Let $S_k(\Gamma_n)$ be the space of Siegel cuspidal forms of weight $k$ with respect to $\Gamma_n=Sp_{2n}(\mathbb{Z})$.

Is it true that $End(S_k(\Gamma_n))$ coincides with the Hecke algebra (i.e. the algebra generated by the Hecke operators)?

I think at some point I heard something like this in the case $n=1$ of classical (elliptic) modular forms, but I might be wrong. If the answer is no, do we know any explicit endomorphism which is not (basically) a Hecke operator?

I am interested in particular for $n=2$. As a follow-up, say that $F_1,\dots,F_m$ is an orthogonal basis of Hecke eigenforms for $S_k(\Gamma_2)$: I am looking for any endomorphism $\phi$ which yields $F_1\in ker(\phi)$.

Examples that come to my mind for $\phi$ are $$\phi: G\mapsto G[T]-\lambda_T(F_1)G$$ or $$\phi_l: G\mapsto \langle G,F_l\rangle F_l$$ for $l>1$, where $\langle\cdot,\cdot\rangle$ is the Petersson inner product, $T$ is any Hecke operator and $\lambda_T(F_1)\in\mathbb{C}$ is the $T$-eigenvalue of $F_1$. Any other idea?

  • 2
    $\begingroup$ The statement for classical modular forms is that the endomorphisms that commute with the Hecke algebra are the Hecke algebra, which because the Hecke algebra is semisimple, is basically the same as that the eigenspaces of the Hecke algebra are one-dimensional, i.e. multiplicity one. So you are looking for counterexamples to multiplicity one. However these won't usually come in the form of explicit endomorphisms. $\endgroup$ – Will Sawin Jun 16 '17 at 15:31

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