Hecke Operators

What are Hecke Operators? These are some operators acting on automorphic forms on some arithmetic groups. Not all discrete subgroups have these operators and existence of these operators is a reason for why the spectrum of these group is different from a generic spectrum and has different statistics compared to the universal distributions expected. It’s crucial to use the fact that these operators commute with the Laplacian to understand the spectrum of Laplacian better.

One of the first application of these operators are to prove multiplicativity of Fourier coefficients of certain $q$ series like the Ramanujan $\tau$ function, which is a immediate consequence of the multiplicative properties of the Hecke operators.

We first treat the case of ${\Gamma = SL_2(\mathbb Z)}$ . Also we don’t bother a lot about weights, characters etc. (Everything can be generalized using appropriate averages over slash operators)

We present some different ways to look at the Hecke operators.

As operators on functions of lattices:

To every lattice ${\Lambda}$ , we define the operator ${T_n}$ on function of lattices as an averaging operator over the sublattices of index ${n}$ .

$\displaystyle T_n (F) (\Lambda) = \sum_{\substack{\Lambda' \subseteq \Lambda \\ [\Lambda: \Lambda'] =n}} F(\Lambda').$

This index condition can also be stated in term of determinants as follows

$\displaystyle T_n (F) = \sum_{ \det \in \Gamma/M_n } F(\gamma z)$

where the sum is over the cosets ${ \Gamma/M_n}$ of matrices of determinant equal to ${n}$ .

If the ${f}$ is a weight ${k}$ form, we define

$\displaystyle T_n (f) (\Lambda) = n^{k-1}\sum_{\substack{\Lambda' \subseteq \Lambda \\ [\Lambda: \Lambda'] =n}} f(\Lambda').$

Also the ${T_p}$ can viewed as the sum over lattices that contain ${\Lambda}$ as an index ${p}$ subgroup.

That is

$\displaystyle T_p (f) (\Lambda) =p^{k-1} \sum_{\substack{\Lambda' \subseteq \Lambda \\ [\Lambda: \Lambda'] =n}} f(\Lambda')= \frac{1}{p} \sum_{\substack{\Lambda \subseteq \Lambda'' \\ [\Lambda'': \Lambda] =p}} f(\Lambda'')$

The sublattices or the representatives of ${ \Gamma/M_n}$ are explicitly given by

$\displaystyle \left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right), ~~ad=n, 0 \le b < d$

So we have the following explicit formulae for the action on the functions. Let ${\langle 1, \tau \rangle}$ denote the lattice:

$\displaystyle T_n(F) (\tau) =\sum_{\substack{ad=n\\ 0 \le b < d}} F\left( \frac{a\tau + b}{d}\right)$

$\displaystyle T_n(f) (\tau) =n^{k-1}\sum_{\substack{ad=n\\ 0 \le b < d}} f\left( \frac{a\tau + b}{d}\right)$

As double coset operators:

If ${F}$ is a automorphic function satisfying ${F(\gamma z)= F(z)}$ (or say ${F(\gamma z) = j(\gamma, z) F(z)}$ ), then

${F(gz)}$ for ${\gamma}$ not in ${\Gamma}$ need not be automorphic. So we try to find restore the modularity by some averaging.

If ${\Gamma g \Gamma}$ is a finite union of the left-cosets ${ \Gamma g'}$ , then we can average over these ${g'}$ to produce a automorphic function. Consider the average

$\displaystyle Tf (z)= \sum_{g'}F(g'z)$

where

$\displaystyle \Gamma g \Gamma = \bigcup_{g'}\Gamma g'.$

Now when we act by ${\gamma}$ , ${g\gamma}$ is mapped to one of the ${g'}$ ‘s mod the ${\Gamma}$ on the left. Hence the average over ${g'}$ is automorphic.

Note that for this to work we need the finiteness of the left cosets for the double coset ${\Gamma g \Gamma.}$ . This finiteness is a very non-generic property, and that’s the reason most groups don’t have Hecke Operators.

More generally, we can also take ${\Gamma_1g\Gamma_2}$ and write it as left cosets of ${\Gamma_1}$ and average to map a ${\Gamma_1}$ invariant function to ${\Gamma_2}$ invariant function.

The Hecke operator ${T_p}$ previously defined turns out to be the Double coset operator

$\displaystyle \Gamma \left[\begin{array}{ll} 1 & 0 \\ 0 & p \end{array}\right] \Gamma$

For ${\Gamma=SL_2(\mathbb Z)}$ , we get

$\displaystyle \Gamma \left[\begin{array}{ll} 1 & 0 \\ 0 & p \end{array}\right] \Gamma=M_p =\left \{ \gamma \in M_{2}(\mathbb{Z}): \quad \det \gamma=p \right\}$

Hence it’s the same operators.

