11
votes

Accepted

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

These things are not trivial at all, but by the time Langlands was writing "Euler products" they were known, and quite familiar to many people at Princeton and Yale, even if not so many other places.
...

10
votes

### Smallest Mazur's good prime

The good primes (not counting $\ell$ itself, if that's allowed to be a good prime) are precisely those that lie both in one of $\ell-2$ reduced residue classes (mod $\ell$) and one of $(p-1)(1-1/\ell)$...

10
votes

Accepted

### Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian?

No, let $G$ be an arbitrary group and take $H=G$ acting on itself by conjugation. Then $H$ becomes the diagonal in $G\rtimes H\cong G\times G$. This is well known to be a Gelfand pair.
PS: The Hecke ...

8
votes

Accepted

### Restriction to the diagonal of Hilbert eigenforms

It is extremely unusual for the restriction of a Hilbert modular form to the diagonal to be an elliptic modular eigenform. It happens occasionally in some small cases (by coincidence, essentially), ...

8
votes

### What is the archimedean Hecke algebra?

The terminology is a bit misleading, and the analogy with the non-archimedean situation is a bit forced.
The goal was/is to have a $\mathfrak g,K$-module be a "Hecke algebra module", for some ...

8
votes

### What is the archimedean Hecke algebra?

Here is the picture, as I understand it, for $\mathrm{GL}_n$; this is described in chapter 8 of Godement-Jacquet (and see also the archimedean theory in Jacquet-Langlands).
Let $F \in \{\mathbb{R},\...

8
votes

Accepted

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

The full reference is
Jean-Loup Waldspurger, "Sur les coefficients de Fourier des formes modulaires de poids demi-entier", (French) Journal de Mathématiques Pures et Appliquées, IX Séries, ...

8
votes

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

This is a really good question. What is known of an answer is somewhat
complicated, and perhaps others have more useful things to say.
As you mention, the Hecke algebra $H$ is categorified by the ...

7
votes

Accepted

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

When I calculated these numbers using Marc van Leeuwen and Fokko du Cloux's software atlas, I got
S_3: 8
S_4: 60
S_5: 482
S_6: 4268
S_7: 41934
S_8: 457782
(I could easily have made some silly ...

7
votes

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

Already in the case of finite symmetric groups, one can find any polynomial with non-negative integral coefficients and constant term 1 as KL polynomial for some pair of group elements. See the ...

7
votes

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

I think your question already contains its own answer.
In classical, "vertical" Iwasawa theory one studies class groups, or other arithetic widgets like elliptic curve Selmer groups, in a ...

6
votes

Accepted

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

A normalized version of your guess is right. First note that the $T_{p^n}$'s satisfy the relation
$$ T_{p^{n+1}} = T_p T_{p^n} - p T_{p^{n-1}} $$
(e.g., Bump Prop 4.6.4). This gives you a recursion ...

6
votes

### Are there Hamilton paths in Cayley graphs of Coxeter groups?

The answer to both questions is yes. See the paper of Conway, Sloane and Wilks called "Gray codes for reflection groups":
http://link.springer.com/article/10.1007/BF01788686

6
votes

Accepted

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

There is a combinatorial rule for the structure constants. It appears in Section 2 of https://arxiv.org/abs/1104.1959, although it is not immediately clear that it is indeed what you are looking for.
...

5
votes

Accepted

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

Let $G$ be a reductive $p$-adic group. First fixing a Haar measure on $G$, you can identify the algebra of distributions of $G$ with the ("big") Hecke algebra $H(G)$ of locally constant complex ...

5
votes

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

A Hecke algebra, in the most common definition, is associated to a pair of groups $G$ and $K$, and is the convolution algebra of bi-$K$-invariant distributions on $G$.
As far as I know the left-...

4
votes

Accepted

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

In the ordinary case, the argument is simple so let me recall it here.
The $p$-divisible group $J$ is an extension of an étale $p$-divisible group $J^{et}$ by a multiplicative $p$-divisible group $J^{...

4
votes

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

Let us consider the case when the centre acts trivially, for simplicity. Let us call $H_p$ to be the polynomial algebra generated by the classical Hecke operators $\{T_{p^r}\mid r\ge 0\}$. Using the ...

4
votes

Accepted

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

The general setting for your question is the theory of types as developped by Bushnell and Kutzko:
Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. ...

4
votes

### Volumes of double cosets $KtK$

Let $t=\varpi^\lambda$ where $\varpi$ is a uniformizer and $\lambda:\mathbb{G}_m\to T$ is a dominant weight. The assumption that $\lambda$ is dominant is harmless as we may conjugate by an appropriate ...

4
votes

Accepted

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

You mean the (commutative, normal for the Petersson inner product thus diagonalizable) complex algebra $\Bbb{T(C)}$ of endomorphisms of the complex vector space $M_k(SL_2(Z))$ ($k$ even) generated by ...

4
votes

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

Interesting observation! Unfortunately, this seems to be a coincidence.
The original, algebraically motivated definition of the Hecke algebra gives the quadratic relation as
$$(T-q)(T+1)=0.$$
For $...

4
votes

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

Here is Sur les coefficients de Fourier des formes modulaires de poids demi-entier
and here is Correspondances de Shimura (not the 1980 paper, but a summary of that paper from 1983)

4
votes

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

If $K$ is a perfect field, $A$ is a finite dimensional $K$-algebra and $F/K$ is an extension, then $M_F$ is semisimple over $A_F$ for any simple $A$-module $M$.
First note that if $J(R)$ denotes the ...

4
votes

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

Yes, it is the only one possible. Your relations are actually wrong - in 1) you should replace -1 by +1. Then in Steinberg every $X_{s_i}$ acts by -1 and 3) immediately implies that $X_{\rho}$ acts ...

4
votes

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

The point of course is that equivalences of categories preserve adjoints, but there is one subtlety as "induction" is a right adjoint whereas the tensor product is a left adjoint. The ...

3
votes

Accepted

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

This isn't quite an answer, but since I cant comment, I'll do it here.
In MAGMA you can ask for HeckeAlgebra of a space of ModularSymbols (or maybe it only works for the cuspidal subspace of such), ...

3
votes

### Traces on Hecke algebras and the Jones polynomial

This is a beautiful question that is not trivial at all. After Jones' paper, Lambropoulou constructed a trace on the generalized Hecke algebra of type B, through which, she obtained the analogue of ...

3
votes

### Motivation for the Kazhdan-Lusztig involution

I'm mostly a combinatorialist who doesn't completely understand this stuff, so I may have something slightly wrong, but...
When $W$ is a Weyl group, the Kazhdan--Lusztig involution is (the $K$-...

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