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Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting representation of $H$ on $H_{*}(G;\mathbb{Z})$, the integral homology of $G$. Is anything in general known about this $H$-representation? For instance, are there nice generators of the homology groups of $G$ on which I might try to describe the action of $H$? Also, I would be grateful for any and all references that might prove relevant.

Thanks!

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up vote 6 down vote accepted

Since $H\leq G$ and $G$ is connected, each $h\in H$ is connected to the identity, hence its action on $G$ is homologous to the identity.

You're not going to get anything interesting here unless you either look at $N_G(H)/H$ acting on $H\backslash G$, or (equivalently) $N_G(H)$ acting on the $H$-equivariant cohomology of $G$, or something like that.

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What about dropping the connected assumption? There are papers by R.L. Taylor in the 1950s on these structures. –  Ronnie Brown Feb 16 '13 at 11:43
    
Hi Allen, thanks for the answer and interesting suggestions! Hi Ronnie, it might prove interesting to remove the connectedness assumption. Thanks for the references! –  Peter Crooks Feb 16 '13 at 16:38
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