The adjoint quotient and its geometry
Title: The adjoint quotient and its geometry
Abstract: The conjugacy classes of n x n matrices feature prominently in classical invariant theory. One particularly well-studied case involves the complex special linear group SL_n acting on the trace-free matrices sl_n by conjugation. The orbits of this action are classified by the trace-free matrices in Jordan canonical form with non-increasing block sizes, where the blocks are read from left-to-right. This classical and purely algebraic fact has geometric implications for the "quotient space" sl_n/SL_n. If defined properly, this space only parametrizes the conjugacy classes of so-called "regular" elements in sl_n. One is thereby motivated to understand the locus of irregular elements and its geometric features.
I will survey the classical underpinnings of this story. If time permits, I will describe an ongoing project with M. Röser concerned with smooth points in the irregular locus of a semisimple Lie algebra.