Let G be a connected, simply connected nilpotent Lie group, H a Lie subgroup, and a an irreducible unitary representation of H. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of Ind(H î G, a). One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is

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... tric in nature and amounts to an extension of the familiar fact that for almost all polynomials over R of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if G is a complex nilpotent Lie group and H a complex Lie subgroup, then the multiplicity is a.e. constant.