This is a symmetric space when equipped by the metric discussed in [this question][1] (it is Riemannian, when $p=2$). The metric is clearly invariant under the action; its other properties can be checked (see the references in Suvrit's nice answer to the linked question, or see one of the papers by Freitas and Friedland, e.g. MR2014882 (2004j:30086) Reviewed 
Friedland, Shmuel(1-ILCC-MS); Freitas, Pedro J.(P-LISBS)
Revisiting the Siegel upper half plane. I. (English summary) 
Linear Algebra Appl. 376 (2004), 19–44.  )


  [1]: https://mathoverflow.net/questions/115192/power-log-distance-between-matrices/115227#115227