# Rigid vectors in $p$-adic principal series representations

Aranya Lahiri and Claus Sorensen

###### Abstract.

In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups
$\mathbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural
$p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_{n}$.
We work with a general connected reductive split group over some $p$-adic
field (with simply connected derived group) and study the
$\mathbb{I}$-analytic vectors in principal series representations. Our main
result is an irreducibility criterion which generalizes results of Clozel and
Ray in the $\text{GL}_{n}$-case.

##  1\. Introduction

In this article we study rigid analytic vectors in continuous $p$-adic
prinicpal series representations of a $p$-adic reductive group $G$. Here
$G={\bf{G}}(L)$ is the locally profinite group of $L$-rational points of a
connected reductive group ${\bf{G}}$ defined over a finite extension
$L/\mathbb{Q}_{p}$. We assume ${\bf{G}}$ is split over $L$, and that the
derived group ${\bf{G}}^{\text{der}}$ is simply connected. Once and for all we
choose a Borel subgroup ${\bf{P}}$ over $L$, and a maximal $L$-split torus
${\bf{T}}\subset{\bf{P}}$. Let $P={\bf{P}}(L)$ and $T={\bf{T}}(L)$ – similarly
for other linear algebraic groups ${\bf{H}}$ over $L$ we will write $H$ for
the group of $L$-points.

The continuous principal series representations of $G$ are defined as follows,
see [Sch06, Ch. 2]. Start with a continuous character $\chi:T\rightarrow
K^{\times}$ taking values in some finite extension $K\supset L$. Inflate
$\chi$ to a character of $P$ by letting $\chi=1$ on the unipotent radical as
usual, and look at the induction in the space $C(G,K)$ of continuous
$K$-valued functions on $G$:

$\text{Ind}_{P}^{G}(\chi)=\\{f\in C(G,K):f(bg)=\chi(b)f(g)\hskip 6.0pt\forall
b\in P\\}.$

(Our normalization is different from [Sch06] where $G$ acts by left
translations. In our setup $G$ acts by right translations, which means
$(g^{\prime}f)(g)=f(gg^{\prime})$ for $g,g^{\prime}\in G$.) One turns
$\text{Ind}_{P}^{G}(\chi)$ into an admissible $K$-Banach representation by
endowing it with the norm $\|f\|=\sup_{g\in C}|f(g)|$ for some choice of
maximal compact subgroup $C\subset G$ for which the Iwasawa decomposition
$G=PC$ holds.

In his paper [Sch06] Schneider expresses optimism about
$\text{Ind}_{P}^{G}(\chi)$ always having a finite composition series, and he
states a precise (topological) irreducibility conjecture [Sch06, Conj. 2.5] –
at least for semisimple and simply connected groups ${\bf{G}}$, in which case
the condition on the character is
$\chi^{-1}\delta\circ\alpha^{\vee}\notin(\cdot)^{\mathbb{Z}_{>0}}$ for all
${\bf{P}}$-positive roots $\alpha$; that is those $\alpha$ appearing in
$\text{Lie}({\bf{P}})$ (and $\delta=\frac{1}{2}\sum_{\alpha>0}\alpha$ as
usual).

The locally analytic analogue of Schneider’s irreducibility conjecture was
proved by Orlik and Strauch in [OS10], as a culmination of work by Feaux de
Lacroix, Frommer, Kohlhaase, Schneider, Teitelbaum, and others. The actual
conjecture is known to be true for $\text{GL}_{2}(\mathbb{Q}_{p})$ but it
remains open in general. Ban and Hundley proved the conjecture for $\chi$ in a
certain cone, see [BH16, Thm. 1.1].

