$\displaystyle:=\mathbb{E}_{\bm{X}}\left[\left(\frac{\mu_{1}^{2}(\bm{X})+\sigma_{1}^{2}(\bm{X})}{e(\bm{X})}+\frac{\mu_{0}^{2}(\bm{X})+\sigma_{0}^{2}(\bm{X})}{1-e(\bm{X})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)\left(\hat{\pi}_{t_{2}}(\bm{X})-\hat{\pi}_{t_{2}-1}(\bm{X})\right)\right],$
$\displaystyle B_{t}$
$\displaystyle:=\mathbb{E}_{\bm{X}}\left[\tau(\bm{X})\left(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})\right)\right]=\Delta(\hat{\pi}_{t};\hat{\pi}_{t-1}).$

Recall $\mu_{1}(\bm{x})$ and $\mu_{0}(\bm{x})$ are the conditional means of
$Y(1)$ and $Y(0)$ given $\bm{X}=\bm{x}$; $\sigma_{1}^{2}(\bm{x})$ and
$\sigma_{0}^{2}(\bm{x})$ are the conditional variances of $Y(1)$ and $Y(0)$
given $\bm{X}=\bm{x}$. Then, we have:

  1. 1.

For any $1\leq t_{1}<t_{2}\leq T$,

$\displaystyle\mathbb{E}\left[\widehat{\Gamma}_{t_{1}t_{2}}\mid\mathcal{H}_{t_{1}}\right]$
$\displaystyle=B_{t_{1}},$ (S17)
$\displaystyle\mathbb{V}\left(\widehat{\Gamma}_{t_{1}t_{2}}\mid\mathcal{H}_{t_{1}}\right)$
$\displaystyle=\frac{1}{B}\left(A_{t_{1}t_{1}}-B_{t_{1}}^{2}\right).$ (S18)

  2. 2.

For any $2\leq j\leq T$,

$\displaystyle\mathbb{E}\left[\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right]$
$\displaystyle=\sum_{t=1}^{j-1}\frac{B_{t}}{T-t},$ (S19)
$\displaystyle\mathbb{V}\left(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right)$
$\displaystyle=\frac{1}{B}\sum_{t_{1}=1}^{j-1}\sum_{t_{2}=1}^{j-1}\frac{A_{t_{1}t_{2}}}{(T-t_{1})(T-t_{2})}-\frac{1}{B}\left(\sum_{t=1}^{j-1}\frac{B_{t}}{T-t}\right)^{2}.$
(S20)

###### Proof.

We first prove Equations (S17) and (S18). Since $\hat{\pi}_{t_{1}}$ and
$\hat{\pi}_{t_{1}-1}$ are known functions conditional on
$\mathcal{H}_{t_{1}}$, and
$\\{(\bm{X}_{i},D_{i},Y_{i})\\}_{i\in\mathcal{B}_{t_{2}}}$ are i.i.d random
variables, we can use the unbiasedness of the IPW estimator and show:

$\displaystyle\mathbb{E}\left[\widehat{\Gamma}_{t_{1}t_{2}}\mid\mathcal{H}_{t_{1}}\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{B}\sum_{i\in\mathcal{B}_{t_{2}}}\left(\frac{Y_{i}D_{i}}{e(\bm{X}_{i})}-\frac{Y_{i}(1-D_{i})}{1-e(\bm{X}_{i})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X}_{i})-\hat{\pi}_{t_{1}-1}(\bm{X}_{i})\right)\
\Biggr{|}\ \mathcal{H}_{t_{1}}\right]$
$\displaystyle=\mathbb{E}_{\bm{Z}}\left[\left(\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)\right]$
$\displaystyle=\mathbb{E}_{\bm{X}}\left[\tau(\bm{X})(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right]$
$\displaystyle=B_{t_{1}},$ (S21)

where $\bm{Z}=\\{\bm{X},D,Y\\}$. This proves Equation (S17). Similarly, for
the variance, we have:

$\displaystyle\mathbb{V}\left(\widehat{\Gamma}_{t_{1}t_{2}}\mid\mathcal{H}_{t_{1}}\right)$
$\displaystyle=\mathbb{V}\left(\frac{1}{B}\sum_{i\in\mathcal{B}_{t_{2}}}\left(\frac{Y_{i}D_{i}}{e(\bm{X}_{i})}-\frac{Y_{i}(1-D_{i})}{1-e(\bm{X}_{i})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X}_{i})-\hat{\pi}_{t_{1}-1}(\bm{X}_{i})\right)\
\biggr{|}\ \mathcal{H}_{t_{1}}\right)$
$\displaystyle=\frac{1}{B}\mathbb{V}_{\bm{Z}}\left(\left(\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)\right)$
$\displaystyle=\frac{1}{B}\mathbb{E}_{\bm{Z}}\left[\left(\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right)^{2}\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)^{2}\right]$
$\displaystyle\ \
-\frac{1}{B}\left\\{\mathbb{E}_{\bm{Z}}\left[\left(\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)\right]\right\\}^{2}.$

Again, a similar calculation yields the following:

$\displaystyle\mathbb{E}_{\bm{Z}}\left[\left(\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right)^{2}\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\bm{Z}}\left[\left\\{\left(\frac{YD}{e(\bm{X})}\right)^{2}+\left(\frac{Y(1-D)}{1-e(\bm{X})}\right)^{2}\right\\}\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\bm{X}}\left[\left(\frac{\mu_{1}(\bm{X})^{2}+\sigma_{1}^{2}(\bm{X})}{e(\bm{X})}+\frac{\mu_{0}(\bm{X})^{2}+\sigma_{0}^{2}(\bm{X})}{1-e(\bm{X})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)^{2}\right]$
$\displaystyle=$ $\displaystyle A_{t_{1}t_{1}}.$

Together with Equation (S21), we have the desired result given in Equation
(S18).

Next, we prove Equations (S19) and (S20). Using the same analytical strategy
as above, we obtain:

$\displaystyle\mathbb{E}\left[\widehat{\Gamma}_{tj}\mid\mathcal{H}_{j-1}\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{B}\sum_{i\in\mathcal{B}_{j}}\left(\frac{Y_{i}D_{i}}{e(\bm{X}_{i})}-\frac{Y_{i}(1-D_{i})}{1-e(\bm{X}_{i})}\right)\left(\hat{\pi}_{t}(\bm{X}_{i})-\hat{\pi}_{t-1}(\bm{X}_{i})\right)\
\Biggr{|}\ \mathcal{H}_{j-1}\right]$
$\displaystyle=\mathbb{E}_{\bm{Z}}\left[\left(\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right)\left(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})\right)\right]$
$\displaystyle=B_{t}.$

Together with the definition of $\widehat{\Gamma}_{j}(T)$, this proves
Equation (S19).

Finally, we have:

$\displaystyle\mathbb{V}\left(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right)\
=\ $
$\displaystyle\mathbb{V}\left(\sum_{t=1}^{j-1}\frac{1}{T-t}\widehat{\Gamma}_{tj}\
\Bigr{|}\ \mathcal{H}_{j-1}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{B}\mathbb{V}_{\bm{Z}}\left(\sum_{t=1}^{j-1}\left\\{\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right\\}\frac{(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X}))}{T-t}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{B}\mathbb{E}_{\bm{X}}\left[\mathbb{V}_{D,Y}\left(\sum_{t=1}^{j-1}\left\\{\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right\\}\frac{(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X}))}{T-t}\
\Bigr{|}\ \bm{X}\right)\right]$
$\displaystyle+\frac{1}{B}\mathbb{V}_{\bm{X}}\left(\mathbb{E}_{D,Y}\left[\sum_{t=1}^{j-1}\left\\{\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right\\}\frac{(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X}))}{T-t}\
\Bigr{|}\ \bm{X}\right]\right).$ (S22)

The standard calculation of the expectation and variance of an IPW estimator
shows:

$\displaystyle\mathbb{E}_{D,Y}\left[\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\
\middle|\ \bm{X}\right]$ $\displaystyle=\tau(\bm{X})$
$\displaystyle\mathbb{V}_{D,Y}\left(\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\
\middle|\bm{X}\right)$
$\displaystyle=\frac{\mu_{1}(\bm{X})^{2}+\sigma_{1}^{2}(\bm{X})}{e(\bm{X})}+\frac{\mu_{0}(\bm{X})^{2}+\sigma_{0}^{2}(\bm{X})}{1-e(\bm{X})}-\tau(\bm{X})^{2}.$

Plugging in these two equations into Equation (S22) yields the desired
variance expression:

$\displaystyle\mathbb{V}\left(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{B}\mathbb{E}_{\bm{X}}\left[\left(\sum_{t=1}^{j-1}\frac{(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X}))}{T-t}\right)^{2}\left(\frac{\mu_{1}(\bm{X})^{2}+\sigma_{1}^{2}(\bm{X})}{e(\bm{X})}+\frac{\mu_{0}(\bm{X})^{2}+\sigma_{0}^{2}(\bm{X})}{1-e(\bm{X})}-\tau(\bm{X})^{2}\right)\right]$
$\displaystyle\ \
+\frac{1}{B}\mathbb{V}_{\bm{X}}\left(\tau(\bm{X})\sum_{t=1}^{j-1}\frac{(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X}))}{T-t}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{B}\mathbb{E}_{\bm{X}}\left[\left(\sum_{t=1}^{j-1}\frac{(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X}))}{T-t}\right)^{2}\left(\frac{\mu_{1}(\bm{X})^{2}+\sigma_{1}^{2}(\bm{X})}{e(\bm{X})}+\frac{\mu_{0}(\bm{X})^{2}+\sigma_{0}^{2}(\bm{X})}{1-e(\bm{X})}\right)\right]$
$\displaystyle\ \
-\frac{1}{B}\mathbb{E}_{\bm{X}}\left[\tau(\bm{X})\sum_{t=1}^{j-1}\frac{(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X}))}{T-t}\right]^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{B}\sum_{t_{1}=1}^{j-1}\sum_{t_{2}=1}^{j-1}\frac{1}{(T-t_{1})(T-t_{2})}A_{t_{1}t_{2}}-\frac{1}{B}\left(\sum_{t=1}^{j-1}\frac{B_{t}}{T-t}\right)^{2}.$

This completes proof. ∎

###  S7.6 Lemma 6

###### Lemma 6

Suppose that Assumption 1, 2, 4 and 5 hold. Then, there exist constants
$K_{A}$ and $K_{B}$ such that for any $t\geq 0$, the following inequalities
hold almost surely:

$\displaystyle t^{1+\delta}|A_{tt}|$ $\displaystyle\leq K_{A},$ (S23)
$\displaystyle t^{1+\delta}|B_{t}|$ $\displaystyle\leq K_{B},$ (S24)
$\displaystyle
t_{1}^{1+\delta}\sup_{t_{2}>t_{1}}\Bigr{|}\sum_{j=t_{1}+1}^{t_{2}}A_{t_{1}j}\Bigr{|}$
$\displaystyle\leq K_{A},$ (S25)

where $\delta$ is as defined in Assumption 2 and $A_{t_{1}t_{2}}$ and $B_{t}$
are defined in Lemma 5.

###### Corollary 2

As a direct result of Lemma 6 and the dominated convergence theorem, the
following equality hold almost surely:

$\displaystyle\lim_{t\rightarrow\infty}|B_{t}|t^{1+\delta/2}$ $\displaystyle\
=\ 0,$ $\displaystyle\lim_{t\rightarrow\infty}|A_{tt}|t^{1+\delta/2}$
$\displaystyle\ =\ 0,$
$\displaystyle\lim_{t_{1}\rightarrow\infty}\sup_{t_{2}>t_{1}}\Bigr{|}\sum_{j=t_{1}+1}^{t_{2}}A_{t_{1}j}\Bigr{|}t_{1}^{1+\delta/2}$
$\displaystyle\ =\ 0.$

Moreover, the following results follow:

$\displaystyle\lim_{t\rightarrow\infty}\mathbb{E}\left[|B_{t}|t^{1+\delta/2}\right]$
$\displaystyle\ =\ 0,$
$\displaystyle\lim_{t\rightarrow\infty}\mathbb{E}\left[|A_{tt}|t^{1+\delta/2}\right]$
$\displaystyle\ =\ 0,$
$\displaystyle\lim_{t_{1}\rightarrow\infty}\mathbb{E}\left[\sup_{t_{2}>t_{1}}\Bigr{|}\sum_{j=t_{1}+1}^{t_{2}}A_{t_{1}j}\Bigr{|}t_{1}^{1+\delta/2}\right]$
$\displaystyle\ =\ 0.$

Proof of Corollary 2 is nearly identical to that of Lemma 1, and hence is
omitted. Now, we present a proof of Lemma 6.

