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2
votes
1answer
163 views

Newform and Galois representation (Shimura-Deligne Reciprocity Law)

Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon ...
1
vote
0answers
107 views

Order of individual Fourier coefficient of a Maass form

Let $D$ be a definite quaternion division algebra over $\mathbb{Q}$ and $\mathcal{O}$ be an Eichler order of $D$. Let $F$ be a Maass form in $L^2(PGL_2(\mathcal{O})\backslash ...
9
votes
0answers
155 views

The Markov trace via Bott-Samelson fibers?

Let $H_n$ be the Hecke algebra of GL(n), i.e., the algebra over $\mathbb{Q}(q)$ with generators $T_1, \ldots, T_{n-1}$ which satisfy the braid relations and also $T^2 = (q-1) T + q$. Recall the ...
0
votes
0answers
58 views

classical specializations of Hida families

Let 𝕋 denote the ordinary Λ-adic Hecke algebra of say tame level N. If $P_{F}$ minimal prime of 𝕋 correspond to hida families F and this family specialise to classical weight one ordinary modular ...
4
votes
2answers
224 views

Gelfand pair and double coset decomposition

Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus ...
6
votes
0answers
325 views

Iwahori-Hecke algebras as endomorphism (or convolution) algebra?

Let $H_n(q,k)$ be the Iwahori Hecke algebra of symmetric group $S_n$ over an algebraically closed field $k$ of characteristic $p>0$, where $q$ is an invertable element in $k$. Assume that $q$ is a ...
4
votes
0answers
88 views

Degeneration of modules over the affine symmetric group and jeu de taquin

Let $H_n$ be the group algebra of the affine Coxeter group of type A (feel free to replace it by the affine Hecke algebra). This is generated by elements $y_i$'s, $i=1,\dots,n$ and transpositions ...
1
vote
0answers
75 views

Explicit generators of maximal ideals in completed Hecke algebras

A particular question: Let M be the subset of Z/3[[x]] consisting of those power series that are reductions of elements of Z[[x]] that arise as expansions of modular forms for Gamma_0 (2). ...
3
votes
3answers
411 views

Definition of Hecke operators

I am confused about the definition of Hecke operators. It will be great if someone provides some references. Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the ...
3
votes
1answer
174 views

Are these powers of a characteristic 3 power series annihilated by certain Hecke operators?

Let D in Z/3[[x]] be sum ((a_n)(x^n)) where the sum runs over all n prime to 6 and a_n is the mod 3 reduction of the number of ideals of norm n in the ring of integers of Q(root(-3)). (So ...
5
votes
0answers
285 views

Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?

The following questions arise from modular form theory. But this theory isn't needed to formulate or understand them, and I'm not using the modular-forms tag. NOTATION Fix an odd prime $N$. Let $$ ...
3
votes
0answers
143 views

Index of the Hecke algebra with operators omitted

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler. Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb ...
8
votes
1answer
260 views

Kazhdan-Lusztig Polynomials and Intersection Cohomology

I hope this question has not been asked before. I would like to know which Ideas led (Deligne), Kazhdan and Lusztig believe, that Kazhdan–Lusztig polynomials can be expressed via intersection ...
10
votes
0answers
332 views

A question about multiplication in $G(\mathbb{C}((t)))$ and Affine Grassmannians

I am sorry to give a bounty to such a crappy question but an answer would help me a lot. I am stuck with the following simple (i guess but) technical problem. Let $G$ be a complex reductive ...
10
votes
1answer
243 views

Reference request: Grothendieck groups of Hecke algebras at root of unity and symmetric functions

Let $\zeta$ be an $\ell^{\text{th}}$ root of unity, and consider $H_n(\zeta)$, the (finite) Hecke algebra of type A. One can consider a dual pair of Hopf algebras arising from this data, denoted ...
10
votes
2answers
283 views

Temperley-Lieb algebras for other Weyl groups?

The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...
2
votes
1answer
182 views

Hecke eigenvalue at p and at p^k

I am interested in the relationship between the Hecke eigenvalue at $p$ and at $p^k$ for $k \geq 2$ in the unramified and ramified situation for modular/Maass forms. More precisely, I know from a ...
2
votes
3answers
455 views

Degenerate affine Hecke Algebra

What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?
2
votes
1answer
121 views

Decomposition of k[Flag(F_q)] as bimodule over GL_n(F_q) , Hecke(q) ?

