Showing papers in "Annales Scientifiques De L Ecole Normale Superieure in 1994"
TL;DR: In this article, the authors investigated the derived category of a differential Z-graded category and deduced a triangulated analogue of a theorem of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories.
Abstract: — We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a "Morita theorem" (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories.
989 citations
TL;DR: In this article, the authors consider properties of closed geodesies in a compact 2-step nilmanifold, where n is a simply connected Lie group with a left invariant metric and r is a discrete cocompact subgroup of N. They show that T\\N, T*N* are equivalent up to isometry and r-almost inner automorphism in the sense of Gordon and Wilson.
Abstract: We consider properties of closed geodesies in a compact nilmanifold T\\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant metric and r is a discrete cocompact subgroup of N. Among other results we show 1) There is an obstruction (resonance) to the density in T\\ (T\\N) of the set of vectors P that are periodic with respect to the geodesic flow. In particular P is not always dense in T\\ (T\\AQ, but P is dense in T\\ (T\\AQ for any F if N is of Heisenberg type. 2) Every free homotopy class of closed curves in T\\N contains a closed geodesic of largest period. Define the maximal length spectrum of FW to be the collection with multiplicities of these largest periods. If T\\N, r* \\N* are compact 2-step nilmanifolds with the same marked maximal length spectrum, then we show that T\\N, T*\\N* are equivalent up to isometry and r-almost inner automorphism in the sense of Gordon and Wilson.
213 citations
TL;DR: In this paper, a Deligne-Lusztig theory for complex characters of a non-connected reductive group over a finite field was developed, based on the theory of Deligne and Lustzig.
Abstract: We develop a Deligne-Lusztig theory for thé complex characters of a non-connected reductive group over a finite field. RÉSUMÉ. Nous développons une théorie de Deligne-Lustzig pour les caractères d'un groupe réductif non connexe sur un corps fini.
111 citations
TL;DR: In this article, a comprehensive account of the theory of the binary additive divisor problem is given, which contains the hitherto best estimates of the error terms in the asymptotic formulas for both the ordinary and the dual versions of the problem.
Abstract: Here we try to develop a comprehensive account of the theory of the binary additive divisor problem. The results contain the hitherto best estimates of the error terms in the asymptotic formulas for both the ordinary and the dual versions of the problem.
111 citations
TL;DR: In this paper, the existence of Lipschitz stratification for subanalytic sets is shown and the authors decompose each compact sub-analytic set into pieces, called L-regular sets.
Abstract: In this paper we show the existence of Lipschitz stratification (in the sense of Mostowski) for subanalytic sets. Such stratification ensures, in particular, bi-Lipschitz triviality of the stratified set along each stratum. In fact, our construction is more precise. We decompose each compact subanalytic set into pieces, called L-regular sets, distinguished by their simple Lipschitz properties. In a way our decomposition is similar to triangulation but technically more complicated. In the course of the proof we develop for subanalytic sets such techniques as: regular projection theorem, subanalytic sets in complex domain and an analog of the Weierstrass preparation theorem for subanalytic functions.
105 citations
TL;DR: In this paper, the existence of non isomorphic tight contact structures on T is proved and it is shown that Lagrangian incompressible embedded tori in T x (R^^O) are homotopic.
Abstract: This paper proves thé existence of non isomorphic tight contact structures on T. It aiso shows that ail Lagrangian incompressible embedded tori in T x (R^^O}) are homotopic.
98 citations
TL;DR: In this paper, the authors show that HL(gI(A), k) is a tensor algebra on thé Hochschild homology of a matrices with fïnite size over an associative, imitai algebra over a base field k => Q. This resuit is a non-commutative analogue of Loday-Quillen's theorem stating that H^gI (A), Â:)=A*HC^_i (A).
