On differential operators on complete symmetric varieties of type A₁ and A₂

2016/09/22 by Dejoncheere, Benoît
#20G05 14 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1609.06998

Abstract

In this paper, we will look at the algebra of global differential operators D_X on wonderful compactifications X of symmetric spaces G/H of type A₁ and A₂. We will first construct a global differential operator on these varieties that does not come from the infinitesimal action of \mathfrakg. We will then focus on type A₂, where we will show that D_X is an algebra of finite type, and that for any invertible sheaf \cal L on X, H⁰(X,\cal L) is either 0 or a simple left D_X,\cal L-module. Finally, we will show with the help of local cohomology that this is still true for higher cohomology groups Hⁱ(X,\cal L).

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