Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Abstract

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over $\mathbb {Q}$, $N_L$ the level of L, and $\mathscr {M}(L)$  be the linear space of $\theta$-series attached to the distinct classes in the genus of L. We prove that, for an odd prime $p|N_L$, if $L_p=L_{p,1}\,\bot \, L_{p,2}$, where $L_{p,1}$ is unimodular, $L_{p,2}$ is (p)-modular, and $\mathbb {Q}_pL_{p,2}$ is anisotropic, then $\mathscr {M}(L;p):=$ $\mathscr {M}(L)$ $+T_{p^2}.$ $\mathscr {M}(L)$  is stable under the Hecke operator $T_{p^2}$. If $L_2$ is isometric to $\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle$ or $\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle$ or $\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle$ with $\varepsilon \in \mathbb {Z}_2^{\times }$ and $\kappa :=\frac{k-1}{2}$, then $\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)$ is stable under the Hecke operator $T_{2^2}$. Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.