Summary: ON THE ADJOINT L-FUNCTION OF THE p-ADIC GSp(4)
MAHDI ASGARI AND RALF SCHMIDT
Abstract. We explicitly compute the adjoint L-function of those L-packets of represen-
tations of the group GSp(4) over a p-adic field of characteristic zero that contain non-
supercuspidal representations. As an application we verify a conjecture of Gross-Prasad
and Rallis in this case. The conjecture states that the adjoint L-function is holomorphic
at s = 1 if and only if the L-packet contains a generic representation.
Let F be a non-archimedean 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 L-parameter, 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 L-parameters to the
non-supercuspidal 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 non-supercuspidal representations of GSp(4, F); see
Table 1 for a complete list of these representations. In a few cases the L-packet of a non-
supercuspidal representation is expected to also contain a supercuspidal representation.
The degree 4 and degree 5 L-factors resulting from the non-supercuspidal local Langlands