Automorphic type (awaiting review)

The automorphic type of a Siegel newform $f \in S_{k,j}(\Gamma)$ of degree 2 is a letter indicating the type of the Arthur parameters of the automorphic representation $\pi$ associated to $f$. These fall into six classes of (packets of ) representations:

(F) - Finite type: (in fact, one-dimensional) representations.

(P) - Saito-Kurokawa type: Their $L$-functions are of the form $L(s,\mu)L(s+1/2,\sigma)L(s-1/2,\sigma)$, where $\mu$ is a cusp form on $\GL(2)$ and $\sigma$ is a Hecke character. Saito-Kurokawa lifts are of type (P).

(Q) - Soudry type (also known as Klingen packets): Their $L$-functions are of the form $L(s+1/2,\mu)L(s-1/2,\mu)$, where $\mu$ is a cusp form on $\GL(2)$.

(B) - Howe-Piatetski-Shapiro type (also known as Borel packets): Their $L$-functions are of the form $L(s+1/2,\sigma_1)L(s-1/2,\sigma_1)L(s+1/2,\sigma_2)L(s-1/2,\sigma_2)$, where $\sigma_1,\sigma_2$ are Hecke characters.

(Y) - Yoshida type: These are functorial (endoscopic) lifts of pairs of automorphic representations $\mu_1,\mu_2$ of $\GL(2)$. Their $L$-functions are of the form $L(s,\mu_1)L(s,\mu_2)$.

(G) - General type: (which contains the remaining representations) These are characterized by admitting a functorial transfer to a cuspidal automorphic representation of $\GL_4$.Their $L$-functions are primitive.