Group representations on Hilbert spaces defined in terms of<ovl><i>∂</i></ovl>_<i>b</i>-cohomology on the Silov boundary of a Siegel domain

Abstract

This surface is of the type introduced in [11], and has an induced ^-complex (as described in that paper) which is, roughly speaking, the residual part (along Σ) of the ^-complex on C. Since the action of N(Q) is complex analytic, it lifts to an action on the spaces E of this complex which commutes with db. Since the action of N(Q) is by translations, the ordinary Euclidean inner product on C is N(Q)invariant, and thus N(Q) acts unitarily in the ZAmetrics on C?(E<) defined by