Hecke operators (awaiting review)

For a positive integer $g$, let $\GSp(2g,\Q)^+=\{\gamma\in\GL(2g,\Q)\, \mid \, \exists\, r(g)\in\Q_{>0}: \gamma^t J\gamma=r(g)J\}$.

Define the semigroup $S=\GSp(2g,\Q)^+\cap M(2g,\Z)$. For a discrete subgroup $\Gamma$ of $\Sp(2g,\Q)$, let $L(\Gamma,S)$ be the free $\C$-module generated by the right cosets $\Gamma\alpha$, where $\alpha\in S$. Note that $\Gamma$ acts on $L(\Gamma,S)$ by right multiplication. The $\C$-algebra $\mathcal{H}(\Gamma)=L(\Gamma,S)^\Gamma$, called the Hecke algebra of $\Gamma$, has the following multiplication:

If $T,T'\in\mathcal{H}(\Gamma)$ with $T=\sum_i c_i\Gamma\alpha_i$ and $T'=\sum_j c_j'\Gamma\alpha'_j$, then $TT'=\sum_{i,j} c_ic_j'\Gamma\alpha_i\alpha_j'$.

For a prime $p$, let $S_p$ be the subsemigroup of $S$ consisting of elements $g$ for which $r(g)$ is a power of $p$. Analogously define $\mathcal{H}_p(\Gamma)=L(\Gamma,S_p)^\Gamma$, the local Hecke algebra at $p$. If $\Gamma=\Sp(2g,\Z)$, then it is known that $\mathcal{H}(\Gamma)\cong\bigotimes_p\mathcal{H}(\Gamma,S_p)$.

We define the following elements of $\mathcal{H}_p(\Gamma)$. Let $1_j$ be the identity matrix of size $j$. We set

• $T(p)=\Gamma\,\mathrm{diag}(1_g,p1_g)\Gamma$

• $T_i(p^2)=\Gamma\,\mathrm{diag}(1_i,p1_{g-i},p^21_i,p1_{g-i})\Gamma$ for $0\leqslant i\leqslant g$

where we identify a double coset with the sum of its single cosets in the Hecke algebra. Also, let $T(p^2)=\sum_{i=0}^gT_i(p^2)$.

Suppose that $\Gamma$ contains $B(N)$, the Borel congruence subgroup of level $N$. Then it is known that $\mathcal{H}_p(\Gamma)$ is generated by $T(p)$ and by $T_i(p^2)$ for $1\leqslant i\leqslant g-1$.

If $g=2$, then $\mathcal{H}_p(\Gamma)$ is generated by $T(p)$ and $T_1(p)$, or alternatively by $T(p)$ and $T(p^2)$.

The Hecke algebra $\mathcal{H}(\Gamma)$ acts on the space $M_\rho(\Gamma)$ of Siegel modular forms by \[ F\vert_{\rho}(\sum c_i\Gamma\alpha_i)=\sum_ic_iF\vert_{\rho}\alpha_i. \] Note that the slash operator can be normalized in various ways. A common normalization is \[ (F\vert_{\rho}\alpha)(\tau)=r(\alpha)^{\lambda_1+\ldots+\lambda_g-\frac{g(g+1)}{2}}\rho(c\tau+d)^{-1}F(\alpha\cdot\tau) \] for $\alpha=\begin{pmatrix}a&b\\c&d\end{pmatrix} \in \GSp(2g,\Q)^+$ and $\rho$ irreducible finite-dimensional representations of $\GL(g,\C)$ of highest weight $\lambda_1\geqslant \lambda_2\geqslant\ldots\geqslant \lambda_g$.

For scalar-valued forms $F\in M_k(\Gamma)$, i.e., $\rho=\det^{\otimes}$, we get \[ (F\vert_{k}\alpha)(\tau)=r(\alpha)^{gk-\frac{g(g+1)}{2}}\det(c\tau+d)^{-k}F(\alpha\cdot\tau) \] for $\alpha=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \GSp(2g,\Q)^+$.

The subspace $S_\rho(\Gamma)$ of cusp forms is preserved under this action.

A Hecke eigenform is a form in $M_{\rho}(\Gamma)$ which is a simultaneous eigenform for all the Hecke operators.