We have detected the v = 1 → 0 S(1) (λ = 2.1218µm) and v = 2 → 1 S(1) (λ = 2.2477µm) lines of H 2 in the Galactic centre, in a 90 × 27 arcsec region between the northeastern boundary of the non-thermal source, Sgr A East, and the giant molecular cloud (GMC) M-0.02-0.07. The detected H 2 v = 1 → 0 S(1) emission has an intensity of 1.6 -21 × 10 −18 W m −2 arcsec −2 and is present over most of the region. Along with the high intensity, the broad line widths (FWHM = 40 -70 km s −1 ) and the H 2 v = 2 → 1 S(1) to v = 1 → 0 S(1) line ratios (0.3 -0.5) can be best explained by a combination of C-type shocks and fluorescence. The detection of shocked H 2 is clear evidence that Sgr A East is driving material into the surrounding adjacent cool molecular gas. The H 2 emission lines have two velocity components at ∼ +50 km s −1 and ∼ 0 km s −1 , which are also present in the NH 3 (3,3) emission mapped by McGary, Coil, & Ho (2001). This two-velocity structure can be explained if Sgr A East is driving C-type shocks into both the GMC M-0.02-0.07 and the northern ridge of McGary, Coil, & Ho (2001).