Poisson geometry of the Grothendieck resolution of a complex semisimple group

2006/10/03 by Evens, Sam, Lu, Jiang-Hua
#14M17 #20G20 #53D17 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.math/0610123

Abstract

We study a Poisson structure π on the Grothendieck resolution X of a complex semi-simple group G and prove that the desingularization map μ:(X,π) → (G,π₀) is Poisson, where π₀ is a Poisson structure such that intersections of conjugacy classes and opposite Bruhat cells BwB_- are Poisson subvarieties. We compute the symplectic leaves of X and show that (X, π) resolves singularities of (G, π₀).

Related

• Poisson structures on complex flag manifolds associated with real forms
• Poisson resolutions
• The Poisson geometry of the conjugation quotient map for simple algebraic groups and deformed Poisson W-algebras
• Double Bruhat cells and symplectic groupoids
• On the splitting of infinitesimal Poisson automorphisms around symplectic leaves