# The Metaplectic Representation is Faithful

Christopher Chang, Simeon Hellsten, Mario Marcos Losada and Sergiu
Novac22footnotemark: 2 St Anne’s College, University of OxfordSt John’s
College, University of OxfordBrasenose College, University of Oxford

(January 2024)

###### Abstract

We develop methods to show that infinite-dimensional modules over the Iwasawa
algebra $KG$ of a uniform pro-$p$ group are faithful and apply them to show
that the metaplectic representation for the group
$G=\exp(p\,\mathfrak{sp}_{2n}(\mathbb{Z}_{p}))$ is faithful. ††Email
addresses<EMAIL_ADDRESS>(C. Chang),
<EMAIL_ADDRESS>(S. Hellsten<EMAIL_ADDRESS>(M. Marcos Losada<EMAIL_ADDRESS>(S. Novac)

##  1 Introduction

Let $p$ be an odd prime and $G$ be a uniform pro-$p$ group. Let $K$ be a
finite extension of $\mathbb{Q}_{p}$ with valuation ring $R\coloneqq\\{x\in
K:|x|_{p}\leq 1\\}$. We are interested in the prime ideals of the Iwasawa
algebra $KG\coloneqq RG\otimes_{R}K$ where $RG\coloneqq\varprojlim R[G/N]$ and
the inverse limit is taken over all open normal subgroups $N\lhd_{o}G$. If $G$
has a closed normal subgroup $N\lhd_{c}G$ such that $G/N$ is uniform, then
$(N-1)KG=\ker(KG\to K(G/N))$ is a prime ideal of $KG$ so we will restrict to
almost simple groups $G$. We are motivated by the following conjecture.

######  Conjecture  —

Let $G$ be an almost simple uniform pro-$p$ group. Then every non-zero prime
ideal of $KG$ has finite codimension.

In particular, if the conjecture is true, then every infinite-dimensional
representation of $KG$ must be faithful. When the Lie algebra of $G$ is of
Type $A$, Section 1 was proven in [Man23], but the methods there do not
generalise to other types. Similar results of this form can also be found for
example in [AW14]. In this paper, we develop methods to approach these types
of questions more generally and we apply them to the case of
$G\coloneqq\exp(p\mathfrak{g}),$ where
$\mathfrak{g}\coloneqq\mathfrak{sp}_{2n}(\mathbb{Z}_{p}).$ We start with the
metaplectic representation from Section 2 and explain how to lift this to an
algebra homomorphism
$\rho:\widehat{U(\mathfrak{g})}_{K}\to\widehat{A_{n}(R)}_{K}$, where
$\widehat{U(\mathfrak{g})}_{K}$ is the affinoid enveloping algebra and
$\widehat{A_{n}(R)}_{K}$ is the completed Weyl algebra, both defined after
Section 2. We can finally embed $KG$ into $\widehat{U(\mathfrak{g})}_{K}$ and
in Section 4.3 we then prove the following.

######  Theorem  —

The metaplectic representation
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KG}}:KG\to\widehat{A_{n}(R)}_{K}$
is injective.

The main tool to do this is the following Gluing Lemma, a special case of the
general version from Section 3. Here,
$U(\mathfrak{n})_{K}=U(\mathfrak{n})\otimes_{\mathbb{Z}_{p}}K$.

######  Proposition  —

Let $\mathfrak{n},\mathfrak{h}$ and
$\mathfrak{g}\coloneqq\mathfrak{n}\oplus\mathfrak{h}$ be finite rank
$\mathbb{Z}_{p}$-Lie algebras with corresponding uniform pro-$p$ groups $N,H$
and $G$ and let $V$ be a $\widehat{U(\mathfrak{g})}_{K}$-module and a
$K-$Banach space, such that $\left(N-1\right)\cdot B_{V}\subseteq pB_{V},$
where $B_{V}\coloneqq\\{v\in V:\left\lVert v\right\rVert\leq 1\\}$. Suppose
there is a subset $\mathcal{V}$ of $V$ and for each $v\in\mathcal{V}$ a
$\widehat{U(\mathfrak{h})}_{K}$-submodule $W_{v}$ of $V$ contained in
$U(\mathfrak{n})_{K}\cdot v$ such that

  * •

$RH$ acts locally finitely on $W\coloneqq\sum_{v\in\mathcal{V}}W_{v}$, meaning
that every cyclic $RH$-submodule of $W$ is finitely generated over $R$.

  * •

$KH$ acts faithfully on $W$.

  * •

For every $v\in\mathcal{V}$ the multiplication map
$KN\otimes_{K}\faktor{U(\mathfrak{n})_{K}}{I_{v}^{\prime}}\to\faktor{\widehat{U(\mathfrak{n})}_{K}}{I_{v}}$
is injective, where
$I_{v}^{\prime}=\operatorname{Ann}_{U(\mathfrak{n})_{K}}(v)$ and
$I_{v}=\operatorname{Ann}_{\widehat{U(\mathfrak{n})}_{K}}(v)$.

Then any $KG$-submodule $V_{0}$ of $V$ containing $W$ is faithful.

This allows us to deduce faithfulness results for a Lie algebra
$\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{h}$ from similar results for
$\mathfrak{n}$ and $\mathfrak{h}$. We do this by exploiting the local
finiteness conditions to replace the action of $\mathfrak{h}$ by an action of
$U(\mathfrak{n})_{K}$, thus turning the problem into a question involving only
the Lie algebra $\mathfrak{n}$.

In Section 4 we then apply this to the metaplectic representation. We first
decompose $\mathfrak{g}=\mathfrak{a}\oplus\mathfrak{b}\oplus\mathfrak{c}$ into
subalgebras $\mathfrak{a},\mathfrak{b}$ and $\mathfrak{c}$ with associated
uniform pro-$p$ groups $A,B$ and $C$, where $\mathfrak{a}$ acts locally
finitely and $\mathfrak{b}$ acts locally nilpotently, and proceed in three
steps.

  * •

In Section 4.1 we start by generalising [AW14, Theorem 3.8] to show that the
multiplication map from the last condition of the Gluing Lemma is indeed
injective for any subalgebra of $\mathfrak{c}$. To do this we use the
generalisation of the Gluing Lemma from Section 3.

  * •

In Section 4.2 we show that the second condition for the Gluing Lemma above
holds, namely that the metaplectic representation is injective when restricted
to $KA$. This involves exploiting symmetries of $\widehat{A_{n}(R)}_{K}$ to be
able to apply the Gluing Lemma as stated above.

  * •

In Section 4.3 we put together the previous results to conclude that the
metaplectic representation is injective.

Finally, in Section 5 we use primary decomposition and a result from [Ard12]
on ideals fixed by a particular class of action to give a different proof for
abelian subalgebras of general Iwasawa algebras, before applying them to $KB$
and $KC$.

The authors believe that the methods developed here can be generalised to an
arbitrary highest weight module for $\mathfrak{sp}_{2n}$ and more generally to
other simple Lie algebras.

###  1.1 Acknowledgements

The authors thank Konstantin Ardakov for his invaluable support, advice and
feedback throughout this project. The first author thanks St Anne’s College,
Oxford for providing accommodation and funding for living expenses for the
duration of this project. The second and last authors thank St John’s College,
Oxford for providing additional funding and accommodation for the duration of
the project. The third author thanks the London Mathematical Society and the
Mathematical Institute for funding this project. The last author also thanks
the Mathematical Institute for funding the project.

##  2 Constructing the Metaplectic Representation

We will use lower-case letters to denote elements in a $\mathbb{Z}_{p}$-Lie
algebra $\mathfrak{g}$ and the corresponding upper-case letter for the
corresponding element in its associated uniform pro-$p$ group
$G\coloneqq\exp(p\mathfrak{g})$. It is a standard fact of Iwasawa algebras
(see for example [Dix+03, §7]) that a general element $\zeta\in KG$ can be
uniquely written as

$\zeta=\sum_{\alpha\in\mathbb{N}_{0}^{d}}\lambda_{\alpha}(\mathbf{G}-1)^{\alpha},\qquad\qquad(\mathbf{G}-1)^{\alpha}\coloneqq(G_{1}-1)^{\alpha_{1}}\cdots(G_{d}-1)^{\alpha_{d}},$

where $\left(G_{1},\dots,G_{d}\right)$ is a topological generating set for $G$
and $\lambda_{\alpha}\in K$ are uniformly bounded with respect to the $p$-adic
valuation on $K$.

For fixed $n\geq 2$, let

$\mathfrak{g}\coloneqq\mathfrak{sp}_{2n}(\mathbb{Z}_{p})=\left\\{\begin{bmatrix}A&B\\\
C&-A^{T}\end{bmatrix}:A,B,C\in M_{n}(\mathbb{Z}_{p}),B^{T}=B,C^{T}=C\right\\}$

with associated uniform pro-$p$ group

$G\coloneqq\exp(p\,\mathfrak{sp}_{2n}(\mathbb{Z}_{p}))=\operatorname{SP}_{2n}(\mathbb{Z}_{p})\cap(I+pM_{2n}(\mathbb{Z}_{p})).$

We will use the following map adapted from [Fol89] to construct an infinite
dimensional $KG$-module.

######  Proposition  (Metaplectic Representation) —

There is a Lie algebra homomorphism $\mathfrak{sp}_{2n}(\mathbb{Z}_{p})\to
A_{n}(\mathbb{Z}_{p})$ given by

$\begin{bmatrix}A&B\\\
C&-A^{T}\end{bmatrix}\longmapsto-\frac{1}{2}\operatorname{tr}(A)-\sum_{1\leq
i,j\leq n}{A_{ij}x_{j}\partial_{i}}+\frac{1}{2}\sum_{1\leq i,j\leq
n}{B_{ij}\partial_{i}\partial_{j}}-\frac{1}{2}\sum_{1\leq i,j\leq
n}{C_{ij}x_{i}x_{j}}.$

Here $A_{n}(\mathbb{Z}_{p})$ is the $n^{th}$ Weyl algebra on generators
$x_{1},\ldots,x_{n}$ and $\partial_{1},\ldots,\partial_{n}$. We will denote
$\textnormal{{x}}^{\alpha}\coloneqq x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}}$ and
$\partial^{\beta}\coloneqq\partial_{1}^{\beta_{1}}\cdots\partial_{n}^{\beta_{n}}$
for all $\alpha,\beta\in\mathbb{N}_{0}^{n}$.

We are interested in knowing if there are any non-zero prime ideals of $KG$ of
infinite codimension. To this end, we lift the above homomorphism to a map on
$KG.$ We tensor first with $R$ to obtain a map of $R$-Lie algebras
$\mathfrak{g}_{R}\coloneqq\mathfrak{sp}_{2n}(R)\to A_{n}(R)$ and lift this to
a map of associative algebras $U(\mathfrak{g}_{R})\to A_{n}(R),$ where
$U(\mathfrak{g}_{R})$ is the universal enveloping algebra.

Now note that both $U(\mathfrak{g}_{R})$ and $A_{n}(R)$ inherit $p$-adic
valuations from $R$. Explicitly, as $K$ is a finite $\mathbb{Q}_{p}$
extension, the valuation on $\mathbb{Q}_{p}$ can be extended to a valuation on
$K,$ which we denote by $v$. Let $d=2n^{2}+n$ be the dimension of
$\mathfrak{g}$ over $\mathbb{Z}_{p}$ and fix a basis $g_{1},\ldots,g_{d}$ of
$\mathfrak{g}$. We denote $\mathbf{g}^{\alpha}\coloneqq
g_{1}^{\alpha_{1}}\cdots g_{d}^{\alpha_{d}}$ for
$\alpha\in\mathbb{N}_{0}^{d}$, and write
$\left\lvert\alpha\right\rvert=\alpha_{1}+\cdots+\alpha_{n}$. Then, for an
element of $U(\mathfrak{g}_{R})$ we define its valuation by

$v_{p}\left(\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}\mathbf{g}^{\alpha}\right)\coloneqq\min_{\alpha\in\mathbb{N}_{0}^{d}}v(c_{\alpha}),$

and similarly, for elements of $A_{n}(R),$

$v_{p}\left(\sum_{\alpha,\beta\in\mathbb{N}_{0}^{n}}c_{\alpha,\beta}\textnormal{{x}}^{\alpha}\partial^{\beta}\right)\coloneqq\min_{\alpha,\beta\in\mathbb{N}_{0}^{n}}v(c_{\alpha,\beta}).$

We then note the algebra homomorphism is continuous with respect to the
induced topologies. Therefore, this extends to a map
$\widehat{U(\mathfrak{g}_{R})}\to\widehat{A_{n}(R)}$ of the $p$-adic
completions.

Then

$\widehat{U(\mathfrak{g}_{R})}\coloneqq\varprojlim_{\lambda\geq
0}\faktor{U(\mathfrak{g}_{R})}{U(\mathfrak{g}_{R})_{\lambda}}=\left\\{\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}\mathbf{g}^{\alpha}:c_{\alpha}\in
R,c_{\alpha}\to 0\text{ as }|\alpha|\to\infty\right\\}$

with $U(\mathfrak{g}_{R})_{\lambda}\coloneqq\left\\{x\in
U(\mathfrak{g}_{R}):v_{p}(x)\geq\lambda\right\\}$ for $\lambda\in\mathbb{R}$,
and similarly

$\widehat{A_{n}(R)}\coloneqq\varprojlim_{\lambda\geq
0}\faktor{A_{n}(R)}{A_{n}(R)_{\lambda}}=\left\\{\sum_{\alpha,\beta\in\mathbb{N}_{0}^{n}}c_{\alpha,\beta}\textnormal{{x}}^{\alpha}\partial^{\beta}:c_{\alpha,\beta}\in
R,c_{\alpha,\beta}\to 0\text{ as }|\alpha|+|\beta|\to\infty\right\\},$

with $A_{n}(R)_{\lambda}\coloneqq\left\\{y\in
A_{n}(R):v_{p}(y)\geq\lambda\right\\}$ for $\lambda\in\mathbb{R}$.

