We construct local models of Shimura varieties and investigate their singularities, with special ... more We construct local models of Shimura varieties and investigate their singularities, with special emphasis on wildly ramified cases. More precisely, with the exception of odd unitary groups in residue characteristic 2 we construct local models, show reducedness of their special fiber, Cohen-Macaulayness and in equi-characteristic also (pseudo-)rationality. In mixed characteristic we conjecture their (pseudo-)rationality. This is based on the construction of parahoric group schemes over two dimensional bases for wildly ramified groups and an analysis of singularities of the attached Schubert varieties in positive characteristic using perfect geometry.
Fix an ideal I ⊂ A. There is an order-preserving correspondence {ideals J ⊆ A containing I} ←→ {i... more Fix an ideal I ⊂ A. There is an order-preserving correspondence {ideals J ⊆ A containing I} ←→ {ideals of A/I}, given by: send an ideal J ⊃ I to its image J in A/I, and send an ideal J ⊆ A/I to its pre-image under the canonical map A → A/I. 1.5. Prime and maximal ideals. A domain is a ring A with the property: 1 = 0 and if x, y ∈ A and xy = 0, then x = 0 or y = 0. Examples are the integers Z, and any ring of polynomial functions over a field. An ideal p ⊂ A is prime if it is proper (p = A) and xy ∈ p implies x ∈ p or y ∈ p. Thus, p is prime if and only if A/p is a domain. An ideal m ⊂ A is maximal if m = A and there is no ideal I satisfying m I A. Equivalently, m is maximal if and only A/m is a field. To see this, check that any ring R having only (0) and R as ideals is a field. Now m is maximal if and only if A/m has no ideals other than (0) and A/m (Prop. 1.4.1), so the result follows on taking R = A/m in the previous statement. In particular, every maximal ideal is prime. Proposition 1.5.1. Maximal ideals exist in any ring A with 1 = 0. Proof. This is a standard application of Zorn's lemma. Let S be the set of all proper ideals in A, ordered by inclusion. Let T = {I α } α∈A be a chain of proper ideals. Then the union ∪ α∈A I α is an ideal which is an upper bound of T in S. Hence by Zorn's lemma S has maximal elements, and this is what we claimed. Let us define Spec(A) to be the set of all prime ideals of A, and Spec m (A) to be the subset consisting of all maximal ideals. These are some of the main objects of study in this course. The nomenclature "spectrum" comes from functional analysis, and will be explained later on. Also, pretty soon we will give the set Spec(A) the structure of a topological space and discuss the foundations of algebraic geometry... 1.6. Operations of contraction and extension. Fix a homomorphism φ : A → B. For an ideal I ⊆ A define its extension I e ⊆ B to the ideal generated by the image φ(I); equivalently, I e = ∩ J J where J ⊆ B ranges over all ideals containing the set φ(I). Dually, for an ideal J ⊆ B define the contraction J c := φ −1 (J), an ideal in A. Note that J prime ⇒ J c prime, so contraction gives a map of sets Spec(B) → Spec(A). On the other hand, contraction does not preserve maximality: consider the contraction of J = (0) under the inclusion Z → Q. Therefore, a homomorphism φ : A → B does not always induce a map of sets Spec m (B) → Spec m (A). As we will see later on, there is a natural situation where φ does induce a map Spec m (B) → Spec m (A): this happens if A, B happen to be finitely generated algebras over a field. This is quite important and is a consequence of Hilbert's Nullstellensatz, one of the first important theorems we will cover. 1.7. Nilradical. Define the nilradical of A by rad(A) := {f ∈ A | f n = 0, for some n ≥ 1}. Check that rad(A) really is an ideal. Elements f satisfying the condition f n = 0 for some n ≥ 1 are called nilpotent. Exercise 2.1.2. Let A be a commutative ring. Show that for X, Y ∈ M n (A), det(XY) = det(X)det(Y). Deduce from this and Cramer's rule that X has an inverse in M n (A) if and only if det(X) ∈ A ×. Corollary 2.1.3 (Improved NAK). If M is f.g. and IM = M , then there exists a ∈ A with a ≡ 1mod I, and aM = 0. Proof. Take φ = id in Lemma 2.1.1, and note that a := 1+a r−1 +• • •+a 0 works. Note that this corollary gives another proof of Prop. 1.11.1: I ⊂ rad m (A) means that a ∈ A × , and so aM = 0 implies M = 0.
