$V_{n}\vcentcolon=\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\otimes_{K}v^{P^{n}}_{Q^{n}}(K)$
(3.14)

is a locally analytic $P^{n}$-representation whose underlying locally convex
vector space is a $K$-Banach space.

######  Lemma 3.2.1.

The transition homomorphisms $V_{n}\rightarrow V_{n+1}$ are injective and
compact, i.e.

$V\vcentcolon=\varinjlim_{n\in{\mathbb{N}}}V_{n}$

is of compact type this way. In particular via the $P^{n}$-actions on the
$V_{n}$, $V$ becomes a locally analytic $({\mathrm{hy}}(G),P_{0})$-module in
the sense of Definition 1.6.18.

###### Proof.

We have seen in Proposition 2.5.2 that the transition maps

$\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n+1})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$

are compact and have dense image. Therefore their transposes are injective and
compact by [67, Lemma 16.4]. Moreover the homomorphisms
$v^{P^{n}}_{Q^{n}}\rightarrow v^{P^{n+1}}_{Q^{n+1}}$ which are induced by the
restriction map
${\mathrm{Ind}}^{P^{n}}_{Q^{n}}(K)\rightarrow{\mathrm{Ind}}^{P^{n+1}}_{Q^{n+1}}(K),\,f\mapsto
f{\\!\,\mid_{P^{n+1}}}$, are injective as well. Hence it follows from [27,
Cor. 1.1.27] that the tensor product $V_{n}\rightarrow V_{n+1}$ of these maps
is injective. The homomorphism $v^{P^{n}}_{Q^{n}}\rightarrow
v^{P^{n+1}}_{Q^{n+1}}$ is compact as a homomorphism between finite-dimensional
$K$-vector spaces by [67, Lemma 16.4]. Therefore [67, Lemma 18.12] implies
that $V_{n}\rightarrow V_{n+1}$ is compact as well.

Finally, we have ${\mathrm{hy}}(P^{n})={\mathrm{hy}}(G)$ because $P^{n}\subset
G$ is an open subgroup. Hence each $V_{n}$ is a locally analytic
$({\mathrm{hy}}(G),P^{n})$-module via Remark 1.6.19, and therefore a locally
analytic $({\mathrm{hy}}(G),P_{0})$-module in particular. It follows that $V$
is a locally analytic $({\mathrm{hy}}(G),P_{0})$-module as well. ∎

######  Remark 3.2.2.

In fact, $V$ is even a locally analytic $P$-representation. For fixed $p\in
P$, we find $n\geq m$ large enough such that $h_{j}P^{n}/Q\subset
pg_{\sigma_{g}(j)}P^{m}/Q$, for all $j=1,\ldots,s_{n}$, like in Remark 3.1.4.
Considering the images under $G/Q\rightarrow G/P$, it follows from $1\in
p^{-1}P^{n}/P$ that $P^{n}/P\subset pP^{m}/P$. This shows that
${\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}\subset
p{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}$ and we obtain a homomorphism

$\varphi_{p}^{t}\colon\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\longrightarrow\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}.$

Moreover, we have the continuous homomorphism $\psi_{p}\colon
v^{P^{m}}_{Q^{m}}\rightarrow v^{P^{n}}_{Q^{n}}$ induced by

${\mathrm{Ind}}^{P^{m}}_{Q^{m}}(K)\longrightarrow{\mathrm{Ind}}^{P^{n}}_{Q^{n}}(K)\,,\quad
f\longmapsto f(p^{-1}{\,\\_\,}).$

The tensor product of these yields the continuous homomorphism
$\tau_{p}=(\varphi_{p}^{t}\otimes\psi_{p})\colon V_{m}\rightarrow V_{n}$. Like
before the collection of these $\tau_{p}$, for all $m\in{\mathbb{N}}$, gives
the action of $p$ on $V$. This $P$-action extends the one of $P_{0}$ and $V$
is a locally analytic $({\mathrm{hy}}(G),P)$-module this way. ∎

######  Lemma 3.2.3.

There is a canonical topological isomorphism of locally analytic
$({\mathrm{hy}}(G),P)$-modules

$V\cong\widetilde{H}^{d-{r}}_{({\mathbb{P}}_{K}^{{r}})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}{\,\widehat{\otimes}_{K}\,}v^{{\mathrm{GL}}_{d-{r}}(K)}_{B_{d-{r}}}.$

Here ${\mathrm{GL}}_{d-{r},K}$ is viewed as a subgroup of the standard Levi
factor ${\mathrm{\bf
L}}\cong{\mathrm{GL}}_{{r}+1,K}\times_{K}{\mathrm{GL}}_{d-{r},K}$ of
${\mathrm{\bf P}}={\mathrm{\bf P}}_{\Delta\setminus\\{\alpha_{r}\\}}$, and
${\mathrm{\bf B}}_{d-{r}}$ denotes the standard (lower) Borel subgroup of
${\mathrm{GL}}_{d-{r},K}$. On $v^{{\mathrm{GL}}_{d-{r}}(K)}_{B_{d-{r}}}$ the
group $P$ acts via inflation, ${\mathrm{hy}}(G)$ acts trivially, and it
carries the finest locally convex topology.

###### Proof.

First note that we have an isomorphism

${\mathrm{Ind}}^{P^{n}}_{Q^{n}}(K)\overset{\cong}{\longrightarrow}{\mathrm{Ind}}^{{\mathrm{GL}}_{d-{r}}({\mathcal{O}}_{K})}_{B_{d-{r}}^{n}}(K)\,,\quad
f\longmapsto f{\\!\,\mid_{{\mathrm{GL}}_{d-{r}}({\mathcal{O}}_{K})}},$

since
$P^{n}/Q^{n}\cong{\mathrm{GL}}_{d-{r}}({\mathcal{O}}_{K})/B_{d-{r}}^{n}$. Here
we view ${\mathrm{GL}}_{d-{r}}({\mathcal{O}}_{K})\subset P_{0}$ as a subgroup.
This yields isomorphisms $v^{P^{n}}_{Q^{n}}\cong
v^{{\mathrm{GL}}_{d-{r}}({\mathcal{O}}_{K})}_{B_{d-{r}}^{n}}$, for all
$n\in{\mathbb{N}}$. Taking the inductive limit over these isomorphisms and
applying Lemma 3.1.3 to the case of
$B_{d-{r}}\subset{\mathrm{GL}}_{d-{r}}(K)$, we obtain an isomorphism
$\varinjlim_{n\in{\mathbb{N}}}v^{P^{n}}_{Q^{n}}\cong
v^{{\mathrm{GL}}_{d-{r}}(K)}_{B_{d-{r}}}$ of locally convex $K$-vector spaces.
Moreover, one computes that this isomorphism is $P$-equivariant when
$\varinjlim_{n\in{\mathbb{N}}}v^{P^{n}}_{Q^{n}}$ carries the $P$-action from
Remark 3.2.2, and $v^{{\mathrm{GL}}_{d-{r}}(K)}_{B_{d-{r}}}$ the one by
inflation.

We recall from Proposition 2.6.1 (ii) that
$\widetilde{H}^{d-{r}}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\cong\varinjlim_{n\in{\mathbb{N}}}\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}$.
By [27, Prop. 1.1.32] we then obtain a $P$-equivariant topological isomorphism

$V=\varinjlim_{n\in{\mathbb{N}}}\Big{(}\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}{\,\otimes_{K}\,}v^{P^{n}}_{Q^{n}}\Big{)}\cong\widetilde{H}^{d-{r}}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}{\,\widehat{\otimes}_{K}\,}v^{{\mathrm{GL}}_{d-{r}}(K)}_{B_{d-{r}}}.$
(3.15)

Finally note that ${\mathrm{hy}}(G)$ acts trivially on $v^{P^{n}}_{Q^{n}}$,
for each $n\in{\mathbb{N}}$, as $Q^{n}\subset P^{n}$ is an open subgroup. On
$\widetilde{H}^{d-{r}}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}$
the ${\mathrm{hy}}(G)$-action is induced by the actions on the
$\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}$.
Therefore (3.15) is ${\mathrm{hy}}(G)$-equivariant. ∎

We come back to the locally analytic $G$-representation
$\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}_{P^{n}}^{G_{0}}(W_{n})$ from
(3.12), for
$W_{n}\vcentcolon=\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{n}}_{Q^{n}}(K)$.
The $K$-Banach spaces $V_{n}$ allow us to exhibit this as a locally convex
$K$-vector space of compact type.

######  Lemma 3.2.4.

There is a canonical topological isomorphism of locally analytic
$G$-representations

$\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(W_{n})\cong\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n}).$
(3.16)

Here the $G$-action on
$\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})$ is
given in the way analogous to Remark 3.1.4. Moreover, the transition maps
${\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\rightarrow{\mathrm{Ind}}^{G_{0}}_{P^{n+1}}(V_{n+1})$
of the right hand side are compact and injective so that the underlying
locally convex $K$-vector space of the above $G$-representation is of compact
type.

###### Proof.

In view of the inductive limit description of Proposition 2.6.1, for
$n\in{\mathbb{N}}$, the homomorphism

$\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\longrightarrow\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n+1})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}$

factors over
$\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}$.
We therefore obtain a commutative diagram of locally analytic
$P^{n+1}$-representations

${\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n-1})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}}$${\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}}$${\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}}$${\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n+1})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}.}$

Combined with the canonical homomorphisms $v^{P^{n}}_{Q^{n}}\rightarrow
v^{P^{n+1}}_{Q^{n+1}}$, this gives factorizations of the transition maps of
both inductive limits in (3.16)

${{\mathrm{Ind}}^{G_{0}}_{P^{n-1}}\Big{(}\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n-1})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\otimes_{K}v^{P^{n-1}}_{Q^{n-1}}\Big{)}}$${{\mathrm{Ind}}^{G_{0}}_{P^{n}}\Big{(}\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\otimes_{K}v^{P^{n}}_{Q^{n}}\Big{)}}$${{\mathrm{Ind}}^{G_{0}}_{P^{n}}\Big{(}\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{n}}_{Q^{n}}\Big{)}}$${{\mathrm{Ind}}^{G_{0}}_{P^{n+1}}\Big{(}\widetilde{H}^{d-{r}}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{n+1})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{n+1}}_{Q^{n+1}}\Big{)}.}$

We conclude that both inductive limits are topologically isomorphic to each
other.

Now consider, for $n\in{\mathbb{N}}$, coset representatives
$h_{1},\ldots,h_{s_{n+1}}$ of $G_{0}/P^{n+1}$ and $g_{1},\ldots,g_{s_{n}}$ of
$G_{0}/P^{n}$ such that $h_{j}$ gets mapped to $g_{i(j)}$ under
$G_{0}/P^{n+1}\rightarrow G_{0}/P^{n}$. Then the transition map
${\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\rightarrow{\mathrm{Ind}}^{G_{0}}_{P^{n+1}}(V_{n+1})$
is given by

$\bigoplus_{i=1}^{s_{n}}g_{i}\bullet
V_{n}\longrightarrow\bigoplus_{j=1}^{s_{n+1}}h_{j}\bullet
V_{n+1}\,,\quad\sum_{i=1}^{s_{n}}g_{i}\bullet
v_{i}\longmapsto\sum_{j=1}^{s_{n+1}}h_{j}\bullet v_{i(j)}.$

Therefore it is injective and compact as the finite direct sum of compact
homomorphisms, by Lemma A.3 (iii). ∎

Our next goal is to interpret
$\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})$ as a
subspace of $C^{\mathrm{la}}(G_{0},V)$. We know from Proposition 1.5.4 that
${\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\cong{\mathrm{Ind}}^{{\mathrm{la}},G_{0}}_{P^{n}}(V_{n})$.
As each $V_{n}$ is a BH-subspace of $V$, we consequently obtain injective
continuous homomorphisms, for all $n\in{\mathbb{N}}$:

$\displaystyle\iota_{n}\colon{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})$
$\displaystyle\longrightarrow C^{\mathrm{la}}(G_{0},V_{n})\longrightarrow
C^{\mathrm{la}}(G_{0},V),$ $\displaystyle\sum_{i=1}^{s_{n}}g_{i}\bullet v_{i}$
$\displaystyle\longmapsto\big{[}g\mapsto p^{-1}.v_{i}\text{ , for $g=g_{i}p$
with $p\in P^{n}$}\big{]},$

Given $f\in C^{\mathrm{la}}(G_{0},V)$, $g\in G_{0}$, and $p\in P_{0}$, we
write $f(g{\,\\_\,}p)$ for the locally analytic function

$f(g{\,\\_\,}p)\colon G_{0}\longrightarrow V\,,\quad h\longmapsto f(ghp).$

We let $\mu(f)$ denote $\mu\in{\mathrm{hy}}(G)$ applied to a function $f\in
C^{\mathrm{la}}(G_{0},V)$ via the pairing (1.26). The
${\mathrm{hy}}(G)$-module action on $V$ is denoted by $\mu\ast v$, for $v\in
V$. Also recall that $\dot{\mu}$ signifies the involution from Lemma 1.6.10.

######  Lemma 3.2.5.

The homomorphism

$\iota\colon\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\longrightarrow
C^{\mathrm{la}}(G_{0},V)$

induced by the $\iota_{n}$ is a closed embedding with

${\mathrm{Im}}(\iota)=\left\\{f\in C^{\mathrm{la}}(G_{0},V)\,\middle|\,\forall
g\in G_{0},p\in
P_{0},\mu\in{\mathrm{hy}}(G):\mu\big{(}f(g{\,\\_\,}p)\big{)}=p^{-1}.\dot{\mu}\ast
f(g)\right\\}.$

###### Proof.

It suffices to show the statement about ${\mathrm{Im}}(\iota)$. Indeed, then
${\mathrm{Im}}(\iota)$ is the intersection of the kernels of the continuous
homomorphisms

$C^{\mathrm{la}}(G_{0},V)\longrightarrow V\,,\quad
f\longmapsto\mu\big{(}f(g{\,\\_\,}p)\big{)}-p^{-1}.\dot{\mu}\ast f(g),$ (3.17)

for $g\in G_{0},p\in P_{0},\mu\in{\mathrm{hy}}(G)$. As $V$ is Hausdorff, these
kernels are closed subspaces, and so is ${\mathrm{Im}}(\iota)$. Since
$C^{\mathrm{la}}(G_{0},V)$ is of compact type by Proposition 1.2.15 (iii),
${\mathrm{Im}}(\iota)$ is of compact type as well using Proposition A.4.
Moreover, the induced homomorphism $\iota$ is a continuous bijection onto its
image. Because
$\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})$ is the
inductive limit of $K$-Banach spaces, we can apply a version of the open
mapping theorem [11, II. §4.6 Cor.] to conclude that $\iota$ is strict.

It remains to show the statement about ${\mathrm{Im}}(\iota)$. Let
$\rho_{n}\colon P^{n}\rightarrow{\mathrm{GL}}(V_{n})$ denote the
representation (3.14) and $\rho_{n,v}\colon P^{n}\rightarrow V_{n}$, for $v\in
V_{n}$, its locally analytic orbit maps. First, consider
$f\in{\mathrm{Im}}(\iota)$. Then there exists $n\in{\mathbb{N}}$ such that
$f\in{\mathrm{Im}}(\iota_{n})$, i.e. $f\in C^{\mathrm{la}}(G_{0},V_{n})$ and
$f(gp)=p^{-1}.f(g)$, for all $g\in G_{0}$, $p\in P^{n}$. Let
$D_{n}\vcentcolon=1+{\pi}^{n}M_{d+1}({\mathcal{O}}_{K})$, and note that
$P^{n}=D_{n}\cdot P_{0}$. It follows that

$f(gxp)=(xp)^{-1}.f(g)=(\rho_{n,f(g)}\circ{\mathrm{inv}})(xp),$ (3.18)

for all $g\in G_{0}$, $x\in D_{n}$, $p\in P_{0}$. We fix $g\in G_{0}$ and
$p\in P_{0}$ for the moment and consider (3.18) as an identity of locally
analytic functions in $x$ on $D_{n}$. The homomorphism

$\displaystyle V_{n}\longrightarrow C^{\rm la}(P^{n},V_{n})\,,\quad
v\longmapsto(\rho_{n,v}\circ{\mathrm{inv}})({\,\\_\,}p),$

is $P^{n}$-equivariant with respect to the left regular representation on
$C^{\rm la}(P^{n},V_{n})$. Therefore, it is equivariant for the
${\mathrm{hy}}(G)$-action as well, and for $v=f(g)$, we obtain

$(\rho_{n,\mu\ast
f(g)}\circ{\mathrm{inv}})({\,\\_\,}p)=\mu\ast\big{(}(\rho_{n,f(g)}\circ{\mathrm{inv}})({\,\\_\,}p)\big{)},$
(3.19)

for all $\mu\in{\mathrm{hy}}(G)$. Finally, we apply $\mu\in{\mathrm{hy}}(G)$
to (3.18) and compute for the functions restricted to $D_{n}$:

$\displaystyle\mu\big{(}f(g{\,\\_\,}p)\big{)}$
$\displaystyle=\mu\big{(}(\rho_{n,f(g)}\circ{\mathrm{inv}})({\,\\_\,}p)\big{)}$
(3.20)
$\displaystyle=\big{(}\dot{\mu}\ast\big{(}(\rho_{n,f(g)}\circ{\mathrm{inv}})({\,\\_\,}p)\big{)}\big{)}(1)$
, by Proposition 1.6.14 $\displaystyle=\big{(}(\rho_{n,\dot{\mu}\ast
f(g)}\circ{\mathrm{inv}})({\,\\_\,}p)\big{)}(1)$ , by (3.19)
$\displaystyle=p^{-1}.\dot{\mu}\ast f(g).$

On the other hand, let $f\in C^{\mathrm{la}}(G_{0},V)$ such that
$\mu\big{(}f(g{\,\\_\,}p)\big{)}=p^{-1}.\dot{\mu}\ast f(g)$, for all $g\in
G_{0}$, $p\in P_{0}$, and $\mu\in{\mathrm{hy}}(G)$. As $G_{0}$ is compact,
there exists some $n\in{\mathbb{N}}$ such that $f$ factors over the BH-space
$V_{n}$ of $V$. Moreover, we find $m\geq n$ such that $f$ is locally analytic
with respect to the finite covering

$G_{0}=\bigcup_{u\in G_{0}/P^{m}}\bigcup_{p\in P^{m}/D_{m}}upD_{m},$ (3.21)

i.e. $f{\\!\,\mid_{upD_{m}}}$ is analytic, for all cosets $upD_{m}$. By
Proposition 1.3.10, $\rho_{n}$ is a locally $K$-analytic map of locally
$K$-analytic manifolds. After increasing $m$, we therefore may assume that
$\rho_{n}\circ{\mathrm{inv}}$ is analytic on each coset of (3.21) as well. In
particular, $(\rho_{n,v}\circ{\mathrm{inv}}){\\!\,\mid_{upD_{m}}}$ is
analytic, for all cosets $upD_{m}$ and all $v\in V_{n}$.

