invariant forms on representations of compact groups

invariant forms on representations of compact groups

Let

G

be a real Lie group. TFAE:

1.

Every real representation of

G

has an

invariant

positive definite form

, and

G

has at least one

faithful representation

.

2.

One faithful representation of

G

has an invariant positive definite form.

3.

G

is

compact

.

Also, any group satisfying these criteria is reductive, and its

Lie algebra

is the

direct sum

of simple algebras and an

abelian

algebra

(such an algebra is often called reductive).

Proof.

(

1

)

⇒

(

2

)

: Obvious.

(

2

)

⇒

(

3

)

: Let

Ω

be the invariant form on a faithful representation

V

. Let then

representation

gives an

embedding

ρ

:

G

→

SO

⁢

(

V

,

Ω

)

, the group of

automorphisms

of

V

preserving

Ω

. Thus,

G

is

homeomorphic

to a

closed subgroup

of

SO

⁢

(

V

,

Ω

)

. Since this group is compact,

G

must be compact as well.

(Proof that

SO

⁢

(

V

,

Ω

)

is compact: By

induction

on

dim

⁡

V

. Let

v

∈

V

be an arbitrary vector. Then there is a map, evaluation on

v

, from

SO

⁢

(

V

,

Ω

)

→

S

dim

⁡

V

-

1

⊂

V

(this is topologically a sphere, since

(

V

,

ω

)

is

isometric

to

ℝ

dim

⁡

V

with the standard norm). This is a a fiber bundle, and the fiber over any point is a copy of

SO

⁢

(

v

⟂

,

Ω

)

, which is compact by the

inductive hypothesis

. Any fiber bundle over a compact base with compact fiber has compact

total space

. Thus

SO

⁢

(

V

,

Ω

)

is compact).

(

3

)

⇒

(

1

)

: Let

V

be an arbitrary representation of

G

. Choose an arbitrary positive definite form

Ω

on

V

.
Then define

Ω

~

⁢

(

v

,

w

)

=

∫

G

Ω

⁢

(

g

⁢

v

,

g

⁢

w

)

⁢

𝑑

g

,

where

d

⁢

g

is Haar measure (normalized so that

∫

G

𝑑

g

=

1

). Since

K

is compact, this gives a well defined form. It is obviously

bilinear

, b

SO

⁢

(

V

,

Ω

)

y the linearity of integration, and

positive definite

since

Ω

~

⁢

(

g

⁢

v

,

g

⁢

v

)

=

∫

G

Ω

⁢

(

g

⁢

v

,

g

⁢

v

)

⁢

𝑑

g

≥

inf

g

∈

G

⁡

Ω

⁢

(

g

⁢

v

,

g

⁢

v

)

>

0

.

Furthermore,

Ω

~

is invariant, since

Ω

~

⁢

(

h

⁢

v

,

h

⁢

w

)

=

∫

G

Ω

⁢

(

g

⁢

h

⁢

v

,

g

⁢

h

⁢

w

)

⁢

𝑑

g

=

∫

G

Ω

⁢

(

g

⁢

h

⁢

v

,

g

⁢

h

⁢

w

)

⁢

d

⁢

(

g

⁢

h

)

=

Ω

~

⁢

(

v

,

w

)

.

For representation

ρ

:

T

→

GL

⁢

(

V

)

of the maximal torus

T

⊂

K

, there exists a representation

ρ

′

of

K

, with

ρ

a

T

-subrepresentation of

ρ

′

. Also, since every

conjugacy class

of

K

intersects

any maximal torus, a representation of

K

is

faithful

if and only if it restricts to a faithful representation of

T

. Since any torus has a faithful representation,

K

must have one as well.

Given that these criteria hold, let

V

be a representation of

G

,

Ω

is positive definite

real form

, and

W

a subrepresentation. Now consider

W

⟂

=

{

v

∈

V

|

Ω

⁢

(

v

,

w

)

=

0

⁢

∀

w

∈

W

}

.

By the positive definiteness of

Ω

,

V

=

W

⊕

W

⟂

. By induction,

V

is

completely reducible

.

Applying this to the

adjoint representation

of

G

on

𝔤

, its Lie algebra,
we find that

𝔤

in the direct sum of simple algebras

𝔤

1

,

…

,

𝔤

n

, in the sense that

𝔤

i

has no proper nontrivial ideals, meaning that

𝔤

i

is simple in the usual sense or it is abelian.
∎

Title

invariant forms on representations of compact groups

Canonical name

InvariantFormsOnRepresentationsOfCompactGroups

Date of creation

2013-03-22 13:23:40

Last modified on

2013-03-22 13:23:40

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

11

Author

bwebste (988)

Entry type

Theorem

Classification

msc 54-00

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