###### Proof.

We first prove Equations (S23) and (S24). Using the definitions of $A_{tt}$
and $B_{t}$ in Lemma 5, we obtain:

$\displaystyle|A_{tt}|$ $\displaystyle\ \leq\
\sup_{\bm{x}\in\mathcal{X}}\left(\frac{\mu_{1}(\bm{x})^{2}+\sigma_{1}^{2}(\bm{x})}{e(\bm{x})}+\frac{\mu_{0}(\bm{x})^{2}+\sigma_{0}^{2}(\bm{x})}{1-e(\bm{x})}\right)\cdot\mathbb{E}_{\bm{X}}\left[\left(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})\right)^{2}\right],$
$\displaystyle|B_{t}|$ $\displaystyle\ \leq\
\sup_{\bm{x}\in\mathcal{X}}|\tau(\bm{x})|\cdot\mathbb{E}_{\bm{X}}\left[|\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})|\right].$

For any $\bm{x}\in\mathcal{X}$ and $t$, we have
$\hat{\pi}_{t}(\bm{x})-\hat{\pi}_{t-1}(\bm{x})\in[-1,1]$, implying
$\mathbb{E}_{\bm{X}}\left[\left(\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})\right)^{2}\right]\leq\mathbb{E}_{\bm{X}}\left[|\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})|\right]$.
Then,

$|A_{tt}|\ \leq\
\sup_{\bm{x}\in\mathcal{X}}\left(\frac{\mu_{1}(\bm{x})^{2}+\sigma_{1}^{2}(\bm{x})}{e(\bm{x})}+\frac{\mu_{0}(\bm{x})^{2}+\sigma_{0}^{2}(\bm{x})}{1-e(\bm{x})}\right)\cdot\mathbb{E}_{\bm{X}}\left[|\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})|\right].$

Assumption 1 and 4 imply:

$\sup_{\bm{x}\in\mathcal{X}}|\tau(\bm{x}^{\prime})|<\infty,\quad\sup_{\bm{x}\in\mathcal{X}}\left(\frac{\mu_{1}(\bm{x})^{2}+\sigma_{1}^{2}(\bm{x})}{e(\bm{x})}+\frac{\mu_{0}(\bm{x})^{2}+\sigma_{0}^{2}(\bm{x})}{1-e(\bm{x})}\right)<\infty.$

By Assumption 2, there exists a constant $R_{1},K_{0},\delta$ such that for
any $t>R_{1}$,
$\mathbb{E}_{\bm{X}}\left[|\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})|\right]t^{1+\delta}\leq
K_{0}$ holds almost surely. Therefore, let

$\displaystyle K_{A}:=$
$\displaystyle\sup_{\bm{x}\in\mathcal{X}}\left(\frac{\mu_{1}(\bm{x})^{2}+\sigma_{1}^{2}(\bm{x})}{e(\bm{x})}+\frac{\mu_{0}(\bm{x})^{2}+\sigma_{0}^{2}(\bm{x})}{1-e(\bm{x})}\right)\cdot\max\left(K_{0},R_{1}^{1+\delta}\right)$
$\displaystyle K_{B}:=$
$\displaystyle\sup_{\bm{x}\in\mathcal{X}}|\tau(\bm{x}^{\prime})|\cdot\max\left(K_{0},R_{1}^{1+\delta}\right),$

then for any $t$, we have $t^{1+\delta}|A_{tt}|\leq K_{A}$ and
$t^{1+\delta}|B_{t}|\leq K_{B}$ almost surely, proving the desired results.

To prove Equation (S25), notice $\forall t_{2}\geq t_{1}+1$,

$\displaystyle\left|\sum_{j=t_{1}+1}^{t_{2}}A_{t_{1}j}\right|$
$\displaystyle=\left|\mathbb{E}\left[\left(\frac{\mu_{1}(\bm{X})^{2}+\sigma_{1}^{2}(\bm{X})}{e(\bm{X})}+\frac{\mu_{0}(\bm{X})^{2}+\sigma_{0}^{2}(\bm{X})}{1-e(\bm{X})}\right)\left(\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right)\left(\hat{\pi}_{t_{2}}(\bm{X})-\hat{\pi}_{t_{1}}(\bm{X})\right)\right]\right|$
$\displaystyle\leq\sup_{\bm{x}\in\mathcal{X}}\left(\frac{\mu_{1}(\bm{x})^{2}+\sigma_{1}^{2}(\bm{x})}{e(\bm{x})}+\frac{\mu_{0}(\bm{x})^{2}+\sigma_{0}^{2}(\bm{x})}{1-e(\bm{x})}\right)\mathbb{E}_{\bm{X}}\left[\left|\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right|\cdot\left|\hat{\pi}_{t_{2}}(\bm{X})-\hat{\pi}_{t_{1}}(\bm{X})\right|\right]$
$\displaystyle\leq\sup_{\bm{x}\in\mathcal{X}}\left(\frac{\mu_{1}(\bm{x})^{2}+\sigma_{1}^{2}(\bm{x})}{e(\bm{x})}+\frac{\mu_{0}(\bm{x})^{2}+\sigma_{0}^{2}(\bm{x})}{1-e(\bm{x})}\right)\mathbb{E}_{\bm{X}}\left[\left|\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right|\right].$
(S26)

Since Equation (S26) does not involve $t_{2}$, we have

$\displaystyle\sup_{t_{2}>t_{1}}\left|\sum_{j=t_{1}+1}^{t_{2}}A_{t_{1}j}\right|$
$\displaystyle\leq\sup_{\bm{x}\in\mathcal{X}}\left(\frac{\mu_{1}(\bm{x})^{2}+\sigma_{1}^{2}(\bm{x})}{e(\bm{x})}+\frac{\mu_{0}(\bm{x})^{2}+\sigma_{0}^{2}(\bm{x})}{1-e(\bm{x})}\right)\cdot\mathbb{E}_{\bm{X}}\left[\left|\hat{\pi}_{t_{1}}(\bm{X})-\hat{\pi}_{t_{1}-1}(\bm{X})\right|\right].$

Therefore,
$t_{1}^{1+\delta}\sup_{t_{2}>t_{1}}\Bigr{|}\sum_{j=t_{1}+1}^{t_{2}}A_{t_{1}j}\Bigr{|}\leq
K_{A}$ holds almost surely. ∎

###  S7.7 Lemma 7

###### Lemma 7

Under Assumptions 1–5, we have, for any $0<h<1$,

$\frac{1}{B}\left|T\sum_{j=2}^{\lfloor
hT\rfloor}\widehat{\mathbb{V}}(\hat{g}_{Tj})-T\sum_{j=2}^{\lfloor
hT\rfloor}\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right|\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0.$

###### Proof.

We will show the following stronger result:

$T\cdot\mathbb{E}\left[\left|\sum_{j=2}^{\lfloor
hT\rfloor}\widehat{\mathbb{V}}(\hat{g}_{Tj})-\sum_{j=2}^{\lfloor
hT\rfloor}\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right|\right]\longrightarrow
0\ \ \text{as}\ \ T\longrightarrow\infty,$

where the expectation is taken over all the observations $\mathcal{H}_{T}$. By
the triangle inequality and Jensen’s inequality, we have:

$\displaystyle T\cdot\mathbb{E}\left[\left|\sum_{j=2}^{\lfloor
hT\rfloor}\widehat{\mathbb{V}}(\hat{g}_{Tj})-\sum_{j=2}^{\lfloor
hT\rfloor}\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right|\right]$
$\displaystyle\leq T\sum_{j=1}^{\lfloor
hT\rfloor}\mathbb{E}\left[\left|\widehat{\mathbb{V}}(\hat{g}_{Tj})-\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right|\right]$
$\displaystyle\leq T\sum_{j=1}^{\lfloor
hT\rfloor}\sqrt{\mathbb{E}\left[\left(\widehat{\mathbb{V}}(\hat{g}_{Tj})-\mathbb{V}_{\mathbf{Z}}(\hat{g}_{Tj}(\bm{Z}))\right)^{2}\right]}$

By the law of iterated expectation, we have:

$\mathbb{E}\left[\left(\widehat{\mathbb{V}}(\hat{g}_{Tj})-\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right)^{2}\right]\
=\
\mathbb{E}\left[\mathbb{E}\left[\left(\widehat{\mathbb{V}}(\hat{g}_{Tj})-\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right)^{2}\
\mid\ \mathcal{H}_{j-1}\right]\right].$

Conditioning on $\mathcal{H}_{j-1}$, $\widehat{\mathbb{V}}(\hat{g}_{Tj})$ is
just the sample variance of $B(T-j+1)$ i.i.d observations, i.e.,

$\widehat{\mathbb{V}}(\hat{g}_{Tj}):=\frac{1}{B(T-j+1)-1}\sum_{k=j}^{T}\sum_{i\in\mathbb{B}_{k}}\left(\hat{g}_{Tj}(\bm{Z}_{i})-\bar{g}_{Tj}\right)^{2}.$

Thus,

$\mathbb{E}\left[\left(\widehat{\mathbb{V}}(\hat{g}_{Tj})-\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right)^{2}\
\biggr{|}\
\mathcal{H}_{j-1}\right]=\mathbb{V}\left(\widehat{\mathbb{V}}(\hat{g}_{Tj})\
\biggr{|}\ \mathcal{H}_{j-1}\right).$

The standard calculation for the variance of a sample variance yields:

$\displaystyle\mathbb{V}\left(\widehat{\mathbb{V}}(\hat{g}_{Tj})\ \Biggr{|}\
\mathcal{H}_{j-1}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{B(T-j+1)}\mathbb{E}_{\bm{Z}}\left[\left(\hat{g}_{Tj}(\bm{Z})-\mathbb{E}_{\bm{Z}}[\hat{g}_{Tj}(\bm{Z})]\right)^{4}\right]-\frac{B(T-j+1)-3}{B(T-j+1)\\{B(T-j+1)-1\\}}\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))^{2}$
$\displaystyle\leq$
$\displaystyle\frac{16}{B(T-j+1)}\mathbb{E}_{\bm{Z}}\left[\hat{g}_{Tj}(\bm{Z})^{4}\right]+\frac{2|B(T-j+1)-3|}{B(T-j+1)\\{B(T-j+1)-1\\}}\mathbb{E}_{\bm{Z}}\left[\hat{g}_{Tj}(\bm{Z})^{4}\right].$
$\displaystyle\leq$
$\displaystyle\frac{16}{B(T-j+1)}\mathbb{E}_{\bm{Z}}\left[\hat{g}_{Tj}(\bm{Z})^{4}\right]+\frac{2}{B(T-j+1)}\mathbb{E}_{\bm{Z}}\left[\hat{g}_{Tj}(\bm{Z})^{4}\right]$
$\displaystyle=$
$\displaystyle\frac{18}{B(T-j+1)}\mathbb{E}_{\bm{Z}}\left[\hat{g}_{Tj}(\bm{Z})^{4}\right].$

The inequality for the first term follows from the fact that for any random
variable $X$, we have:

$\mathbb{E}[|X-\mathbb{E}[X]|^{4}]=\mathbb{E}[(|X|+|\mathbb{E}[X]|)^{4}]\leq\mathbb{E}[(2|X|)^{4}]=\mathbb{E}[16|X|^{4}].$

Similarly, the second term follows from the following general inequality for
random variable $X$:

$\mathbb{V}(X)^{2}=(\mathbb{E}[X^{2}]-\mathbb{E}[X]^{2})^{2}\leq\mathbb{E}[X^{2}]^{2}+\mathbb{E}[X]^{4}\leq
2\mathbb{E}[X^{4}]$

Since $j\leq\lfloor hT\rfloor\leq hT$, we have $T-j+1\geq(1-h)T$. This
implies:

$\mathbb{V}(\widehat{\mathbb{V}}(\hat{g}_{Tj}))\leq\frac{18}{(1-h)BT}\mathbb{E}_{\bm{Z}}[\hat{g}_{Tj}(\bm{Z})^{4}].$

Applying Lemma 4 and the overlap assumption (Assumption 1), we have:

$\displaystyle|\hat{g}_{Tj}(\bm{Z})|$
$\displaystyle=\left|\frac{YD}{e(\bm{X})}-\frac{Y(1-D)}{1-e(\bm{X})}\right|\cdot\left|\sum_{t=1}^{j-1}\frac{\hat{\pi}_{t}(\bm{X})-\hat{\pi}_{t-1}(\bm{X})}{T-t}\right|$
$\displaystyle\leq\left|\frac{Y}{c}+\frac{Y}{c}\right|\cdot\frac{1}{T-j+1}$
$\displaystyle\leq\frac{2}{c}\frac{|Y|}{(1-h)T}$

where the last inequality is because of $j\leq\lfloor hT\rfloor$. Therefore,

$\mathbb{E}_{\bm{Z}}\left[\hat{g}_{Tj}(\bm{Z})^{4}\right]\leq\mathbb{E}\left[|Y|^{4}\left(\frac{2}{c}\right)^{4}\left(\frac{1}{(1-h)T}\right)^{4}\right]\leq\frac{16K_{4}}{c^{4}(1-h)^{4}T^{4}}$

where the second inequality utilizes the moment condition about the outcome
(Assumption 5). Therefore, we can bound the conditional variance as follows:

$\mathbb{V}(\widehat{\mathbb{V}}(\hat{g}_{Tj})\mid\mathcal{H}_{j-1})\leq\frac{288K_{4}}{c^{4}(1-h)^{5}BT^{5}}.$

This implies:

$T\sum_{j=1}^{\lfloor
hT\rfloor}\sqrt{\mathbb{E}\left[\left(\widehat{\mathbb{V}}(\hat{g}_{Tj})-\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right)^{2}\right]}\leq
T^{2}\sqrt{\frac{288K_{4}}{c^{4}(1-h)^{5}BT^{5}}}=\frac{\sqrt{288K_{4}/B}}{c^{2}(1-h)^{5/2}T^{1/2}}\longrightarrow
0.$

