Dirichlet series and automorphic forms on unitary groups
Author:
Tobias Orloff
Journal:
Trans. Amer. Math. Soc. 290 (1985), 431456
MSC:
Primary 11F55
MathSciNet review:
792806
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Abstract: In a special case our unitary group takes the form Here is a skewHermitian matrix with entries in an imaginary quadratic number field . We suppose that has signature . This group acts naturally on the symmetric domain If with the ring of integers in , then an automorphic form with respect to has an expansion . The functions are theta functions. Given another automorphic form with an expansion we define a Dirichlet series . Here is a certain positive definite inner product on the space of theta functions. The series is obtained as an integral of Rankin type: with a subgroup of "translations". The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over . In the case our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type.
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DOI:
http://dx.doi.org/10.1090/S00029947198507928060
PII:
S 00029947(1985)07928060
Article copyright:
© Copyright 1985
American Mathematical Society
