Hecke algebra

Let $f$ be a modular form for $\mathrm{\Gamma }$ a congruence subgroup of ${\text{SL}}_2(\mathbb{Z})$.

$$f(z)=\sum _{n=0}^{\infty }{a}_n{q}^n$$(1)

where $q=e^{2i\pi \tau }$.

For $m\in \mathbb{N}$, let $T_{m}f(z)=\sum _{n=0}^{\infty }{b}_n{q}^n$ with :

$$b_n=\sum _{d|\gcd (m,n)}^{}{d}^{k-1}{a}_{mn/d^2}$$(2)

In particular, for $p$ a prime, $T_{p}f(z)=\sum _{n=0}^{\infty }{b}_n{q}^n$ with;

$$b_n=a_{pn}+p^{k-1}{a}_{n/p}$$(3)

where $a_{n/p}=0$ if $n$ is not divisible by $p$.

The operator $T_n$ is a linear operator on the space of modular forms called a Hecke operator.

The Hecke operators leave the space of modular forms and cusp forms invariantand turn out to be self-adjoint for a scalar product called the Peterssonscalar product. In particular they have real eigenvalues. Hecke operatorsalso satisfy multiplicative properties that are best summarized by the formalidentity:

$$\sum _{n=1}^{\infty }{T}_n{n}^{-s}=\prod _{𝑝}{(1-T_p{p}^{-s}+p^{k-1-2s})}^{-1}$$(4)

That equation in particular implies that $T_m{T}_n=T_n{T}_m$ whenever $\gcd (n,m)=1$.

The set of all Hecke operators is usually denoted $𝕋$ and is called the Hecke algebra.

0.1 Group algebra example

Definition 0.1Let $G_{lcd}$ be a locally compact totally disconnected group; then theHecke algebra $\mathrm{H}\mathit{}\mathrm{(}{G}_{l\mathit{}c\mathit{}d}\mathrm{)}$ of the group $G_{lcd}$ is defined as the convolution algebra oflocally constant complex-valued functions on $G_{lcd}$ with compact support.

Such $\mathscr{H}(G)$ algebras play an important role in the theory ofdecomposition of group representations into tensor products.