The authors consider the extended Hecke groups H(γq) generated by T(z) = -1 / z, S(z) = -1(z +γq) and R(z) = 1 / z with A, = 2 cos(π/q) for q≥3 an integer. In this paper, the even subgroup He(γq), the second comm...The authors consider the extended Hecke groups H(γq) generated by T(z) = -1 / z, S(z) = -1(z +γq) and R(z) = 1 / z with A, = 2 cos(π/q) for q≥3 an integer. In this paper, the even subgroup He(γq), the second commutator subgroup H''(γq) and the principal congruence subgroups Hp(λq) of the extended Hecke groups .H(γq) are studied. Also, relations between them are given.展开更多
Hecke groups are an important tool in subgroups of Hecke groups play an important rule investigating functional equations, and congruence in research of the solutions of the Dirichlet series. When q, m are two primes,...Hecke groups are an important tool in subgroups of Hecke groups play an important rule investigating functional equations, and congruence in research of the solutions of the Dirichlet series. When q, m are two primes, congruence subgroups and the principal congruence subgroups of level m of the Hecke group H（√q） have been investigated in many papers. In this paper, we generalize these results to the case where q is a positive integer with q ≥ 5, √q ￠ Z and m is a power of an odd prime.展开更多
Let K be an algebraic number field and OK its ring of integers.For any prime ideal p,the group(OK/p) of the reduced residue classes of integers is cyclic.We call any element of a generator of the group(OK/p) a primiti...Let K be an algebraic number field and OK its ring of integers.For any prime ideal p,the group(OK/p) of the reduced residue classes of integers is cyclic.We call any element of a generator of the group(OK/p) a primitive root modulo p.Stimulated both by Shoup's bound for the rational improvement and Wang and Bauer's generalization of the conditional result of Wang Yuan in 1959,we give in this paper a new bound for the least primitive root modulo a prime ideal p under the Grand Riemann Hypothesis for algebraic number field.Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.展开更多
Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representati...Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affne Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affne Weyl groups, but that is the theme of another paper.展开更多
Shimura has studied the structure of the Hecke ring R(Δ,Γ) in the case Γ=SLn(Z)andΔΓis a subsemigroup of G =GLn(Q).Letpbe a prime and letΓ =GLn(Zp) Δ G =GLn( p),whereΔis a semigroup. In this paper, we shall ...Shimura has studied the structure of the Hecke ring R(Δ,Γ) in the case Γ=SLn(Z)andΔΓis a subsemigroup of G =GLn(Q).Letpbe a prime and letΓ =GLn(Zp) Δ G =GLn( p),whereΔis a semigroup. In this paper, we shall determine the structure of the Hecke ringR(Δ,Γ)and its some properties.展开更多
We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalizatio...We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.展开更多
Some derived categories and their deformed versions are used to develop a theory of the ramifications of field studied in the geometrical Langlands program to obtain the correspondences between moduli stacks and solut...Some derived categories and their deformed versions are used to develop a theory of the ramifications of field studied in the geometrical Langlands program to obtain the correspondences between moduli stacks and solution classes represented cohomologically under the study of the kernels of the differential operators studied in their classification of the corresponding field equations. The corresponding D-modules in this case may be viewed as sheaves of conformal blocks (or co-invariants) (images under a version of the Penrose transform) naturally arising in the framework of conformal field theory. Inside the geometrical Langlands correspondence and in their cohomological context of strings can be established a framework of the space-time through the different versions of the Penrose transforms and their relation between them by intertwining operators (integral transforms that are isomorphisms between cohomological spaces of orbital spaces of the space-time), obtaining the functors that give equivalences of their corresponding categories.(For more information,please refer to the PDF version.)展开更多
The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter gr...The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter group is found. The new base is described by using the notion of cell datum of Graham and Lehrer and the notion of norm.展开更多
Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to giv...Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L2(г\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.展开更多
This is a continuation of our previous work. We classify all the simple ?q(D n )-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenf n(q) ≠ 0, this yields a classification of all the simple ? q (D...This is a continuation of our previous work. We classify all the simple ?q(D n )-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenf n(q) ≠ 0, this yields a classification of all the simple ? q (D n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ(1) = λ(2),wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?q,1(B n )- module Dλ remains irreducible on restriction to ?q(D n ).展开更多
Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x...Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.展开更多
Let A be a tame Hecke algebra of type A. A new minimal projective bimodule resolution for A is constructed and the dimensions of all the Hochschild homology groups and cyclic homology groups are calculated explicitly.
