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Pawel Kasprzak - One of the best experts on this subject based on the ideXlab platform.
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rieffel deformation of homogeneous spaces
Journal of Functional Analysis, 2011Co-Authors: Pawel KasprzakAbstract:Let G1⊂G be a closed subgroup of a locally compact group G and let X=G/G1 be the quotient space of Left Cosets. Let X=(C0(X),ΔX) be the corresponding G-C∗-algebra where G=(C0(G),Δ). Suppose that Γ is a closed abelian subgroup of G1 and let Ψ be a 2-cocycle on the dual group Γˆ. Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ-C∗-algebra XΨ – the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G1 obtaining the closed quantum subgroup G1Ψ⊂GΨ, which in turn, by the results of S. Vaes, leads to the GΨ-C∗-algebra GΨ/G1Ψ. In this paper we show that GΨ/G1Ψ≅XΨ. We also consider the case where Γ⊂G is not a subgroup of G1, for which we cannot construct the subgroup G1Ψ. Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ-simple object in the category of GΨ-C∗-algebras.
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rieffel deformation of homogeneous spaces
arXiv: Operator Algebras, 2010Co-Authors: Pawel KasprzakAbstract:Let H be a closed subgroup of a locally compact group G and let X=G/H be the quotient space of Left Cosets. Let C*X be the corresponding G-C*-algebra of continuous functions on X, vanishing at infinity. Suppose that L is a closed abelian subgroup of H and let f be a 2-cocycle on the dual group of L. Let G(f) be the Rieffel deformation of G. Using these data we may construct G(f)-C*-algebra C*X(f) - the Rieffel deformation of C*X. On the other hand we may perform the Rieffel deformation of the subgroup H obtaining the closed quantum subgroup H(f) of G(f) which in turn, by the results of Vaes, leads to the G(f)-C*-algebra G(f)/H(f). In this paper we show that G(f)/H(f) and C*X(f) are isomorphic G(f)-C*-algebras. We also consider the case where L is a subgroup of G but not of H, for which we cannot construct the subgroup H(f). Then C*X(f) cannot be identified with a quantum quotient. What may be shown is that it is a G(f)-simple object in the category of G(f)-C*-algebras.
Runde Volker - One of the best experts on this subject based on the ideXlab platform.
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Cohen–Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca–Herz algebras
Elsevier Inc., 2007Co-Authors: Runde VolkerAbstract:AbstractLet G be a locally compact group, and let R(G) denote the ring of subsets of G generated by the Left Cosets of open subsets of G. The Cohen–Host idempotent theorem asserts that a set lies in R(G) if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We apply our Cohen–Host type theorems to the study of the Figà-Talamanca–Herz algebras Ap(G) with p∈(1,∞). For arbitrary G, we characterize those closed ideals of Ap(G) that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those G such that Ap(G) is 1-amenable for some—and, equivalently, for all—p∈(1,∞): these are precisely the abelian groups
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Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Fig\`a-Talamanca-Herz algebras
2006Co-Authors: Runde VolkerAbstract:Let $G$ be a locally compact group, and let ${\cal R}(G)$ denote the ring of subsets of $G$ generated by the Left Cosets of open subsets of $G$. The Cohen--Host idempotent theorem asserts that a set lies in ${\cal R}(G)$ if and only if its indicator function is a coefficient function of a unitary representation of $G$ on some Hilbert space. We prove related results for representations of $G$ on certain Banach spaces. We apply our Cohen--Host type theorems to the study of the Fig\`a-Talamanca--Herz algebras $A_p(G)$ with $p \in (1,\infty)$. For arbitrary $G$, we characterize those closed ideals of $A_p(G)$ that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those $G$ such that $A_p(G)$ is 1-amenable for some -- and, equivalently, for all -- $p \in (1,\infty)$: these are precisely the abelian groups.Comment: 20 pages; LaTeX2e; one reference exchange
Roman Bezrukavnikov - One of the best experts on this subject based on the ideXlab platform.
