Unitary structures on cohomology

Patton, C. M.; Rossi, H.

doi:10.1090/S0002-9947-1985-0787963-6

Unitary structures on cohomology
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Trans. Amer. Math. Soc. 290 (1985), 235-258 Request permission

Abstract:

Let ${{\mathbf {C}}^{p + q}}$ be endowed with a hermitian form $H$ of signature $(p,q)$. Let ${M_r}$ be the manifold of $r$-dimensional subspaces of ${{\mathbf {C}}^{p + q}}$ on which $H$ is positive-definite and let $E$ be the determinant bundle of the tautological bundle on ${M_r}$. We show (starting from the oscillator representation of ${\text {SU}}(p,q))$ that there is an invariant subspace of ${H^{r(p - r)}}({M_r},\mathcal {O}(E(p + k)))$ which defines a unitary representation of ${\text {SU}}(p,q)$. For $W \in {M_p},\operatorname {Gr}(r,W)$ is the subvariety of $r$-dimensional subspaces of $W$. Integration over $\operatorname {Gr}(r,W)$ associates to an $r(p - r)$-cohomology class $\alpha$, a function $P(\alpha )$ on ${M_p}$. We show that this map is injective and provides an intertwining operator with representations of ${\text {SU}}(p,q)$ on spaces of holomorphic functions on Siegel space

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