10 ELDAR STRAUME

from the weight system Q(O), cf. (1.6) and (1.7) of Chapter II, so all cases with

c(O) = dim^O - dim (G/G

0

) 3

are readily found by some straightforward estimates of dimensions.

Next, assume O is reducible. By direct checking there are just a few combinations

0 = 0 ^ + O2, with c(O^) + c(32) - 3, leading to c(3) 3, namely, when G = SO(n),

G2 or Spin(n), 5 n 9, see Table II and III.

Case (ii) O =

x¥l

® 42 g g 4 ^ , q 2, O irreducible.

Let K(n) = O(n), U(n) or Sp(n), with standard representation 8

n

on F

n

=

lRn,

d

n

or

Hn,

respectively. The isotropy types of the tensor product linear group

( 5 ) (K(m) x K(n),5

m

% 5

n

), m n ,

are given by the family

( 6 ) A[K(t-|) x • • • x K(t

r

)] x K(m-t)x K(n-t), 0 Z tj = t m,

where A[ ] denotes the diagonal imbedding A -» (A, A) of the "blocks" K(tj) into

K(t)x K(t) c K(m)x K(n). So the principal isotropy type (G^) corresponds to tj = 1,

t = m in (6), consequently

( 7 ) c(5

m

®

F

8n) = m (m n)

In particular, any linear group (G,0) = (G-j x 02,^ 1 %^2)

w i t n m =

d i m p ^ ^

n =

dim[p4/2,

is a subgroup of the group in (5), so m c(O).

Assume first q = 2, G semisimple and m 2. Since dim^O dim G + 3, it is

straightforward to verify that, in addition to the semisimple component of (5), there

are just two more cases, namely (SO(3)xSpin(7),p3®A7) and (Sp(1)xSp(n),

S

3

v

1

®

[ H

v

n

) .

Assume q 3, G semisimple. Since dim G increases "additively" with q, whereas

dim 3 increases "multiplicatively, the ratio (dim G):(dim O) is generally too small.

By checking "critical" cases such as (8) below, we find that c(O) 3 is only possible

when O = P3 ^ ( v j (% v^) = P3 (% P4, where G = SO(3)xSO(4) is the effective

group. Note for example