and generic, comparison of the results shows that $$L(s,\varphi_{\pi})=L(s,\pi)$$. Now, for generic $$\pi$$, the authors prove an expression for $$L(s,\pi)$$ in terms of the eigenvalues of two specific Hecke operators on the newform and the eigenvalue of $$\pi(u_n)$$, where $$u_n$$ is a certain element of the normalizer of $$K({\mathfrak p}^n)$$ and $$n$$ is the level of $$\pi$$. Next, suppose $$\pi$$ is a non-generic representation for which there exists $$n\geq 0$$ such that $$V(n)\neq 0$$. Then $$\pi$$ is non-supercuspidal and it can be checked that $$L(s,\varphi_{\pi})$$ is given by the same expression in the eigenvalues as above.
The proofs require many explicit computations and case-by-case verifications. Tables are given not only for $$L$$-parameters, $$L$$- and $$\varepsilon$$-factors, dimension of the spaces $$V(n)$$, eigenvalues, but also for some representation-technical items, such as Jacquet modules.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields 11-02 Research exposition (monographs, survey articles) pertaining to number theory 22-02 Research exposition (monographs, survey articles) pertaining to topological groups
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