Let W be a classical Weyl group and ∏ be the corresponding system of simple roots. For w∈ W, let R(w)={α∈∏|l(ws_α)【l(w)}, where we denote by s_α the simple reflection in the hyperplane orthogonal to α for α...Let W be a classical Weyl group and ∏ be the corresponding system of simple roots. For w∈ W, let R(w)={α∈∏|l(ws_α)【l(w)}, where we denote by s_α the simple reflection in the hyperplane orthogonal to α for α∈∏ and by l(w)the minimal length of an expression of w as a product of simple reflections. To any Weyl group one can associate a展开更多
We consider generalizations of the Radon-Schmid transform on coherent DG/H-Modules, with the intention of obtaining the equivalence between geometric objects (vector bundles) and algebraic objects (D-Modules) characte...We consider generalizations of the Radon-Schmid transform on coherent DG/H-Modules, with the intention of obtaining the equivalence between geometric objects (vector bundles) and algebraic objects (D-Modules) characterizing conformal classes in the space-time that determine a space moduli [1] on coherent sheaves for the securing solutions in field theory [2]. In a major context, elements of derived categories like D-branes and heterotic strings are considered, and using the geometric Langlands program, a moduli space is obtained of equivalence between certain geometrical pictures (non-conformal world sheets [3]) and physical stacks (derived sheaves), that establishes equivalence between certain theories of super symmetries of field of a Penrose transform that generalizes the implications given by the Langlands program. With it we obtain extensions of a cohomology of integrals for a major class of field equations to corresponding Hecke category.展开更多
Atkin and Lehner studied the theory of new forms of the space S<sub>2k</sub>（N） of cusp forms with group Γ<sub>0</sub>（N） and weight 2k and proved that S<sub>2k</sub>（N）=S<...Atkin and Lehner studied the theory of new forms of the space S<sub>2k</sub>（N） of cusp forms with group Γ<sub>0</sub>（N） and weight 2k and proved that S<sub>2k</sub>（N）=S<sub>2k</sub><sup>n</sup>ew（N）⊕S<sub>2k</sub><sup>o</sup>ld（N）and there exists a basis in S<sub>2k</sub><sup>n</sup>ew（N） which are eigenvectors for all Hecke operators but there exists a basis in S<sub>2k</sub><sup>o</sup>ld（N） which are eigenvectors for only those Hecke operators T（p）((p,展开更多
is the disjoint union of for all , where is the set of all roots of primitive second degree equations , with reduced discriminant equal to k2m. We study the action of two Hecke groups—the full modular group and the g...is the disjoint union of for all , where is the set of all roots of primitive second degree equations , with reduced discriminant equal to k2m. We study the action of two Hecke groups—the full modular group and the group of linear-fractional transformations on . In particular, we investigate the action of on for finding different orbits.展开更多
文摘The authors consider the extended Hecke groups H(γq) generated by T(z) = -1 / z, S(z) = -1(z +γq) and R(z) = 1 / z with A, = 2 cos(π/q) for q≥3 an integer. In this paper, the even subgroup He(γq), the second commutator subgroup H''(γq) and the principal congruence subgroups Hp(λq) of the extended Hecke groups .H(γq) are studied. Also, relations between them are given.
基金NSF of China(No.10571180)the Guangdong Provincial Natural Science Foundation(No.04009801)
文摘Hecke groups are an important tool in subgroups of Hecke groups play an important rule investigating functional equations, and congruence in research of the solutions of the Dirichlet series. When q, m are two primes, congruence subgroups and the principal congruence subgroups of level m of the Hecke group H（√q） have been investigated in many papers. In this paper, we generalize these results to the case where q is a positive integer with q ≥ 5, √q ￠ Z and m is a power of an odd prime.
基金supported by National Natural Science Foundation of China (Grant Nos.10671056,10801105)
文摘Let K be an algebraic number field and OK its ring of integers.For any prime ideal p,the group(OK/p) of the reduced residue classes of integers is cyclic.We call any element of a generator of the group(OK/p) a primitive root modulo p.Stimulated both by Shoup's bound for the rational improvement and Wang and Bauer's generalization of the conditional result of Wang Yuan in 1959,we give in this paper a new bound for the least primitive root modulo a prime ideal p under the Grand Riemann Hypothesis for algebraic number field.Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.
基金partially supported by Natural Sciences Foundation of China (10671193)
文摘Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affne Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affne Weyl groups, but that is the theme of another paper.
基金supported by National Natural Science Foundation of China(Grant No. 10731070)the Doctoral Program of Higher Educationthe Fundamental Research Funds for the Central University
文摘In the present paper we determine the representation type of the 0-Hecke algebra of a finite Coxeter group.
