Generic Newton points and the Newton poset in Iwahori-double cosets

Abstract

We 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.

Cordial elements and dimensions of affine Deligne–Lusztig varieties

“…The new strategy in this paper is as follows. Instead of using minimal length elements as the starting point, we use the cordial elements introduced by Milićević and Viehmann in [MV20] as the starting point. In Section 4, we construct a new family of cordial elements.…”

“…It is mentioned in [MV20] that fully characterising the cordial elements is fairly difficult. In [MV20, Theorem 1.2], some interesting families of cordial elements are provided.…”

Maximal Newton Points and the Quantum Bruhat Graph

“…Although each of the two techniques presents different challenges when x lies outside of the "shrunken" Weyl chambers, one might hope that an interpolation between these two methods might result in a complete picture for the non-emptiness problem, complementing our detailed knowledge of the basic case established in [GHN15]. In joint work with Viehmann, the author identifies a family of elements x such that the Newton poset N(G) x is saturated, from which one deduces geometric information: formulas for codimensions of the Newton strata in the affine Schubert cell IxI, as well as their equidimensionality [MV19].…”

Closure relations of Newton strata in Iwahori double cosets

“…Using an example of a non-equidimensional affine Deligne-Lusztig variety given in [3,5] together with [14,Cor. 3.11], one sees that in general, the N [b],x ⊆ I x I are no longer pure of any fixed codimension.…”

“…Besides this, very little was known previously. On the one hand, there are examples of double cosets where the whole pattern of closures is known (in particular for the group SL 3 by [1] and for so-called cordial elements by [14]). In all of these examples, we still have equality in (1.…”