Tilting modules over the Auslander algebra of k[T]/T^t with applications to spherical twists and parabolic group actions

The Auslander algebra $A_ t$ of the truncated polynomial ring plays an important role for several applications. Mostly we are interested in tilting modules over this algebra. Certain tilting modules, the good ones, have been
classified in a joint work with Brüstle, Ringel and Röhrle more than ten years ago. They are closely related to the Richardson orbit for a parabolic subgroup in the General Linear Group. Later it turned out, that the classification of all tilting modules is closely related to coherent sheaves on a chain of $(-2)$-curves in a rational surface. Each $(-2)$-curve defines a spherical twist, an
autoequivalence of the derived category. Going back to modules over $A_t$ we can use the spherical twists to obtain all tilting modules from the good ones. In our main result we explain the classification of all tilting
modules and give further applications.