LOW COHOMOGENEITY ACTIONS 13

(b) r q : By Lemma 1.2, the linear group (G,0) extends to a linear group (G,I)

in the same c-equivalence class, where G = T9 x

G1.

(G,0) is a group of the type

discussed in subcase (a), or it is actually splitting. However, although r ( 0 ) may be

disconnected, we are looking for those subtori Tr eT°l which lead to connected diagrams

r(O), 0 = 0 |(T

r

xG'). An inclusion

Tr

e T°l is described by the "twist coefficients"

which appear in the diagram r(O). This procedure leads to the remaining linear groups

of Table III. To illustrate the above ideas we shall give two examples.

Example 1.3 : r = 2, q = k = 3. r ( 0 ) splits into three components, each of type

o (n) or c x ^ O

and one factor U(1) of T°l = U(1)3 is associated with each component. An inclusion

T

2

a T

3

is given by three

T2-weights

coj, say coj = a-frj + b#2 where {#j, i = 1,2 }

are unit weights with respect to a decomposition T

2

=

U(1)2.

To obtain r(E) from

r(O) each of the three simplices of r(O) defines a subdiagram of r(3), and they have

two vertices of type o in common. For example, a simplex o (n) of r(O) gives rise

to the diagram

n) or o (n) if b = 0

o

where the left picture indicates a 2-simplex, and the right one is a 1-simplex. r(O) is

the union of these three diagrams, and the coefficients a, b, . . should have values so that

the union is connected. The groups #45 through #47 in Table III are of this type.

Example 1.4 r ( 0 ) = o [n] o (#6a, Table III)

A circle group T

1

c T

2

is given by (atf-j -

b^2)-L,

a 0, (a,b) = 1. If b = 0 then

r(O) = o [ n l O #6b' T a b l e M| )

If a = b ( = 1) then (G,0) = (U(n),2p,n), but here c(O) = 4. For a * b and nonzero,

again c(O) = 3 and a, b are the "twist coefficients". Note the special case a = -b = 1

(#6c, Table III) :

1 _

T(O) = ^ 3 Z 3 ^

n ]

' ° = P2 ®IR [M-nllR = l m ®0 M-nllR

+

I *H ®0 M-nllR-

-1

) :