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.
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