Abstract: — Leibnitz algebras are algebras whose product, denoted [., . ], satisfies a certain form of Jacobi's identity, without any symmetry assumption. So, ail Lie algebras are Leibniz. A (co)homological theory is defîned for thèse algebras, and can be considered as non-commutative Lie (co)homology. Thé standard géométrie interprétations in low degrees (H classifies extensions, and so forth...) remain true. We apply thèse functors to thé Lie algebra gl (A) of matrices with fïnite size over A an associative, imitai algebra over a base field k => Q, and show that HL^(gI(A), k) is thé tensor algebra on thé Hochschild homology ofA. This resuit is a non-commutative analogue of Loday-Quillen's theorem stating that H^(gI(A), Â:)=A*HC^_i (A).
90 citations
TL;DR: In this paper, the authors studied the equivariant periodic cyclic homology of the algebra C°° (M) in terms of global Equivariant differential forms on the fixed point set M.
Abstract: Let G be a compact Lie group, and let M be a compact manifold on which G acts smoothly In this paper, we give a description of the equivariant periodic cyclic homology HP^ (C°° (M)) of C°° (M) as the cohomology of global equivariant differential forms on M: these are sections of a sheaf over the group G, whose stalk at g (E G is the complex of equivariant differential forms on the fixed-point set M, with action of the centralizer 0 By the isomorphism HP^ (C°° (M)) '= K^ (M) (g)R (G) R (G) with equivariant K-theory [where R (G) is the space of smooth functions on G invariant under the adjoint action], we also obtain a de Rham description of equivariant K-theory Let G a compact Lie group, and let M be a compact manifold on which G acts smoothly Let R°° (G) be the ring C°° (G)° of smooth conjugation invariant functions on the group G; it is an algebra over the representation ring R(G) of G, since R(G) maps into R°° (G) by the character map Then there is an equivariant Chem character ch^: K^ (M) = K^ (C°° (M)) HP^ (C°° (M)) from the equivariant K-theory of M to the periodic cyclic homology HP^ (C°° (M)) of the algebra C°° (M) of smooth functions on M This map induces an isomorphism HP^ (C°° (M)) ^ K^ (M)0R(G) R (G); furthermore, there are graded-commutative products on both HP^ (C°° (M)) and K^ (M) such that the Chem character map is a ring homomorphism These results are due to Block [3] (although he works with a crossed product involving algebraic functions instead of smooth ones), and Brylinski [5] In this paper, we will study the equivariant cyclic homology of the algebra C°° (M) in terms of equivariant differential forms on M; this extends the description which HochschildKostant-Rosenberg gave of the Hochschild homology of C°° (M) in terms of differential () This paper is dedicated to the memory of Ellen Block ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE 0012-9593/94/047$ 400/© Gauthier-Villars 494 J BLOCK AND E GETZLER forms on M, which was extended by Connes to cyclic homology Let us give a rough idea of how this works If c= /o^—^/ fc^ fi^C°°(M) and ^ G C°° (G), we define a map from the Lie algebra 0 of G to the space of A:-forms f^(M) on M, by the formula X^(expX) / fod(e-f,)/\'-/\d{e-fk)dt^"dtk J^k Here, A/: is the ^-simplex { ( ^ i , , 4)|0 < ^i ^ t 2 ^ tk < 1} c R^ This definition extends to define a map from C (M^ x G) to C (5, (M)), which moreover commutes with the actions of G on these two spaces: these actions are defined as follows: on C°° (M^ x G) by (hc) ( x o , , X k \ g ) = c(h~ xo,,h~ Xk\h~ gh), and on C (5, (M)) by ( fa o ; ) (X) = L ^ i C c ; ( a d ( f a ) X ) Thus, we obtain a map from C^ (C°° (M)) = C (M^ x G)° to C (0, (M))° = (C[fl] 0 (M)) ^[s] C (0)° This map is just one component of our equivariant Hochschild-Kostant-Rosenberg map; the other components correspond to other points of G, and define maps from C^ (C°° (M)) to C (^, (M))', where M is the fixed-point set of g acting on M, G is the fixedpoint set of g acting by conjugation on G (in other words the centralizer of g) and Q is the Lie algebra of G In the above notation, this map is induced by sending /o • • • fk to X e ^ ^ ^ ( ^ e x p X ) / /od(et l x A)A••Ad(et f c x ^) |M^l•••d4 ^Afc We call this map Og It turns out that the