Description of G-bundles over G-spaces with quasi-free proper action of discrete group II

According [1, p.1] the bundle ξ separates as the sum of its G-subbundles:

where the index runs over all (unitary) irreducible representations ρ k : H-→U(V k ) of the group H and, as a H-bundle ξ k can be presented as the tensor product:

where the action of the group H over the bundles η k is trivial, V k denotes the trivial bundle with fiber V k and with fiberwise action of the group H, defined using the linear representation ρ k .

The particular case ξ = η k V k was described in the previous article [1]. According [1, p.14] the bundle ξ k can be obtained as the inverse image of a mapping

where

is the group of equivariant automorphisms of the space X k as a vector G-bundle over the base G 0 . So, the bundle ξ can be given by a mapping

Consider the vector bundle over the discrete base G 0

Define a fiberwise action G × X ρ → X ρ by the formula

(2)

Definition 1 The bundle X ρ -→G 0 with the just defined action is called the canonical model for the representation ρ.

By Aut G (X ρ ) we denote the group of equivariant automorphisms of the canonical model X ρ as a vector G-bundle over the base G 0 with fiber k (F k ⊗ V k ) and canonical action of the group G.

Lemma 1 There exists a monomorphism

Proof.As before, an element of the group Aut G (X ρ ) is an equivariant mappingA a such that the pair (A a , a) defines a commutative diagram

By the lemma 1 [1] applied to group of automorphisms Aut G (X ρ ), for A a ∈ Aut G (X ρ ), we have

Note that this restriction is G-equivariant.

This is clearly a homomorphism: it is a product of restrictions over invariant subspaces. Lets prove that it is injective.

In order to prove that the image of i is closed, note that it coincides with those automorphisms which commute with the inclusion

its restriction defines an element in Aut G (X ρ ) and the diagram

where

is the epimorphism of lemma 2 [1, p. 9] and

Corollary 1 It takes place an exact sequence of groups

using the monomorphisms

. This means that, for every k, there is a

Denote by Vect G (M, ρ) the category of G-equivariant vector bundles ξ over the base M with quasi-free action of the group G over the base and normal stationary subgroup H < G.

Then, by lemma 1 and the observations on p. 13 in [1] in terms of homotopy we have

Vect

Denote by Bundle(X, L) the category of principal L-bundles over the base X.

Proof. We already have a monomorphism

with the property that there exist h α,k (x) ∈ G 0 such that

does not depends on k. Lets show that can be found transition functions with the property that

for some cocycle a αβ (x).

Because the group G 0 is discrete, for an atlas of connected charts with connected intersections, we can assume that pr k •Ψ αβ (x) = a αβ,k and h α,k (x) = h α,k ∈ G 0 do not depend on x and, therefore,

Vect G (M, ρ).

(

Proof.We will follow the proof theorem 3 in [1, p. 14]. Given a bundle ξ ∈ Bundle(X, Aut G (X ρ )) with transition functions

we obtain transition functions

defining an element M ∈ Bundle(X, G 0 ) together with a projection ξ-→M . Changing the fibers Aut G (X ρ ) by X ρ , we obtain an action of the group G, that reduces over the base to the factor group G 0 .

Lets rewrite this in terms of homotopy.

Corollary 2 If the space X is compact, then

2 The case when the subgroup H < G is not normal

Consider an equivariant vector G-bundle ξ over the base

Let H < G be a finite subgroup. Assume that M is the set of fixed points of the conjugation class of this subgroup, more accurately

and that there is no more fixed points of the conjugation class of H in the total space of the bundle ξ; here we have denoted by M H the set of fixed points of the action of the subgroup H over the space M , N (H) the normalizer of the group H in G and we are using the equality gM H = M gHg -1 and the fact that lHl -1 = gHg -1 if and only if g -1 l ∈ N (H). Lets denote by F ξ the family of subgroups of G having non-trivial fixed points in the total space of the bundle ξ, i.e.

This is a partial ordered set by inclusions and is closed under the action of the group G by conjugation 1 . Also, the action

preserves the order.

Definition 2 We will say that H < G is the unique, up to conjugation, maximal subgroup for the G-bundle ξ if every conjugate gHg -1 is maximal in F ξ and there is no more maximal elements in this family.

In this section will assume in any case, that H < G is the unique, up to conjugation, maximal subgroup.

If there is an x ∈ M H ∩ M gHg -1 then, the point x is fixed under the action of the subgroup generated by H and gHg -1 , but this group is not contained in any of the subgroups of the form lH -1 l, l ∈ G.

Lemma 3 If the condition (7) holds, then the G-bundle ξ can be presented as a disjoint union of pair-wise isomorphic bundles with quasi-free action over the base.

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