  We derive the central stellar velocity dispersion function for quiescent
galaxies in 280 massive clusters with $\log (M_{200} / M_{\odot}) > 14$ in
IllustrisTNG300. The velocity dispersion function is an independent tracer of
the dark matter mass distribution of subhalos in galaxy clusters. Based on the
IllustrisTNG cluster catalog, we select quiescent member subhalos with a
specific star formation rate $< 2 \times 10^{-11}$ yr${^-1}$ and stellar mass
$\log (M_{*} / M_{\odot}) > 9$. We then simulate fiber spectroscopy to measure
the stellar velocity dispersion of the simulated galaxies; we compute the
line-of-sight velocity dispersions of star particles within a cylindrical
volume that penetrates the core of each subhalo. We construct the velocity
dispersion functions for quiescent subhalos within $R_{200}$. The simulated
cluster velocity dispersion function exceeds the simulated field velocity
dispersion function for $\log \sigma_{*} > 2.2$, indicating the preferential
formation of large velocity dispersion galaxies in dense environments. The
excess is similar in simulations and in the observations. We also compare the
simulated velocity dispersion function for the three most massive clusters with
$\log (M_{200} / M_{\odot}) > 15$ with the observed velocity dispersion
function for the two most massive clusters in the local universe, Coma and
A2029. Intriguingly, the simulated velocity dispersion functions are
significantly lower for $\log \sigma_{*} > 2.0$. This discrepancy results from
1) a smaller number of subhalos with $\log (M_{*} / M_{\odot}) > 10$ in TNG300
compared to the observed clusters, and 2) a significant offset between the
observed and simulated $M_{*} - \sigma_{*}$ relations. The velocity dispersion
function offers a unique window on galaxy and structure formation in
simulations.