Therefore, as $T\rightarrow\infty$,

$0\leq T\cdot\mathbb{E}\left[\left|\sum_{j=1}^{\lfloor
hT\rfloor}\widehat{\mathbb{V}}(\hat{g}_{Tj})-\sum_{j=1}^{\lfloor
hT\rfloor}V(\hat{g}_{Tj})\right|\right]\leq T\sum_{j=1}^{\lfloor
hT\rfloor}\sqrt{\mathbb{E}\left[\left(\widehat{\mathbb{V}}(\hat{g}_{T,j})-\mathbb{V}_{\mathbf{Z}}(\hat{g}_{Tj}(\mathbf{Z}))\right)^{2}\right]}\longrightarrow
0.$

As $L_{1}$ convergence implies convergence in probability, we have:

$\frac{1}{B}\left|T\cdot\sum_{j=2}^{\lfloor
hT\rfloor}\widehat{\mathbb{V}}(\hat{g}_{Tj})-T\sum_{j=2}^{\lfloor
hT\rfloor}\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right|\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0.$

∎

###  S7.8 Lemma 8

###### Lemma 8

Under Assumptions 1–5, we have:

$\frac{T}{B}\sum_{j=\lfloor
hT\rfloor+1}^{T}{\mathbb{V}}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0,\quad\frac{T}{B}\sum_{j=\lfloor
hT\rfloor+1}^{T}\widehat{\mathbb{V}}(\hat{g}_{Tj})\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0.$

###### Proof.

By the law of iterated expectation, we have:

$0\leq\mathbb{E}\left[\sum_{j=\lfloor
hT\rfloor+1}^{T}\widehat{\mathbb{V}}(\hat{g}_{Tj})\right]=\mathbb{E}\left[\sum_{j=\lfloor
hT\rfloor+1}^{T}\mathbb{E}\left[\widehat{\mathbb{V}}(\hat{g}_{Tj})\mid\mathcal{H}_{j-1}\right]\right]\
\leq\ \mathbb{E}\left[\sum_{j=\lfloor
hT\rfloor+1}^{T}\mathbb{V}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right]$

where the last inequality follows from the fact that
$\widehat{\mathbb{V}}(\hat{g}_{T,j})$ is the sample variance of $B(T-j+1)$
i.i.d observations from the $j,j+1,\ldots,T$-th batches. For $j<T$ or $j=T$
and $B>1$, the sample variance is unbiased for the true variance as
$B(T-j+1)>1$. For $j=T$ and $B=1$, the expectation of sample variance is less
than the true variance:

$\mathbb{E}[\hat{V}(\hat{g}_{TT})\mid\mathcal{H}_{T-1}]=0\leq\mathbb{V}_{\bm{Z}}(\hat{g}_{TT}(\bm{Z})).$

To prove these two statements, we will show the following stronger result:

$\displaystyle\mathbb{E}\left[\frac{T}{B}\sum_{j=\lfloor
hT\rfloor+1}^{T}{\mathbb{V}}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))\right]\rightarrow
0\ \ \text{as}\ \ T\rightarrow\infty.$ (S27)

This $L_{1}$ convergence result implies the convergence in probability results
of this lemma.

We begin by noting the following equality:

$\frac{T}{B}\sum_{j=\lfloor
hT\rfloor+1}^{T}{\mathbb{V}}_{\bm{Z}}(\hat{g}_{Tj}(\bm{Z}))=T\sum_{j=\lfloor
hT\rfloor+1}^{T}\mathbb{V}(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}).$

By Lemma 5, we have:

$\displaystyle\mathbb{V}\left(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right)$
$\displaystyle=\frac{1}{B}\sum_{t_{1}=1}^{j-1}\sum_{t_{2}=1}^{j-1}\frac{A_{t_{1}t_{2}}}{(T-t_{1})(T-t_{2})}-\frac{1}{B}\left(\sum_{t=1}^{j-1}\frac{B_{t}}{T-t}\right)^{2}$
$\displaystyle=\frac{1}{B}\sum_{t=1}^{j-1}\frac{A_{tt}-B_{t}^{2}}{(T-t)^{2}}+\frac{1}{B}\sum_{1\leq
t_{1}<t_{2}\leq
j-1}\frac{A_{t_{1}t_{2}}-B_{t_{1}}B_{t_{2}}}{(T-t_{1})(T-t_{2})}.$

Thus,

$\displaystyle\sum_{j=2}^{T}\mathbb{V}\left(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right)$
$\displaystyle=\sum_{t=1}^{T-1}\frac{A_{tt}-B_{t}^{2}}{T-t}+\sum_{1\leq
t_{1}<t_{2}\leq T-1}\frac{2(A_{t_{1}t_{2}}-B_{t_{1}}B_{t_{2}})}{T-t_{1}}$
$\displaystyle\sum_{j=2}^{\lfloor
hT\rfloor}\mathbb{V}\left(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right)$
$\displaystyle=\sum_{t=1}^{\lfloor
hT\rfloor-1}\frac{A_{tt}-B_{t}^{2}}{T-t}+\sum_{1\leq t_{1}<t_{2}\leq\lfloor
hT\rfloor-1}\frac{2(A_{t_{1}t_{2}}-B_{t_{1}}B_{t_{2}})}{T-t_{1}}.$

Therefore,

$T\sum_{j=\lfloor
hT\rfloor+1}^{T}\mathbb{V}\left(\widehat{\Gamma}_{j}(T)\mid\mathcal{H}_{j-1}\right)=T\left\\{\sum_{t=\lfloor
hT\rfloor}^{T-1}\frac{A_{tt}-B_{t}^{2}}{T-t}+\sum_{t_{2}=\lfloor
hT\rfloor}^{T-1}\sum_{t_{1}=1}^{t_{2}-1}\frac{2(A_{t_{1}t_{2}}-B_{t_{1}}B_{t_{2}})}{T-t_{1}}\right\\}.$
(S28)

Lemma 6 implies $|A_{tt}-B_{t}^{2}|\leq(K_{A}+K_{B}^{2})t^{-1-\delta}$ almost
surely. Therefore, the first term of Equation (S28) can be shown to converge
to zero almost surely:

$\displaystyle T\sum_{t=\lfloor
hT\rfloor}^{T-1}\left|\frac{A_{tt}-B_{t}^{2}}{T-t}\right|$
$\displaystyle\leq(K_{A}+K_{B}^{2})\sum_{t=\lfloor
hT\rfloor}^{T-1}\frac{T}{(T-t)t^{1+\delta}}$
$\displaystyle\leq(K_{A}+K_{B}^{2})\left(\sum_{t=\lfloor
hT\rfloor}^{T-1}\frac{1}{t^{1+\delta}}+\sum_{t=\lfloor
hT\rfloor}^{T-1}\frac{1}{(T-t)t^{\delta}}\right)\longrightarrow
0\quad\text{almost surely},$

where the convergence result follows because each summation term in the
parenthese converges to zero by Lemma 3.

We similarly analyze the second term of Equation (S28). Equation (S4) in the
proof of Condition C2 shows that for $t_{2}>R_{1}$,

$\left|\sum_{t_{1}=1}^{t_{2}-1}\frac{A_{t_{1}t_{2}}-B_{t_{1}}B_{t_{2}}}{T-t_{1}}\right|\leq\frac{K_{B}\sup_{\bm{x}\in\mathcal{X}}|\tau(\bm{x})|+K_{A}K_{0}}{(T-t_{2}+1)t_{2}^{1+\delta}}\
\ \text{almost surely}.$

Thus, when $\lfloor hT\rfloor>R_{1}$, using Lemma 3, we can show:

$\displaystyle T\left|\sum_{t_{2}=\lfloor
hT\rfloor}^{T-1}\sum_{t_{1}=1}^{t_{2}-1}\frac{A_{t_{1}t_{2}}-B_{t_{1}}B_{t_{2}}}{T-t_{1}}\right|$
$\displaystyle\leq$ $\displaystyle\sum_{t=\lfloor
hT\rfloor}^{T-1}\frac{(K_{B}\sup_{\bm{x}\in\mathcal{X}}|\tau(\bm{x})|+K_{A}K_{0})T}{(T-t+1)t^{1+\delta}}$
$\displaystyle\leq$
$\displaystyle\left(K_{B}\sup_{\bm{x}\in\mathcal{X}}|\tau(\bm{x})|+K_{A}K_{0}\right)\left(\sum_{t=\lfloor
hT\rfloor}^{T-1}\frac{1}{t^{1+\delta}}+\sum_{t=\lfloor
hT\rfloor}^{T-1}\frac{1}{(T-t)t^{\delta}}\right)\longrightarrow$
$\displaystyle 0\ \ \text{almost surely}.$

Hence, the proof is complete. ∎

##  Appendix S8 Additional details and results of the simulation studies

###  S8.1 Implementation details

#### Simulation environment.

We conducted simulations on a machine with Intel i7-12700K CPU and Ubuntu
22.10 operating system. All the simulations were done with Python 3.9.13. For
the results based on ridge regressions and neural networks, we used an Nvidia
RTX 3090 GPU for computational efficiency.

#### Data generating processes.

We use the ACIC 2016 data challenge datasets (Dorie et al.,, 2019) to generate
simulated data. Under each of the 77 simulation setups, we first sample the
individual covariates from the empirical distribution of the covariates that
are part of the corresponding ACIC 2016 challenge dataset. We then randomly
assign each observation to either the treatment or control condition using
Bernoulli distribution with the success probability of 0.5. We then use the
package created by Dorie et al., (2019) to generate the potential outcomes for
each observation. Finally, the observed outcome is obtained based on the
treatment status and the outcome model. In the original ACIC 2016 datasets,
different covariates variables have varying scales. For simplicity, we
standardize each covariate so that it has zero mean and unit variance. We
generate a total of 3,000 Monte Carlo replicates for each simulation setting.

#### Implementation of policy learning algorithms.

Causal forest is implemented using the python package econml. We use the
default hyperparameters in the package and set the number of trees as $100$,
the maximum depth of each tree as $3$. The ridge regression and neural
networks are implemented using Pytorch, For ridge regression, we set the
hyperparameter of ridge penalty to 0.1.

For neural networks, we use a 3-layer fully connected neural network. Assuming
the input feature dimension is $p$, we set the number of neurons in the first
hidden layer as $p/4$ and the number of neurons in the second hidden layer as
$p/16$. We set the non-linear activation function as the sigmoid function. In
the training, we use a stochastic gradient descent (SGD) optimizer with
$L_{2}$ loss for 3000 epochs where a learning rate starts with 0.01 and decays
exponentially in each epoch with a decay rate of $0.999$.

For S-learners, we regress the outcome on all the covariates (including a
constant for an intercept) and all two-way interactions between the covariates
and the treatment variable. For M-learners, we first transform the outcome
using the known propensity score, and then regress this transformed outcome on
all covariates.

Policy Learning Method | Evaluation Method | Value | Absolute Bias | S.E. | Coverage
---|---|---|---|---|---
S-Learner(LR) | Cram | 3.888 | 0.015 | 0.725 | 0.948
| Sample Split | 3.859 | 0.011 | 1.268 | 0.948
S-Learner(NN) | Cram | 3.925 | 0.011 | 0.764 | 0.948
| Sample Split | 3.902 | 0.013 | 1.261 | 0.949
M-learner(LR) | Cram | 3.547 | 0.028 | 0.817 | 0.947
| Sample Split | 3.485 | 0.008 | 1.160 | 0.948
M-learner(NN) | Cram | 3.544 | 0.018 | 0.984 | 0.941
| Sample Split | 3.514 | 0.012 | 1.176 | 0.949
CausalForest | Cram | 4.032 | 0.023 | 0.709 | 0.949
| Sample Split | 4.019 | 0.041 | 1.337 | 0.946
Table S2: Average performance of cramming and sample-splitting (80–20% split)
across all 77 DGPs.

#### Average performance.

Table S2 presents the average performance of cramming and sample-splitting
across 77 data generating processes.

###  S8.2 Adjustment for the finite sample bias

As discussed in the Section 3, the cram method evaluates the policy learned in
the first $T-1$ batches rather than $T$ batches. Though theoretically the
difference is negligible when $T\rightarrow\infty$, it may affect the finite
sample performance. In Figure 3 of Section 4, we find that the cram method has
a slightly negative bias.

(a) Percentage improvement in the policy value

(b) Percent improvement in standard error

(c) Standardized bias

(d) Coverage of 95% confidence intervals

Figure S2: Simulation Results Analogous to Figure 3. The only difference is
that for this simulation study, the final learned policy does not use the last
batch.

We can eliminate this bias by simply letting $\hat{\pi}_{T-1}$ be the final
policy (rather than $\hat{\pi}_{T}$) and use the last batch only for
evaluation not learning. Figure S2 presents the results based on the same
simulation setups as those used for Figure 3. We find that the cram method no
longer suffers from a small bias.

###  S8.3 Comparison with 60–40% sample-splitting

(a) Percentage improvement in the policy value

(b) Percent improvement in standard error

(c) Standardized bias

(d) Coverage of 95% confidence intervals

Figure S3: Summary Results Analogous to Figure 3. The only difference is that
the comparison is against the 60–40% sample-splitting rather than the 80–20%
sample-splitting.