基金National Natural Science Foundation of China ( No.11071110)Natural Science Foundation of Jiangsu Province, China(No.BK2010362)
文摘Shimura has studied the structure of the Hecke ring R(Δ,Γ) in the case Γ=SLn(Z)andΔΓis a subsemigroup of G =GLn(Q).Letpbe a prime and letΓ =GLn(Zp) Δ G =GLn( p),whereΔis a semigroup. In this paper, we shall determine the structure of the Hecke ringR(Δ,Γ)and its some properties.
文摘We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.
文摘Some derived categories and their deformed versions are used to develop a theory of the ramifications of field studied in the geometrical Langlands program to obtain the correspondences between moduli stacks and solution classes represented cohomologically under the study of the kernels of the differential operators studied in their classification of the corresponding field equations. The corresponding D-modules in this case may be viewed as sheaves of conformal blocks (or co-invariants) (images under a version of the Penrose transform) naturally arising in the framework of conformal field theory. Inside the geometrical Langlands correspondence and in their cohomological context of strings can be established a framework of the space-time through the different versions of the Penrose transforms and their relation between them by intertwining operators (integral transforms that are isomorphisms between cohomological spaces of orbital spaces of the space-time), obtaining the functors that give equivalences of their corresponding categories.(For more information,please refer to the PDF version.)
文摘The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter group is found. The new base is described by using the notion of cell datum of Graham and Lehrer and the notion of norm.
文摘Let G be the semidirect product A1 ■ A of the adeles and the norm 1 ideles of a global field k. Let г be the discrete subgroup kx ■ k. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L2(г\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.
基金the Tianyuan Math. Foundation of China (Grant No. TY10126011) the China Post-doctoral Science Foundation given to the first author.
文摘This is a continuation of our previous work. We classify all the simple ?q(D n )-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenf n(q) ≠ 0, this yields a classification of all the simple ? q (D n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ(1) = λ(2),wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?q,1(B n )- module Dλ remains irreducible on restriction to ?q(D n ).
基金This work is supported by the National Natural Science Foundation of China （Grant No. 10701048）
文摘Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.
文摘Let A be a tame Hecke algebra of type A. A new minimal projective bimodule resolution for A is constructed and the dimensions of all the Hochschild homology groups and cyclic homology groups are calculated explicitly.
基金Project supported by the National Natural Science Foundation of China.
文摘Let W be a classical Weyl group and ∏ be the corresponding system of simple roots. For w∈ W, let R(w)={α∈∏|l(ws_α)【l(w)}, where we denote by s_α the simple reflection in the hyperplane orthogonal to α for α∈∏ and by l(w)the minimal length of an expression of w as a product of simple reflections. To any Weyl group one can associate a
文摘We consider generalizations of the Radon-Schmid transform on coherent DG/H-Modules, with the intention of obtaining the equivalence between geometric objects (vector bundles) and algebraic objects (D-Modules) characterizing conformal classes in the space-time that determine a space moduli [1] on coherent sheaves for the securing solutions in field theory [2]. In a major context, elements of derived categories like D-branes and heterotic strings are considered, and using the geometric Langlands program, a moduli space is obtained of equivalence between certain geometrical pictures (non-conformal world sheets [3]) and physical stacks (derived sheaves), that establishes equivalence between certain theories of super symmetries of field of a Penrose transform that generalizes the implications given by the Langlands program. With it we obtain extensions of a cohomology of integrals for a major class of field equations to corresponding Hecke category.
文摘Atkin and Lehner studied the theory of new forms of the space S<sub>2k</sub>（N） of cusp forms with group Γ<sub>0</sub>（N） and weight 2k and proved that S<sub>2k</sub>（N）=S<sub>2k</sub><sup>n</sup>ew（N）⊕S<sub>2k</sub><sup>o</sup>ld（N）and there exists a basis in S<sub>2k</sub><sup>n</sup>ew（N） which are eigenvectors for all Hecke operators but there exists a basis in S<sub>2k</sub><sup>o</sup>ld（N） which are eigenvectors for only those Hecke operators T（p）((p,
文摘is the disjoint union of for all , where is the set of all roots of primitive second degree equations , with reduced discriminant equal to k2m. We study the action of two Hecke groups—the full modular group and the group of linear-fractional transformations on . In particular, we investigate the action of on for finding different orbits.
基金The project was supported by NNSF of China and SF of Minisitry of Education of China
文摘In this note,we give a relationship between the Mordell conjecture on the Pellianequation and the action of the Hecke operator on the Hirzebruch sum.
