Toggle rows/columns by clicking on the red/green fields. Rows Columns G type finite Y name stable Q Arthur parameter $\psi$ paramodular P Arthur parameter ingredients global base point $\pi_\psi$ B contribution to $L^2_{\rm disc}(G(F)\backslash G(\A),\chi)$ cuspidality condition for $\pi\in\Pi_\psi$ F parity condition spin L-function $L(s,\pi_\psi,\rho_4)$ tempered standard L-function $L(s,\pi_\psi,\chi,\rho_5)$

## Arthur packets for GSp(4)

Decomposition of the space $L^2_{\rm disc}(G(F)\backslash G(\A),\chi)$ for $G=\GSp(4)$, where $F$ is a number field, $\A$ its ring of adeles, and $\chi$ is the central character.
Grab and drag elements of top row to change column order.
type name Arthur parameter ingredients contribution to $L^2_{\rm disc}$ parity condition tempered finite stable para global base point $\pi\in\Pi_\psi$ cuspidal spin L-function standard L-function
(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 $L(s+\frac32,\sigma)L(s+\frac12,\sigma)L(s-\frac12,\sigma)L(s-\frac32,\sigma)$ $L(s+2,\chi)L(s+1,\chi)L(s,\chi)L(s-1,\chi)L(s-2,\chi)$