Differential operators and theta series
Author:
Solomon Friedberg
Journal:
Trans. Amer. Math. Soc. 287 (1985), 569589
MSC:
Primary 11F25; Secondary 11F11
MathSciNet review:
768726
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Abstract: Let be a modular form on a congruence subgroup of not necessarily holomorphic, but an eigenfunction of the weight Casimir operator. Maass introduced differential operators (coming from the complexified universal enveloping algebra) which raise and lower by the weight of such a form and shift the eigenvalue. Here we introduce differential operators on hyperbolic space analogous to the Maass operators. These change by the weight of a modular form for an imaginary quadratic field. Theorem. The Maass operators and the hyperbolic space operators are intertwined by the imaginary quadratic DoiNaganuma (base change) lifting. That is, the following diagram is commutative: Using similar techniques for the dual pair , we give a simple proof that the Shimura correspondences preserve holomorphicity (for weight ) and an explanation for this property directly in terms of the theta series (Weil representation) integral kernel. We also establish similar results for the real quadratic DoiNaganuma lifting.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507687264
PII:
S 00029947(1985)07687264
Keywords:
Theta function,
modular form,
DoiNaganuma lifting,
Shimura correspondence,
Maass operator,
hyperbolic space
Article copyright:
© Copyright 1985 American Mathematical Society