Abstract. We study the local factor at p of the semi-simple zeta function of a Shimura variety of... more Abstract. We study the local factor at p of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at p by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.
We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Be... more We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson-Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an earlier paper, which encode the data of the Jordan-Hoelder series. We prove the multiplicity functions are polynomials in q, and we give a sharp bound for their degrees. Our results apply as well to the nearby cycles in the p-adic deformation of Laumon-Haines-Ngô, and also to Wakimoto sheaves.
Our aim here is to give a fairly self-contained exposition of some basic facts about the Iwahori-... more Our aim here is to give a fairly self-contained exposition of some basic facts about the Iwahori-Hecke algebra H of a split p-adic group G, including Bernstein’s presentation and description of the center, Macdonald’s formula, the Casselman-Shalika formula, and the Lusztig-Kato formula. There are no new results here, and the same is essentially true of the proofs. We have been strongly influenced by the notes [1] of a course given by Bernstein. In the spirit of Bernstein’s work, we approach the material with an emphasis on the “universal unramified principal series ” module M = Cc(AON\G/I), which is a right module over the Iwahori-Hecke algebra H = Cc(I\G/I). We use M to develop the theory of intertwining operators in a purely algebraic framework. Once this framework is established, we adapt it to produce rather efficient proofs of the above results, following closely at times earlier proofs. In particular, in our treatment of Macdonald’s formula and the Casselman-Shalika formula, w...
Let G be an unramified group over a p-adic field. This article introduces a base change homomorph... more 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].
We elaborate the theory of the stable Bernstein center of a p-adic group G, and apply it to state... more 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.
We prove the test function conjecture of Kottwitz and the first named author for local models of ... more 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≥ 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.
We study the Jordan-Hoelder series for nearby cycles on certain Shimura varieties and Rapoport-Zi... more We study the Jordan-Hoelder series for nearby cycles on certain Shimura varieties and Rapoport-Zink local models, and on finite-dimensional pieces of Beilinson's deformation of the affine Grassmannian to the affine flag variety (and their p-adic analogues). We give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups, and we also provide an algorithm to compute multiplicities, in terms of the affine Hecke algebra.
We study the fibers of Mirkovic-Vilonen convolution morphisms. We prove their equidimensionality ... more 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.
We give a new description of the set Adm(μ) of admissible alcoves as an intersection of certain &... more 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.
This article constructs the Satake parameter for any irreducible smooth J-spherical representatio... more 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.
This survey article explains the construction of Rapoport-Zink local models and their use in unde... more This survey article explains the construction of Rapoport-Zink local models and their use in understanding various questions relating to the singularities in the reduction modulo p of certain Shimura varieties with parahoric level structure at p.
We give explicit formulae for certain elements occurring in the Bernstein presentation of an affi... more We give explicit formulae for certain elements occurring in the Bernstein presentation of an affine Hecke algebra, in terms of the usual Iwahori- Matsumoto generators. We utilize certain minimal expressions for said elements and we give a sheaf-theoretic interpretation for the existence of these minimal expressions.
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group.... more 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.
This article concerns properties of mixed ℓ-adic complexes on varieties over finite fields, relat... more This article concerns properties of mixed ℓ-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 c...
We introduce the notion of ideal triangle in the Bruhat-Tits building associated to a split group... more 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.
We study the structure constants defining two related rings: the spherical Hecke algebra of a spl... more We study the structure constants defining two related rings: the spherical Hecke algebra of a split connected reductive group over a non-Archimedean local field, and the representation ring of the Langlands dual group.
We study the singularities of integral models of Shimura varieties and moduli stacks of shtukas w... more 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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Papers by Thomas Haines