Selfadjoint extension of the Casimir operator

Abstract

In 1.2.6 and 1.5.7 we discussed the selfadjoint extension of the differential operator L _r . This concerned the modular case. The extension was an operator in a Hilbert space H(r) for r ∈ ℝ. Its eigenfunctions were stated to be modular forms, and $\frac{{|r|}}{2}\left( {1 - \frac{{|r|}}{2}} \right)$ its smallest eigenvalue. In this chapter we prove these statements, in the more general setting of Part I. We work in a Hilbert space H(x,l) depending on a unitary character x of $\widetilde \Gamma$, and a (real) weight l suitable for x. In Section 6.1 we define this Hilbert space as a completion of the space of all smooth x-l-equivariant functions with compact support in Y.