  We show that the algebras describing blocks of the category of cuspidal
weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are
one-parameter (respectively multi-parameter) deformations of certain Brauer
tree algebras. We explicitly determine these deformations both graded and
ungraded. The algebras we deform also appear as special centralizer subalgebras
of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in
the context of highest weight representations of the Virasoro algebra, in the
context of rational representations of the general linear group and Schur
algebras and in the study of the Milnor fiber of Kleinian singularities.
