# Questions tagged [hecke-algebras]

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116
questions

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### Hecke algebra relation versus $\operatorname{SL}_2$ trace relation

The quadratic relation in the (type $A$) Hecke algebra is $(T-t)(T+t^{-1}) =0$, which can be rewritten as
$$
T-T^{-1} = t-t^{-1}$$
Suppose $A \in \operatorname{SL}_2(\mathbb{Q})$ with eigenvalues $a,...

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159 views

### Algebra of Hecke operators on $M_k(\mathrm{SL}_2\mathbb{Z})$ is an integral domain?

Let $M_k(\mathrm{SL}_2\mathbb{Z})$ be the space of modular forms of (integer) weight for the full modular group. Let $\mathbf{H}$ denote the Algebra generated by the Hecke operators $T_n$. Is $\mathbf{...

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86 views

### Classical Hecke operators and Hecke algebra of type $A_1$

What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices ...

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133 views

### Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic

Let $(W,S)$ be a Coxeter system. Let $q=(q_s)_{s\in S} \in \mathbb{R}^{\text{#}S}$ be a tuple of positive real numbers with $q_s=q_t$ whenever $s$ and $t$ are conjugate to each other. Follwing Davis', ...

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204 views

### On the status of some conjectures mentioned/used in Harish Chandra's 1970 lecture notes

In van Dijk's notes of Harish Chandra's lectures on harmonic analysis, several conjectures are mentioned throughout, such as in Part 1, section 4 of van Dijk's notes
Conjecture I : Let $\omega$ be ...

**6**

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**1**answer

214 views

### Cusp forms have an orthonormal basis of eigenfunctions for all Hecke operators

I am reading Langlands' pape Euler Products and have a few questions. Let $G$ be a split adjoint semisimple group over $\mathbb Q$. If $p$ is a place of $\mathbb Q$, finite or infinite, let $G_{\...

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313 views

### How does one compute the Hecke algebra acting on modular forms?

I asked this on mathstackexchange, but got no answer.
Let $N\geq1$ be an integer, and let $\mathbb{T}$ be the Hecke algebra acting on the cusp forms of weight k and level $\Gamma_0(N)$.
Then $\...

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114 views

### Homogenous Hermitian form on the KLR algebra

Does there exist a homogenous conjugate-linear automorphism of the KLR algebra? I want to be able to define a homogenous Hermitian form on the (Specht) modules of the (cyclotomic) KLR algebra.

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57 views

### Flag variety over quaternions and its Hecke algebra

Consider cell decomposition of flag variety of $\mathbb{H}^{n}$ into orbits under the action of $B(\mathbb{H})$ - group of upper triangular matrices with coefficients in $\mathbb{H}$. I think cells ...

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110 views

### Volumes of double cosets $KtK$

Let $G$ be the group of rational points of a connected reductive group $\mathbb G$ defined over a non-archimedean local field $F$. For simplicity sake, I assume that $\mathbb G$ is split over $F$. Let ...

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470 views

### Smallest Mazur's good prime

Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Following Mazur, we say that a prime $q$ is a $\textit{good prime}$ if $\ell$ does not divide $q-1$ and $q$ is not a $\ell$th ...

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37 views

### Admissible (unitary) spherical representation $sl(2,Q_p)$. Does dimension fixed point vector increase proportional index

I am using terminology of Cartier's Harmonic analysis on trees. Take $\pi$ be one of the irreducible principal or complementary (unitary) spehrical series of $Sl(2, Q_p)$. Let $K$ be the maximal ...

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274 views

### Basic theorem on induction for representations of $p$-adic groups

I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it ...

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26 views

### Decomposition numbers of characteristic 0 Ariki-Koike algebras for q=0 or q generic

Consider a complex Ariki-Koike algebra $H:=H(q;Q_1,\dots, Q_\ell)$ for some invertible elements $q, Q_1,
\dots, Q_\ell$ of $\mathbb{C}$, i.e. an associative $\mathbb{C}$-algebra with generators $T_0, \...

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45 views

### Branching rule for degenerate cyclotomic Hecke algebras

Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight ...

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171 views

### Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p))$ and Hecke operators

I was reading James Cogdell's notes here on automorphic representations and came to the following claim about the spherical Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p), \operatorname{GL}...

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149 views

### Basis for Annular Skein Algebra

Background/Notation:
Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis
$\{T_{w}\}_{w\in S_{...

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238 views

### Examples of non-trivial Kazhdan-Lusztig polynomials

I'm looking for examples of non-trivial Kazhdan-Lusztig polynomials, specifically in the case where the Coxeter system is a Weyl group.
For example, the simplest polynomial with non-trivial $q$-...

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213 views

### Categorifying skein algebras?

We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov ...

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183 views

### Understanding the Hecke Algebra via Different Constructions

I'm reading through this paper by Pouchin, and in it he makes a clam that some function space is isomorphic to the Hecke algebra. I'm trying to understand this and could really use some help.
Let $G =...

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39 views

### Why are the definitions of i-good nodes of a multipartition equivalent?

Let $e\geq 2$ and $0\leq i\leq e-1$.
For a multipartition $\lambda\vdash_\ell n$ of $n $ one can define the notion of an $i$-good box of the Young diagram of $\lambda$. But there are seemingly ...

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929 views

### What is the archimedean Hecke algebra?

Let $\mathbf G$ be a connected, reductive group over $\mathbb Q$. For each nonarchimedean place $v$, let $K_v$ be a maximal compact subgroup of $\mathbf G(\mathbb Q_v)$. The space $\mathscr H(\mathbf ...