Question: What is decomposition of the representation k[Flag(F_q)] as bimodule over GL_n(F_q) , Hecke(q) ? (Let k=Complex numbers. Further question: is there any change for char k = p ? ) Remark: ...
15
votes
3answers
508 views

Kazhdan-Luzstig Polynomials and Lower Intervals in the Bruhat Order

I have read in a number of places that the lower Bruhat interval $[e, w]$ is rank-symmetric if and only if the KL-polynomial $P_{e, w}(q) = 1$. All of the proofs I've come across use "rationally ...
3
votes
0answers
313 views

local deformation rings and Hecke algebras

Let $\bar{\rho}_p$ be a two-dimensional irreducible local Galois representation of $Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ on a $k$-vector space, where $k$ is a finite extension of $\mathbb{F}_p$. ...
2
votes
2answers
341 views

description of an endomorphism algebra

Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus). I ...
11
votes
2answers
570 views

Why are there no triple affine Hecke algebras?

This question arised after I recently stumbled upon the paper "Triple groups and Cherednik algebras". Doubly affine Hecke algebras are sort of a natural object to consider after finite and affine ...
3
votes
3answers
305 views

2nd eigenvalues for cusp forms for $\Gamma_0(4)$

Let f be a newform for $\Gamma_0(4)$ with a trivial character. I guess that the eigenvalue $\lambda(2)$ for $T_2$ is 0. I want to know that this is known result or not. If so, could you explain or ...
9
votes
3answers
385 views

Are there Hamilton paths in Cayley graphs of Coxeter groups?

Hi everyone. I want to optimize certain computation on finite Coxeter groups $(W,S)$. Basically I compute the matrices $\rho(T_w)$ for all $w\in W$ of a matrix representation $H\to K^{d\times d}$ of ...
2
votes
0answers
193 views

When are parabolic Kazhdan-Lusztig polynomials nonzero?

Let $W$ be a Coxeter group with simple reflections $S$ and let $J \subseteq S$. Let $P^J_{\tau, \sigma}$ be the parabolic Kazhdan-Lusztig polynomials in the case $u = q$ in the sense of On Some ...
4
votes
0answers
215 views

When is a Hecke algebra not a bialgebra?

Let $\mathcal{H}_q(d)$ denote the Iwahori-Hecke algebra of type $A$ over a field of characteristic zero. When $q = 1$, this is just the group algebra of the symmetric group on $d$ letters. In this ...
4
votes
1answer
252 views

Victor Miller basis for higher $N$ // why is this bilinear form perfect?

Hello. I am trying to understand the proof of Thm 9.23 in http://wstein.org/books/modform/modform/newforms.html#congruences-between-newforms . Let $S_k(\Gamma)$ be the cusp forms for a subgroup ...
4
votes
2answers
363 views

Hecke Operators for $\Gamma_1(N)$ *with* character?

Hello. I wonder whether there are hecke operators for modular forms for $\Gamma = \Gamma_1(N)$ with additive character $\chi : \mathbb{Z}_N \mapsto \mathbb{C}^{\times}$. There is a somewhat ...
1
vote
1answer
192 views

Algorithm for the cell multiplication rule for GL(n,F)

Consider $F$ a non archimedean field and let $o$ be its ring of integer Let $B$ be the Iwahori subgroup of $GL_n(F)$ (resp. $GL_n(o)$) and let $N$ be the normalizer of the diagonal matrices ...
1
vote
1answer
180 views

Formula for the “integral form” action of Iwahori-Hecke algebra on the standard basis for Specht modules

Is there a formula somewhere in the literature for the action of the generators $T_1,\ldots,T_{n-1}$ of the Iwahori-Hecke algebra on the standard basis of its Specht modules? It is well-known that the ...
6
votes
2answers
636 views

Hecke algebra and $H^*(G/B)$

Given a complex reductive group, with Weyl group $W$, one can associate to it lots of "algebras of size $|W|$". For example $B$ equivariant functions on $G/B$ with convolution, grothendieck groups of ...
10
votes
1answer
427 views

Traces on Hecke algebras and the Jones polynomial

In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr_z$ on the Iwahori-Hecke algebras $H(q,n)$ of type ...
5
votes
1answer
242 views

Efficient enumeration of Bruhat intervals

Hi everyone. I'm currently programming some stuff for Hecke algebras. My current implementations have several bottlenecks and I'd like to improve that as much as I can so that I can use stuff like ...
8
votes
2answers
489 views

Subexpressions of reduced words in Coxeter groups

Let $\underline{w} = [s_1, s_2, \dots ,s_n]$ be a reduced expression in a Coxeter group $W$. Given $x$ in $W$ one can consider the set $\Pi(\underline{w},x)$ consisting of all subexpressions of ...
1
vote
1answer
490 views

