18 A. W. KNAPP

is exact. But

Hom1(U(q), (t\mu)* 0 V)

~

Hom1(U(u) 0 U(l), (t\mu)* 0 V)

~

Hom1(U(l), Homc(U(u), (t\mu)* 0 V))

~

Hornc(U(u), (t\mu)* 0 V)

~

Hornc(U(u) 0

t\mu,

V)

=

Hornc(Xm, V),

where

Xm

is the Koszul (projective) resolution of C in C(u, 1). (See [9, Theorem

4.6).)

It

is easy to check that the differentials for (5.13) are the ones induced from

the differentials for

Xm,

and hence (5.13) is exact. This completes the proof.

REFERENCES

1.

R.

Aguilar-Rodriguez, Connections between representations of Lie groups and sheaf coho-

mology, Ph.D. dissertation, Harvard University, 1987.

2. L. Barchini, A. W. Knapp, and R. Zierau, Intertwining opemtors into Dolbeault cohomology

representations, J. Func. Anal. 107 (1992), 302-341.

3. R. J. Baston and M. G. Eastwood, The Penrose Transform: Its Intemction with Repre-

sentation Theory, Oxford University Press, Oxford, 1989.

4.

R.

Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203-248.

5. R. Godement, Sur les relations d'orthogonalite de V. Bargmann, I and II, C. R. Acad.

Sci. Paris 225 (1947), 521-523 and 657-659.

6. P. Griffiths and W. Schmid, Locally homogeneous complex manifolds, Acta Math. 123

(1969), 253-302.

7. Harish-Chandra, Discrete series for semisimple Lie groups II, Acta Math. 116 (1966),

1-111.

8. A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on

Examples, Princeton University Press, Princeton, 1986.

9. ___ , Lie Groups, Lie Algebms, and Cohomology, Princeton University Press, Prince-

ton, 1988.

10. A. W. Knapp and D. A. Vogan, Duality theorems in relative Lie algebm cohomology,

duplicated notes, Cornell University and Massachusetts Institute of Technology, 1986.

11. B. Kostant, Orbits, symplectic structures, and representation theory, Proceedings of the

U.S.-Japan Seminar on Differential Geometry, Kyoto, 1965.

12. R. P. Langlands, Dimension of spaces of automorphic forms, Algebraic Groups and Dis-

continuous Subgroups, Proc. Symp. in Pure Math. 9, American Mathematical Society,

Providence, 1966, pp. 253-257.

13. W. Schmid, Homogeneous complex manifolds and representations of semisimple Lie

groups, Ph.D. dissertation, University of California, Berkeley, 1967, Representation The-

ory and Harmonic Analysis on Semisimple Lie Groups, Math. Surveys and Monographs,

American Mathematical Society, Providence, 1989, pp. 223-286.

14. ___ , On the realization of the discrete series of a semisimple Lie group, Rice University

Studies, Vol. 56, No. 2, 1970, pp. 99-108.

15. ___ , L

2

-cohomology and the discrete series, Ann. of Math. 103 (1976), 375-394.

16. N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, 1951.

17. J. A. Tirao and J. A. Wolf, Homogeneous holomorphic vector bundles, Indiana U. Math.

J. 20 (1970), 15-31.

18. D. A. Vogan, Representations of Real Reductive Lie Groups, Birkhauser, Boston, 1981.

19. R. 0. Wells, Differential Analysis on Complex Manifolds, Springer-Verlag, New York,

1980.

20. R. 0. Wells and J. A. Wolf, Poincare series and automorphic cohomology on flag domains,

Ann. of Math. 105 (1977), 397-448.