correct way to describe the situation is by means of sheaves on G, with the topology given by open sets invariant under conjugation; all of our sheaves will be equivariant In Section 1, we define a sheaf whose stalk at g G G is the space of germs at 0 of maps from Q to f2* (M) invariant under the centralizer G In Section 2, we introduce the equivariant cyclic chains; these are just smooth functions on M^ x G which are invariant under the action of G: c (xo,, X k , g) = c (h~ XQ, , h~ xj,, h~ gh) for all h G G It is easy to see how to define the sheaf C, (C (M), G) of equivariant ^-chains over G: the space of sections C^ (C (M)) over the invariant open set U is the space of invariant smooth functions on M^ x U 4' SERIE TOME 27 1994 N° 4 EQUIVARIANT CYCLIC HOMOLOGY 495 The maps {ag\g G G} assemble to define a map of sheaves a : C (C°° (M), G) -^ (M, G) The main result of this paper is the following equivariant generalization of the theorems of Hochschild-Kostant-Rosenberg and Connes; in a sense, we are completing the program of Baum-Brylinski-MacPherson THEOREM The map a defines a quasi-isomorphism of complexes of sheaves a: (C(C°°(M), G), & + z t B ) ^ ( ^ ( M , G), i+ud) Taking the homology of both sides, we see that HP^ (C (M)) ^ IT (^ (M), d + ), where A^ (M) = F(G, O* (M, G)) is the space of global equivariant differential forms In combination with the result relating equivariant K-theory with equivariant periodic cyclic homology, we obtain the following theorem: K^ (M) 0R (G) R°° (G) ^ K (^ (M), d + ) This work is heavily influenced by the papers of Baum-Brylinski-MacPherson [I], Berline-Vergne [2], and Brylinski [4] We would like to thank M Vergne and the referee for a number of helpful suggestions The paper was written while the first author was at MIT and at the Courant Institute The second author would like to thank the MSRI and the ENS for their hospitality during the writing of parts of this paper Both authors are partially funded by the NSF
86 citations
TL;DR: In this article, a characterization of orbital intégrais on real reductive Lie groups is given, which is used to prove a scalar Paley-Wiener type theorem for such functions.
Abstract: We give a characterization of orbital intégrais and stable orbital intégrais of indefinitely differentiable functions of compact support on real reductive Lie groups. This is used to prove a scalar Paley-Wiener type theorem for such functions. We aiso study orbital intégrais on symmetric spaces of type GC/GK.
40 citations
TL;DR: In this article, a general formula for the formal degrees of those square integrable representations of a p-adic Chevalley group which have both an Iwahori-fixed vector and a Whittaker model is given.
Abstract: We give a general formula for the formal degrees of those square integrable representations of a p-adic Chevalley group which have both an Iwahori-fixed vector and a Whittaker model. They are worked out in detail for small groups, using a computer. The results are interpreted in terms of L-packets.
38 citations
TL;DR: In this article, the authors studied the coherent cohomology of automorphic vector bundles, restricted to the toroidal boundary strata of Shimura varieties associated to maximal rational parabolic subgroups.
Abstract: We study the coherent cohomology of automorphic vector bundles, restricted to the toroidal boundary strata of Shimura varieties associated to maximal rational parabolic subgroups. The cohomology is computed in terms of coherent cohomology of the Shimura varieties attached to the boundary components. The main result concerns the restriction of a global coherent cohomology class to the boundary stratum associated with the maximal parabolic P; it is shown that, in terms of Dolbeault cohomology with growth conditions, this restriction is given by taking the constant term along the unipotent radical of P. This result is used to show that certain non-holomorphic, absolutely convergent Eisenstein series define rational global (coherent) cohomology classes. The main technical construction is a comparison between the (simplicial) Dolbeault complex associated to a complex torus embedding and the (simplicial) de Rham complex associated to its \"real part\".