Now, we tensor with $K$ and obtain a map
$\rho:\widehat{U(\mathfrak{g})}_{K}\to\widehat{A_{n}(R)}_{K}$, where

$\widehat{U(\mathfrak{g})}_{K}\coloneqq\widehat{U(\mathfrak{g}_{R})}\otimes_{R}K=\left\\{\sum_{\alpha\in\mathbb{N}_{0}^{n}}c_{\alpha}\mathbf{g}^{\alpha}:c_{\alpha}\in
K,c_{\alpha}\to 0\text{ as }|\alpha|\to\infty\right\\}$

and

$\widehat{A_{n}(R)}_{K}\coloneqq\widehat{A_{n}(R)}\otimes_{R}K=\left\\{\sum_{\alpha,\beta\in\mathbb{N}_{0}^{n}}c_{\alpha,\beta}\textnormal{{x}}^{\alpha}\partial^{\beta}:c_{\alpha,\beta}\in
K,c_{\alpha,\beta}\to 0\text{ as }|\alpha|+|\beta|\to\infty\right\\}.$

We also define $U(\mathfrak{g})_{K}\coloneqq
U(\mathfrak{g}_{R})\otimes_{R}K\subseteq\widehat{U(\mathfrak{g})}_{K}$.
Finally, as noted in [Man23, Corollary 2.5.4] $KG$ embeds into
$\widehat{U(\mathfrak{g})}_{K}$ via $g\mapsto e^{pg}$ for $g\in G$, so we can
restrict $\rho$ along this embedding to obtain a map
$\rho|_{KG}:KG\to\widehat{A_{n}(R)}_{K}$. The aim is to show that
$\ker{\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KG}}=0$.

Let us also mention the valuations defined on $U(\mathfrak{g}_{R})$ and
$A_{n}(R)$ extend in the natural way to valuations on
$\widehat{U(\mathfrak{g})}_{K},$ and $\widehat{A_{n}(R)}_{K}$ respectively.
That these are indeed valuations follows from [AW14, Lemma 5.2] together with
[ST02, Remark 4.6] for $\widehat{U(\mathfrak{g})}_{K}$ and [Pan07, Lemma
1.2.4] for the Weyl algebra (and by continuity for the completed Weyl algebra
as well).

From now on, we fix the following basis for $\mathfrak{g}$. Let
$e_{ij}=[\delta_{iI}\delta_{jJ}]_{IJ}$ for $1\leq i,j\leq 2n$ denote the
$2n\times 2n$ unit matrices. Then, for $1\leq i,j\leq n$ we let

$\displaystyle a_{ij}$ $\displaystyle=e_{ij}-e_{j+n,i+n}$ $\displaystyle
b_{ij}$ $\displaystyle=e_{i,j+n}+e_{j,i+n}$ $\displaystyle c_{ij}$
$\displaystyle=e_{i+n,j}+e_{j+n,i}$ $\displaystyle\rho(a_{ij})$
$\displaystyle=-\frac{1}{2}\delta_{ij}-x_{j}\partial_{i}$
$\displaystyle\rho(b_{ij})$ $\displaystyle=\partial_{i}\partial_{j}$
$\displaystyle\rho(c_{ij})$ $\displaystyle=-x_{i}x_{j}$

so note that in particular $b_{ij}=b_{ji}$ and $c_{ij}=c_{ji}$. We record here
the commutation relations for later reference.

######  Lemma  —

For all $1\leq i,j,k,l\leq n$ we have

$[a_{ij},a_{kl}]=\delta_{jk}a_{il}-\delta_{il}a_{kj},\qquad[a_{ij},b_{kl}]=\delta_{jk}b_{il}+\delta_{jl}b_{ik},\qquad[a_{ij},c_{kl}]=-\delta_{il}c_{jk}-\delta_{ik}c_{jl},$
$[b_{ij},c_{kl}]=\delta_{jk}a_{il}+\delta_{jl}a_{ik}+\delta_{ik}a_{jl}+\delta_{il}a_{jk},\qquad[b_{ij},b_{kl}]=0,\qquad[c_{ij},c_{kl}]=0.$

###### Proof.

For all $1\leq i,j,k,l\leq 2n$ we have

$[e_{ij},e_{kl}]=\delta_{jk}e_{il}-\delta_{il}e_{kj}$

which gives the relations above. ∎

In particular, we have the subalgebras

$\mathfrak{a}=\langle a_{ij}:1\leq i,j\leq
n\rangle_{\mathbb{Z}_{p}}\qquad\mathfrak{b}=\langle b_{ij}:1\leq i\leq j\leq
n\rangle_{\mathbb{Z}_{p}}\qquad\mathfrak{c}=\langle c_{ij}:1\leq i\leq j\leq
n\rangle_{\mathbb{Z}_{p}}$

and denote by $A,B$ and $C$ the corresponding uniform pro-$p$ groups. We
present the root space decompositions of $\mathfrak{sp}_{4}$ and of
$\mathfrak{sp}_{6}$ as illustrative examples. In particular, it is easy to see
from these that $\mathfrak{b}$ and $\mathfrak{c}$ are abelian.

Figure 1: Root space decomposition of $\mathfrak{sp}_{4}$, and image under
$\rho$

Figure 2: Root space decomposition of $\mathfrak{sp}_{6}$, and image under
$\rho$

In order to translate results between the different subalgebras of
$\mathfrak{g}$ we will make use of the following Fourier transforms of
$\widehat{A_{n}(R)}_{K}$. For $1\leq i,j\leq n$, let
$\tau_{i}:\widehat{A_{n}(R)}_{K}\to\widehat{A_{n}(R)}_{K}$ be the continuous
extension of the automorphism on $A_{n}(R)_{K}$ given by

$\displaystyle\tau_{i}(x_{j})=\begin{cases}\partial_{j}&\text{ if }i=j\\\
x_{j}&\text{ if }i\neq j\end{cases}$
$\displaystyle\tau_{i}(\partial_{j})=\begin{cases}-x_{j}&\text{ if }i=j\\\
\partial_{j}&\text{ if }i\neq j,\end{cases}$

and note that $\tau_{i}$ is still an automorphism.

######  Lemma  —

For every $1\leq i\leq n$, there is an automorphism
$\sigma_{i}:\widehat{U(\mathfrak{g})}_{K}\to\widehat{U(\mathfrak{g})}_{K}$
making the following diagram commute:

${\widehat{U(\mathfrak{g})}_{K}}$${\widehat{U(\mathfrak{g})}_{K}}$${{\widehat{A_{n}(R)}}_{K}}$${{\widehat{A_{n}(R)}}_{K}}$$\scriptstyle{\rho}$$\scriptstyle{\rho}$$\scriptstyle{\sigma_{i}}$$\scriptstyle{\tau_{i}}$

Explicitly,

$\sigma_{i}(a_{jk})=\begin{cases}-a_{ii}&\text{ if }j=k=i\\\ -c_{ik}&\text{ if
}j=i,k\neq i\\\ -b_{ji}&\text{ if }j\neq i,k=i\\\ a_{jk}&\text{ otherwise
}\end{cases}\qquad\sigma_{i}(b_{jk})=\begin{cases}-c_{ii}&\text{ if }j=k=i\\\
a_{ki}&\text{ if }j=i,k\neq i\\\ a_{ji}&\text{ if }j\neq i,k=i\\\
b_{jk}&\text{ otherwise }\end{cases}$
$\sigma_{i}(c_{jk})=\begin{cases}-b_{ii}&\text{ if }j=k=i\\\ a_{ik}&\text{ if
}j=i,k\neq i\\\ a_{ij}&\text{ if }j\neq i,k=i\\\ c_{jk}&\text{ otherwise.
}\end{cases}$

###### Proof.

From Section 2, $\tau_{i}$ preserves the image $\rho(\mathfrak{g})$, and
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{\mathfrak{g}}}$
is injective, so we can pull
${\left.\kern-1.2pt\tau_{i}\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{A_{n}(R)}}$
back to an automorphism of $\mathfrak{g}$ and complete it to get an
automorphism $\sigma_{i}$ of $\widehat{U(\mathfrak{g})}_{K}$ that makes the
above diagram commute. Since $\rho(\sigma_{i}(x))=\tau_{i}(\rho(x))$ for all
$x\in\mathfrak{g}$ we obtain the explicit formulas above. ∎

We also let the total Fourier transform on $\widehat{A_{n}(R)}_{K}$ be
$\tau\coloneqq\tau_{1}\circ\cdots\circ\tau_{n}$ with the corresponding
automorphism on $\widehat{U(\mathfrak{\mathfrak{g}})}_{K}$ be
$\sigma\coloneqq\sigma_{1}\circ\cdots\circ\sigma_{n}$.

##  3 The Gluing Lemma

In this section we develop the main tool for proving the injectivity of
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KG}}$,
partly reducing the problem to showing it is injective when restricted to
various subalgebras of $KG$. Note that setting $T=R$ gives Section 1.

######  Proposition  (Gluing Lemma) —

Let

  * •

$\mathfrak{n},\mathfrak{h}$ and
$\mathfrak{g}\coloneqq\mathfrak{n}\oplus\mathfrak{h}$ be finite rank
$\mathbb{Z}_{p}$-Lie algebras with corresponding uniform pro-$p$ groups $N,H$
and $G$.

  * •

$T=\bigcup_{d\in\mathbb{Z}}F_{d}T$ be an associative $R$-algebra with a
$\mathbb{Z}$-filtration by $R$-modules, and $T_{K}\coloneqq T\otimes_{R}K$.

  * •

$V$ be a $\widehat{U(\mathfrak{g})}_{K}\otimes_{K}T_{K}$-module and a
$K$-Banach space, such that $\left(N-1\right)\cdot B_{V}\subseteq pB_{V},$
where $B_{V}\coloneqq\\{v\in V:\left\lVert v\right\rVert\leq 1\\}$.

  * •

$\mathcal{V}$ be a subset of $V$ and for each $v\in\mathcal{V}$, let $W_{v}$
be a $\widehat{U(\mathfrak{h})}_{K}\otimes_{K}T_{K}$-submodule of $V$
contained in $U(\mathfrak{n})_{K}\cdot v$.

  * •

$V_{0}$ be a $KG\otimes_{K}T_{K}$-submodule of $V$ containing
$W\coloneqq\sum_{v\in\mathcal{V}}W_{v}$.

Suppose the following conditions hold:

  * •

For each $d\in\mathbb{Z}$ and $w\in W$, $RH\otimes_{R}F_{d}T\cdot w$ is
finitely generated over $R$.

  * •

$KH\otimes_{K}T_{K}$ acts faithfully on $W$.

  * •

For every $v\in\mathcal{V}$, the multiplication map

$KN\otimes_{K}\faktor{U(\mathfrak{n})_{K}}{I_{v}^{\prime}}\to\faktor{\widehat{U(\mathfrak{n})}_{K}}{I_{v}}$

is injective, where
$I_{v}^{\prime}\coloneqq\operatorname{Ann}_{U(\mathfrak{n})_{K}}(v)$ and
$I_{v}\coloneqq\operatorname{Ann}_{\widehat{U(\mathfrak{n})}_{K}}(v)$.

Then $V_{0}$ is a faithful $KG\otimes_{K}T_{K}$-module.

###### Proof.

Let $(n_{1},\ldots,n_{k})$ be a basis for $\mathfrak{n}$ and
$(N_{1},\ldots,N_{k})$ the corresponding topological generating set for $N$.
Note that it is enough to show that $V_{0}$ is faithful as an
$RG\otimes_{R}T$-module. Take
$\zeta\in\operatorname{Ann}_{RG\otimes_{R}T}(V_{0})$ and note that we can
write it as
$\zeta=\sum_{\alpha\in\mathbb{N}_{0}^{k}}\left(\mathbf{N}-1\right)^{\alpha}\zeta_{\alpha}$
where $\zeta_{\alpha}\in RH\otimes_{R}F_{d}T$ for some $d\in\mathbb{Z}$ which
does not depend on $\alpha$. Fix $v\in\mathcal{V}$ and $w\in W_{v}$. Now since

$RH\otimes_{R}F_{d}T\cdot w\subseteq U(\mathfrak{n})_{K}\cdot v$

is finitely generated over $R$, we can choose $u_{1},\ldots,u_{r}\in
U(\mathfrak{n})$ such that

$RH\otimes_{R}F_{d}T\cdot w\subseteq\langle u_{i}\cdot v:1\leq i\leq
r\rangle_{K}$

and $u_{1}+I_{v}^{\prime},\ldots,u_{r}+I_{v}^{\prime}$ are $K$-linearly
independent in $\faktor{U(\mathfrak{\mathfrak{n}})_{K}}{I_{v}^{\prime}}\cong
U(\mathfrak{\mathfrak{n}})_{K}\cdot v$. Moreover, by taking common
denominators for the coefficients of $u_{i}\cdot v$ in a finite generating set
for $RH\otimes_{R}F_{d}T\cdot w$ over $R$, we see that

$RH\otimes_{R}F_{d}T\cdot w\subseteq p^{-m}\langle u_{i}\cdot v:1\leq i\leq
r\rangle_{R}$

for some $m>0$ and so we can write

$\zeta_{\alpha}\cdot w=\sum_{i=1}^{r}\lambda_{i,\alpha}u_{i}\cdot v$

for some $\lambda_{i,\alpha}\in K$ which are uniformly bounded in $\alpha$ and
$i$. The condition $\left(N-1\right)\cdot B_{V}\subseteq pB_{V}$ then ensures