In this article we restrict $\text{Ind}_{P}^{G}(\chi)$ to a pro-$p$ Iwahori
subgroup $I$ (for $p>\\!\\!>0$ large enough) and investigate the subspace of
rigid analytic vectors $\text{Ind}_{P}^{G}(\chi)_{\mathbb{I}-\text{an}}$ as
defined in [Eme17, Df. 3.3.1]. Here $\mathbb{I}$ is a rigid analytic
$\mathbb{Q}_{p}$-group with $\mathbb{I}(\mathbb{Q}_{p})=I$, and as a rigid
analytic space $\mathbb{I}$ is a closed ball.

As a trivial first step the Bruhat-Tits decomposition gives an isomorphism of
$I$-representations

$\text{Ind}_{P}^{G}(\chi)_{\mathbb{I}-\text{an}}\overset{\sim}{\longrightarrow}\bigoplus_{w\in
W}\text{Ind}_{I\cap wPw^{-1}}^{I}(w\chi)_{\mathbb{I}-\text{an}}$

where $W=W({\bf{G}},{\bf{T}})$ denotes the Weyl group and
$(w\chi)(t)=\chi(w^{-1}tw)$ for $w\in W$. The main problem is to understand
each summand $\text{Ind}_{I\cap wPw^{-1}}^{I}(w\chi)_{\mathbb{I}-\text{an}}$,
which by [Eme17, Prop. 3.3.7] agrees with the rigid analytic induction of
$w\chi$ (the space of rigid functions $f:I\rightarrow K$ satisfying the usual
transformation property; see (6.1) in section 6 plus section 9 in the main
text for more details). Of course, for all these spaces to be non-trivial we
must assume the character $\chi$ is rigid analytic when restricted to
$T^{1}=I\cap T$. This means the function $\chi:T^{1}\rightarrow
K^{\times}\hookrightarrow K$ lies in the space
$\mathcal{C}^{\text{rig}}(T^{1},K)$ introduced in definition 4.4.

The first half of our paper is concerned with promoting $I$ to a rigid
analytic group $\mathbb{I}$. Our convention will be that our choice of $I$
corresponds to the ${\bf{P}}$-negative roots. (This part is potentially
confusing; to aid the reader we point out that $P$ is being denoted by $B^{-}$
in the main body of the text.) See (2.1). We show that $I$ admits a natural
$p$-valuation $\omega$ provided $p-1>e(L/\mathbb{Q}_{p})h_{G}$ where $h_{G}$
is the Coxeter number of ${\bf{G}}$ and $e(L/\mathbb{Q}_{p})$ is the
ramification index; see Proposition 3.4. Moreover $(I,\omega)$ is saturated,
which is what allows us to associate a rigid analytic group $\mathbb{I}$ with
$\mathbb{Q}_{p}$-points $I$. This is presumably well-known but not easily
accessible in the literature. We give the details in Section 4. We should
emphasize there is some overlap between our work and [CR18], [Ray20a]. In
[CR18] Cornut and Ray give a minimal set of generators for $I$ in any split
reductive group ${\bf{G}}$, but they do not directly use $p$-valuations. In
[Ray20a] Ray gives an ordered basis for $(I,\omega)$ in the case of
$\text{SL}_{n}(\mathbb{Z}_{p})$ and $\text{GL}_{n}(\mathbb{Z}_{p})$. He is
working with $\omega$ as defined in [Sch11, Ex. 23.3] which leads to massive
matrix calculations. Part of our original motivation for writing this paper
was to give a more conceptual approach to $(I,\omega)$ and its ordered bases
for any Chevalley group.

Having defined the rigid analytic pro-$p$ Iwahori group $\mathbb{I}$ for
$p-1>e(L/\mathbb{Q}_{p})h_{G}$ it now makes sense to ask for the
$\mathbb{I}$-analytic vectors of any $K$-Banach representation of $I$. Here is
our main result for principal series representations, which gives an
irreducibility criterion analogous to [OS10, Thm. 3.4.12] in the rigid
setting.