To show that $f\in{\mathrm{Im}}(\iota_{m})$, we fix $g\in G_{0}$ and $p\in
P^{m}$ and write $p=xp_{0}$, for $x\in D_{m}$, $p_{0}\in P_{0}$. Then
$f(g{\,\\_\,}p_{0}){\\!\,\mid_{D_{m}}}$ and
$(\rho_{n,v}\circ{\mathrm{inv}})({\,\\_\,}p_{0}){\\!\,\mid_{D_{m}}}$ are
analytic, for all $v\in V_{n}$. Indeed, let $u^{\prime}p^{\prime}D_{m}$ by the
coset of (3.21) containing $gp_{0}$. Because $f$ is analytic on
$u^{\prime}p^{\prime}D_{m}$, $f(g{\,\\_\,}p_{0})$ is analytic on
$g^{-1}u^{\prime}p^{\prime}D_{m}p_{0}^{-1}$. But using that $D_{m}$ is normal
in $G_{0}$ being the kernel of the reduction homomorphism, we see that this
last set is equal to $D_{m}$. Similarly $(\rho_{n,v}\circ{\mathrm{inv}})$ is
analytic on the coset $g^{-1}u^{\prime}p^{\prime}D_{m}$ so that
$(\rho_{n,v}\circ{\mathrm{inv}})({\,\\_\,}p_{0})$ is analytic on $D_{m}$. If
we apply $\mu\in{\mathrm{hy}}(G)$ to the analytic function
$(\rho_{n,v}\circ{\mathrm{inv}})({\,\\_\,}p_{0}){\\!\,\mid_{D_{m}}}$, we
compute analogously to (3.20)

$p_{0}^{-1}.\dot{\mu}\ast
f(g)=\mu\big{(}(\rho_{n,f(g)}\circ{\mathrm{inv}})({\,\\_\,}p_{0})\big{)}.$

Combining this with the assumption
$\mu\big{(}f(g{\,\\_\,}p)\big{)}=p^{-1}.\dot{\mu}\ast f(g)$, we see that the
functions on $D_{m}$ satisfy

$\displaystyle\mu\big{(}f(g{\,\\_\,}p_{0})\big{)}=\mu\big{(}(\rho_{n,f(g)}\circ{\mathrm{inv}})({\,\\_\,}p_{0})\big{)},$

for all $\mu\in{\mathrm{hy}}(G)$. By Proposition 1.6.12, this implies that the
locally analytic germs at $1$ of $f(g{\,\\_\,}p_{0})$ and
$(\rho_{n,f(g)}\circ{\mathrm{inv}})({\,\\_\,}p_{0})$ agree. As both functions
are analytic on all of $D_{m}$, we conclude that they agree there, and we
have, for all $g\in G_{0}$, $p=xp_{0}\in D_{m}\cdot P_{0}=P^{m}$:

$f(gp)=f(gxp_{0})=(\rho_{n,f(g)}\circ{\mathrm{inv}})(xp_{0})=(xp_{0})^{-1}.f(g)=p^{-1}.f(g).$

∎

Because $V$ is of compact type, by Proposition 1.2.15 (iii) we have the
topological isomorphism $C^{\rm la}(G_{0},K){\,\widehat{\otimes}_{K}\,}V\cong
C^{\rm la}(G_{0},V)$ induced by $f\otimes v\mapsto f({\,\\_\,})\,v$. Under
this identification, the homomorphisms (3.17) are given by

$C^{\mathrm{la}}(G_{0},K){\,\widehat{\otimes}_{K}\,}V\longrightarrow V\,,\quad
f\otimes
v\longmapsto\mu\big{(}f(g{\,\\_\,}p)\big{)}\,v-f(g)\,p^{-1}.\dot{\mu}\ast v.$

Therefore, $\iota$ fits into the sequence of continuous homomorphism

$\displaystyle
0\longrightarrow\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}_{P^{n}}^{G_{0}}(V_{n})\overset{\iota}{\longrightarrow}C^{\mathrm{la}}(G_{0},K){\,\widehat{\otimes}_{K}\,}V$
$\displaystyle\overset{\psi}{\longrightarrow}\prod_{g\in G_{0},p\in
P_{0},\mu\in{\mathrm{hy}}(G)}V$ (3.22) $\displaystyle f\otimes v$
$\displaystyle\longmapsto\Big{(}\mu\big{(}f(g{\,\\_\,}p)\big{)}\,v-f(g)\,p^{-1}.\dot{\mu}\ast
v\Big{)}_{g,p,\mu}$

which is algebraically exact, and $\iota$ is strict.

We want to consider the strong dual of this sequence (3.22). Note that by [67,
Rmk. 16.1 (ii)], the homomorphisms of this dualized sequence are continuous
again. By [67, Prop. 9.11] there is a topological isomorphism

$\bigoplus_{\begin{subarray}{c}g\in G_{0}\\\ p\in P_{0}\\\
\mu\in{\mathrm{hy}}(G)\end{subarray}}V^{\prime}_{b}\overset{\cong}{\longrightarrow}\Bigg{(}\prod_{\begin{subarray}{c}g\in
G_{0}\\\ p\in P_{0}\\\
\mu\in{\mathrm{hy}}(G)\end{subarray}}\\!\\!V\,\,\Bigg{)}^{\prime}_{b}\,,\quad\sum\ell_{g,p,\mu}\longmapsto\left[(v_{g,p,\mu})\mapsto\sum\ell_{g,p,\mu}(v_{g,p,\mu})\right].$

Under this isomorphism, the transpose of $\psi$ is given by

$\displaystyle\psi^{t}\colon\bigoplus_{g\in G_{0},p\in
P_{0},\mu\in{\mathrm{hy}}(G)}V^{\prime}_{b}$
$\displaystyle\longrightarrow\big{(}C^{\rm
la}(G_{0},K){\,\widehat{\otimes}_{K}\,}V\big{)}^{\prime}_{b}$
$\displaystyle\sum\ell_{g,p,\mu}$ $\displaystyle\longmapsto\left[f\otimes
v\mapsto\sum\mu\big{(}f(g{\,\\_\,}p)\big{)}\,\ell_{g,p,\mu}(v)-f(g)\,\ell_{g,p,\mu}(p^{-1}.\dot{\mu}\ast
v)\right].$

By applying Corollary 1.4.5 at various points, we have

$\displaystyle\mu\big{(}f(g{\,\\_\,}p)\big{)}$
$\displaystyle=\mu\big{[}h_{1}\mapsto f(gh_{1}p)\big{]}$
$\displaystyle=\mu\big{[}h_{1}\mapsto\delta_{g}\big{[}h_{2}\mapsto\delta_{p}\big{[}h_{3}\mapsto
f(h_{2}h_{1}h_{3})\big{]}\big{]}\big{]}$
$\displaystyle=\delta_{g}\big{[}h_{2}\mapsto\mu\big{[}h_{1}\mapsto\delta_{p}\big{[}h_{3}\mapsto
f(h_{2}h_{1}h_{3})\big{]}\big{]}\big{]}$
$\displaystyle=(\delta_{g}\ast\mu\ast\delta_{p})(f).$

Furthermore, by the definition of the contragredient action of $P_{0}$ and
${\mathrm{hy}}(G)$ on $V^{\prime}_{b}$:

$\ell_{g,p,\mu}(p^{-1}.\dot{\mu}\ast v)=(p.\ell_{g,p,\mu})(\dot{\mu}\ast
v)=(\mu\ast p.\ell_{g,p,\mu})(v).$

Moreover, both $V$ and $C^{\mathrm{la}}(G_{0},K)$ are reflexive with their
strong duals being reflexive Fréchet spaces. Hence by [67, Prop. 20.13] and
[67, Cor. 20.14], we have a topological isomorphism

$D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}\overset{\cong}{\longrightarrow}\big{(}C^{\rm
la}(G_{0},K){\,\widehat{\otimes}_{K}\,}V\big{)}^{\prime}_{b}\,,\quad\delta\otimes\ell\longmapsto\big{[}f\otimes
v\mapsto\delta(f)\,\ell(v)\big{]}.$

All in all, we see that the strong dual of (3.22) is the complex

$\displaystyle\bigoplus_{g\in G_{0},p\in
P_{0},\mu\in{\mathrm{hy}}(G)}V^{\prime}_{b}$
$\displaystyle\overset{\psi^{t}}{\longrightarrow}D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}\overset{\iota^{t}}{\longrightarrow}\Big{(}\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}_{P^{n}}^{G_{0}}(V_{n})\Big{)}^{\prime}_{b}\longrightarrow
0$ (3.23) $\displaystyle\sum\ell_{g,p,\mu}$
$\displaystyle\longmapsto\sum\delta_{g}\ast\mu\ast\delta_{p}\otimes\ell_{g,p,\mu}-\delta_{g}\otimes\mu\ast
p.\ell_{g,p,\mu}.$

As $\iota$ is a closed embedding, the Hahn–Banach Theorem [67, Cor. 9.4]
implies that $\iota^{t}$ is surjective. It follows from the open mapping
theorem [67, Prop. 8.6] that $\iota^{t}$ is strict. Moreover, by [11, IV. §4.1
Prop. 2] we have ${\mathrm{Ker}}(\iota^{t})={\mathrm{Im}}(\iota)^{\perp}$
where

${\mathrm{Im}}(\iota)^{\perp}\vcentcolon=\big{\\{}\ell\in
D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}\,\big{|}\,\forall
v\in{\mathrm{Im}}(\iota):\ell(v)=0\big{\\}}.$

Since ${\mathrm{Im}}(\iota)^{\perp}={\mathrm{Ker}}(\psi)^{\perp}$ by the
algebraic exactness of (3.22), Lemma A.11 implies that
${\mathrm{Ker}}(\iota^{t})={\mathrm{Ker}}(\psi)^{\perp}\subset\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$.
As ${\mathrm{Im}}(\psi^{t})\subset{\mathrm{Ker}}(\iota^{t})$ and
${\mathrm{Ker}}(\iota^{t})$ is closed, we conclude that
${\mathrm{Ker}}(\iota^{t})=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$.

Under the equivalence of Proposition 1.4.10 (ii), the homomorphism $\iota^{t}$
becomes a homomorphism of $D(G_{0})$-modules when
$D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}$ carries the
$D(G_{0})$-module structure via multiplication on the left in the first
factor. We therefore obtain a topological isomorphism of $D(G_{0})$-modules

$\Big{(}\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}_{P^{n}}^{G_{0}}(V_{n})\Big{)}^{\prime}_{b}\cong\big{(}D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}\big{)}\big{/}{\mathrm{Ker}}(\iota^{t})\cong\big{(}D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}\big{)}\big{/}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}.$

The submodule ${\mathrm{Im}}(\psi^{t})$ in turn is generated by the elements

$\delta_{g}\ast\mu\ast\delta_{p}\otimes\ell-\delta_{g}\otimes\mu\ast
p.\ell\quad\text{, for $g\in G_{0}$, $\mu\in{\mathrm{hy}}(G)$, $p\in P_{0}$,
$\ell\in V^{\prime}_{b}$.}$

Recall from Proposition 1.6.16 that $D({\mathfrak{g}},P_{0})$ is generated by
the elements of the form $\mu\ast\lambda$, for $\mu\in{\mathrm{hy}}(G)$,
$\lambda\in D(P_{0})$. Together with the density of the Dirac distributions,
it follows that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
is equal to the closure of the $D(G_{0})$-submodule generated by the vectors

$\delta\ast\nu\otimes\ell-\delta\otimes\nu\ast\ell\quad\text{, for $\delta\in
D(G_{0})$, $\nu\in D({\mathfrak{g}},P_{0})$, $\ell\in V^{\prime}_{b}$,}$

where $V^{\prime}_{b}$ is a separately continuous
$D({\mathfrak{g}},P_{0})$-module via Corollary 1.6.20. Using Remark 1.5.9 we
conclude the following:

######  Proposition 3.2.6.

There is a canonical topological isomorphism of $D(G_{0})$-modules

$\bigg{(}\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}_{P^{n}}^{G_{0}}(V_{n})\bigg{)}^{\prime}_{b}\cong
D(G_{0}){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{0})}\,}V^{\prime}_{b}.$

###  3.3. The Subquotients of
$H^{0}({\mathcal{X}},{\mathcal{E}})^{\prime}_{b}$ as Locally Analytic
${\mathrm{GL}}_{d+1}(K)$-Representations

On the level of locally analytic $G_{0}$-representations, the above
Proposition 3.2.6 already is a description of the term

$\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}\left(\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{n}}_{Q^{n}}\right)$

occuring in Theorem 3.1.5. However, we want to extend this to a description as
locally analytic $G$-representations or equivalently as $D(G)$-modules.

######  Lemma 3.3.1.

There is a topological isomorphism of $D(G_{0})$-$D(P)$-bi-modules

$D(G_{0}){\,\otimes_{D(P_{0}),\iota}\,}D(P)\overset{\cong}{\longrightarrow}D(G)\,,\quad\mu\otimes\nu\longmapsto\mu\ast\nu,$

(cf. [73, Lemma 6.1 (i)] for the statement on the algebraic level).

For a separately continuous $D({\mathfrak{g}},P)$-module $M$, natural
inclusion $D(G_{0})\hookrightarrow D(G)$ induces a topological isomorphism

$D(G_{0}){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{0}),\iota}\,}M\overset{\cong}{\longrightarrow}D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P),\iota}\,}M\,,\quad\delta\otimes\ell\mapsto\delta\otimes\ell,$
(3.24)

of $D(G_{0})$-modules.

###### Proof.

For (i), essentially the proof from [73, Lemma 6.1 (i)] for the algebraic
statement applies: The Iwasawa decomposition $G=G_{0}P$ with $G_{0}\cap
P=P_{0}$ (see [16, §3.5]) gives a disjoint covering $G=\bigcup_{p\in
P_{0}\backslash P}G_{0}p$ by compact open subsets. In view of Proposition
1.4.2 (iii) this yields a topological isomorphism

$\displaystyle D(G)\cong\bigoplus_{p\in P_{0}\backslash
P}D(G_{0})\ast\delta_{p}$

of $D(G_{0})$-$D(P)$-bi-modules, and one of $D(P_{0})$-$D(P)$-bi-modules

$\displaystyle D(P)\cong\bigoplus_{p\in P_{0}\backslash
P}D(P_{0})\ast\delta_{p}.$

Moreover, there is a topological isomorphism [46, Lemma 1.2.13]

$D(G_{0}){\,\otimes_{K,\iota}\,}\bigg{(}\bigoplus_{p\in P_{0}\backslash
P}D(P_{0})\ast\delta_{p}\bigg{)}\cong\bigoplus_{p\in P_{0}\backslash
P}D(G_{0}){\,\otimes_{K,\iota}\,}D(P_{0})\ast\delta_{p}$

of $D(G_{0})$-$D(P)$-bi-modules. Passing to the quotients we obtain the
topological isomorphism

$\displaystyle D(G_{0}){\,\otimes_{D(P_{0}),\iota}\,}D(P)$ $\displaystyle\cong
D(G_{0}){\,\otimes_{D(P_{0}),\iota}\,}\bigg{(}\bigoplus_{p\in P_{0}\backslash
P}D(P_{0})\ast\delta_{p}\bigg{)}$ $\displaystyle\cong\bigoplus_{p\in
P_{0}\backslash
P}D(G_{0}){\,\otimes_{D(P_{0}),\iota}\,}D(P_{0})\ast\delta_{p}$
$\displaystyle\cong\bigoplus_{p\in P_{0}\backslash
P}D(G_{0})\ast\delta_{p}\cong D(G).$

of $D(G_{0})$-$D(P)$-bimodules.

For (ii), clearly the continuous homomorphism
$D(G_{0}){\,\otimes_{K,\iota}\,}M\rightarrow D(G){\,\otimes_{K,\iota}\,}M$
induces the continuous homomorphism (3.24) by passing to the homomorphism
between the quotients

$D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0}),\iota}\,}M\longrightarrow
D(G){\,\otimes_{D({\mathfrak{g}},P),\iota}\,}M$

and completing. Moreover, using the statement of (i) and Lemma 1.5.11 (ii) we
have a topological isomorphism of $D(G_{0})$-modules

$D(G){\,\otimes_{K,\iota}\,}M\cong\big{(}D(G_{0}){\,\otimes_{D(P_{0}),\iota}\,}D(P)\big{)}{\,\otimes_{K,\iota}\,}M\cong
D(G_{0}){\,\otimes_{D(P_{0}),\iota}\,}\big{(}D(P){\,\otimes_{K,\iota}\,}M\big{)}.$

Together with the well-defined continuous homomorphism of $D(G_{0})$-modules

$\displaystyle
D(G_{0})\otimes_{D(P_{0}),\iota}\big{(}D(P){\,\otimes_{K,\iota}\,}M\big{)}$
$\displaystyle\longrightarrow
D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0}),\iota}\,}M,$
$\displaystyle\delta\otimes\lambda\otimes\ell$
$\displaystyle\longmapsto\delta\otimes\lambda.\ell.$

we obtain $D(G){\,\otimes_{K,\iota}\,}M\rightarrow
D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0}),\iota}\,}M$. This homomorphism
factors over the quotient as

$D(G){\,\otimes_{D({\mathfrak{g}},P),\iota}\,}M\longrightarrow
D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0}),\iota}\,}M,$

and one verifies that the completion of the latter homomorphism is an inverse
to (3.24). ∎

######  Theorem 3.3.2.

Let ${\mathcal{E}}$ be a ${\mathrm{\bf G}}$-equivariant vector bundle on
${\mathbb{P}}_{K}^{d}$. For the terms that occur on the right hand side of the
description of the subquotients in Theorem 3.1.5, there are topological
isomorphisms of $D(G)$-modules

$\displaystyle\bigg{(}\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{n}_{d-q}}_{Q^{n}_{d-q}}\Big{)}\bigg{)}^{\prime}_{b}$
$\displaystyle\qquad\cong
D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{d-q}),\iota}\,}\bigg{(}\widetilde{H}^{q}_{({\mathbb{P}}_{K}^{d-q})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}){\,\widehat{\otimes}_{K}\,}\Big{(}v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}\Big{)}^{\prime}_{b}\bigg{)},$

for $q=1,\ldots,d$. Here $P_{d-q}$ acts via inflation from the subgroup
${\mathrm{GL}}_{q}(K)$ of its standard Levi factor $L_{d-q}$ on
$v^{{\mathrm{GL}}_{q}(K)}_{B_{q}(K)}$, and ${\mathrm{hy}}(G)$ acts trivially
there101010Since $v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}$ carries the finest locally
convex topology, $\Big{(}v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}\Big{)}^{\prime}$
equals the algebraic dual. The smooth dual of
$v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}$ is a $K$-subspace thereof, but this
inclusion is not an equality in general..

###### Proof.

We keep the simplified notation with ${r}=d-q\in\\{0,\ldots,d-1\\}$ fixed and
${\mathrm{\bf P}}\vcentcolon={\mathrm{\bf P}}_{{d-q}}$, ${\mathrm{\bf
Q}}\vcentcolon={\mathrm{\bf Q}}_{{d-q}}$. Combining the topological
isomorphism

$\bigg{(}\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\bigg{)}^{\prime}_{b}\cong
D(G_{0}){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{0})}\,}V^{\prime}_{b}$

from Proposition 3.2.6 which was obtained via the topological embedding

$\iota\colon\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\lhook\joinrel\longrightarrow
C^{\mathrm{la}}(G_{0},K){\,\widehat{\otimes}_{K}\,}V$

with the statement of Lemma 3.3.1 (ii), already gives a topological
isomorphism of $D(G_{0})$-modules

$\omega\colon\Big{(}\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\Big{)}^{\prime}_{b}\overset{\cong}{\longrightarrow}D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P),\iota}\,}V^{\prime}_{b}.$

It remains to show that $\omega$ is $D(G)$-linear.

To do so we first construct a $G$-equivariant homomorphism

$\tilde{\iota}\colon\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\longrightarrow
C^{\mathrm{la}}(G,V)$

which is compatible with $\iota$ and the restriction map

$({\,\\_\,}){\\!\,\mid_{G_{0}}}\colon C^{\mathrm{la}}(G,V)\longrightarrow
C^{\mathrm{la}}(G_{0},V)\,,\quad f\longmapsto f{\\!\,\mid_{G_{0}}},$

in the sense that $({\,\\_\,}){\\!\,\mid_{G_{0}}}\circ\tilde{\iota}=\iota$.
Let $f\in{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})$ correspond to the locally
analytic function $f\colon G_{0}\rightarrow V_{n}$ so that
$f(gp)=p^{-1}.f(g)$, for all $g\in G_{0}$, $p\in P^{n}$. We define an
associated function $\tilde{f}\in C^{\mathrm{la}}(G,V)$ as follows. For $g\in
G$ with Iwasawa decomposition $g=g_{0}p$, with $g_{0}\in G_{0}$, $p\in P$, we
set

$\tilde{f}(g)=p^{-1}.f(g_{0})\in V$

where $p^{-1}$ acts on $f(g_{0})\in V$ as explained in Remark 3.2.2. This
gives a well-defined function $\tilde{f}\colon G\rightarrow V$, because, for a
different decomposition $g=g^{\prime}_{0}p^{\prime}$ with
$g^{\prime}_{0}=g_{0}p^{-1}_{0}$, $p^{\prime}=p_{0}p$, for some $p_{0}\in
P_{0}$, we have

$\tilde{f}(g^{\prime}_{0}p^{\prime})=(p^{\prime})^{-1}.f(g^{\prime}_{0})=p^{-1}.p_{0}^{-1}.f(g_{0}p^{-1}_{0})=p^{-1}.f(g_{0})=\tilde{f}(g)$

as $p_{0}\in P^{n}$.

The function $\tilde{f}$ is locally analytic: For fixed $g=g_{0}p$ with $g\in
G_{0}$, $p\in P$, we consider the open neighbourhood $G_{0}p$ of $g$. There
$\tilde{f}{\\!\,\mid_{G_{0}p}}\colon hp\mapsto\tilde{f}(hp)=p^{-1}.f(h)$ is
locally analytic by Proposition 1.2.14 (i) since $f$ is locally analytic. In
total we obtain the sought homomorphism

$\tilde{\iota}\colon\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\longrightarrow
C^{\mathrm{la}}(G,V)\,,\quad f\longmapsto\tilde{f},$

with $({\,\\_\,}){\\!\,\mid_{G_{0}}}\circ\tilde{\iota}=\iota$.

