Pseudo Casimir operator
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, . 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.
KeywordsEisenstein Series Automorphic Form Selfadjoint Operator Casimir Operator Selfadjoint Extension
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