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190 views

### Iwahori-Hecke algebra of $GL_2$

I had make an old post about this, but since seeing it now shows that I had no idea how to explain my question (because I didn't understand it) I'll try to do it normally here.
So I am studying this ...

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210 views

### Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero

I am interested in how many pairs of permutations $(u,w)$ in $S_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is non-zero? where $\mu_{u,w}=[q^{\frac{l(w)-l(u)-1}{...

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534 views

### Hecke algebra of GL(2,F)

I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to ...

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112 views

### What is the kernel of the action of the Iwahori-Hecke algebra?

The Iwahori-Hecke algebra $H_n(q)$ acts on the $n$th tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$. What is the kernel of this action? Does anyone know a reference?
I'm happy ...

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401 views

### Relation between Hecke operators and coefficient of L-functions

This question has its seed in this one by Gory, which found an enlightening answer but one of the comments kept me wondering. I am beginning to discover Hecke operators, and there appears to be an ...

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312 views

### A spin extension of a Coxeter group?

Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$.
Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) ...

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160 views

### Semisimple vs ordinary trace of Frobenius on nearby cycles of affine flag variety

Let $G \rightarrow X$ be a parahoric group scheme over a curve, with parahoric level structure at $x_0$. Gaitsgory essentially showed that the nearby cycles functor $R\Psi$ takes perverse sheaves on ...

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137 views

### Krull dimension of Hecke algebra (level 1) for p = 2, 3

Let $p$ be a prime and consider the ($p$-deprived) Hecke algebra $\mathbb{T}$ which the projective limit of the Hecke $\mathbb{Z}_p$-algebras $\mathbb{T}_k$ which act on modular forms of level $1$ ...

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56 views

### Reference request: Hecke agebra over non-commutative rings

I think the title sums it up quite well: Is it a useful idea to define the Iwahori-Hecke algebra over a non-commutative $k$-algebra? If so, what shape should the relations attain?
Bonus question: ...

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47 views

### symmetric polynomials for Super Hecke Clifford algebra

Fix a natural number $n$. In https://arxiv.org/abs/1107.1039, §3.5, Kang/Kashiwara/Tsuchioka define a (version of a) Hecke Clifford superalgebra. It is the superalgebra with the following generators:
...

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### Noncommutative cohomology of flag varieties

Consider the Grassmannian $Gr(n, N)$ of $n$-dimensional subspaces of $\mathbf C^N$. Its cohomology ring is isomorphic to $\mathbf C[x_1, \ldots, x_n, \bar x_1, \ldots \bar x_{N-n}]/I_{n,N}$, wherethe ...

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119 views

### About left cell of a permutation

I am reading a paper Cellular algebras by J.J. Graham, G.I. Lehrer. I do not understand the follwing words labelled by yellow.
First, I know Robinson-Schensted correspondence of a permutation in the ...

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269 views

### Modularity of endomorphism algebras

This question is about comparing Hecke algebras and endomorphism algebras.
Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...

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164 views

### Quantum Schur-Weyl duality for quantum affine algebras of other types

In the paper by Chari and Pressley, it is proved that the there is functor from the category $C_m$ of finite dimensional representations of the affine Hecke algebra of $GL(m)$ to the category $D_n$ of ...

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31 views

### Cellular basis of $KW(B_2)$

Take $K$ a field. Let $W(B_2)$ be the coxeter group of type $B_2$ with a set $\{s_0,s_1\}$ of generators. Then $W(B_2)=\{1, s_0,s_1,s_0s_1,s_1s_0,s_0s_1s_0, s_1s_0s_1,s_1s_0s_1s_0\}$. I know that the ...

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264 views

### Does Gorensteinness of $\mathbb{T}_{\mathfrak{m}}$ imply multiplicity one?

Let $N$ be a prime and let $\mathbb{T} \subset \mathrm{End}(J_0(N))$ be the Hecke algebra generated over $\mathbb{Z}$ by $U_N$ and the operators $T_p$ for primes $p \nmid N$. Fix a maximal ideal $\...

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### How to translate multi-segments to Drinfeld polynomials?

Let $\hat{H}_m=\hat{H}_m(q)$ be the Iwahori-Hecke algebra of $GL_m$, see for example, Section 2. The simple $\hat{H}_m$-modules are parametrized by Zelevinsky's multi-segments, See Section 2.2 of the ...

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295 views

### Kazhdan-Lusztig basis of the symmetric group algebra $\mathbb{K}S_n$

Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$ and $\mathbb{K}$ a field. Denote by $\mathbb{K}S_n$ the symmetric group algebra.
What's Kazhdan-Lusztig basis of $\mathbb{K}S_n$? I ...

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### Eigenvariety and Hecke algebra

Let $h^{n,ord}(Np^\infty)$ be the cuspidal nearly ordinary Hecke algebra of tame level $N$. For $N \geq 4$, we know that the Hecke algebra is the generic fibre of the Hecke-Hilbert Eigenvariety and so ...

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### Deligne-Scholl's motives for modular forms: Hecke operators vs. transposed Hecke operators

EDIT: I moved my original question down to the bottom. The question here at the top is related, at least I suppose that the same phenomenon is behind both of them.
In the article "Valeurs de ...

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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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307 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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267 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 $\mathbb{Z}(W)$ has a geometric meaning. On the other hand, the Hecke algebra is a ...