Brauer group of the moduli spaces of stable vector bundles of fixed determinant over a smooth curve

Let $X$ be an irreducible smooth projective curve, defined over an algebraically closed field $k$, of genus at least three and $L$ a line bundle on $X$. Let ${\mathcal{M}}_X(r,L)$ be the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$ with $r\ge 2$. We prove that the Brauer group $\mathrm{B}\mathrm{r}({\mathcal{M}}_X(r,L))$ is cyclic of order ${\rm g.c.d.}(r,{\rm degree}(L))$\mathrm{.}Wealsoprovethat${\rm Br}($\mathcal{M}$_X(r,L))$ is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in \cite{BBGN} under the assumption that $k=\mathbb{C}$.