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Let $K$ be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of $G{L}_{n+1}\left(K\right)$ and the space of harmonic cochains defined on the Bruhat-Tits building of $G{L}_{n+1}\left(K\right)$, in the sense of E. de Shalit [11]. We deduce, applying the results of a paper of P. Schneider and U. Stuhler [9], that there exists a $G{L}_{n+1}\left(K\right)$-equivariant isomorphism between the cohomology group of the Drinfeld symmetric space and the space of harmonic cochains.