Recovering the Alexander Polynomial from Ocneanu's HOMFLYPT

Let $H_q(n)$ denote the Hecke algebra associated to the symmetric group $S(n)$: this is the $\mathbb{Z}[q^{\pm 1/2}]$ algebra generated by $T_1, \ldots, T_{n-1}$ satisfying the braid relations along ...
5
votes
1answer
305 views

Parabolic convolution of perverse sheaves in terms of the Hecke algebra

It is "well-known" that the Hecke algebra $\mathcal{H}$ can be thought of as the Grothendieck group for the category of perverse sheaves on $G/B$, where the product in $\mathcal{H}$ corresponds to ...
1
vote
2answers
397 views

Difference between orthogonal form and seminormal form

Frequently in the literature on Hecke algebras for the symmetric group and their generalisations, one encounters references to Young's seminormal form and Young's orthogonal form. I have a good ...
7
votes
2answers
522 views

Hecke algebra generated by a single element

Let $\mathbb{T}_{\mathbb{Z}}$ be a $\mathbb{Z}$-module generated by Hecke operators $T_n$ acting on the space of cups forms $S_{k}(\Gamma,\mathbb{C})$ for the congruent subgroup satisfying ...
3
votes
2answers
503 views

What is known about the centraliser of the Hecke algebra in the affine Hecke algebra?

This question is a sequel to 66602 The Hecke algebra is the quotient of the group algebra of the braid group of type $A_n$ by quadratic relations and the affine Hecke algebra is the quotient of the ...
1
vote
0answers
195 views

On the decomposition of two representations of the Iwahori-Hecke algebra of type A_n

Consider the Iwahori Hecke algebra $H_q(n)$ of the symmetric group $S(n)$, the $\Bbb{Z}[q^{1/2},q^{-1/2}]$ algebra with generators $T_i$, $1 \leq i \leq n-1$, subject to the braid relations and the ...
2
votes
0answers
161 views

Notion of positivity in $\mathbb{Q}(q)$

When dealing with an element $P \in \mathbb{Z}[q]$, a natural combinatorial question is to ask whether $P$ has non-negative coefficients. For instance, this is true if $P$ is a Kazhdan-Lusztig ...
1
vote
1answer
426 views

Reference for Hecke algebra version of Young's orthogonal basis

In the paper Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Arun Ram defines a seminormal basis as follows: given a chain of split semisimple $K$-algebras $K\cong H_0 \subseteq ...
4
votes
2answers
894 views

Double coset representatives and structure of hecke algebras

Let $GL_n(F_q)$ be the general linear group over finite field $F_q$ and $B_n$ be its borel subgroup consisting of all upper triangular matrices. Then the double cosets $B_n\backslash GL_n(F_q)/B_n$ ...
10
votes
2answers
968 views

Adjoint of Atkin-Lehner's U_p

If $\Gamma=\Gamma_1(N)$, or $\Gamma=\Gamma_0(N)$, the Hecke operator $[\Gamma diag(1,l) \Gamma]$ for $l$ a prime (acting on the space of cusp forms of level $\Gamma$ and some weight $k$) is in general ...
5
votes
4answers
863 views

Hecke-algebras in your field of mathematics

(How) do Hecke-algebras arise naturally in your field of mathematics and why are they important? How would you define them and how do you think about them? e.g. generators and relations, functions ...
4
votes
4answers
897 views

Bernstein's presentation for the Hecke Algebra

Any one know of any good references for reading about the Bernsteins Presentation of the Iwahori Hecke Algebra? I need some notes which has an example or two. It would really help.
4
votes
1answer
457 views

Is there a way to define Hecke operators “inherently” as certain endomorphisms of the Jacobian?

From the Eichler-Shimura relation, we have a formula for $T_p$ when we reduce $\textrm{End}(\textrm{Jac}(X))$ mod $p$. Explicity, $T_p=\textrm{Frob}_p+p\textrm{Frob}_p^{-1}$. Is there a way to define ...
3
votes
2answers
739 views

Representations of finite Coxeter groups

What is reference for complex irreducible representations of Hecke algebra of finite Coxeter groups (say generic case q =1)? I am interested in knowing its Wedderburn decomposition. So want explicit ...
11
votes
2answers
1k views

Is Soergel's proof of Kazhdan-Lusztig positivity for Weyl groups independent of other proofs?

Let $(W, S)$ be a Coxeter system. Soergel defined a category of bimodules $B$ over a polynomial ring whose split Grothendieck group is isomorphic to the Hecke algebra $H$ of $W$. Conjecturally, the ...