But the good thing about this perspective is that, now we can define Hecke operators for any group.

$\displaystyle \Gamma =\Gamma_1(N) =\left\{\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \in S L_{2}(\mathbb{Z}):\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \equiv\left[\begin{array}{ll} 1 & * \\ 0 & 1 \end{array}\right] \quad \bmod N\right\},$

then we have

$\displaystyle \Gamma \left[\begin{array}{ll} 1 & 0 \\ 0 & p \end{array}\right] \Gamma=M_p =\left \{ \gamma \in M_{2}(\mathbb{Z}): \gamma \equiv\left[\begin{array}{ll} 1 & * \\ 0 & p \end{array}\right] \quad \bmod N, \det \gamma=p \right\}$

Thus we get the following expressions for Hecke operators ${T_p}$ on congruence subgroups.

$\displaystyle T_{p} f(\tau)= \begin{cases}\frac{1}{p}\displaystyle \sum_{j=0}^{p-1} f\left(\frac{\tau+j}{p}\right) & \text { if } p \mid N \\ \frac{1}{p} \displaystyle \sum_{j=0}^{p-1} f\left(\frac{\tau+j}{p}\right)+p^{k-1} f(p \tau) & \text { if } p \nmid N\end{cases}$

Another view point: (Commensurate subgroups)

The decomposition of ${\Gamma g \Gamma}$ into left cosets of ${\Gamma}$ is equivalent to finding the representatives of ${\Gamma/\Gamma_g}$ where ${\Gamma_g=g^{-1}\Gamma g \cap \Gamma}$ is the intersection of ${\Gamma}$ and it’s conjugate. In fact, if we have

$\displaystyle \Gamma=\bigcup_{\gamma \in \Gamma_{g} \backslash \Gamma} \Gamma_{g} \gamma_{i}$

then we have

$\displaystyle \Gamma g \Gamma=\bigcup_{\gamma \in \Gamma_{g} \backslash \Gamma} \Gamma g \gamma_{i}$

Thus the finiteness of left cosets ${ \Gamma \backslash \Gamma g \Gamma }$ is equivalent to the finiteness of ${\Gamma_{g} \backslash \Gamma}$ , that is the intersection ${\Gamma_g}$ of ${\Gamma}$ and it’s conjugate ${g^{-1}\Gamma g}$ is of finite index in ${\Gamma}$ . So ${\Gamma}$ and ${g^{-1}\Gamma g}$ have to be “commensurate”.

So to find Hecke operators we need ${g}$ ‘s such that ${\Gamma}$ and ${g^{-1}\Gamma g}$ are commensurate. Not all ${\Gamma}$ have such elements ${g}$ . Having these is a very special property and is not always true. The arithmetic groups like ${SL_2(\mathbb Z)}$ or ${\Gamma_1(N)}$ are very non-generic, that why we found Hecke operators. In the case of ${SL_2(\mathbb Z)}$ all the elements ${g \in SL_2(\mathbb Q)}$ form the “commensurate” subgroup (check that ${g}$ ‘s with ${| \Gamma_{g} \backslash \Gamma| <\infty}$ for a group)

As correspondences:

The double coset operator formally helps us to map ${\Gamma_2 \tau}$ to ${\sum_{i} \Gamma_1 g_i \tau.}$ That is we have a map from divisor on ${\mathbb H/\Gamma_2}$ to the divisor on ${\mathbb H/\Gamma_1}$

Or we can just think of it as multivalued function

$\displaystyle \Gamma_2 \tau \rightarrow \{ \Gamma_1 g_i \tau \}.$

In the lattice picture for ${SL_2(\mathbb Z)}$ , we are mapping one lattice to all the lattices of index ${n}$ .

This multivalued function is captured by it’s graph in ${\mathbb H/\Gamma_2 \times \mathbb H/\Gamma_1}$ . The projection from the graph to the first factor is unique for a single valued function, but now has a degree because of the multi-valuedness. (all those that map to the same point).

