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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.

By a Hecke Algebra we will usually mean an Iwahori Hecke algebra. It is a term for a class of algebras. They often appear as convolution algebras or as double coset spaces. For p-adic algebraic groups Hecke algebras often play a role similar a Lie algebra plays in the complex case (the Lie algebra still exists, but is too small). Representations of Hecke algebras led to discovery of quantum groups by Michio Jimbo. Michael Freedman proposed Hecke algebras as a foundation for topological quantum computation.

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Eigenvalues of adjoints of Hecke operators

I am trying to work through Diamond and Shurman's A First Course in Modular Forms but am stuck in one of the exercises. Exercise 5.11.2 asks us to show that given a normalized eigenform $f\in S_k(\Gamma_1(N))$ with eigenvalues $a_p$ for Hecke…
Devang Agarwal
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1-dim representations of the affine Hecke algebra for $G = \text{SL}_2$

I want to count the number of (isomoprhism clases) of one-dimensional representations of the affine Hecke algebra for $G = \text{SL}_2$. I'm doing it in two ways: (1) by explicitly looking at generators and relations, and (2) by looking at the…
user148177
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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 and its closed subgroup. While from here, the…
MAS
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Universality of Hecke algebra of a finite group

I am solving an assignment problem on the Hecke algebra of a finite group, and looking for an idea that might help find a right direction. Given a pair of finite groups $G\geq K$, the Hecke algebra $\mathcal{H}_{G,K}$ can be defined as $$…
Hyeongmuk LIM
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Why is the Galois conjugate of a modular eigenform another eigenform?

In this question, $k \geq 2$ and $N \geq 1$ are integers. We consider the space $S = \mathcal{S}_k(\Gamma_1(N))^{new}$ of modular forms of weight $k$ for $\Gamma_1(N)$ (the same question can certainly be asked for $k = 1$ but I think that this MO…
Aphelli
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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 from $G$ to $V$, which has the $G$-representation…
Adam Higgins
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Isomorphism of Hecke algebra $H(G_1 \times G_2)$ with $H(G_1) \otimes H(G_2)$

Let $G$ be a topological group of totally disconnected (td) type. This means that the identity of $G$ has a fundamental system of neighborhoods consisting of open compact subgroups. Then $G$ is locally compact, and has a Haar measure. We give…
D_S
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De Shalit's lemma in R = T.

In Wiles' celebrated paper where any semi-stable elliptic curve $E$ over ${\Bbb Q}$ is modular, Theorem $0.3.$ therein assumes that either $E$ is good or multiplicative reduction at $3$. This condition seems to restrict the Hecke algebra ${\Bbb…
Pierre MATSUMI
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Directed system of irreducible modules

I have a question on a paper in the Corvallis proceedings on automorphic forms. Background: Let $G$ be a topological group of td type. This means that $G$ is Hausdorff, and every neighborhood of the identity of $G$ contains a compact open…
D_S
  • 35,843
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Order of sublattice.

I am reading serre's book A Course in Arithmetic"", and want to understand the Hecke operator. Let $\omega_1,\omega_2\in \mathbb{C},$ and $\frac{\omega_1}{\omega_1}\in \mathbb{H},$ where $\mathbb{H}$ stands for upper half plane, we define a lattice…
VerMoriarty
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Is there a synthetic definition of the $0$-Hecke monoid of $S_n$?

Background. Let $n$ be a nonnegative integer, and let $S_n$ denote the $n$-th symmetric group. The $0$-Hecke monoid $H_0\left(S_n\right)$ is defined to be the monoid given by generators $t_1, t_2, \ldots, t_{n-1}$ and relations $t_i^2 = t_i$ for…
darij grinberg
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Eigenvalues of Hecke operators for Siegel eigenforms are algebraic

Cross-posted to MO Let $F$ be a Siegel modular form for $\text{Sp}_4(\mathbb{Z})$ of genus two. Let it also be an eigenform for all the Hecke operators $T_n$, $n\geq1$ (genus two). Are the eigenvalues corresponding to these Hecke operators…
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Hecke algebra of $(\mathrm{GL}(2, \mathbb{Q}), \mathrm{GL}(2, \mathbb{Z}))$

In the Wikipedia article about the Hecke algebra of a locally compact group, it is noted that if we take $(\mathrm{GL}(2, \mathbb{Q}), \mathrm{GL}(2, \mathbb{Z}))$ as the pair $(G,K)$ of a unimodular locally compact topological group $G$ and a…
cxrlo
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Two definitions of the Jacquet functor: why are they the same?

Let $G$ be (the rational points of) a connected, reductive group over a local field $F$. Let $P$ be a parabolic subgroup of $G$ with unipotent radical $N$ and Levi subgroup $M$. The inclusion $M \subseteq P$ induces an isomorphism of $M$ with…
D_S
  • 35,843
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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 that if $G$ is abelian then this is not possible.…
Soumyadip Sarkar
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