In Section 4, we used 80–20% split for sample-splitting. Here, we repeat the
same simulation study using 60-40% split for comparison. All other aspects of
the simulation setup remain identical. Figure S3 show that when compared with
60–40% sample-splitting, the cram method improves the value of the learned
policy even more than 80–20% sample-splitting. This make sense because the
60–40% split uses less training data. On the other hand, the improvement in
the standard error is less substantial and there are few setups where cram
method has a larger standard error in the train-test splitting. These results
illustrate the trade-off of sample splitting. The cram method does not suffer
from this tradeoff.


# Generalised Whittaker models as instances of relative Langlands duality II:
Plancherel density and global periods

Wee Teck Gan  and  Bryan Wang Peng Jun  Department of Mathematics, National
University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In an earlier paper of the authors, a general family of instances of the
relative Langlands duality of Ben-Zvi$-$Sakellaridis$-$Venkatesh [BZSV] were
proposed and studied in the setting of branching problems for smooth
representations. In this paper, we show the numerical conjectures of [BZSV]
for the local Plancherel density, as well as an application to their
conjectures on global periods, for this general family of instances.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Spectral decomposition
  4. 4 Inner product
  5. 5 Plancherel density of basic function
  6. 6 Global period

##  1\. Introduction

This is a sequel to an earlier paper [GW] of the authors. There, the main aim
was to explicate the recently-proposed relative Langlands duality of [BZSV] in
the (local) setting of branching problems for smooth representations, and to
propose many (new) cases in which such duality could be verified.

However, [GW] only treated the conjectures of [BZSV] in the smooth setting,
and did not treat some other conjectures there, such as the conjectural
precise formula for the local Plancherel density [BZSV, Section 9] as well as
the conjecture about global period [BZSV, Section 14]. The aim of the present
paper is to treat these remaining issues in one representative new case
introduced in [GW, Section 7]. The method employed can be similarly applied
without much difficulty to the other cases (but we limit to this case for ease
of exposition).

###  1.1. Background

Let us now recall the relevant context and background as laid out in [BZSV]
and [GW]. One of the central themes of the relative Langlands program is to
characterize the non-vanishing of certain periods of automorphic forms on a
reductive group $G$, and when the period is non-zero, to relate it to certain
special values of automorphic $L$-functions. For a long time, however, this
relation between periods and $L$-functions has been relatively poorly
understood, with a disparate collection of instances known, but no systematic
framework with which to understand this relationship.

A systematic (i.e. general) study of the relations between periods and
$L$-functions was first laid out in the book [SV], using the framework of
spherical varieties. There, the important observation was further made, that
the factorisation of the global period functionals (associated to a spherical
variety $X$), into corresponding local functionals, should be facilitated by
local Plancherel functionals obtained from the spectral decomposition of
$L^{2}(X)$. In particular, at unramified places, the evaluation of these
Plancherel functionals on the (unit) spherical vector should give rise to
(special values of) corresponding local $L$-factors. This forms the basis for
relating the global period functional to a corresponding $L$-value.

However, the framework of [SV] was still not entirely satisfactory in some
respects. On one hand, as also explained in [GW], there are several important
examples exhibiting analogous phenomena, such as models with a Whittaker-
twisted component (arising from a nilpotent orbit), or Howe duality/theta
correspondence, which do not fall (strictly) into the framework of [SV]. On
the other hand, the framework of [SV] also does not explain which (and why) a
particular $L$-function should be related to a given period, so that the
period-$L$-function relationship still remained relatively mysterious.

The framework recently proposed in [BZSV] addresses these issues. On one hand,
the periods should be indexed by certain Hamiltonian $G$-varieties, known as
hyperspherical varieties (generalising spherical varieties), which encompass
the aforementioned examples of Whittaker-twisted models and theta
correspondence. On the other hand, the corresponding $L$-functions should also
be indexed by analogous hyperspherical $G^{\vee}$-varieties (where $G^{\vee}$
is the Langlands dual group of $G$). Key to this is the insight that there
should exist an involutive duality of hyperspherical varieties (in the spirit
of Langlands duality), so that the period associated to a hyperspherical
$G$-variety $M_{1}$ is related to the $L$-function associated to its dual
hyperspherical $G^{\vee}$-variety $M_{2}$, and vice versa. Furthermore, a
similar structure should exist at the local (unramified) level: the Plancherel
density associated to (the quantization of) $M_{1}$ should be related to the
local $L$-factor associated to $M_{2}$, and vice versa.

###  1.2. Results of [GW]

Let us recall briefly the results of the prequel [GW]. For this, we first
recall that a hyperspherical $G$-variety $M$ can be built out of the following
two pieces of data:

  * •

a morphism $\iota:H\times\mathrm{SL}_{2}\longrightarrow G$, such that
$H\hookrightarrow Z_{G}(\iota(\mathrm{SL}_{2}))$ is a spherical subgroup;

  * •

a symplectic representation $S$ of $H$.

In [GW], we consider special cases of hyperspherical varieties where:

  * •

$G$ is a symplectic or orthogonal group;

  * •

$H=Z_{G}(\iota(\mathrm{SL}_{2}))$;

  * •

$S=0$.

Under these hypotheses, the associated hyperspherical $G$-variety $M$ just
depends on a unipotent conjugacy class $e$ in $G$. In this setting, we
classified in [GW, Section 5] those $e$ for which $M=M_{e}$ hyperspherical. In
particular, if $e$ corresponds to partitions of hook type, then $M_{e}$ is
hyperspherical. For this general family of examples, we determined in [GW]
their hyperspherical dual $M_{e}^{\vee}$, by describing the smooth local
period problem associated to $M_{e}$ in terms of the geometry of a dual
hyperspherical $G^{\vee}$-variety $M_{e}^{\vee}$. Indeed, $M_{e}^{\vee}$ is of
the same type as $M_{e}$, namely is associated to a nilpotent orbit in
$G^{\vee}$ of hook type, so that

$M_{e}^{\vee}=M_{e^{\vee}}\quad\text{for some $e^{\vee}\in G^{\vee}$.}$

Moreover, we determine $e^{\vee}$ in terms of $e$.

With these notions introduced, we now specify the representative family of
examples from [GW, Section 7.1] that we will work with in this paper:

  * •

$G=\mathrm{O}_{2k}$ is an even orthogonal group;

  * •

the nilpotent conjugacy class $e^{\vee}\in G^{\vee}$ corresponds to the
partition $[2k-2a-1,1^{2a+1}]$, so that the data defining the hyperspherical
$G^{\vee}$-variety $M_{1}=M_{e^{\vee}}$ consists of the datum

$\iota_{1}:\mathrm{O}_{2a+1}\times\mathrm{SL}_{2}\rightarrow\mathrm{O}_{2k},\quad\text{with
$S=0$;}$

  * •

the nilpotent conjugacy class $e\in G=\mathrm{O}_{2k}$ corresponds to the
partition $[2a-1,1^{2k-2a+1}]$, so that the data defining the hyperspherical
$G$-variety $M_{2}=M_{e}$ consists of the datum

$\iota_{2}:\mathrm{O}_{2k-2a+1}\times\mathrm{SL}_{2}\rightarrow\mathrm{O}_{2k},\quad\text{
with $S=0$.}$

Then, for this particular representative family, the main result in [GW] can
be loosely formulated as:

######  Theorem′ 1.1.

[GW, Section 7.1] The hyperspherical varieties $M_{1}$ and $M_{2}$ defined
above are dual under the expected [BZSV]-duality of hyperspherical varieties.
More precisely:

  * (i)

the irreducible smooth representations (of Arthur type) of
$G^{\vee}=\mathrm{O}_{2k}$ which occur as submodules in the (smooth)
quantization of $M_{1}$ have A-parameters factoring through $\iota_{2}$ and
can be constructed as local theta lifts from generic representations of
$\mathrm{Sp}_{2k-2a}$;

  * (ii)

the irreducible smooth representations (of Arthur type) of $G=\mathrm{O}_{2k}$
which occur as submodules in the (smooth) quantization of $M_{2}$ have
A-parameters factoring through $\iota_{1}$ and can be constructed as local
theta lifts from generic representations of $\mathrm{Sp}_{2a}$.

However, as mentioned earlier, the corresponding conjectures in [BZSV]
concerning the Plancherel decomposition and global periods were not studied in
[GW], and it is the aim of this paper to address these problems.

###  1.3. Main result

The main result of the paper is (Theorem 5.5):

######  Theorem 1.2.

The numerical conjecture for the local Plancherel density [BZSV, Proposition
9.2.1] is true for the hyperspherical dual pair $(M_{1},M_{2})$.

In other words, the local Plancherel density associated to (the quantization
of) $M_{2}$ corresponds to a local $L$-value attached to $M_{1}$, and vice
versa.

We now explain the precise meaning of this last statement:

‘the Plancherel density associated to $M_{2}$ is related to a local $L$-value
attached to $M_{1}$’.

Denote $G=\mathrm{O}_{2k}$, so that $M_{2}$ is a $G$-variety, and $M_{1}$ is a
$G^{\vee}$-variety. Note first that $M_{2}$ is a twisted cotangent bundle of a
certain $G$-variety

$X=\mathrm{O}_{2k-2a+1}U,\overline{\chi_{\gamma}}\backslash G,$

for $U$ a unipotent subgroup of $G$ (corresponding to the nilpotent orbit
defining $M_{2}$) and $\chi_{\gamma}$ a unitary character of $U$ (we will
define $X$ more precisely in Section 2).

Our first main result (Theorem 3.6) concerns the spectral decomposition of
$L^{2}(X)$, which is the quantization of $M_{2}$, and shows that the spectral
measure on $L^{2}(X)$ is given by the pushforward, via theta lifting, of the
(Whittaker-)Plancherel measure on $G_{X}:=\mathrm{Sp}_{2a}$, or in other
words, via (roughly speaking) functorial lifting through the map

$\mathrm{O}_{2a+1}\times\mathrm{SL}_{2}\rightarrow\mathrm{O}_{2k}$

defining $M_{1}$. At least on the level of unramified representations, the
theta lifting is certainly known to facilitate such a functorial lifting, as
we already saw in Theorem’ 1.1. In other words, Theorem 3.6 is the
$L^{2}$-version of Theorem’ 1.1.

Our second main result (Theorem 5.6) concerns the Plancherel density of the
basic function $\phi_{0}$ on $X$ (coming from the spectral measure of
$L^{2}(X)$). This is a measure on $\widehat{G}_{\text{unramified}}$ (the
unramified unitary dual of $G$), and hence (via the Satake isomorphism)
identified with a measure on $A^{\vee}/W$, where $A^{\vee}$ is the maximal
torus in $\mathrm{O}_{2k}$ and $W$ its Weyl group.

We would also like to describe this measure in terms of the data of $M_{1}$.
Denote $G_{X}^{\vee}=\mathrm{O}_{2a+1}$, and denote by $A_{X}^{\vee(1)}$ the
maximal compact subgroup of the torus $A_{X}^{\vee}$ in $G_{X}^{\vee}$.

In view of the first main result, this measure should be given as a
pushforward of a certain measure on $A_{X}^{\vee(1)}$, via the map

$\displaystyle A_{X}^{\vee(1)}$ $\displaystyle\rightarrow A^{\vee}$
$\displaystyle(-)$ $\displaystyle\mapsto(-)\cdot q^{-\rho_{L(X)}},$

where here $\rho_{L(X)}$ is $\frac{1}{2}$ of the (co)weight for
$\mathrm{O}_{2k}$ corresponding to the map
$\mathfrak{sl}_{2}\rightarrow\mathfrak{so}_{2k}$ (coming from the map
$\mathrm{O}_{2a+1}\times\mathrm{SL}_{2}\rightarrow\mathrm{O}_{2k}$ defining
$M_{1}$) and $q$ is the cardinality of the residue field.

To describe the measure on $A_{X}^{\vee(1)}$, we recall that we have an
isomorphism

$M_{1}\cong V_{X}\times^{G_{X}^{\vee}}G^{\vee},$

for $V_{X}$ a finite-dimensional ${\mathbb{Z}}$-graded representation of
$G_{X}^{\vee}$. We view $V_{X}$ as a
$G_{X}^{\vee}\times\mathbb{G}_{m}$-representation, with $\mathbb{G}_{m}$
acting via the ${\mathbb{Z}}$-grading. The symplectic representation $S$ in
the definition of $M_{1}$ is a direct summand of $V_{X}$ (in degree 1) in
general.

The second main result is then that this measure on $A_{X}^{\vee(1)}$ is given
by

$\frac{1}{|W_{X}|}\frac{\det(I-t|_{{\mathfrak{g}}_{X}^{\vee}/\mathfrak{a}_{X}^{\vee}})}{\det(I-(t,q^{-1/2})|_{V_{X}})}dt,$

for $dt$ the usual Haar measure on $A_{X}^{\vee(1)}$.

In more informal terms, the local unramified Plancherel density associated to
$M_{2}$ is therefore related to a local $L$-value

$\prod_{i\in{\mathbb{Z}}}L(i/2,V_{X}^{(i)})$

associated to $M_{1}$.

###  1.4. Method of proof

We now say a few words on the method of proof. The main tool employed, as in
[GW], is that of the theta correspondence, in particular the $L^{2}$-theta
correspondence [Sa]. A rather general framework for addressing such problems
concerning Plancherel decomposition (and global periods), using theta
correspondence, was first laid out in a paper of the first author and Wan
[GW2], and indeed we largely follow the general framework there.

