# Questions tagged [hecke-algebras]

The hecke-algebras tag has no usage guidance.

150
questions

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### How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)

Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...

4
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### Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...

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### How is Taylor-Wiles patching "horizontal Iwasawa theory"?

I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the ...

3
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### Taylor-Wiles systems for higher dimensional deformation rings

Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module.
A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...

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### Automorphisms of Iwahori/affine Hecke algebras

Has there been any serious study of automorphisms of extended affine Hecke algebras? Has anyone determined the automorphism group of say, type A extended affine Hecke algebras? I ask because the ...

3
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101
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### List of techniques that have been used to prove topological properties of locus in the deformation ring or the Hecke algebra

My question is maybe going to be a bit vague. My apologies if so.
The setting:
Let $\overline{\rho}$ be a residual representation and $R$ be a deformation ring of $\overline{\rho}$.
Let $\mathbb{T}$ ...

5
votes

2
answers

315
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### Parabolic induction, and tensoring (Iwahori/affine) Hecke algebras

In several places, such as [1] and [2], it seems to be implicitly known that (normalized) parabolic induction corresponds to tensoring affine Hecke algebras. I say implicitly because both [1] and [2] ...

4
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### Eigenvalue of Iwahori Hecke Algebra element for the Steinberg

In Iwahori-Matsumoto's paper the Iwahori Hecke Algebra for $G=GL_n(F)$ is generated by $X_{s_0}, X_{s_i},i\in\{0,...,n-1\}$ and $ X_{\rho}$ with the relations:
$
1) (X_{s_{i}}-q)(X_{s_{i}}+1)=0\:,\;\;...

0
votes

1
answer

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### Analog of self-conjugate representation of symmetric group for Hecke algebra

Consider a symmetric group $S_n$. It is generated by generators $\sigma_1\dotsc\sigma_{n-1}$ that satisfy the following relations:
Square relations: $\sigma_k^2=1,\qquad k=1\ldots n-1$.
Braid ...

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### A list of $R=T$ theorems for $\mathbf{GSp}_4$

I know of only two cases of theorem $R=T$ where $T$ is a Hecke algebra acting on an automorphic forms (or representations) space of $\mathbf{GSp}_4$, the first one was proved by A.Genestier and J....

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115
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### Property of simplicity and semi-simplicity under base change of base field

Suppose $K$ is a field of characteristic $0$ and $A$ is a $K$-algebra. Let $F$ be a field extension of $K$ and let $M$ be an $A$-module. What can we say about simplicity or semi-simplicity of $A_F$-...

4
votes

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answer

154
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### Spectral projection of an eigenvalue associated to a generator of Hecke algebras

In his paper "Hecke Algebras of type $A_n$ (Inv. Math. 1988, EUDML link) and subfactors", in section 2 "Orthogonal representations...", Wenzl takes the usual third relation of a ...

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### Admissibility of representations induced from Hecke algebra for covering groups

Assume $G$ is a semisimple algebraic group and $B$ is an Iwahori subgroup. Let $(r,E)$ be a representation of $H(G,B)$ which is an Iwahori-Hecke algebra, then Borel proved that $C_{c}(G/B)\otimes_{H}E$...

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398
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### p-adic Hecke operators in the Iwahori-Hecke algebra $C_c(J\backslash G(F)/J)$

$\DeclareMathOperator\ch{ch}$Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. I shall use $\kappa(F)$ to denote ...

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### Has anyone met this "$q$-character" table for $S_4$?

Is anyone aware of the following $q$-character table for the
symmetric group $S_4$?
\begin{array}{|c|c|c|c|c|c|}
\hline
\mathrm{conj}\backslash\mathrm{rep}
& 2+1+1 & 3+1 & ...

1
vote

1
answer

180
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### Commutative subalgebra of Iwahori-Hecke algebra

I am currently reading the (french) book [1] by Matsumoto. In (2.1.11) the following statement (reformulated in my own words/notation) appears:
Let $(W,S)$ be a Coxeter system, let $R$ be a ...

7
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255
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### Naive Schur-Weyl duality for the 0-Hecke algebra

The 0-Hecke algebra $\mathcal{H}_n(0)$ is the $\Bbb{C}$-algbra generated by elements $T_1,
\dots, T_{n-1}$ satisfying the braid relations
and the idempotency relations $T_i^2 = T_i$.
It is known that $...

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### LLT polynomials and graded Specht modules

In the paper ``Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras And Unipotent Varieties'' Lascoux, Leclerc and Thibon define polynomials which quantise the Schur functions. These $...

