group representations on hilbert spaces defined in ... - Project Euclid
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GROUP REPRESENTATIONS ON HILBERT SPACES DEFINED IN TERMS OF 36-COHOMOLOGY ON THE SILOV BOUNDARY OF A SIEGEL DOMAIN H. Rossi AND M. VERGNE Let Q be a O-valued quadratic form on Cm. Let N(Q) be the 2-step nilpotent group defined on Rn X Cm by the group law (x, u)-(x'f u') = (x + %' + 2 Im Q(u, u'),u + u') . Then N(Q) has a faithful representation as a group of complex affine transformations of Cn+m as follows: g>(z, u) = (z + Xo) + ί(2Q(u, u0) + Q(u, u0), uQ + u0) ,
where g = (xQ, u0). The orbit of the origin is the surface Σ = {{z, u) e Cn+m; Imz = Q{u, u)}.
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 n+m . Since the action of N(Q) is complex analytic, it lifts to an action on the spaces Eq of this complex which commutes with db. Since the action of N(Q) is by translations, the ordinary Euclidean inner product on Cn+m is N(Q)invariant, and thus N(Q) acts unitarily in the ZAmetrics on C?(E
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