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2
votes
0answers
100 views

Finite-dimensional representations of DAHA of rank 1

DAHA of rank 1 is defined by the relation $$ (T - t^{1/2})(T + t^{-1/2})=0~, \quad TXT=X^{-1}~, \quad TY^{-1}T=Y~, \quad Y^{-1}X^{-1}YXT^2q^{1/2}=1 .$$ To understand its representations, it is useful ...
0
votes
0answers
21 views

A presentation of degenerated double affine Hecke algebra of type A_1

I really want to understand the definition of degenerated double affine Hecke algebra of type A_1 (simple presentation). What are the generators and relations for type A_1? Thank you very much for ...
2
votes
1answer
153 views

p-adic modular forms, Hecke algebra, deformation theory and modular curves.

Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation ...
4
votes
1answer
175 views

Finite-dimensional representations of DAHA

It is shown by Berest-Etingof-Ginzburg that there exist finite-dimensional irreducible representations of rational Cherednik algebra $H_c(S_n)$ of $A_{n-1}$ type if and only if the deformation ...
3
votes
0answers
82 views

Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that ...
3
votes
0answers
78 views

Is there a geometric interpretation of the Hecke algebra of a Weyl group?

The Weyl group $W$ can be realized as a fundamental group of a topological space. So that its group algebra $\mathbb{C}(W)$ has a geometric meaning. On the other hand, the Hecke algebra is a ...
4
votes
1answer
95 views

Reference request for $pro-p$ Iwahori subgroup of $GL_n(F)$

I am searching for a book/lecture notes/articles where I can find the definition and properties of the $pro-p$ Iwahori subgroup of $GL_n(F)$,(with examples if possible) the Iwahori decomposition of ...
4
votes
1answer
154 views

Smoothness of Hecke algebras

First I will introduce some notation and definitions. Fix a level $N$ (take $N=1$ if it makes things easier) and a prime $p$. Let $k$ be a finite field of characteristic $p$ and let $\mathcal{C}$ be ...
8
votes
1answer
289 views

Arithmetic Points are Dense on a Hida Family

I am reading the paper "Constancy of the Adjoint L-invariant" by H. Hida (http://www.math.ucla.edu/~hida/ConstP.pdf). Correct me if I'm wrong, but I've read/heard that the arithmetic points $p \in ...
5
votes
1answer
147 views

Do Iwahori-Hecke algebras come from cohomology classes?

Let $W$ be a Coxeter group. The Iwahori-Hecke algebra $H_q(W)$ is a deformation of $k W$. Question: is there some way to interpret the deformation $H_q(W)$ as a cohomology class? It doesn't ...
5
votes
1answer
125 views

The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on ...
15
votes
1answer
490 views

How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, … algebras?

There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique. What about just traces on separate algebras? That is, take one of ...
5
votes
0answers
81 views

divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1

For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...
4
votes
0answers
162 views

Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?

The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...
1
vote
0answers
166 views

Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
4
votes
0answers
73 views

Highest (short) roots and commutation relations in (twisted) DAHA

I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...
7
votes
1answer
424 views

Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc. My question is ...
3
votes
1answer
299 views

What does the defect of a block measure?

In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by Geck-Jacon or ...
2
votes
1answer
252 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
135 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 ...
10
votes
0answers
229 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 ...
4
votes
2answers
270 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
471 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
114 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
121 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). ...
4
votes
3answers
501 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
184 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
298 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 $$ ...
4
votes
1answer
339 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
341 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
382 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
293 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
371 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
256 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
749 views

Degenerate affine Hecke Algebra

What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?
2
votes
1answer
147 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
572 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
373 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
400 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 ...
13
votes
2answers
891 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
315 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 ...
10
votes
3answers
501 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
270 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
235 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
333 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 ...
5
votes
2answers
398 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
198 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
204 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
735 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
586 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 ...