# Tag Info

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. ...
• 22.8k

### 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 $(p-1)(1-1/\ell)$...
• 12.7k
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 ...
• 15.4k
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), ...
• 36.6k

### 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 ...
• 22.8k

• 10.6k

### 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 ...
• 1,692
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. ...
• 6,279

### 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 ...
• 2,386
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 ...
• 3,415