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

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. [ABSTRACT FROM AUTHOR]