Next we want to show that $\tilde{\iota}$ is $G$-equivariant with respect to
the left-regular $G$-action111111As $G$ is not compact, the left-regular
$G$-representation is not locally analytic but it is continuous nevertheless.
on $C^{\mathrm{la}}(G,V)$. To this end, let
$f\in{\mathrm{Ind}}^{G_{0}}_{P^{m}}(V_{m})$ be given by
$\sum_{i=1}^{s_{m}}g_{i}\bullet v_{i}$ so that $f(g_{i}p)=p^{-1}.v_{i}$, for
$p\in P^{m}$. As usual $g_{1},\ldots,g_{s_{m}}$ denote coset representatives
of $G_{0}/P^{m}$.

We fix $g\in G$, and want to show that
$\tilde{\iota}(g.f)=g.\tilde{\iota}(f)$. Let $n\geq m$ like in Remark 3.1.4 so
that $P^{n}/Q\subset p_{g,j}P^{m}/Q$ with
$p_{g,j}\vcentcolon=h_{j}^{-1}gg_{\sigma_{g}(j)}$. Then we have seen that

$g.f=g.\bigg{(}\sum_{i=1}^{s_{m}}g_{i}\bullet
v_{i}\bigg{)}=\sum_{j=1}^{s_{n}}h_{j}\bullet\tau_{p_{g,j}}(v_{\sigma_{g}(j)})\in{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n}).$

Now consider $h\in G$, and let $j\in\\{1,\ldots,s_{n}\\}$ such that $h\in
h_{j}P^{n}/P$, i.e. $h=h_{j}p_{(n)}p$, for some $p_{(n)}\in P^{n}$, $p\in P$.
We compute that

$\displaystyle\big{(}\tilde{\iota}(g.f)\big{)}(h)$
$\displaystyle=\big{(}\tilde{\iota}(g.f)\big{)}(h_{j}p_{(n)}p)=p^{-1}.\big{(}(g.f)(h_{j}p_{(n)})\big{)}$
$\displaystyle=p^{-1}.p_{(n)}^{-1}.\tau_{p_{g,j}}(v_{\sigma_{g}(j)})=(\tau_{p^{-1}}\circ\tau_{p_{(n)}^{-1}}\circ\tau_{p_{g,j}})(v_{\sigma_{g}(j)}).$

On the other hand, we have

$g^{-1}h=(g_{\sigma_{g}(j)}p_{g,j}^{-1}h_{j}^{-1})(h_{j}p_{(n)}p)=g_{\sigma_{g}(j)}p^{-1}_{g,j}p_{(n)}p.$

Hence

$\displaystyle\big{(}g.\tilde{\iota}(f)\big{)}(h)$
$\displaystyle=\tilde{\iota}(f)(g^{-1}h)=\tilde{\iota}(f)(g_{\sigma_{g}(j)}p^{-1}_{g,j}p_{(n)}p)$
$\displaystyle=\big{(}p_{g,j}^{-1}p_{(n)}p\big{)}^{-1}.f(g_{\sigma_{g}(j)})=(\tau_{p^{-1}}\circ\tau_{p_{(n)}^{-1}}\circ\tau_{p_{g,j}})(v_{\sigma_{g}(j)})$

This shows that indeed $\tilde{\iota}$ is $G$-equivariant.

We now use the injective continuous integration homomorphism from Proposition
1.4.7 (i) together with [67, Cor. 18.8] to obtain

$C^{\mathrm{la}}(G,V)\longrightarrow{\mathcal{L}}_{b}(D(G),V)\cong
D(G)^{\prime}_{b}{\,\widehat{\otimes}_{K,\pi}\,}V.$

As $C^{\mathrm{la}}(G,V)$ is reflexive by Corollary 1.2.16 this yields an
injective continuous homomorphism

$\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}}(V_{n})\overset{\tilde{\iota}}{\longrightarrow}C^{\mathrm{la}}(G,V)\longrightarrow
C^{\mathrm{la}}(G,K){\,\widehat{\otimes}_{K,\pi}\,}V$

that we continue to call $\tilde{\iota}$.

Moreover, let $G=\bigcup_{i\in I}g_{i}G_{0}$ be a disjoint covering, for coset
representatives $g_{i}$ of $G/G_{0}$, so that
$C^{\mathrm{la}}(G,K)\cong\prod_{i\in I}C^{\mathrm{la}}(g_{i}G_{0},K)$ and
$D(G)\cong\bigoplus_{i\in I}D(g_{i}G_{0},K)$. Then there are topological
isomorphisms

$\displaystyle\big{(}C^{\mathrm{la}}(G,K){\,\widehat{\otimes}_{K,\pi}\,}V\big{)}^{\prime}_{b}$
$\displaystyle\cong\bigg{(}\prod_{i\in
I}C^{\mathrm{la}}(g_{i}G_{0},K){\,\widehat{\otimes}_{K}\,}V\bigg{)}^{\prime}_{b}$
, by [13, Lemma 2.1 (iii)] $\displaystyle\cong\bigoplus_{i\in
I}\big{(}C^{\mathrm{la}}(g_{i}G_{0},K){\,\widehat{\otimes}_{K}\,}V\big{)}^{\prime}_{b}$
, by [67, Prop. 9.11] $\displaystyle\cong\bigoplus_{i\in
I}D(g_{i}G_{0},K){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}$ , by [27, Prop.
1.1.32 (ii)] $\displaystyle\cong
D(G){\,\widehat{\otimes}_{K,\iota}\,}V^{\prime}_{b}.$ , by [46, Cor. 1.2.14]

The resulting $G$-equivariant isomorphism fits into the commutative square

${\big{(}C^{\mathrm{la}}(G_{0},K){\,\widehat{\otimes}_{K}\,}V\big{)}^{\prime}_{b}}$${\big{(}C^{\mathrm{la}}(G,K){\,\widehat{\otimes}_{K,\pi}\,}V\big{)}^{\prime}_{b}}$${D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}}$${D(G){\,\widehat{\otimes}_{K,\iota}\,}V^{\prime}_{b}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

where the horizontal homomorphism on the top is the transpose of
$({\,\\_\,}){\\!\,\mid_{G_{0}}}$, and the bottom one is induced by the
embedding $D(G_{0})\hookrightarrow D(G)$. We finally arrive at the commutative
diagram

${D(G_{0}){\,\widehat{\otimes}_{K}\,}V^{\prime}_{b}}$${D(G_{0}){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{0})}\,}V^{\prime}_{b}}$${\big{(}C^{\mathrm{la}}(G_{0},K){\,\widehat{\otimes}_{K}\,}V\big{)}^{\prime}_{b}}$${\Big{(}\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}_{P^{n}}^{G_{0}}(V_{n})\Big{)}^{\prime}_{b}}$${\big{(}C^{\mathrm{la}}(G,K){\,\widehat{\otimes}_{K,\pi}\,}V\big{)}^{\prime}_{b}}$${D(G){\,\widehat{\otimes}_{K,\iota}\,}V^{\prime}_{b}}$${D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P),\iota}\,}V^{\prime}_{b}.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\iota^{t}}$$\scriptstyle{\cong}$$\scriptstyle{\omega}$$\scriptstyle{\cong}$$\scriptstyle{\tilde{\iota}^{t}}$

Since $\iota^{t}$ is surjective, so is $\tilde{\iota}^{t}$. It follows that
the induced topological isomorphism $\omega$ is $G$-equivariant because all
the other homomorphisms in the lower “square” are. We conclude that $\omega$
is $D(G)$-linear by using the density of the Dirac distributions in $D(G)$. ∎

###  3.4. The Functors ${\mathcal{F}}_{P}^{G}$ of Orlik–Strauch

In this section we want to relate our description from Theorem 3.3.2 to the
functors ${\mathcal{F}}_{P}^{G}$ introduced by Orlik and Strauch in [58]. To
this end we suppose that the non-archimedean local field $K$ is of mixed
characteristic, i.e. a finite extension of ${\mathbb{Q}}_{p}$. We begin by
recapitulating the definition of the functors ${\mathcal{F}}_{P}^{G}$, but for
simplicity only under the assumption that the field of definition $L$ agrees
with the field of coefficients $K$. We normalize the absolute value of $K$
such that ${\lvert{p}\rvert}=p^{-1}$.

Let ${\mathrm{\bf G}}$ be a connected split reductive group over $K$. We fix a
split maximal torus and a Borel subgroup ${\mathrm{\bf T}}\subset{\mathrm{\bf
B}}\subset{\mathrm{\bf G}}$, as well as a standard parabolic subgroup
${\mathrm{\bf P}}\supset{\mathrm{\bf B}}$ with Levi decomposition
${\mathrm{\bf P}}={\mathrm{\bf L}}_{\mathrm{\bf P}}{\mathrm{\bf
U}}_{\mathrm{\bf P}}$ with ${\mathrm{\bf T}}\subset{\mathrm{\bf
L}}_{\mathrm{\bf P}}$. We assume that ${\mathrm{\bf G}}$ and the above
subgroups already are defined over ${\mathcal{O}}_{K}$. Let $G={\mathrm{\bf
G}}(K)$, $P={\mathrm{\bf P}}(K)$, etc. denote the associated locally
$K$-analytic Lie groups and write $G_{0}={\mathrm{\bf G}}({\mathcal{O}}_{K})$,
$P_{0}={\mathrm{\bf P}}({\mathcal{O}}_{K})$, etc. by abuse of notation.
Furthermore let ${\mathfrak{g}}={\rm Lie}({\mathrm{\bf G}})$,
${\mathfrak{p}}={\rm Lie}({\mathrm{\bf P}})$, etc. denote the corresponding
Lie algebras. We consider the following subcategories of modules for the
universal enveloping algebra $U({\mathfrak{g}})$.

######  Definition 3.4.1 ([58, §2.5]).

Let ${\mathcal{O}}^{\mathfrak{p}}$ be the full subcategory of
$U({\mathfrak{g}})$-modules $M$ satisfying $M$ is finitely generated as a
$U({\mathfrak{g}})$-module,

viewed as an ${\mathfrak{l}}_{\mathrm{\bf P}}$-module, $M$ is the direct sum
of finite-dimensional simple modules,

the action of ${\mathfrak{u}}_{\mathrm{\bf P}}$ on $M$ is locally finite, i.e.
for every $m\in M$, the $K$-vector subspace $U({\mathfrak{u}}_{\mathrm{\bf
P}})m\subset M$ is finite-dimensional.

Let ${\rm Irr}({\mathfrak{l}}_{\mathrm{\bf P}})^{\rm fd}$ be the set of
isomorphism classes of finite-dimensional irreducible
${\mathfrak{l}}_{\mathrm{\bf P}}$-representations. We define
${\mathcal{O}}_{\mathrm{alg}}^{\mathfrak{p}}$ to be the full subcategory of
${\mathcal{O}}^{\mathfrak{p}}$ of $U({\mathfrak{g}})$-modules $M$ such that
for a decomposition

$M=\bigoplus_{{\mathfrak{a}}\in{\rm Irr}({\mathfrak{l}}_{\mathrm{\bf P}})^{\rm
fd}}M_{\mathfrak{a}}$

into the ${\mathfrak{a}}$-isotypic components as in (2), we have: If
$M_{\mathfrak{a}}\neq(0)$ then ${\mathfrak{a}}$ is the Lie algebra
representation induced by a finite-dimensional algebraic ${\mathrm{\bf
L}}_{\mathrm{\bf P}}$-representation.

Note that for ${\mathfrak{p}}={\mathfrak{b}}$,
${\mathcal{O}}\vcentcolon={\mathcal{O}}^{\mathfrak{b}}$ is the adaptation of
the classical category ${\mathcal{O}}$ introduced by Bernstein, Gelfand and
Gelfand for semi-simple Lie algebras over the complex numbers.

The functor ${\mathcal{F}}_{P}^{G}$ from [58, §3,4] is now defined as follows:
Let $M\in{\mathcal{O}}^{\mathfrak{p}}_{\mathrm{alg}}$ and let $V$ be a smooth
admissible representation of the Levi subgroup $L_{\mathrm{\bf P}}\subset P$
on a $K$-vector space. We regard $V$ as a smooth $P$-representation via
inflation and endow it with the finest locally convex topology so that it
becomes a locally analytic $P$-representation of compact type, see [75, §2].
By the conditions on $M\in{\mathcal{O}}^{\mathfrak{p}}$, there exists a
finite-dimensional $U({\mathfrak{p}})$-submodule $W\subset M$ which generates
$M$ as a $U({\mathfrak{g}})$-module, i.e. there is a short exact sequence of
$U({\mathfrak{g}})$-modules

$0\longrightarrow{\mathfrak{d}}\longrightarrow
U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}})}\,}W\longrightarrow
M\longrightarrow 0.$

Then the ${\mathfrak{p}}$-representation $W$ uniquely lifts to the structure
of an algebraic ${\mathrm{\bf P}}$-representation on $W$ [58, Lemma 3.2].
There is a pairing (cf. [58, (3.2.2)])

$\displaystyle\langle{\,\\_\,},\\!{\,\\_\,}\rangle_{C^{\mathrm{la}}(G,V)}\colon
D(G){\,\otimes_{D(P)}\,}W\times{\mathrm{Ind}}^{G}_{P}(W^{\prime}{\,\otimes_{K}\,}V)$
$\displaystyle\longrightarrow C^{\mathrm{la}}(G,V),$
$\displaystyle(\delta\otimes w,f)$
$\displaystyle\longmapsto\big{[}g\mapsto\big{(}\delta\ast_{r}({\mathrm{ev}}_{w}\circ
f)\big{)}(g)\big{]}.$

Here we use the identification
$W^{\prime}{\,\otimes_{K}\,}V\cong{\mathcal{L}}_{b}(W,V)$ from [67, Cor. 18.8]
and denote the evaluation homomorphism by
${\mathrm{ev}}_{w}\colon{\mathcal{L}}_{b}(W,V)\rightarrow V$, $h\mapsto h(w)$.
Moreover, “$\ast_{r}$” signifies the $D(G)$-module action on
$C^{\mathrm{la}}(G,V)$ induced from the right regular action of $G$ (see
Example 1.3.7 (ii)). Via the injective map

$U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}})}\,}W\lhook\joinrel\longrightarrow
D(G){\,\otimes_{D(P)}\,}W$

one may consider the subspace of
${\mathrm{Ind}}^{G}_{P}(W^{\prime}{\,\otimes_{K}\,}V)$ annihilated by
${\mathfrak{d}}$ and define [58, (4.4.1)]

${\mathcal{F}}^{G}_{P}(M,V)\vcentcolon={\mathrm{Ind}}^{G}_{P}(W^{\prime}{\,\otimes_{K}\,}V)^{\mathfrak{d}}=\big{\\{}f\in{\mathrm{Ind}}^{G}_{P}(W^{\prime}{\,\otimes_{K}\,}V)\,\big{|}\,\forall{\mathfrak{z}}\in{\mathfrak{d}}:\langle{\mathfrak{z}},f\rangle_{C^{\mathrm{la}}(G,V)}=0\big{\\}}.$

The resulting ${\mathcal{F}}^{G}_{P}(M,V)$ is an admissible121212In the sense
of [71, §6]. locally analytic $G$-representation which even is strongly
admissible131313Meaning that as a representation of any (equivalently, of one)
compact open subgroup $H\subset G$, it is strongly admissible in the sense of
[74, §3], i.e. its strong dual is finitely generated as a $D(H)$-module. if
$V$ is of finite length [58, Prop. 4.8]. This construction yields an exact bi-
functor

${\mathcal{F}}^{G}_{P}\colon{\mathcal{O}}^{\mathfrak{p}}_{\mathrm{alg}}\times{\rm
Rep}_{K}^{{\mathrm{sm}},{\rm adm}}(L_{\mathrm{\bf P}})\longrightarrow{\rm
Rep}_{K}^{{\mathrm{la}},{\rm
adm}}(G)\,,\quad(M,V)\longmapsto{\mathcal{F}}_{P}^{G}(M,V),$

which is contravariant in $M$ and covariant in $V$, see [58, Prop. 4.7].

In the case that $V=K$ is the trivial representation, there is another
description of ${\mathcal{F}}^{G}_{P}(M)={\mathcal{F}}^{G}_{P}(M,K)$, for
$M\in{\mathcal{O}}_{\mathrm{alg}}^{\mathfrak{p}}$. Since $M$ is the union of
finite-dimensional $U({\mathfrak{p}})$-submodules, via lifting each of those
to an algebraic ${\mathrm{\bf P}}$-representation one obtains a $D(P)$-module
structure on $M$, cf. [58, §3.4]. This yields a unique
$D({\mathfrak{g}},P)$-module structure on $M$ such that the two actions of
$U({\mathfrak{p}})$ agree and the Dirac distributions $\delta_{p}\in D(P)$ act
like the group elements $p\in P$ on $M$ ([58, Cor. 3.6]). Then there are
isomorphisms of $D(G)$\- resp. $D(G_{0})$-modules [58, Prop. 3.7]

${\mathcal{F}}^{G}_{P}(M)^{\prime}\cong
D(G){\,\otimes_{D({\mathfrak{g}},P)}\,}M\cong
D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0})}\,}M.$ (3.25)

By [71, Thm. 5.1] the locally analytic distribution algebra $D(G_{0})$ is a
Fréchet–Stein algebra since $G_{0}$ is compact. It holds that
$D(G){\,\otimes_{D({\mathfrak{g}},P)}\,}M$ is a coadmissible $D(G)$-module
[58, Prop. 3.7]. Recall that any coadmissible module over a Fréchet–Stein
algebra can be endowed with a canonical Fréchet topology [71, §3]. We note
that with this topology (3.25) is a topological isomorphism.

######  Remark 3.4.2.

It is expected that a description similar to (3.25) holds for the case of non-
trivial $V$ as well. In fact, in [1] Agrawal and Strauch consider functors
defined by

$\check{{\mathcal{F}}}_{P}^{G}(M,V)\vcentcolon=D(G){\,\otimes_{D({\mathfrak{g}},P)}\,}\big{(}{\rm
Lift}(M,\log){\,\otimes_{K}\,}V^{\prime}\big{)}$

to generalize ${\mathcal{F}}_{P}^{G}$ to the case of $M$ being an element of
the extension closure ${\mathcal{O}}_{\mathrm{alg}}^{{\mathfrak{p}},\infty}$
of ${\mathcal{O}}_{\mathrm{alg}}^{{\mathfrak{p}}}$. They show that the
resulting $D(G)$-modules $\check{{\mathcal{F}}}_{P}^{G}(M,V)$ are
coadmissible, see [1, Thm. 4.2.3].

We now come back to the concrete situation of ${\mathrm{\bf
G}}={\mathrm{GL}}_{d+1,K}$. As mentioned towards the end of Section 3.1, Orlik
shows in [56, Lemma 1.2.1] that the $U({\mathfrak{g}})$-module
$\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is contained in ${\mathcal{O}}_{\mathrm{alg}}^{{\mathfrak{p}}_{d-q}}$.
Moreover, let ${\mathrm{\bf B}}_{q}={\mathrm{\bf B}}\cap{\mathrm{GL}}_{q,K}$
denote the induced Borel subgroup of
${\mathrm{GL}}_{q,K}\hookrightarrow{\mathrm{\bf L}}_{{\mathrm{\bf P}}_{d-q}}$.
Then the Steinberg representation $v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}$ is an
irreducible smooth representation of ${\mathrm{GL}}_{q}(K)$ ([17, Thm. 2
(a)]), and hence of $P_{d-q}$. Orlik then obtains a description of the locally
analytic $G$-representation (3.13) as being isomorphic to

${\mathcal{F}}^{G}_{P_{d-q}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}),v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}\Big{)}.$
(3.26)

The other term
$H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}{\,\otimes_{K}\,}v^{G}_{Q_{d-q}}$
of the extension (3.12) is a strongly admissible locally analytic
$G$-representation as well, cf. the proof of [58, Lemma 2.4]. It follows that
the extension $(V^{q}/V^{q-1})^{\prime}_{b}$ of the two terms is strongly
admissible. Since the homomorphisms between these (strongly) admissible
representations in Orlik’s description are necessarily strict (see [71, Prop.
6.4 (ii)]), we can conclude that there is a topological isomorphism between
(3.26) and the strong dual of

$\displaystyle
D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{d-q}),\iota}\,}\bigg{(}\widetilde{H}^{q}_{({\mathbb{P}}_{K}^{d-q})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}){\,\widehat{\otimes}_{K}\,}\Big{(}v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}\Big{)}^{\prime}_{b}\bigg{)}$
(3.27)

from our description in Theorem 3.3.2 by the uniqueness of the quotient.

We recall from Corollary 2.5.5 that
$\widetilde{H}^{q}_{({\mathbb{P}}_{K}^{d-q})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is the completion of its subspace
$\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.
Hence (3.27) is nothing but the Hausdorff completion of

$D(G){\,\otimes_{D({\mathfrak{g}},P_{d-q}),\iota}\,}\bigg{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}){\,\otimes_{K,\pi}\,}\Big{(}v^{{\mathrm{GL}}_{q}(K)}_{B_{q}}\Big{)}^{\prime}_{b}\bigg{)}.$
(3.28)

Therefore it is natural to ask how this Hausdorff completion relates to the
coadmissible abstract $D(G)$-module underlying (3.28) when one considers the
latter with its canonical Fréchet topology. We answer this question in
Corollary 3.4.9 by showing that they agree, i.e. that (3.28) already is
complete and its locally convex topology is the same as the canonical Fréchet
topology.

