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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Unitary structures on cohomology


Authors: C. M. Patton and H. Rossi
Journal: Trans. Amer. Math. Soc. 290 (1985), 235-258
MSC: Primary 22E45; Secondary 32F10, 32L10
MathSciNet review: 787963
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0787963-6
PII: S 0002-9947(1985)0787963-6
Article copyright: © Copyright 1985 American Mathematical Society