Degenerating products of flag varieties and applications to the Breuil–Mézard conjecture

Abstract

We consider closed subschemes in the affine grassmannian obtained by degenerating e-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For G=GL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G= {\ ext {GL}}_2$$\\end{document}, and cocharacters small relative to the characteristic, we relate the cycles of these degenerations to the representation theory of G. We then show that these degenerations smoothly model the geometry of (the special fibre of) low weight crystalline subspaces inside the Emerton–Gee stack classifying p-adic representations of the Galois group of a finite extension of Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}_p$$\\end{document}. As an application we prove new cases of the Breuil–Mézard conjecture in dimension two.