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We report deep optical integral-field spectroscopy with the MUSE of theluminous radio-quiet QSO PG1307+085 (z=0.154) obtained during the commissioningof the instrument. Given the high sensitivity and spatial resolution deliveredby MUSE, we are able to resolve the compact ($r_e$~1.3") elliptical hostgalaxy. After careful spectroscopic deblending of the QSO and host galaxyemission, we infer a stellar velocity dispersion of $155\pm19$km/s. This placesPG1307+085 local $M_{BH}-\sigma_*$ relation within the intrinsic scatter butoffset towards a higher black hole mass with respect to the mean relation. Theobservations with MUSE also reveal a large extended ENLR around PG1307+085reaching out to 30kpc. In addition, we detect a faint bridge of ionized gastowards the most massive galaxy of the galaxy group being just 20" (50kpc)away. Previous long-slit spectroscopic observations missed most of theseextended features due to a miss-aligned slit. The ionized gas kinematics doesnot show any evidence for gas outflows on kpc scales despite the high QSOluminosity of $L_\mathrm{bol}>10^{46}$ erg/s. Based on the ionized gasdistribution, kinematics and metallicity we discuss the origin of the ENLR withrespect to its group environments including minor mergers, ram-pressurestripping or filamentary gas accretion as the most likely scenarios. Weconclude that PG1307+085 is a normal elliptical host in terms of the scalingrelations, but that the gas is most likely affected by the environment throughgravity or ambient pressure. It is possible that the ongoing interaction withthe environment, mainly seen in the ionized gas, is also be responsible fordriving sufficient gas to feed the black hole at the centre of the galaxy.

The MUSE view of QSO PG1307+085: An elliptical galaxy on the $M_{BH}-\sigma_*$ relation interacting with its group environment