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.
65
questions
2
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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 ...
- 724
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vote
0
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30
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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 ...
- 93k
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0
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19
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Multiplication in Hecke algebra for $\mathrm{GL}_2$
Let $G = \mathrm{GL}_2(F)$ for a $p$-adic field $F$, and let $K = \mathrm{GL}_2(\mathcal O_F)$ for the ring of integers $\mathcal O_F$ of $F$. I want to make the multiplication in the Hecke algebra $\...
- 590
1
vote
0
answers
28
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Convolution of characteristic functions on double cosets
I'm trying to verify formula (5.1) in Getz's and Hahn's An Introduction to Automorphic Representations, §5.2, p. 138. The goal is to calculate the product of basis elements in the Hecke algebra.
The ...
- 590
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votes
0
answers
18
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Embedding contragradient of Hecke algebra into smooth functions on group
Let $G$ be a unimodular locally profinite group (locally compact t.d.). The Hecke algebra $H(G)$ is the convolution algebra of locally constant, compactly supported distributions on $G$ and $C^\infty (...
3
votes
1
answer
78
views
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 ...
- 111
0
votes
0
answers
54
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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 ...
- 53
0
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0
answers
59
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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
...
0
votes
0
answers
50
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Hecke operator on a constant function
I would like to understand how the Hecke operator for the SL(2,Z) group works.
In particular its action on, for example a constant function.
Would the following developpement be right?
$T_n(f(\tau)) :=...
- 125
0
votes
0
answers
79
views
Connection between Operator Theory and Number Theory
I was wondering if there is any connection between number theory and operator theory. Especially the applications of Hardy spaces, de branges-Rovnyak spaces, Dirichlet spaces in number theory. For ...
- 17
6
votes
1
answer
148
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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 $\...
- 806
1
vote
0
answers
83
views
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 ...
- 149
0
votes
0
answers
44
views
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 ...
0
votes
0
answers
38
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Gauss sum involved in the Hecke action on classical Hilbert modular forms
Let $F=\mathbb{Q}(\sqrt{D})$ be a real quadratic field and consider the classical Hilbert modular forms over $F$. Let $\varepsilon_0>1$ be the fundamental unit of $F$ and write $d=\varepsilon_0\...
- 73
1
vote
0
answers
44
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Correspondence for $U_p$ operator
I've seen in many places that one can define the usual Hecke operator $T_p$ via correspondences by $(\pi_1)_*(\pi_2^*(\bullet))$ in the diagram
$$Y \xleftarrow{\pi_1} Y_0(p) \xrightarrow{\pi_2} Y$$
...
- 240
0
votes
1
answer
61
views
A property of simultaneous eigenforms
While reading the following theorem for Apostol's modular functions and dirichlet series in number theory, I have a question:
(Theorem 6.14, page 130)
Assume that k is even and $k\geq 4$. If the space ...
- 67
3
votes
0
answers
118
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Orthogonal basis with rational coefficients (Petersson inner product)
Let us consider the Petersson inner product $\left<-,-\right>:S_2(\Gamma_0(N))\times S_2(\Gamma_0(N))\rightarrow \mathbb{C}$,
$$\left<f,g\right>:=\int_{\Gamma_0(N)\backslash \mathbb{H}}\...
- 103
1
vote
0
answers
45
views
Generators of the monodromy generating braid groups
I'm reading the article 'Complex reflection groups, braid groups and Hecke algebras', by Broué, Malle and Rouquier, but I need some help with the 'generators of the monodromy' they defined and that ...
- 61
1
vote
0
answers
70
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p-adic Hecke Operators
p-adic Hecke Operators in the Iwahori-Hecke Algebra $C_c(J\backslash G(F)/J)$
Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a ...
- 189
2
votes
1
answer
73
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On axiomatic definition of affine root systems
In Macdonald's book Affine Hecke Algebras and Orthogonal Polynomials, chapter 1 introduces affine root systems. I will recall the definition here: Let $E$ be a non-zero real Euclidean space (finite ...
- 247
2
votes
0
answers
520
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Strong multiplicity one theorem for new forms of different levels
Let $f, g$ be newforms of level $N_f, N_g$ (possibly different) with $N_f, N_g |N$, and assume that they have same eigenvalue for Hecke operators $T_p$ with $(p, N) = 1$. Then $N_f = N_g$ and $g = cf$ ...
- 15.3k
4
votes
1
answer
127
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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 ...
- 50
6
votes
0
answers
407
views
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 ...
- 35.9k
5
votes
0
answers
73
views
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 ...
- 883
7
votes
0
answers
140
views
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(\...
- 336
2
votes
1
answer
121
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Image of the induced representation $\mathrm{Ind}_K^G 1$ inside the group algebra $\mathbb{C}[G]$
Set-up:
Consider a finite group $G$ and a subgroup $K$. Given these, we can look at the induced representation $\mathrm{Ind}_K^G 1$ (here, $1$ is the trivial $1$-dim complex representation of $K$). As ...