$0=\zeta\cdot w=\left[\sum_{1\leq i\leq
r}\sum_{\alpha\in\mathbb{N}_{0}^{k}}\lambda_{i,\alpha}(\mathbf{N}-1)^{\alpha}u_{i}\right]\cdot
v$

so

$\sum_{1\leq i\leq
r}\sum_{\alpha\in\mathbb{N}_{0}^{k}}\lambda_{i,\alpha}(\mathbf{N}-1)^{\alpha}u_{i}\in
I_{v}$

and by the injectivity of the multiplication map we have

$\sum_{1\leq i\leq
r}\left[\sum_{\alpha\in\mathbb{N}_{0}^{k}}\lambda_{i,\alpha}(\mathbf{N}-1)^{\alpha}\right]\otimes_{K}(u_{i}+I_{v}^{\prime})=0.$

Since $u_{i}+I_{v}^{\prime}$ are linearly independent over $K$, this can only
be the case if

$\sum_{\alpha\in\mathbb{N}_{0}^{k}}\lambda_{i,\alpha}(\mathbf{N}-1)^{\alpha}=0$

in $KN$ for all $1\leq i\leq r$. But then $\lambda_{i,\alpha}=0$ so
$\zeta_{\alpha}\cdot w=0$. By linearity this is true for all $w\in W$ and
since $KH\otimes_{K}T_{K}$ acts faithfully on $W$ we get that
$\zeta_{\alpha}=0$ for all $\alpha\in\mathbb{N}_{0}^{k}$. Then $\zeta=0$,
$V_{0}$ is faithful as an $RG\otimes_{R}T$-module, and so also as a
$KG\otimes_{K}T_{K}$-module. ∎

##  4 Faithfulness of the Metaplectic Representation

Our next goal is to show that
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KG}}:KG\to\widehat{A_{n}(R)}_{K}$
is injective. In order to do this, we will repeatedly apply Section 3 with the
$K$-algebras

$V\coloneqq K\langle
X_{1}^{\pm},\ldots,X_{n}^{\pm}\rangle=\left\\{\sum_{\alpha\in\mathbb{Z}^{n}}\lambda_{\alpha}X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\lambda_{\alpha}\in K,\lambda_{\alpha}\to 0\text{ as
}\left\lvert\alpha\right\rvert\to\infty\right\\},$

$V_{0}\coloneqq K\langle
X_{1},\ldots,X_{n}\rangle=\left\\{\sum_{\alpha\in\mathbb{N}_{0}^{n}}\lambda_{\alpha}X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\lambda_{\alpha}\in K,\lambda_{\alpha}\to 0\text{ as
}\left\lvert\alpha\right\rvert\to\infty\right\\}$

which are naturally also $\widehat{A_{n}(R)}_{K}$-modules, and so also
$\widehat{U(\mathfrak{g})}_{K}$-modules along the map
$\rho:\widehat{U(\mathfrak{g})}_{K}\to\widehat{A_{n}(R)}_{K}$. Here,
$|\alpha|\coloneqq|\alpha_{1}|+|\alpha_{2}|+...+|\alpha_{n}|$ for
$\alpha\in\mathbb{Z}^{n}$.

Note that both $V$ and $V_{0}$ are $K$-Banach spaces with the norm induced
from the $p$-adic valuation:

$v_{p}\left(\sum_{\alpha\in\mathbb{Z}^{n}}\lambda_{\alpha}X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}\right)\coloneqq\inf_{\alpha\in\mathbb{Z}^{n}}v(\lambda_{\alpha})$

and, from the definition of $\rho$, we have $g\cdot B_{V}\subseteq B_{V},$ for
any $g\in\mathfrak{g}$ so $\left(G-1\right)\cdot B_{V}\subseteq pB_{V}$ since
$e^{pg}-1\in p\,\widehat{U(\mathfrak{g}_{R})}$.

It follows from [AW13, Theorem 7.3] that $\widehat{A_{n}(R)}_{K}$ is a simple
ring, so $V_{0}$ and $V$ are faithful $\widehat{A_{n}(R)}_{K}$-modules. This
can also be seen explicitly as follows, where

$\operatorname{Ann}_{\widehat{A_{n}(R)}_{K}}(S)\coloneqq\left\\{\zeta\in\widehat{A_{n}(R)}_{K}:\zeta\cdot
s=0\text{ for all }s\in S\right\\}$

for a subset $S\subseteq V_{0}$.

######  Lemma  —

For $K[X_{1},\ldots,X_{n}]\subseteq V_{0}$ we have
$\operatorname{Ann}_{\widehat{A_{n}(R)}_{K}}K[X_{1},\ldots,X_{n}]=0$. In
particular, $V_{0}$ and $V$ are faithful $\widehat{A_{n}(R)}_{K}$-modules.

###### Proof.

Take

$\zeta=\sum_{\alpha,\beta\in\mathbb{N}_{0}^{n}}\lambda_{\alpha,\beta}x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}}\partial_{1}^{\beta_{1}}\cdots\partial_{n}^{\beta_{n}}\in\operatorname{Ann}_{\widehat{A_{n}(R)}_{K}}K[X_{1},\ldots,X_{n}].$

We consider the lexicographic order $<$ on $\mathbb{N}_{0}^{n}$ and argue by
induction on $\beta\in\mathbb{N}_{0}^{n}$ that $\lambda_{\alpha,\beta}=0$ for
all $\alpha\in\mathbb{N}_{0}^{n}$. Indeed, for $\beta=0$ we have

$0=\zeta\cdot
1=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\lambda_{\alpha,0}X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}},$

so $\lambda_{\alpha,0}=0$ for all $\alpha\in\mathbb{N}_{0}^{n}$. More
generally, note that for $\beta,\gamma\in\mathbb{N}_{0}^{n}$ with
$\gamma<\beta$ we have

$\partial_{1}^{\beta_{1}}\cdots\partial_{n}^{\beta_{n}}\cdot
X_{1}^{\gamma_{1}}\cdots X_{n}^{\gamma_{n}}=0.$

By induction, fix $\gamma\in\mathbb{N}_{0}^{n}$ such that for any
$\beta\in\mathbb{N}_{0}^{n}$ with $\beta<\gamma$ we have
$\lambda_{\alpha,\beta}=0$. Then

$\zeta\cdot X_{1}^{\gamma_{1}}\cdots
X_{n}^{\gamma_{n}}=\gamma_{1}!\cdots\gamma_{n}!\sum_{\alpha\in\mathbb{N}_{0}^{n}}\lambda_{\alpha,\gamma}X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}},$

so $\lambda_{\alpha,\gamma}=0$ for all $\alpha\in\mathbb{N}_{0}^{n}$. The
conclusion then follows. ∎

In particular, whenever we have a $K$-algebra $S$ with an algebra homomorphism
$\varphi:S\to\widehat{A_{n}(R)}_{K}$, we have that $V_{0}$ is a faithful
$S$-module along $\varphi$ if and only if $\varphi$ is injective.

###  4.1 The Multiplication Map

The image $\rho\left(\widehat{U(\mathfrak{c})}_{K}\right)$ embeds into $V$ by
the map $x_{i}\mapsto X_{i}$, for $1\leq i\leq n$, and under this
identification, the action of $\widehat{U(\mathfrak{c})}_{K}$ on $V$ is given
by multiplication $x\cdot v=\rho(x)v$. Then since $V$ is a domain, for any
non-zero $v\in V$ and subalgebra $\mathfrak{c}^{\prime}\subseteq\mathfrak{c}$
we have that

$\displaystyle\operatorname{Ann}_{U(\mathfrak{c^{\prime}})_{K}}(v)$
$\displaystyle=\ker{\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{U(\mathfrak{c^{\prime}})_{K}}},$
$\displaystyle\operatorname{Ann}_{\widehat{U(\mathfrak{c^{\prime}})}_{K}}(v)$
$\displaystyle=\ker{\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{\widehat{U(\mathfrak{c^{\prime}})}_{K}}},$

and so in particular they are independent of the non-zero $v\in V$. There is a
special class of subalgebras for which these annihilators are zero.

######  Lemma  —

Let $I\subseteq\\{1,\ldots,n\\}^{2}$ such that at most one of $(i,j)$ and
$(j,i)$ is in $I$ for each $i,j\in\\{1,\ldots,n\\}$, and suppose
$f=(f_{1},\ldots,f_{n}):\mathbb{N}_{0}^{I}\to\mathbb{N}_{0}^{n}$ given by

$f_{k}((\alpha_{ij})_{(i,j)\in I})=\sum_{(i,j)\in
I}\alpha_{ij}(\delta_{ik}+\delta_{kj})$

is injective. Then
$\ker{\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{\widehat{U(\mathfrak{c^{\prime}})}_{K}}}=0$
for $\mathfrak{c}^{\prime}=\left\langle c_{ij}:(i,j)\in
I\right\rangle_{\mathbb{Z}_{p}}$ and so

$KC^{\prime}\otimes_{K}\rho\left(U(\mathfrak{c^{\prime}})_{K}\right)\to\rho\left(\widehat{U(\mathfrak{c^{\prime}})}_{K}\right)$

is injective.

###### Proof.

Take

$\zeta=\sum_{\alpha\in\mathbb{N}_{0}^{I}}\lambda_{\alpha}\left(\prod_{(i,j)\in
I}c_{ij}^{\alpha_{ij}}\right)\in\ker{\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{\widehat{U(\mathfrak{c^{\prime}})}_{K}}}$

for some $\lambda_{\alpha}\in K$ uniformly bounded and note that by
construction

$0=\rho(\zeta)=\sum_{\alpha\in\mathbb{N}_{0}^{I}}(-1)^{|\alpha|}\lambda_{\alpha}\mathbf{x}^{f(\alpha)}.$

Since $f$ is injective, then $\lambda_{\alpha}=0$ for all
$\alpha\in\mathbb{N}_{0}^{I}$ and the first part follows. Finally, by [AW14,
Theorem 3.2] we have that
$KC^{\prime}\otimes_{K}U(\mathfrak{c^{\prime}})_{K}\to\widehat{U(\mathfrak{c^{\prime}})}_{K}$
is injective so the last part also follows. ∎

We will then show by induction that

$KC\otimes_{K}\rho(U(\mathfrak{c})_{K})\to\rho\left(\widehat{U(\mathfrak{c})}_{K}\right)$

is injective, so it is convenient to introduce the following subalgebras for
$1\leq k\leq n$.

$\displaystyle\mathfrak{c}^{k}$ $\displaystyle\coloneqq\langle c_{ij}:1\leq
i,j\leq k\rangle_{\mathbb{Z}_{p}}\oplus\langle c_{1j}:k+1\leq j\leq
n\rangle_{\mathbb{Z}_{p}},$ $\displaystyle\mathfrak{c}_{+}^{k}$
$\displaystyle\coloneqq\sigma_{k}\cdots\sigma_{n}\left(\mathfrak{c}^{k}\right),$
$\displaystyle\mathfrak{c}_{-}^{k}$
$\displaystyle\coloneqq\sigma_{1}\cdots\sigma_{k-1}\left(\mathfrak{c}^{k}\right)$
$\displaystyle\tilde{\mathfrak{c}}^{k}$ $\displaystyle\coloneqq\langle
c_{ik}:1\leq i\leq k\rangle_{\mathbb{Z}_{p}}\oplus\langle c_{1j}:k+1\leq j\leq
n\rangle_{\mathbb{Z}_{p}},$ $\displaystyle\tilde{\mathfrak{c}}_{+}^{k}$
$\displaystyle\coloneqq\sigma_{k}\cdots\sigma_{n}\left(\tilde{\mathfrak{c}}^{k}\right),$
$\displaystyle\tilde{\mathfrak{c}}_{-}^{k}$
$\displaystyle\coloneqq\sigma_{1}\cdots\sigma_{k-1}\left(\tilde{\mathfrak{c}}^{k}\right)$

Note that these are all abelian by Section 2, and also that
$\tilde{\mathfrak{c}}^{k}\subseteq\mathfrak{c}^{k}$,
$\tilde{\mathfrak{c}}_{+}^{k}\subseteq\mathfrak{c}_{+}^{k}$,
$\tilde{\mathfrak{c}}_{-}^{k}\subseteq\mathfrak{c}_{-}^{k}$ and
$\mathfrak{c}^{n}=\mathfrak{c}$. By Section 4.1, generators of the images of
these subalgebras under $\rho$ are shown below for $2\leq k\leq n$.