TL;DR: In this paper, the properties of helices on the level of the Grothendieck group Ko(X) were investigated, where the semiorthogonal bases of KQ were viewed as a Z-module.
Abstract: — The paper deals with the properties of helices on the level of the Grothendieck group Ko(X). For Fano threefolds with Pic X^Z (the simplest from the point of view of helix theory) there are considered the semiorthogonal bases of Ky (X), viewed as a Z-module, which arise as a natural generalization of images in KQ of foundations of helices. For these threefolds equations are derived, which play the role analogous to that of Markov equation for helices on P. With the help of these equations the semiorthogonal bases of KQ are classified modulo action of mutations, i. e. the \"Ko-constructivity\" of helices is proved.
TL;DR: In this paper, it was shown that if two non-degenerate hypersurfaces in projective space have the same quotient of cubic to quadratic forms, then they are projectively congruent.
Abstract: — Using the method of moving frames, we prove that if two nondegenerate hypersurfaces in projective space have the same quotient of cubic to quadratic forms, then they are projectively congruent.
TL;DR: In this article, the authors consider a subalgrebra G? of thé algebra of germs at 0 of real C functions on (IR^) and extend to holomorphic functions in some open set of C with Gevrey conditions near thé origin.
Abstract: We consider a subalgrebra G? of thé algebra of germs at 0 of real C functions on (IR^)\"; thèse germs are analytic on thé open quadrant (R''*)\" and extend to holomorphic functions in some open set of C\", with Gevrey conditions near thé origin. We first defîne this algebra and study its properties (in particular, each germ is determined by its formai expansion at thé origin). Then, we consider a germ of \"semianalytic set\" X at thé origin ono^)\" which is defined by a fînite number of functions of Gj? and we prove for X thé same results as for thé usual analytic situation: X admits locally a fînite number ofconnected components ; if X^O, X contains a small Gevrey arc; every function of Gj? satisfîes a (Lojasiewicz inequality. At last, we can mix thèse results with Khovanskii's theory to get a large class of analytic algebras which verify good topological or metric properties. Dans cet article, on considère une sous-algèbre (notée ô^) de l'algèbre des fonctions réelles, C au voisinage de 0 dans (ll^)\", analytiques dans (R'^*)\"; cette algèbre, précisée ultérieurement, est «quasi-analytique», i.e. l'application G^9/->/e(R[[xi,..., xj] qui à / associe son développement asymptotique en 0, est injective. Les résultats principaux sont alors les suivants : THÉORÈME I. Soit X^UO^^O^)\" i/ij-W^O} un germe semi-analytique défini i J à l'aide d'un nombre fini de fonctions f^eG^. (1) X admet localement un nombre fini de composantes connexes i.e. si X est un représentant de X et si D^ est la boule euclidienne de W '. ||x||0 (« arc Gevrey » signifie que chacune de ses composantes est Gevrey, i.e. appartient à G^). THÉORÈME II. — Tout feQ^ vérifie une inégalité de Lojasiewicz par rapport au germe de ses zéros (si f est un représentant de f et si X=/~ (0), il existe des constantes C>0 et a>0 telles que \\f(x)\\^Cd(x, X), V^e^)\" assez voisin de 0). Dans les deux premiers paragraphes, on étudie l'algèbre ^=Gi des « fonctions multisommables dans la direction R^ » (cf. [l], [4], [6], [11] pour diverses approches de ces ANNALES SCIENTIFIQUES DE L'ÉCOLE NORMALE SUPÉRIEURE. — 0012-9593/94/02/S 4.00/ © Gauthier-Villars
TL;DR: Gauthier-Villars as discussed by the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1994, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: In this paper, it was shown that the Grothendieck group of J-modules is an isomorphism if qo is not a root of 1 or q = 1.
