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^
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