Intersection cohomology of Vinberg-Popov varieties

2025/07/22 by Dancer, Andrew, Martens, Johan, Proudfoot, Nicholas
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2507.16492

Abstract

The Vinberg-Popov variety of a simply connected reductive algebraic group G is a singular affine variety that contains the basic affine space G/U as a Zariski open subset. It is defined as the spectrum of the ring of functions on G/U, and can also be identified with the universal symplectic implosion for the maximal compact subgroup of G. We provide a recursive procedure for computing the intersection cohomology of this variety, with an emphasis on the case where G = SL_n.

Citations

• Fourier transforms on the basic affine space of a quasi-split group
• The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
• The Kazhdan-Lusztig polynomial of a matroid
• Differential operators on G/U and the affine Grassmannian
• Symplectic implosion
• Symplectic Implosion
• Gluing of perverse sheaves on the basic affine space
• On the Schwartz space of the basic affine space

Related

• Intersection Cohomology of reductive varieties
• Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations
• Intersection cohomology groups of instanton moduli spaces and cotangent bundles of affine flag varieties
• Intersection cohomology of representation spaces of surface groups
• The intersection cohomology of the Satake compactification of \mathcal A_g for g≤ 4