The presence of OH (1720 MHz) masers and the absence of counterparts at 1665/1667 MHz has proved to be a clear diagnostic of shocked molecular gas associated with Galactic supernova remnants. This suggests that shocked molecular gas should be associated with the OH (1720 MHz) masers that have been detected in the circumnuclear disk (CND) and Sgr A East at the Galactic center. In order to test this hypothesis, we observed the H2 1-0 S(1) and Brγ lines using NICMOS on the Hubble Space Telescope (HST) and the University of New South Wales Infrared Fabry-Perot (UNSWIRF) etalon at the Anglo-Australian Telescope (AAT), near the regions where OH (1720 MHz) masers are detected in the CND and Sgr A East. We present the distribution of H2 in the north and south lobes of the CND and in Sgr A East. H2 emission accompanies almost all the maser spots detected at the Galactic center. In particular, we find a striking filamentary structure near the northwest of the CND and evidence that shocked molecular gas is associated with the 70 km s-1 molecular cloud at the Galactic center. We argue that the emission from the CND could arise in gas heated by the dissipation of the random motion of clumps by collisions or the dissipation of turbulence in a more homogeneous medium. In addition, highly redshifted gas of up to 140 km s-1 close to the eastern edge of the Sgr A East shell is detected. These observations combined with OH (1720 MHz) results suggest that the H2 gas is shocked and accelerated by the expansion of Sgr A East into the 50 and the 70 km s-1 clouds and into the lobes of the CND.