# Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?

Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $$\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$$ gives rise to a primitive L-function.

Is the converse true? That is, assuming Ramanujan conjecture, if a degree $$n$$ primitive L-function comes from an automorphic representation of $$\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$$, is this representation necessarily irreducible and cuspidal?

If $$\pi$$ is not irreducible, say $$\pi=\pi_1\oplus\pi_2$$, then $$L(s,\pi)=L(s,\pi_1)L(s,\pi_2)$$.
If $$\pi$$ is not cuspidal, then by Langlands' theory of Eisenstein series, there is a nontrivial partition $$n=\sum n_j$$ and cuspidal irreducible representations $$\pi_j$$ of $$\mathrm{GL}_{n_j}(\mathbb{A}_\mathbb{Q})$$ such that $$\pi$$ is parabolically induced from $$\prod\pi_j$$ on $$\prod\mathrm{GL}_{n_j}(\mathbb{A}_\mathbb{Q})$$ as a Levi subgroup of $$\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$$. Then, $$L(s,\pi)=\prod L(s,\pi_j)$$.
• It's not true without the Ramanujan conjecture if by "primitive $L$-function" you mean "primitive element of the Selberg class whose Dirichlet series coefficients satisfy $a(n) \ll_{\varepsilon} n^{\varepsilon}$", which is what I imagine the OP meant. – Peter Humphries Mar 21 at 1:25
• @PeterHumphries: I guess I thought of primitivity within automorphic $L$-functions or some extended Selberg class. But you are absolutely right. – GH from MO Mar 21 at 3:32