<STRONG>Local Newforms for GSp(4) describes a theory of new- and oldforms for representations of GSp(4) over a non-archimedean local field. This theory considers vectors fixed by the paramodular groups, and singles out certain vectors that encode canonical information, such as L-factors and epsilon-factors, through their Hecke and Atkin-Lehner eigenvalues. While there are analogies to the GL(2) case, this theory is novel and unanticipated by the existing framework of conjectures. An appendix includes extensive tables about the results and the representation theory of GSp(4).
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1918
Brooks Roberts
Ralf Schmidt
Local Newforms for GSp(4)
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Authors Brooks Roberts
Ralf Schmidt
Department of Mathematics University of Idaho Moscow ID 83844-1103 USA e-mail:
[email protected]
Department of Mathematics University of Oklahoma Norman OK 73019-0315 USA e-mail:
[email protected]
About the diagram. The diagram illustrates natural bases for the new- and oldforms in a generic representation π of GSp(4, F) with trivial central character. The solid dot in the first row is the newform at level Nπ . The solid dots and circles of the k-th row represent vectors in a natural basis for the oldforms at level Nπ + k. Thus, the dimension of the paramodular vectors at level Nπ is 1, the dimension at level Nπ + 1 is 2, the dimension at level Nπ + 2 is 4, and so on. The basis at a particular level is obtained from the newform by application of