######  Theorem 1.1.

Let $G$ be a $p$-adic reductive group over $L$ as above. Assume
$p-1>e(L/\mathbb{Q}_{p})h_{G}$. Let $\chi:T\rightarrow K^{\times}$ be a
continuous character which is rigid analytic when restricted to $T^{1}=I\cap
T$, and consider the Verma module $\mathcal{V}_{d\chi}$ associated with its
derivative $d\chi$ (see Section 7 for more details). Then
$\mathcal{V}_{d\chi}$ is simple if and only if the $I$-representation

$\text{Ind}_{I\cap wPw^{-1}}^{I}(w\chi)_{\mathbb{I}-\text{an}}$

is topologically irreducible for some $w\in W$ – or equivalently all $w\in W$.
If so, these representations are mutually non-isomorphic as $w$ varies.

There is a standard criterion for the simplicity of $\mathcal{V}_{d\chi}$ due
to Bernstein-Gelfand-Gelfand, see [Dix96, Thm. 7.6.24]. It is the
infinitesimal version of the criterion in the irreducibility conjecture
discussed above. In Section 10 we work out the irreducibility criterion
explicitly in the case of $\text{Sp}_{4}(\mathbb{Q}_{p})$ for $p>5$.

The $\text{GL}_{2}(\mathbb{Q}_{p})$-case of Theorem 1.1 is [Clo18, Thm. 3.7],
which continued the line of research begun by Robert in [Rob84] and [Rob85].
As part of his thesis Ray extended Clozel’s argument to
$\text{GL}_{n}(\mathbb{Q}_{p})$ and $w=1$. This is the main result of his
paper [Ray20b, Thm. 3.9]. In the last paragraph of that paper Ray expresses
hope that the proof can be adapted to the case $w\neq 1$. This is essentially
what we do. By reworking the argument of Clozel and Ray, and setting it up in
a more streamlined fashion, we are able to avoid the heavy matrix calculations
in [Ray20b] and deal with the case $w\neq 1$ with very little extra effort.
Here are some of the main differences between [Ray20b] and this article:

  * •

The coordinates in [Ray20b] are given by the matrix entries (ordered
lexicographically). This seems a bit artificial to us, and it results in
cumbersome matrix calculations. Instead, for an arbitrary split group, our
coordinates arise from a natural ordered basis in the sense of Lazard. This
lets us argue more conceptually.

  * •

[Ray20b, Lem. 3.4] gives an explicit Iwahori factorization. We do not need to
be explicit. In Remark 4.3 we explain how the Iwahori factorization (as
$p$-adic manifolds say) can be boosted to an isomorphism of rigid analytic
spaces – without explicitly inverting the multiplication map.

  * •

The proof in [Ray20b, p. 479] juggles with $n$ operators $T_{1},\ldots,T_{n}$
corresponding to the $n$ cocharacters
$x\mapsto\text{diag}(1,\ldots,x,\ldots,1)$. We work with a single operator
$H_{\mu}$ corresponding to cocharacter $\mu$ adapted to the choice of Weyl
element $w$ in the sense that $\langle w\alpha,\mu\rangle>0$ for all positive
roots $\alpha$. In the $\text{GL}_{n}$-case, for $w=1$ our cocharacter $\mu$
can be taken to be $x\mapsto\text{diag}(1,x,x^{2},\ldots,x^{n-1})$ for
instance.

Our interest in Theorem 1.1 stems primarily from the globally analytic variant
of base change introduced by Clozel in [Clo17] and [Clo18]. In smooth
representation theory, base change is reflected by a homomorphism between
Hecke algebras. Clozel instead defines a homomorphism between globally
analytic distribution algebras such as $\mathcal{D}^{\text{an}}(\mathbb{I},K)$
in the notation of [Eme17, Cor. 5.1.8], and he seems to suggest that this map
should be compatible with classical Langlands base change – at least for
unramified extensions.

