I am trying to calculate the following integral:
$$\int_{\mathrm{SU}(2)}\,\mathrm{d}g\,\delta(g^{2}),$$
where $\mathrm{d}g$ denotes the normalized Haar measure on $\mathrm{SU}(2)$ and where $\delta(g)$ denotes the "delta-function" on $\mathrm{SU}(2)$, which is the distribution defined via
$$\int_{\mathrm{SU}(2)}\,\mathrm{d}g\,f(g)\delta(gh^{-1})=f(h).$$
Using the Peter-Weyl decomposition, one can formally write the delta-functionn as
$$\delta(g):=\sum_{j\in\mathbb{N}_{0}/2}(2j+1)\chi^{j}(g)$$
where $\chi^{j}$ denotes the characters of the spin-$j$ representations, i.e. the unique (up to unitary equivalence) unitary irreducible representations of dimension $2j+1$. Now, I was told that one can evaluate the integral explicitely. However, I have some struggles to do so.
My attempt: My plan was to use the Euler-angle representation of elements in $\mathrm{SU}(2)$ together with the fact that $\chi^{j}$ are class functions. Now, according to wikipedia, the characters only depened on the angle $\beta$ and are for $g=g(\alpha,\beta,\gamma)\in\mathrm{SU}(2)$ given by
$$\chi^{j}(g)=\frac{\mathrm{sin}((2j+1)\beta/2)}{\mathrm{sin}(\beta/2)}$$
However, my problems already start when trying to write down $\chi^{j}(g^{2})$ since it is hard to find a general expression for the Euler-angles of $g^{2}$ in terms of the Euler-angles of $g$...
Any ideas?
EDIT: I was able to compute the integrals, however, I think I get the wrong results. See the post in MathOverflow.