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So according to http://www-users.math.umn.edu/~garrett/m/v/SL2C.pdf I can write the Haar measure of $SL(2,\mathbb{C})$ as $$d\mu = \sinh^2(r) dr dk dk'$$ where $r$ runs over nonnegative real numbers and $k,k' \in SU(2)$. Does this mean that I can decompose the volume of $SL(2,\mathbb{C})$ as volume of $S^3$ times some infinite residual volume? Namely, $$\rm{vol} SL(2,\mathbb{C})= \rm{vol} S^3\times vol_R$$ where $\rm vol_R = \int_0^\infty \sinh^2(r)dr$?

$\mathbf{Update}$: I have a path integral of the form $$ Z= \int_M\frac{d\mu}{{\rm vol}G}\exp\left[-\int_{S^1} dt{L[\phi(t),d\phi(t)/dt,d^2\phi(t)/dt^2]}\right]. $$ where $G=SL(2,\mathbb{C})$, $t\in [-\pi,\pi]$ parametrizes a circle and $\phi(t)$ is a complex-valued periodic function with possible singularities for some values of $t$, defined on a non-compact space of $M=\rm{Diff}(S^1)_\mathbb{C}$ and finally $L$ is a functional of $\phi$ and its derivatives. Now the path integral here diverges like $\lim_{\epsilon\rightarrow 0}\exp\log[\sinh(\epsilon)]$ and I need to cancel that divergence with the volume of $SL(2,\mathbb{C})$ in the manner I explained above.

Thank you,

AB

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    $\begingroup$ What does it mean to ask whether two infinite measures are equal? $\endgroup$
    – LSpice
    Commented Apr 26, 2017 at 17:08
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    $\begingroup$ For every choice of left Haar measure on a non-compact locally compact group, the volume is infinite. Indeed, for every compact neighborhood $K$ of 1, there exists an infinite sequence of pairwise disjoint left translates of $K$ and since $K$ has positive measure it follows that the total volume is $\infty$. $\endgroup$
    – YCor
    Commented Apr 26, 2017 at 17:30
  • $\begingroup$ @LSpice Since I am taking a space and moding out a SL(2,C), I need to see how volumes cancel out. So I need to know what this infinite volume is made of. $\endgroup$
    – Alireza Behtash
    Commented Apr 26, 2017 at 19:58

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