The local Langlands conjecture for GSp(4)

Abstract

We prove the local Langlands conjecture for GSp 4 (F ) where F is a nonarchimedean local field of characteristic zero.

The Local Langlands Conjecture for Sp(4)

“…The hypotheses (H1) and (H2) are contained in the Main Theorem of [GT1]. For the hypothesis (H3), note that (H2) already implies that I(π) ⊂ I(φ) if π ∈ L φ .…”

“…Moreover, we already know from [GT1] that A φ naturally indexes the elements in the L-packet L φ (which has size 1 or 2). Thus, it is completely natural to insist that for a character η of A φ , the constituents of π η | Sp 4 correspond to those elements of A ϕ which restrict to η.…”

“…For the rest of this section, we assume the following working hypotheses, which we have verified for the case n = 2 in [GT1].…”

“…For each ̟ ∈ Π(Sp 4 ) one can find π ∈ Π(GSp 4 ) such that the restriction π| Sp 4 contains ̟. If φ : W D F → GSp 4 (C) is the L-parameter of π attached by [GT1], then the composite ϕ := std(φ) of φ with the projection std gives the L-parameter for ̟. After checking that this is well-defined, the main point is to verify that the fiber L ϕ of L over ϕ has size #A ϕ .…”

“…Here, note that the left vertical arrow is not a map at all: it is a correspondence defined by the subset of Π(GSp 4 ) × Π(Sp 4 ) consisting of pairs (π, ̟) such that ̟ is a constituent of the restriction of π to Sp 4 . Moreover, the map L is given by the local Langlands correspondence for GSp 4 given in [GT1,Main Theorem] and the right vertical arrow is defined by composition with the natural projection std : GSp 4 (C) → PGSp 4 (C) ∼ = SO 5 (C), which gives the standard representation of GSp 4 (C).…”