As already noted, our Theorem 1.1 bears a close resemblance to [OS10, Thm.
3.4.12] in the locally analytic setup. The latter asserts that the strong dual
of the locally analytic induction $\big{(}\text{Ind}_{I\cap
wPw^{-1}}^{I}(w\chi)_{\text{la}}\big{)}_{b}^{\prime}$ is a simple
$\mathcal{D}^{\text{la}}(I,K)$-module if $\mathcal{V}_{d\chi}$ is simple. This
more or less immediately gives the irreducibility of the induction by
employing the duality theory of Schneider and Teitelbaum established in
[ST02], [ST03]. There does not seem to be a good duality theory in the rigid
analytic setup. For instance, as pointed out in [Clo18, App. A.2] the
distribution algebras $\mathcal{D}^{\text{an}}(\mathbb{I},K)$ are non-
Noetherian in general, which renders the Fréchet-Stein techniques of [ST03]
useless in the rigid analytic setup. We tried unsuccessfully to mimic the
approach of [OS10], but Theorem 1.1 really does seem to require a direct proof
along the lines of Clozel and Ray rather than passing to duals.

##  2\. Notation

Throughout the paper the following notation will remain in force. Let $p>2$ be
a prime. We work over a fixed finite extension $L/\mathbb{Q}_{p}$ with
valuation ring $\mathcal{O}$ and maximal ideal
$\mathfrak{m}=(\varpi)\subset\mathcal{O}$. The absolute ramification index
$e(L/\mathbb{Q}_{p})$ will just be denoted by $e$. We let
$\ell=[L:\mathbb{Q}_{p}]$.

We fix a connected reductive group ${\bf{G}}$ which is split over $L$, and
choose an $L$-split maximal torus ${\bf{T}}\subset{\bf{G}}$. Let
$(X^{*}({\bf{T}}),\Phi,X_{*}({\bf{T}}),\Phi^{\vee})$ be the associated root
datum. Once and for all we pick a system of positive roots $\Phi^{+}$ with
simple roots $\Delta\subset\Phi^{+}$ and let $\Phi^{-}=-\Phi^{+}$. We let
$\text{ht}(\cdot)$ be the height function on $\Phi$, and we recall the Coxeter
number of ${\bf{G}}$ is $h=1+\text{ht}(\theta)$ where $\theta$ is the highest
root. To each $\alpha\in\Phi$ there is attached a unipotent root subgroup
${\bf{U}}_{\alpha}$ normalized by ${\bf{T}}$.

We are mostly interested in representations of the $p$-adic group
$G={\bf{G}}(L)$ with the usual locally profinite topology. Similarly
$T={\bf{T}}(L)$ and we have the maximal compact subgroup $T^{0}$ and its Sylow
pro-$p$-subgroup $T^{1}$. We let $U_{\alpha}={\bf{U}}_{\alpha}(L)$ and note
that there is an isomorphism
$u_{\alpha}:L\overset{\sim}{\longrightarrow}U_{\alpha}$ such that
$tu_{\alpha}(x)t^{-1}=u_{\alpha}(\alpha(t)x)$ for all $t\in T$ and $x\in L$.
This gives a filtration of $U_{\alpha}$ by the subgroups
$U_{\alpha,r}=u_{\alpha}(\mathfrak{m}^{r})$ for varying $r\in\mathbb{Z}$.

We consider the full Iwahori subgroup $J$ corresponding to $\Phi^{+}$.
Multiplication defines a homeomorphism

$\prod_{\alpha\in\Phi^{-}}U_{\alpha,1}\times
T^{0}\times\prod_{\alpha\in\Phi^{+}}U_{\alpha,0}\overset{\sim}{\longrightarrow}J.$

Our main interest however is the pro-$p$ Iwahori subgroup $I$ which is the
Sylow pro-$p$ subgroup of $J$. Again multiplication gives a homeomorphism

(2.1) $\prod_{\alpha\in\Phi^{-}}U_{\alpha,1}\times
T^{1}\times\prod_{\alpha\in\Phi^{+}}U_{\alpha,0}\overset{\sim}{\longrightarrow}I.$

Here the products over $\Phi^{-}$ and $\Phi^{+}$ are ordered in an arbitrarily
chosen way. For the latter see the proof of [OS19, Lem. 2.3] and Lemma 2.1,
part i, in [OS19].