However, in [GW2], the relevant spectral decomposition is shown using a
different method (first done in [GG]), which, while the natural argument in
that context, does not generalise to the more general cases we consider. Our
approach in this paper differs in this key regard; we show the spectral
decomposition in a way which is not only more general, but which is also
parallel to the treatment of the smooth local and global aspects of the
problem. This parallel in the treatment of the smooth local, $L^{2}$-local and
global aspects via theta correspondence was not previously noticed in [GW2],
and is in a sense the main new contribution of this paper. Furthermore, the
generality of the method means that it can be similarly applied without much
difficulty to the other cases considered in [GW, Section 7], and may
potentially find application to other new cases beyond those considered in
[GW] (since, for instance, it may be easily adapted to the case of unitary
dual pairs).

Finally, as an application of the local (unramified) results shown in this
paper, and continuing with the general framework of [GW2] in the global
setting, one is also able to show the numerical conjecture of [BZSV]
concerning global periods, as stated in the simplest case (i.e. that
considered in [SV]) where the $\mathrm{SL}_{2}$-type of the automorphic form
and the hyperspherical (dual) space coincide, for automorphic forms on $G$
which are cuspidal and in the image of global theta lifting from (cuspidal)
automorphic representations of the relevant dual group $G_{X}$. We refer to
Theorem 6.1 for the precise statement.

In conclusion, this paper and its predecessor [GW] have achieved the full
analog of the results of [GW2] (shown for the rank 1 spherical variety
$\mathrm{O}_{k}\backslash\mathrm{O}_{k+1}$) in the context of the
hyperspherical varieties of $\mathrm{O}_{2k}$ associated to hook type
partitions.

##  2\. Preliminaries

Throughout, let $F$ be a non-Archimedean local field (of characteristic 0)
with residual characteristic not 2, with ring of integers ${\mathfrak{o}}_{F}$
and uniformizer $\varpi_{F}$. Denote by $\kappa_{F}$ the residue field of $F$
and $q$ its cardinality. Fix a non-trivial unitary character
$\psi:F\rightarrow{\mathbb{C}}^{\times}$ with conductor ${\mathfrak{o}}_{F}$.
Then $\psi$ determines an additive self-dual Haar measure on $F$, under which
${\mathfrak{o}}_{F}$ has volume 1, which we fix throughout.

In this paper, all classical groups are split, unless stated otherwise.

###  2.1. The case under consideration

Recall that the main aim of this paper is to show the following duality,
conjectured in [GW, Section 7], in the setting of the local (unramified)
Plancherel measure, as explicated in [BZSV, Proposition 9.2.1].

######  Theorem′ 2.1.

[GW, Section 7.1] The hyperspherical varieties $M_{1},M_{2}$ defined
respectively by

  * •

the datum $\mathrm{O}_{2a+1}\times\mathrm{SL}_{2}\rightarrow\mathrm{O}_{2k}$,
corresponding to the nilpotent orbit with partition $[2k-2a-1,1^{2a+1}]$, and
trivial $S$;

  * •

and the datum
$\mathrm{O}_{2k-2a+1}\times\mathrm{SL}_{2}\rightarrow\mathrm{O}_{2k}$,
corresponding to the nilpotent orbit with partition $[2a-1,1^{2k-2a+1}]$, and
trivial $S$.

are dual under the expected [BZSV]-duality of hyperspherical varieties.

In this paper, we will focus on the Plancherel measure associated to $M_{2}$
(and describe it in terms of $M_{1}$); the dual problem involving the
Plancherel measure associated to $M_{1}$ is completely similar.

Similarly, the other cases considered in [GW, Section 7] can be handled in a
similar manner.

###  2.2. Notation

We will use all the notation and set-up of [W, Section 2], specialised to the
case of the dual pair $(G,G^{\prime})$, with $G^{\prime}=\mathrm{Sp}_{2a}$ the
isometry group of a $2a$-dimensional symplectic space $V^{\prime}$, and
$G=\mathrm{O}_{2k}$ the isometry group of a $2k$-dimensional orthogonal space
$V$ which we assume to be the direct sum of $k$ hyperbolic planes. Note that,
in the notation of the introduction, we have $G^{\prime}=G_{X}$.

######  Remark 2.2.

Strictly speaking, this specialisation is not needed until Section 5, but we
do it now to fix ideas and streamline the exposition.

We also specialise to the case of the partitions (and corresponding nilpotent
orbits) $\gamma^{\prime}=[2a]$, $\gamma=[2a-1,1^{2k-2a+1}]$ (both even
nilpotent orbits). Furthermore we take the nilpotent element $e^{\prime}$ in
$\mathfrak{sp}_{2a}$ to be the ‘standard’ regular nilpotent element.

To be precise, this means that we have a decomposition

$V^{\prime}=V^{\prime}_{-(2a-1)}\oplus\cdots\oplus V^{\prime}_{-1}\oplus
V^{\prime}_{1}\oplus\cdots\oplus V^{\prime}_{2a-1},$

into one-dimensional subspaces, such that
$V^{\prime}_{-(2a-1)}\oplus\cdots\oplus V^{\prime}_{-1}$ and
$V^{\prime}_{1}\oplus\cdots\oplus V^{\prime}_{2a-1}$ are isotropic subspaces,
and there are basis vectors $v^{\prime}_{i}\in V^{\prime}_{i}$, with $\langle
v^{\prime}_{i},v^{\prime}_{-i}\rangle=1$ for all $i>0$. Then
$e^{\prime}\in\mathrm{End}(V^{\prime})$ sends $v^{\prime}_{i}\mapsto
v^{\prime}_{i+2}$ for $i=-(2a-1),\dots,2a-3$, and $v^{\prime}_{2a-1}\mapsto
0$.

We have the similar decomposition

$V=V_{-(2a-2)}\oplus\cdots\oplus V_{-2}\oplus V_{0}\oplus
V_{2}\oplus\cdots\oplus V_{2a-2},$

with $\dim V_{0}=2k-2a+2$, $V_{-(2a-2)}\oplus\cdots\oplus V_{-2}$ and
$V_{2}\oplus\cdots\oplus V_{2a-2}$ isotropic, $V_{i}\oplus V_{-i}$ a
hyperbolic plane (in the obvious way) for all $i\neq 0$, and the (split) form
on $V$ restricting to a non-degenerate (split) form on $V_{0}$.

Set $r=2a-2$. As in [W, Section 2.2.2], define a flag $(\overline{V}_{t})$ of
subspaces of $V$ by

$\overline{V}_{t}:=\bigoplus_{j=-r}^{t}V_{j},$

then we have a parabolic $P=MU$ which is the stabiliser of this flag. One has,
as in [W, Section 2.2], a unitary character $\chi_{\gamma}$ of $U$.

Denote, for $0\leq m\leq r$,

$V_{(m)}:=V_{-m}\oplus\cdots\oplus V_{m},$

and $G_{(m)}=G(V_{(m)})$, the isometry group of $V_{(m)}$.

For $1\leq m\leq r$, denote by $P_{m}$ the stabiliser of $V_{-m}$ in
$G_{(m)}$; it is a parabolic subgroup of $G_{(m)}$ with $P_{m}=M_{(m)}U_{m}$,
where $M_{(m)}=M_{m}\times G_{(m-1)}$ with $M_{m}\cong\mathrm{GL}(V_{-m})$. If
$U_{(m)}:=U\cap G_{(m)}$, then one has

$U_{(m)}=\big{(}(U_{1}\ltimes U_{2})\ltimes\cdots\big{)}\ltimes U_{m}.$

One may define $\chi_{\gamma,m}=\chi_{\gamma}|_{U_{m}}$, and then clearly
$\chi_{\gamma}=\chi_{\gamma,1}\cdots\chi_{\gamma,r}$.

For each $m$, the Lie algebra $\mathfrak{u}_{m}$ has a vector space
decomposition

$\mathfrak{u}_{m}\cong\mathfrak{q}_{m}\oplus\mathfrak{z}_{m}\cong\mathrm{Hom}(V_{(m-1)},V_{-m})\oplus\mathfrak{z}_{m}.$

Here $\mathfrak{z}_{m}$ is a Lie subalgebra and is central in
$\mathfrak{u}_{m}$, and

$\mathfrak{z}_{m}=\\{z\in\mathrm{Hom}(V_{m},V_{-m})|z^{\ast}=-z\\}\subseteq{\mathfrak{g}}_{-2m},$

while the isomorphism $\mathrm{Hom}(V_{(m-1)},V_{-m})\cong\mathfrak{q}_{m}$ is
provided by

$q\mapsto q-q^{\ast}.$

One has a corresponding short exact sequence

$1\rightarrow Z_{m}\rightarrow U_{m}\rightarrow Q_{m}\rightarrow 1.$

One makes similar definitions for the group $G^{\prime}$, except that
$P^{\prime}_{m}$ is the stabiliser of $V^{\prime}_{m}$ in $G^{\prime}_{(m)}$,
and we have the identification (again as vector spaces)

$\mathfrak{q}^{\prime}_{m}\cong\mathrm{Hom}(V^{\prime}_{m},V^{\prime}_{(m-1)}).$

As in [W, Section 2.4], write $\omega_{(l),(m)}$ for the Weil representation
associated to the dual pair $(G_{(l)},G^{\prime}_{(m)})$; it is realised on
the space of Schwarz functions on a Lagrangian $Y_{(l),(m)}$ of the symplectic
vector space $\mathrm{Hom}(V_{(l)},V^{\prime}_{(m)})$.

We choose Lagrangian subspaces once and for all in the following way. Choose a
Lagrangian $Y_{(0),(0)}$ of $\mathrm{Hom}(V_{(0)},V^{\prime}_{(0)})$. Then for
each $0\leq j\leq r$, set

$Y_{(j),(j+1)}=\mathrm{Hom}(V_{(j)},V^{\prime}_{j+1})\oplus Y_{(j),(j)}$

and for each $1\leq j\leq r$, set

$Y_{(j),(j)}=\mathrm{Hom}(V_{-j},V^{\prime}_{(j)})\oplus Y_{(j-1),(j)}.$

Finally, as in [W, Proposition 2.3], we have an element
$f\in\mathrm{Hom}(V,V^{\prime})$ facilitating the moment map transfer of
nilpotent orbits, with

$f=\bigoplus_{j=-r}^{r}f_{j}$

(here $r=2a-2$) and $f_{j}\in\mathrm{Hom}(V_{j},V^{\prime}_{j+1})$ for each
$j$. We have that $f_{0}^{\ast}v^{\prime}_{-1}$ is of norm 1 in $V_{0}$, and
$(f_{0}^{\ast}v^{\prime}_{-1})^{\perp}=\mathrm{ker}f_{0}$. Recall that $L$ is
the isometry group of this subspace
$(f_{0}^{\ast}v^{\prime}_{-1})^{\perp}=\mathrm{ker}f_{0}$ of dimension
$2k-2a+1$.

Denote

$X:=LU,\overline{\chi_{\gamma}}\backslash G,$

and recall that $M_{2}$ is a twisted cotangent bundle of $X$.

###  2.3. Measures

We take on the groups $G=G_{(2a-2)},\dots,G_{(0)},L$ (which are smooth group
schemes over ${\mathfrak{o}}_{F}$), invariant, integral volume forms which are
residually non-vanishing. This defines a Haar measure on $X$ (also referred to
as ‘Tamagawa measure’ in [BZSV]), under which

$\mathrm{vol}(X({\mathfrak{o}}_{F}))=q^{-\dim X}|X(\kappa_{F})|.$

This differs from the normalisation ultimately used in [BZSV, Section 9], but
we shall return to this relatively minor issue later (in Section 5).

In general, on all spaces we will work with the measure associated to an
invariant, integral volume form which is residually non-vanishing, or
‘Tamagawa measure’.

##  3\. Spectral decomposition

The aim of this section, which is the technical heart of this paper, is to
show a spectral decomposition for (the inner product on)
$L^{2}(X)=L^{2}(LU,\overline{\chi_{\gamma}}\backslash G)$.

###  3.1. Smooth setting

Recall that the smooth Weil representation
$\omega^{\infty}=\omega_{(r),(r+1)}^{\infty}$ (where here, $r=2a-2$)
associated to the dual pair $(G^{\prime},G)$, is realised on the space
$\mathscr{S}(Y_{(r),(r+1)})$ of Schwarz functions on the Lagrangian
$Y_{(r),(r+1)}$ of $\mathrm{Hom}(V,V^{\prime})$.

Recall also that the main result of [GZ] is the following (in the smooth
setting)

######  Theorem 3.1.

The map

$\displaystyle p:\omega^{\infty}=\mathscr{S}(Y_{(r),(r+1)})$
$\displaystyle\rightarrow\mathscr{S}(LU,\overline{\chi_{\gamma}}\backslash G)$
$\displaystyle\Phi$
$\displaystyle\mapsto\phi:=\big{(}g\mapsto(g\cdot\Phi)(f)\big{)}$

is surjective, and induces the isomorphism
$(\omega^{\infty})_{U^{\prime},\chi_{\gamma^{\prime}}}\cong\mathscr{S}(LU,\overline{\chi_{\gamma}}\backslash
G)$.

Write $\phi=p(\Phi)$, $\phi_{1}=p(\Phi_{1})$, etc. throughout.

