Dimensions of Newton Strata in the Adjoint Quotient of Reductive Groups
نویسنده
چکیده
In [Cha00] Chai made a conjecture on the codimensions of Newton strata in Shimura varieties, which then led Rapoport [Rap05] to his conjecture on dimensions of affine Deligne-Lusztig varieties inside affine Grassmannians. The main goal of this paper is to show that exactly the same codimensions arise in a simpler context, that of the Newton stratification in the adjoint quotient of a reductive group. Along the way we study this stratification and then introduce the notion of defect, which we use to rewrite the codimension formula without having to use the greatest integer function. The influence of the work of Chai and Rapoport on this paper is obvious. Less obvious is the influence of some joint work (as yet unpublished) with M. Goresky and R. MacPherson on codimensions of root-valuation strata in Lie algebras over Laurent power series fields. It is an especially great pleasure to acknowledge the influence of Goresky and MacPherson in a paper like this one, dedicated to Bob MacPherson on his 60th birthday. In addition I would like to thank T. Haines and M. Sabitova for some very helpful comments on an earlier version of this paper.
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