For the first ingredient used to this end, we return to the general setup of
connected split reductive ${\mathrm{\bf G}}$ considered at the beginning of
this section. We fix on $U({\mathfrak{g}})\subset D(G)$ the subspace topology,
and likewise on all subalgebras of $U({\mathfrak{g}})$. As $U({\mathfrak{g}})$
already is contained in $D(G_{0})$ and the latter is a $K$-Fréchet algebra,
$U({\mathfrak{g}})$ and its subalgebras become (jointly continuous) locally
convex $K$-algebras this way.

We want to give a concrete description of this topology of
$U({\mathfrak{g}})$. Let ${\mathfrak{x}}_{1},\ldots,{\mathfrak{x}}_{s}$ be a
$K$-basis of ${\mathfrak{g}}$, and $\varepsilon>0$. Via the associated PBW
basis for $U({\mathfrak{g}})$ consisting of
${\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{s}^{k_{s}}$, for
${\underline{k}}\in{\mathbb{N}}_{0}^{s}$, we define the norm

$\bigg{\lvert}\sum_{{\underline{k}}\in{\mathbb{N}}_{0}^{s}}a_{{\underline{k}}}\,{\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{s}^{k_{s}}\bigg{\rvert}_{\varepsilon}\vcentcolon=\sup_{{\underline{k}}\in{\mathbb{N}}_{0}^{s}}{\lvert{a_{{\underline{k}}}}\rvert}\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{{\underline{k}}}\rvert}}$
(3.29)

on $U({\mathfrak{g}})$ where
${\lvert{{\underline{k}}}\rvert}\vcentcolon=k_{1}+\ldots+k_{s}$.

######  Lemma 3.4.3.

The topology on $U({\mathfrak{g}})$ prescribed by the family of norms
${\lvert{{\,\\_\,}}\rvert}_{\varepsilon}$, for $0<\varepsilon<1$, equals the
topology defined by regarding $U({\mathfrak{g}})$ as a subspace of $D(G)$.

###### Proof.

By [74, Lemma 2.4], the subspace topology of $U({\mathfrak{g}})\subset
D_{e}(G)\subset D(G)$ is defined via the family of norms

$\bigg{\lVert}\sum_{{\underline{k}}\in{\mathbb{N}}_{0}^{s}}a_{{\underline{k}}}\,{\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{s}^{k_{s}}\bigg{\rVert}_{\varepsilon}\vcentcolon=\sup_{{\underline{k}}\in{\mathbb{N}}_{0}^{s}}{\lvert{a_{{\underline{k}}}\,{\underline{k}}!}\rvert}\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{{\underline{k}}}\rvert}},$

for $\varepsilon>0$, where
${\underline{k}}!\vcentcolon=(k_{1}!)\cdots(k_{s}!)$. Here
$\lVert{\,\\_\,}\rVert_{\varepsilon}$ yields a topology finer than the one of
$\lVert{\,\\_\,}\rVert_{\varepsilon^{\prime}}$, for
$\varepsilon\leq\varepsilon^{\prime}$. We immediately find that
$\lVert{\,\\_\,}\rVert_{\varepsilon}\leq{\lvert{{\,\\_\,}}\rvert}_{\varepsilon}$
since ${\lvert{{\underline{k}}!}\rvert}\leq 1$.

On the other hand, we obtain by Legendre’s formula for the $p$-adic valuation
of $n!$ that $v_{p}(n!)\leq\frac{n}{p-1}$, for $n\in{\mathbb{N}}$. Hence

${\lvert{{\underline{k}}!}\rvert}=p^{-v_{p}({\underline{k}}!)}\geq\Big{(}p^{\frac{1}{p-1}}\Big{)}^{-{\lvert{{\underline{k}}}\rvert}}.$

It follows that
$\lVert{\,\\_\,}\rVert_{\varepsilon}\geq\lvert{\,\\_\,}\rvert_{p^{\frac{1}{p-1}}\varepsilon}$,
and we conclude that the topology defined by the family
$(\lvert{\,\\_\,}\rvert_{\varepsilon})_{0<\varepsilon<1}$ is equal to the
topology defined by the family
$(\lVert{\,\\_\,}\rVert_{\varepsilon})_{0<\varepsilon}$. ∎

######  Remark 3.4.4.

In [63, Thm. 2.1] Schmidt shows that there is a natural isomorphism of
topological $K$-algebras between the completion of $U({\mathfrak{g}})$ with
respect to the family of norms ${\lvert{{\,\\_\,}}\rvert}_{\varepsilon}$, for
$0<\varepsilon<1$, and the Arens–Michael envelope
$\widehat{U}({\mathfrak{g}})$ of $U({\mathfrak{g}})$. The latter is defined as
the Hausdorff completion of $U({\mathfrak{g}})$ with respect to all
submultiplicative seminorms on $U({\mathfrak{g}})$. Using that the completion
of $U({\mathfrak{g}})\subset D(G)$ with respect to the subspace topology is
its closure $D_{e}(G)$ in $D(G)$ (see Corollary 1.6.13), it follows that there
is a natural isomorphism of topological $K$-algebras
$\widehat{U}({\mathfrak{g}})\cong D_{e}(G)$ as well.

Recalling that we fix the subspace topology on $U({\mathfrak{g}})\subset
D(G)$, we now consider a finitely generated $U({\mathfrak{g}})$-module $M$
(e.g. $M\in{\mathcal{O}}_{\mathrm{alg}}^{\mathfrak{p}}$). By assumption we
then find some $n\in{\mathbb{N}}$ and an epimorphism of
$U({\mathfrak{g}})$-modules

$U({\mathfrak{g}})^{\oplus n}\relbar\joinrel\twoheadrightarrow M.$ (3.30)

######  Lemma 3.4.5.

When $M$ is endowed with the quotient topology via (3.30), it becomes a
locally convex $U({\mathfrak{g}})$-module. Its Hausdorff completion is a
nuclear $K$-Fréchet space.

Any homomorphism $f\colon M\rightarrow M^{\prime}$ between finitely generated
$U({\mathfrak{g}})$-modules is continuous and strict when $M$ and $M^{\prime}$
carry the quotient topology induced by some epimorphism as above. In
particular, this topology on $M$ does not depend on the choice of the
epimorphism (3.30).

###### Proof.

For (i), since the quotient topology on $M$ is locally convex, we only have to
show that the multiplication $U({\mathfrak{g}})\times M\rightarrow M$ is
continuous. To do so, we consider the commutative diagram

${U({\mathfrak{g}})\times U({\mathfrak{g}})^{\oplus
n}}$${U({\mathfrak{g}})^{\oplus n}}$${U({\mathfrak{g}})\times M}$${M}$

where the vertical maps are open by our choice of topology on $M$. But the
multiplication map for the finite free $U({\mathfrak{g}})$-module
$U({\mathfrak{g}})^{\oplus n}$ is continuous which implies that the
multiplication map for $M$ is as well.

The Hausdorff completion of $U({\mathfrak{g}})$ is its closure $D_{e}(G)$ in
$D(G)$ which is a nuclear $K$-Fréchet space. In particular, $D_{e}(G)$ is
hereditarily complete, see the discussion after [27, Def. 1.1.39]. Hence the
strict epimorphism (3.30) induces a strict epimorphism

$D_{e}(G)^{\oplus n}\relbar\joinrel\twoheadrightarrow\widehat{M}$

onto the Hausdorff completion of $M$ by [13, Cor. 2.2]. As a quotient of a
nuclear $K$-Fréchet space $\widehat{M}$ then is one itself, see [67, Prop.
8.3] and [67, Prop. 19.4 (ii)].

For (ii), we argue similarly to [7, 3.7.3 Prop. 2]. Consider an epimorphism
$\varphi\colon U({\mathfrak{g}})^{\oplus n}\twoheadrightarrow M$ of
$U({\mathfrak{g}})$-modules which endows $M$ with its topology. Then the
homomorphism $\varphi^{\prime}\vcentcolon=f\circ\varphi$ is continuous because
the addition and multiplication maps of the locally convex
$U({\mathfrak{g}})$-module $M^{\prime}$ are. As $\varphi$ is open by
definition, it follows that $f$ is continuous. Furthermore,
$M/{\mathrm{Ker}}(f)$ and ${\mathrm{Im}}(f)$ are isomorphic as abstract
$U({\mathfrak{g}})$-modules. The homomorphisms between these finitely
generated $U({\mathfrak{g}})$-modules that arrange this isomorphism are
continuous. Therefore $M/{\mathrm{Ker}}(f)$ and ${\mathrm{Im}}(f)$ are
topologically isomorphic, i.e. $f$ is strict. ∎

######  Lemma 3.4.6.

The multiplication map $U({\mathfrak{g}})\times D(P_{0})\rightarrow D(G_{0})$,
$(\mu,\delta)\mapsto\mu\ast\delta$ induces a topological isomorphism

$U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}D(P_{0})\overset{\cong}{\longrightarrow}D({\mathfrak{g}},P_{0})$

of $U({\mathfrak{g}})$-$D(P_{0})$-bi-modules.

###### Proof.

Since the convolution product is jointly continuous here, it induces a
continuous homomorphism
$U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}D(P_{0})\rightarrow
D(G_{0})$ of $U({\mathfrak{g}})$-$D(P_{0})$-bi-modules. By [65, Lemma 4.1]
this homomorphism is injective with image being precisely
$D({\mathfrak{g}},P_{0})$.

On the other hand, let ${\mathfrak{u}}^{-}$ denote the Lie algebra of the
opposite unipotent radical ${\mathrm{\bf U}}^{-}$ of ${\mathrm{\bf P}}$. Then
the direct sum decomposition
${\mathfrak{g}}={\mathfrak{u}}^{-}\oplus{\mathfrak{p}}$ yields an isomorphism
$U({\mathfrak{g}})\cong
U({\mathfrak{u}}^{-}){\,\otimes_{K}\,}U({\mathfrak{p}})$. As mentioned in
Remark 3.4.4 the completion of $U({\mathfrak{g}})$ with respect to the
subspace topology $U({\mathfrak{g}})\subset D(G)$ is topologically isomorphic
to the Arens–Michael envelope $\widehat{U}({\mathfrak{g}})$ of
$U({\mathfrak{g}})$ considered in [64, §3.2]. Hence we obtain a commutative
diagram

${\widehat{U}({\mathfrak{u}}^{-}){\,\widehat{\otimes}_{K,\pi}\,}\widehat{U}({\mathfrak{p}})}$${\widehat{U}({\mathfrak{g}})}$${U({\mathfrak{u}}^{-}){\,\otimes_{K,\pi}\,}U({\mathfrak{p}})}$${U({\mathfrak{g}})}$$\scriptstyle{\cong}$

where the vertical maps are the canonical embeddings, and the upper map is a
topological isomorphism by [64, Lemma 3.2.4]. It follows from [49, Lemma 2]
that the bijective continuous homomorphism
$U({\mathfrak{u}}^{-}){\,\otimes_{K,\pi}\,}U({\mathfrak{p}})\rightarrow
U({\mathfrak{g}})$ is strict and therefore a topological isomorphism.

Via this topological isomorphism, Lemma 1.5.11 (ii), and the exactness of the
projective tensor product (see [13, Lemma 2.1 (ii)]) we obtain the topological
embedding

$U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}D(P_{0})\cong
U({\mathfrak{u}}^{-}){\,\otimes_{K,\pi}\,}D(P_{0})\lhook\joinrel\longrightarrow
D(U^{-}_{0}){\,\otimes_{K,\pi}\,}D(P_{0})\lhook\joinrel\longrightarrow
D(U^{-}_{0}){\,\widehat{\otimes}_{K,\pi}\,}D(P_{0}).$

The group multiplication induces an isomorphism $U^{-}_{0}\times P_{0}\cong
G_{0}$ which yields a topological isomorphism

$D(U^{-}_{0}){\,\widehat{\otimes}_{K,\pi}\,}D(P_{0})\cong D(U^{-}_{0}\times
P_{0})\cong D(G_{0}).$

This isomorphism is given by the multiplication in $D(G_{0})$ like in the
definition of the convolution product in the proof of Proposition 1.4.3. Hence
we obtain the commutative square

${D(U^{-}_{0}){\,\widehat{\otimes}_{K,\pi}\,}D(P_{0})}$${D(G_{0})}$${U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}D(P_{0})}$${D({\mathfrak{g}},P_{0})}$$\scriptstyle{\cong}$

where the vertical maps are strict monomorphisms. Again, [49, Lemma 2] implies
that
$U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}D(P_{0})\rightarrow
D({\mathfrak{g}},P_{0})$ is strict as well, and thus a topological isomorphism
as claimed. ∎

######  Proposition 3.4.7.

Let $M\in{\mathcal{O}}^{\mathfrak{p}}_{\mathrm{alg}}$ be endowed with the
topology coming from an epimorphism (3.30). Moreover, let $V$ be a strongly
admissible smooth $P$-representation endowed with the finest locally convex
topology and considered as a locally analytic representation of $P$. Then
there is a topological isomorphism of separately continuous $D(G)$-modules

$D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P),\iota}\,}\big{(}M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\big{)}\cong
D(G){\,\otimes_{D({\mathfrak{g}},P)}\,}\big{(}M{\,\otimes_{K}\,}V^{\prime}\big{)}$
(3.31)

in the sense that
$D(G){\,\otimes_{D({\mathfrak{g}},P),\iota}\,}\big{(}M{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\big{)}$
defined according to Definition 1.5.8 is complete, and its topology agrees
with the canonical Fréchet topology induced from its underlying (abstract)
$D(G)$-module being coadmissible.

###### Proof.

By the assumptions on $M\in{\mathcal{O}}^{\mathfrak{p}}_{\mathrm{alg}}$, we
may find a finite-dimensional $U({\mathfrak{p}})$-module $W$ and an
epimorphism $\varphi\colon
U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}})}\,}W\twoheadrightarrow M$ of
$U({\mathfrak{g}})$-modules. Since
$U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}W\cong
U({\mathfrak{g}})^{\oplus{\mathrm{dim}}_{K}(W)}$ is a quotient of
$U({\mathfrak{g}}){\,\otimes_{K,\pi}\,}W$, the composition

$U({\mathfrak{g}}){\,\otimes_{K,\pi}\,}W\relbar\joinrel\twoheadrightarrow
U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}W\overset{\varphi}{\relbar\joinrel\twoheadrightarrow}M$

is a strict epimorphism by Lemma 3.4.5 (i). Hence it follows from [49, Lemma 2
(3)] that $\varphi$ is a strict epimorphism, too.

Using Lemma 3.4.6 and Lemma 1.5.11 there is a topological isomorphism

$\displaystyle D({\mathfrak{g}},P_{0}){\,\otimes_{D(P_{0}),\pi}\,}W$
$\displaystyle\cong\big{(}U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}D(P_{0})\big{)}{\,\otimes_{D(P_{0}),\pi}\,}W$
$\displaystyle\cong
U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}\big{(}D(P_{0}){\,\otimes_{D(P_{0}),\pi}\,}W\big{)}$
$\displaystyle\cong U({\mathfrak{g}}){\,\otimes_{U({\mathfrak{p}}),\pi}\,}W$

which maps $\mu\ast\delta\otimes w$ to $\mu\otimes\delta\ast w$, for $\mu\in
U({\mathfrak{g}})$, $\delta\in D(P_{0})$, $w\in W$. Therefore $\varphi$
induces a strict epimorphism

$D({\mathfrak{g}},P_{0}){\,\otimes_{D(P_{0}),\pi}\,}W\relbar\joinrel\twoheadrightarrow
M$

which one checks to be $D({\mathfrak{g}},P_{0})$-linear via the method of the
proof of Proposition 1.6.16.

As taking the projective tensor product is exact (see [13, Lemma 2.1 (ii)]),
we obtain a $D({\mathfrak{g}},P_{0})$-linear strict epimorphism

$D({\mathfrak{g}},P_{0}){\,\otimes_{D(P_{0}),\pi}\,}\big{(}W{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\big{)}\cong\big{(}D({\mathfrak{g}},P_{0}){\,\otimes_{D(P_{0}),\pi}\,}W\big{)}{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\relbar\joinrel\twoheadrightarrow
M{\,\otimes_{K,\pi}\,}V^{\prime}_{b}$

using Lemma 1.5.11 (ii) once again. Here we extend the trivial
$U({\mathfrak{p}})$-action on $V^{\prime}_{b}$ (recall that $V$ is a smooth
representation) to the trivial $U({\mathfrak{g}})$-action. We note that this
extended action together with the given $P$-action on $V$ satisfies the
condition (2) of Definition 1.6.18.

Moreover, the locally convex $D(P_{0})$-module
$W{\,\otimes_{K,\pi}\,}V^{\prime}_{b}$ is finitely generated as an abstract
$D(P_{0})$-module, cf. the proof of [1, Prop. 4.1.5] and [1, Prop. 6.4.1].
Such an epimorphism $D(P_{0})^{\oplus n}\twoheadrightarrow
W{\,\otimes_{K,\pi}\,}V^{\prime}_{b}$, for some $n\in{\mathbb{N}}$, is
necessarily strict by the open mapping theorem [67, Prop. 8.6] since
$W{\,\otimes_{K,\pi}\,}V^{\prime}_{b}$ is a $K$-Fréchet space. Therefore we
obtain a commutative diagram

${D({\mathfrak{g}},P_{0}){\,\otimes_{K,\pi}\,}D(P_{0})^{\oplus
n}}$${D({\mathfrak{g}},P_{0}){\,\otimes_{K,\pi}\,}\big{(}W{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\big{)}}$${D({\mathfrak{g}},P_{0})^{\oplus
n}}$${D({\mathfrak{g}},P_{0}){\,\otimes_{D(P_{0}),\pi}\,}\big{(}W{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\big{)}}$

where the top homomorphism is a strict epimorphism by [13, Lemma 2.1 (ii)]. It
follows from [49, Lemma 2] that the bottom epimorphism is strict as well.

In total we thus arrive at a strict epimorphism (cf. [1, Prop. 4.1.5] for the
statement that the abstract module is finitely presented)

$\psi\colon D({\mathfrak{g}},P_{0})^{\oplus
n}\relbar\joinrel\twoheadrightarrow M{\,\otimes_{K,\pi}\,}V^{\prime}_{b}.$

Using the exactness of the projective tensor product again, we obtain the
commutative diagram

${D(G_{0}){\,\otimes_{K,\pi}\,}D({\mathfrak{g}},P_{0})^{\oplus
n}}$${D(G_{0}){\,\otimes_{K,\pi}\,}\big{(}M{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\big{)}}$${D(G_{0})^{\oplus
n}}$${D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0}),\pi}\,}\big{(}M{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\big{)}}$$\scriptstyle{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$

where all maps except a priori the bottom one are strict epimorphisms.
Similarly to before one argues that the epimorphism $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
is strict. On the other hand, the (abstract) $D(G_{0})$-module
$D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0})}\,}\big{(}M{\,\otimes_{K}\,}V^{\prime}_{b}\big{)}$
is coadmissible by [1, Thm. 4.2.3]. Therefore $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
also is strict when
$D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0})}\,}\big{(}M{\,\otimes_{K}\,}V^{\prime}_{b}\big{)}$
carries its canonical Fréchet topology (see [71, §3]). By the uniqueness of
the quotient we conclude that this Fréchet topology agrees with the topology
on
$D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0}),\pi}\,}(M{\,\otimes_{K,\pi}\,}V^{\prime}_{b})$
from Definition 1.5.8. In particular, the latter already is complete so that
taking the Hausdorff completion is redundant:

$D(G_{0}){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{0})}\,}\big{(}M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\big{)}=D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0}),\pi}\,}\big{(}M{\,\otimes_{K,\pi}\,}V^{\prime}_{b}\big{)}.$

Note here that since $\widehat{M}$ is a $K$-Fréchet space by Lemma 3.4.5 (i),
$M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\cong\widehat{M}{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}$
is so as well by [67, Lemma 19.10 (i)] and the discussion after [67, Prop.
17.6]. Therefore the projective and inductive tensor product of $D(G_{0})$ and
$M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}$ over
$D({\mathfrak{g}},P_{0})$ agree. Finally, the Iwasawa decomposition yields the
isomorphism

$D(G){\,\otimes_{D({\mathfrak{g}},P)}\,}\big{(}M{\,\otimes_{K}\,}V^{\prime}\big{)}\cong
D(G_{0}){\,\otimes_{D({\mathfrak{g}},P_{0})}\,}\big{(}M{\,\otimes_{K}\,}V^{\prime}\big{)}$

of $D(G_{0})$-modules (cf. [1, §4.2.2]), and Lemma 3.3.1 (ii) yields the
topological isomorphism

$D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P),\iota}\,}\big{(}M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\big{)}\cong
D(G_{0}){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{0})}\,}\big{(}M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\big{)}$

of separately continuous $D(G_{0})$-modules. Together they give the
topological isomorphism (3.31) of separately continuous $D(G)$-modules. ∎

We return to the concrete setting of ${\mathrm{\bf G}}={\mathrm{GL}}_{d+1,K}$
and the parabolic subgroup ${\mathrm{\bf P}}={\mathrm{\bf P}}_{r}$, for
${r}\in\\{0,\ldots,d-1\\}$, from Theorem 3.3.2. We want to apply Proposition
3.4.7 to show that (3.28) is complete and its locally convex topology agrees
with the canonical Fréchet topology.

