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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