In deep memory of Armand (Borel)

A reminiscence from his daughter

A reminiscence from one of his students

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

A reminiscence from one of his colleagues

Local Fourier transforms, analogous to the l-adic local Fourier transforms [14], are constructed for connections over k((t)). Following a program of Katz [12], a meromorphic connection on a curve is shown to ber igid, i.e. determined by local data at the singularities, if and only if a certain infinitesimal rigidity conditionis satisfied. As in [12],the argument uses local Fourier transforms to prove an invariance result for the rigidity index under global Fourier transform. A key technical tool is the notion of good lattice pairs for a connection[5].

The sets $\gamma H_{Y}$, with $Y$ fixed, as $\gamma$ ranges over $\Gamma$ are disjoint (when not identical), and their union is stable under $\Gamma$ as is its complement in $H$. The quotient of this complement by $\Gamma$ is compact. As far as I know, it was Jim Arthur who first generalized this result explicitly to arbitrary arithmetical quotients (in 1977), although I think it's fair to say that this generalization was already implicit in Satake's work on compactifications of arithmetic quotients. In Arthur's generalization the subsets of the partition are parametrized by $\Gamma$-conjugacy classes of rational parabolic subgroups, which is also how Satake's rational boundary components are parametrized. Of course Arthur did this work with the intention of using it in dealing with his extension of the Selberg trace formula, but subsequently it has also been useful in other contexts. In this note, which is largely expositoray, I will explain Arthur's partition for $GL_{n}(\bb{Z})$, applying ideas almost entirely due to Harder, Stuhler, and Grayson, and including a self-contained account of their work.

In this survey we review some results concerning negatively curved exotic structures (DIFF and PL) and its (unexpected) implications on the limitations of some analytic methods in geometry. Among these methods are the harmonic map method and the Ricci flow method. First in section 1 we mention certain results about the rigidity of negatively curved manifolds. In section 2 and 3 we survey some results concerning the limitations of the harmonic map technique and the natural map technique for negatively curved manifolds. Finally,in section 4, we mention some limitations of the Ricci flow method for pinched negatively curved manifolds. We are grateful to J-F. Lafont and R. Spatzier for the useful information they provided to us and to E.Gasparim for suggesting some improvements in the text. We are also grateful to the referee for pointing out certain inaccuracies.

It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write "will," have important consequences for the Langlands functoriality program as well as for the study of the Galois representations on the l-adic cohomology of Shimura varieties. At present, full stabilization is still only known for SL(2) and U(3) and their inner forms [LL,R]. The automorphic and arithmetic consequences of stabilization for U(3) form the subject of the influential volume [LR]. Under somewhat restrictive hypotheses, one can sometimes derive the expected corollaries of the stable trace formula. Examples of such "pseudo-stabilization" include Kottwitz' analysis in [K2] of the zeta functions of certain "simple" Shimura varieties attached to twisted forms of unitary groups over totally real fields, and the proof in [L1] of stable cyclic base change of automorphic representations which are locally Steinberg at at least two places. These conditional results have been used successfully to provide non-trivial examples of compatible systems of l-adic representations attached to certain classes of automorphic representations of GL(n) [C3], and of non-trivial classes of cohomology of S-arithmetic groups [BLS, L1]. Conditional results also suffice for important local applications, such as the local Langlands conjecture for GL(n) [HT, He]. The present article develops a technique for obtaining conditional base change and functorial transfer. Let Un be a unitary group over a number field F attached to a quadratic extension E/F. The technique applies to quadratic base change from Un to GL(n)E, and to transfer between inner forms of unitary groups. Roughly speaking, if p is an automorphic representation of U which is locally supercuspidal at two places of F split in E, then the expected consequences of the stable trace formula hold for p; in particular p admits a base change to a cuspidal automorphic representation of GL(n)E (Theorem 2.2.2). Slightly more general results are available when F is totally real and E is totally imaginary, and when p is of cohomological type. Automorphic descent from GL(n)E to Un can be proved under analogous hypotheses (Theorem 2.4.1, Theorem 3.1.2). Finally, we prove transfer between distinct inner forms of unitary groups (Jacquet-Langlands transfer) under quite general local hypotheses (Theorem 2.1.2 and, in a more precise form, Theorem 3.1.6 and Proposition 3.1.7). As in [L1], all results are obtained from the simple version of the Arthur-Selberg trace formula, in which non-elliptic and non-cuspidal terms are absent.