$x_{1}^{2}$ | $\cdots$ | $x_{1}x_{k-1}$ | $x_{1}x_{k}$ | $\cdots$ | $x_{1}x_{n}$
---|---|---|---|---|---
| $\ddots$ | $\vdots$ | $\vdots$ |  |
|  | $x_{k-1}^{2}$ | $x_{k-1}x_{k}$ |  |
|  |  | $x_{k}^{2}$ |  |
(a) Generators of $\rho(\mathfrak{c}^{k})$

$x_{1}x_{k}$ | $\cdots$ | $x_{1}x_{n}$
---|---|---
$\vdots$ |  |
$x_{k-1}x_{k}$ |  |
$x_{k}^{2}$ |  |
(b) Generators of $\rho(\tilde{\mathfrak{c}}^{k})$

$x_{1}^{2}$ | $\cdots$ | $x_{1}x_{k-1}$ | $x_{1}\partial_{k}$ | $\cdots$ | $x_{1}\partial_{n}$
---|---|---|---|---|---
| $\ddots$ | $\vdots$ | $\vdots$ |  |
|  | $x_{k-1}^{2}$ | $x_{k-1}\partial_{k}$ |  |
|  |  | $\partial_{k}^{2}$ |  |
(c) Generators of $\rho(\mathfrak{c}_{+}^{k})$

$x_{1}\partial_{k}$ | $\cdots$ | $x_{1}\partial_{n}$
---|---|---
$\vdots$ |  |
$x_{k-1}\partial_{k}$ |  |
$\partial_{k}^{2}$ |  |
(d) Generators of $\rho(\tilde{\mathfrak{c}}_{+}^{k})$

$\partial_{1}^{2}$ | $\cdots$ | $\partial_{1}\partial_{k-1}$ | $\partial_{1}x_{k}$ | $\cdots$ | $\partial_{1}x_{n}$
---|---|---|---|---|---
| $\ddots$ | $\vdots$ | $\vdots$ |  |
|  | $\partial_{k-1}^{2}$ | $\partial_{k-1}x_{k}$ |  |
|  |  | $x_{k}^{2}$ |  |
(e) Generators of $\rho(\mathfrak{c}_{-}^{k})$

$\partial_{1}x_{k}$ | $\cdots$ | $\partial_{1}x_{n}$
---|---|---
$\vdots$ |  |
$\partial_{k-1}x_{k}$ |  |
$x_{k}^{2}$ |  |
(f) Generators of $\rho(\tilde{\mathfrak{c}}_{-}^{k})$

######  Lemma  —

We have

$\displaystyle\mathfrak{c}_{+}^{k}$ $\displaystyle=\begin{cases}\langle
b_{1i}:1\leq i\leq n\rangle_{\mathbb{Z}_{p}}&\text{ if }k=1,\\\ \langle
c_{ij}:1\leq i,j\leq k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{ki}:1\leq
i\leq k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{i1}:k+1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}\oplus\langle b_{kk}\rangle_{\mathbb{Z}_{p}}&\text{
if }2\leq k\leq n;\end{cases}$ $\displaystyle\mathfrak{c}_{-}^{k}$
$\displaystyle=\begin{cases}\langle c_{1i}:1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}&\text{ if }k=1,\\\ \langle b_{ij}:1\leq i,j\leq
k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{ik}:1\leq i\leq
k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{1i}:k+1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}\oplus\langle c_{kk}\rangle_{\mathbb{Z}_{p}}&\text{
if }2\leq k\leq n;\end{cases}$ $\displaystyle\tilde{\mathfrak{c}}_{+}^{k}$
$\displaystyle=\begin{cases}\langle b_{1i}:1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}&\text{ if }k=1,\\\ \langle a_{ki}:1\leq i\leq
k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{i1}:k+1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}\oplus\langle b_{kk}\rangle_{\mathbb{Z}_{p}}&\text{
if }2\leq k\leq n;\end{cases}$ $\displaystyle\tilde{\mathfrak{c}}_{-}^{k}$
$\displaystyle=\begin{cases}\langle c_{1i}:1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}&\text{ if }k=1,\\\ \langle a_{ik}:1\leq i\leq
k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{1i}:k+1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}\oplus\langle c_{kk}\rangle_{\mathbb{Z}_{p}}&\text{
if }2\leq k\leq n.\end{cases}$

###### Proof.

For $k=1$ by definition we have

$\mathfrak{c}^{1}=\tilde{\mathfrak{c}}^{1}=\mathfrak{c}_{-}^{1}=\tilde{\mathfrak{c}}_{-}^{1}=\langle
c_{1i}:1\leq i\leq n\rangle_{\mathbb{Z}_{p}}$

and by Section 2 we have

$\displaystyle c_{11}$
$\displaystyle\xmapsto{\sigma_{n}}\cdots\xmapsto{\sigma_{2}}c_{11}\xmapsto{\sigma_{1}}-b_{11},$
$\displaystyle c_{1i}$
$\displaystyle\xmapsto{\sigma_{n}}\cdots\xmapsto{\sigma_{i+1}}c_{1i}\xmapsto{\sigma_{i}}a_{i1}\xmapsto{\sigma_{i-1}}\cdots\xmapsto{\sigma_{2}}a_{i1}\xmapsto{\sigma_{1}}-b_{1i}$
$\displaystyle\text{ if }2\leq i\leq n,$

giving the desired equalities for $\mathfrak{c}_{+}^{1}$ and
$\tilde{\mathfrak{c}}_{+}^{1}.$

Now for $2\leq k\leq n$, by Section 2 we have

$\displaystyle c_{ij}$
$\displaystyle\xmapsto{\sigma_{n}}\cdots\xmapsto{\sigma_{k}}c_{ij}$
$\displaystyle\text{ if }1\leq i,j\leq k-1,$ $\displaystyle c_{ki}$
$\displaystyle\xmapsto{\sigma_{n}}\cdots\xmapsto{\sigma_{k-1}}c_{ki}\xmapsto{\sigma_{k}}a_{ki}$
$\displaystyle\text{ if }1\leq i\leq k-1$ $\displaystyle c_{kk}$
$\displaystyle\xmapsto{\sigma_{n}}\cdots\xmapsto{\sigma_{k-1}}c_{kk}\xmapsto{\sigma_{k}}-b_{kk}$
$\displaystyle c_{1i}$
$\displaystyle\xmapsto{\sigma_{n}}\cdots\xmapsto{\sigma_{i+1}}c_{1i}\xmapsto{\sigma_{i}}a_{i1}\xmapsto{\sigma_{i-1}}\cdots\xmapsto{\sigma_{k}}a_{i1}$
$\displaystyle\text{ if }k+1\leq i\leq n$

giving the desired equalities for $\mathfrak{c}_{+}^{k}$ and
$\tilde{\mathfrak{c}}_{+}^{k}$. Moreover

$\displaystyle c_{ij}$
$\displaystyle\xmapsto{\sigma_{k-1}}\cdots\xmapsto{\sigma_{j+1}}c_{ij}\xmapsto{\sigma_{j}}a_{ji}\xmapsto{\sigma_{j-1}}\cdots\xmapsto{\sigma_{i+1}}a_{ji}\xmapsto{\sigma_{i}}-b_{ij}\xmapsto{\sigma_{i-1}}\cdots\xmapsto{\sigma_{1}}-b_{ij}$
$\displaystyle\text{ if }1\leq i<j\leq k-1,$ $\displaystyle c_{ii}$
$\displaystyle\xmapsto{\sigma_{k-1}}\cdots\xmapsto{\sigma_{i+1}}c_{ii}\xmapsto{\sigma_{i}}-b_{ii}\xmapsto{\sigma_{i-1}}\cdots\xmapsto{\sigma_{1}}-b_{ii}$
$\displaystyle\text{ if }1\leq i\leq k-1,$ $\displaystyle c_{ki}$
$\displaystyle\xmapsto{\sigma_{k-1}}\cdots\xmapsto{\sigma_{i+1}}c_{ki}\xmapsto{\sigma_{i}}a_{ik}\xmapsto{\sigma_{i-1}}\cdots\xmapsto{\sigma_{1}}a_{ik}$
$\displaystyle\text{ if }1\leq i\leq k-1$ $\displaystyle c_{kk}$
$\displaystyle\xmapsto{\sigma_{k-1}}\cdots\xmapsto{\sigma_{1}}c_{kk}$
$\displaystyle c_{1i}$
$\displaystyle\xmapsto{\sigma_{k-1}}\cdots\xmapsto{\sigma_{2}}c_{1i}\xmapsto{\sigma_{1}}a_{1i}$
$\displaystyle\text{ if }k+1\leq i\leq n$

giving the equalities for $\mathfrak{c}_{-}^{k}$ and
$\tilde{\mathfrak{c}}_{-}^{k}$. ∎

######  Lemma  —

For $2\leq k\leq n$, the map
$K\tilde{C}_{+}^{k}\otimes_{K}\rho\left(U\left(\mathfrak{c}_{+}^{k}\right)_{K}\right)\to\widehat{A_{n}(R)}_{K}$
is injective.

###### Proof.

Fix $2\leq k\leq n$ and let

$\displaystyle\mathfrak{a}_{+}^{k}$
$\displaystyle\coloneqq\mathfrak{a}\cap\mathfrak{c}_{+}^{k}=\mathfrak{a}\cap\tilde{\mathfrak{c}}_{+}^{k}=\langle
a_{ki}:1\leq i\leq k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{i1}:k+1\leq
i\leq n\rangle_{\mathbb{Z}_{p}},$ $\displaystyle\mathfrak{a}_{-}^{k}$
$\displaystyle\coloneqq\mathfrak{a}\cap\mathfrak{c}_{-}^{k}=\mathfrak{a}\cap\tilde{\mathfrak{c}}_{-}^{k}=\langle
a_{ik}:1\leq i\leq k-1\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{1i}:k+1\leq
i\leq n\rangle_{\mathbb{Z}_{p}},$

where the equalities follow by Section 4.1. We proceed in three steps.

Step 1:
$\operatorname{Ann}_{\widehat{U\left(\mathfrak{a}_{+}^{k}\right)}_{K}}K[X_{1},\ldots,X_{n}]=0$.

First note the action is well-defined since $\mathfrak{a}_{+}^{k}$ acts by
homogeneous operators of degree zero. Now take

$\xi=\sum_{\alpha,\beta}\mu_{\alpha,\beta}a_{k1}^{\alpha_{1}}\ldots
a_{k,k-1}^{\alpha_{k-1}}a_{k+1,1}^{\beta_{k+1}}\ldots
a_{n1}^{\beta_{n}}\in\operatorname{Ann}_{\widehat{U(\mathfrak{a}_{+}^{k})}_{K}}K[X_{1},\ldots,X_{n}]$

where the sum is over
$\alpha=(\alpha_{1},\ldots,\alpha_{k-1})\in\mathbb{N}_{0}^{k-1}$ and
$\beta=(\beta_{k+1},\ldots,\beta_{n})\in\mathbb{N}_{0}^{n-k}$ and
$\mu_{\alpha,\beta}\in K$ satisfy $\mu_{\alpha,\beta}\to 0$ as
$|\alpha|+|\beta|\to\infty$; in particular, they are uniformly bounded. Recall
that $\rho(a_{ij})=-x_{j}\partial_{i}$ so for any
$\gamma=(\gamma_{1},\ldots,\gamma_{n})\in\mathbb{N}_{0}^{n}$ we have

$0=\xi\cdot X_{1}^{\gamma_{1}}\cdots
X_{n}^{\gamma_{n}}=\sum_{\alpha,\beta}c_{\alpha,\beta}^{(\gamma)}X_{1}^{\gamma_{1}+\alpha_{1}+|\beta|}X_{2}^{\gamma_{2}+\alpha_{2}}\cdots
X_{k-1}^{\gamma_{k-1}+\alpha_{k-1}}X_{k}^{\gamma_{k}-|\alpha|}X_{k+1}^{\gamma_{k+1}-\beta_{k+1}}\cdots
X_{n}^{\gamma_{n}-\beta_{n}},$

where

$c_{\alpha,\beta}^{(\gamma)}=(-1)^{|\alpha|+|\beta|}\mu_{\alpha,\beta}\frac{\gamma_{k}!\cdots\gamma_{n}!}{(\gamma_{k}-|\alpha|)!(\gamma_{k+1}-\beta_{k+1})!\cdots(\gamma_{n}-\beta_{n})!}$

whenever $|\alpha|\leq\gamma_{k}$ and $\beta_{j}\leq\gamma_{j}$ for $k+1\leq
j\leq n$, and $c_{\alpha,\beta}^{(\gamma)}=0$ otherwise. Since

$(\alpha,\beta)\in\mathbb{N}_{0}^{n-1}\mapsto(\gamma_{1}+\alpha_{1}+|\beta|,\alpha_{2}+\gamma_{2},\ldots,\alpha_{k-1}+\gamma_{k-1},\gamma_{k}-|\alpha|,\gamma_{k+1}-\beta_{k+1},\ldots,\gamma_{n}-\beta_{n})\in\mathbb{N}_{0}^{n}$

is injective for any $\gamma\in\mathbb{N}_{0}^{n}$ we get that
$c_{\alpha,\beta}^{(\gamma)}=0$ for all
$\alpha\in\mathbb{N}_{0}^{k-1},\beta\in\mathbb{N}_{0}^{n-k}$ and
$\gamma\in\mathbb{N}_{0}^{n}$. Then $\mu_{\alpha,\beta}=0$ whenever
$|\alpha|\leq\gamma_{k}$ and $\beta_{j}\leq\gamma_{j}$ for all $k+1\leq j\leq
n$. Since $\gamma$ is arbitrary we get $\xi=0$.

Step 2: $K[X_{1},\ldots,X_{n}]$ is faithful as a
$KA_{+}^{k}\otimes_{K}K[x_{1},\ldots,x_{k-1},\partial_{k},\ldots,\partial_{n}]$-module.

