  We fix integers $k> 0$ and $n>0$. For a $k$-punctured Riemann surface $\Sigma
\setminus \{ p_1,\ldots,p_k \}$ and a $k$-tuple
$\boldsymbol{\mu}=(\mu^1,\ldots,\mu^k)$ of partitions of $n$, we can define the
character variety of type $\boldsymbol{\mu}$. In this paper, we consider the
case where $\Sigma=\mathbb{P}^1$ and $\boldsymbol{\mu}$ is indivisible (i.e.
$\mathrm{g.c.d.}(\boldsymbol{\mu})=1$). For the case, we prove the purity
conjecture due to Hausel, that is, the pure parts of the mixed Hodge structures
of the character variety is isomorphic to the ordinary rational cohomology
groups of the quiver variety of type $\boldsymbol{\mu}$.