Let $I\subset\\{0,\ldots,d\\}$ be a non-empty subset, and let
$U_{I}\subset{\mathbb{P}}_{K}^{d}$ denote the intersection of the
corresponding principal open subsets as considered in Section 2.4. Via the
countable admissible covering

$U_{I}^{\mathrm{rig}}=\bigcup_{\begin{subarray}{c}0<\varepsilon<1\\\
\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}\end{subarray}}U_{I,\varepsilon},$

and Lemma 2.4.1 we regard ${\mathcal{E}}(U_{I})$ as a subspace of
${\mathcal{E}}(U_{I}^{\mathrm{rig}})=\varprojlim_{\varepsilon\searrow
0}{\mathcal{E}}(U_{I,\varepsilon})$. Then the topology of
${\mathcal{E}}(U_{I})$ is induced by the family of norms
${\lvert{{\,\\_\,}}\rvert}_{\varepsilon}$, for $0<\varepsilon<1$, defined in
(2.19).

######  Proposition 3.4.8.

For every non-empty subset $I\subset\\{0,\ldots,d\\}$, there exist
$m\in{\mathbb{N}}$ and a surjective homomorphism of
$U({\mathfrak{g}})$-modules

$\varphi\colon U({\mathfrak{g}})^{\oplus
m}\relbar\joinrel\twoheadrightarrow{\mathcal{E}}(U_{I})$

such that $\varphi$ is continuous and strict when $U({\mathfrak{g}})\subset
D(G)$ and ${\mathcal{E}}(U_{I})\subset{\mathcal{E}}(U_{I}^{\mathrm{rig}})$
carry their respective subspace topologies.

######  Corollary 3.4.9.

The $U({\mathfrak{g}})$-module
$\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is finitely generated, and its locally convex topology induced by an
epimorphism from a finite free $U({\mathfrak{g}})$-module via (3.30) agrees
with the subspace topology
$\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\subset\widetilde{H}^{d-{r}}_{({\mathbb{P}}_{K}^{{r}})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
from Corollary 2.5.5. Consequently, the separately continuous $D(G)$-module
(3.28) already is complete and its topology agrees with the canonical Fréchet
topology.

###### Proof of Corollary 3.4.9.

Recall that we defined the Čech complex
$C^{\bullet}({\mathcal{U}},{\mathcal{E}})$ for the covering ${\mathcal{U}}$
given by

${\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}=\bigcup_{i={r}+1}^{d}U_{i}.$

We fix on $C^{q}({\mathcal{U}},{\mathcal{E}})$, for $q\geq 0$, the locally
convex topology given via the subspace topologies of
${\mathcal{E}}(U_{I})\subset{\mathcal{E}}(U_{I}^{\mathrm{rig}})$, for all non-
empty $I\subset\\{{r}+1,\ldots,d\\}$. Then we have seen in Corollary 2.5.5 (or
rather in the proof of Proposition 2.5.2) that the thereby induced topology on
$\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
agrees with the topology of the latter as a subspace of
$\widetilde{H}^{d-{r}}_{({\mathbb{P}}_{K}^{{r}})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

On the other hand, the epimorphisms of $U({\mathfrak{g}})$-modules onto
${\mathcal{E}}(U_{I})$ from Proposition 3.4.8 yield $m_{q}\in{\mathbb{N}}$ and
epimorphisms

$\varphi_{q}\colon U({\mathfrak{g}})^{\oplus
m_{q}}\relbar\joinrel\twoheadrightarrow
C^{q}({\mathcal{U}},{\mathcal{E}})\quad\text{, for all $q\geq 0$,}$ (3.32)

which are strict with regard to the subspace topologies of
$U({\mathfrak{g}})\subset D(G)$ and
${\mathcal{E}}(U_{I})\subset{\mathcal{E}}(U_{I}^{\mathrm{rig}})$. Therefore
the topology on $C^{q}({\mathcal{U}},{\mathcal{E}})$ we fixed above agrees
with the one induced via (3.32) by the uniqueness of the quotient.

Since $U({\mathfrak{g}})$ is noetherian and the differentials of
$C^{\bullet}({\mathcal{U}},{\mathcal{E}})$ are $U({\mathfrak{g}})$-linear, the
subspace $Z^{q}({\mathcal{U}},{\mathcal{E}})\subset
C^{q}({\mathcal{U}},{\mathcal{E}})$ of Čech cocycles is a finitely generated
$U({\mathfrak{g}})$-module, too. Lemma 3.4.5 (ii) implies that its topology
induced by some epimorphism from a finite free $U({\mathfrak{g}})$-module
agrees with its topology as a subspace of
$C^{q}({\mathcal{U}},{\mathcal{E}})$. We conclude that
$H^{q}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}})$ is
finitely generated as a $U({\mathfrak{g}})$-module, and its topology via an
epimorphism from a finite free $U({\mathfrak{g}})$-module agrees with its
topology as a subquotient of $C^{q}({\mathcal{U}},{\mathcal{E}})$. Arguing
similarly for
$\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$,
we find that its topology induced via some epimorphism from a finite free
$U({\mathfrak{g}})$-module agrees with the topology as a subspace of
$\widetilde{H}^{d-{r}}_{({\mathbb{P}}_{K}^{{r}})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

Proposition 3.4.7 applied to
$M=\widetilde{H}^{d-{r}}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
and $V=v^{{\mathrm{GL}}_{d-{r}}(K)}_{B_{d-{r}}}$ then shows the last
statement. ∎

###### Proof of Proposition 3.4.8.

We begin by fixing a suitable $K$-basis of ${\mathfrak{g}}$. For
$\alpha_{u,v}\in\Phi$ with $(u,v)\in\\{0,\ldots,d\\}^{2}$, $u\neq v$, let
$L_{(u,v)}$ be the standard generator of the root space
${\mathfrak{g}}_{\alpha_{u,v}}$. Moreover, let $L_{0},\ldots,L_{d}$ be the
standard basis of ${\mathfrak{t}}$. Then the $L_{(u,v)}$ together with the
$L_{i}$ constitute a $K$-basis of ${\mathfrak{g}}$. The action of
$U({\mathfrak{g}})$ on ${\mathcal{O}}(U_{I})$ is given by [56, (1.5)]

$\displaystyle L_{(u,v)}^{k}.X^{\underline{\mu}}$
$\displaystyle=\frac{\mu_{v}!}{(\mu_{v}-k)!}\,X^{{\underline{\mu}}+k\alpha_{u,v}},$
$\displaystyle L^{k}_{j}.X^{\underline{\mu}}$
$\displaystyle=\mu_{j}^{k}\,X^{\underline{\mu}},$

for $k\in{\mathbb{N}}_{0}$, where we use the convention that
$\frac{\mu_{v}!}{(\mu_{v}-k)!}=\mu_{v}\,(\mu_{v}-1)\cdots(\mu_{v}-k+1)$.

Let the fixed subset $I\subset\\{0,\ldots,d\\}$ be
$I=\\{i_{1},\ldots,i_{r}\\}$ with pairwise distinct $i_{j}$. Let $E$ be a
finitely generated graded $K[X_{0},\ldots,X_{d}]$-module such that
${\mathcal{E}}$ is associated with $E$, e.g.

$E=\bigoplus_{k\in{\mathbb{N}}_{0}}\Gamma\big{(}{\mathbb{P}}_{K}^{d},{\mathcal{E}}(k)\big{)},$

cf. [77, Tag 0BXD]. As ${\mathcal{E}}(U_{I})$ is the homogeneous localization
$(E_{X_{i_{1}}\cdots X_{i_{r}}})^{0}$ which is a finitely generated
${\mathcal{O}}(U_{I})$-module, we find homogeneous elements
$v_{1},\ldots,v_{n}\in E$ and $l_{1},\ldots,l_{n}\in{\mathbb{N}}_{0}$ with
$\deg(v_{k})=rl_{k}$ such that ${\mathcal{E}}(U_{I})$ is generated by

$\frac{v_{k}}{(X_{i_{1}}\cdots X_{i_{r}})^{l_{k}}}\quad\text{, for
$k=1,\ldots,n$,}$

as an ${\mathcal{O}}(U_{I})$-module. For $N\vcentcolon=\max_{k=1}^{n}l_{k}$,
we set $E^{\prime}\vcentcolon=\bigoplus_{l=0}^{N}E_{rl}$ and may assume that
$v_{1},\ldots,v_{n}$ is a $K$-basis of $E^{\prime}$. Since ${\mathrm{\bf G}}$
acts on $E$ by homogeneous endomorphisms, see [50, §2.3.1], so does
${\mathfrak{g}}$ and $E^{\prime}$ is a $U({\mathfrak{g}})$-submodule of $E$.
Moreover, for $\varepsilon>0$ small enough, we may suppose that the estimate

$\big{\lVert}L_{(u,v)}.v_{k}\big{\rVert}_{E^{\prime}}\leq\frac{1}{\varepsilon}\quad\text{,
for all $(u,v)\in\\{0,\ldots,d\\}^{2}$ with $u\neq v$, and $k=1,\ldots,n$,}$
(3.33)

for the maximum norm on $E^{\prime}$ with respect to the basis
$v_{1},\ldots,v_{n}$ holds.

We recall the notation

$\displaystyle\Lambda_{I}$
$\displaystyle\vcentcolon=\bigg{\\{}{\underline{\mu}}\in{\mathbb{Z}}^{d+1}\bigg{|}\sum_{j=0}^{d}\mu_{j}=0,\forall
j\in\\{0,\ldots,d\\}\setminus I:\mu_{j}\geq 0\bigg{\\}},$
$\displaystyle\Delta_{d}$
$\displaystyle\vcentcolon=\left\\{{\underline{\mu}}\in{\mathbb{Z}}^{d+1}\,\middle|\,\forall
j=0,\ldots,d:{\lvert{\mu_{j}}\rvert}\leq d\right\\}.$

As a candidate for the sought $U({\mathfrak{g}})$-linear epimorphism $\varphi$
we consider

$\displaystyle\varphi\colon
U({\mathfrak{g}})^{(I)}\vcentcolon=\bigoplus_{\begin{subarray}{c}{\underline{\nu}}\in\Lambda_{I}\cap\Delta_{d}\\\
k=1,\ldots,n\end{subarray}}U({\mathfrak{g}})\,e_{{\underline{\nu}},k}\longrightarrow{\mathcal{E}}(U_{I})\,,\quad{\mathfrak{X}}_{{\underline{\nu}},k}\,e_{{\underline{\nu}},k}\longmapsto{\mathfrak{X}}_{{\underline{\nu}},k}.\bigg{(}X^{\underline{\nu}}\,\frac{v_{k}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}\bigg{)}.$

Let ${\lvert{{\,\\_\,}}\rvert}_{\varepsilon}$ denote the norm on
$U({\mathfrak{g}})$ defined in (3.29) in terms of the $K$-basis of
${\mathfrak{g}}$ chosen in the beginning of the proof. Then it follows from
Lemma 3.4.3 that the topology on the locally convex direct sum
$U({\mathfrak{g}})^{(I)}$ is induced by the family of norms

$\bigg{\lvert}\sum_{\begin{subarray}{c}{\underline{\nu}}\in\Lambda_{I}\cap\Delta_{d}\\\
k=1,\ldots,n\end{subarray}}{\mathfrak{X}}_{{\underline{\nu}},k}\,e_{{\underline{\nu}},k}\bigg{\rvert}_{\varepsilon}\vcentcolon=\max_{\begin{subarray}{c}{\underline{\nu}}\in\Lambda_{I}\cap\Delta_{d}\\\
k=1,\ldots,n\end{subarray}}\lvert{\mathfrak{X}}_{{\underline{\nu}},k}\rvert_{\varepsilon}\quad\text{,
for $0<\varepsilon<1$.}$

######  Lemma 3.4.10.

The map $\varphi$ is a continuous homomorphism of locally convex $K$-vector
spaces.

###### Proof.

It suffices to show that, for all $v\in{\mathcal{E}}(U_{I})$, the map
$U({\mathfrak{g}})\rightarrow{\mathcal{E}}(U_{I})$,
${\mathfrak{X}}\mapsto{\mathfrak{X}}.v$, is continuous. This in turn follows
once we see that, for all $v\in{\mathcal{E}}(U_{I})$ and $0<\varepsilon<1$,
there exists $0<\delta<1$ such that

$\lvert{\mathfrak{X}}.v\rvert_{\varepsilon}\leq\lvert{\mathfrak{X}}\rvert_{\delta}\lvert
v\rvert_{\varepsilon}\quad\text{, for all ${\mathfrak{X}}\in
U({\mathfrak{g}})$.}$ (3.34)

We denote the $K$-basis elements of ${\mathfrak{g}}$ fixed earlier by

$\big{\\{}{\mathfrak{x}}_{1},\ldots,{\mathfrak{x}}_{s}\big{\\}}\vcentcolon=\big{\\{}L_{(u,v)}\,\big{|}\,\alpha_{u,v}\in\Phi\big{\\}}\cup\big{\\{}L_{0},\ldots,L_{d}\big{\\}}.$

Moreover, let $w_{1},\ldots,w_{m}$ be an ${\mathcal{O}}(U_{I})$-basis of
${\mathcal{E}}(U_{I})$ which defines the family of norms
${\lvert{{\,\\_\,}}\rvert}_{\varepsilon}$ as in (2.19). For a given
$0<\varepsilon<1$, we choose $\delta>0$ small enough such that
$\delta\leq\varepsilon$ and

$\lvert{\mathfrak{x}}_{j}w_{l}\rvert_{\varepsilon}\leq\frac{1}{\delta}\quad\text{,
for all $j=1,\ldots,s$ and $l=1,\ldots,m$.}$

We first want to show that

$\lvert{\mathfrak{x}}.f\rvert_{\varepsilon}\leq\frac{1}{\varepsilon}\,\lvert
f\rvert_{\varepsilon}\quad\text{, for all
${\mathfrak{x}}\in\\{{\mathfrak{x}}_{1},\ldots,{\mathfrak{x}}_{s}\\}$ and
$f\in{\mathcal{O}}(U_{I})$.}$ (3.35)

To this end it suffices to consider monomials $f=X^{\underline{\mu}}$, for
${\underline{\mu}}\in\Lambda_{I}$. For ${\mathfrak{x}}=L_{j}$, we have

$\lvert
L_{j}.X^{\underline{\mu}}\rvert_{\varepsilon}=\lvert\mu_{j}\,X^{\underline{\mu}}\rvert_{\varepsilon}={\lvert{\mu_{j}}\rvert}\lvert
X^{\underline{\mu}}\rvert_{\varepsilon}\leq\lvert
X^{\underline{\mu}}\rvert_{\varepsilon}\leq\frac{1}{\varepsilon}\,\lvert
X^{\underline{\mu}}\rvert_{\varepsilon}.$

Moreover, for ${\mathfrak{x}}=L_{(u,v)}$, we compute

$\displaystyle\lvert
L_{(u,v)}.X^{\underline{\mu}}\rvert_{\varepsilon}=\lvert\mu_{v}\,X^{{\underline{\mu}}+\alpha_{u,v}}\rvert_{\varepsilon}={\lvert{\mu_{v}}\rvert}\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}}+\alpha_{u,v})}\rvert}}\leq\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}}+\alpha_{u,v})}\rvert}}.$

The sought inequality $\lvert
L_{(u,v)}.X^{\underline{\mu}}\rvert_{\varepsilon}\leq\big{(}\frac{1}{\varepsilon}\big{)}^{1+{\lvert{\max(0,{\underline{\mu}})}\rvert}}=\frac{1}{\varepsilon}\lvert
X^{\underline{\mu}}\rvert_{\varepsilon}$ then follows from

$\max(0,\mu_{u}+1)+\max(0,\mu_{v}-1)\leq 1+\max(0,\mu_{u})+\max(0,\mu_{v}).$

Now, we turn towards
${\mathcal{E}}(U_{I})\cong{\mathcal{O}}(U_{I}){\,\otimes_{K}\,}{\mathcal{E}}(x_{i})$.
To prove (3.34) it suffices to show the following inequalities on the PBW
basis of $U({\mathfrak{g}})$:

$\bigg{\lvert}({\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{s}^{k_{s}}).\bigg{(}\sum_{l=1}^{m}f_{l}\,w_{l}\bigg{)}\bigg{\rvert}_{\varepsilon}\leq\bigg{(}\frac{1}{\delta}\bigg{)}^{{\lvert{{\underline{k}}}\rvert}}\max_{l=1}^{m}\,\lvert
f_{l}\rvert_{\varepsilon}=\lvert{\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{s}^{k_{s}}\rvert_{\delta}\,\bigg{\lvert}\sum_{l=1}^{m}f_{l}\,w_{l}\bigg{\rvert}_{\varepsilon},$

for all ${\underline{k}}\in{\mathbb{N}}_{0}^{s}$ and
$\sum_{l=1}^{m}f_{l}\,w_{l}\in{\mathcal{E}}(U_{I})$. We do so via induction on
${\lvert{{\underline{k}}}\rvert}$. With the base case being clear, let
${\underline{k}}\in{\mathbb{N}}^{s}_{0}$ with
${\lvert{{\underline{k}}}\rvert}>0$, and
$\sum_{l=1}^{m}f_{l}\,w_{l}\in{\mathcal{E}}(U_{I})$. Let
$j\in\\{1,\ldots,s\\}$ be maximal such that $k_{j}>0$ and $k_{j^{\prime}}=0$,
for all $j^{\prime}>j$. By the Leibniz product rule we have

$\displaystyle\bigg{\lvert}{\mathfrak{x}}_{j}.\bigg{(}\sum_{l=1}^{m}f_{l}\,w_{l}\bigg{)}\bigg{\rvert}_{\varepsilon}$
$\displaystyle=\bigg{\lvert}\sum_{l=1}^{m}({\mathfrak{x}}_{j}.f_{l})\,w_{l}+f_{l}\,({\mathfrak{x}}_{j}.w_{l})\bigg{\rvert}_{\varepsilon}\leq\max_{l=1}^{m}\,\max\big{(}\,\lvert{\mathfrak{x}}_{j}.f_{l}\rvert_{\varepsilon}\,,\,\lvert
f_{l}\rvert_{\varepsilon}\,\lvert{\mathfrak{x}}_{j}.w_{l}\rvert_{\varepsilon}\big{)}$
$\displaystyle\leq\max_{l=1}^{m}\,\max\bigg{(}\frac{1}{\varepsilon}\,\lvert
f_{l}\rvert_{\varepsilon}\,,\,\lvert
f_{l}\rvert_{\varepsilon}\,\frac{1}{\delta}\bigg{)}\leq\frac{1}{\delta}\max_{l=1}^{m}\,\lvert
f_{l}\rvert_{\varepsilon}.$

Here we have additionally used (3.35), $\delta\leq\varepsilon$, and that
$\lvert{\mathfrak{x}}_{j}.w_{l}\rvert_{\varepsilon}\leq\frac{1}{\delta}$, for
all $l=1,\ldots,m$. Together with the induction hypothesis we then conclude
that

$\displaystyle\bigg{\lvert}({\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{s}^{k_{s}}).\bigg{(}\sum_{l=1}^{m}f_{l}\,w_{l}\bigg{)}\bigg{\rvert}_{\varepsilon}$
$\displaystyle=\Bigg{\lvert}({\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{j}^{k_{j}-1}).\Bigg{(}{\mathfrak{x}}_{j}.\bigg{(}\sum_{l=1}^{m}f_{l}\,w_{l}\bigg{)}\Bigg{)}\Bigg{\rvert}_{\varepsilon}$
$\displaystyle\leq\lvert{\mathfrak{x}}_{1}^{k_{1}}\cdots{\mathfrak{x}}_{j}^{k_{j}-1}\rvert_{\delta}\,\bigg{\lvert}{\mathfrak{x}}_{j}.\bigg{(}\sum_{l=1}^{m}f_{l}\,w_{l}\bigg{)}\bigg{\rvert}_{\varepsilon}\leq\bigg{(}\frac{1}{\delta}\bigg{)}^{{\lvert{{\underline{k}}}\rvert}}\max_{l=1}^{m}\,\lvert
f_{l}\rvert_{\varepsilon}.$

∎

We also introduce, for ${\underline{\mu}}\in\Lambda_{I}$, the auxiliary
constants

$A({\underline{\mu}})\vcentcolon=\prod_{k=0}^{N}\Bigg{(}\prod_{\begin{subarray}{c}j=0\\\
j\in
I\end{subarray}}^{d}\frac{1}{{\lvert{\min(0,\mu_{j}+1-k)!}\rvert}}\cdot\prod_{\begin{subarray}{c}j=0\\\
j\notin
I\end{subarray}}^{d}\frac{1}{{\lvert{\min(0,\mu_{j}+1)!}\rvert}}\Bigg{)}.$

Here and later we use the conventions

$\displaystyle(-n)!$
$\displaystyle\vcentcolon=(-n)(-n+1)\cdots(-1)=(-1)^{n}n!\quad\text{, for
$n\in{\mathbb{N}}_{0}$, and}$ $\displaystyle\max(0,{\underline{\mu}})$
$\displaystyle\vcentcolon=\big{(}\max(0,\mu_{0}),\ldots,\max(0,\mu_{d})\big{)}.$

######  Lemma 3.4.11.

For all $l=0,\ldots,N$, ${\underline{\mu}}\in\Lambda_{I}$, and
$u,v\in\\{0,\ldots,d\\}$ with $u\neq v$, $\mu_{u}>1$, $\mu_{v}<-1$, it holds
that

$\frac{1}{{\lvert{\mu_{v}+1-l}\rvert}}\,A({\underline{\mu}}-\alpha_{u,v})\leq
A({\underline{\mu}}).$

There are constants $C>0$, $a\geq 1$ such that $A({\underline{\mu}})\leq
Ca^{\lvert{\max(0,{\underline{\mu}})}\rvert}$, for all
${\underline{\mu}}\in\Lambda_{I}$.

