By Irving Kaplansky

ISBN-10: 0226424545

ISBN-13: 9780226424545

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Konarski, Properties of projective orbits of actions of affine algebraic groups, to appear. CKos] B. Kostant, Lie group representations on polynomial rings, Amer. J. of Math. 8£, 327-404. I. Lieberman, Rationality and holomorphic vector fields, to. appear. _ Holomorphic vector fields on projective varieties, Proc. Symp. in Pure Math 30 (1977), 273-276. J J. B. Carrell 37 CM] Y. Matsushima, Holomorphic vector fields on compact Kaehler manifolds . Regional Conference Series In Math. No. 7, A. S. (1971) - CMS] J.

I) the geometric realization homeomorphic to the unit sphere (ii) Then: S the rational homology in H#(D |r| of r is E. =S = : H ^ is 44 Characters of Groups of Lie Type given by: H^CD • 0 except in dimensions and 0 and n-1 HQ(r) = Hnn(r) s $ as rational vector spaces. (iii) the rational homology group the trivial representation of W, group H r n_i( ) HQ(D affords and the homology affords the sign representation We shall give a sketch of the proof. e. 3a } be the fundamental system of roots such that the reflections and let C }- .

In the character theory, of G Lj, X 5 € Irr(G) I T X, for some u of LT, for J c R, J ^ are uniquely deter- Thus the main problems are to find all cuspi- dal characters, and to decompose the induced characters IjX from cuspidal irreducible characters of L,. 4) DEFINITION. endomorphism 5 •* c* The duality operation is a of ch(EG) 2- defined by ( - l ) l J ' l TT T TT£ , C € c h ( E G ) . 3). The fact that operation in ii) ch(lCG) C •*• 5* ch(IDW) (see is a duality will be proved in § 3- The dual of the trivial character is the Stein- berg character of G, 1 * = StQ, where a character of degree |G| , the StQ € Irr(G) is p-part of the order of G.

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