The Bott-Borel-Weil Theorem for direct limit groups

Authors:
Loki Natarajan, Enriqueta Rodríguez-Carrington and Joseph A. Wolf

Journal:
Trans. Amer. Math. Soc. **353** (2001), 4583-4622

MSC (1991):
Primary 222E30, 22E65; Secondary 22C05, 32C10, 46G20, 22E70

DOI:
https://doi.org/10.1090/S0002-9947-01-02452-7

Published electronically:
July 3, 2001

MathSciNet review:
1650034

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Abstract | References | Similar Articles | Additional Information

Abstract: We show how highest weight representations of certain infinite dimensional Lie groups can be realized on cohomology spaces of holomorphic vector bundles. This extends the classical Bott-Borel-Weil Theorem for finite-dimensional compact and complex Lie groups. Our approach is geometric in nature, in the spirit of Bott's original generalization of the Borel-Weil Theorem. The groups for which we prove this theorem are strict direct limits of compact Lie groups, or their complexifications. We previously proved that such groups have an analytic structure. Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. One of the technical problems here is that, in general, the limit Lie algebras will have root systems but need not have root spaces, so we need to develop machinery to handle this somewhat delicate situation.

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Additional Information

**Loki Natarajan**

Affiliation:
Department of Mathematics 0112, University of California at San Diego, La Jolla, California 92093

Email:
loki@euclid.ucsd.edu

**Enriqueta Rodríguez-Carrington**

Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854

Email:
carringt@math.rutgers.edu

**Joseph A. Wolf**

Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, California 94720–3840

Email:
jawolf@math.berkeley.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02452-7

Keywords:
Direct limit,
direct limit Lie group,
diagonal Lie algebra,
diagonal embedding,
Borel--Weil Theorem,
Bott--Borel--Weil Theorem,
direct limit representation,
inverse limit representation,
direct limit cohomology,
inverse limit cohomology,
infinite dimensional Lie group

Received by editor(s):
May 6, 1997

Received by editor(s) in revised form:
July 6, 1998, and April 26, 2000

Published electronically:
July 3, 2001

Additional Notes:
LN: research partially supported by NSF Grant DMS 92 08303.

ERC: research partially supported by PSF–CUNY Grant 6–66386.

JAW: research partially supported by NSF Grants DMS 93 21285 and DMS 97 05709.

Article copyright:
© Copyright 2001
American Mathematical Society