###### Proof.

Writing ${\underline{\mu}}^{\prime}\vcentcolon={\underline{\mu}}-\alpha_{u,v}$
in the setting of (i), it suffices to show that the rational number

$\displaystyle\frac{1}{\mu_{v}+1-l}\,\prod_{k=0}^{N}\Bigg{(}\prod_{\begin{subarray}{c}j=0\\\
j\in
I\end{subarray}}^{d}\frac{\min(0,\mu_{j}+1-k)!}{\min(0,\mu^{\prime}_{j}+1-k)!}\cdot\prod_{\begin{subarray}{c}j=0\\\
j\notin
I\end{subarray}}^{d}\frac{\min(0,\mu_{j}+1)!}{\min(0,\mu^{\prime}_{j}+1)!}\Bigg{)}$

in fact is an integer. We may focus on the terms in the product for $j=u$ and
$j=v$. But using $\mu_{u}>1$ and $\mu_{v}<-1$, we indeed find that

$\displaystyle\frac{1}{\mu_{v}+1-l}\,\prod_{k=0}^{N}\frac{\min(0,\mu_{v}+1-k)!}{\min(0,\mu^{\prime}_{v}+1-k)!}\cdot\frac{\min(0,\mu_{u}+1)!}{\min(0,\mu^{\prime}_{u}+1)!}$
$\displaystyle\qquad=\frac{1}{\mu_{v}+1-l}\,\prod_{k=0}^{N}\frac{(\mu_{v}+1-k)!}{(\mu_{v}+2-k)!}\cdot\frac{0!}{0!}=\prod_{\begin{subarray}{c}k=0\\\
k\neq l\end{subarray}}^{N}(\mu_{v}+1-k)\in{\mathbb{Z}}.$

For (ii), we again use the estimate $v_{p}(n!)\leq\frac{n}{p-1}$ of the
$p$-adic valuation with $n\in{\mathbb{N}}_{0}$. This yields, for $j\in I$,

$\displaystyle
v_{p}\big{(}\min(0,\mu_{j}+1-k)!\big{)}\leq\frac{-\min(0,\mu_{j}+1-k)}{p-1}\leq\frac{-\min(0,\mu_{j})+k}{p-1}.$

It follows from similar estimates for $j\notin I$ and from
$\sum_{j=0}^{d}\mu_{j}=0$ that

$\displaystyle
v_{p}\Bigg{(}\prod_{k=0}^{N}\Bigg{(}\prod_{\begin{subarray}{c}j=0\\\ j\in
I\end{subarray}}^{d}\min(0,\mu_{j}+1-k)!\cdot\prod_{\begin{subarray}{c}j=0\\\
j\notin I\end{subarray}}^{d}\min(0,\mu_{j}+1)!\Bigg{)}\Bigg{)}$
$\displaystyle\qquad\leq\frac{1}{p-1}\sum_{k=0}^{N}\bigg{(}rk-\sum_{j=0}^{d}\min(0,\mu_{j})\bigg{)}=\frac{1}{p-1}\bigg{(}r\,\frac{N(N+1)}{2}+(N+1){\lvert{\max(0,{\underline{\mu}})}\rvert}\bigg{)}.$

Hence we conclude that

$\displaystyle A({\underline{\mu}})\leq
p^{\frac{1}{p-1}\big{(}r\,\frac{N(N+1)}{2}+(N+1){\lvert{\max(0,{\underline{\mu}})}\rvert}\big{)}}$

which shows the claim with the constants $C=p^{\frac{rN(N+1)}{2(p-1)}}$ and
$a=p^{\frac{N+1}{p-1}}$. ∎

######  Lemma 3.4.12.

Let $0<\varepsilon<1$ be small enough such that it satisfies the condition of
(3.33). Then, for all ${\underline{\mu}}\in\Lambda_{I}$ and
$k\in\\{1,\ldots,n\\}$, there exists ${\mathfrak{Y}}_{{\underline{\mu}},k}\in
U({\mathfrak{g}})^{(I)}$ such that
$\displaystyle\varphi\big{(}{\mathfrak{Y}}_{{\underline{\mu}},k}\big{)}=X^{\underline{\mu}}\,\frac{v_{k}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}$,

$\displaystyle\big{\lvert}{\mathfrak{Y}}_{{\underline{\mu}},k}\big{\rvert}_{\varepsilon}\leq
A({\underline{\mu}})\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{\lvert{\max(0,{\underline{\mu}})}\rvert}$.

###### Proof.

We use induction on
$\lVert{\underline{\mu}}\rVert\vcentcolon=\sum_{j=0}^{d}{\lvert{\mu_{j}}\rvert}$.
For ${\underline{\mu}}\in\Delta_{d}$, we can take
${\mathfrak{Y}}_{{\underline{\mu}},k}\vcentcolon=1\,e_{{\underline{\mu}},k}$
and the claim is trivial. Therefore we may assume that there is
$u\in\\{0,\ldots,d\\}$ such that ${\lvert{\mu_{u}}\rvert}>d$. If $\mu_{u}>0$,
it follows from

$0=\sum_{j=0}^{d}\mu_{j}=\mu_{u}+\sum_{\begin{subarray}{c}j=0\\\ j\neq
u\end{subarray}}^{d}\mu_{j}$

that $\sum_{j=0,j\neq u}^{d}\mu_{j}<-d$, and hence that there exists $v\in I$,
$v\neq u$ such that $\mu_{v}<-1$. Arguing similarly for the case $\mu_{u}<0$,
we thus may assume that there are $u\in\\{0,\ldots,d\\}$, $v\in I$, $u\neq v$,
such that $\mu_{u}>1$, $\mu_{v}<-1$.

We now set
${\underline{\mu}}^{\prime}\vcentcolon={\underline{\mu}}-\alpha_{u,v}={\underline{\mu}}-\epsilon_{u}+\epsilon_{v}$
so that
$\lVert{\underline{\mu}}^{\prime}\rVert=\lVert{\underline{\mu}}\rVert-2$. For
$k\in\\{1,\ldots,n\\}$, we express $L_{(u,v)}.v_{k}$ in the $K$-basis
$v_{1},\ldots,v_{n}$ of $E^{\prime}$:

$L_{(u,v)}.v_{k}=\sum_{k^{\prime}=1}^{n}a_{k^{\prime}}v_{k^{\prime}}\quad\text{,
for $a_{1},\ldots,a_{n}\in K$,}$

so that $\max_{k^{\prime}=1}^{n}{\lvert{a_{k^{\prime}}}\rvert}=\lVert
L_{(u,v)}.v_{k}\rVert_{E^{\prime}}\leq\frac{1}{\varepsilon}$ using (3.33).
Note that for the degrees of these homogeneous elements we have
$l_{k^{\prime}}=l_{k}$, for all $k^{\prime}\in\\{1,\ldots,n\\}$ with
$a_{k^{\prime}}\neq 0$, since ${\mathfrak{g}}$ acts on $E^{\prime}$ by
homogeneous endomorphisms. Moreover, by the induction hypothesis for
${\underline{\mu}}^{\prime}$, for all $k^{\prime}=1,\ldots,n$, there exist
${\mathfrak{Y}}_{{\underline{\mu}}^{\prime},k^{\prime}}\in
U({\mathfrak{g}})^{(I)}$ as specified in the statement of the lemma. We set

${\mathfrak{Y}}_{{\underline{\mu}},k}\vcentcolon=\frac{1}{\mu_{v}+1-l_{k}}\bigg{(}L_{(u,v)}{\mathfrak{Y}}_{{\underline{\mu}}^{\prime},k}-\sum_{k^{\prime}=1}^{n}a_{k^{\prime}}{\mathfrak{Y}}_{{\underline{\mu}}^{\prime},k^{\prime}}\bigg{)}.$

Applying the Leibniz product rule we then find that

$\displaystyle
L_{(u,v)}.\bigg{(}X^{{\underline{\mu}}^{\prime}}\frac{v_{k}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}\bigg{)}$
$\displaystyle=\bigg{(}L_{(u,v)}.\frac{X^{{\underline{\mu}}^{\prime}}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}\bigg{)}v_{k}+\frac{X^{{\underline{\mu}}^{\prime}}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}\big{(}L_{(u,v)}.v_{k}\big{)}$
$\displaystyle=(\mu_{v}+1-l_{k})\frac{X^{{\underline{\mu}}}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}v_{k}+\sum_{k^{\prime}=1}^{n}a_{k^{\prime}}X^{{\underline{\mu}}^{\prime}}\frac{v_{k^{\prime}}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k^{\prime}}}}.$

From this it follows that

$\varphi\big{(}{\mathfrak{Y}}_{{\underline{\mu}},k}\big{)}=\frac{X^{{\underline{\mu}}}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}v_{k}.$

Moreover, the estimates from the induction hypothesis for the
${\mathfrak{Y}}_{{\underline{\mu}}^{\prime},k^{\prime}}$ yield

$\displaystyle{\lvert{{\mathfrak{Y}}_{{\underline{\mu}},k}}\rvert}_{\varepsilon}$
$\displaystyle\leq\frac{1}{{\lvert{\mu_{v}+1-l_{k}}\rvert}}\max\Big{(}{\lvert{L_{(u,v)}{\mathfrak{Y}}_{{\underline{\mu}}^{\prime},k}}\rvert}_{\varepsilon}\,,\,\max_{k^{\prime}=1}^{n}{\lvert{a_{k^{\prime}}}\rvert}{\lvert{{\mathfrak{Y}}_{{\underline{\mu}}^{\prime},k^{\prime}}}\rvert}_{\varepsilon}\Big{)}$
$\displaystyle\leq\frac{1}{{\lvert{\mu_{v}+1-l_{k}}\rvert}}\max\Bigg{(}A({\underline{\mu}}^{\prime})\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}}^{\prime})}\rvert}+1},\Big{(}\max_{k^{\prime}=1}^{n}{\lvert{a_{k^{\prime}}}\rvert}\Big{)}A({\underline{\mu}}^{\prime})\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}}^{\prime})}\rvert}}\Bigg{)}$
$\displaystyle\leq\frac{1}{{\lvert{\mu_{v}+1-l_{k}}\rvert}}A({\underline{\mu}}^{\prime})\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}}^{\prime})}\rvert}+1}$
$\displaystyle\leq
A({\underline{\mu}})\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}})}\rvert}}.$

In the last step we also have used Lemma 3.4.11 (i) and that
${\lvert{\max(0,{\underline{\mu}}^{\prime})}\rvert}={\lvert{\max(0,{\underline{\mu}})}\rvert}-1$.
∎

After these preparations we now show that the epimorphism $\varphi$ is strict.
It suffices to prove that $\varphi$ is an open map. Hence, let $U\subset
U({\mathfrak{g}})^{(I)}$ be an open neighbourhood of $0$ and we want to show
that $\varphi(U)$ is an open neighbourhood of $0$. By [59, Thm. 3.3.6] there
exist $0<\varepsilon_{1},\ldots,\varepsilon_{t}<1$, for some
$t\in{\mathbb{N}}$, and $R>0$ such that

$\big{\\{}{\mathfrak{X}}\in
U({\mathfrak{g}})^{(I)}\big{|}\max_{i=1}^{t}\,\lvert{\mathfrak{X}}\rvert_{\varepsilon_{i}}<R\big{\\}}\subset
U.$

We may suppose that
$\varepsilon\vcentcolon=\varepsilon_{1}\leq\ldots\leq\varepsilon_{t}$ so that
the former set is equal to

$B^{({\lvert{{\,\\_\,}}\rvert}_{\varepsilon}),-}_{R}(0)\vcentcolon=\big{\\{}{\mathfrak{X}}\in
U({\mathfrak{g}})^{(I)}\,\big{|}\,\lvert{\mathfrak{X}}\rvert_{\varepsilon}<R\big{\\}}$

as
$\lvert{\,\\_\,}\rvert_{\varepsilon^{\prime}}\leq\lvert{\,\\_\,}\rvert_{\varepsilon}$,
for $\varepsilon\leq\varepsilon^{\prime}$. It follows that
$\varphi\big{(}B^{({\lvert{{\,\\_\,}}\rvert}_{\varepsilon}),-}_{R}(0)\big{)}\subset\varphi(U)$.
We moreover may assume that $\varepsilon>0$ is sufficiently small as in
(3.33).

Then let $\delta\vcentcolon=a^{-1}\varepsilon$ where $a\geq 1$ is the constant
from Lemma 3.4.11 (ii). Let $\lVert{\,\\_\,}\rVert_{\delta}$ denote the
quotient norm on ${\mathcal{E}}(U_{I})$ induced via

$\bigoplus_{k=1}^{n}{\mathcal{O}}(U_{I})\,e_{k}\relbar\joinrel\twoheadrightarrow{\mathcal{E}}(U_{I})\,,\quad
X^{\underline{\mu}}\,e_{k}\longmapsto
X^{\underline{\mu}}\frac{v_{k}}{(X_{i_{1}}\cdots X_{i_{r}})^{l_{k}}},$

with ${\mathcal{O}}(U_{I})\subset{\mathcal{O}}(U_{I,\delta})$ carrying the
Gauss norm ${\lvert{{\,\\_\,}}\rvert}_{\delta}$, see Remark 2.4.2. Note that
the norm $\lvert{\,\\_\,}\rvert_{\delta}$ on ${\mathcal{E}}(U_{I})$ (i.e. the
one given by a fixed ${\mathcal{O}}(U_{I})$-basis as in (2.19)) is equivalent
to $\lVert{\,\\_\,}\rVert_{\delta}$ by [7, 3.7.3 Prop. 3]. To prove the
proposition it therefore suffices to show that

$B_{C^{-1}R}^{(\lVert{\,\\_\,}\rVert_{\delta}),-}(0)\subset\varphi\big{(}B_{R}^{({\lvert{{\,\\_\,}}\rvert}_{\varepsilon}),-}(0)\big{)}$
(3.36)

where $C>0$ is the constant from Lemma 3.4.11 (ii). Indeed, then
$B_{C^{-1}R}^{(\lVert{\,\\_\,}\rVert_{\delta}),-}(0)\subset\varphi(U)$ is an
open neighbourhood of $0$ with respect to $\lvert{\,\\_\,}\rvert_{\delta}$ on
${\mathcal{E}}(U_{I})$, and hence with respect to the subspace topology of
${\mathcal{E}}(U_{I})\subset{\mathcal{E}}(U_{I}^{\mathrm{rig}})$ as well.

To show (3.36), let $v\in{\mathcal{E}}(U_{I})$ with $\lVert
v\rVert_{\delta}<\frac{R}{C}$. By the definition of the quotient norm
$\lVert{\,\\_\,}\rVert_{\delta}$, there exist $a_{{\underline{\mu}},k}\in K$
such that

$\displaystyle v=\sum_{\begin{subarray}{c}{\underline{\mu}}\in\Lambda_{I}\\\
k=1,\ldots,n\end{subarray}}a_{{\underline{\mu}},k}\,X^{\underline{\mu}}\frac{v_{k}}{(X_{i_{1}}\cdots
X_{i_{r}})^{l_{k}}}\qquad\text{and}\qquad\sup_{\begin{subarray}{c}{\underline{\mu}}\in\Lambda_{I}\\\
k=1,\ldots,n\end{subarray}}{\lvert{a_{{\underline{\mu}},k}}\rvert}\bigg{(}\frac{1}{\delta}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}})}\rvert}}<\frac{R}{C}.$

For all ${\underline{\mu}}\in\Lambda_{I}$ and $k=1,\ldots,n$, we find
${\mathfrak{Y}}_{{\underline{\mu}},k}\in U({\mathfrak{g}})^{(I)}$ as specified
in Lemma 3.4.12. Then

${\mathfrak{Y}}\vcentcolon=\sum_{\begin{subarray}{c}{\underline{\mu}}\in\Lambda_{I}\\\
k=1,\ldots,n\end{subarray}}a_{{\underline{\mu}},k}\,{\mathfrak{Y}}_{{\underline{\mu}},k}\in
U({\mathfrak{g}})^{(I)}$

satisfies $\varphi({\mathfrak{Y}})=v$. Moreover, it follows from Lemma 3.4.11
(ii) that

$\displaystyle\lvert{\mathfrak{Y}}\rvert_{\varepsilon}$
$\displaystyle\leq\sup_{\begin{subarray}{c}{\underline{\mu}}\in\Lambda_{I}\\\
k=1,\ldots,n\end{subarray}}{\lvert{a_{{\underline{\mu}},k}}\rvert}{\lvert{{\mathfrak{Y}}_{{\underline{\mu}},k}}\rvert}_{\varepsilon}$
$\displaystyle\leq\sup_{\begin{subarray}{c}{\underline{\mu}}\in\Lambda_{I}\\\
k=1,\ldots,n\end{subarray}}{\lvert{a_{{\underline{\mu}},k}}\rvert}A({\underline{\mu}})\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}})}\rvert}}\leq
C\sup_{\begin{subarray}{c}{\underline{\mu}}\in\Lambda_{I}\\\
k=1,\ldots,n\end{subarray}}{\lvert{a_{{\underline{\mu}},k}}\rvert}\bigg{(}\frac{1}{\delta}\bigg{)}^{{\lvert{\max(0,{\underline{\mu}})}\rvert}}<R$

showing that
$v\in\varphi\big{(}B_{R}^{({\lvert{{\,\\_\,}}\rvert}_{\varepsilon}),-}(0)\big{)}$.
∎

Finally, we allow $K$ to be a finite extension of a non-archimedean local
field $L$ of arbitrary characteristic, and return to the setting of a
connected split reductive group ${\mathrm{\bf G}}$ over $L$ as considered in
the beginning of this section. Let ${\mathrm{\bf P}}$ a standard parabolic
subgroup of ${\mathrm{\bf G}}$. The preceding considerations as well as the
generalization of the functors ${\mathcal{F}}^{G}_{P}$ in the $p$-adic
situation due to Agrawal and Strauch [1] motivate the following definition.

######  Definition 3.4.13 (cf. [1, Def. 4.2.1]).

For a separately continuous $D({\mathfrak{g}},P,K)$-module $M$ and an
admissible smooth $L_{\mathrm{\bf P}}$-representations $V$ on a $K$-vector
space considered with the finest locally convex topology, we define the
functor

$\widetilde{{\mathcal{F}}}^{G}_{P}(M,V)\vcentcolon=D(G,K){\,\widehat{\otimes}_{D({\mathfrak{g}},P,K),\iota}\,}\big{(}M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\big{)}$

which takes values in the category of separately continuous $D(G,K)$-modules.

Like before $V^{\prime}_{b}$ is a nuclear $K$-Fréchet space, for such an
admissible smooth $L_{\mathrm{\bf P}}$-representation $V$. If the Hausdorff
completion $\widehat{M}$ of $M$ is a nuclear $K$-Fréchet space as well, it
follows from the discussion after [67, Prop. 17.6] and [67, Prop. 19.11, Prop.
20.4] that
$M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}=\widehat{M}{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}$
is a nuclear $K$-Fréchet space, too. Since Lemma 3.3.1 (ii) yields a
topological isomorphism

$D(G){\,\widehat{\otimes}_{D({\mathfrak{g}},P),\iota}\,}\big{(}M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\big{)}\cong
D(G_{0}){\,\widehat{\otimes}_{D({\mathfrak{g}},P_{0})}\,}\big{(}M{\,\widehat{\otimes}_{K,\pi}\,}V^{\prime}_{b}\big{)},$

it follows from [67, Prop. 19.4 (ii)] that
$\widetilde{{\mathcal{F}}}^{G}_{P}(M,V)$ is a nuclear $K$-Fréchet space. Hence
the strong dual $\widetilde{{\mathcal{F}}}^{G}_{P}(M,V)^{\prime}_{b}$ is a
locally analytic $G$-representation of compact type in this case, by
Proposition 1.4.10.

