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I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

Let's consider the moduli space of representations of $\pi=\pi_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let $Y=\operatorname{Hom}(\pi,G)/\\!/G$...

This is a somewhat speculative question, so bear with that (or not, as is your preference). Let $X$ be a smooth projective variety, and let $\omega_X$ be its canonical sheaf. The Euler class of ...