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.

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.

Reviewer: J. G. M. Mars (Utrecht)

##### 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 |