explicit integrals over a Lie group

Question

I am looking for families of invariant integrals $\int_G dg f(g)$ (where $dg$ is a Haar measure) over a semisimple Lie group that can be evaluated in closed form, together with references where I can find proofs.

I don't know yet what I'll need, but $f$ can be quite involved, so I want to study the techniques that are available for deriving such formulas. My primary interest is in $SU(n)$ for general $n$, but results on other groups are likely to be instructive, too.

Examples involving f=1: http://mathoverflow.net/questions/84848/volume-of-compact-simple-lie-groups-under-the-natural-euclidean-embedding http://mathoverflow.net/questions/203836/volume-of-the-unitary-group — David Roberts, May 15 '15 at 15:35
And for some other functions http://mathoverflow.net/questions/36025/explicit-computations-using-the-haar-measure/36031#36031 — David Roberts, May 15 '15 at 15:38

score 8 · Accepted Answer · answered May 15 '15 at 16:18

8

Integration with respect to the Haar measure on unitary, orthogonal and symplectic group will tell you pretty much all we know for integrals of polynomial functions. (For a more recent paper, see Elementary derivation of Weingarten functions of classical Lie groups.)

There is not much more. Regrettably, even the integral over U(N) of a rational function has no closed-form expression.

answered May 15 '15 at 16:18

Carlo Beenakker

177,695

what about polynomials times simple exponentials? – Arnold Neumaier May 16 '15 at 15:14
1

for that purpose this celebrated formula can be useful: https://terrytao.wordpress.com/2013/02/08/the-harish-chandra-itzykson-zuber-integral-formula/ – Carlo Beenakker May 16 '15 at 15:59

explicit integrals over a Lie group

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