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The authors consider the Hodge Laplacian on the Heisenberg group H n , endowed with a left-invariant and U(n) -invariant Riemannian metric. For 0 k 2n 1 , let k denote the Hodge Laplacian restricted to k -forms. In this paper they address three main, related questions: (1) whether the L 2 and L p -Hodge decompositions, 1 (2) whether the Riesz transforms d 12 k are L p -bounded, for 1<< ; (3) how to prove a sharp Mihilin-Hoermander multiplier theorem for k , 0 k 2n 1 .