######  Remark 3.4.14.

Let $M$ be an abstract module for the (algebraic) distribution algebra ${\rm
Dist}({\mathrm{\bf G}}){\,\otimes_{L}\,}K={\mathrm{hy}}(G)$. An obvious
question is under which algebraic conditions on the module $M$ (such as
$M\in{\mathcal{O}}_{\mathrm{alg}}^{\mathfrak{p}}$ in the $p$-adic case) this
lifts to a separately continuous $D({\mathfrak{g}},P)$-module structure on
$M$. Here analogues of the BGG category ${\mathcal{O}}$ in the setting of
${\rm char}(K)>0$ should play a role. For recent developments in regard to
these analogues, see [2, 3, 31, 55].

##  Appendix A Non-Archimedean Functional Analysis

Here we collect some results of non-archimedean functional analysis. They are
all (slight generalizations of) statements that can be found in the literature
or adaptions from the archimedean setting. Let $K$ be a spherically complete
non-archimedean field with ring of integers ${\mathcal{O}}_{K}=\\{x\in
K\mid{\lvert{x}\rvert}\leq 1\\}$.

We recall the definition of a compact continuous homomorphism between locally
convex $K$-vector spaces [59, Ch. 8.8 p. 334]:

######  Definition A.1.

A continuous homomorphism $f\colon V\rightarrow W$ between Hausdorff locally
convex $K$-vector spaces is compact if there exists a complete compactoid
subset $X\subset W$ (cf. [59, Def. 3.8.1]) such that $f^{-1}(X)\subset V$ is a
neighbourhood of $0$ in $V$.

A continuous homomorphism $f\colon V\rightarrow W$ between Hausdorff locally
convex $K$-vector spaces is semicompact if there exists a compactoid Banach
disk $B\subset W$ (cf. [59, p. 414]) such that $f^{-1}(B)\subset V$ is a
neighbourhood of $0$ in $V$.

######  Remarks A.2.

If $f$ is compact, then $f$ is semicompact. If $W$ is quasi-complete, then the
converse holds as well [59, Ch. 8.8 p. 334].

This definition is equivalent to the one in [67, §16]. There $f$ is defined to
be compact if there exists an open lattice $L\subset V$ such that the closure
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{f}\subset W$ is bounded and c-compact.

###### Proof of (ii).

For an open lattice $L\subset V$, by [67, Prop. 12.7] $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
is bounded and c-compact if and only if it is compactoid and complete. Hence
$X\vcentcolon=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
yields a complete compactoid subset such that $f^{-1}(X)\supset L$ is a
neighbourhood of $0$ in $V$.

On the other hand, given a complete compactoid $X\subset W$ such that
$f^{-1}(X)$ is a neighbourhood of $0$ in $V$, i.e. $f^{-1}(X)$ contains an
open lattice $L\subset V$, it follows that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{f}\subset\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}$.
But this implies that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
itself is compactoid by [59, Thm. 3.8.4], and complete by [67, Rmk. 7.1 (iv),
(v)]. ∎

######  Lemma A.3.

Consider the following commutative square of Hausdorff locally convex
$K$-vector spaces:

${V}$${W}$${V^{\prime}}$${W^{\prime}.}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$
(A.1)

If $g^{\prime}$ is compact and $f^{\prime}$ a strict monomorphism, then $g$ is
compact.

In the commutative square (A.1), if $g$ is compact and $f$ a strict
epimorphism, then $g^{\prime}$ is compact.

Finite products of compact homomorphisms are compact.

###### Proof.

For (i), it follows from [67, Rmk. 16.7 (i)] that $f^{\prime}\circ
g=g^{\prime}\circ f$ is compact if $g^{\prime}$ is. Since
${\mathrm{Im}}(f^{\prime})\cong V^{\prime}$, [67, Rmk. 16.7 (ii)] implies that
$g$ is compact.

In the situation of (ii), we again know from [67, Rem. 16.7 (i)] that
$g^{\prime}\circ f=f^{\prime}\circ g$ is compact. Then by definition there
exists an open lattice $L\subset V$ such that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
is bounded and c-compact. But as $f$ is a strict epimorphism, $f(L)$ is an
open lattice in $W$ which shows that $g^{\prime}$ is compact.

For (iii), we consider compact homomorphisms $f\colon V\rightarrow W$ and
$f^{\prime}\colon V^{\prime}\rightarrow W^{\prime}$ with open lattices
$L\subset V$, $L^{\prime}\subset V^{\prime}$ such that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
is bounded and c-compact in $W$ and the same holds for $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
in $W^{\prime}$. Then $L\times L^{\prime}$ is an open lattice in $V\times
V^{\prime}$ and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{f}\times\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
is bounded in $W\times W^{\prime}$. Moreover, $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{f}\times\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
is c-compact by [67, Prop. 12.2]. ∎

######  Proposition A.4 ([74, Prop. 1.2 (i)]).

Let $V$ be a locally convex $K$-vector space of compact type. If $U\subset V$
is a closed subspace then $U$ and $V/U$ are of compact type again.

###### Proof.

Let $(V_{n})_{n\in{\mathbb{N}}}$ be an inductive sequence of $K$-Banach spaces
with injective and compact transition maps such that
$V\cong\varinjlim_{n\in{\mathbb{N}}}V_{n}$. By [23, Thm. 3.1.16], $U$ with its
subspace topology is topologically isomorphic to the inductive limit of the
sequence $(U_{n})_{n\in{\mathbb{N}}}$ where $U_{n}\vcentcolon=U\cap V_{n}$.
Then the transition maps of $(U_{n})_{n\in{\mathbb{N}}}$ are compact again
[67, Rmk. 16.7]. Therefore $U$ is of compact type.

Furthermore, by Lemma A.3 (ii), the induced maps $V_{n}/U_{n}\rightarrow
V_{n+1}/U_{n+1}$ are injective and compact. Hence
$\varinjlim_{n\in{\mathbb{N}}}V_{n}/U_{n}$ is of compact type. Moreover,
taking the inductive limit over the short strictly exact sequences
$0\rightarrow U_{n}\rightarrow V_{n}\rightarrow V_{n}/U_{n}\rightarrow 0$ we
arrive at the sequence of continuous homomorphisms

$0\longrightarrow U\longrightarrow
V\longrightarrow\varinjlim_{n\in{\mathbb{N}}}V_{n}/U_{n}\longrightarrow 0$

which is exact as a sequence of $K$-vector spaces. It follows from the open
mapping theorem [11, II. §4.6 Cor.] that the continuous surjection
$V\rightarrow\varinjlim_{n\in{\mathbb{N}}}V_{n}/U_{n}$ is strict. Therefore
$\varinjlim_{n\in{\mathbb{N}}}V_{n}/U_{n}\rightarrow V/U$ is a topological
isomorphism. ∎

######  Proposition A.5 ([74, Thm. 1.3]).

The strong dual of a locally convex $K$-vector space of compact type is a
nuclear Fréchet space, and the strong dual of a nuclear $K$-Fréchet space is
of compact type.

###### Proof.

If $V$ is a locally convex $K$-vector space of compact type, it follows from
[23, Thm. 3.1.7 (vii),(viii)] that $V^{\prime}_{b}$ is a nuclear $K$-Fréchet
space.

Conversely, let $V$ be a nuclear $K$-Fréchet space. For a decreasing
neighbourhood base of $0$ consisting of lattices $(L_{n})_{n\in{\mathbb{N}}}$,
one obtains an inductive sequence
$(V^{\prime}_{L_{n}^{\circ}})_{n\in{\mathbb{N}}}$ of certain $K$-Banach spaces
$V^{\prime}_{L_{n}^{\circ}}$ with
$V^{\prime}=\bigcup_{n\in{\mathbb{N}}}V^{\prime}_{L_{n}^{\circ}}$, see [23,
Def. 2.5.2]. Then [23, Prop. 3.1.13] says that this sequence is semicompact;
even compact by Remark A.2 (ii). Note that if $K$ is spherically complete
every locally convex $K$-vector space is polar [59, Thm. 4.4.3 (i)].

Moreover, by [59, Cor. 8.5.3], $V$ in particular is reflexive, and therefore
$V^{\prime}_{b}$ is barrelled by [59, Thm. 7.4.11 (i)]. (If $K$ is spherically
complete, barrelled and polarly barrelled are equivalent [59, Thm. 7.1.9
(ii)].) Now [23, Cor. 2.5.9] implies that the inductive limit topology on
$\bigcup_{n\in{\mathbb{N}}}V^{\prime}_{L_{n}^{\circ}}$ agrees with the strong
topology of $V^{\prime}$. Hence $V^{\prime}_{b}$ is of compact type. ∎

######  Lemma A.6.

Let $(V_{n})_{n\in{\mathbb{N}}}$ be a projective sequence of locally convex
$K$-vector spaces and $W$ a normed $K$-vector space. Then the canonical
continuous homomorphism

$\varinjlim_{n\in{\mathbb{N}}}{\mathcal{L}}_{b}(V_{n},W)\longrightarrow{\mathcal{L}}_{b}\bigg{(}\varprojlim_{n\in{\mathbb{N}}}V_{n},W\bigg{)}$
(A.2)

is surjective. If all projections
${\mathrm{pr}}_{n}\colon\varprojlim_{n\in{\mathbb{N}}}V_{n}\rightarrow V_{n}$
have dense image, or if all $V_{n}$ are Hausdorff and the transition
homomorphisms $V_{n+1}\rightarrow V_{n}$ are compact, then (A.2) even is
bijective.

###### Proof.

First consider
$f\in{\mathcal{L}}\big{(}\varprojlim_{n\in{\mathbb{N}}}V_{n},W\big{)}$, and
let $B_{W}\vcentcolon=\left\\{w\in W\,\middle|\,\lVert w\rVert_{W}\leq
1\right\\}$ denote the unit ball of $W$ so that
$f^{-1}(B_{W})\subset\varprojlim_{n\in{\mathbb{N}}}V_{n}$ is open. By the
definition of the initial topology of $\varprojlim_{n\in{\mathbb{N}}}V_{n}$,
there exist integers $n_{1},\ldots,n_{r}\in{\mathbb{N}}$ and open lattices
$L_{n_{i}}\subset V_{n_{i}}$, $i=1,\ldots,r$, such that

${\mathrm{pr}}_{n_{1}}^{-1}(L_{n_{1}})\cap\ldots\cap{\mathrm{pr}}_{n_{r}}^{-1}(L_{n_{r}})\subset
f^{-1}(B_{W}).$

Let $m\geq n_{1},\ldots,n_{r}$, and note that
${\mathrm{Ker}}({\mathrm{pr}}_{m})\subset{\mathrm{Ker}}({\mathrm{pr}}_{n_{i}})\subset{\mathrm{pr}}_{n_{i}}^{-1}(L_{n_{i}})$,
for $i=1,\ldots,r$. Hence we find that
${\mathrm{Ker}}({\mathrm{pr}}_{m})\subset f^{-1}(B_{W})$. As
${\mathrm{Ker}}({\mathrm{pr}}_{m})$ is a $K$-subvector space, it follows that
${\mathrm{Ker}}({\mathrm{pr}}_{m})\subset{\mathrm{Ker}}(f)$. Therefore $f$
factors over $V_{m}$ via ${\mathrm{pr}}_{m}$. This shows that (A.2) is
surjective.

If all projections ${\mathrm{pr}}_{n}$ have dense image, then the
homomorphisms

${\mathcal{L}}(V_{n},W)\longrightarrow{\mathcal{L}}\bigg{(}\varprojlim_{n\in{\mathbb{N}}}V_{n},W\bigg{)}\,,\quad
f\longmapsto f\circ{\mathrm{pr}}_{n},$

are injective because $W$ is Hausdorff. As (A.2) is the direct limit of these
homomorphisms, its injectivity follows.

If all $V_{n}$ are Hausdorff and the transition maps are compact, then there
exists a projective system $(U_{n})_{n\in{\mathbb{N}}}$ such that
$\varprojlim_{n\in{\mathbb{N}}}V_{n}$ and
$\varprojlim_{n\in{\mathbb{N}}}U_{n}$ are topologically isomorphic, and the
canonical projections of the latter have dense image [67, p. 93]. Moreover, we
have
$\varinjlim_{n\in{\mathbb{N}}}{\mathcal{L}}_{b}(V_{n},W)\cong\varinjlim_{n\in{\mathbb{N}}}{\mathcal{L}}_{b}(U_{n},W)$
by functoriality, and can conclude using the previous case. ∎

######  Proposition A.7 ([74, Prop. 1.5]).

Let $V$ be a locally convex $K$-vector space of compact type, expressed as
$V=\varinjlim_{n\in{\mathbb{N}}}V_{n}$, for a sequence of $K$-Banach spaces
$V_{n}$ with compact and injective transition maps. Moreover, let $W$ be a
$K$-Banach space. Then the canonical continuous homomorphism

$\varinjlim_{n\in{\mathbb{N}}}\big{(}V_{n}{\,\widehat{\otimes}_{K}\,}W\big{)}\longrightarrow
V{\,\widehat{\otimes}_{K}\,}W$

is bijective.

###### Proof.

To ease the notation, we will simply denote the strong dual of a locally
convex $K$-vector space $U$ by $U^{\prime}$. Using [67, Cor. 18.8] and the
fact that $V$ is reflexive, we have a topological isomorphism

$V{\,\widehat{\otimes}_{K}\,}W\overset{\cong}{\longrightarrow}V^{\prime\prime}{\,\widehat{\otimes}_{K}\,}W\overset{\cong}{\longrightarrow}{\mathcal{L}}_{b}(V^{\prime},W)\,,\quad
v\otimes w\longmapsto\big{[}\ell\mapsto\ell(v)\,w\big{]}.$

For each $V_{n}$, the duality map and [67, Lemma 18.1] at least give a
continuous homomorphism

$V_{n}{\,\widehat{\otimes}_{K}\,}W\longrightarrow
V_{n}^{\prime\prime}{\,\widehat{\otimes}_{K}\,}W\longrightarrow{\mathcal{L}}_{b}(V_{n}^{\prime},W)\,,\quad
v\otimes w\longmapsto\big{[}\ell\mapsto\ell(v)\,w\big{]},$

so that we arrive at a commutative diagram of continuous homomorphism

${\varinjlim_{n\in{\mathbb{N}}}(V_{n}{\,\widehat{\otimes}_{K}\,}W)}$${V{\,\widehat{\otimes}_{K}\,}W}$${\varinjlim_{n\in{\mathbb{N}}}{\mathcal{L}}_{b}(V_{n}^{\prime},W)}$${{\mathcal{L}}_{b}(V^{\prime},W).}$$\scriptstyle{\cong}$
(A.3)

By [67, Lemma 16.4 (ii)] the projective system
$(V_{n}^{\prime})_{n\in{\mathbb{N}}}$ is compact, and therefore the lower map
in (A.3) is a bijection by Lemma A.6.

Hence the claim follows if we show that the left homomorphism is an
isomorphism. But by [67, Lemma 16.4 (iii)], the transition maps
$V_{n}^{\prime\prime}\rightarrow V_{n+1}^{\prime\prime}$ factor over
$V_{n+1}\subset V_{n+1}^{\prime\prime}$ which gives

$\varinjlim_{n\in{\mathbb{N}}}(V_{n}{\,\widehat{\otimes}_{K}\,}W)\overset{\cong}{\longrightarrow}\varinjlim_{n\in{\mathbb{N}}}\big{(}V_{n}^{\prime\prime}{\,\widehat{\otimes}_{K}\,}W\big{)}.$

Moreover, the image of $V_{n}^{\prime\prime}{\,\widehat{\otimes}_{K}\,}W$ in
${\mathcal{L}}_{b}(V_{n}^{\prime},W)$ precisely is the subspace of compact
homomorphisms ${\mathcal{C}}(V_{n}^{\prime},W)$ by [67, Prop. 18.11]. It
follows from [67, Rmk. 16.7 (i)] that the transition maps
${\mathcal{L}}_{b}(V_{n}^{\prime},W)\rightarrow{\mathcal{L}}_{b}(V_{n+1}^{\prime},W)$
factor over
${\mathcal{C}}(V_{n+1}^{\prime},W)\subset{\mathcal{L}}_{b}(V_{n+1}^{\prime},W)$
because they are given by precomposition with the compact maps
$V_{n+1}^{\prime}\rightarrow V_{n}^{\prime}$. Hence

$\varinjlim_{n\in{\mathbb{N}}}\big{(}V_{n}^{\prime\prime}{\,\widehat{\otimes}_{K}\,}W\big{)}\cong\varinjlim_{n\in{\mathbb{N}}}{\mathcal{C}}(V_{n}^{\prime},W)\overset{\cong}{\longrightarrow}\varinjlim_{n\in{\mathbb{N}}}{\mathcal{L}}_{b}(V^{\prime}_{n},W)$

is a topological isomorphism, too. ∎

######  Lemma A.8 (cf. [70, Lemma 1.3]).

Let $X$ be a locally compact topological space. Let $V$ and $W$ be locally
convex $K$-vector spaces, and assume that $V$ is barrelled. If $\beta\colon
X\times V\rightarrow W$ is a separately continuous map and
$\beta(x,{\,\\_\,})\colon V\rightarrow W$ is $K$-linear, for every $x\in X$,
then $\beta$ is jointly continuous.

###### Proof.

By the linearity of the $\beta(x,{\,\\_\,})$ it suffices to show that $\beta$
is continuous at $(x,0)$, for every $x\in X$. Fix $x_{0}\in X$, and let
$U\subset X$ be a compact neighbourhood of $x_{0}$. We claim that
$H\vcentcolon=\left\\{\beta(x,{\,\\_\,})\,\middle|\,x\in
U\right\\}\subset{\mathcal{L}}_{s}(V,W)$ is bounded. Assuming this claim for
the moment, it follows from the Banach-Steinhaus theorem [67, Prop. 6.15] that
$H$ is equicontinuous. Hence, for any open lattice $M\subset W$, there exists
an open lattice $L\subset V$ such that $\beta(U,L)\subset M$. This shows that
$\beta$ is continuous at $(x_{0},0)$.

To show the claim, recall that the seminorms of ${\mathcal{L}}_{s}(V,W)$ are
$q_{v,p}$, for $v\in V$ and $p$ a continuous seminorm of $W$, and defined by
$q_{v,p}(f)\vcentcolon=p(f(v))$, for $f\in{\mathcal{L}}_{s}(V,W)$ [67, Example
1 after 6.6]. For such a seminorm, we have

$\sup_{f\in H}q_{v,p}(f)=\sup_{x\in U}p(\beta(x,v))<\infty$

because the image of the compact subset $U$ under the continuous map
$p\circ\beta({\,\\_\,},v)$ is bounded. It follows that
$H\subset{\mathcal{L}}_{s}(V,W)$ is bounded. ∎

######  Lemma A.9.

Let $(V_{i})_{i\in I}$ and $W$ be locally convex $K$-vector spaces. Then there
exists a natural topological isomorphism

${\mathcal{L}}_{b}\bigg{(}\bigoplus_{i\in
I}V_{i},W\bigg{)}\overset{\cong}{\longrightarrow}\prod_{i\in
I}{\mathcal{L}}_{b}(V_{i},W)\,,\quad f\longmapsto(f{\\!\,\mid_{V_{i}}})_{i\in
I}.$

(For the case of $W=K$, see [67, Prop. 9.10])

###### Proof.

Let $\iota_{i}\colon V_{i}\rightarrow V\vcentcolon=\bigoplus_{i\in I}V_{i}$
denote the canonical embeddings, and ${\mathrm{pr}}_{i}\colon V\rightarrow
V_{i}$ the canonical projections. By [11, III. §3 Ex. 5 on p. III.41] there is
a natural topological isomorphism

${\mathcal{L}}_{\mathcal{B}}(V,W)\overset{\cong}{\longrightarrow}\prod_{i\in
I}{\mathcal{L}}_{b}(V_{i},W)\,,\quad f\longmapsto(f{\\!\,\mid_{V_{i}}})_{i\in
I},$

where ${\mathcal{B}}$ is the family $\left\\{\iota_{i}(B_{i})\,\middle|\,i\in
I,B_{i}\subset V_{i}\text{ bounded}\right\\}$ of bounded sets of $V$. We want
to show that the ${\mathcal{B}}$-topology coincides with the strong topology
on ${\mathcal{L}}(V,W)$. In view of [67, Lemma 6.5], it suffices to show that
for a given bounded subset $B\subset V$, there exist
$\iota_{i_{1}}(B_{1}),\ldots,\iota_{i_{m}}(B_{m})\in{\mathcal{B}}$ such that
$B$ is contained in the closure of the ${\mathcal{O}}_{K}$-module generated by
$\iota_{i_{1}}(B_{1})\cup\ldots\cup\iota_{i_{m}}(B_{m})$.