Abstract: We show that the lowest based ring of an affine Weyl group W is very interesting to understand some simple representations of the corresponding Hecke algebra H^^o^C*) even when q^ is a root of 1. Let Hq be the Hecke algebra (over C) attached by Iwahori and Matsumoto [IM] to an affine Weyl group W and to a parameter ^eC*. When qo is not a root of 1 or q^= 1, the simple H^-modules have been classified (see [KL2]). However we know little about the simple H^-modules when qo is a root of 1. In this paper we give some discussion to the representations of H^ with qo a root of 1. Namely, let J be the asymptotic Hecke algebra defined in [L3, III]. There exists a natural injection (|)^:H^-^J. Let K(J) [resp. K(H^)] be the Grothendieck group of J-modules (resp. H^-modules) of finite dimension over C, then ^^ induces a surjective homomorphism (<|)^)^ :K(J)-^K(H^), when qo is not a root of 1 or ^=1, (^o)* ls an isomorphism (loc. cit.). For each two-sided cell c of W, we can define the direct summand K(J,) [resp. K(HJJ of K(J) [resp. K(Hj]. Thus (^^ induces a homomorphism ((j^o)^ , c : K (^c) K (H^)cThe map ^qo)^,c remains surjective and is an isomorphism if qo is not a root of 1 or q^= 1. In this paper we mainly discuss the map ^ ^o)*, co' ^ere CQ is the lowest two-sided cell of W. (
TL;DR: In this article, it was shown that a constant number of eigenvalues less than A of thé image under a unitary irréductible représentation TT of a sublaplacian A of a stratified Lie algebra with a constant C > 1 depending oniy on (S: 1/CNo (À/C), where No (A) = p,{l ç.
Abstract: We prove that thé number N (A) of eigenvalues less than A of thé image under a unitary irréductible représentation TT of thé sublaplacian A of a stratified niipotent Lie algebra (& vérifies, with a constant C > 1 depending oniy on (S: 1/CNo (À/C) ^ N (À) ^ CNo {C A) where No (A) = p,{l ç. o (7r), |||/||| ^ A }, o (7r) being thé Kirillov orbit of TT, p, thé canonical measure on thé orbit, and 1 1 1 • 1 1 1 homogeneous norm. We aiso obtain an \"approximate diagonalization\" ofthe image under TT ofan homogeneous operator on (S, and use it to give a proof of conjectures, formulated in thé book of Heiffer and Nourrigat, about limit sets of représentations. Thé tools in thé proof are thé Wick calculus, and a construction of cohérent states associâtes with thé représentation TT. 0. Introduction Les travaux de A. Melin [15] et D. Manchon [13] associent, à toute représentation TT d'un groupe niipotent, un calcul pseudo-différentiel qui fournit l'inverse approché [15], ou les projecteurs spectraux approchés [14], de l'image par TT de certains opérateurs invariants sur le groupe. Une catégorie d'opérateurs reste cependant exclue de ces calculs : il s'agit de l'image d'un opérateur homogène P, invariant à gauche, tel que TT (P) vérifie une «estimation L maximale» sans que P lui-même soit hypoelliptique sur le groupe. Comme le montrent les articles [20] et [2l], ces opérateurs servent pourtant dans la preuve d'estimations sous-elliptiques, et il semble donc utile de construire un calcul qui leur soit applicable. A défaut de pouvoir utiliser facilement un calcul pseudo-différentiel, nous proposons, pour l'étude des opérateurs 7r(P) évoqués ci-dessus, d'adapter le classique «calcul de Wick». Dans le cas du groupe commutatif G == R, notons Yi, • • • , Yn une base de son algèbre de Lie G et, pour tout À > 0, considérons la représentation TT\\ de Q dans S (R) définit par : (0.1) ^(^.)=A-^( l ^ j ^n ) . ANNALES SCIENTIFIQUES DE L'ÉCOLE NORMALE SUPÉRIEURE. 