The hecke-algebras tag has no usage guidance.

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### Restriction to the diagonal of Hilbert eigenforms

Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?

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### Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian?

Let $G$ be a finite group and $K \subset G$ a subgroup. Then $(G,K)$ is a Gelfand pair if the double coset Hecke algebra $\mathbb{C}(K \backslash G / K)$ is commutative.
Let $H$ be a subgroup of $...

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### References of an operator $T: V \otimes V \to V \otimes V$

Let $V$ be a vector space with a basis $v_1, \ldots, v_n$ and let $X_{ij} = v_i \otimes v_j$. Then $X_{ij}, i,j=1,\ldots, n$, is a basis of $V \otimes V$. Let $T: V \otimes V \to V \otimes V$ be the ...

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### 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 ...

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### 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 ...

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180 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 $\bar{\...

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180 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 ...

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### 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 $W\subset\...

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### 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 ...

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119 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 ...

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### 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 ...

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312 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 ...

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### 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 ...

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### 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 $l^2(\...

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### 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 them,...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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433 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 ...

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### 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 ...

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256 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 {\mathrm{...

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### 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 PGL_2(D\otimes\mathbb{R}))...

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232 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 ...

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277 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 $\...

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### 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 ...

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### 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 $s_i$...

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### 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). (...

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### 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 ...

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### 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 D=x+2(x^7)+2(...

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### 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
$$
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $G(\...

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377 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 ...

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### 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 ...

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### Degenerate affine Hecke Algebra

What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?

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### 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: ...

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### 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 ...

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### 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 $\mathbb{F}_p$. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 (...