For instance in case of ${\Gamma_0(N)}$ the correspondence is basically the map

The first map ${\alpha}$ is basically the reduction ${ \Gamma_0(Np) \tau \rightarrow \Gamma_0(N)\tau}$ and the second map ${\beta}$ is ${\tau \rightarrow p \tau}$ obtain by reduction after conjugating , that is it’s obtained by reduction using the isomorphism

$\displaystyle \mathbb H/ \Gamma_{0}(p N) \stackrel{\sim}{\longrightarrow} \mathbb H /\left(\begin{array}{ll} p & 0 \\ 0 & 1 \end{array}\right) \Gamma_{0}(p N)\left(\begin{array}{ll} p & 0 \\ 0 & 1 \end{array}\right)^{-1} .$

In terms of Fourier expansion:

If we have the fourier expansion

$f(q) =\sum_{n} A_n q^n$ ,

the expansion of ${T_p(f)}$ is given by

$\displaystyle T_n (f) (q) = \sum_{ad=n} a^{k-1} A_{d m} q^{a m}$

In particular for ${n=p}$ , we have

$\displaystyle T_p (f) (q) = \sum A_{p m} q^{m} + p^{k-1}\sum A_m q^{pm}.$

This is what allows us to get multiplicative relations for eigenfunctions from the multiplicative relations on the operators.

The algebra structure:

We have the following relations for Hecke operators on ${SL_2{\mathbb Z}}$ .

$\displaystyle \begin{aligned} T_{n m} &=T_{n} T_{m} & & \text { if } \text{gcd}(n, m)=1 \\ T_{p^{r+1}} &=T_{p^{r}} T_{p}-p^{k-1} T_{p^{r-1}} & & \text { if } r \geq 1 \text { and } p \text { prime. } \end{aligned}$

The first relation is basically the fact that for a sublattice ${\Lambda'}$ of index ${mn}$ of a lattice ${\Lambda}$ , there is a unique lattice $\displaystyle \Lambda' \subseteq \Lambda'' \subseteq \Lambda$ such that $\displaystyle [\Lambda: \Lambda''] =n, [\Lambda'': \Lambda']=m.$

The other relation for prime powers also follows by looking at whether ${\Lambda'' =p\Lambda}$ is contained in between the lattices ${\Lambda}$ and ${\Lambda'}$ or not.

In terms of the double cosets

If we have two double cosets

$\displaystyle \Gamma g \Gamma = \bigcup_{g'}\Gamma g_i, ~\Gamma h \Gamma = \bigcup_{g'}\Gamma h_j$ ,

the product ${[\Gamma g \Gamma][\Gamma h \Gamma ]}$ acts on ${f}$ by

$\displaystyle [\Gamma g \Gamma][\Gamma h \Gamma ] (f) (z) = \sum_{g_i, h_j} f(g_ih_j z)$

More Examples:

We will now consider the case of ${\Gamma=SL_n(\mathbb Z)}$ . Even here the double cosets ${\Gamma g \Gamma}$ with ${g \in SL_n (\mathbb Q)}$ give commensurability (finiteness ) and hence give Hecke operators on the automorphic forms.

The distinct double cosets can be parametrized by diagonal matrices

$\displaystyle \text{diag}\left(d_{1}, \ldots, d_{n}\right), \quad \text { where } d_{i}>0, d_{i} \mid d_{i+1}$

In the case of ${SL_2(\mathbb Z)}$ , ${SL_2(\mathbb Z) \left(\begin{array}{ll} 1 & 0 \\ 0 & p \end{array}\right)SL_2(\mathbb Z)}$ corresponds to the Hecke operator ${T_p}$ . We also get the same operator for

$\displaystyle SL_2(\mathbb Z) \left(\begin{array}{ll} 1 & 0 \\ 0 & p \end{array}\right)SL_2(\mathbb Z) = SL_2(\mathbb Z) \left(\begin{array}{ll} p & 0 \\ 0 & 1 \end{array}\right)SL_2(\mathbb Z).$

If we restrict all the ${d_i}$ to be powers of ${p}$ , we generate the “local” Hecke algebra. But the algebra can be generated by elements of the form

$\displaystyle g=\text{diag}\left(1, 1, 1\ldots,1, p, p, \ldots p \right)$

That is all the double cosets operators $\Gamma g \Gamma$ with $g= \text{diag}\left(d_{1}, \ldots, d_{n}\right)$ , $d_{i}>0, d_{i} \mid d_{i+1}$ , and ${d_i}$ powers of ${p}$ , can be written as rational function functions of the operators corresponding to the elements ${ \text{diag}\left(1, 1, 1\ldots,1, p, p, \ldots p \right) }$ , and these generators are algebraically independent.

For the case of symplectic group ${Sp_{2g}(\mathbb Z)}$ , the algebra is generated by operators

$\displaystyle T(p) = T(\underbrace{1, \ldots, 1}_{g} ; \underbrace{p, \ldots, p}_{g})$

and

$\displaystyle T_i(p^2)=(\underbrace{1, \ldots, 1}_{g-i}, \underbrace{p, \ldots, p}_{i}, \underbrace{p^{2}, \ldots, p^{2}}_{g-i}, \underbrace{p, \ldots, p}_{i}).$

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