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-dimensional (in fact, one-dimensional) representations. These correspond to Siegel-Eisenstein series.

(Q), (P), (B) - representations induced from cuspidal representations of the three proper parabolic subgroups (Klingen, Siegel and Borel respectively). For example, Klingen-Eisenstein series are of type (Q) and Saito-Kurokawa lifts are of type (P).

(Y) - Yoshida type, the irreducible quotient of the tensor product of two distinct cuspidal automorphic representations of ${\rm GL}_2$, corresponding to Yoshida lifts.

(G) - General type, which contains the remaining representations. These are characterized by admitting a functorial transfer to a cuspidal automorphic representation of ${\rm GL}_4$