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Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, preprint math.RT/0201256
2012Co-Authors: Roman BezrukavnikovAbstract:Abstract. This paper is a continuation of [1]. In [1] we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar “Langlands dual ” description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik. Acknowledgements. I am greatful to all the people mentioned in acknowledgements in [1]. I also thank Eric Sommers for stimulating interest. The author is supported by NSF and Clay Institute. 1. Statements 1.1. Recollection of notations and set-up. We keep the set-up and notations of [1]. In particular, Fℓ is the affine flag variety of a split simple group G over a field k which is either finite or algebraically closed; Wf is the Weyl group of G, and W is the extended affine Weyl group; fW f ⊂ fW ⊂ W are the sets of minimal length representatives of respectively 2-sided and Left Cosets of Wf in W; PI is the category of Iwahori equivariant perverse sheaves on Fℓ is the category whose objects are mixed Iwahori equivariant perverse sheaves on Fℓ, and morphisms are weight 0 geometric morphisms, i.e. weight 0 morphisms between the pull-backs of sheaves to Fℓ ¯ k (the notation is (the notation used only for an algebraically closed k); while P mix I used for finite k only). Lw, w ∈ W are irreducible objects of PI, or irreducible self-dual objects of P mix on the cardinality of k. The Serre quotient categories f PI, f PI f P mi
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Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, preprint
2012Co-Authors: Roman BezrukavnikovAbstract:Abstract. This paper is a continuation of [1]. In [1] we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar “Langlands dual ” description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik. Acknowledgements. I am greatful to all the people mentioned in acknowledgements in [1]. I also thank Eric Sommers for stimulating interest. The author is supported by NSF and Clay Institute. 1. Statements 1.1. Recollection of notations and set-up. We keep the set-up and notations of [1]. In particular, Fℓ is the affine flag variety of a split simple group G over a field k which is either finite or algebraically closed; Wf is the Weyl group of G, and W is the extended affine Weyl group; fW f ⊂ fW ⊂ W are the sets of minimal length representatives of respectively 2-sided and Left Cosets of Wf in W; PI is the category of Iwahori equivariant perverse sheaves on Fℓ is the category whose objects are mixed Iwahori equivariant perverse sheaves on Fℓ, and morphisms are weight 0 geometric morphisms, i.e. weight 0 morphisms between the pull-backs of sheaves to Fℓ ¯ k (the notation is (the notation used only for an algebraically closed k); while P mix I used for finite k only). Lw, w ∈ W are irreducible objects of PI, or irreducible self-dual objects of P mix on the cardinality of k. The Serre quotient categories f PI, f PI f P mi
Vankov Kirill - One of the best experts on this subject based on the ideXlab platform.
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Algèbres de Hecke, séries génératices et applications
2008Co-Authors: Vankov Kirill, Pantchichkine AlexisAbstract:Le résultat principal dans le travail présenté est le calcul explicite de la série génératrice des opérateurs de Hecke dans l'algèbre de Hecke locale pour les groupes symplectiques de genre 3 et 4. L'algorithme est basé sur l'isomorphisme de Satake, qui permet de réaliser toutes les opérations dans l'algèbre des polynômes à plusieurs variables. C'est la première fois que cette expression est calculée pour le genre 4. Pour obtenir le résultat principal, une méthode de calcul symbolique a été développée. Cette approche algorithmique s'applique à d'autres types de séries de Hecke. En particulier, nous formulons et prouvons un analogue du Lemme de Rankin pour le genre 2. Nous avons aussi calculé les séries génératrices des carrés symétriques et des cubes symétriques.Se basant sur nos résultats nous formulons une conjecture de modularité pour les convolutions des fonctions L spineurs associées aux formes modulaires de Siegel. Nous considérons d'autres conjectures importantes liées aux formes modulaires de Siegel et à leurs fonctions L. Nous utilisons ces constructions pour calculer les facteurs algébriques rationnels aux valeurs critiques de la fonction L spineur attachée à F12 de Miyawaki. A notre connaissance c'est le premier exemple d'une fonction L-spineur de forme parabolique de Siegel de degré 3, dont certaines valeurs spéciales peuvent être calculées explicitement.Finalement, nous appliquons la théorie des algèbres de Hecke pour construire des cryptosystèmes algébriques sur ensembles finis de classes à gauches dans l'algèbre de Hecke. Nous utilisons une relation entre les classes à gauches et les points sur certains variétés algébriques projectives.The main result in presented work consists of explicit computation of the generating power series of Hecke operators in local Hecke algebra for the symplectic groups of genus 3 and 4. The computation algorithm is based on the Satake isomorphism, which allows to carry out all operations in the algebra of polynomials in multiple variables. This is the first time when this expression was computed in genus 4. In order to obtain the main result, the method of symbolic computation was developed. This algorithmic approach is also applied to other types of Hecke series. In particular, we formulate and prove the analog of Rankin's Lemma in higher genus. We also computed the symmetric squares and symmetric cubes generating series.Based on our computational results we formulate a modularity lifting conjecture for convolutions of L-functions attached to Siegel modular forms. We review other important conjectures related to Siegel modular forms and their L-functions. We use these constructions to compute the rational algebraic factors in critical values of the spinor L-function attached to F12 of Miyawaki. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree 3, when the special values can be computed explicitly. Finally, we apply the theory of Hecke algebras to constructions of algebraic cryptosystems on some finite sets of Left Cosets in Hecke algebra. We use a relation between Left Cosets and points on certain projective algebraic varieties.GRENOBLE1-BU Sciences (384212103) / SudocSudocFranceF
