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

TL;DR

This paper studies the irreducibility of representations of certain differential operator algebras on line bundles over a specific projective variety, revealing new operators that influence these representations.

Contribution

It introduces a novel second-order differential operator on the wonderful compactification of PGL_3 that affects the irreducibility of the associated algebra representations.

Findings

The space of global sections is either irreducible or zero under the algebra's action.

A new second-order differential operator is constructed that is not derived from automorphisms.

The irreducibility of representations is established for the specific variety considered.

Abstract

For a projective variety $X$ and a line bundle $L$ over $X$, one considers the $L-$twisted global differential operator algebra $\call{D}_L(X)$ 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 $\mathrm{PGL}_3$, one proves that the space $H^0(X,L)$ is an irreducible representation of the algebra $\call{D}_L(X)$ 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 $\mathrm{Aut}(X)$.