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Topic: Haar measure
Replies: 2   Last Post: May 4, 2011 8:25 AM

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 Jyrki Lahtonen Posts: 880 Registered: 12/8/04
Re: Haar measure
Posted: May 4, 2011 8:25 AM

On 4.5.2011 8:50, dushya wrote:
> Hi,
> Is Haar measure over unitary group U(N) invariant under inversion? I
> have only very vague idea of what Haar measure over a group means; It
> is usually constructed so as to be invariant under right (and/or left)
> multiplication by elements of the group; but are there any conditions
> on a group (/ top group) under which its Haar measure is invariant
> under inversion? in particular what's the case for U(N)?

For compact groups it should be the case. An argument could go something
like the following. If m is a left (or right) invariant measure, then m
precomposed with the inverse mapping is a right (or left) invariant
measure (because the inverse of a product is the product of inverses in
the opposite order). In a compact connected group a left invariant
measure is also right invariant, so putting these pieces together gives
us the claim.

Cheers,

Jyrki

Date Subject Author
5/4/11 dushya
5/4/11 A N Niel
5/4/11 Jyrki Lahtonen