An exceptional Siegel–Weil formula and poles of the Spin L-function of

Abstract

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$

“…As π has trivial central character, this is equivalent to being tempered. The first part of the statement then follows from [18,Theorem 1.1]. We now discuss the formula for the residue of the Spin L-function.…”

“…As a special case of Langlands functoriality, one then expects that if L S (s, π, Spin) has a simple pole at s = 1, π is a functorial lift from either G 2 or G c 2 , where recall that G c 2 denotes the form of G 2 which is compact at ∞ and split at all finite places of Q. We invite the reader to consult [21], [17], [48], and [18] for results in this direction. In particular, using results of [11], [18] and [41], one deduces the following.…”

“…We invite the reader to consult [21], [17], [48], and [18] for results in this direction. In particular, using results of [11], [18] and [41], one deduces the following. Proof.…”

“…Proof. By the description of the action in (18) and the linear independence of the coordinates (x, y, z) of the representative of the open orbit, one sees that U 0 acts freely on it. Hence, the result follows from showing that any element in U H acts on the triple (x, y, z) as an element of U 0 and vice versa.…”

The Local Langlands Conjecture for

“…Then by the Arthur multiplicity formula for established by Xu, there exists a cuspidal automorphic representation satisfying: for all or ; . In addition, one has an equality of partial Spin L-functions: In particular, since is nonzero at (as is associated with a generic A-parameter), we see that has a pole at . By [GS20, Thm. 1.1], has nonzero global theta lift to a form of over k .…”