Abstract
AbstractWe consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.
Highlights
Besides being a natural analog of the classical theory, these varieties play an important role when studying the special fiber of both Shimura varieties and moduli spaces of shtukas
The defined affine Deligne-Lusztig varieties are equidimensional of known dimension, and the closure of a Newton stratum is equal to the union of all Newton strata associated with [ ′] ≤ [ ]; compare [37]
The Newton strata are not equidimensional [11, Section 5], it is unknown under which conditions the closure of any Newton stratum is a union of strata, and we do not even have a general conjecture describing the set ( )
Summary
We consider affine Deligne-Lusztig varieties, which are an analogue of the above in affine flag varieties. This necessary condition is far from sufficient, and a complete description of ( ) is known only in very special cases Whenever it is non-empty, N[ ], := [ ] ∩ is shown in [34] to be the set of geometric points of a locally closed reduced subscheme of : namely, the Newton stratum associated with [ ]. Another natural (and unsolved) question is to describe the closure of N[ ], in.
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