##  3\. A $p$-valuation on pro-$p$ Iwahori

In this section we will define a $p$-valuation $\omega$ on $I$ (in the sense
of Lazard) provided $p-1>eh$ and exhibit an ordered basis for $(I,\omega)$.
The strategy is to emulate [Sch11, Ex. 23.3] and conjugate $I$ into a small
enough principal congruence subgroup of ${\bf{G}}(E)$ for a large enough
finite extension $E/L$. This approach is based on the following well-known
observation.

######  Lemma 3.1.

Let $E/\mathbb{Q}_{p}$ be a finite extension and suppose $\mathcal{G}$ is an
$\mathcal{O}_{E}$-subgroup scheme of $\text{GL}_{N}$ for some $N$. For
$r=1,2,3,\ldots$ consider the congruence subgroup $K_{r}$ defined as

$K_{r}=\ker\big{(}\mathcal{G}(\mathcal{O}_{E})\longrightarrow\mathcal{G}(\mathcal{O}_{E}/\mathfrak{m}_{E}^{r})\big{)}.$

Then for $r>e_{E}/(p-1)$ the expression

$\omega(g)=\frac{1}{e_{E}}\cdot\sup\\{n\in\mathbb{N}:g\in K_{n}\\}$

defines a $p$-valuation on $K_{r}$. (Here $e_{E}=e(E/\mathbb{Q}_{p})$ and
$\mathcal{O}_{E}$ denotes the valuation ring of $E$; by convention
$\omega(1)=\infty$.)

###### Proof.

The $\text{GL}_{N}$-case is [Sch11, Ex. 23.2]. There $\omega$ is defined by
$\omega(g)=\min_{i,j}v(a_{ij}-\delta_{ij})$ for a matrix $g=(a_{ij})$ where
$v$ is the additive valuation on $E$ with $v(p)=1$, but this is easily checked
to agree with our definition. For a subgroup scheme
$\mathcal{G}\hookrightarrow\text{GL}_{N}$ just note that
$K_{r}=\mathcal{G}(\mathcal{O}_{E})\cap\tilde{K}_{r}$ where $\tilde{K}_{r}$
denotes the $r$th congruence subgroup of $\text{GL}_{N}$. ∎

We will apply this to the following $\mathcal{G}$. As in [OS19, Sect. 2.1.6]
we fix a hyperspecial vertex $x_{0}$ in the apartment of the semisimple
Bruhat-Tits building of $G$ corresponding to ${\bf{T}}$. We consider the
associated smooth affine group scheme $\mathcal{G}_{x_{0}}$ over $\mathcal{O}$
and its identity component $\mathcal{G}_{x_{0}}^{\circ}$. As discussed in
[OS19, Sect. 7.2.2] the $r^{\text{th}}$ congruence subgroup of
$\mathcal{G}_{x_{0}}^{\circ}$ is generated by all the $U_{\alpha,r}$ for
varying $\alpha\in\Phi$, and a subgroup $T^{r}\subset T$ defined in terms of
the Neron model of ${\bf{T}}$. See [OS19, Prop. 7.9].

Instead of $\mathcal{G}_{x_{0}}^{\circ}$ we work with the base change
$\mathcal{G}=\mathcal{G}_{x_{0}}^{\circ}\times_{\text{Spec}(\mathcal{O})}\text{Spec}(\mathcal{O}_{E})$,
where $E/L$ is a yet unspecified finite extension. As above we let $K_{r}$
denote the $r^{\text{th}}$ congruence subgroup of
$\mathcal{G}(\mathcal{O}_{E})$.

######  Lemma 3.2.

Assume $p-1>eh$. Then there exists a pair $(E,t)$ consisting of a finite
extension $E/L$ and an element $t\in{\bf{T}}(E)$ such that $tIt^{-1}\subset
K_{r}$ where $r$ is the smallest integer $>e_{E}/(p-1)$.

