Paper: Furusawa Theta lift from SO(2n+1) to MP(2n)

##  Main theorem 
![](https://leanote.com/api/file/getImage?fileId=5d7220d0ab64412991006582)

The condition that $\pi$ has a Bessel model of that type is equvalent to $L(1/2,\pi)\neq 0$.

Question: 
1. Rallis inner product formula ? 
 
2. ![](https://leanote.com/api/file/getImage?fileId=5d7220d0ab64412991006586)

**The principle** 
A period $\mathcal P_1$ of tne theta lift $\Theta(\pi)$ of a cuspidal representation $\pi$ is related to a corresponding period $\mathcal P_2$ on $\pi$
## $SO(2n+1)$to $MP(2n)$
1. Global theorem: 
The Whittaker period of $\Theta(\pi)$ is related to the Bessel period of $\pi$. 
![](https://leanote.com/api/file/getImage?fileId=5d7220d0ab64412991006585)
2. Local theorem 
The local analog is to calculate the whittaker model for $\Theta(\pi)$. 
Lemma: $(\omega_{V,W,\psi})_{N_W,\psi_a}\cong ?$ Bessel model

For proving this, using the model $S(V\otimes Y^{\vee})$, where the unipotent radical of $Mp$ acting by characters.

Know case, for $SO(2,1)$ to $MP(2)$
$$(\omega_{V,W,\psi})_{N_V,\psi_a}\cong ind_{Q(V_a)}^{O(V)} 1$$

Question: Also have three cases $n<m. n=m, n>m $ Wee teck know 
## $MP(2n)$ to $SO(2n+1)$

1. Global theorem 
The Whittaker period of $\Theta(\sigma)$ is related to the Whittaker period of $\sigma$.
![](https://leanote.com/api/file/getImage?fileId=5d7220d0ab64412991006583)
![](https://leanote.com/api/file/getImage?fileId=5d7220d0ab64412991006584)
2. Local theorem: 
When $n<m$, $$(\omega_{V,W,\psi_{\lambda}})_{N_V,\psi}=0$$
When $m=n$
 $$(\omega_{V,W,\psi})_{N_V,\psi_{\lambda}}\cong ind_{N_W}^{Mp(W)} \psi_{\lambda}$$
For any $\lambda$
When $n>m$
 $$(\omega_{V,W,\psi})_{N_W,\psi_{\lambda}}\cong Fourier-jacobi \,\,model ??$$
Remarks: 
(1): For proving this, use the model $S(X^{\vee}\otimes W+e\otimes Y^{\vee})$, where the unipotent radical of $SO(V)$ acting by characters. 
3. Corollary. When $m=n$
For $\sigma \in Irr(Mp(W))$,
$$
Hom_{\mathbb C}(ind_{N_W}^{Mp(W)}\psi_{\lambda}, 1)\cong Hom_{N_{V}}(\omega_{V,W,\psi},\psi_{\lambda})
$$
implies 
$$
Hom_{Mp(W)}(ind_{N_W}^{Mp(W)}\psi_{\lambda}, \sigma)\cong Hom_{N_{V}}(\Theta(\sigma),\psi_{\lambda})
$$
and 
$$
Hom_{Mp(W)}(ind_{N_W}^{Mp(W)}\psi_{\lambda}, \sigma)\cong Hom_{Mp(W)}(\sigma^{\vee}, Ind_{N_W}^{Mp(W)}\psi_{\lambda}^{-1})\cong Hom_{N_W}(\sigma^{\vee},\psi_{\lambda}^{-1})\cong Hom_{N_W}(\sigma,\psi_{\lambda}^{-1})
$$
 
