 
Summary: ON THE ADJOINT LFUNCTION OF THE pADIC GSp(4)
MAHDI ASGARI AND RALF SCHMIDT
Abstract. We explicitly compute the adjoint Lfunction of those Lpackets of represen
tations of the group GSp(4) over a padic field of characteristic zero that contain non
supercuspidal representations. As an application we verify a conjecture of GrossPrasad
and Rallis in this case. The conjecture states that the adjoint Lfunction is holomorphic
at s = 1 if and only if the Lpacket contains a generic representation.
1. Introduction
Let F be a nonarchimedean local field of characteristic zero and let WF be the Weil
Deligne group of F. The conjectural local Langlands correspondence for the group GSp(4, F)
assigns to each irreducible admissible representation of GSp(4, F) an Lparameter, i.e.,
an equivalence class of admissible representations
: WF  GSp(4, C).
It was shown in [RS, Sect. 2.4] that there is a unique way to assign Lparameters to the
nonsupercuspidal irreducible, admissible representations of GSp(4, F) such that certain
desired properties of the local Langlands correspondence hold. In this sense the local Lang
lands correspondence is known for the nonsupercuspidal representations of GSp(4, F); see
Table 1 for a complete list of these representations. In a few cases the Lpacket of a non
supercuspidal representation is expected to also contain a supercuspidal representation.
The degree 4 and degree 5 Lfactors resulting from the nonsupercuspidal local Langlands