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135
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### Barr's element and the type-A Iwahori–Hecke algebra

Let $\mathfrak{S}_n$ denote the symmetric group on $n$ letters. Barr's element $\mathcal{S}(n) \in \Bbb{R}\bigl[ \mathfrak{S}_n \bigr]$ is defined as
\begin{equation}
\mathcal{S}(n) := \ \sum_{i=1}^{n-...

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### Hecke algebras with modified braid relations

I'm studying certain games on graphs on $n$ vertices and it turns out that I get a matrix algebra with generators $T_i , i=1,\dotsc,n$ satisfying a certain type of relations. Here the $T_i$ are ...

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### On the order of the head of product of two simple modules over Quiver Hecke Algebras

My question is:
We assume the underlying quiver is a Dynkin quiver. Let $L(\lambda)$ and $L(\mu)$ be two simple modules over Quiver Hecke algebra $R$ where $\lambda$ and $\mu$ are two Konstant ...

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54
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### Block sum for degenerate affine Hecke algebras

The degenerate affine Hecke algebra $H_k$ over a field $F$ is the algebra with generators $s_1,\ldots,s_{k-1}$ and $x_1,\ldots,x_k$, subject to the following relations:
$s_is_js_i=s_js_is_j$ for $i=j\...

3
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### compactly induction of smooth modules over Hecke algebras

Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...

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### Hidden grading on $kS_n$

Brundan and Kleshchev showed that if $k$ is a field of characteristic $p$, then the group ring $kS_n$ of the symmetric group $S_n$ admits an integer grading which is nontrivial in the sense that it is ...

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### When is an affine Hecke algebra of type A, a quantum Lie algebra?

An affine Hecke algebra of type $A_{k-1}$ is an unital, universal, associative C-algebra generated by the elements $T_1, \dots, T_{k-1}$, $X_1, \dots , X_k$, $X_1^{-1}, \dots , X_k^{-1}$ subject to ...

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### Positivity of Iwahori–Hecke algebra characters on the Kazhdan-Lusztig basis

$\DeclareMathOperator\tr{tr}$I'm interested in the irreducible characters of a finite Iwahori–Hecke algebra evaluated at the Kazhdan–Lusztig basis. These are Laurent polynomials.
Are the coefficients ...

6
votes

1
answer

193
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### Positivity of Schur elements in Iwahori-Hecke algebras

I'm interested in finite Iwahori-Hecke algebras.
If $\mathcal{H}$ is such a Hecke algebra, defined over $\mathbb{Z}[q^{\pm 1/2}]$, and $\Lambda$ an irreductible representation, there is the notion of ...

8
votes

3
answers

602
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### Motivation for the Kazhdan-Lusztig involution

I would like to know about the motivation behind the Kazhdan–Lusztig involution on an Iwahori–Hecke algebra.
I'll borrow the conventions from Libedinsky's Gentle introduction to Soergel bimodules I: ...

7
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2
answers

879
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### Reference of J.L. Waldspurger's paper on Shimura correspondence

I want to find reference of Waldspurger's paper referred at "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (...

4
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### Relationship between Hecke algebra and center of universal enveloping algebra (and the Harish-Chandra isomorphism)

Let $G$ be a semisimple Lie group of noncompact type and let $K$ be a maximal compact subgroup. Let $\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{k}$ be the Cartan decomposition coming from some ...

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### Promoting representation of a subgroup to representation of an action groupoid

Suppose I have a group $G$, a subgroup $K \leq G$, and a representation $(\sigma, V)$ of $K$.
There is a natural left action of $G$ on $X := G/K \times G/K$ given by $g \cdot (g'K,g''K) = (gg'K,gg''K)$...

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### Kazhdan-Lusztig basis elements appearing in product with distinguished involution

My apologies if the below is too malformed to make sense.
Let $(W,S)$ be the affine Weyl group of a reductive group $G$, and let $\{C_w\}$ be the Kazhdan-Lusztig $C$-basis (an answer in terms of the $...

8
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1
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376
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### Structure constants for the double coset algebra of a Young subgroup

Fix a Young subgroup $H_\lambda \subseteq \mathcal S_n$, where $\lambda \vdash n$ is a partition of $n$ with $k$ blocks. Inside the group algebra $\mathbb C[\mathcal S_n]$, consider the idempotent
$$\...

8
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241
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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,...

1
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1
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224
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### 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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### 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 ...

3
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### 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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### 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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### 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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1
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### 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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### 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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### 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 ...

4
votes

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answer

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### 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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### 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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### 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 ...

5
votes

2
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502
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### 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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### 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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### 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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### 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}...

3
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1
answer

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### 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_{...