In this paper we show that the classical notion of association of projective point sets,[DO], Chapter III, is a special case of a general duality between weight varieties (i.e. torus quotients of flag manifolds) of a reductive group G induced by the action of the Chevalley involution on the set of these quotients. We compute the dualities explicitly on both the classical and quantum levels for the case of the weight varieties associated to GL_n(C). In particular we obtain the following formula as a special case. Let r = (r₁, ...,r_n)be an n-tuple of positive real numbers and M_r(CP^m)be the moduli space of semistable weighted(by r)configurations of n points in CP^m modulo projective equivalence, see for example[FM]. Let LAMBDA be the vector in Rⁿ with all components equal to [Sigma]_ir _i/(m + 1). Then M_r(CP ^m) is isomorphic to M_LAMBDA-rr (CP^n-m-2) (the meaning of is isomorphic to depends on r and will be explained below, see Theorem 1.6). We conclude by studying "self-duality" i.e. those cases where the duality isomorphism carries the torus quotient into itself. We characterize when such a self-duality is trivial, i.e. equal to the identity map. In particular we show that all self-dualities are nontrival for the weight varieties associated to the exceptional groups. The quantum version of this problem, i.e. determining for which self-adjoint representations V of G the Chevalley involution acts as a scalar on the zero weight space V[0], is important in connection with the irreducibility of the representations of Artin groups of Lie type which are obtained as the monodromy of the Casimir connection, see [MTL], and will be treated in [HMTL].

A conjecture of Sarnak [Sa] states that the counting function fo the cuspidal spectrum satisfies Weyl's law. This conjecture has been established in some special cases. First of all, it was Selberg [Se] who proved it for congruence subgroups of $SL(2,Z)$ and $\sigma = 1$. Other cases for which the conjecture has been established are Hilbert modular groups [Ef], congruence subgroups of $SO(n,1)$ [Rez], $SL(3,Z)$ [Mil], and in particular, the conjecture was proved in [Mu2] for principal congruence subgroups of $SL(n,Z)$ and arbitrary $\sigma$ ... The purpose of this paper is to prove a weaker result which holds for every $G$. Recall that an upper bound, which holds for arbitrary $G$ and $\Gamma$, is already known thanks to Donnelly [Do] ... By working with a simple form of the trace formula, we shall get, for a general $G$, a lower bound that depends on the choice of a set $S$ of primes containing at least two finite primes. For every such set $S$ we shall define a certain constant $c_{S}(\Gamma}$ less than or equal to 1, which is non zero for $\Gamma$ deep enough.

In [6], G. Laumon defined a set of functors, called by him local Fourier transformations, which allow to analyze the local structure of the l-adic Fourier transform of a constructible l-adic sheaf G on the affine line in terms of the local behaviour of G at infinity and at the points where it is not lisse. These local Fourier transforms play a major role in his cohomological interpretation of the local constants and in his proof of the product formula (see loc. cit. and [5]). In this article, we are concerned with differential systems defined over a field K of characteristic zero. We define a set of functors which allow to prove a stationary phase formula, expressing the formal germ at infinity of the D-module theoretic Fourier transform of a holonomic K[t]h.ti-module M in terms of the formal germs defined by M at its singular points and at infinity. These functors might therefore be regarded as formal analogues of Laumon's local Fourier transformations ... We will use some results from D-module theory for which we refer e.g. to [14], [15]. Our proof of the formal stationary phase formula follows the leitfaden of the one given by C. Sabbah in [15] for modules with regular singularities. I thank C. Sabbah, B. Malgrange, G. Christol and W. Messing for their useful remarks. I thank also G. Lyubeznik and S. Sperber for their invitation to the University of Minnesota, during which part of this work was done.

In this paper we show that the convolution product of "almost characters" of a connected reductive group over a finite field is given by "structure constants" whose leading coefficients can be interpreted in K-theoretic terms and in particular are natural numbers. I wish to thank M. Finkelberg and V. Ostrik for comments on an earlier version of this paper.

The purpose of the article is two-fold. First of all, we will show that in general gap rigidity already fails in the complex topology. More precisely, we show that gap rigidity fails for $\Delta^2, \Delta \ti 0$ by constructing a sequence of ramified coverings f_i : S_i goes to T_i between hyperbolic compact Riemann surfaces such that, with respect to norms defined by the Poincare metrics. Since any bounded symmetric domain of rank greater than or equal to 2 contains a totally geodesic bidisk, this implies that gap rigidity fails in general on any bounded symmetric domain of rank greater than or equal to 2. Our counter examples make it all the more interesting to find sufficient conditions for pairs(OMEGA,D)for which gap rigidity holds. This will be addressed in the second part of the article, where for irreducible, we generalize the results for holomorphic curves in [Mok2002]to give a sufficient condition for gap rigidity to hold for (OMEGA,D)in the Zariski topology.

