Nilpotent commuting varieties and support varieties
Event details
| Date | 13.10.2015 |
| Hour | 15:15 › 17:00 |
| Speaker | Paul Levy (Lancaster) |
| Location | |
| Category | Conferences - Seminars |
Abstract :
Commuting varieties are classical objects of study in Lie theory. Historically the first result was the proof by Motzkin-Taussky in 1955 (established independently by Gerstenhaber) that the variety of pairs of commuting n x n matrices is irreducible. Richardson extended this to an arbitrary reductive Lie algebra in characteristic zero. More recently there has been interest in the subvariety of pairs of commuting nilpotent elements. One highlight was Premet's proof that this nilpotent commuting variety of a reductive Lie algebra is equidimensional, and is irreducible for \gl_n. Here I will explore two variations on this theme. First of all, I will introduce some generalisations of the nilpotent commuting variety, for which we confine the first coordinate to a fixed nilpotent orbit closure. The main task here is to determine the irreducible components and their dimension. This is joint work with N. Ngo. Secondly, I will summarize some recent results on the variety of r-tuples of commuting nilpotent elements of \gl_n or \sp_{2n}. This is joint work with N. Ngo and K. Sivic. The main applications of this work are to cohomology and representation theory of Frobenius kernels of simple algebraic groups (the support varieties of the title). These applications are somewhat more esoteric than the main subject matter, so I will reserve most of the details for the second part of the talk.
Commuting varieties are classical objects of study in Lie theory. Historically the first result was the proof by Motzkin-Taussky in 1955 (established independently by Gerstenhaber) that the variety of pairs of commuting n x n matrices is irreducible. Richardson extended this to an arbitrary reductive Lie algebra in characteristic zero. More recently there has been interest in the subvariety of pairs of commuting nilpotent elements. One highlight was Premet's proof that this nilpotent commuting variety of a reductive Lie algebra is equidimensional, and is irreducible for \gl_n. Here I will explore two variations on this theme. First of all, I will introduce some generalisations of the nilpotent commuting variety, for which we confine the first coordinate to a fixed nilpotent orbit closure. The main task here is to determine the irreducible components and their dimension. This is joint work with N. Ngo. Secondly, I will summarize some recent results on the variety of r-tuples of commuting nilpotent elements of \gl_n or \sp_{2n}. This is joint work with N. Ngo and K. Sivic. The main applications of this work are to cohomology and representation theory of Frobenius kernels of simple algebraic groups (the support varieties of the title). These applications are somewhat more esoteric than the main subject matter, so I will reserve most of the details for the second part of the talk.
Practical information
- Informed public
- Free
Organizer
- Donna Testerman