###  3.2. Plancherel theorems

Recall now the spectral decomposition of the Weil representation $\omega$ as
first explicated in [Sa] and later also in [GW2, Section 5], which may
(equivalently) be formulated as a spectral decomposition of the inner product
$\langle\cdot,\cdot\rangle_{\omega}$ on $\omega$, that is,

$\langle\Phi_{1},\Phi_{2}\rangle_{\omega}=\int_{\widehat{G^{\prime}}}\langle\theta_{\sigma}(\Phi_{1}),\theta_{\sigma}(\Phi_{2})\rangle_{\sigma\otimes\theta(\sigma)}\mathop{d\mu_{G^{\prime}}(\sigma)}$

for $\Phi_{1},\Phi_{2}\in\omega^{\infty}$, where $\widehat{G^{\prime}}$
denotes (as usual) the unitary dual of $G^{\prime}$, $d\mu_{G^{\prime}}$ the
Plancherel measure of $G^{\prime}$ (associated to the Haar measure on
$G^{\prime}$), and $\theta_{\sigma}$ is the canonical map

$\theta_{\sigma}:\omega^{\infty}\rightarrow\sigma\otimes\theta_{\psi}(\sigma)$

obtained from the smooth theta correspondence. (Recall that we fix the
additive character $\psi$, and so suppress it from all notation.)

Recall also the Whittaker-Plancherel theorem for
$(U^{\prime},\chi_{\gamma^{\prime}})\backslash G^{\prime}$ (as explicated in
[GW2, Section 3.2] for instance), which furnishes canonical Whittaker
functionals

$l_{\sigma}\in\mathrm{Hom}_{U^{\prime}}(\sigma,\chi_{\gamma^{\prime}}),$

with

$l_{\sigma}\otimes\overline{l_{\sigma}}:v_{1}\otimes
v_{2}\mapsto\int_{U^{\prime}}^{\ast}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle
u^{\prime}\cdot v_{1},v_{2}\rangle_{\sigma}du^{\prime},$

where ∗ here indicates that the integral should be suitably regularised as in
[LM].

From Theorem 3.1, it is clear that the below commutative square can be
completed by a map

$\alpha_{\theta(\sigma)}:\mathscr{S}(LU,\overline{\chi_{\gamma}}\backslash
G)\rightarrow\theta(\sigma).$

The question now is whether this (family of) maps also facilitates the
spectral decomposition of $X$, or equivalently the inner product
$\langle\cdot,\cdot\rangle_{X}$. This is addressed in the next subsection.

${\mathscr{S}(Y_{(r),(r+1)})}$${\mathscr{S}(LU,\overline{\chi_{\gamma}}\backslash
G)}$${\sigma\otimes\theta(\sigma)}$${\theta(\sigma)}$${p}$${\theta_{\sigma}}$${\alpha_{\theta(\sigma)}}$${l_{\sigma}}$

This is the key point of departure from the previous work of [GW2] on
Plancherel decomposition (à la Sakellaridis-Venkatesh). There, the spectral
decomposition of $L^{2}(X)$ was first shown in a different manner, which does
not generalise to the cases we consider here, and only afterwards compared
with the above commutative diagram. Here, we show such a spectral
decomposition in a more general manner which is in fact parallel to the
(smooth) local and global aspects of the problem.

###  3.3. Key argument

In the next Section 4, we will show that

######  Proposition 3.2.

For $\Phi_{1},\Phi_{2}\in\omega^{\infty}$, one has

$\langle\phi_{1},\phi_{2}\rangle_{X}=\int_{U^{\prime}}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle
u^{\prime}\cdot\Phi_{1},\Phi_{2}\rangle_{\omega}du^{\prime}.$

Therefore, assuming this proposition for the moment, we see that in order to
analyse the inner product $\langle\cdot,\cdot\rangle_{X}$, it is enough to
consider the integral on the RHS, and we do so first.

From the above spectral decomposition of the Weil representation $\omega$, we
have that

$\int_{U^{\prime}}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle
u^{\prime}\cdot\Phi_{1},\Phi_{2}\rangle_{\omega}du^{\prime}=\int_{U^{\prime}}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\int_{\widehat{G^{\prime}}}\langle\theta_{\sigma}(u^{\prime}\cdot\Phi_{1}),\theta_{\sigma}(\Phi_{2})\rangle_{\sigma\otimes\theta(\sigma)}\mathop{d\mu_{G^{\prime}}(\sigma)}\mathop{du^{\prime}}$

We would now like to justify an interchange in the order of integration.

To that end, consider first the integral

$\displaystyle\int_{\widehat{G^{\prime}}}\int_{U^{\prime}}^{\ast}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle\theta_{\sigma}(u^{\prime}\cdot\Phi_{1}),\theta_{\sigma}(\Phi_{2})\rangle_{\sigma\otimes\theta(\sigma)}\mathop{du^{\prime}}\mathop{d\mu_{G^{\prime}}(\sigma)}$
$\displaystyle=\int_{\widehat{G^{\prime}}}\langle
l_{\sigma}(\theta_{\sigma}(\Phi_{1})),l_{\sigma}(\theta_{\sigma}(\Phi_{2}))\rangle_{\theta(\sigma)}\mathop{d\mu_{G^{\prime}}(\sigma)}$

We now make two key observations:

  * •

Since $\Phi_{1},\Phi_{2}\in\omega^{\infty}$ (hence fixed by a compact open
subgroup), by [LM, Proposition 2.3], the inner integral over $U^{\prime}$ may
(and should be interpreted as being) replaced by an integral over any compact
open subgroup $U_{x}^{\prime}$ containing a fixed compact open subgroup
$U_{0}^{\prime}$ of $U^{\prime}$, which depends only on $\Phi_{1},\Phi_{2}$
(and not on $\sigma$).

  * •

Also since $\Phi_{1},\Phi_{2}\in\omega^{\infty}$ (hence fixed by a compact
open subgroup), the integral over $\widehat{G^{\prime}}$ is compactly
supported (cf. [Wa, Théorème VIII.1.2]).

Therefore (the integral exists and) we may interchange the order of
integration; in other words, we obtain

######  Proposition 3.3.

We have

$\int_{U^{\prime}}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle
u^{\prime}\cdot\Phi_{1},\Phi_{2}\rangle_{\omega}\mathop{du^{\prime}}=\int_{\widehat{G^{\prime}}}\langle
l_{\sigma}(\theta_{\sigma}(\Phi_{1})),l_{\sigma}(\theta_{\sigma}(\Phi_{2}))\rangle_{\theta(\sigma)}\mathop{d\mu_{G^{\prime}}(\sigma)}.$

######  Remark 3.4.

The interchange in order of integration could also be justified by a similar
argument to that in the proof of [SV, Theorem 6.3.4].

Combining Propositions 3.2 and 3.3, we obtain:

######  Theorem 3.5.

We have

$\displaystyle\langle\phi_{1},\phi_{2}\rangle_{X}$
$\displaystyle=\int_{\widehat{G^{\prime}}}\langle
l_{\sigma}(\theta_{\sigma}(\Phi_{1})),l_{\sigma}(\theta_{\sigma}(\Phi_{2}))\rangle_{\theta(\sigma)}\mathop{d\mu_{G^{\prime}}(\sigma)}$
$\displaystyle=\int_{\widehat{G^{\prime}}}\langle\alpha_{\theta(\sigma)}(\phi_{1}),\alpha_{\theta(\sigma)}(\phi_{2})\rangle_{\theta(\sigma)}\mathop{d\mu_{G^{\prime}}(\sigma)}$

for all $\phi_{1},\phi_{2}\in\mathscr{S}(X)$.

In other words, we have obtained a spectral decomposition of the inner product
on $X=LU,\overline{\chi_{\gamma}}\backslash G$, which we now see is
facilitated by the family of maps obtained from the commutative square in the
previous subsection 3.2 (together with the Plancherel measure of
$G^{\prime}$).

We may also state this as

######  Theorem 3.6.

We have

$L^{2}(X)\cong\int_{\widehat{G^{\prime}}}\theta_{\psi}(\sigma)\cdot\dim\mathrm{Hom}_{U^{\prime}}(\sigma,\chi_{\gamma^{\prime}})\mathop{d\mu_{G^{\prime}}(\sigma)},$

so that the spectral measure on $L^{2}(X)$ is given by the pushforward, via
theta lifting, of the (Whittaker-)Plancherel measure on $G^{\prime}$, which is
supported on the tempered, $\psi$-generic irreducible representations of
$G^{\prime}$.

Furthermore, the resulting family of projections

$\alpha_{\theta(\sigma)}:\mathscr{S}(X)\rightarrow\theta(\sigma)$

is that defined by the commutative diagram of Section 3.2.

######  Remark 3.7.

Although we have worked exclusively with the non-Archimedean case (as is also
done in [BZSV]), it is to be expected that the analogous results concerning
spectral decomposition are true as well in the Archimedean case, albeit with
greater analytic difficulties to be considered.

##  4\. Inner product

The aim of this section (which may be thought of as an appendix to Section 3)
is to show the following (earlier stated as Proposition 3.2):

######  Proposition 4.1.

For $\Phi\in\omega^{\infty}$, one has

$\int_{U^{\prime}}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle
u^{\prime}\cdot\Phi_{1},\Phi_{2}\rangle_{\omega}du^{\prime}=\langle\phi_{1},\phi_{2}\rangle_{X}.$

###### Proof.

First, we remark that the integral over $U^{\prime}$ is not necessarily
convergent in general. But in this case, the matrix coefficient of the Weil
representation lies in the Harish-Chandra Schwarz space of $G^{\prime}$, and
so the integral over $U^{\prime}$ does at least converge. However, for our
purposes, we would like to say something stronger.

From the first key observation in the preceding section (which is by [LM,
Proposition 2.3]), the integral in fact stabilises, in the sense that there is
a compact open subgroup $U_{0}^{\prime}$ of $U^{\prime}$ (depending only on
$\Phi_{1},\Phi_{2}$) such that for all compact open subgroups $U_{x}^{\prime}$
containing $U_{0}^{\prime}$, the integral over $U_{x}^{\prime}$ exists and
takes the same value for all $U_{x}^{\prime}$.

######  Remark 4.2.

We remark that this (and the argument in the preceding section) is in essence
similar to the argument used in the proof of [LM, Proposition 2.11].

Therefore we may replace the integral over $U^{\prime}$ with an integral over
$U_{x}^{\prime}$ for any such (sufficiently large) compact open subgroup
$U_{x}^{\prime}$. In what follows we do not notate $U_{x}^{\prime}$ but simply
understand the integral over $U^{\prime}$ to be stabilised in the sense just
explained.

Furthermore, note that by definition

$\langle
u^{\prime}\cdot\Phi_{1},\Phi_{2}\rangle_{\omega}=\int_{Y_{(r),(r+1)}}(u^{\prime}\cdot\Phi_{1})(y)\overline{\Phi_{2}(y)}dy.$

Now since $\Phi_{2}$ is compactly supported, we may freely interchange the
integrals over $U_{x}^{\prime}$ and $Y_{(r),(r+1)}$ throughout the
computation.

The computation is formally analogous to that carried out in [W] (and so we do
not repeat some of the details which are formally similar). The idea is to
compute the integral over $U^{\prime}$ by inductively, or iteratively,
computing the integrals first over $U^{\prime}_{r+1}$, and so on, down to
$U^{\prime}_{1}$. (Recall again here $r=2a-2$.)

Therefore, we first compute

(4.1)
$\int_{U^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle
u^{\prime}\cdot\Phi_{1},\Phi_{2}\rangle_{\omega}du^{\prime},$

Recall that we have a short exact sequence

$1\rightarrow Z^{\prime}_{r+1}\rightarrow U^{\prime}_{r+1}\rightarrow
Q^{\prime}_{r+1}\rightarrow 1,$

with $\chi_{\gamma^{\prime},r+1}$ trivial on $Z^{\prime}_{r+1}$ (for $r\geq
1$).

So we may write

$\displaystyle(\ref{KeyInductive})$
$\displaystyle=\int_{Q^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime},r+1}(q^{\prime})}\int_{Z^{\prime}_{r+1}}\int_{Y_{(r),(r+1)}}(z^{\prime}q^{\prime}\cdot\Phi_{1})(y)\overline{\Phi_{2}(y)}\mathop{dy}\mathop{dz^{\prime}}\mathop{dq^{\prime}}$
$\displaystyle=\int_{Q^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime},r+1}(q^{\prime})}\int_{Z^{\prime}_{r+1}}$
$\displaystyle\int_{\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})\backslash\\{0\\}}\int_{Y_{(r),(r)}}(z^{\prime}q^{\prime}\cdot\Phi_{1})(F_{r})(y_{(r),(r)})\overline{\Phi_{2}(F_{r})(y_{(r),(r)})}\mathop{dy_{(r),(r)}}\mathop{dF_{r}}\mathop{dz^{\prime}}\mathop{dq^{\prime}}$

######  Remark 4.3.

Note that $\dim V^{\prime}_{r+1}=1$, so that the non-zero elements in
$\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})$ are precisely those with maximal
rank.

We first consider the integral over $Z^{\prime}_{r+1}$. From the action of
$Z^{\prime}_{r+1}$, we have

$(z^{\prime}q^{\prime}\cdot\Phi_{1})(F_{r})(y_{(r),(r)})=\psi(\mathrm{Tr}(z^{\prime}F_{r}F_{r}^{\ast}/2))(q^{\prime}\cdot\Phi_{1})(F_{r})(y_{(r),(r)}).$

Now the map

$\displaystyle q_{r}:\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})\backslash\\{0\\}$
$\displaystyle\rightarrow F$ $\displaystyle F_{r}$ $\displaystyle\mapsto
F_{r}F_{r}^{\ast}$

is a submersion (away from the origin of
$\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})$).