Take

$\zeta=\sum_{\alpha,\beta}\zeta_{\alpha,\beta}x_{1}^{\alpha_{1}}\cdots
x_{k-1}^{\alpha_{k-1}}\partial_{k}^{\beta_{k}}\cdots\partial_{n}^{\beta_{n}}\in\operatorname{Ann}_{KA_{+}^{k}\otimes_{K}K[x_{1},\ldots,x_{k-1},\partial_{k},\ldots,\partial_{n}]}K[X_{1},\ldots,X_{n}]$

where the sum is over
$\alpha=(\alpha_{1},\ldots,\alpha_{k-1})\in\mathbb{N}_{0}^{k-1}$ and
$\beta=(\beta_{k},\ldots,\beta_{n})\in\mathbb{N}_{0}^{n-k+1}$, with only
finitely many $\zeta_{\alpha,\beta}\in KA_{+}^{k}$ non-zero. In particular,
there are $d_{1},d_{2}\in\mathbb{N}_{0}$ such that $\zeta_{\alpha,\beta}=0$
whenever $|\alpha|\geq d_{1}$ or $|\beta|\geq d_{2}$. Now for any monomial
$f\in K[X_{1},\ldots,X_{n}]$ we have

$0=X_{1}^{d_{1}+d_{2}}\partial_{k}^{d_{1}}\zeta\cdot
f=\sum_{\alpha,\beta}(-1)^{d_{1}+\left\lvert\beta\right\rvert}\zeta_{\alpha,\beta}a_{k1}^{\beta_{k}+d_{1}-(\alpha_{2}+\cdots+\alpha_{k-1})}a_{k2}^{\alpha_{2}}\cdots
a_{k,k-1}^{\alpha_{k-1}}a_{k+1,1}^{\beta_{k+1}}\cdots a_{n,1}^{\beta_{n}}\cdot
X_{1}^{d_{2}+|\alpha|-|\beta|}f.$

Fix $d\geq-d_{2}$. Since $\mathfrak{a}_{+}^{k}$ acts by homogeneous operators
of degree $0$, looking at the terms of total degree $d+d_{2}+\deg(f)$ in
$X_{1}^{d_{1}+d_{2}}\partial_{k}^{d_{1}}\zeta\cdot f$ we see that

$0=X_{1}^{d_{2}+d}\left(\sum_{|\alpha|-|\beta|=d}(-1)^{d_{1}+\left\lvert\beta\right\rvert}\zeta_{\alpha,\beta}a_{k1}^{\beta_{k}+d_{1}-(\alpha_{2}+\cdots+\alpha_{k-1})}a_{k2}^{\alpha_{2}}\cdots
a_{k,k-1}^{\alpha_{k-1}}a_{k+1,1}^{\beta_{k+1}}\cdots a_{n,1}^{\beta_{n}}\cdot
f\right).$

But since $K[X_{1},\ldots,X_{n}]$ is a domain and $f$ is an arbitrary monomial
we have that

$\sum_{|\alpha|-|\beta|=d}(-1)^{d_{1}+\left\lvert\beta\right\rvert}\zeta_{\alpha,\beta}a_{k1}^{\beta_{k}+d_{1}-(\alpha_{2}+\cdots+\alpha_{k-1})}a_{k2}^{\alpha_{2}}\cdots
a_{k,k-1}^{\alpha_{k-1}}a_{k+1,1}^{\beta_{k+1}}\cdots
a_{n,1}^{\beta_{n}}\in\operatorname{Ann}_{\widehat{U(\mathfrak{a}_{+}^{k})}_{K}}K[X_{1},\ldots,X_{n}]=0.$

Finally by [AW14, Theorem 3.2] we get $\zeta_{\alpha,\beta}=0$ whenever
$|\alpha|-|\beta|=d$. Since $d$ is arbitrary we get $\zeta=0$.

Step 3: $V_{0}$ is faithful as a
$K\tilde{C}_{+}^{k}\otimes_{K}\rho(U(\mathfrak{c}_{+}^{k})_{K})$-module.

By applying the automorphism $\sigma\otimes_{K}\tau$ it is enough to show that
$V_{0}$ is faithful as a
$K\tilde{C}_{-}^{k}\otimes_{K}\rho(U(\mathfrak{c}_{-}^{k})_{K})$-module.

We apply Section 3 with

$\displaystyle\mathfrak{n}$ $\displaystyle=\langle c_{ik}:1\leq i\leq
n\rangle_{\mathbb{Z}_{p}},$ $\displaystyle\mathfrak{h}$
$\displaystyle=\mathfrak{a}_{-}^{k},$ $\displaystyle T$
$\displaystyle=\rho\left(U\left(\mathfrak{c}_{-}^{k}\otimes_{\mathbb{Z}_{p}}R\right)\right),$
$\displaystyle\mathcal{V}$
$\displaystyle=\left\\{X_{k}^{-2nr}:r\in\mathbb{N}_{0}\right\\}\subseteq V,$

where we filter $T\subseteq A_{n}(R)$ by total degree. Also note that
$\mathfrak{n}\oplus\mathfrak{h}$ is a subalgebra since by Section 2 we have

$\displaystyle[c_{ik},a_{jk}]$ $\displaystyle=\delta_{ij}c_{kk}\text{ for
}1\leq i\leq n,\,1\leq j\leq k-1$ $\displaystyle[c_{ik},a_{1j}]$
$\displaystyle=\delta_{i1}c_{jk}\text{ for }1\leq i\leq n,\,k+1\leq j\leq n$

and that $\tilde{\mathfrak{c}}_{-}^{k}\subseteq\mathfrak{n}\oplus\mathfrak{h}$
by Section 4.1. For $v_{r}=X_{k}^{-2nr}\in\mathcal{V}$ let

$W_{v_{r}}\coloneqq\langle X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\,\alpha_{1},\ldots,\alpha_{k-1}\leq
r,\,\alpha_{1}+\alpha_{i}\leq r\text{ for }k+1\leq i\leq n,\,|\alpha|\text{
even}\rangle_{K}\subseteq U(\mathfrak{\mathfrak{n}})_{K}\cdot X_{k}^{-2nr}$

where the inclusion follows since

$X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}=(-1)^{nr+\frac{\left\lvert\alpha\right\rvert}{2}}c_{1k}^{\alpha_{1}}\cdots
c_{k-1,k}^{\alpha_{k-1}}c_{kk}^{\alpha_{k}+nr-\frac{\left\lvert\alpha\right\rvert}{2}}c_{k+1,k}^{\alpha_{k+1}}\cdots
c_{nk}^{\alpha_{n}}\cdot X_{k}^{-2nr}$

for $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}_{0}^{n}$ with
$|\alpha|$ even and $\alpha_{1},\ldots,\alpha_{k-1}\leq r$,
$\alpha_{1}+\alpha_{i}\leq r$ for $k+1\leq i\leq n$, since in particular
$|\alpha|\leq 2nr$. Each $W_{v_{r}}$ is both a
$\widehat{U(\mathfrak{h})}_{K}$-submodule since it is stable by the actions of
$x_{k}\partial_{1},\ldots,x_{k}\partial_{k-1}$ and
$x_{k+1}\partial_{1},\ldots,x_{n}\partial_{1}$, and a $T_{K}$-submodule as it
is also stable under the action of $\partial_{i}\partial_{j}$ for $1\leq
i,j\leq k-1$ and under the action of $x_{k}^{2}$. Then $W=\langle
X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}:|\alpha|\text{ even}\rangle_{K}$
and we have

  * •

For $f\in W$ we have

$RH\otimes_{R}F_{d}T\cdot f\subseteq\left\\{g\in K[X_{1},\ldots,X_{n}]:\deg
g\leq d+\deg f,\,v_{p}(g)\geq v_{p}(f)\right\\},$

which is finitely generated as an $R$-module, hence so is
$RH\otimes_{R}F_{d}T\cdot f$ as $R$ is noetherian.

  * •

$KH\otimes_{K}T_{K}$ acts faithfully on $W$ since if $\zeta\in
KH\otimes_{K}T_{K}$ annihilates $W$, it also annihilates
$K[X_{1},\ldots,X_{n}]=W+\partial_{1}\cdot W$ since $\partial_{1}$ commutes
with the image of $KH\otimes_{K}T_{K}$ in $\widehat{A_{n}(R)}_{K}$, and then
$\zeta=0$ by Step 2, after applying the automorphism $\sigma\otimes_{K}\tau$.

  * •

For every $v\in\mathcal{V}$, the multiplication map
$KN\otimes_{K}\rho\left(U(\mathfrak{n})_{K}\right)\to\rho\left(\widehat{U(\mathfrak{n})}_{K}\right)$
is injective by Section 4.1 as

$(\alpha_{1k},\ldots,\alpha_{nk})\in\mathbb{N}_{0}^{n}\mapsto(\alpha_{1k},\ldots,\alpha_{k-1,k},\alpha_{kk}+|\alpha|,\alpha_{k+1,k},\ldots,\alpha_{nk})\in\mathbb{N}_{0}^{n}$

is injective.

The result now follows from Section 3 ∎

######  Proposition  —

The multiplication map
$KC\otimes_{K}\rho\left(U(\mathfrak{c})_{K}\right)\to\rho\left(\widehat{U(\mathfrak{c})}_{K}\right)$
is injective.

###### Proof.

We show by induction that the multiplication map
$KC^{k}\otimes_{K}\rho\left(U(\mathfrak{c^{\textit{k}}})_{K}\right)\to\rho\left(\widehat{U(\mathfrak{c^{\textit{k}}})}_{K}\right)$
is injective for $1\leq k\leq n$. The base case $k=1$ follows from Section 4.1
as the map

$(\alpha_{11},\ldots,\alpha_{1n})\in\mathbb{N}_{0}^{n}\mapsto(2\alpha_{11}+\alpha_{12}+\cdots+\alpha_{1n},\alpha_{12},\ldots,\alpha_{1n})\in\mathbb{N}_{0}^{n}$

is injective. Now, for the induction, fix $2\leq k\leq n$ and note that it is
enough to show that $V_{0}$ is faithful over the algebra
$KC^{k}\otimes_{K}\rho\left(U(\mathfrak{c}^{k})_{K}\right)$. Applying the
automorphism $\sigma_{k}\cdots\sigma_{n}$ this is equivalent to showing that
$V_{0}$ is faithful over the algebra
$KC_{+}^{k}\otimes_{K}\rho\left(U(\mathfrak{c}_{+}^{k})_{K}\right)$. The
following pictures provide an illustration of this argument for
$\mathfrak{sp}_{4}$.

(a) Action of $\sigma_{2}$

(b) Applying Section 3

We then apply Section 3 with

$\displaystyle\mathfrak{n}$ $\displaystyle=\mathfrak{c}^{k-1},$
$\displaystyle\mathfrak{h}$ $\displaystyle=\tilde{\mathfrak{c}}_{+}^{k},$
$\displaystyle T$
$\displaystyle=\rho\left(U(\mathfrak{c}_{+}^{k}\otimes_{\mathbb{Z}_{p}}R)\right)\subseteq
R[x_{1},\ldots,x_{k-1},\partial_{k},\ldots,\partial_{n}],$
$\displaystyle\mathcal{V}$
$\displaystyle=\left\\{X_{1}^{-2nr}:r\in\mathbb{N}_{0}\right\\}\subseteq V,$

where we filter $T$ by total degree, $\mathfrak{n}\oplus\mathfrak{h}$ is a
subalgebra by Section 2 and
$\mathfrak{c}_{+}^{k}\subseteq\mathfrak{n}\oplus\mathfrak{h}$ by Section 4.1.
For $v_{r}=X_{1}^{-2nr}\in\mathcal{V}$ let

$W_{v_{r}}\coloneqq\langle X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\,\alpha_{k},\ldots,\alpha_{n}\leq
r,\,\left\lvert\alpha\right\rvert\text{ is even}\rangle_{K}\subseteq
U(\mathfrak{\mathfrak{n}})_{K}\cdot X_{1}^{-2nr}$

where the inclusion follows since

$X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}=(-1)^{nr+\left\lvert\alpha\right\rvert/2}c_{11}^{nr+\alpha_{1}-\left\lvert\alpha\right\rvert/2}c_{12}^{\alpha_{2}}\cdots
c_{1n}^{\alpha_{n}}\cdot X_{1}^{-2nr}$

for $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}_{0}^{n}$ with
$\alpha_{i}\leq r$ for all $i\geq k$ and $\left\lvert\alpha\right\rvert$ even.

These are both a $\widehat{U(\mathfrak{h})}_{K}$-submodule and a
$T_{K}$-submodule as both $\mathfrak{h}=\tilde{\mathfrak{c}}_{+}^{k}$ and
$\mathfrak{c}_{+}^{k}$ act by homogeneous operators of even degree, which can
only increase the degrees of $X_{1},\ldots,X_{k-1}$. Then $W=\langle
X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\left\lvert\alpha\right\rvert\text{ is even}\rangle_{K}$
and we have:

  * •

For $f\in W$ we have

$RH\otimes_{R}F_{d}T\cdot f\subseteq\left\\{g\in K[X_{1},\ldots,X_{n}]:\deg
g\leq d+\deg f,\,v_{p}(g)\geq v_{p}(f)\right\\},$

which is finitely generated as an $R$-module, hence so is
$RH\otimes_{R}F_{d}T\cdot f$ as $R$ is noetherian.

  * •

$KH\otimes_{K}T_{K}$ acts faithfully on $W$ by Section 4.1.

  * •

For every $v\in\mathcal{V}$, the multiplication map is injective by the
inductive hypothesis.

The result now follows from Section 3. ∎

###  4.2 The subalgebra $KA$

The next step is to show that $KA$ acts faithfully on $V_{0}$, so that
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KA}}$ is
injective. We start by looking at a Cartan subalgebra
$\mathfrak{a}_{0}\coloneqq\langle a_{ii}:1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}$ of $\mathfrak{g}$.

######  Lemma  —

$KA_{0}$ acts faithfully on $K[X_{1},\dots,X_{n}]\subseteq V_{0}$.