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras(g,t), we construct, via cohomological induction, the fundamental series F ^. (p,E) of generalized Harish-Chandra modules. We then use F^. (p,E) to characterize any simple generalized Harish-Chandra module with generic minimal t-type. More precisely, we prove that any such simple(g,t)-module of finite type arises as the unique simple submodule of an appropriate fundamental series module F^s (p,E) in the middle dimension s. Under the stronger assumption that t contains a semisimple regular element of g, we prove that any simple(g,t)-module with generic minimal t-type is necessarily of finite type, and hence obtain a reconstruction theorem for a class of simple(g,t)-module which can a priori have infinite type. We also obtain generic general versions of some classical theorems of Harish-Chandra, such as the Harish-Chandra admissibility theorem. The paper is concluded by examples, in particular we compute the genericity condition on a ttype for any pair (g,t)with t about equal to sl(2).

Early experiences with classical (holomorphic) cusp forms, which initially started with the Ramanujan t-function, and later extended to even Maass cusp forms (cf. [70];[78], last paragraph) on the upper half plane, suggested that their Fourier coefficients ap at a prime p must be bounded by 2p(k-1)/2, where k is the weight (cf. [25, 96]). This is what is classically called the Ramanujan-Petersson conjecture. Its archimedean counterpart, the Selberg conjecture [79], states that the positive eigenvalues of the hyperbolic Laplacian on the space of cuspidal functions (functions vanishing at all the cusps) on a hyperbolic Riemann surface parametrized by a congruence subgroup must all be at least 1/4 (cf. [76, 79, 94]). While for the holomorphic modular cusp forms, this is a theorem ([25], also see [8, 12]), the case of Maass forms is far from resolved and both conjectures are yet unsettled and out of reach. Satake [78] was the first to observe that both conjectures can be uniformly formulated. More precisely, if one considers the global cuspidal representation attached to a given cuspidal eigenfunction, then all its local components must be tempered. This means that their matrix coefficients must all belong to $L2+(PGL2(Qp))$ for all $e > 0$ and every prime p of Q. We note that here we are allowing p = 8 and letting $Q_{/inf} = R$. It is now generally believed that the conjecture in its general form should be valid for GLn over number fields to the effect that all the local components of an irreducible (unitary) cuspidal representation of GLn(AF ) must be tempered (modulo center).

An orientable hyperbolic 3-manifold is isometric to the quotient of hyperbolic 3-space H3 by a discrete torsion free subgroup G of the group Iso(H3)0 of orientation -- preserving isometries of H3. The latter group is isomorphic to the (connected) group PGL2(C), the real Lie group SL2(C) modulo its center $\pm 1$. Generally, a discrete subgroup of PGL2(C) is called a Kleinian group. The group G is said to have finite covolume if H3/G has finite volume, and is said to be cocompact if H3/G is compact. Among hyperbolic 3-manifolds, the ones originating with arithmetically defined Kleinian groups form a class of special interest. Such an arithmetically defined 3-manifold H3/G is essentially determined (up to commensurability) by an algebraic number field k with exactly one complex place, an arbitrary (but possibly empty) set of real places and a quaternion algebra Dover k which ramifies (at least) at all real places of k. These arithmetic Kleinian groups fall naturally into two classes ...

This paper has as its primary purpose a clear exposition of the idea in [21] which we describe as transfer between real forms of a semi-simple Lie group over $\bb{C}$. The basic point is that there are representations of one real form that can be more easily understood in the context of another real from. An interesting example is a minimal representation (that is annihilated by the Joseph ideal) of a split group over $\bb{R}$. In the case when the complexi.cation admits a Hermitian symmetric real form, minimal representations of that real form are part of the "analytic continuation of the holomorphic discrete series"(cf.[5]) ...

We establish a procedure for constructing compatible mixed Hodge structures for the cohomology of various topological compactifications of locally symmetric varieties, notably ones that are not algebraic varieties. This is carried out in full for the case of the reductive Borel-Serre compactification, and conditionally for the excentric compactifications.