###### Proof.

Let $E/L$ be arbitrary for now. We apply [OS19, Prop. 7.9] to the base change
${\bf{G}}\times_{\text{Spec}(L)}\text{Spec}(E)$ and the facet $\\{x_{0}\\}$.
This tells us the multiplication map gives a homeomorphism

$\prod_{\alpha\in\Phi^{-}}U_{\alpha,r}^{E}\times
T_{E}^{r}\times\prod_{\alpha\in\Phi^{+}}U_{\alpha,r}^{E}\overset{\sim}{\longrightarrow}K_{r}.$

Here the superscripts/subscript $E$ indicate we are are now working over $E$.
For instance $U_{\alpha,r}^{E}=u_{\alpha}^{E}(\mathfrak{m}_{E}^{r})$ where
$u_{\alpha}^{E}:E\overset{\sim}{\longrightarrow}{\bf{U}}_{\alpha}(E)$ extends
$u_{\alpha}$. By conjugating the analogous factorization (2.1) of $I$ by
$t\in{\bf{T}}(E)$ we see that the inclusion $tIt^{-1}\subset K_{r}$ holds if
and only if

  * 1.

For $\alpha\in\Phi^{-}$,

$tU_{\alpha,1}t^{-1}\subset
U_{\alpha,r}^{E}\Longleftrightarrow\alpha(t)\mathfrak{m}\subset\mathfrak{m}_{E}^{r}\Longleftrightarrow-v(\alpha(t))<\frac{1}{e}-\frac{1}{p-1}.$

  * 2.

For the middle factor,

$tT^{1}t^{-1}=T^{1}\subset
T_{E}^{r}\Longleftrightarrow\mathfrak{m}\subset\mathfrak{m}_{E}^{r}\Longleftrightarrow
p-1>e.$

  * 3.

For $\alpha\in\Phi^{+}$,

$tU_{\alpha,0}t^{-1}\subset
U_{\alpha,r}^{E}\Longleftrightarrow\alpha(t)\mathcal{O}\subset\mathfrak{m}_{E}^{r}\Longleftrightarrow
v(\alpha(t))>\frac{1}{p-1}.$

We remind the reader that $v(p)=1$. These inequalities can be summarized as
saying that

$\frac{1}{p-1}<v(\alpha(t))<\frac{1}{e}-\frac{1}{p-1}$

for all $\alpha\in\Phi^{+}$. It remains to construct a pair $(E,t)$ satisfying
all these inequalities. We will do this by means of an auxiliary cocharacter
defined as follows. Start by enlarging $\Delta$ to a $\mathbb{Q}$-basis for
$X^{*}({\bf{T}})_{\mathbb{Q}}$ and choose a linear form
$X^{*}({\bf{T}})_{\mathbb{Q}}\rightarrow\mathbb{Q}$ sending all elements of
$\Delta$ to one. This is given by pairing with some element $\mu_{0}\in
X_{*}({\bf{T}})_{\mathbb{Q}}$. Thus
$\langle\alpha,\mu_{0}\rangle=\text{ht}(\alpha)$ for all $\alpha\in\Phi$. Pick
an $a\in\mathbb{N}$ for which $a\mu_{0}\in X_{*}({\bf{T}})$ and call this
cocharacter $\mu$. Now suppose $E/L$ is large enough to contain an element
$c\neq 0$ with $v(c)=\frac{1}{aeh}$. We claim $t=\mu(c)$ works. Indeed

$v(\alpha(t))=v(c^{\langle\alpha,\mu\rangle})=\langle\alpha,\mu\rangle
v(c)=a\text{ht}(\alpha)\cdot\frac{1}{aeh}=\frac{\text{ht}(\alpha)}{eh}$

and $1\leq\text{ht}(\alpha)\leq h-1$ for all $\alpha\in\Phi^{+}$. It therefore
suffices to check that

$\frac{1}{p-1}<\frac{1}{eh}<\frac{1}{h-1}\cdot\bigg{(}\frac{1}{e}-\frac{1}{p-1}\bigg{)},$

and as one readily verifies this amounts to our assumption that $p-1>eh$. ∎

Combining the two preceding lemmas endows $I$ with a $p$-valuation $\omega$ by
restricting and conjugating the one on $K_{r}$. We wish to give an intrinsic
description of $\omega$ independent of the choice of $(E,t)$. First we
introduce the following notion.