###### Proof.

We actually show that $\widehat{U(\mathfrak{a}_{0})}_{K}$ acts faithfully on
$K[X_{1},\dots,X_{n}]$, and note that the action is well-defined as
$\widehat{U(\mathfrak{a}_{0})}_{K}$ acts on $V_{0}$ by homogeneous operators
of degree 0. Let

$\zeta=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\lambda_{\alpha}a_{11}^{\alpha_{1}}\cdots
a_{nn}^{\alpha_{n}}\in\operatorname{Ann}_{\widehat{U(\mathfrak{a}_{0})}_{K}}K[X_{1},\dots,X_{n}]$

for $\lambda_{\alpha}\in K$ with $\lambda_{\alpha}\to 0$ as
$|\alpha|\to\infty$. Then for each $\beta\in\mathbb{N}_{0}^{n}$, since
$\rho(a_{ii})=-\frac{1}{2}-x_{i}\partial_{i}$ for $1\leq i\leq n$ we have

$0=\zeta\cdot X_{1}^{\beta_{1}}\cdots
X_{n}^{\beta_{n}}=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\lambda_{\alpha}\left(\prod_{i=1}^{n}\left(-\frac{1}{2}-\beta_{i}\right)^{\alpha_{i}}\right)X_{1}^{\beta_{1}}\cdots
X_{n}^{\beta_{n}}=f\left(-\frac{1}{2}-\beta_{1},\ldots,-\frac{1}{2}-\beta_{n}\right)X_{1}^{\beta_{1}}\cdots
X_{n}^{\beta_{n}}$

where

$f(X_{1},\ldots,X_{n})=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\lambda_{\alpha}X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}\in K\langle X_{1},\ldots,X_{n}\rangle.$

Then by [AW14, Lemma 4.7] it follows that $\lambda_{\alpha}=0$ for all
$\alpha\in\mathbb{N}_{0}^{n}$, and hence $\zeta=0$. ∎

######  Proposition  —

${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KA}}$ is
injective.

###### Proof.

For $1\leq k\leq n$ let

$\mathfrak{a}_{k}\coloneqq\langle a_{ij}:1\leq i,j\leq
k\rangle_{\mathbb{Z}_{p}}\oplus\langle a_{ii}:k+1\leq i\leq
n\rangle_{\mathbb{Z}_{p}}\qquad\mathfrak{b}_{k}\coloneqq\langle b_{ik}:1\leq
i\leq k-1\rangle_{\mathbb{Z}_{p}}\qquad\mathfrak{c}_{k}\coloneqq\langle
c_{ik}:1\leq i\leq k-1\rangle_{\mathbb{Z}_{p}}$
$\mathfrak{a}_{k}^{-}\coloneqq\mathfrak{a}_{k-1}\oplus\mathfrak{c}_{k}\qquad\mathfrak{a}_{k}^{+}\coloneqq\sigma(\mathfrak{a}_{k}^{-})=\mathfrak{a}_{k-1}\oplus\mathfrak{b}_{k}\qquad\mathfrak{a}_{k}^{\pm}\coloneqq\sigma_{k}(\mathfrak{a}_{k})=\mathfrak{a}_{k-1}\oplus\mathfrak{b}_{k}\oplus\mathfrak{c}_{k},$

where the equalities follow by Section 2 and note that these are subalgebras
by Section 2. We prove by induction on $k$ for $1\leq k\leq n$ that
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KA_{k}}}$
is injective. By Section 4 this is equivalent to showing that $V_{0}$ is
faithful as a $KA_{k}$-module. The case $k=1$ follows then from Section 4.2 as
$\mathfrak{a}_{1}=\mathfrak{a}_{0}$. Now fix $2\leq k\leq n$ and suppose
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KA_{k-1}}}$
is injective. By applying the automorphism $\sigma_{k}$, it is then enough to
show that $KA_{k}^{\pm}$ acts faithfully on $V_{0}$, which we do in two steps.
The following pictures provide an illustration of this argument for
$\mathfrak{sp}_{4}$.

(a) Action of $\sigma_{1}$

$\to$

(b) Using Section 3 to show
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KA_{k}^{\pm}}}$
is injective

Step 1: The restriction
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KA_{k}^{+}}}$
is injective.

We apply Section 1 with

$\displaystyle\mathfrak{n}$ $\displaystyle=\mathfrak{c}_{k},$
$\displaystyle\mathfrak{h}$ $\displaystyle=\mathfrak{a}_{k-1},$
$\displaystyle\mathcal{V}$ $\displaystyle=\\{X_{k}^{r_{k}}\cdots
X_{n}^{r_{n}}:r_{k}\in\mathbb{Z},r_{k+1},\ldots,r_{n}\in\mathbb{N}_{0}\\}\subseteq
V$

and note that $\mathfrak{n}\oplus\mathfrak{h}=\mathfrak{a}_{k}^{-}$. For
$v=X_{k}^{r_{k}}\cdots X_{n}^{r_{n}}\in\mathcal{V}$ let

$W_{v}\coloneqq\langle X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\,\alpha_{k}=\alpha_{1}+\cdots+\alpha_{k-1}+r_{k},\,\alpha_{i}=r_{i}\text{
for }i>k\rangle_{K}\subseteq U(\mathfrak{n})_{K}\cdot X_{k}^{r_{k}}\cdots
X_{n}^{r_{n}},$

where the inclusion follows since

$X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}=(-1)^{\alpha_{1}+\cdots+\alpha_{k-1}}c_{1k}^{\alpha_{1}}\cdots
c_{k-1,k}^{\alpha_{k-1}}\cdot X_{k}^{r_{k}}\cdots X_{n}^{r_{n}}$

for all $\alpha\in\mathbb{N}_{0}^{n}$ with
$\alpha_{k}=\alpha_{1}+\cdots+\alpha_{k-1}+r_{k}$ and $\alpha_{i}=r_{i}$ for
$i>k$. These are $\widehat{U(\mathfrak{h})}_{K}$-submodules since
$\mathfrak{a}_{k-1}$ acts by homogeneous elements of degree 0 that cannot
change the individual degrees in any of $X_{k},\ldots,X_{n}$ nor the total
degree in $X_{1},\ldots,X_{k-1}$. Then $W=K[X_{1},\ldots,X_{n}]$ and we have

  * •

$RH$ acts locally finitely on $W$ since for $f\in W$,

$RH\cdot f\subseteq\left\\{g\in K[X_{1},\ldots,X_{n}]:\deg g=\deg
f,\,v_{p}(g)\geq v_{p}(f)\right\\},$

which is finitely generated as an $R$-module, hence so is $RH\cdot f$ as $R$
is noetherian.

  * •

$KH$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ since by the induction
hypothesis
$\ker{\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KH}}=0$
and $\rho(KH)$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ by Section 4.

  * •

The multiplication map is injective by Section 4.1 (or alternatively by
Section 4.1).

So $V_{0}$ is faithful as a $KA_{k}^{-}$-module, and applying the automorphism
$\sigma$ we get that $KA_{k}^{+}$ acts faithfully on $V_{0}$ as well. Then
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KA_{k}^{+}}}$
is injective.

Step 2: $V_{0}$ is a faithful $KA_{k}^{\pm}$-module.

Now we apply Section 1 again, replacing $\mathfrak{h}=\mathfrak{a}_{k-1}$ with
$\mathfrak{h}=\mathfrak{a}_{k}^{+}=\mathfrak{a}_{k-1}\oplus\mathfrak{b}_{k}$
and keeping everything else unchanged. That is,

$\displaystyle\mathfrak{n}$ $\displaystyle=\mathfrak{c}_{k},$
$\displaystyle\mathfrak{h}$ $\displaystyle=\mathfrak{a}_{k}^{+}$
$\displaystyle\mathcal{V}$ $\displaystyle=\\{X_{k}^{r_{k}}\cdots
X_{n}^{r_{n}}:r_{k}\in\mathbb{Z},r_{k+1},\ldots,r_{n}\in\mathbb{N}_{0}\\}\subseteq
V.$

and

$W_{v}\coloneqq\langle X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\,\alpha_{k}=\alpha_{1}+\cdots+\alpha_{k-1}+r_{k},\,\alpha_{i}=r_{i}\text{
for }i>k\rangle_{K}\subseteq U(\mathfrak{n})_{K}\cdot X_{k}^{r_{k}}\cdots
X_{n}^{r_{n}}.$

Note that $\mathfrak{n}\oplus\mathfrak{h}=\mathfrak{a}_{k}^{\pm}$ and that
$W_{v}$ are still $\widehat{U(\mathfrak{h})}_{K}$-submodules since the action
of $\mathfrak{b}_{k}$ keeps $\alpha_{k}-(\alpha_{1}+\cdots+\alpha_{k-1})$
invariant. Then we have

  * •

$RH$ still acts locally finitely on $W$ since for $f\in W$,

$RH\cdot f\subseteq\left\\{g\in K[X_{1},\ldots,X_{n}]:\deg g\leq\deg
f,\,v_{p}(g)\geq v_{p}(f)\right\\},$

which is finitely generated as an $R$-module, hence so is $RH\cdot f$ as $R$
is noetherian.

  * •

$KH$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ since by the previous step
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KH}}$ is
injective, and $\rho(KH)$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ by
Section 4.

  * •

The multiplication map is injective by Section 4.1 (or alternatively by
Section 4.1).

So $V_{0}$ is faithful as a $KA_{k}^{\pm}$-module, completing the induction. ∎

###  4.3 Final Gluing

We can now glue all three subalgebras together to prove the main result.

######  Theorem  —

${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KG}}$ is
injective.

###### Proof.

Let $\mathfrak{s}\coloneqq\mathfrak{a}\oplus\mathfrak{c}$ and
$\mathfrak{t}\coloneqq\mathfrak{a}\oplus\mathfrak{b}$ and note that these are
subalgebras by Section 2. The following pictures provide an illustration of
this argument for $\mathfrak{sp}_{4}$.

$\includegraphics[page=10,raise={-3986129sp}]{images.pdf}\qquad\to\qquad\quad\includegraphics[page=11,raise={-4121799sp}]{images.pdf}$

We first show that $V_{0}$ is faithful as a $KS$-module by applying Section 1
with

$\displaystyle\mathfrak{n}=\mathfrak{c},\qquad\mathfrak{h}=\mathfrak{a},\qquad\mathcal{V}=\\{1,X_{1}^{-1}\\}\subseteq
V,$ $\displaystyle W_{1}=\langle X_{1}^{\alpha_{1}}\ldots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\left\lvert\alpha\right\rvert\text{
is even}\rangle_{K}\subseteq U(\mathfrak{n})_{K}\cdot 1,$ $\displaystyle
W_{X_{1}^{-1}}=\langle X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\left\lvert\alpha\right\rvert\text{
is odd}\rangle_{K}\subseteq U(\mathfrak{n})_{K}\cdot X_{1}^{-1},$

where the inclusions follow as $\\{X_{i}X_{j}:1\leq i,j\leq n\\}$ generates
$W_{1}$. Note that $\mathfrak{n}\oplus\mathfrak{h}=\mathfrak{s}$ and that
$W_{1}$ and $W_{X_{1}^{-1}}$ are $\widehat{U(\mathfrak{h})}_{K}$-modules since
the elements of $\mathfrak{a}$ act by homogeneous operators of degree $0$.
Then $W=K[X_{1},\ldots,X_{n}]$ and we have

  * •

$RH$ acts locally finitely on $W$ since for $f\in W$,

$RH\cdot f\subseteq\left\\{g\in K[X_{1},\ldots,X_{n}]:\deg g=\deg
f,\,v_{p}(g)\geq v_{p}(f)\right\\},$

which is finitely generated as an $R$-module, hence so is $RH\cdot f$ as $R$
is noetherian.

  * •

$KH$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ since by Section 4.2
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KH}}$ is
injective, and $\rho(KH)$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ by
Section 4.

  * •

The multiplication map is injective by Section 4.1.

Then $V_{0}$ is faithful as a $KS$-module and by applying the automorphism
$\sigma$ it is also faithful as a $KT$-module so
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KT}}$ is
injective.

We finally apply Section 3 again, only changing $\mathfrak{h}$. In particular,

$\displaystyle\mathfrak{n}=\mathfrak{c},\qquad\mathfrak{h}=\mathfrak{t},\qquad\mathcal{V}=\\{1,X_{1}^{-1}\\}\subseteq
V,$ $\displaystyle W_{1}=\langle X_{1}^{\alpha_{1}}\ldots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\left\lvert\alpha\right\rvert\text{
is even}\rangle_{K}\subseteq U(\mathfrak{n})_{K}\cdot 1,$ $\displaystyle
W_{X_{1}^{-1}}=\langle X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}:\alpha\in\mathbb{N}_{0}^{n},\left\lvert\alpha\right\rvert\text{
is odd}\rangle_{K}\subseteq U(\mathfrak{n})_{K}\cdot X_{1}^{-1},$

Note that $\mathfrak{n}\oplus\mathfrak{h}=\mathfrak{g}$ and $W_{1}$ and
$W_{X_{1}^{-1}}$ are still $\widehat{U(\mathfrak{t})}_{K}$-submodules since
the elements of $\mathfrak{b}$ act by homogeneous operators of even non-
positive degree. Then:

  * •

$RH$ acts locally finitely on $W=K[X_{1},\ldots,X_{n}]$ since for $f\in W$,

$RH\cdot f\subseteq\left\\{g\in K[X_{1},\ldots,X_{n}]:\deg g\leq\deg
f,\,v_{p}(g)\geq v_{p}(f)\right\\},$

which is finitely generated as an $R$-module, hence so is $RH\cdot f$ as $R$
is noetherian.

  * •

$KH$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ since
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KH}}={\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KT}}$
is injective, and $\rho(KH)$ acts faithfully on $K[X_{1},\ldots,X_{n}]$ by
Section 4.

  * •

The multiplication map is injective by Section 4.1.

