Microlocal Approach to Lusztig’s Symmetries

The chapter 'Microlocal Approach to Lusztig’s Symmetries' delves into the intricate relationship between Lusztig’s symmetries and Coxeter categories. It introduces a topological reformulation of Coxeter categories, which provides a fresh perspective on the subject. The authors explore the vanishing cycles and Lusztig’s symmetries, presenting conjectures and detailed analyses. The chapter also includes an example in type A2, comparing the action of Lusztig’s symmetries in weight spaces with the monodromy action in vanishing cycles of perverse sheaves. The authors propose a Coxeter structure on the category of factorizable sheaves and conjecture that this structure is equivalent to the Coxeter structure on the category of integrable modules over Lusztig’s small quantum group. The chapter concludes with a discussion of iterated specialization and microlocalization, highlighting the importance of these techniques in the study of Lusztig’s symmetries.