Complexity of recognizing whether two $\omega$-regular expressions represent the same language
It's quite easy to reduce the equivalence problem for regular expressions to the equivalence problem for $\omega$-regular expressions, so you have EXPSPACE hardness.

For EXPSPACE completeness, observe that given two $\omega$ regular expressions $r_1$ and $r_2$, you can translate them to equivalent NBWs $A_1,A_2$, where the size of $A_i$ is at most single-exponential in the size of $r_1$. Then, checking whether $L(A_1)=L(A_2)$ can be done in PSPACE (nontrivial, but well known).