So $V_{0}$ is faithful as a $KG$-module and hence
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KG}}$ is
injective. ∎

##  5 Abelian subalgebras

In this section, let $G$ be any uniform pro-$p$ group, and $H\leq G$ a
torsion-free abelian pro-$p$ group. We give conditions for maps out of $KH$
with infinite-dimensional image to be injective, first in terms of closed
subgroups of $\operatorname{Aut}(H)$, and then in terms of the normaliser of
$\mathfrak{h}$ in $\mathfrak{g}$.

###  5.1 Invariant ideals in commutative algebras

Let $H$ have topological generating set $H_{1},\ldots,H_{d}$. We equip $KH$
with the $p$-adic filtration

$w_{p}\left(\sum_{\alpha\in\mathbb{N}_{0}^{d}}\lambda_{\alpha}\left(\mathbf{H}-1\right)^{\alpha}\right)=\inf_{\alpha\in\mathbb{N}_{0}^{d}}v(\lambda_{\alpha})$

where
$(\mathbf{H}-1)^{\alpha}=(H_{1}-1)^{\alpha_{1}}\cdots(H_{d}-1)^{\alpha_{d}}$
and the $\lambda_{\alpha}\in K$ are uniformly bounded. Note that $w_{p}$ is
complete and separated with $\operatorname{gr}KH\cong kH[s,s^{-1}]$, where $s$
is the image of the uniformiser of $K$.

We endow $\operatorname{Aut}(H)$ with the congruence topology as defined in
[Dix+03, §5.2], and let $\Gamma\leq\operatorname{Aut}(H)$ be a closed subgroup
such that

  1. (i)

$\mathcal{L}(H)$ is an irreducible $\mathcal{L}(\Gamma)$-module;

  2. (ii)

$[\Gamma,H]\coloneqq\\{\gamma(h)h^{-1}:\gamma\in\Gamma,\,h\in H\\}\subseteq
H^{p}$.

Here $\mathcal{L}$ denotes the corresponding $\mathbb{Q}_{p}$-Lie algebra as
in [Dix+03, §9.5]. Note that (ii) forces $\Gamma$ to be isomorphic to a closed
subgroup of the first congruence subgroup $\Gamma_{1}\coloneqq
1+M_{d}(\mathbb{Z}_{p})$ of $\operatorname{GL}_{d}(\mathbb{Z}_{p})$, which is
uniform by [Dix+03, Theorem 5.2]. In particular then, $\Gamma$ is a finitely
generated pro-$p$ group, and every
$\Gamma^{l}\coloneqq\left\langle\gamma^{l}:\gamma\in\Gamma\right\rangle\leq\operatorname{Aut}(H)$
is also a closed subgroup by [Mar94].

We extend the action of $\Gamma$ first to $RH$ by linearity and then to $KH$,
as the $\Gamma$-action does not affect the $w_{p}$-filtration. Similarly,
$\Gamma$ acts on the local ring $kH$, and is exactly chosen so that [Ard12,
Corollary 8.1] gives

######  Lemma  —

For $l\in\mathbb{N}_{0}$, the only $\Gamma^{l}$-invariant prime ideals of $kH$
are 0 and the maximal ideal $\mathfrak{m}=(H_{1}-1,\ldots,H_{d}-1)$.

This uses the observation that $\mathcal{L}(\Gamma^{l})=\mathcal{L}(\Gamma)$,
which follows directly from the definition of $\mathcal{L}$. This maximality
condition lifts to a condition on $\Gamma$-invariant ideals in $KH$.

######  Proposition  —

Let $I$ be an non-zero, $\Gamma$-invariant ideal in $KH$. Then $I$ has finite
$K$-codimension.

###### Proof.

If $I=KH$ this is trivial, so suppose not. Consider the
$\operatorname{gr}K$-module $\operatorname{gr}KH$, and the non-zero, proper,
$\Gamma$-invariant, graded submodule $\operatorname{gr}I$. Identifying
$\operatorname{gr}KH\cong kH[s,s^{-1}]$, we see that $s$ is a unit, and so
each graded component of $\operatorname{gr}I$ is equal. Thus we may write
$\operatorname{gr}I=J[s,s^{-1}]$ for some ideal $J\subseteq kH$.

As $kH$ is noetherian, $\textnormal{rad}(J)=\bigcap_{i=1}^{m}\mathfrak{p}_{i}$
for some non-zero prime ideals $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{m}$. As
$\textnormal{rad}(J)$ is also $\Gamma$-invariant and $\Gamma$ acts by
automorphisms, each element of $\Gamma$ permutes the $\mathfrak{p}_{i}$, so we
can find an $l\geq 1$ such that $\Gamma^{l}$ fixes all of the
$\mathfrak{p}_{i}$. So by Section 5.1, $\textnormal{rad}(J)=\mathfrak{m}$.

Thus we can find an $r\geq 0$ such that $\mathfrak{m}^{r}\subseteq J$, and
since $\faktor{kH}{\mathfrak{m}^{n}}$ is finite-dimensional over $k$ so is
$\faktor{kH}{J}$. Now note that

$\faktor{\operatorname{gr}KH}{\operatorname{gr}I}\cong\faktor{kH[s,s^{-1}]}{J[s,s^{-1}]}\cong\left(\faktor{kH}{J}\right)[s,s^{-1}]$

so $\faktor{\operatorname{gr}KH}{\operatorname{gr}I}$ is finite-dimensional
over $\operatorname{gr}K\cong k[s,s^{-1}]$. By [LO96, Theorem I.4.2.4] we have
$\faktor{\operatorname{gr}KH}{\operatorname{gr}I}\cong\operatorname{gr}\left(\faktor{KH}{I}\right)$
so $\operatorname{gr}\left(\faktor{KH}{I}\right)$ is generated over
$\operatorname{gr}K$ by finitely many elements of degree 0. Finally, by [LO96,
Theorem I.5.7], $\faktor{KH}{I}$ has finite $K$-dimension. ∎

This immediately gives the following corollary.

######  Corollary  —

Let $T$ be a filtered $K$-algebra, and let $\psi:KH\to T$ a filtered
$K$-algebra homomorphism such that $\psi(KH)$ is not finite $K$-dimensional.
If $\ker\psi$ is $\Gamma$-invariant, then $\psi$ is injective.

###  5.2 Abelian subalgebras

A key example of such automorphism groups $\Gamma$ comes from conjugation.
Suppose now that $H$ is subgroup of a uniform pro-$p$ group $G$, and let
$\mathfrak{h},\mathfrak{g}$ be their associated $\mathbb{Z}_{p}$-Lie algebras.
Let $\mathfrak{n}\subseteq\mathfrak{g}$ be a subalgebra contained in the
normaliser $\\{x\in\mathfrak{g}:[x,y]\in\mathfrak{h}\text{ for
}y\in\mathfrak{h}\\}$ of $\mathfrak{h}$, and $N$ its associated pro-$p$ group.
We assume that $\mathfrak{g}/\mathfrak{n}$ is torsion-free as a
$\mathbb{Z}_{p}$-module so that by [Dix+03, Proposition 7.15], $N\leq_{c}G$ is
a closed uniform subgroup.

######  Proposition  —

Let $T$ be a filtered $K$-algebra, and let $\psi:KG\to T$ be a filtered
$K$-algebra homomorphism such that $\psi(KH)$ is not finite $K$-dimensional.
If $\mathcal{L}(H)$ is an irreducible $\mathcal{L}(N)$-module, then
${\left.\kern-1.2pt\psi\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KH}}$ is
injective.

###### Proof.

We first show that the conjugation homomorphism
$\varphi:N\to\operatorname{Aut}(H)$ by $x\mapsto(\varphi_{x}:y\mapsto
xyx^{-1})$ is continuous. To this end, suppose a net
$(x_{\lambda})_{\lambda\in\Lambda}$ converges to $1_{N}$ in $N$. It suffices
to prove $\varphi_{x_{\lambda}}$ converges to $\operatorname{id}_{H}$ in the
congruence topology; that is, that for any open normal subgroup
$H^{\prime}\lhd_{o}H$, there is a $\mu_{H^{\prime}}\in\Lambda$ such that
$[\varphi_{x_{\lambda}},H]\subseteq H^{\prime}$ for
$\lambda\geq\mu_{H^{\prime}}$.

Fix some $H^{\prime}\lhd_{o}H$. For any $y\in H$,
$\varphi_{x_{\lambda}}(y)y^{-1}=[x_{\lambda},y]$ converges to $1_{H}$ in $H$,
so there is a $\mu_{y}\in\Lambda$ such that $[x_{\lambda},y]\in H^{\prime}$
for $\lambda\geq\mu_{y}$. If $h\in H^{\prime}$, we also see that
$[x_{\lambda},hy]\in H^{\prime}$ by normality, so $[x_{\lambda},g]\in
H^{\prime}$ whenever $g\in H^{\prime}y$, $\lambda\geq\mu_{y}$. As $H$ is
profinite, $H^{\prime}$ has finite index in $H$, so it is enough to take
$\mu_{N^{\prime}}\geq\mu_{y_{i}}$, $1\leq i\leq m$, for some finite collection
$y_{1},\ldots,y_{m}$ of coset representatives of $H/H^{\prime}$.

We check that $\Gamma=\varphi(N)$ satisfies conditions (i) and (ii).

  1. (i)

$\mathcal{L}(\varphi):\mathcal{L}(N)\to\mathcal{L}(\varphi(N))$ is surjective
by [Dix+03, §9, Ex. 7], so $\mathcal{L}(H)$ is irreducible as an
$\mathcal{L}(\varphi(N))$-module.

  2. (ii)

$[\mathfrak{n},\mathfrak{h}]\leq\mathfrak{h}\cap p\mathfrak{g}=p\mathfrak{h}$
as $\mathfrak{g}$ is a powerful Lie algebra and $\mathfrak{g}/\mathfrak{h}$ is
torsion-free. Then the Campbell-Baker-Hausdorff formula guarantees that
$[\varphi(N),H]=[N,H]\leq H^{p}$.

Finally, $\ker\psi$ is a (two-sided) ideal of $KG$ and the conjugation action
of $N$ on $G$ fixes $H$, so both $\ker\psi$ and $KH$ are $N$-invariant. Thus
$\ker{\left.\kern-1.2pt\psi\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KH}}=KH\cap\ker\psi$
is $N$-invariant, and we can apply Section 5.1 with $\Gamma=\varphi(N)$. ∎

We apply Section 5.2 with $G=\exp(p\mathfrak{sp}_{2n}(\mathbb{Z}_{p}))$,
$H=C$, $N=A$ and $\psi=\rho$ as in Section 2 to obtain the injectivity of the
restrictions
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KB}}$ and
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KB}}$.
Note then $\mathfrak{g}=p\mathfrak{sp}_{2n}(\mathbb{Z}_{p})$,
$\mathfrak{h}=p\mathfrak{c}$, $\mathfrak{n}=p\mathfrak{c}$ in the notation of
Section 2.

######  Corollary  —

${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KC}}$ and
${\left.\kern-1.2pt\rho\mathchoice{\vphantom{\big{|}}}{}{}{}\right|_{KB}}$ are
injective.

###### Proof.

$KC$ acts on the Tate algebra $K\langle X_{1},X_{2},...X_{n}\rangle$ via
$\rho$ with

$(C_{12}-1)\cdot
X_{1}^{n_{1}}X_{2}^{n_{2}}=\left(\exp\left(-px_{1}x_{2}\right)-1\right)\cdot
X_{1}^{n_{1}}X_{2}^{n_{2}}=-pX_{1}^{n_{1}+1}X_{2}^{n_{2}+1}+\frac{p^{2}}{2}X_{1}^{n_{1}+2}X_{2}^{n_{2}+2}+\cdots$

which shifts the total trailing degree by $2$, and hence the actions of
$(C_{12}-1)^{n}$ for $n\in\mathbb{N}$ are linearly independent. Then the image
of $\rho(KC)$ is infinite dimensional.

By Section 2, $p\mathfrak{a}$ is contained in the normaliser of
$p\mathfrak{c}$. $p\mathfrak{a}\otimes_{\mathbb{Z}_{p}}\bar{\mathbb{Q}}_{p}$
contains a copy of $\mathfrak{sl}_{n}(\bar{\mathbb{Q}}_{p})$, so we can view
$p\mathfrak{c}\otimes_{\mathbb{Z}_{p}}\bar{\mathbb{Q}_{p}}$ as a
$\mathfrak{sl}_{n}(\bar{\mathbb{Q}}_{p})$-module by restriction of scalars.
Now under this action, $p\mathfrak{c}$ is just a highest weight module with
integral dominant weight, hence simple. Then
$\mathcal{L}(C)=p\mathfrak{c}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ is simple
for $\mathfrak{sl}_{n}(\mathbb{Q}_{p})$ and therefore also for
$\mathcal{L}(A)=p\mathfrak{a}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

The argument for $H=B$ is completely analogous. ∎

## References

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  * [Dix+03] John D Dixon, Marcus PF Du Sautoy, Avinoam Mann and Dan Segal  “Analytic Pro-p Groups”  Cambridge University Press, 2003  DOI: 10.1017/CBO9780511470882
  * [Pan07] A Pangalos  “p-adic Weyl algebras”, 2007  URL: https://d-nb.info/987679023/34
  * [Ard12] Konstantin Ardakov  “Prime ideals in nilpotent Iwasawa algebras”  In _Inventiones mathematicae_ 190.2 Springer, 2012, pp. 439–503  DOI: 10.1017/CBO9780511470882
  * [AW13] Konstantin Ardakov and Simon Wadsley  “On irreducible representations of compact p-adic analytic groups”  In _Annals of Mathematics_ JSTOR, 2013, pp. 453–557
  * [AW14] Konstantin Ardakov and Simon Wadsley  “Verma modules for Iwasawa algebras are faithful”  In _Münster J. of Math. 7 (2014), 5–26_ 7, 2014, pp. 5–26  DOI: urn:nbn:de:hbz:6-22359595785
  * [Man23] Stephen Mann  “Faithfulness of generalised Verma modules for Iwasawa algebras”  In _Journal of Algebra_ 614, 2023, pp. 417–457  DOI: 10.1016/j.jalgebra.2022.09.018


# TF.Learn: TensorFlow’s High-level Module for Distributed Machine Learning

Yuan Tang<EMAIL_ADDRESS>
Uptake Technologies, Inc., 600 W. Chicago Ave, Suite 620, Chicago, IL 60654

###### Abstract

TF.Learn is a high-level Python module for distributed machine learning inside
TensorFlow (Abadi et al., 2015). It provides an easy-to-use Scikit-learn
(Pedregosa et al., 2011) style interface to simplify the process of creating,
configuring, training, evaluating, and experimenting a machine learning model.
TF.Learn integrates a wide range of state-of-art machine learning algorithms
built on top of TensorFlow’s low level APIs for small to large-scale
supervised and unsupervised problems. This module focuses on bringing machine
learning to non-specialists using a general-purpose high-level language as
well as researchers who want to implement, benchmark, and compare their new
methods in a structured environment. Emphasis is put on ease of use,
performance, documentation, and API consistency.

