William M. McGovern

On the spaltenstein-steinberg map for classical lie algebras

We compute the fibers of the canonical map from Weyl group elements to orbital varieties for a Lie algebra of type B,C, or D. We also compute one irreducible component of the associated variety of any simple highest weight module in any classical type, thereby sharpening a result of Joseph and partially proving a conjecture of Tanisaki.

A triangularity result for associated varieties of highest weight modules

We prove a conjecture of Tanisaki in the classical case, which says roughly that the matrix of orbital varieties occurring in the associated variety of a simple highest weight module is upper triangular.

Upper semicontinuity of KLV polynomials for certain blocks of Harish-Chandra modules

We show that the coefficients of Kazhdan–Lusztig–Vogan polynomials attached to certain blocks of Harish-Chandra modules satisfy a monotonicity property relative to the closure order on K-orbits in the flag variety.

Closures of -orbits in the flag variety for *(2)

We characterize the Sp_{2n} orbits in the flag variety for SL_{2n} with rationally smooth closure via a pattern avoidance criterion, also showing that the singular and rationally singular loci of such orbit closures coincide.

The Mathematical Work of David A. Vogan, Jr.

Over four decades David Vogan’s groundbreaking work in representation theory has changed the face of the subject. We give a brief summary here.