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.