U(p,q) may be represented on
Hp−1(P+,𝒪(λ)) where P+ is an open orbit of U(p,q) in CPp+q−1 and λ is a
homogeneous holomorphic line bundle. Although it is not their definition, the
twistor elementary states turn out to be the U(p) ×U(q)-finite vectors. We
show that Hp−1(P+,𝒪(λ)) has a natural Fréchet space topology in which
these states are dense. Using this, we show that a certain Hermitian product
defined on Hp−1(P+,𝒪(λ)) is positive definite and hence complete a twistor
construction of a family of unitary representations of U(p,q), namely the ladder
representations. Though these representations are well-studied by other means, we
feel that their realization on cohomology is especially natural and merits special
investigation.