Therefore the integration over
$\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})\backslash\\{0\\}$ may be performed
first by integrating over the fibers of $q_{r}$, followed by integration along
the base, and by a completely analogous argument to that in [GW2, p.161-162],
one has

$\displaystyle(\ref{KeyInductive})$
$\displaystyle=\int_{Q^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime},r+1}(q^{\prime})}$
$\displaystyle\int_{\begin{subarray}{c}F_{r}\in\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})\backslash\\{0\\}\\\
F_{r}F_{r}^{\ast}=0\end{subarray}}\int_{Y_{(r),(r)}}(q^{\prime}\cdot\Phi_{1})(F_{r})(y_{(r),(r)})\overline{\Phi_{2}(F_{r})(y_{(r),(r)})}\mathop{dy_{(r),(r)}}\mathop{dF_{r}}\mathop{dq^{\prime}}$

For the sake of completeness, we reproduce the argument here. Observe that it
is no harm to first restrict to those $\Phi_{2}$ which are (locally constant
and) compactly supported away from the origin of
$\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})$, and then deduce the same for general
$\Phi_{2}$ by a density argument.

By the observation at the start of the entire proof, we replace the integral
over $Z^{\prime}_{r+1}$ with an integral over a sufficiently large compact
open subgroup of $Z^{\prime}_{r+1}$, which we may identify with
$\varpi_{F}^{-n}{\mathfrak{o}}_{F}$ for sufficiently large $n$.

Now if $F_{r}F_{r}^{\ast}\not\in\varpi_{F}^{n}{\mathfrak{o}}_{F}$, then
$\psi(\mathrm{Tr}(z^{\prime}F_{r}F_{r}^{\ast}/2))$ is a non-trivial character
(in $z^{\prime}$), hence the entire integral is zero. On the other hand, if
$F_{r}F_{r}^{\ast}\in\varpi_{F}^{n}{\mathfrak{o}}_{F}$, then the integral over
$z^{\prime}$ simply gives the volume of $\varpi_{F}^{-n}{\mathfrak{o}}_{F}$,
which is $q^{n}$.

Thus

$\displaystyle(\ref{KeyInductive})$
$\displaystyle=\int_{Q^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime},r+1}(q^{\prime})}$
$\displaystyle
q^{n}\int_{q_{r}^{-1}(\varpi_{F}^{n}{\mathfrak{o}}_{F})}\int_{Y_{(r),(r)}}(q^{\prime}\cdot\Phi_{1})(F_{r})(y_{(r),(r)})\overline{\Phi_{2}(F_{r})(y_{(r),(r)})}\mathop{dy_{(r),(r)}}\mathop{dF_{r}}\mathop{dq^{\prime}}$

But the integral over fibers of $q_{r}$ is a locally constant function on the
base. Therefore we have for all sufficiently large $n$,

$\displaystyle(\ref{KeyInductive})$
$\displaystyle=\int_{Q^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime},r+1}(q^{\prime})}$
$\displaystyle
q^{n}\mathrm{vol}(\varpi_{F}^{n}{\mathfrak{o}}_{F})\int_{q_{r}^{-1}(0)}\int_{Y_{(r),(r)}}(q^{\prime}\cdot\Phi_{1})(F_{r})(y_{(r),(r)})\overline{\Phi_{2}(F_{r})(y_{(r),(r)})}\mathop{dy_{(r),(r)}}\mathop{dF_{r}}\mathop{dq^{\prime}},$

and from $\mathrm{vol}(\varpi_{F}^{n}{\mathfrak{o}}_{F})=q^{-n}$ we get the
desired.

Now the $F_{r}\in\mathrm{Hom}(V_{(r)},V^{\prime}_{r+1})\backslash\\{0\\}$ with
$F_{r}F_{r}^{\ast}=0$ form a single orbit under the $G=G_{(r)}$-action, and in
fact we already have a representative $f_{r}$ in this orbit coming from the
moment map transfer. (This $f_{r}$ is trivial when restricted to
$V_{-r}\oplus\cdots\oplus V_{r-1}$, and restricts to an isomorphism
$V_{r}\xrightarrow{\sim}V^{\prime}_{r+1}$.) The stabiliser of $f_{r}$ in
$G_{(r)}$ is $G_{(r-1)}U_{r}$.

Therefore, considering the action of $g\in G_{(r)}$, we have

$\displaystyle(\ref{KeyInductive})$
$\displaystyle=\int_{Q^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime},r+1}(q^{\prime})}$
$\displaystyle\int_{G_{(r-1)}U_{r}\backslash
G_{(r)}}\int_{Y_{(r),(r)}}\big{(}\omega_{(r),(r)}(g_{r}^{-1})(g_{r}q^{\prime}\cdot\Phi_{1})(f_{r})\big{)}(y_{(r),(r)})$
$\displaystyle\cdot\overline{\big{(}\omega_{(r),(r)}(g_{r}^{-1})(g_{r}\cdot\Phi_{2})(f_{r})\big{)}(y_{(r),(r)})}\mathop{dy_{(r),(r)}}\mathop{dg_{r}}\mathop{dq^{\prime}}$
$\displaystyle=\int_{Q^{\prime}_{r+1}}\overline{\chi_{\gamma^{\prime},r+1}(q^{\prime})}$
$\displaystyle\int_{G_{(r-1)}U_{r}\backslash
G_{(r)}}\int_{Y_{(r),(r)}}(g_{r}q^{\prime}\cdot\Phi_{1})(f_{r})(y_{(r),(r)})\overline{(g_{r}\cdot\Phi_{2})(f_{r})(y_{(r),(r)})}\mathop{dy_{(r),(r)}}\mathop{dg_{r}}\mathop{dq^{\prime}}$

where here we have used the unitarity of $\omega_{(r),(r)}$.

The integral over $Q^{\prime}_{r+1}$ is handled in exactly the same way (and
is in fact simpler, in some sense), by considering the action of
$Q^{\prime}_{r+1}$ and again employing a completely analogous argument to that
in [GW2, p.161-162]. One obtains

$\displaystyle(\ref{KeyInductive})$
$\displaystyle=\int_{G_{(r-1)}U_{r}\backslash
G_{(r)}}\int_{Y_{(r-1),(r)}}(g_{r}\cdot\Phi_{1})(f_{r})(f_{-r})(y_{(r-1),(r)})$
$\displaystyle\cdot\overline{(g_{r}\cdot\Phi_{2})(f_{r})(f_{-r})(y_{(r-1),(r)})}\mathop{dy_{(r-1),(r)}}\mathop{dg_{r}}$

(Here, we simply need to perform a pointwise integration over
$\mathrm{Hom}(V_{-r},V^{\prime}_{(r)})$, rather than fiberwise integration
using a submersion $q_{r}$, which explains why the situation here is simpler.)

Now one again proceeds iteratively to compute the integral over
$U^{\prime}_{r},\dots,U^{\prime}_{1}$, as in [W], and the computation proceeds
in completely the same way as what we have just done for $U^{\prime}_{r+1}$.

At the ‘base case’ of $U^{\prime}_{1}$, there are certain differences (which
in fact simplify the computation).

First, $Q^{\prime}_{1}$ is trivial, hence we need only consider the integral
over $Z^{\prime}_{1}$; now $\chi_{\gamma^{\prime},1}$ is no longer trivial on
$Z^{\prime}_{1}$, and so one obtains an integral over the $F_{0}$ with

$F_{0}F_{0}^{\ast}=e^{\prime}\in\mathrm{Hom}(V^{\prime}_{-1},V^{\prime}_{1})$

(instead of $F_{0}F_{0}^{\ast}=0$). Then $G_{(0)}$ acts transitively on the
set of such $F_{0}$, and we have the representative $f_{0}$ with stabiliser
$L$.

In conclusion, we therefore obtain

$\displaystyle\int_{U^{\prime}}\overline{\chi_{\gamma^{\prime}}(u^{\prime})}\langle
u^{\prime}\cdot\Phi_{1},\Phi_{2}\rangle_{\omega}du^{\prime}$
$\displaystyle=\int_{LU\backslash
G_{(r)}}(g\cdot\Phi_{1})(f)\cdot\overline{(g\cdot\Phi_{2})(f)}\mathop{dg},$

which is as desired.

∎

##  5\. Plancherel density of basic function

We would now like to compute the Plancherel density of the ‘basic function’
$\phi_{0}$ on $X$, that is, the characteristic function of
$X({\mathfrak{o}}_{F})$.

By a similar argument as in [W, Section 4.7], note that
$\phi_{0}=p(\Phi_{0})$, where $\Phi_{0}\in\omega^{\infty}$ is the
characteristic function of $Y_{(r),(r+1)}({\mathfrak{o}}_{F})$.

Therefore, by Theorem 3.5, what is pertinent is the computation of

$\langle
l_{\sigma}(\theta_{\sigma}(\Phi_{0})),l_{\sigma}(\theta_{\sigma}(\Phi_{0}))\rangle_{\theta(\sigma)}$

for (tempered, $\psi$-generic) $G^{\prime}({\mathfrak{o}}_{F})$-unramified
$\sigma\in\widehat{G^{\prime}}$.

For each such $\sigma$, fix a $G^{\prime}({\mathfrak{o}}_{F})$-invariant
vector $v_{0}\in\sigma$ with $\langle v_{0},v_{0}\rangle_{\sigma}=1$ and a
$G({\mathfrak{o}}_{F})$-invariant vector $w_{0}\in\theta(\sigma)$ with
$\langle w_{0},w_{0}\rangle_{\theta(\sigma)}=1$. Suppose
$\theta_{\sigma}(\Phi_{0})=c_{\sigma}v_{0}\otimes w_{0}$. Then

$\langle
l_{\sigma}(\theta_{\sigma}(\Phi_{0})),l_{\sigma}(\theta_{\sigma}(\Phi_{0}))\rangle_{\theta(\sigma)}=|c_{\sigma}|^{2}\cdot|l_{\sigma}(v_{0})|^{2}.$

Both these quantities have been computed in [LR, Proposition 3] and [LM,
Proposition 2.14] respectively. In particular, we have

$|c_{\sigma}|^{2}=Z(\Phi_{0},\Phi_{0},v_{0},v_{0})=\mathrm{vol}(G^{\prime}({\mathfrak{o}}_{F}))\frac{L(k-a,\sigma,\mathrm{std})}{\zeta_{F}(k)\cdot\prod_{j=1}^{a}\zeta_{F}(2k-2j)}$

where $Z$ denotes the local doubling zeta integral (note here that we are in
the second case of [LR, Remark 3]), and

$|l_{\sigma}(v_{0})|^{2}=\frac{\Delta_{G^{\prime}}(1)}{L(1,\sigma,\mathrm{Ad})},$

where $\Delta_{G^{\prime}}$ is the $L$-factor of the dual to the motive
associated to $G^{\prime}$ (introduced by Gross in [Gr]).

Observe that
$\mathrm{vol}(G^{\prime}({\mathfrak{o}}_{F}))=\Delta_{G^{\prime}}(1)^{-1}$, so
that

$\langle
l_{\sigma}(\theta_{\sigma}(\Phi_{0})),l_{\sigma}(\theta_{\sigma}(\Phi_{0}))\rangle_{\theta(\sigma)}=\frac{L(k-a,\sigma,\mathrm{std})}{L(1,\sigma,\mathrm{Ad})\cdot\zeta_{F}(k)\cdot\prod_{j=1}^{a}\zeta_{F}(2k-2j)}.$

Now, recall that the choice of normalisation of measure in [BZSV] is that of
the ‘Tamagawa measure’ on $X$ divided by the ‘Tamagawa volume’ of
$G({\mathfrak{o}}_{F})$, which in this case (considering the exponents of
$G=\mathrm{O}_{2k}$) is

$\Delta_{G}(1)^{-1}=(\zeta_{F}(k)\cdot\prod_{j=1}^{k-1}\zeta_{F}(2k-2j))^{-1}.$

In conclusion, we see that one obtains:

######  Proposition 5.1.

We have

(5.1)
$\langle\phi_{0},\phi_{0}\rangle^{\prime}_{X}=\int_{\widehat{G^{\prime}}_{\text{unramified}}}\frac{L(k-a,\sigma,\mathrm{std})\cdot\zeta_{F}(2k-2a-2)\cdots\zeta_{F}(4)\zeta_{F}(2)}{L(1,\sigma,\mathrm{Ad})}\mathop{d\mu_{G^{\prime}}(\sigma)},$

Here, ′ indicates the renormalised (as in [BZSV]) measure on $X$.

It remains now to verify that this matches the expectations for the
hyperspherical dual $M_{1}$.

In the notation of [BZSV], for the hyperspherical dual $M_{1}$, we have
$(S=)S_{X}=0$, and a routine computation (using the fact that one has
$\mathrm{ad}=\wedge^{2}\mathrm{std}$ as representations of
$\mathfrak{so}_{2k}$) shows that

######  Lemma 5.2.

We have

$V_{X}=\mathfrak{so}_{2a+1}^{\perp}\cap\mathfrak{so}_{2k}^{e}=\mathrm{std}[2k-2a]\oplus\mathrm{triv}[4k-4a-4,\cdots,8,4]$

as $\mathrm{O}_{2a+1}$-representations.

