Table of Contents:
  • Introduction
  • Generalities on B/B and C/Q
  • Specifics for the clasical groups
  • The tangent space and smoothness
  • Root system description of T (W, T)
  • Rational smoothness and Kazhdan-Lusztig theory
  • Nil-Hecke ring and the singular locus of X(W)
  • Patterns, smoothness, and rational smoothness
  • Minuscule and cominuscule G/P
  • Rank two results
  • Related combinatorial results
  • Related varieties
  • Addendum.