Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture

We prove the Mirković–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.

[1]  Jens Carsten Jantzen,et al.  Nilpotent Orbits in Representation Theory , 2004 .

[2]  L. Rider,et al.  The affine Grassmannian and the Springer resolution in positive characteristic , 2014, Compositio Mathematica.

[3]  I. Mirkovic,et al.  Geometric Langlands duality and representations of algebraic groups over commutative rings , 2004, math/0401222.

[4]  N. Lauritzen,et al.  Frobenius splitting of cotangent bundles of flag varieties , 1998, math/9809039.

[5]  Basic Theory,et al.  Kac-Moody Groups , 2002 .

[6]  I. Mirkovic,et al.  Perverse Sheaves on affine Grassmannians and Langlands Duality , 1999, math/9911050.

[7]  Wolfgang Soergel,et al.  Koszul Duality Patterns in Representation Theory , 1996 .

[8]  C. Mautner,et al.  Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture , 2015, Journal of the European Mathematical Society.

[9]  D. Arinkin,et al.  Perverse coherent sheaves , 2009, 0902.0349.

[10]  N. Chriss,et al.  Representation theory and complex geometry , 1997 .

[11]  Myron Minn-Thu-Aye.  Multiplicity formulas for perverse coherent sheaves on the nilpotent cone , 2013 .

[12]  Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone , 2001, math/0102039.

[13]  P. Achar,et al.  Constructible sheaves on affine Grassmannians and geometry of the dual nilpotent cone , 2011, 1102.2821.

[14]  I. Ágoston,et al.  Standardly Stratified Algebras and Tilting , 2000 .

[15]  P. Achar.  Perverse coherent sheaves on the nilpotent cone in good characteristic , 2011, 1109.2813.

[16]  Tilting exercises , 2003, math/0301098.

[17]  W. Soergel.  On the relation between intersection cohomology and representation theory in positive characteristic , 2000 .

[18]  E Cline At Worcester,et al.  Journal Fur Die Reine Und Angewandte Mathematik Finite Dimensional Algebras and Highest Weight Categories ') , 2022 .

[19]  Properly Stratified Algebras and Tilting , 2005, math/0502238.

[20]  T. A. Springer.  Some arithmetical results on semi-simple lie algebras , 1966 .

[21]  Shrawan Kumar,et al.  Kac-Moody Groups, their Flag Varieties and Representation Theory , 2002 .

[22]  V. Ginsburg.  Perverse sheaves and *-actions , 1991 .

[23]  W. Soergel.  Kategorie , perverse Garben und Moduln über den Koinvarianten zur Weylgruppe , 1990 .

[24]  Xinwen Zhu,et al.  Integral homology of loop groups via Langlands dual groups , 2009, 0909.5487.

[25]  Quantum groups, the loop Grassmannian, and the Springer resolution , 2003, math/0304173.

[26]  G. Williamson,et al.  Parity Sheaves , 2009, 0906.2994.

[27]  G. Williamson,et al.  Parity sheaves and tilting modules , 2014, 1403.1647.

[28]  Robert Steinberg,et al.  Representations of Algebraic Groups , 1963, Nagoya Mathematical Journal.

[29]  Sheaves on a loop group and langlands duality , 1995, alg-geom/9511007.

[30]  D. Juteau.  Modular representations of reductive groups and geometry of affine Grassmannians , 2008, 0804.2041.

[31]  Masaki Kashiwara,et al.  Sheaves on Manifolds , 1990 .

[32]  G. Williamson,et al.  Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties , 2010, 1008.0719.

[33]  V. Dlab.  Properly stratified algebras , 2000 .