Let $|jm\rangle,m=-j,-j+1,...,j-1,j$ be the standard (unique up to a phase choice) orthonormal basis of the $d=(2j+1)$-dim complex "unitary" irrep $u: \text{su}(2)\rightarrow \text{gl}(V)$ of Lie algebra $\text{su}(2)$, i.e., $u(X)^* =-u(X)$. Then what is $\exp(i\pi J_2) |jm\rangle$? where $J_2 = u(\sigma_2/2)$ and $\sigma_2$ is the standard Pauli matrix (with slight abuse of notation, I am also using $u$ to denote its complexification).
I asked this in physics stackexchange and provided an incomplete answer, where I was able to prove that $$ \exp(i\pi J_2) |jm\rangle = \pm (-1)^{j-m}|j,-m\rangle $$ However, I can't figure out whether it should be $+$ or $-$.
EDIT: $|jm\rangle,m=-j,-j+1,...,j-1,j$ is the unique (up to phase choice) orthonormal basis such that
\begin{align} J_3 |jm\rangle &=m|jm\rangle \\ J_{\pm} |jm\rangle &= \sqrt{j(j+1)-m(m\pm 1)} |j,m\pm1\rangle \end{align}
EDIT 2: In physics stackexchange, I added a note that we can use the property of characters to show that if $j=l$ is an integer, then we must have $+1$, so my guess for half-integers $j=s$, we should have $+1$ since that it the case for $j=1/2$, though I'm not sure of the details for general half-integer $j=s$.
EDIT 3: (After a year or so) In physics stackexchange, I have provided a complete solution showing that $$ \exp(i\pi J_2) |jm\rangle = (-1)^{j-m}|j,-m\rangle $$ The main idea is that for $j=1/2$, we can show that it has $+$ and for all integer $j$, we have $+$. Now if $j$ is a half-integer and $V_j$ is the corresponding irrep, then $$ V_j \otimes V_{1/2} = V_{j+1/2} \oplus V_{j-1/2} $$ And the rest follows from this decomposition (see link for more detail).