0012-9593/94/067$ 4.00/© Gauthier-Villars 708 P. LÉVY-BRUHL ET J. NOURRIGAT Le calcul de Wick est très classique dans ce cas. On choisit une fonction ^ e S (R\") dont la nonne L est égale, par exemple, à (27^)/, et on pose, pour tout (y, T]} G R : (0.2) VW Çx) = À\"/ e ̂ -^ ̂ (A (x y)). Pour simplifier, posons z == (rc, ^), w == (î/, 77), et ck = dxd^. Pour toute fonction / € L (R) et pour tout À > 0, on peut écrire, au sens faible : (0.3) / / (f^^^dz et 11/H =f |(/,^)|^. jR2n JR2rz L'inégalité suivante joue un rôle essentiel : il existe M > 0, indépendant de ^ € R^ et de À > 0, tel que : (0.4) / |(̂ , ̂ x)\\dw ^ M, Mz e R, VA > 0. JR2^ Grâce à (0.3), on voit que, si a(^, À) est une fonction mesurable, bornée sur R\" x R+, l'opérateur A\\ défini par : A\\f = \\ a{z, À)(/, ^^)^z\\dz JR^ est uniformément borné dans ^(R\"), La fonction a(z, A) est appelée le «symbole d'anti-Wick» de l'opérateur A\\, les fonctions ^zx sont des «paquets d'ondes» [2], ou des « états cohérents » et, pour certains choix particuliers de la fonction ^?, la fonction T f {z, A) = (/, ̂ z\\) est liée à la transformée de Bargmann, ou de Fourier-Bros-Iagolnitzer [25], de/. La construction d'états cohérents vérifiant (0.3) et obtenus, à partir d'une seule fonction ^, par une famille d'opérateurs unitaires liés à la représentation étudiée, a été étendue à de nombreuses situations par Perelomov [23], Unterberger (cf. par exemple [26]), et Ali-Antoine-Gazeau [1]. Signalons que la notion d'« états cohérents» diffère de celle des ondelettes, adaptée au cas des groupes niipotents par Lemarié [9]. Dans la section 1, nous montrons que l'existence d'une famille d'éléments (^) dans un espace de Hilbert H, dépendant de manière mesurable d'un paramètre z dans un espace mesuré Z, tels que la norme de ^z soit une constante K > 0 et tels qu'on ait, pour tout / ç jEf, les égalités (0.3), peut avoir des conséquences en théorie spectrale. En effet, si un opérateur autoadjoint positif P de domaine D (P) c H vérifie les inégalités suivantes, pour tout / € D (P) : (0.5) / s(z)\\{^ ̂ d(z) ̂ (P/, / )+A| | /H 2 . J z (0.6) W,/)^ / S{z)\\{f^^d{z). J z [où s{z) et S(z} sont des fonctions positives mesurables sur Z], et si S (z) est une «fonction poids» au sens suivant : 4 SÉRIE TOME 27 1994 N° 6 REPRÉSENTATIONS DE GROUPES NILPOTENTS 709 '-' /.(s^r'̂ î -. ^ (où A est une constante), alors le nombre N (A) de valeurs propres de P, répétées selon leur multiplicité, qui sont inférieures à À, est encadré par la « formule de Weyl » suivante :
TL;DR: The main result of as mentioned in this paper is that separation holds for M if and only if every semialgebraic subset of M described by global analytic inequalities can also be described by s global Nash inequalities.
Abstract: Let M superset-of R be a compact Nash manifold, and N (M) [resp O(M)] its ring of global Nash (resp analytic) functions A global Nash (resp analytic) set is the zero set of finitely many global Nash (resp analytic) functions, and we have the usual notion of irreducible set Then we say that separation holds for M if every Nash irreducible set is analytically irreducible The main result of this paper is that separation holds if and only if every semialgebraic subset of M described by s global analytic inequalities can also be described by s global Nash inequalities In passing, we also prove that when separation holds, every Nash function on a Nash set extends to a global Nash function on M
TL;DR: Gauthier-Villars as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1994, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.