LOW COHOMOGENEITY ACTIONS 9

Next, a summand Oj which is a representation of only one factor K = G; or U(1) is a

O-simplex, and is depicted by a loop . Hence, a 0-simplex is not a vertex! Here a label

p e Z

+

indicates that dim^Oj = p if K is simple, and otherwise K = U(1) and Oj =

[ (n^P]^. If there is no label then the loop indicates the "standard" representation.

Finally, we also encounter 1-simplices like S

2

5

n

® (J), A

2

5

n

8 |, An® ty. Then the

simplex (or edge) is labelled S, A or A in the obvious way. Since

S3V!

is a 2-

dimensional Sp(1 ^representation over the quaternions fH,the tensor product in

Example 1.1(a) below has a label 2 near the corresponding vertex Sp(1).

Examples 1.1

(a) • {n} : S ^ j ^ V p

(b)

P

C ^ {m} : [OI^PJIR + [( ^ ®cvm]iR

(c) {m] B_o_9 [n] : [v

m

®([( ^ ) P ]

R

+ [( ^ % ^

n

]

K

(d) © P-o-*— *-(7) : P m ^ ^ ^ P l i R + C p

2

) ^

K

A

7

Remark If G is semisimple, then (G,0) is nonsplitting if and only if r(O) is

connected. However, in general r(O) depends on the decomposition

Zr

=

U(1)r

of the

central torus

Zr.

We say (G,0) is nonsplitting if no choice of decomposition can make

r(0) disconnected.

(C ) Classification We shall describe the procedure leading to a classification of

all (G,E) with c(O) 3. There are several case by case considerations. By (1) we may

and shall assume (G,0) is nonsplitting, and we shall discuss the following three cases

separately :

(i) G simple,

(ii) G nonsimple, I irreducible,

(iii) G nonsimple, O reducible.

Case (i) Determine the list of all real irreducible representations O of simple

groups G satisfying dim O dim G + d. Here we take d = 3, although at this point it is

also a good idea to include the cases d = 4 (resp. 6) if O has a complex (resp.

quaternionic) form, since these will be needed in Case (ii), cf. (9) below. (It is really

a routine exercise to identify all simple linear groups of cohomogeneity 8, but we

shall not tabulate them here.)

Clearly, only d = 3 can possibly lead to a G-representation with c(O) 3. Note that

there is an algorithm for calculating the dimension of a principal isotropy group G^