I. INVARIANT CONES IN K-MODULES
In this first section we collect some material concerning the geometry of
wedges in vector spaces and invariant wedges in finite dimensional modules of
compact groups. A rather complete reference for the geometry of wedges is the
first chapter in [HHL89]. We recall the definitions and propositions which are
needed later and sketch some of the elementary proofs. We will see in Section III
that it is possible to classify invariant wedges in Lie algebras by their intersections
with certain subalgebras which are the set of fixed points of an action of a compact
group of automorphisms of the Lie algebra. This is where the theory of wedges
in K-modules is needed. Throughout this section L denotes a finite dimensional
vector space.
Definition 1,1. Let L be a finite dimensional vector space. A subset W is
called a wedge if it is a closed convex cone. The vector space H(W) := W fl W
is called the edge of the wedge. We say that W is pointed if the edge of W is
trivial and that W is generating if W W = L. We denote the dual of L with
L. The dual wedge W* C L is the set of all functionals which are non-negative
on W. We set
algint W := {x G W : u(x) 0 for all u e W* \ H(W*)}.
m
Proposition 1.2. We identify the dual of L with L. Then the following
assertions hold for a wedge W C L:
(1) (W*)* = W.
(2) W is generating iff W* is pointed and conversely W is pointed iff W* is
generating.
(3) UJ e algint W* iff w(x) 0 for all x G W \ H(W) and algint W is the
interior of W with respect to the subspace W W of L.
(4) For a family {W{)i^i of wedges in L we have that
(f| Wiy
= ]TW?
and (£,
W
V
=
fl
W
i-
iei iei iei iei
Proof. (1) [HHL89, 1.1.4]
(2) [HHL89, 1.1.7]
(3) [HHL89, 1.2.21]
(4) [HHL89, 1.1.6]
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