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ALGÈBRES DE HECKE, SÉRIES GÉNÉRATRICES ET APPLICATIONS
HAL CCSD, 2008Co-Authors: Vankov KirillAbstract:The main result in presented work consists of explicit computation of the generating power series of Hecke operators in local Hecke algebra for the symplectic groups of genus 3 and 4. The computation algorithm is based on the Satake isomorphism, which allows to carry out all operations in the algebra of polynomials in multiple variables. This is the first time when this expression was computed in genus 4. In order to obtain the main result, the method of symbolic computation was developed. This algorithmic approach is also applied to other types of Hecke series. In particular, we formulate and prove the analog of Rankin's Lemma in higher genus. We also computed the symmetric squares and symmetric cubes generating series.Based on our computational results we formulate a modularity lifting conjecture for convolutions of L-functions attached to Siegel modular forms. We review other important conjectures related to Siegel modular forms and their L-functions. We use these constructions to compute the rational algebraic factors in critical values of the spinor L-function attached to F12 of Miyawaki. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree 3, when the special values can be computed explicitly.Finally, we apply the theory of Hecke algebras to constructions of algebraic cryptosystems on some finite sets of Left Cosets in Hecke algebra. We use a relation between Left Cosets and points on certain projective algebraic varieties.Le résultat principal dans le travail présenté est le calcul explicite de la série génératrice des opérateurs de Hecke dans l'algèbre de Hecke locale pour les groupes symplectiques de genre 3 et 4. L'algorithme est basé sur l'isomorphisme de Satake, qui permet de réaliser toutes les opérations dans l'algèbre des polynômes à plusieurs variables. C'est la première fois que cette expression est calculée pour le genre 4. Pour obtenir le résultat principal, une méthode de calcul symbolique a été développée. Cette approche algorithmique s'applique à d'autres types de séries de Hecke. En particulier, nous formulons et prouvons un analogue du Lemme de Rankin pour le genre 2. Nous avons aussi calculé les séries génératrices des carrés symétriques et des cubes symétriques.Se basant sur nos résultats nous formulons une conjecture de modularité pour les convolutions des fonctions L spineurs associées aux formes modulaires de Siegel. Nous considérons d'autres conjectures importantes liées aux formes modulaires de Siegel et à leurs fonctions L. Nous utilisons ces constructions pour calculer les facteurs algébriques rationnels aux valeurs critiques de la fonction L spineur attachée à F12 de Miyawaki. A notre connaissance c'est le premier exemple d'une fonction L-spineur de forme parabolique de Siegel de degré 3, dont certaines valeurs spéciales peuvent être calculées explicitement.Finalement, nous appliquons la théorie des algèbres de Hecke pour construire des cryptosystèmes algébriques sur ensembles finis de classes à gauches dans l'algèbre de Hecke. Nous utilisons une relation entre les classes à gauches et les points sur certains variétés algébriques projectives
Sun Zhi-wei - One of the best experts on this subject based on the ideXlab platform.
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Finite covers of groups by Cosets or subgroups
2006Co-Authors: Sun Zhi-weiAbstract:This paper deals with combinatorial aspects of finite covers of groups by Cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be Left Cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its proper subsystems does. We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\geq m+f([G:\bigcap_{i=1}^kG_i])$, where $f(\prod_{t=1}^r p_t^{\alpha_t})=\sum_{t=1}^r\alpha_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $\alpha_1,...,\alpha_r$ are nonnegative integers. When all the $a_i$ are the identity element of $G$ and all the $G_i$ are subnormal in $G$, we prove that there is a composition series from $\bigcap_{i=1}^kG_i$ to $G$ whose factors are of prime orders. The paper also includes some other results and two challenging conjectures.Comment: 19 page
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On the Herzog–Schönheim conjecture for uniform covers of groups
Elsevier Inc., 2004Co-Authors: Sun Zhi-weiAbstract:AbstractLet G be any group and a1G1,…,akGk (k>1) be Left Cosets in G. In 1974 Herzog and Schönheim conjectured that if A={aiGi}i=1k is a partition of G then the (finite) indices n1=[G:G1], …, nk=[G:Gk] cannot be pairwise distinct. In this paper we show that if A covers all the elements of G the same number of times and G1,…,Gk are subnormal subgroups of G not all equal to G, then M=max1⩽j⩽k|{1⩽i⩽k:ni=nj}| is not less than the smallest prime divisor of n1⋯nk; moreover, min1⩽i⩽klogni=O(Mlog2M) where the O-constant is absolute
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Exactm-covers of Groups by Cosets
Academic Press., 2001Co-Authors: Sun Zhi-weiAbstract:AbstractLet G be a group covered by its Left Cosets a1G1,⋯ , akGkexactly m times. It is known that [ G:∩i=1⊇kGi] ⩽k!. When all the Giare subnormal in G and∩i=1⊇kGi=H, we are able to determine the least value of k in terms of m, G, H. For any i= 1,⋯ , k, providing G/(Gi)Gis solvable we show that k⩾m+f([ G: Gi]) and hence [ G: Gi] ⩽ 2⊇k−m, where f(n) =∑s=1⊇rαs(ps− 1) if p1⊇α1⋯pr⊇αris the standard factorization of n. These extend some previous results on disjoint covers