Sony A6000 has a Sony E lens mount and currently there are 215 native lenses available for this mount.
The latest released lenses that are compatible with the A6000 are the Samyang 14mm F2.8, the Samyang 8mm F2.8 Fisheye and the Samyang 85mm F1.4.
Sony A6000 doesn't have a sensor based image stabilization system but
43 of these lenses already comes with optical image stabilization.
Since Sony A6000 has a APS-C sensor, it has a focal length multiplier of 1.5x so you have to multiply lenses original focal length with this multiplier to find the Full Frame equivalent focal length of the lens when mounted on A6000.
You may also also be interested in these:
Sony A6000 Lenses with Image Stabilization
Sony A6000 Lenses with Weather Sealings