Department of Mathematics

This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over non-archimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the { μ } \{ \mu \} -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 \Sigma _0 , the Knop root system Σ ~ 0 \widetilde {\Sigma }_0 and the Macdonald root system Σ 1 \Sigma _1 , in terms of Galois actions on the absolute roots Φ \Phi ; 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. The latter gives an explicit form of the test function conjecture for general Shimura varieties with parahoric level structure.

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.

Abstract. Let G be an unramified group over a p-adic field F, and let E/F be a finite unramified extension field. Let θ denote a generator of Gal(E/F). This paper concerns the matching, at all semi-simple elements, of orbital integrals on G(F) with θ-twisted orbital integrals on G(E). More precisely, suppose φ belongs to the center of a parahoric Hecke algebra for G(E). This paper introduces a base change homomorphism φ 7 → bφ taking values in the center of the corresponding parahoric Hecke algebra for G(F). It proves that the functions φ and bφ are associated, in the sense that the stable orbital integrals (for semi-simple elements) of bφ can be expressed in terms of the stable twisted orbital integrals of φ. In the special case of spherical Hecke algebras (which are commutative) this result becomes precisely the base change fundamental lemma proved previously by Clozel [Cl90] and Labesse [Lab90]. As has been explained in [H05], the fundamental lemma proved in this paper is a key i...

These lectures describe Hecke algebra isomorphisms and types for depth-zero principal series blocks, a.k.a. Bernstein components Rs(G) for s = sχ = [T, e χ]G, where χ is a depth-zero character on T (O). (Here T is a split maximal torus in a p-adic group G.) We follow closely the treatment of A. Roche [Ro] with input from D. Goldstein [Gol] and L. Morris [Mor]. We give an elementary proof that (I, ρχ) is a type for sχ, in the sense of Bushnell-Kutzko [BK]. This is a very special case of a result of Roche [Ro]. Our method is to imitate Casselman’s proof of Borel’s theorem on unramified principal series (the case χ = 1 of the present theorem). In contrast to the situation for general principal series blocks (see [Ro]), in the depth-zero case there is no restriction on the residual characteristic of F .

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.

This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. Rapoport conjectured a formula for the dimensions of the varieties Xμ(b) in the affine Grassmannian. We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b. In the affine flag manifold, we prove a formula that reduces the dimension question for Xx(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits with orbits of the unipotent radical. Calculations using this formula allow us to verify a conjecture of Reuman in many new cases, and to make progress toward a generalization of his conjecture. Cet article concerne les dimensions de certaines variétés de Deligne-Lusztig affines, dans la Grassmannienne affine et dans la variété de drapeaux affine. Rapoport a conjecturé une formule pour les dimensi...

Let F be a local field (or any function field k(($))), with ring of integers OF . The main object of this manuscript is to provide a first step in defining Rapoport-Zink “local models” Mμ̌ attached to an arbitrary split reductive OF -group G and an arbitrary dominant coweight μ̌ for G. As it stands at this time, this paper proposes a definition for Mμ̌ when F = k(($)) and char(k) = 0 (the “equi-char 0” case).

Schubert varieties in classical flag varieties are known to be normal by the work of Ramanan-Ramanathan, Seshadri, Anderson and Mehta-Srinivas. The normality of Schubert varieties in affine flag varieties in characteristic zero was proved by Kumar, Mathieu and Littelmann in the Kac-Moody setting, and in positive characteristic $p > 0$ for simply connected split groups by Faltings. Pappas-Rapoport generalized Faltings' result to show that Schubert varieties are normal whenever $p$ does not divide $|\pi_1(G_{\textrm{der}})|$. This covers many cases, but the case of $\text{PGL}_n$ for $p \mid n$ is not covered, for instance. In this article, we use a variation of arguments of Faltings and Pappas-Rapoport to handle general Chevalley groups over $\mathbb{Z}$. In addition, for general Chevalley groups $G$ and parahoric subgroups $\mathcal{P} \subset LG$, we determine exactly when the partial affine flag variety $\text{Fl}_{\mathcal{P}} \otimes_{\mathbb{Z}}\mathbb{F}_p$ is reduced, ...

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, including Bernstein's presentation and description of the center, Macdonald's formula, the CasselmanShalika formula, and the Kato-Lusztig 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. The reader may find in The following notation will be used throughout this paper. We work over a padic field F with valuation ring O and prime ideal P = (π). We denote by k the residue field O/P and by q the cardinality of k. Consider a split connected reductive group G over F , with split maximal torus A and Borel subgroup B = AN containing A. We writeB = AN for the Borel subgroup containing A that is opposite to B. We assume that G, A, N are defined over O. We write K for G(O) and

Abstract. 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

In section 2.2 of [H09], there is a minor misstatement that this note will correct and clarify. It has no effect on the main results of [H09], but nevertheless this corrigendum seems necessary in order to avoid potential confusion. Also, I take this opportunity to point out a related typographical error in [BT2], section 5.2.4,

Abstract. This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. Rapoport conjectured a formula for the dimensions of the varieties Xµ(b) in the affine Grassmannian. We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b. In the affine flag manifold, we prove a formula that reduces the dimension question for Xx(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits with or-bits of the unipotent radical. Calculations using this formula allow us to verify a conjecture of Reuman in many new cases, and to make progress toward a generalization of his conjecture. Cet article concerne les dimensions de certaines variétés de Deligne-Lusztig affines, dans la Grassmannienne affine et dans la variéte ́ de drapeaux affine. Rapoport a conjecture ́ une formul...

Abstract. Fix a split connected reductive group G over a field k, and a positive integer r. For any r-tuple of dominant coweights µi of G, we consider the restriction mµ • of the r-fold convolution morphism of Mirkovic-Vilonen [MV1, MV2] to the twisted product of affine Schubert varieties corresponding to µ•. We show that if all the coweights µi are minuscule, then the fibers of mµ • are equidimensional varieties, with dimension the largest allowed by the semi-smallness of mµ •. We derive various consequences: the equivalence of the non-vanishing of Hecke and representation ring structure constants, and a saturation property for these structure constants, when the coweights µi are sums of minuscule coweights. This complements the saturation results of Knutson-Tao [KT] and Kapovich-Leeb-Millson [KLM]. We give a new proof of the P-R-V conjecture in the “sums of minuscules ” setting. Finally, we generalize and reprove a result of Spaltenstein pertaining to equidimensionality of certain...

Abstract. A construction of Bernstein associates to each cocharacter of a split p-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz ' conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of Gln; one example can be used to verify Kottwitz ' conjecture for a special class of Shimura varieties (the \Drinfeld case"). In addition we prove some general facts concerning the support of Bernstein functi...