0
votes
1
answer
110
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Hecke algebra relations
From Chriss-Ginzburg book "Representation theory and complex geometry", it is written that the relation in the affine Hecke algebra $$ T_s e^{s(\lambda)} - e^{\lambda} T_s = (1-q) \frac{e^{\...
- 831
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votes
0
answers
30
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Is it possible for $f(z)$ and $f(\sigma z)$ to both be Hecke eigenfunctions?
Let $f(z)$ be an automorphic/modular form on $\Gamma_0(p)/\mathbb{H}$, where $p$ is some prime.
I know that the Hecke operators $T_n$ act on this space of functions whenever $(n,p)=1$.
Assume that $f(...
- 1,270
0
votes
0
answers
70
views
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?
- 1,270
7
votes
1
answer
345
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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 ...
- 10.8k
3
votes
0
answers
69
views
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 ...
- 307
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vote
0
answers
24
views
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, ...
- 229
2
votes
1
answer
585
views
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 ...
- 95
2
votes
1
answer
49
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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 ...
- 4,628
1
vote
1
answer
258
views
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 ...
- 1,177
3
votes
0
answers
227
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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 $\mathfrak{gl}_n$...
- 31
6
votes
2
answers
244
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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 ...
- 2,271
2
votes
0
answers
608
views
Hecke operators self-adjoint with respect to Petersson inner product
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 ...
- 680
2
votes
1
answer
285
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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
0
answers
32
views
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 $H$-...
- 461
2
votes
2
answers
661
views
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 ...
2
votes
1
answer
203
views
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 ...
- 34.7k
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votes
0
answers
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Conventions for Kazhdan Lusztig-polynomials and the form of the KL-conjecture
This is a follow-up question to Reference request: Presentations of Hecke algebras.
In the common definition
$$H_t = \langle T_1,\dotsc, T_n \mid \text{braid relations, and for all i: $T_i^2 = t + (t-...
- 2,264
4
votes
1
answer
281
views
Two definitions of the Jacquet functor: why are they the same?
Let $G$ be (the rational points of) a connected, reductive group over a local field $F$. Let $P$ be a parabolic subgroup of $G$ with unipotent radical $N$ and Levi subgroup $M$. The inclusion $M \...
- 34.7k
2
votes
0
answers
70
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$H(G') \otimes_{H(G)} V$ and $f^{-1} \mathcal G \otimes_{f^{-1}\mathcal O_Y} \mathcal O_X$, the connection
1 . Let $f: (X,\mathcal O_X) \rightarrow (Y,\mathcal O_Y)$ be a morphism of ringed spaces, and let $\mathcal G$ be a sheaf of $\mathcal O_Y$-modules. Define the inverse image $f^{\ast} \mathcal G$ ...
- 34.7k
5
votes
0
answers
149
views
Directed system of irreducible modules
I have a question on a paper in the Corvallis proceedings on automorphic forms. Background: Let $G$ be a topological group of td type. This means that $G$ is Hausdorff, and every neighborhood of the ...
- 34.7k
6
votes
1
answer
191
views
Isomorphism of Hecke algebra $H(G_1 \times G_2)$ with $H(G_1) \otimes H(G_2)$
Let $G$ be a topological group of totally disconnected (td) type. This means that the identity of $G$ has a fundamental system of neighborhoods consisting of open compact subgroups. Then $G$ is ...
- 34.7k
3
votes
0
answers
282
views
Hecke operators acting on the Jacobian $J(X_1(N))$
I'm stuck with p.238 of Diamond's book on modular forms. If $\{f_j\}_{j=1}^g$ is the eigenform basis of $S_2(\Gamma_0(N))$, fixing $p_0 \in X_0(N)$ we have an holomorphic map $$\phi : X_0(N) \to J(X_0(...
- 78.7k
1
vote
0
answers
57
views
Finite group with $BN$-pair, if $s_i$ and $s_j$ are conjugate in $W$, why do they have the same index parameter?
Suppose $G$ is a finite group with $BN$-pair, and $(W,S)$ is its Coxeter system.
Iwahori's theorem that the corresponding Hecke algebra $\mathcal{H}$ has a standard basis $\{a_w:w\in W\}$ where $a_w=...
- 1,673
1
vote
1
answer
76
views
Why is the index map on the Hecke algebra $\mathcal{H}(G,H,1_H)$ an algebra homomorphism?
Suppose $H\leq G$ are finite groups, $k$ a splitting field for $G$, and $\mathcal{H}(G,H,1_H)=ekGe$ the corresponding Hecke algebra, where $e=\frac{1}{|H|}\sum_{h\in H}h$ is the idempotent ...
- 1,673