Quantization of the action of U( k, l) on [formula omitted]2( K + l)

Abstract

The natural action of U( k, l) on C k + l leaves invariant a real skew non-degenerate bilinear form B, which turns C k + l into a symplectic manifold ( M, ω). The polarization F of M defined by the complex structure of C k + l is non-positive. If L is the prequantization complex line bundle carried by ( M, ω), then U( k, l) acts on the space U of square-integrable L ⊗ ΛF ∗ forms on M, leaving invariant the natural non-degenerate, but non-definite, inner product ((·, ·)) on U . The polarization F also defines a closed, densely defined covariant differential ∇̃ on U which is U(k, l)-invariant. Let ⊥ denote orthocomplementation with respect to ((·, ·)). It is shown that the restriction of ((·, ·)) to the U(k, l)-stable subspace Ĥ (Ker ∇̃) ∩ (Im ∇̃) ⊥ is semi-definite and that the unitary representation of Uk, l on the Hilbert space H arising from Ĥ by dividing out null vectors is unitarily equivalent to the representation of U( k, l) obtained from the tensor product of the metap ectic and Det −1 2 representations of MU( k, l), the double cover of U( k, l).