The circumstellar dust disk of the Herbig Ae star AB Aur has been found to exhibit complex spiral-like structures in the near-IR image obtained with the Subaru Telescope . We present maps of the disk in both ^ { 12 } CO ( 3-2 ) and dust continuum at 345 GHz with the SMA at an angular resolution of 1 ^ { \prime \prime } .0 \times 0.7 ^ { \prime \prime } ( 144AU \times 100AU ) . The continuum emission traces a dust disk with a central depression and a maximum overall dimension of 450AU ( FWHM ) . This dust disk exhibits several distinct peaks that appear to coincide with bright features in the near-IR image , in particular the brightest inner spiral arm . The CO emission traces a rotating gas disk of size 530AU \times 330AU with a deprojected maximum velocity of 2.8 km s ^ { -1 } at 450 AU . In contrast to the dust disk , the gas disk exhibits an intensity peak at the stellar position . Furthermore , the CO emission in several velocity channels traces the innermost spiral arm seen in the near-IR . We compare the observed spatial-kinematic structure of the CO emission to a simple model of a disk in Keplerian rotation , and find that only the emission tracing the main spiral arm clearly lies outside the confines of our model . This emission has a net outward radial motion compared with the radial velocity predicted by the model at the location of the main spiral arms . The disk of AB Aur is therefore quite different from the Keplerian disks seen around many Herbig Ae stars . The spiral-like structures of the disk with non-Keplerian motions we revealed in ^ { 12 } CO ( 3-2 ) , together with the central depression of the dust disk , may be explained to be driven by the possible existence of a giant planet forming in the disk .