Work/BackwardsC3G2

This is a discussion page, related to the project Backwards lifting from $C_3$ to $G_2$:

=A mistake?=

A space of "Shalika periods" is obtained from the minimal representation of $E_8$, as described here:
 * Begin with the minimal representation $\Pi_8$ of $E_8$.
 * Restrict it to a representation of $F_4 \times G_2$.
 * Consider the Whittaker functor for $(N_2, \psi)$, where $N_2$ is the unipotent radical of a Borel subgroup of $G_2$.
 * Taking such Whittaker co-invariants of $\Pi_8$ yields a representation of $F_4$.
 * This representation of $F_4$ is essentially $Ind_{S_4}^{F_4} \psi_4$, where $S_4$ is the semidirect product of $G_2$ and $U_4$, and $\psi_4$ is the appropriate character of $U_4$. Call this the space of "Shalika periods" (in analogy with the space you consider in Section 3.1).

The is the following:
 * Suppose that $\sigma$ is a generic (supercuspidal) irreducible representation of PGSp_6.
 * Let $J(\sigma)$ be as defined in Lemma 4.1.
 * Let $\tau$ be a generic supercuspidal constituent of \Theta_7(\sigma).
 * Suppose that $J(\sigma) \boxtimes \tau$ occurs in $\Pi_8$, as claimed in Proposition 4.2.
 * Let $Wh(\tau)$ be the $(N_2, \psi)$-coinvariants (Whittaker) of $\tau$.
 * Then, by genericity of $\tau$, we find that $Wh(\tau)$ is nonzero.
 * We find that $J(\sigma) \boxtimes Wh(\tau)$ is a quotient of the space of Shalika periods, as a representation of $F_4$.

Thus, we obtain a surjective $F_4$-intertwining homomorphism: $$Ind_{S_4}^{F_4} \psi_4 \twoheadrightarrow J(\sigma).$$ By Frobenius reciprocity, this yields a nonzero $S_4 = G_2 U_4$-intertwining map: $$\psi_4 \rightarrow J(\sigma).$$ Considering the $(U_4, \psi_4)$-coinvariants yields a nonzero $G_2$-intertwining map: $$\CC \rightarrow J(\sigma)_{U_4, \psi_4}.$$