Correction to
The geometry and structure of isotropy irreducible homogeneous spaces
by
JOSEPH A. WOLF
University o f California Berkeley, CA, U.S.A.
(Acta Math. 120 (1968), 59-148)
Professors McKenzie Wang and Wolfgang Ziller pointed out to me that Theorem 4.1 omits the spaces Sp (n)/Sp (1) x SO (n) and SO (4n)/Sp (1) x Sp (n), which are isotropy irreducible for n > 1. The gap in the proof is in the argument of Case 2 on page 69, where it is assumed that the representation ql is nontrivial, which is the case only for p l > 1.
Since Sp (2)/Sp (1)xSO (2)=Sp (2)/U(2), which is hermitian symmetric, the correct state- ment is:
4.1. THEOREM. The only simply connected nonsymmetric coset spaces G/K o f compact connected Lie groups, where (a) G acts effectively, (b) G is a classical group, (c) rank (G)>rank (K), (d) K is not simple, and (e) K acts R-irreducibly on the tangent space, are the following:
(1) S U ( p q ) / S U ( p ) x S U ( q ) , p > l , q > l , pq>4. Here G=SU(pq)/Zm where m = l c m ( p , q ) , K={SU(p)/Zp}X{SU(q)/Zq}, and the isotropy representation is the tensor product o f the adjoint representations o f SU (p) and SU (q).
(2) S p ( n ) / S p ( 1 ) x S O ( n ) , n>2. Here G=Sp(n)/Z2. I f n is even then K=(Sp(1)/Z2} x {SO(n)/Z2} and the isotropy representation is given by
2 2 2
o|174 if n = 4 , by
/Sn > 4 .
9 t - 8 4 8 2 8 8 A c t a Mathematica 152. Imprim~ le 17 Avril 1984
142 J.A. wouF
I f n is odd then K = {Sp (1)/Z2} • SO (n) and the isotropy representation is given by
2 2
O|
(3) SO (4n)/Sp (1) x Sp (n), n > 1. Here G= SO (4n)/Zz, K= {Sp (1)/Z2} x {Sp (n)/Z2}, and the isotropy representation is given by
2 I
o | if n = 2 , by
2 I
o| /f n > 2.
The table on pages 107-110 should, accordingly, be modified by insertion o f two families o f spaces, c o r r e s p o n d i n g to items 2 and 3 o f the c o r r e c t e d T h e o r e m 4.1. T h e table information which is not contained in the statement o f T h e o r e m 4.1 above, is:
G/ K Z N~( K)/ZK K c G Conditions
I I
Sp(n)/Sp(l)xSO(n) (1} {1} O| n odd, n>2
1 I I
{1} S 3 0 | 1 7 4 n=4
I I
{1} Z 2 O| O n even, n>4
1 I
SO(4n)/Sp(1)xSp(n) {1} {1} O| n>l
T h e r e is no change in the rest o f the paper. In particular, Sections 13 and 14 are unchanged b e c a u s e , in the additional cases noted here, the isotropy r e p r e s e n t a t i o n is absolutely irreducible so t h e r e is no invariant almost c o m p l e x structure, and its Sp(1) f a c t o r acts b y a 3-dimensional r e p r e s e n t a t i o n so there is no invariant quaternionic structure.
Received May 18, 1983