######  Definition 3.3.

Let $H$ be a $p$-valuable group and let $(H_{1},\ldots,H_{s})$ be a list of
subgroups such that the multiplication map

$m:H_{1}\times\cdots\times H_{s}\longrightarrow H$

is bijective. We say that a $p$-valuation $\omega$ on $H$ is compatible with
the tuple $(H_{1},\ldots,H_{s})$ if

$\omega(h_{1}\cdots h_{s})=\min\\{\omega(h_{1}),\ldots,\omega(h_{s})\\}$

for all $h_{i}\in H_{i}$ with $i=1,\ldots,s$.

In this situation one can construct an ordered basis for $(H,\omega)$ by
concatenating ordered bases for each factor $(H_{i},\omega|_{H_{i}})$ as
$i=1,\ldots,s$. For the definition of an ordered basis of a $p$-valued group
we refer the reader to [Sch11, p. 182].

Here is our characterization of the $p$-valuation we have defined on $I$.

######  Proposition 3.4.

Assume $p-1>eh$. Then there exists a unique $p$-valuation $\omega$ on $I$
satisfying the following properties:

  * (a)

$\omega$ is compatible with the Iwahori factorization (2.1) of $I$, cf.
Definition 3.3.

  * (b)

$\omega(u_{\alpha}(x))=v(x)+\frac{\text{ht}(\alpha)}{eh}$ where
$\begin{cases}\text{$x\in\mathfrak{m}$ if $\alpha\in\Phi^{-}$}\\\
\text{$x\in\mathcal{O}$ if $\alpha\in\Phi^{+}$.}\end{cases}$

  * (c)

$\omega(t)=\frac{1}{e}\cdot\sup\\{n\in\mathbb{N}:t\in T^{n}\\}$ for $t\in
T^{1}$.

(Recall that $T^{n}$ denotes the $n^{\text{th}}$ congruence subgroup of $T$;
in part (a) the compatibility of $\omega$ holds for any ordering of the roots
$\Phi^{\pm}$.) Moreover $(I,\omega)$ is saturated (i.e., every $g\in I$ with
$\omega(g)\geq\frac{p}{p-1}$ lies in $I^{p}$).

###### Proof.

Uniqueness is clear (as $\omega$ is determined by its restriction to each
factor of the Iwahori decomposition). We need to show the $\omega$ we defined
satisfies (a) through (c). Fix $(E,t)$ as in the proof of Lemma 3.2. Then
$\omega(g)=\frac{1}{e_{E}}\cdot\sup\\{n\in\mathbb{N}:tgt^{-1}\in K_{n}\\}$ for
$g\in I$. Consider the Iwahori factorization

$g=\bigg{(}\prod_{\alpha\in\Phi^{-}}u_{\alpha}\bigg{)}\cdot
t_{1}\cdot\bigg{(}\prod_{\alpha\in\Phi^{+}}u_{\alpha}\bigg{)}$

