# Questions tagged [hecke-algebras]

This tag is for questions regarding the Hecke algebra or, Iwahori Hecke algebra, which is the algebra generated by Hecke operators.

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### A question on representation of nilpotent groups

Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$. I can see ...
1 vote
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### Associativity of Hecke Operators in Bump

I was looking through Bump's Automorphic Forms and Representations and was thinking about his initial description of Hecke algebras. In short, for $\alpha \in \mathrm{GL}(2, \mathbb{Q})^+$, the double ...
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### How to Construct the Theory of Hecke Operators for Maass Forms

I'm trying to find a construction for the theory of Hecke operators for Maass forms that is analgous to the double coset operator construction for modular forms. For modular forms of weight $k$, this ...
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### The $SL_{2}(\mathbb{Z})$ double coset of diagonal matrix

I have a trouble proving that: For $k\in \mathbb{N}$, the double coset \begin{align*} SL_{2}(\mathbb{Z})\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} SL_{2}(\mathbb{Z})\end{align*} is ...
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### About elements in affine extended Weyl group

I'm not sure how to prove a statement about extended Weyl groups. Let $V$ be a finite vector space over $\mathbb{R}$, with a positive definite symmetric bilinear form (·,·), R ⊂ V be a reduced ...
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1 vote
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### Fricke involution’s effect on character

I’m using the definition $W_Nf(\tau)=i^kN^{-k/2}\tau^{-k}f(-1/N\tau)$. Now suppose $f\in M_k(\Gamma_1(N),\chi)$, show $W_Nf\in M_k(\Gamma_1(N),\chi^{-1})$. I know this is pure calculation but I’m ...
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### Is there a known description for the composition of two double coset operators?

Let $\Gamma$ and $\Gamma'$ be congruence subgroups of $SL_2(\mathbb{Z})$, and $\alpha\in GL_2^+(\mathbb{Q})$. Then one defines a "double coset operator" or "change of automorphy ...
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### A Hecke-Maass eigenbasis for the space of Maass cusp newforms

I heard that the space of Maass cusp newforms on $\Gamma_0(N)/\mathbb{H}$ has a basis of Hecke eigenforms. Would anyone happen to know of a reference of this fact? Or, even better, how to prove it?
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### Which Hecke algebra is used in representation theory?

Which Hecke algebra is used in representation theory or more specifically in the study of Langlands's conjecture ? From here, the Hecke algebra is constructed from a locally compact topological group ...
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### Why do we care whether Hecke algebras are complete intersections?

The title really says it all. I get the impression that proving certain Hecke algebras are complete intersections is a crucial step in the proof of Fermat's Last Theorem. But how do you use a result ...
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1 vote
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### If two algebras are generated by the same elements, then are they isomorphic?

The $0$-Hecke algebra $\mathcal{H}_0(S_{n+1})$ is generated by $\{ h_1, \ldots,h_n\}$ satisfying i) $h_i^2=-h_i$ ii) $h_ih_j=h_jh_i$ if $|j-i| > 1$ iii) $h_ih_{i+1}h_i=h_{i+1}h_ih_{i+1}$. Now, ...
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### Definition of Diamond Operator

I'm studying modular forms, but I can't understand the definition of diamond operator. Why can I define for all $\alpha$ with $\delta \equiv d$? I can't understand the reason why two different matrix ...
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### Submodule of $R$-Invariants for a Hecke Pair $(R,S)$

I'll just define a Hecke pair here for completeness: Definition: Let $S$ be a monoid and $R$ a group contained in $S$. We call the pair $(R,S)$ a Hecke pair if every $R$-double coset of $S$ is a ...
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### Left and right coset representatives of $\text{SL}_2(\mathbb{Z})$ action

Let \begin{align} \Gamma=\text{SL}_2(\mathbb{Z})=\bigg\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}: a,b,c,d\in \mathbb{Z}, \;ad-bc=1\bigg\}, \end{align} the group of integer matrices with ...
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### Reference Request: Jimbo's Proof of Quantum Schur-Weyl Duality

In his seminal 1986 paper "A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke Algebra, and the Yang-Baxter Equation", Jimbo asserted (Proposition 3) that the quantum group associated to $\mathfrak{gl}_n$...
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### Q Exercise 4, Hecke Algebras - Daniel Bump

I'm struggling with exercise 4 in Bump's Stanford Hecke Algebra notes linked here It states the following: Let $G$ be a finite group and $V,W$ vector spaces. Let $C(G,V)$ denote the spaces of maps ...
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I am working through Murty's 'Problems in the Theory of Modular Froms', but I am stuck in the proof of Hecke operators being self adjoint with respect to the Petersson inner product. Before describing ...
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### 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 ...
1 vote
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### Algebra-module over a set, is there such a thing?

I'm reading a paper and $H$ is an algebra and $M$ is a set. And then they define $M|H$ to be the "$H$-module $M$". Is it obvious what this is or is it bad math description? My guess is the free $H$-...
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### Hecke Bound for Cusp - Modular Forms

The problem statement, all variables and given/known data i have a questions on the piece of lecture notes attached: Relevant equations The attempt at a solution I agree 2) of proposition 2.12 ...
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### Some questions on the Hecke algebra in Casselman's notes

Let $G$ be a connected, reductive group over a $p$-adic field $F$, which is unramified in the sense that $G$ is quasisplit to split over an unramified extension of $F$. Then $G$ is necessarily ...
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