Inverse problem, does not fit the theorem in Gan&Savin 
so we have 
$$
Hom_{N_{V}}(\Theta(\sigma),\psi_{\lambda})\cong Hom_{N_W}(\sigma,\psi_{\lambda}^{-1})
$$
(1): If $\sigma$ is $\psi_{\lambda}$ generic, then $\Theta(\sigma)$ is $\psi_{\lambda}$ generic. So if $\Theta(\sigma)=\theta(\sigma)$, then $\theta_{\sigma}$ is generic. In particular, if $\sigma$ is temepered, then $\theta_{\sigma}$ is generic. 
(2): If $\pi\in Irr(SO(V^{+})$ is $\psi_{\lambda}$ generic, and if $\theta(\pi)\neq 0$, then $\sigma=\theta(\pi)=\Theta(\pi)$ is $\psi_{\lambda}$ generic. 
Prove: Let $\sigma=\theta(\pi)\neq 0$, then $\Theta(\sigma)$ is $\psi_{\lambda}$ generic as $\pi=\theta(\sigma)$ is generic, so $\sigma$ is $\psi_{\lambda}$ generic since
$$
Hom_{N_{V}}(\Theta(\sigma),\psi_{\lambda})\cong Hom_{N_W}(\sigma,\psi_{\lambda})
$$
$\Theta(\sigma)=\theta(\sigma)$ follows from standard module conjecture and for $\sigma$ tempered, $\Theta(\sigma)=\theta(\sigma)$. 
Say $$\sigma=\tau_1|det|^{s_1}\times \cdots \times \tau_{r}|det|^{s_r}\times \sigma_{0} \quad s_1>s_2\cdots, s_r>0$$ then 
$$\theta(\sigma)= \tau_1|det|^{s_1}\times \cdots \times \tau_{r}|det|^{s_r}\times \theta(\sigma_{0})$$, Since $\pi=\theta(\tau)$ is generic, by 
standard module conjecture, $I(\tau_1|det|^{s_1}\times \cdots \times \tau_{r}|det|^{s_r}\times \theta(\sigma_{0}))$ is irreducible and $$\Theta(\sigma)=\theta(\sigma)=I(\tau_1|det|^{s_1}\times \cdots \times \tau_{r}|det|^{s_r}\times \theta(\sigma_{0}))$$
Similarly, $\Theta(\pi)=\theta(\pi)$
Remark: 
(1) In (1), it is not true that $\theta(\sigma)$ is $\psi_{\lambda}$ generic, examples: $n=1, \sigma=\omega_{\psi}^{e}$ be the even Weil representation of $Mp(W)$, then $\theta(\sigma)$ is the trivial representation of $SO(V^{+})$, which is not generic. 
(2): As there is only one orbit for generic characters of $SO(V)$, we have if If $\pi\in Irr(SO(V^{+})$ is $\psi$ generic, then $\sigma=\theta(\pi)=\Theta(\pi)$ is $\psi_{\lambda}$ generic for any $\lambda\in F^{*}$.

## Back to global 
Let $\Pi$ be an automorphic cuspidal generic representation of $SO(2n+1)$, then 
(1) $\Theta_{V,W_{2k},\psi}(\Pi)=0$ for $k<n$. This is by local ? 
(2) If $\Sigma= \Theta_{V,W_{2n},\psi}(\Pi)\neq 0$, then $\Sigma$ is $\psi_{\lambda}$ generic for any $\lambda\in F^{*}$. This is because $\Theta(\Sigma)=\Pi$ is generic for any $\psi_{\lambda}$. So $\Pi$ has the Bessel model of the type $(R_{\lambda},1,\psi)$ for any $\lambda$, in particular it has the Bessel model of type $(R_{1},1,\psi)$, which is equivalent to $L(1/2,\Pi)\neq 0$. Converse, if $L(1/2,\Pi)\neq 0$, then $\Pi$ has the Bessel model of type $(R_{1},1,\psi)$, hence $\Sigma$ is $\psi_{1}$ generic, so $\Sigma\neq 0$.

Corollary: Let $\Pi$ be automorphic cuspidal generic representation of $SO(2n+1)$, If $\Pi$ has a Bessel model of special type $(R_{\lambda},1,\psi)$ for some $\lambda \in F^{*}$, then $\Pi$ has the Bessel model of special type $(R_{\lambda},1,\psi)$ for any $\lambda$

Remark
1. Because we do not know the back and forth of a automorphic representation $\pi$ is itself, we only know that this is not orthogonal to $\pi$, so the above assumption is not enough. See the oringanal paper for the assupmtions. 
2. The paper define the global theta as 
$$
\theta_{\psi,V,W}: \omega_{\psi,V,W}\otimes  \pi\longrightarrow \Theta_{\psi,V,W}(\pi)
$$
So the $\Theta_{\psi,V,W}(\pi)=\Theta_{\psi,V,W}(\pi^{\vee})$ in our setting, so he will have some $\psi^{-1}$ appear.

Mixiu

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