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For a projective variety $X$ and a line bundle $L$ over $X$, one considersthe $L-$twisted global differential operator algebra $\call{D}_L(X)$ whichnaturally operates on the space of global sections $H^0(X,L)$. In the casewhere $X$ is the wonderful compactification of the group $\mathrm{PGL}_3$, oneproves that the space $H^0(X,L)$ is an irreducible representation of thealgebra $\call{D}_L(X)$ or zero. For that, one introduces a $2-$orderdifferential operator which is defined over whole $X$ but which does not arisefrom the infinitesimal action of the automorphism group $\mathrm{Aut}(X)$.

Repr\'esentations irr\'eductibles de certaines alg\`ebres d'op\'erateurs diff\'erentiels