| (G) |
general type, $\GL(4)$ type |
$\mu\boxtimes\nu(1)$ |
$\mu$ is a $\chi$-self-dual, symplectic, unitary, cuspidal, automorphic representation of $\GL(4,\A)$. |
$\displaystyle\bigoplus_{\psi\in\mathbf{(G)}}\;\bigoplus_{\pi\in\Pi_\psi}\pi$ |
none |
$\bullet$ |
$\bullet$ |
$\bullet$ |
$\bullet$ |
globally generic |
always |
$L(s,\mu)$ (primitive) |
$\displaystyle\frac{L(s,\mu,\Lambda^2)}{L(s,\chi)}$ |
| (Y) |
Yoshida type |
$(\mu_1\boxtimes\nu(1))\boxplus(\mu_2\boxtimes\nu(1))$ |
$\mu_1$ and $\mu_2$ are distinct, unitary, cuspidal, automorphic representations of $\GL(2,\A)$ with the same central character $\chi$. |
$\displaystyle\bigoplus_{\psi\in\mathbf{(Y)}}\;\bigoplus_{\{\pi\in\Pi_\psi:\:\langle\cdot,\pi\rangle=1\}}\pi$ |
$(-1)^n=1,$ where $n=\#\{v\,|\,\pi_v\text{ is non-generic}\}.$ |
$\bullet$ |
$\bullet$ |
|
|
globally generic |
if $\pi\in L^2$ |
$L(s,\mu_1)L(s,\mu_2)$ |
$L(s,\mu_1\times\mu_2)L(s,\chi)$ |
| (Q) |
Klingen packets, Soudry type |
$\mu\boxtimes\nu(2)$ |
$\mu$ is a $\chi$-self-dual, unitary, cuspidal, automorphic representation of $\GL(2,\A)$ with central character $\omega_\mu\neq\chi$. ($\mu$ is then of orthogonal type.) |
$\displaystyle\bigoplus_{\psi\in\mathbf{(Q)}}\;\bigoplus_{\pi\in\Pi_\psi}\pi$ |
none |
|
|
$\bullet$ |
|
Langlands quotient of $|\cdot|\omega_\mu^{-1}\chi\rtimes|\cdot|^{-1/2}\mu$ |
if $\pi\neq\pi_\psi$ |
$L(s+\frac12,\mu)L(s-\frac12,\mu)$ |
$\displaystyle L(s+1,\omega_\mu)L(s-1,\omega_\mu)\frac{L(s,\mu\times\mu)}{L(s,\chi)}$ |
| (P) |
Siegel packets, Saito-Kurokawa type |
$(\mu\boxtimes\nu(1))\boxplus(\sigma\boxtimes\nu(2))$ |
$\mu$ is a unitary, cuspidal, automorphic representation of $\GL(2,\A)$ with central character $\omega_\mu=\chi$, and $\sigma$ is a Hecke character with $\sigma^2=\chi$. |
$\displaystyle\bigoplus_{\psi\in\mathbf{(P)}}\;\bigoplus_{\{\pi\in\Pi_\psi:\:\langle\cdot,\pi\rangle=\varepsilon(1/2,\sigma^{-1}\mu)\}}\pi$ |
$(-1)^n=\varepsilon(1/2,\sigma^{-1}\mu),$ where $n=\#\{v\,|\,\pi_v$ is not the base point in $\Pi_{\psi_v}\}.$ |
|
$\bullet$ |
|
$\bullet$ |
Langlands quotient of $|\cdot|^{1/2}\sigma^{-1}\mu\rtimes\sigma|\cdot|^{-1/2}$ |
if ($\pi\in L^2$ and $\pi\neq\pi_\psi$) or if ($\pi=\pi_\psi$ and $\varepsilon(1/2,\sigma^{-1}\mu)=1$ and $L(1/2,\sigma^{-1}\mu)=0$) |
$L(s,\mu)L(s+\frac12,\sigma)L(s-\frac12,\sigma)$ |
$L(s+\frac12,\sigma\mu)L(s-\frac12,\sigma\mu)L(s,\chi)$ |
| (B) |
Borel packets, Howe - Piatetski-Shapiro type |
$(\sigma_1\boxtimes\nu(2))\boxplus(\sigma_2\boxtimes\nu(2))$ |
$\sigma_1$ and $\sigma_2$ are distinct unitary Hecke characters with $\sigma_1^2=\sigma_2^2=\chi$. |
$\displaystyle\bigoplus_{\psi\in\mathbf{(B)}}\;\bigoplus_{\{\pi\in\Pi_\psi:\:\langle\cdot,\pi\rangle=1\}}\pi$ |
$(-1)^n=1,$ where $n=\#\{v\,|\,\pi_v$ is not the base point in $\Pi_{\psi_v}\}.$ |
|
|
|
|
Langlands quotient of $|\cdot|\sigma_1\sigma_2^{-1}\times\sigma_1\sigma_2^{-1}\rtimes|\cdot|^{-1/2}\sigma_2$ |
if $\pi\in L^2$ and $\pi\neq\pi_\psi$ |
$L(s+\frac12,\sigma_1)L(s-\frac12,\sigma_1)L(s+\frac12,\sigma_2)L(s-\frac12,\sigma_2)$ |
$L(s+1,\sigma_1\sigma_2)L(s,\sigma_1\sigma_2)^2L(s-1,\sigma_1\sigma_2)L(s,\chi)$ |
| (F) |
finite-dimensional |
$\sigma\boxtimes\nu(4)$ |
$\sigma$ is a unitary Hecke character with $\sigma^2=\chi$. |
$\displaystyle\bigoplus_{\psi\in\mathbf{(F)}}\;\sigma$ |
none |
|
$\bullet$ |
$\bullet$ |
|
Langlands quotient of $|\cdot|^2\times|\cdot|\rtimes\sigma|\cdot|^{-3/2}$ |
never |
$L(s+\frac32,\sigma)L(s+\frac12,\sigma)L(s-\frac12,\sigma)L(s-\frac32,\sigma)$ |
$L(s+2,\chi)L(s+1,\chi)L(s,\chi)L(s-1,\chi)L(s-2,\chi)$ |