Unramified Group Research Articles

Connected components of affine Deligne–Lusztig varieties for unramified groups

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

The Twisted Satake Transform and the Casselman–Shalika Formula for Quasi-Split Groups

Abstract We prove the Casselman–Shalika formula for unramified groups over a non-archimedean local field by studying the action of the spherical Hecke algebra on the space of compact spherical Whittaker functions via the twisted Satake transform. This method provides a conceptual explanation of the appearance of characters of a dual group in the Casselman–Shalika formula.

Local models for Weil-restricted groups

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

The metaplectic Casselman-Shalika formula

This paper studies spherical Whittaker functions for central extensions of reductive groups over local fields. We follow the development of Chinta and Offen to produce a metaplectic Casselman-Shalika formula for tame covers of all unramified groups.

The Dimension of Affine Deligne–Lusztig Varieties in the Affine Grassmannian

In this paper we calculate the dimension of affine Deligne-Lusztig varieties in the affine Grassmannian of unramified groups. We also examine the irreducible components of ADLVs in the superbasic case and the $J_b(F)$-action on them.

Base change for Bernstein centers of depth zero principal series blocks

Let G be an unramified group over a p-adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for G and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with Γ1(p)-level structure initiated by M. Rapoport and the author in [HR2].

The base change fundamental lemma for central elements in parahoric Hecke algebras

Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.

On the Hodge-Newton decomposition for split groups

The main purpose of this paper is to prove a group-theoretic generalization of a theorem of Katz on isocrystals. Along the way we reprove the group-theoretic generalization of Mazur's inequality for isocrystals due to Rapoport-Richartz, and generalize from split groups to unramified groups a result of Kottwitz-Rapoport which determines when an affine Deligne-Lusztig subset of the affine Grassmannian is non-empty.

A geometric construction of the Iwahori–Hecke algebra for unramified groups

This paper concerns constructing the Iwahori-Hecke algebra for certain p-adic groups. It can be regarded as a natural extension of the ideas laid down in [KL2]; the objective (and conclusion) of that paper was the proof of Deligne-Langlands conjecture for Hecke algebras arising from split p-adic groups (with connected centers). The key observation of that paper was the geometric construction of the Iwahori-Hecke algebra for a split p-adic group.