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

For a projective variety $X$ and a line bundle $L$ over $X$, one considers the $L-$twisted global differential operator algebra $\text{call}{D}_L(X)$ which naturally operates on the space of global sections $H^0(X,L)$. In the case where $X$ is the wonderful compactification of the group ${\mathrm{P}\mathrm{G}\mathrm{L}}_3$, one proves that the space $H^0(X,L)$ is an irreducible representation of the algebra $\text{call}{D}_L(X)$ or zero. For that, one introduces a $2-$order differential operator which is defined over whole $X$ but which does not arise from the infinitesimal action of the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(X)$.