bd) cz + d \bd - Math Berkeley - University of California, Berkeley

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Theorem 5.7 had been suspected by P. A. Griffiths in the case where GQC\PX is compact and G is simple; he felt that one should be able to join points of GQ(X).
THE ACTION OF A REAL SEMISIMPLE GROUP ON A COMPLEX FLAG MANIFOLD. I: ORBIT STRUCTURE AND HOLOMORPHIC ARC COMPONENTS 1 BY JOSEPH A. WOLF Table of Contents 1. Introduction Chapter I. Decomposition of a complexflagmanifold into real group orbits 2. Basic facts on the orbit structure 3. The closed orbit 4. Open orbits: construction, covering and counting 5. Open orbits: coset space structure and holomorphic functions 6. Open orbits: invariant measure 7. Integrable orbits Chapter II. Decomposition of a real group orbit into complex analytic pieces. 8. Holomorphic arc components 9. Global conditions for the components of an orbit Chapter III. Hermitian symmetric spaces 10. Hermitian symmetric spaces: orbit structure and holomorphic arc components 11. Hermitian symmetric spaces: compact sub varieties and Siegel domain realizations References

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1. Introduction. This paper describes a topic that is of some interest in Lie groups and in differential and algebraic geometry. The topic shows good promise of being the "correct" context for explicit realization of those series of irreducible unitary representations of semisimple Lie groups t h a t come into the Plancherel formula, so it probably is also of interest in harmonic analysis. We start with an example. Let X be the Riemann sphere, viewed as CKJ { } via stereographic projection. Then the group G of all holomorphic automorphisms of X consists of the linear fractional transformations : z-•

(

\b

d)

j

cz + d

det (

\b

)

d)

Detailed version of an address delivered at the Riverside meeting of the American Mathematical Society on November 16, 1968, by invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings; received by the editors May 21, 1969. 1 Research partially supported by N.S.F. Grants GP-2439, GP-5798 and GP-8008, and by an Alfred P. Sloan Research Fellowship. 1121

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that expresses G as a complex matrix group SL(2, C)/{ ± / } . Let Go = SL(2, R ) / { ± / } , subgroup of elements represented by real matrices, so G is the complexification of Go. The complex group G is transitive on X but the real group Go is not. In fact X is the union of two open Go-orbits Go(VÏ) = {zEC:Imz>0}

and GoC-V" 11 !) = {* GIC: lm z < 0}

and the closed Go-orbit Go(0) = { z e C : I m 2 = 0 } U { o o J . The Cayley transform

1 /

Vz-Ï\

1

of X relates these orbits, so they are G0(#o)

an

d

GQ(C2X0)

open,

G0(cx0) closed,

x0 = y/— 1.

The series of irreducible unitary representations of Go are the "discrete series, " the "principal series" and the "complementary series. " Discrete series representations are the square integrable ones; they can be realized on the spaces of L2-holomorphic sections of Gohomogeneous holomorphic line bundles over the open orbits. Principal series representation can be realized on L 2 (relative to invariant measure for the rotation subgroup of G0) sections of certain Go-homogeneous complex line bundles over the closed orbit. The complementary series does not contribute to Plancherel measure. In general we start with a complex manifold X = G/P, where G is a complex semisimple Lie group and P is a complex subgroup of the sort called parabolic. The latter means any of the following conditions, which are equivalent: X = G/P is compact, X is a compact simply connected kaehler manifold, the complex manifold X is a projective variety, X is a closed G-orbit in a projective representation. Then X is a complex flag manifold. Let Go be a real form of G, i.e. a real Lie group whose complexification is G. Then Go acts on X as a subgroup of G. It turns out that there are only a finite number of

1969]

ACTION OF A REAL SEMISIMPLE GROUP

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Go-orbits on X and that there is just one closed orbit. The decomposition of X into Go-orbits, and especially the structure of the open orbits, is studied in some detail in Chapter I. We then take an arbitrary orbit Go(x)QX and partition it into "maximal complex analytic pieces" that we call its holomorphic arc components. If z^Go(x) then S[z] denotes the holomorphic arc component of Go(x) through z and N[Z],0 denotes the identity component of the Lie subgroup {g£Go: gS[Z) = S[Z]} ; then S[Z] = N[Z],0(z) so it is a real submanifold of X. The best situation is that in which some (hence every) S[Z](ZGo(x) carries a positive iV^.o-invariant Radon measure; then we say that Go(*0 is measurable. If Go(x) is measurable and z£Go(aO then S[Z] is a complex submanifold of X, in fact is an open iV[*],o-orbit on the complex flag submanifold N[z]tQ(z) of X, and its invariant measure is the volume element of an N[z\ , 0 -invariant, possibly indefinite signature, kaehler metric. For example, the closed Go-orbit on X is always measurable, and we have a method for deciding measurability of the open Go-orbits. Holomorphic arc components and global conditions on them such as measurability of the orbit, are studied in Chapter II. In Chapter III we work out the hermitian symmetric case in complete detail, extending earlier work of Korânyi-Wolf [2], WolfKorânyi [IS] and Takeuchi [9]. This works well because every orbit is measurable. The Riemann sphere, described above, is the simplest of the hermitian symmetric cases; in fact its extreme simplicity can be misleading, so we look at the example of a general complex Grassmann manifold. Let X be the complex Grassmann manifold consisting of the kplanes through the origin in Cw. If {»i, • • • , » * } C C n is linearly independent then v\ A • • • NokÇiX denotes its span. The complex general linear group acts on X by giviA • • • At>*-»2(i>i)A • • • Agfa); there the scalars act trivially so the resulting group of transformations is the complex Lie group G = GL(», C)/{aI: a^O} = SL(», C)/{e 2 ™'»/} of complex dimension n2 — 1. Let {«1, • • • , « * } be a fixed basis of O and #0 = 01 A * • * AejcÇzX our base point. Then the isotropy subgroup of G at Xo is the complex subgroup P = {g G SL(n, C):gx0 = * 0 }/{e 2 " v / "/}

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of complex dimension k2+(n — k)2+nk — 1. That exhibits I as a complex flag manifold2 G/P of complex dimension k(n—k). Consider the hermitian form ( ]QL ZV», ]T)* wje^) = — X a st where fliA • • • Ai>k-t is a subspace of eiA • • • A^* and WiA ' * • A ^ is a subspace of e*+iA • • • A^n. If xÇzX has signature (& — t, /), its orthogonal projection to ei A • • ' Atffc is a subspace t>i A * • • Avk-u and its orthogonal projection to £*+iA • • • A^n is some WiA • • • Awù by definition fit(x)=ViA ' ' • Avk-tAwiA • • • Aw*. It is not too hard to check that this defines a holomorphic fibre bundle with total space and base as mentioned, whose fibre over k(