12

ELDAR STRAUME

Henceforth, let G = T

r

x G\ where T

r

=

U(1)r

is the central torus and

G1

is semi-

simple. Let q be the number of irreducible summands Oj such that Oj I T r is nontrivial,

so clearly 1 r q k 3. We want to find those "simplest" diagrams r(O), i.e.,

which cannot be further simplified, and in particular cannot split into disconnected

pieces by a reparametrization of Tr.

Lemma 1.2 Let *F be a (real) faithful representation of the torus

Tr,

and let

Z = {±coi, • • -,±coq } be the subset of the weight system £}(¥) consisting of all different

pairs (±oo) in

^(VF).

TExpress each COJ as an integer vector with respect to the integer

lattice, and let A be the integer matrix with rows COJ. Then the following hold:

(i) If r = q, then A is unimodular.

(ii) If r q, then A can be extended to a unimodular matrix by adding q - r columns.

The lemma is essentially a reformulation of the fact that n(aj) = 1, where (COJ) is

the kernel of the homomorphism COJ :

Tr

- U(1). We omit the proof.

Now, let Gj, j = 1, • •, q, be the row vectors of A when ¥ = J | T

r

in Lemma 1.2. They

can be regarded as unit weights of a torus extension TQ 3

Tr.

(The central torus is "too

small" if r q, and then it has to be "saturated"). There are two obvious subcases.

(a) r = q : The above choice of unit weights defines a reparametrization of

Tr

=

U(1)r

such that for each factor U(1) there is a unique summand Oj with Oj|U(1)

nontrivial. r(O) looks like one of the following diagrams :

CH

r = 1

where the box indicates a subdiagram whose vertices are simple groups, and r(O) is

connected. We find that the case r = 3 is impossible. There is one diagram when r = 2

and then the box is the vertex SU(n). For r = 1 there are several diagrams; a typical

one is

o {n} •

where (G,0) = (U(1)xSp(n)xSp(1),[|i

l

®0V

n

][

R

+ V p ® , ^ ) . Here c(O) = 2, 3 for

n = 1, n 1, respectively.