Sony A6000 has a Sony E lens mount and currently there are 364 native lenses available for this mount.
The latest released lenses that are compatible with the A6000 are the Sony FE 100-400mm f5.6-8, the Tamron 12-20mm f2.8 and the Viltrox 26mm f2.8 FE.
Sony A6000 doesn't have a sensor based image stabilization system but
55 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.
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Sony A6000 Lenses with Image Stabilization
Sony A6000 Lenses with Weather Sealings