To do so, we proceed similarly to the proof of [67, Prop. 9.10], and fix a
bounded subset $B\subset V$. By [59, Thm. 3.6.4 (ii)], all
${\mathrm{pr}}_{i}(B)\subset V_{i}$ are bounded and there exists a finite
subset $J\subset I$ such that ${\mathrm{pr}}_{i}(B)\subset\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$,
for all $i\in I\setminus J$. We define the bounded subsets
$B_{j}\vcentcolon={\mathrm{pr}}_{j}(B)$, for $j\in J$. For given $v\in B$, we
write $v=\sum_{i\in I}\iota_{i}(v_{i})$ where $v_{i}\in V_{i}$. We then have
$\sum_{j\in J}\iota_{j}(v_{j})\in\sum_{j\in J}\iota_{j}(B_{j})$.

To deal with $\sum_{i\in I\setminus J}\iota_{i}(v_{i})$, let $q$ be a
continuous seminorm of $V$. By [67, Lemma 5.1 (ii)], $q\circ\iota_{i}$ is a
continuous seminorm of $V_{i}$. Hence

$q(\iota_{i}(v_{i}))=(q\circ\iota_{i})({\mathrm{pr}}_{i}(v))=0,$

for all $i\in I\setminus J$, as ${\mathrm{pr}}_{i}(B)\subset\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
for those $i$. It follows that, for $i\in I\setminus J$,

$\iota_{i}(v_{i})\in\bigcap_{q}{\mathrm{Ker}}(q)=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$

where we take the intersection over all continuous seminorms $q$ of $V$. This
shows that $v$ is contained in the closure of $\sum_{j\in J}\iota_{j}(B_{j})$
in $V$. ∎

######  Proposition A.10 (cf. [14, Prop. 1]).

Let $K$ be a complete field with absolute value ${\lvert{{\,\\_\,}}\rvert}$.
Let $U\subset K^{n}$ be open and let $f\colon U\longrightarrow E$ be a
function into a $K$-Banach space $E$. Let $V$ and $W$ be $K$-Banach spaces and
let

$E\times V\longrightarrow W\,,\quad(u,v)\longmapsto\langle u,v\rangle,$ (A.4)

be a continuous $K$-bilinear map which induces an isometric embedding of $E$
into the $K$-Banach space ${\mathcal{L}}_{b}(V,W)$, i.e. such that, for all
$u\in E$,

${\lvert{u}\rvert}=\left\|\langle
u,\\_\,\rangle\right\|_{{\mathcal{L}}_{b}(V,W)}\vcentcolon=\sup_{v\in
V\setminus\\{0\\}}\frac{{\lvert{\langle
u,v\rangle}\rvert}}{{\lvert{v}\rvert}}\,.$

Then, for $z_{0}\in U$, the following are equivalent: The function $f$ is
analytic in some open neighbourhood of $z_{0}$.

For all $v\in V$, there exists an open neighbourhood of $z_{0}$ such that the
function

$f_{v}\colon U\longrightarrow W\,,\quad z\longmapsto\langle f(z),v\rangle,$

is analytic there.

###### Proof.

First assume that there exists $r\in{\mathbb{R}}_{>0}$ such that $f$ is given
by the power series

$f(z)=\sum_{{\underline{i}}\in{\mathbb{N}}^{n}_{0}}a_{{\underline{i}}}\,(z-z_{0})^{\underline{i}}\quad\text{,
for $x\in B_{r}^{n}(z_{0})$,}$

for certain $a_{{\underline{i}}}\in E$. For $v\in V$, it follows from the
continuity of (A.4) that

$f_{v}(z)=\langle
f(z),v\rangle=\sum_{{\underline{i}}\in{\mathbb{N}}^{n}_{0}}\langle
a_{{\underline{i}}},v\rangle\,(z-z_{0})^{\underline{i}}\quad\text{, for all
$z\in B_{r}^{n}(z_{0})$.}$

This shows that (i) implies (ii).

Conversely, for all $v\in V$, let $f_{v}$ be analytic in a neighbourhood of
$z_{0}$. This means that there exists a radius $r^{(v)}\in{\mathbb{R}}_{>0}$
such that $f_{v}$ is given by the convergent power series

$f_{v}(z)=\sum_{{\underline{i}}\in{\mathbb{N}}^{n}_{0}}a_{\underline{i}}^{(v)}(z-z_{0})^{\underline{i}}\quad\text{,
for all $z\in B^{n}_{r^{(v)}}(z_{0})$,}$

for certain $a_{\underline{i}}^{(v)}\in W$. Hence there is a constant
$C^{(v)}>0$ such that

$\displaystyle{\lvert{a_{\underline{i}}^{(v)}}\rvert}\leq
C^{(v)}\left(\frac{1}{r^{(v)}}\right)^{{\lvert{{\underline{i}}}\rvert}}\,,$
(A.5)

for all ${\underline{i}}\in{\mathbb{N}}^{n}_{0}$. As an intermediate step, we
want to show:

Claim: There are $r>0$ and $C>0$ such that, for all
${\underline{j}}\in{\mathbb{N}}^{n}_{0}$, the maps

$b_{\underline{j}}\colon V\longrightarrow W\,,\quad v\longmapsto
a_{\underline{j}}^{(v)},$

are $K$-linear and continuous with

$\displaystyle\lVert{b_{\underline{j}}}\rVert_{{\mathcal{L}}_{b}(V,W)}\leq
C\left(\frac{1}{r}\right)^{{\lvert{{\underline{j}}}\rvert}}\,.$ (A.6)

###### Proof of the Claim.

The bilinearity of $\langle{\,\\_\,},{\,\\_\,}\rangle$ implies that

$V\longrightarrow\mathrm{Map}(U,W)\,,\quad v\longmapsto f_{v},$

is $K$-linear. The linearity of $b_{\underline{j}}$ then follows from the
identity theorem for the coefficients of convergent power series [12, 3.2.1
resp. 4.2.1].

To proof the continuity and the bound (A.6), we endow ${\mathbb{N}}^{n}_{0}$
with the lexicographical order, i.e. ${\underline{i}}<{\underline{j}}$ if and
only if $i_{k}<j_{k}$ for the smallest $k\in\\{1,\ldots,n\\}$ where $i_{k}\neq
j_{k}$. We proceed by induction on ${\underline{j}}\in{\mathbb{N}}^{n}_{0}$
with respect to this order.

Fix ${\underline{j}}\in{\mathbb{N}}^{n}_{0}$ and assume that the claim holds
for all ${\underline{i}}<{\underline{j}}$ (a vacuous assumption if
${\underline{j}}=(0,\ldots,0)$). Hence there are $r_{\underline{j}}>0$ and
$C_{\underline{j}}>0$ such that

${\lvert{a_{\underline{i}}^{(v)}}\rvert}\leq
C_{\underline{i}}\left(\frac{1}{r_{\underline{j}}}\right)^{{\lvert{{\underline{i}}}\rvert}}{\lvert{v}\rvert}\quad\text{,
for all $v\in V$ and all ${\underline{i}}<{\underline{j}}$.}$

Therefore the power series
$\sum_{{\underline{i}}<{\underline{j}}}a_{\underline{i}}^{(v)}h_{1}^{i_{1}}\cdots
h_{n}^{i_{n}}$ converges, for all $(h_{1},\ldots,h_{n})\in
B^{n}_{r_{\underline{j}}}(0)$ and all $v\in V$. Moreover, for any $z\in U$,
the linear map $V\rightarrow W$, $v\mapsto f_{v}(z)=\langle f(z),v\rangle$, is
continuous. Hence, for any $h\in K^{n}$ with $z_{0}+h\in
B^{n}_{r_{\underline{j}}}(z_{0})\cap U$, the operator

$T_{h}\colon V\longrightarrow W\,,\quad
v\longmapsto\frac{1}{h_{1}^{j_{1}}\cdots
h_{n}^{j_{n}}}\bigg{(}f_{v}(z_{0}+h)-\sum_{{\underline{i}}<{\underline{j}}}a_{\underline{i}}^{(v)}h^{\underline{i}}\bigg{)},$

is $K$-linear and continuous, too. For fixed $v\in V$, we may assume $h\in
B_{r^{(v)}}^{n}(0)$, and we compute:

$\displaystyle\lim_{h_{n}\rightarrow 0}\ldots\lim_{h_{1}\rightarrow
0}T_{h}(v)=$ $\displaystyle\lim_{h_{n}\rightarrow
0}\ldots\lim_{h_{1}\rightarrow 0}\,\,\,\frac{1}{h_{1}^{j_{1}}\cdots
h_{n}^{j_{n}}}\bigg{(}\sum_{{\underline{i}}\in{\mathbb{N}}^{n}_{0}}a_{\underline{i}}^{(v)}h^{\underline{i}}\,-\sum_{{\underline{i}}<{\underline{j}}}a_{\underline{i}}^{(v)}h^{\underline{i}}\bigg{)}$
$\displaystyle=$ $\displaystyle\lim_{h_{n}\rightarrow
0}\ldots\lim_{h_{1}\rightarrow
0}\,\,\,\sum_{{\underline{i}}\geq{\underline{j}}}a_{\underline{i}}^{(v)}h_{1}^{i_{1}-j_{1}}\cdots
h_{n}^{i_{n}-j_{n}}$ $\displaystyle=$ $\displaystyle\lim_{h_{n}\rightarrow
0}\ldots\lim_{h_{2}\rightarrow
0}\,\,\sum_{\begin{subarray}{c}{\underline{i}}\geq{\underline{j}}\\\
i_{1}=j_{1}\end{subarray}}a_{\underline{i}}^{(v)}h_{2}^{i_{2}-j_{2}}\cdots
h_{n}^{i_{n}-j_{n}}$ $\displaystyle\vdots$ $\displaystyle=$
$\displaystyle\,\,\,a_{\underline{j}}^{(v)}=b_{{\underline{j}}}(v).$

By the Banach–Steinhaus theorem [11, III. §4.2 Cor. 2] we conclude that
$b_{\underline{j}}$ is continuous as the pointwise limit of the operators
$T_{h}$.

It remains to show the bound (A.6) for all
${\underline{i}}\leq{\underline{j}}$. Define, for $k,l\in{\mathbb{N}}$,

$B^{k,l}_{\underline{j}}=\big{\\{}v\in
V\,\big{|}\,\forall{\underline{i}}\leq{\underline{j}}\colon{\lvert{a_{\underline{i}}^{(v)}}\rvert}\leq
k\,l^{{\lvert{{\underline{i}}}\rvert}}\big{\\}}.$

Then each $B_{\underline{j}}^{k,l}$ is closed in $V$ by the continuity of
$b_{\underline{i}}$, for ${\underline{i}}\leq{\underline{j}}$. By the
pointwise bounds (A.5), we have
$V=\bigcup_{k,l\in{\mathbb{N}}}B_{\underline{j}}^{k,l}$. Hence it follows from
Baire’s theorem [9, Ch. IX. §5.3 Thm. 1] that some $B_{\underline{j}}^{k,l}$
contains an open ball $B_{\varepsilon}(v_{0})$ of radius $\varepsilon>0$
centred at some $v_{0}\in B_{\underline{j}}^{k,l}$. Hence, for all $v\in V$
with ${\lvert{v}\rvert}<\varepsilon$ and all
${\underline{i}}\leq{\underline{j}}$, we have

${\lvert{a_{\underline{i}}^{(v)}}\rvert}={\lvert{a_{\underline{i}}^{(v+v_{0})}-a_{\underline{i}}^{(v_{0})}}\rvert}\leq{\lvert{a_{\underline{i}}^{(v+v_{0})}}\rvert}+{\lvert{a_{\underline{i}}^{(v_{0})}}\rvert}\leq
2k\,l^{{\lvert{{\underline{i}}}\rvert}}\,.$

We fix some $\varpi\in K$ with $0<{\lvert{\varpi}\rvert}<1$. For any fixed
$v\in V\setminus\\{0\\}$, let $m\in{\mathbb{Z}}$ such that
${\lvert{\varpi}\rvert}^{m}<\frac{\varepsilon}{{\lvert{v}\rvert}}\leq{\lvert{\varpi}\rvert}^{m-1}$.
Then

$\displaystyle{\lvert{a_{\underline{i}}^{(v)}}\rvert}=\frac{{\lvert{a_{\underline{i}}^{(\varpi^{m}v)}}\rvert}}{{\lvert{\varpi^{m}}\rvert}}\leq\frac{2k}{\varepsilon\,{\lvert{\varpi}\rvert}}l^{{\lvert{{\underline{i}}}\rvert}}\,{\lvert{v}\rvert}$

which implies

$\left\|b_{\underline{i}}\right\|_{{\mathcal{L}}_{b}(V,W)}=\sup_{v\in
V\setminus\\{0\\}}\frac{{\lvert{a_{\underline{i}}^{(v)}}\rvert}}{{\lvert{v}\rvert}}\leq\frac{2k}{\varepsilon\,{\lvert{\varpi}\rvert}}l^{{\lvert{{\underline{i}}}\rvert}}\,,$

for all ${\underline{i}}\leq{\underline{j}}$. ∎

It follows from the claim that the power series
$\sum_{{\underline{j}}\in{\mathbb{N}}^{n}_{0}}b_{\underline{j}}\,(z-z_{0})^{\underline{j}}$
defines an analytic function with values in ${\mathcal{L}}_{b}(V,W)$, on some
open neighbourhood $U^{\prime}\subset U$ of $z_{0}$. On $U^{\prime}$, this
power series agrees with $f$ viewed as a map

$f\colon U^{\prime}\longrightarrow{\mathcal{L}}_{b}(V,W)\,,\quad
z\longmapsto\langle f(z),{\,\\_\,}\rangle.$

Indeed, for $z\in U^{\prime}$ and $v\in V$:

$\displaystyle\bigg{(}\sum_{{\underline{j}}\in{\mathbb{N}}^{n}_{0}}b_{\underline{j}}\,(z-z_{0})^{\underline{j}}\bigg{)}(v)=\sum_{{\underline{j}}\in{\mathbb{N}}^{n}_{0}}b_{\underline{j}}(v)\,(z-z_{0})^{{\underline{j}}}=\sum_{{\underline{j}}\in{\mathbb{N}}^{n}_{0}}a_{\underline{j}}^{(v)}(z-z_{0})^{\underline{j}}=f_{v}(z)=\langle
f(z),v\rangle.$

Hence
$f(z)=\sum_{{\underline{j}}\in{\mathbb{N}}^{n}_{0}}b_{\underline{j}}\,(z-z_{0})^{\underline{j}}$
is analytic on $U^{\prime}$ as a map into ${\mathcal{L}}_{b}(V,W)$.

Finally, $E$ can be identified with a closed subspace of
${\mathcal{L}}_{b}(V,W)$ by assumption. By the same reasoning as in the proof
of the claim, the coefficients $b_{\underline{j}}$ of $f$ can be computed as
the limits of sequences in $E$. Therefore $b_{\underline{j}}\in E$, for all
${\underline{j}}\in{\mathbb{N}}^{n}_{0}$, and $f\colon U^{\prime}\rightarrow
E$ is given by a convergent power series. ∎

######  Lemma A.11.

Let $f\colon V\rightarrow W$ be a continuous homomorphism between locally
convex $K$-vector spaces, and assume that $V$ is Hausdorff. Then in
$V^{\prime}$ we have the equality

${\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}^{s}={\mathrm{Ker}}(f)^{\perp}\vcentcolon=\big{\\{}\ell\in
V^{\prime}\,\big{|}\,\forall v\in{\mathrm{Ker}}(f):\ell(v)=0\big{\\}}$

where ${\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}^{s}$ denotes the closure of the image of the transpose $f^{t}\colon
W^{\prime}\rightarrow V^{\prime}$ in $V^{\prime}_{s}$.

Moreover, if in addition $V$ is semi-reflexive, then
${\mathrm{Ker}}(f)^{\perp}\subset\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
in $V^{\prime}_{b}$.

###### Proof.

Taking the transpose twice

${\mathcal{L}}(V,W)\longrightarrow{\mathcal{L}}(W^{\prime}_{s},V^{\prime}_{s})\longrightarrow{\mathcal{L}}\big{(}(V^{\prime}_{s})^{\prime}_{s},(W^{\prime}_{s})^{\prime}_{s}\big{)},$

we see that $f$ still defines a continuous homomorphism when $V$ and $W$ carry
the respective weak topologies $V_{s}=(V^{\prime}_{s})^{\prime}_{s}$ and
$W_{s}=(W^{\prime}_{s})^{\prime}_{s}$. Then the statement
${\mathrm{Ker}}(f)^{\perp}={\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}^{s}$ is part of [11, II. §6.4 Cor. 2].

It is a consequence of the Hahn–Banach theorem that the closed subspace
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}\subset V^{\prime}_{b}$ is weakly closed as well [59, Thm. 5.2.1]. If we
assume that $V$ is semi-reflexive, i.e. the duality homomorphism
$V\rightarrow(V^{\prime}_{b})^{\prime}$ is bijective, then the topology on the
weak dual of $V$ agrees with the weak topology of the strong dual:
$V^{\prime}_{s}=(V^{\prime}_{b})_{s}$, see [23, Thm. 7.4.9]. Since
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
now is a closed subset of $V^{\prime}_{s}$ which contains
${\mathrm{Im}}(f^{t})$, it follows that ${\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}^{s}\subset\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$.
∎

######  Lemma A.12.

Let $V$ be a locally convex $K$-vector space and $f\colon V\rightarrow W$ a
homomorphism of $K$-vector spaces. If $W$ is given the locally convex final
topology with respect to $f$, then $f$ is strict.

###### Proof.

Note that $f$ as a homomorphism between locally convex $K$-vector spaces is
continuous if and only if the induced algebraic isomorphism $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{f}\colon V/{\mathrm{Ker}}(f)\rightarrow{\mathrm{Im}}(f)$ is continuous
with respect to the quotient respectively subspace topology.

We assume by the way of contradiction that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{f}$
is not a homeomorphism if $W$ carries the locally convex final topology with
respect to $f$. In this case we find an open lattice $L\subset V$ such that
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{f}(L+{\mathrm{Ker}}(f))$ is a lattice in ${\mathrm{Im}}(f)$ which is not
open. By extending we obtain a lattice $M\subset W$ such that
$M\cap{\mathrm{Im}}(f)=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{f}(L+{\mathrm{Ker}}(f))$. But $f^{-1}(M)=L+{\mathrm{Ker}}(f)$ is an open
lattice of $V$. Therefore $M$ must be an open lattice by the definition of the
topology on $W$. This yields a contradiction. ∎

######  Lemma A.13 (Snake lemma for quasi-abelian categories [49]141414Note
that the notion of “semiabelian” categories used in [49] agrees with the one
of “quasi-abelian” categories from [76], cf. [49, p. 511]. ).

Consider the commutative diagram

${0}$${V^{\prime}}$${V}$${V^{\prime\prime}}$${0}$${0}$${W^{\prime}}$${W}$${W^{\prime\prime}}$${0}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

of continuous homomorphisms between locally convex $K$-vector spaces with
strictly exact rows. Then the induced
${\mathrm{Ker}}$-${\mathrm{Coker}}$-sequence

$0\longrightarrow{\mathrm{Ker}}(\alpha)\overset{\varepsilon}{\longrightarrow}{\mathrm{Ker}}(\beta)\overset{\zeta}{\longrightarrow}{\mathrm{Ker}}(\gamma)\overset{\delta}{\longrightarrow}{\mathrm{Coker}}(\alpha)\overset{\tau}{\longrightarrow}{\mathrm{Coker}}(\beta)\overset{\theta}{\longrightarrow}{\mathrm{Coker}}(\gamma)\longrightarrow
0$ (A.7)

of continuous homomorphisms is exact in ${\mathrm{Ker}}(\alpha)$,
${\mathrm{Ker}}(\beta)$, ${\mathrm{Coker}}(\beta)$, and
${\mathrm{Coker}}(\gamma)$. Furthermore, $\varepsilon$ and $\theta$ are
strict. We moreover have: If $\beta$ is strict, then (A.7) is exact in
${\mathrm{Ker}}(\gamma)$ and ${\mathrm{Coker}}(\alpha)$, and $\delta$ is
strict.

If $\alpha$ is strict, then (A.7) is exact in ${\mathrm{Ker}}(\beta)$ and
${\mathrm{Ker}}(\gamma)$, and $\zeta$ is strict.