# Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an explicit statement in the literature of the $S^1$-equivariant Poincare series of $\Omega SU(2)$?

• I don't know about the literature (though I would look at [Harada-Henriques-Holm]), but I wonder if you can get somewhere by starting with the $T^2$-equivariant Poincare series and forgetting the loop action. – Allen Knutson Jan 23 '14 at 22:56
• Thanks! I think I have found a means to a solution. Since $\Omega SU(2)$ has rational cohomology only in even degrees, the spectral sequence for $\Omega SU(2)\rightarrow(\Omega SU(2))_{S^1}\rightarrow BS^1$ collapses on its second page. Therefore, $P_{S^1}(\Omega SU(2))=P(\Omega SU(2))P_{S^1}(pt)$. The ordinary Poincare series of $\Omega SU(2)$ is $\frac{1}{1-t^2}$, so that $P_{S^1}(\Omega SU(2))=\frac{1}{(1-t^2)^2}$. I think the same sort of argument would work for the $T^2$-action. – Peter Crooks Jan 25 '14 at 9:59