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Pseudo Casimir operator

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Abstract

In this chapter we consider — in a more general context the operator α A discussed in 1.6.2, and its resolvent. We call a A the pseudo Casimir operator, as it is the generalization of the pseudo Laplacian of Colin de Verdière, [12]. We use the resolvent to prove a central result of this book, Theorem 9.4.1. This theorem gives the existence of meromorphic families of automorphic forms that satisfy certain conditions on their Fourier terms. The theorem implies that — generically on an open neighborhood in v × ℂ of any point in v r x ℂ there are as many families of automorphic forms as the Maass-Selberg relation allows, see Section 9.5. In the context of Section 1.6, Theorem 9.4.1 amounts to the construction of \(^a\widetilde E\) (r, s) in 1.6.4.

Keywords

  • Eisenstein Series
  • Automorphic Form
  • Selfadjoint Operator
  • Casimir Operator
  • Selfadjoint Extension

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© 1994 Springer Basel AG

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Bruggeman, R.W. (1994). Pseudo Casimir operator. In: Families of Automorphic Forms. Modern Birkhäuser Classics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0336-2_9

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  • DOI: https://doi.org/10.1007/978-3-0346-0336-2_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0346-0335-5

  • Online ISBN: 978-3-0346-0336-2

  • eBook Packages: Springer Book Archive