Lecture 18: Deligne-simpson Problem

The Deligne-Simpson problem asks to find a condition on conjugacy classes C1, . . . , Ck ⊂ Matn(C) such that there are matrices Yi ∈ Ci with (1) ∑k i=1 Yi = 0, (2) and there are no proper subspaces in C stable under all Yi. Crawley-Boevey reduced this problem to checking if there is an irreducible representation in Rep(Π(Q), v) for suitable Q, λ, v produced from C1, . . . , Ck. Recall, Section 4.1 of Lecture 17, that this is equivalent to v ∈ Σλ, where Σλ ⊂ Z0 is a combinatorially defined set. Crawley-Boevey’s approach was strongly motivated the Kraft-Procesi construction who proved that the closures of conjugacy classes of matrices are normal. The proof easily reduces to the case of nilpotent orbits. Kraft and Procesi realized their closures as certain quotients that are special cases of Nakajima quiver varieties. This allowed them to prove the normality. In the first section we will recall the necessary background from Invariant theory, a field that studies quotients under group actions. Then we will explain the Kraft-Procesi construction. Finally, we will explain Crawley-Boevey’s approach to the DS problem.