(Recall that the grading on
$\mathfrak{so}_{2a+1}^{\perp}\cap\mathfrak{so}_{2k}^{e}$ is given by adding 2
to the weights of the adjoint $\mathfrak{sl}_{2}$-action on
$\mathfrak{so}_{2k}$.)

######  Remark 5.3.

Recall that, as in the introduction, the significance of the vector space
$V_{X}$ is that one has an isomorphism $M_{1}\cong
V_{X}\times^{G^{\prime\vee}}G^{\vee}$.

Now note that the point(s) of evaluation of the $L$-factor associated to
$V_{X}$ are (expected to be) given by $\frac{1}{2}$ times the grading on
$V_{X}$. Proposition 5.1 taken together with Lemma 5.2 therefore already shows
that the obtained Plancherel density is in remarkable agreement with the
general conjecture of [BZSV].

In order to verify that one has a precise agreement with the explicit
numerical conjecture in the form given in [BZSV, Proposition 9.2.1], it
remains to make two further observations.

First, the fact that the Plancherel measure is given as the pushforward under
multiplication by $q^{-\rho_{L(X)}}$ is a direct consequence of [Ga, Theorem
8.1(b)]. Of course, this is simply a reflection of the expected
characterisation of the spectral decomposition as functorial lifting along the
map $\mathrm{O}_{2a+1}\times\mathrm{SL}_{2}\rightarrow\mathrm{O}_{2k}$. Here,
$\rho_{L(X)}$ is $\frac{1}{2}$ of the (co)weight for $\mathrm{O}_{2k}$
corresponding to the map $\mathfrak{sl}_{2}\rightarrow\mathfrak{so}_{2k}$.

Second, the Plancherel formula of [BZSV, Proposition 9.2.1] is given in terms
of the Haar measure $dt$ on the maximal compact subgroup $A_{X}^{\vee(1)}$ of
the torus $A_{X}^{\vee}$ in $G_{X}^{\vee}=\mathrm{O}_{2a+1}$, via the Satake
isomorphism. (Recall that we have $G^{\prime}=G_{X}$.)

However, cf. also the group case of the same conjecture (which has essentially
been shown by Macdonald [Ma] and later also by Casselman [Ca]), we may express
the Plancherel measure $d\mu_{G^{\prime}}(\sigma)$ (on unramified
representations) as

$d\mu_{G^{\prime}}(\sigma)=\frac{1}{|W_{X}|}\frac{L(1,\sigma,\mathrm{Ad})}{L(0,\sigma,{\mathfrak{g}}_{X}^{\vee}/\mathfrak{a}_{X}^{\vee})}dt.$

######  Remark 5.4.

The astute reader would no doubt notice that there is a potential issue with
normalisations to be addressed; namely, the group case of the same conjecture
entails a renormalisation which gives $G^{\prime}({\mathfrak{o}}_{F})$ a
volume equal to the reciprocal of the ‘Tamagawa volume’
$\mathrm{vol}(G^{\prime}({\mathfrak{o}}_{F}))$ that we have given it in this
paper. In other words we should have

$\frac{1}{\mathrm{vol}(G^{\prime}({\mathfrak{o}}_{F}))}=\frac{1}{|W_{X}|}\int_{A_{X}^{\vee(1)}}\frac{L(1,\sigma,\mathrm{Ad})}{L(0,\sigma,{\mathfrak{g}}_{X}^{\vee}/\mathfrak{a}_{X}^{\vee})}dt.$

However the (Harish-Chandra-)Plancherel formula for $G^{\prime}$ (with its
Tamagawa measure) gives

$\mathrm{vol}(G^{\prime}({\mathfrak{o}}_{F}))=\int_{\widehat{G^{\prime}}_{\text{unramified}}}\mathrm{vol}(G^{\prime}({\mathfrak{o}}_{F}))^{2}d\mu_{G^{\prime}}(\sigma),$

and comparing the two shows that we have the correct normalisation for the
Tamagawa measure on $G^{\prime}$.

The above two observations, taken together with the main Proposition 5.1, show

######  Theorem 5.5.

The numerical conjecture for the Plancherel density [BZSV, Proposition 9.2.1]
is true for the hyperspherical dual pair $M_{1}$ and $M_{2}$.

To be precise, what we have shown is:

######  Theorem 5.6.

The Plancherel density of the basic function $\phi_{0}$, which is a measure on
$\widehat{G}_{\text{unramified}}$, and hence (via Satake) identified with a
measure on $A^{\vee}/W$, where $A^{\vee}$ is the maximal torus in
$\mathrm{O}_{2k}$ and $W$ its Weyl group, is given as the pushforward via

$\displaystyle A_{X}^{\vee(1)}$ $\displaystyle\rightarrow A^{\vee}$
$\displaystyle(-)$ $\displaystyle\mapsto(-)\cdot q^{-\rho_{L(X)}}$

of the measure on $A_{X}^{\vee(1)}$ given by

$\frac{1}{|W_{X}|}\frac{\det(I-t|_{{\mathfrak{g}}_{X}^{\vee}/\mathfrak{a}_{X}^{\vee}})}{\det(I-(t,q^{-1/2})|_{V_{X}})}dt.$

##  6\. Global period

The global period conjecture, at least as stated in the case when the (Arthur)
$\mathrm{SL}_{2}$-type of the automorphic form (for the $M_{2}$-period)
coincides with that of $M_{1}$ [BZSV, Equation (14.24)], can now be verified
in exactly the same way as in [GW2, Section 12.14], conditional on the
corresponding conjecture for the Whittaker period on general symplectic groups
$G^{\prime}=\mathrm{Sp}_{2a}$ [LM, Conjecture 3.3].

Fix a number field $k$ with ring of adeles $\mathbb{A}$ and a non-trivial
unitary additive character
$\psi=\otimes_{v}\psi_{v}:k\backslash\mathbb{A}\rightarrow{\mathbb{C}}^{\times}$,
and for any algebraic group $G$ over $k$, denote as usual $[G]:=G(k)\backslash
G(\mathbb{A})$. For $f$ a cuspidal automorphic form on $G$, denote the period
integral

$P_{\gamma}(f)=\int_{[L]}\int_{[U]}f(ul)\cdot\overline{\chi_{\gamma}(u)}\mathop{du}\mathop{dl},$

taking as usual Tamagawa measure on all adelic groups.

######  Theorem 6.1.

Assume [LM, Conjecture 3.3], and that one has local multiplicity one for
$LU,\overline{\chi_{\gamma}}\backslash G$ (this is the less serious
hypothesis).

Let $\Sigma$ be a globally $\psi$-generic cuspidal automorphic representation
of $G^{\prime}=\mathrm{Sp}_{2a}$ with corresponding Langlands parameter
$\phi$, and with global theta lift $\Pi:=\Theta(\Sigma)$ to $G$ which is
cuspidal. Suppose further that $\Theta(\Pi)$ is isomorphic to $\Sigma$ (this
is the case if $\Theta(\Pi)$ is cuspidal, for instance).

Let $T$ be a finite set of places (including the Archimedean ones) outside
which all data are unramified. Then the restriction of the functional
$P_{\gamma}$ to $\Theta(\Sigma)$ factorises as

$|P_{\gamma}(f)|^{2}=\frac{1}{|S_{\phi}|}\frac{\Delta_{G}^{T}(1)}{\Delta_{L}^{T}(1)^{2}}\frac{\prod_{i}L^{T}(i/2,\Sigma,V_{X}^{(i)})}{L^{T}(1,\Sigma,\mathrm{Ad})}\prod_{v\in
T}|l^{\prime}_{\Pi_{v}}(f_{v})|^{2}$

for $f=\otimes_{v}f_{v}\in\otimes_{v}\Pi_{v}$ and where $l^{\prime}_{\Pi_{v}}$
are the (normalised, as in [GW2], so that $l^{\prime}_{\Pi_{v}}(w_{0,v})=1$
for almost all $v$) canonical functionals obtained from the Plancherel
decomposition. Note that we have $\Pi_{v}\cong\theta_{\psi_{v}}(\Sigma_{v})$.

###### Proof.

As noted above, the proof is completely analogous to that in [GW2], since the
local spectral decomposition has of course been shown in this paper, and the
necessary global computation [GW2, Proposition 12.2] has been performed in
[W].

We will give a sketch (of the key argument) here, omitting the other formal
details which can be found in [GW2].

Define, in the global setting, the global theta lift

$\displaystyle B^{\text{Aut}}_{\Pi}:\omega\otimes\overline{\Pi}$
$\displaystyle\rightarrow\Sigma$ $\displaystyle\Phi\otimes\overline{f}$
$\displaystyle\mapsto\Big{(}g^{\prime}\mapsto\int_{[G]}\theta(\Phi)(g^{\prime}g)\overline{f(g)}dg\Big{)}.$

The image of $B^{\text{Aut}}_{\Pi}$ is $\Sigma$, by multiplicity one for
$G^{\prime}=\mathrm{Sp}_{2a}$ (at least for tempered A-packets, since $\Sigma$
is globally generic) [Ar].

Then the main result of [W] (cf. Section 4.6 there) can be stated as:

(6.1) $P_{\gamma^{\prime}}(B^{\text{Aut}}_{\Pi}(\Phi,f))=\langle
p(\Phi),\beta^{\text{Aut}}_{\Pi}(f)\rangle_{X(\mathbb{A})},$

where $p$ is defined as in Section 3, and here

$\displaystyle\beta^{\text{Aut}}_{\Pi}:\Pi$ $\displaystyle\rightarrow
C^{\infty}(X(\mathbb{A}))$ $\displaystyle f$
$\displaystyle\mapsto\Big{(}g\mapsto P_{\gamma}(g\cdot f)\Big{)}.$

On the other hand, the spectral decomposition shown in this paper provides a
local analog of this result.

Indeed, we have, in the local setting, for any $w\in\theta(\sigma)$,

$\displaystyle\langle\alpha_{\theta(\sigma)}(p(\Phi)),w\rangle_{\theta(\sigma)}$
$\displaystyle=\langle
l_{\sigma}(\theta_{\sigma}(\Phi)),w\rangle_{\theta(\sigma)}$
$\displaystyle=l_{\sigma}(B_{\theta(\sigma)}(\Phi,w))$

where here

$\displaystyle B_{\theta(\sigma)}:\omega\otimes\overline{\theta(\sigma)}$
$\displaystyle\rightarrow\sigma$ $\displaystyle\Phi\otimes w$
$\displaystyle\mapsto\langle\theta_{\sigma}(\Phi),w\rangle_{\theta(\sigma)}$

is the local version of the global theta lifting $B^{\text{Aut}}_{\Pi}$.

We have the dual or adjoint

$\beta_{\theta(\sigma)}:\theta(\sigma)\rightarrow C^{\infty}(X)$

to the family of maps $\alpha_{\theta(\sigma)}$ which facilitates the spectral
decomposition of $X$, and

(6.2) $l_{\sigma}(B_{\theta(\sigma)}(\Phi,w))=\langle
p(\Phi),\beta_{\theta(\sigma)}(w)\rangle_{X}.$

Note that $\beta_{\theta(\sigma)}$ is the local version of
$\beta^{\text{Aut}}_{\Pi}$, and that
$l_{\theta(\sigma)}=\text{ev}_{\text{id}_{G}}\circ\beta_{\theta(\sigma)}$.

To summarise, we have two main results (6.1) and (6.2), with (6.2) the local
analog of the global (6.1).

By multiplicity one, we have factorisations

$P_{\gamma^{\prime}}=c(\Sigma)\cdot\prod_{v}^{\ast}l_{\Sigma_{v}},$
$B_{\Pi}^{\text{Aut}}=b(\Sigma)\cdot\prod_{v}^{\ast}B_{\Pi_{v}},$
$P_{\gamma}=c(\Theta(\Sigma))\cdot\prod_{v}^{\ast}l_{\Pi_{v}},$

and hence also

$\beta_{\Pi}^{\text{Aut}}=c(\Theta(\Sigma))\cdot\prod_{v}^{\ast}\beta_{\Pi_{v}},$

for constants $c(\Sigma),b(\Sigma),c(\Theta(\Sigma))$, and where ∗ denotes
suitable (standard) regularisation as in [GW2, Section 12.12].

Comparing (6.1) and (6.2), it is clear that we should have

$c(\Sigma)\cdot b(\Sigma)=\overline{c(\Theta(\Sigma))}.$

Now [LM, Conjecture 3.3] gives $|c(\Sigma)|^{2}=\frac{1}{|S_{\phi}|}$, and as
in [GW2, Proposition 12.4] (by the Rallis inner product formula), we have
$|b(\Sigma)|=1$.

Therefore $|c(\Theta(\Sigma))|^{2}=\frac{1}{|S_{\phi}|}$, as desired.

Finally, we note that, in the local unramified computation,

$|l_{\theta(\sigma)}(w_{0})|^{2}=\mathrm{vol}(X({\mathfrak{o}}_{F}))^{-2}\cdot\langle\alpha_{\theta(\sigma)}(\phi_{0}),\alpha_{\theta(\sigma)}(\phi_{0})\rangle_{\theta(\sigma)},$

with the latter quantity already computed in Section 5 (but here, we do not do
the renormalisation of [BZSV]). This explains the extra factor of
$\frac{\Delta_{G}^{T}(1)}{\Delta_{L}^{T}(1)^{2}}$ in the final expression.

∎

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