Keywords: machine learning, deep learning, distributed system, open-source,
Python

##  1 Introduction

TensorFlow, a general-purpose numerical computation library open-sourced by
Google in November 2015, has flexible implementation and architecture enables
users to focus on building the computation graph and deploy the model with
little efforts on heterogeous platforms such as mobile devices, hundreds of
machines, or thousands of computational devices.

TensorFlow is generally very straightforward to use in a sense that most of
the researchers in the research area without experience of using this library
could understand what’s happening behind the code blocks. TensorFlow provides
a good backbone for building different shapes of machine learning
applications.

However, there’s a large number of potential users, including some
researchers, data scientists, and students who may be familiar with many
machine learning algorithms already but who have never got involved in deep
learning research and applications, may found it really hard to start building
deep learning models using TensorFlow.

TF.Learn is a simplified interface for TensorFlow, to get people started on
predictive analytics and data mining. It helps smooth the transition from the
Scikit-learn world of one-liner machine learning into the more open world of
building different shapes of machine learning models. Users can start by using
familiar fit/predict style and slide into utilizing TensorFlow APIs as getting
more comfortable. It’s Scikit-learn compatible so users can also benefit from
Scikit-learn features like GridSearch and Pipeline.

With such high-level interface and building blocks provided by TF.Learn,
researchers can focus on building their model specifications and architecture
that are suitable for their applications. For example, Google has recently
open-sourced the implementation for wide and deep learning recommender system
using TF.Learn (Cheng et al., 2016). This greatly increases reproducibility
and productivity in research.

##  2 Main Features

TF.Learn fully utilizes Python’s object oriented characteristics. Every
classes are as modular, reusable, and extensible as possible. This allows
users or developers to easily extend the package and implement new machine
learning algorithms by using existing modular components as well as
TensorFlow’s low-level APIs that serve as building blocks of machine learning
algorithms, such as metrics, layers, losses, optimizers, etc. The followings
are the main features provided by TF.Learn.

_Estimator_. An estimator is a rule for calculating an estimate of a given
quantity. Estimators are used to train and evaluate TensorFlow models. Each
estimator is an implementation of a particular type of machine learning
algorithm. They currently support regression and classification problems. A
list of available estimators include LinearRegressor/Classifier,
DNNRegressor/Classifier, DNNLinearCombinedRegressor/Classifier,
TensorForestEstimator, SVM, LogisticRegressor, as well as generic Estimator
that can be used to construct a custom model for either classification or
regression problems. This provides a wide range of state-of-art machine
learning algorithms and the building blocks needed for users to construct
their own algorithms. Estimators module also utilizes the graph actions module
in TF.Learn that contains all the necessary and complicated logics for
distributed training, inference, evaluation of a model that are built on top
of TensorFlow’s low-level APIs such as Supervisor and Coordinator. These
details are hidden away from users side so users can focus on utilizing the
simplified interface to conduct their research. All estimators can then be
distributed using multiple machines and devices and all extended estimators
get this functionality for free.

_RunConfig_. This class specifies the run-time configurations for an Estimator
run, it provides necessary parameters such as number of cores to be used, the
fraction of GPU memory to be used, etc. It also contains a ClusterConfig that
specifies the configurations for a distributed run, which configures the
tasks, clusters, master nodes, parameter servers, etc.

_DataFrame_. Similar to libraries like Pandas (McKinney, 2012), a high-level
DataFrame module was included in TF.Learn to facilitate many common data
reading/parsing tasks from various resources such as tensorflow.Examples,
pandas.DataFrame, etc. It also includes functions like FeedingQueueRunner to
fetch data batches and put them in a queue so training and data feeding can be
performed asynchronously in different threads to avoid wasting a lot of time
waiting for data batches to get fetched. This is very useful especially in
case of using virtual GPU cloud computing services such as Google Cloud.

_SessionRunHook_. SessionRunHooks are useful to track training, report
progress, request early stopping and more. SessionRunHooks use the observer
pattern and notify when a session starts being used, before and after a call
to the session.run(), and when the session is closed. There are several pre-
defined hooks that serves as monitors such as StopAtStepHook that requests
stop based on certain conditions, CheckpointSaverHook that saves the
checkpoints, etc.

_Experiment_. Experiment is a class containing all information needed to train
a model. After an experiment is created by passing an Estimator and inputs for
training and evaluation, an Experiment instance knows how to invoke training
and evaluation loops in a sensible fashion for distributed training. Users can
configure things like minimum evaluation frequency, evaluation delay seconds,
continuous evaluation throttle seconds, etc., that controls experiments in
different settings such as local or distributed settings. Export strategy can
also be specified if the model needs to be exported into a format that
TensorFlow Serving (https://github.com/tensorflow/serving) accepts to deploy
and serve new algorithms and experiments in production, while keeping the same
server architecture and APIs.

##  3 Examples

To illustrate basic functionalities of TF.Learn module, we start with building
a basic deep neural network model. First of all, we load example iris data
from TF.Learn’s datasets module and use Scikit-learn’s cross-validated
splitting function to split the dataset into $80\%$ training and $20\%$
validation. Next, we use TF.Learn helper function to infer the real valued
columns from the dataset that we can then use to pass into DNNClassifier to
build our deep neural network model with 3 fully connected layers with 10, 20,
10 units respectively in each layer. Here we also specify the number of
classes to to indicate that this is multi-class classifications. Other allowed
types are multi-class single label classification, binary classification,
multi-label classification, regression, etc. We then call the familiar Scikit-
learn-like functions like fit() to specify training data and the number of
training steps, as well as predict() to make predictions on validation data.
Note that we can specify as_iterable to indicate whether we want to return the
predictions as Python’s iterator, which can be really useful in situations
like making predictions for streaming data. Finally, we can use Scikit-learn’s
metrics module to calculate the score such as accuracy.

    from sklearn import cross_validation
    from sklearn import metrics
    import tensorflow as tf
    from tensorflow.contrib import learn

    iris = learn.datasets.load_dataset(‘iris’)
    x_train, x_test, y_train, y_test = cross_validation.train_test_split(
        iris.data, iris.target, test_size=0.2, random_state=42)

    feature_columns = learn.infer_real_valued_columns_from_input(x_train)
    classifier = learn.DNNClassifier(
        feature_columns=feature_columns, hidden_units=[10, 20, 10], n_classes=3)

    classifier.fit(x_train, y_train, steps=200)
    predictions = list(classifier.predict(x_test, as_iterable=True))
    score = metrics.accuracy_score(y_test, predictions)

We can also build our customized model function to define our own losses,
model architecture, predictions, etc. Here we firstly convert the target to a
one-hot tensor of shape (length of features, 3) and with a on-value of 1 for
each one-hot vector of length 3. We then define our fully connected layers
using TensorFlow’s helper function to stack three same layers without
repeating the code for three times respectively of size 10, 20, and 10 with
each layer having a dropout probability of 0.1. Next we compute each logit for
each class and the softmax cross-entropy loss. Then we use TensorFlow’s
function to create training operation, which is the optimizer to use for each
training step and finally we return the acceptable key-value pairs that will
be used during model fitting. We can then pass this customized model function
to the Estimator class so it will get all the distributed training logics and
advanced data queueing for free. Users can focus on building their different
shapes of deep learning and machine learning models.

    def my_model(features, target):
      target = tf.one_hot(target, 3, 1, 0)
      features = layers.stack(features, layers.fully_connected, [10, 20, 10],
                              normalizer_fn=layers.dropout,
                              normalizer_params=‘keep_prob’: 0.9)
      logits = layers.fully_connected(features, 3, activation_fn=None)
      loss = tf.contrib.losses.softmax_cross_entropy(logits, target)
      train_op = tf.contrib.layers.optimize_loss(
          loss, tf.contrib.framework.get_global_step(),
          optimizer=‘Adagrad’,
          learning_rate=0.1)
      return ({
          ‘class’: tf.argmax(logits, 1),
          ‘prob’: tf.nn.softmax(logits)}, loss, train_op)

    classifier = learn.Estimator(model_fn=my_model)
    classifier.fit(x_train, y_train, steps=1000)

Here we only demonstrate the basic functionalities. Users can find more
examples, tutorials, as well as API guide on TensorFlow’s website
(https://www.tensorflow.org/).

##  4 Availability, Documentation, Maintenance, and Code Quality Control

The TF.Learn source code is available under Apache-2.0 license and hosted on
Github (https://github.com/tensorflow/tensorflow) currently located under
tensorflow/contrib/learn folder. Stable releases of the code are regularly
published on Github releases tab and all stable releases will have pre-built
binary distributions available. We also provide very detailed tutorials for
main features of TF.Learn and API documentation on TensorFlow’s website
(https://www.tensorflow.org/). We use Bazel (https://bazel.build/) for
continuous integration tests and continuous delivery on Jenkins at
https://ci.tensorflow.org/ and have nightly binary distributions available.

##  5 Comparison to Similar Frameworks

There are many high-level Python libraries, for instance, Lasagne (Battenberg
et al., 2016) and Keras (Chollet, 2016). Both of these two libraries are used
widely in data science competitions, especially Kaggle
(https://www.kaggle.com/). Lasagne, initiated by a team of deep learning and
music information retrieval researchers and based solely on Theano back-end,
is a lightweight library with a simple API that allows for fast prototyping of
neural network models. Keras is a very flexible wrapper that can switch back-
end to either Theano or TensorFlow in a sense that it hides all Theano and
TensorFlow code behind its API so users don’t need to understand what’s
actually behind the scenes but rather focusing on prototyping their models and
at the end choosing the back-end that’s more suitable for their own use cases.
It greatly promotes reproducible research and researchers can spend time on
the actual model shapes. Whenever a user wants to use some new ops from
TensorFlow, he needs to register it in Keras’ back-end module for TensorFlow
and wrap it around to in cope with Keras’ high-level APIs. On the other hand,
with TF.Learn, users can directly use TensorFlow’s new ops and build a custom
model functions that can be directly inserted into TF.Learn’s Estimator object
as illustrated in previous sections. A lot of existing functionalities and
features also become available for free, such as early stopping, dropout,
custom learning rate decay, class weights, and multi-output
regression/classification without much additional efforts. Users can focus on
building their custom graphs to meet the needs. Additionally, TensorFow’s
distributed power on heterogeneous systems also comes for free without any
barrier to streamline TensorFlow graph creation, advanced asynchronous data
queueing and batching, model building and testing.

There are also similar toolkits available in other languages, such as packages
in R language (R Core Team, 2016), which is popular in the field of machine
learning and statistics, including mlr (Bischl et al., 2016) and caret (from
Jed Wing et al., 2016) that provide unified interface to many machine learning
algorithms. Each framework has its own pros and cons and we highly suggest
users to conduct careful comparisons for the best framework for each different
use cases.

##  6 Conclusions and Future Development

TF.Learn, as a high-level Python module for distributed machine learning
inside TensorFlow, provides an easy-to-use Scikit-learn style interface to
simplify the process of creating, configuring, training, evaluating, and
experimenting a machine learning model. TF.Learn integrates a wide range of
state-of-art machine learning algorithms built on top of TensorFlow’s low
level APIs for small to large-scale supervised and unsupervised problems. It
hides all the grundy details of distributed implementation so users can focus
on prototyping their own customized machine learning architectures and later
deploy to production without much additional efforts.

In the future, this module will continue to be highly community-driven and
keep moving towards more user-friendly, flexible, extensible, and modular
interface so a larger group of users could be benefited.

Acknowledgments

We would like to acknowledge the co-creator of TF.Learn Illia Polosukhin from
Google Research and Martin Wicke from Google TensorFlow team for their
continuous support and contributions as well as many other developers from
Google TensorFlow team and people from open-source community who submit bug
reports, feature requests, and provide really valuable feedback in each phase
of the development.

## References

  * Abadi et al. (2015) Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng.  TensorFlow: Large-scale machine learning on heterogeneous systems, 2015\.  URL http://tensorflow.org/.  Software available from tensorflow.org.
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  * Bischl et al. (2016) Bernd Bischl, Michel Lang, Lars Kotthoff, Julia Schiffner, Jakob Richter, Erich Studerus, Giuseppe Casalicchio, and Zachary M. Jones.  mlr: Machine learning in r.  _Journal of Machine Learning Research_ , 17(170):1–5, 2016.  URL http://jmlr.org/papers/v17/15-066.html.
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