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Thomas J. Haines

Publications and source records attributed to Thomas J. Haines.

At least 19 recordsLinked to original sources
arXiv2026

Alcove walk models for parabolic Mirković-Vilonen intersections and branching to Levi subgroups

This article establishes alcove walk models for intersections of Schubert varieties and partially semi-infinite orbits in the affine Grassmannian of a split reductive group (we call such intersections parabolic Mirković-Vilonen intersections). More precisely, we describe explicit cellular pavings of these intersections, indexed by certain positively-folded alcove walks. We prove a parametrization of the irreducible components of maximal possible dimension, in terms of alcove walks of maximal possible dimension. We then deduce a new combinatorial description of branching to Levi subgroups of irreducible highest weight representations, and in particular we give a new algorithm for computing the characters of such representations.

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arXiv2025

Cellular pavings of fibers of convolution morphisms

This article proves, in the case of split groups over arbitrary fields, that all fibers of convolution morphisms attached to parahoric affine flag varieties are paved by products of affine lines and affine lines minus a point. This applies in particular to the affine Grassmannian and to the convolution morphisms in the context of the geometric Satake correspondence. The second part of the article extends these results over $\mathbb Z$. Those in turn relate to the recent work of Cass-van den Hove-Scholbach on the geometric Satake equivalence for integral motives, and provide some alternative proofs for some of their results.

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arXiv2025

On the cohomology of simple Shimura varieties with non quasi-split local groups

We study the Scholze test functions for bad reduction of simple Shimura varieties at a prime where the underlying local group is any inner form of a product of Weil restrictions of general linear groups. Using global methods, we prove that these test functions satisfy a vanishing property of their twisted orbital integrals, and we prove that the pseudostabilization base changes of such functions exist (even though the local group need not be quasi-split) and can be expressed in terms of explicit distributions in the stable Bernstein center. We then deduce applications to the stable trace formula and local Hasse-Weil zeta functions for these Shimura varieties.

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arXiv2022

On the normality of Schubert varieties: remaining cases in positive characteristic

We study the geometry of equicharacteristic partial affine flag varieties associated to tamely ramified groups $G$ in characteristics $p>0$ dividing the order of the fundamental group $π_1(G_{\text{der}})$. We obtain that most Schubert varieties are not normal and provide an explicit criterion for when this happens. Apart from this, we show, on the one hand, that loop groups of semisimple groups satisfying $p \mid \lvert π_1(G_{\text{der}})\rvert$ are not reduced, and on the other hand, that their integral realizations are ind-flat. Our methods allow us to classify all tamely ramified Pappas-Zhu local models of Hodge type which are normal.

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arXiv2020

Smoothness of Schubert varieties in twisted affine Grassmannians

We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of ${\rm Spin}_8$ ("ramified triality") provides an input needed in the article by He-Pappas-Rapoport classifying Shimura varieties with good or semi-stable reduction.

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arXiv2020

Equidimensionality of convolution morphisms and applications to saturation problems

We study the fibers of Mirkovic-Vilonen convolution morphisms. We prove their equidimensionality when all the coweights in question are minuscule, and some related statements. We give applications to saturation problems for structure constants of Hecke and representation rings. An erratum has been added to correct insufficiencies in the proofs of Lemma 9.4 and Corollary 9.5 as they appeared in the published version.

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arXiv2020

The Test Function Conjecture for Local Models of Weil-restricted groups

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geq 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.

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arXiv2019

Normality and Cohen-Macaulayness of parahoric local models

We study the singularities of integral models of Shimura varieties and moduli stacks of shtukas with parahoric level structure. More generally our results apply to the Pappas-Zhu and Levin mixed characteristic parahoric local models, and to their equal characteristic analogues. For any such local model we prove under minimal assumptions that the entire local model is normal with reduced special fiber and, if $p>2$, it is also Cohen-Macaulay. This proves a conjecture of Pappas and Zhu, and shows that the integral models of Shimura varieties constructed by Kisin and Pappas are Cohen-Macaulay as well.

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arXiv2018

Dualities for root systems with automorphisms and applications to non-split groups

This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over nonarchimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the $\{ μ\}$-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Tits échelonnage root system $Σ_0$, the Knop root system $\widetildeΣ_0$, and the Macdonald root system $Σ_1$, in terms of Galois actions on the absolute roots $Φ$; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis.

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arXiv2017

Vertexwise criteria for admissibility of alcoves

We give a new description of the set $Adm(μ)$ of admissible alcoves as an intersection of certain "obtuse cones" of alcoves, and we show this description may be given by imposing conditions vertexwise. We use this to prove the vertexwise admissibility conjecture of Pappas-Rapoport-Smithling. The same idea gives simple proofs of two ingredients used in the proof of the Kottwitz-Rapoport conjecture on existence of crystals with additional structure.

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arXiv2017

Frobenius semisimplicity for convolution morphisms

This article concerns properties of mixed $\ell$-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple; this conjecture would imply that a strong form of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber is valid over finite fields. We prove our conjecture for (generalized) convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields, and we prove allied Frobenius semisimplicity results for the intersection cohomology groups of twisted products of Schubert varieties. We offer two proofs for these results: one is based on the paving by affine spaces of the fibers of certain convolution morphisms, the other involves a new schematic theory of "big cells" adapted to partial affine flag varieties, and combines Deligne's theory of weights with a suitable contracting $\mathbb G_m$-action on those big cells. Both proofs rely on our general result that the intersection complex of the image of a proper map of varieties over a finite field is a direct summand of the direct image of the intersection complex of the domain. With suitable reformulations, the main results are valid over any algebraically closed ground field.

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arXiv2014

On Satake parameters for representations with parahoric fixed vectors

This article constructs the Satake parameter for any irreducible smooth $J$-spherical representation of a $p$-adic group, where $J$ is any parahoric subgroup. This parametrizes such representations when $J$ is a special maximal parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in formulating a geometric Satake isomorphism for such groups over local function fields.

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arXiv2014

The stable Bernstein center and test functions for Shimura varieties

We elaborate the theory of the stable Bernstein center of a $p$-adic group $G$, and apply it to state a general conjecture on test functions for Shimura varieties due to the author and R. Kottwitz. This conjecture provides a framework by which one might pursue the Langlands-Kottwitz method in a very general situation: not necessarily PEL Shimura varieties with arbitrary level structure at $p$. We give a concrete reinterpretation of the test function conjecture in the context of parahoric level structure. We also use the stable Bernstein center to formulate some of the transfer conjectures (the "fundamental lemmas") that would be needed if one attempts to use the test function conjecture to express the local Hasse-Weil zeta function of a Shimura variety in terms of automorphic $L$-functions.

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arXiv2011

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 $Γ_1(p)$-level structure initiated by M. Rapoport and the author in [HR2].

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arXiv2010

Ideal triangles in Euclidean buildings and branching to Levi subgroups

We introduce the notion of ideal triangle in the Bruhat-Tits building associated to a split group -- it is analogous to the usual notion of triangle, but one vertex is "at infinity" in a certain direction. We prove that the algebraic variety of based ideal triangles with prescribed side-lengths is naturally isomorphic to a suitable variety of genuine triangles. From theorems pertaining to genuine triangles, we deduce saturation theorems related to branching to Levi subgroups and to the constant term homomorphisms.

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arXiv2010

Affine Deligne-Lusztig varieties in affine flag varieties

This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.

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