Vol. 65, No. 1, 1976

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Group representations on Hilbert spaces defined in terms of b-cohomology on the Silov boundary of a Siegel domain

Hugo Rossi and Michele Vergne

Vol. 65 (1976), No. 1, 193–207

Let Q be a Cn-valued quadratic form on Cm. Let N(Q) be the 2-step nilpotent group defined on Rn ×Cm by the group law

(x,u) ⋅(x′,u′) = (x + x′ + 2Im 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 +x0) +i(2Q(u,u0)+ Q(u,u0),u0 + u0),

where g = (x0,u0). The orbit of the origin is the surface

Σ = {(z,u) ∈ C    ;Im z = Q(u,u)}.

This surface is of the type introduced in [11], and has an induced b-complex (as described in that paper) which is, roughly speaking, the residual part (along Σ) of the -complex on Cn+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 b. 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 L2-metrics on C0(Eq) defined by

     -- 2  ∫      2
∥ΣaIduI∥ =    Σ|aI| dV

where dV is ordinary Lebesgue surface measure. In this way we obtain unitary representations ρq of N(Q) on the square-integrable cohomology spaces Hq(E) of the induced b-complex.

Mathematical Subject Classification 2000
Primary: 22E45
Secondary: 32M15
Received: 8 January 1976
Published: 1 July 1976
Hugo Rossi
Michele Vergne