according to (2.1). Conjugating by $t$ we find that $tgt^{-1}\in K_{n}$ if and
only if all $tu_{\alpha}t^{-1}\in U_{\alpha,n}^{E}$ and $t_{1}\in T_{E}^{n}$.
The largest $n$ with this property is the minimum of
$\min_{\alpha\in\Phi}\omega(u_{\alpha})$ and $\omega(t_{1})$. This shows (a).
The formula in (b) follows almost immediately from the relation
$v(\alpha(t))=\frac{\text{ht}(\alpha)}{eh}$ as noted in the proof of Lemma
3.2. Since ${\bf{T}}$ is split the proof of (c) easily reduces to the
$\text{GL}_{1}$-case. When $t_{1}\in 1+\mathfrak{m}$ just note that the
largest $n$ for which $t_{1}\in 1+\mathfrak{m}_{E}^{n}$ is
$n=e(E/L)v_{L}(t_{1}-1)$. In other words
$\omega(t_{1})=\frac{1}{e}v_{L}(t_{1}-1)$ as claimed. (Here $v_{L}$ denotes
the additive valuation on $L$ with $v_{L}(\varpi)=1$.)

To show saturation it suffices to find an ordered basis consisting of elements
with $p$-valuation $\leq\frac{p}{p-1}$, see [Sch11, Prop. 26.11]. We start by
writing down a basis for $U_{\alpha,0}$ for all $\alpha\in\Phi^{+}$. Pick a
$\mathbb{Z}_{p}$-basis $(b_{1},\ldots,b_{\ell})$ for $\mathcal{O}$. Then
$(u_{\alpha}(b_{1}),\ldots,u_{\alpha}(b_{\ell}))$ is a basis for
$U_{\alpha,0}$. Indeed any element $u\in U_{\alpha,0}$ can be factored
uniquely as

$u=u_{\alpha}(x)=\prod_{i=1}^{\ell}u_{\alpha}(b_{i})^{x_{i}}$

where $x=\sum_{i=1}^{\ell}x_{i}b_{i}$ with coefficients
$x_{i}\in\mathbb{Z}_{p}$. To verify that
$\omega(u)=\min_{i=1,\ldots,\ell}\\{v(x_{i})+\omega(u_{\alpha}(b_{i}))\\}$ it
suffices to note the relations $v(x)=\min_{i=1,\ldots,\ell}\\{v(x_{i})\\}$ and
$v(b_{i})=0$. Minor modifications of the argument when $\alpha\in\Phi^{-}$
show that $U_{\alpha,1}$ has basis $(u_{\alpha}(\varpi
b_{1}),\ldots,u_{\alpha}(\varpi b_{\ell}))$. Finally, to find a basis for
$T^{1}$ we first consider the $\text{GL}_{1}$-case $1+\mathfrak{m}$. Since
$p-1>e$ the exp and log give isomorphisms
$\mathfrak{m}^{n}\overset{\sim}{\longrightarrow}1+\mathfrak{m}^{n}$ for all
$n\geq 1$. It follows easily that $(e^{\varpi b_{1}},\ldots,e^{\varpi
b_{\ell}})$ is a basis for $1+\mathfrak{m}$. (Here $e^{\varpi
b_{i}}=\exp(\varpi b_{i})$ and $e$ should not be confused with the
ramification index.) In general pick a basis $\mathcal{B}$ for
$X_{*}({\bf{T}})$ and concatenate all tuples $(\mu(e^{\varpi
b_{1}}),\ldots,\mu(e^{\varpi b_{\ell}}))$ as $\mu\in\mathcal{B}$ varies. On
these basis elements the $p$-valuation takes the values

  * •

$\omega(u_{\alpha}(\varpi b_{i}))=\frac{1}{e}+\frac{\text{ht}(\alpha)}{eh}$
for $\alpha\in\Phi^{-}$.

  * •

$\omega(\mu(e^{\varpi b_{i}}))=\frac{1}{e}$ for $\mu\in\mathcal{B}$.

  * •

$\omega(u_{\alpha}(b_{i}))=\frac{\text{ht}(\alpha)}{eh}$ for
$\alpha\in\Phi^{+}$.

We just need to check all these numbers are at most $\frac{p}{p-1}$. For
example $\frac{\text{ht}(\alpha)}{eh}\leq\frac{h-1}{eh}<\frac{1}{e}\leq
1<\frac{p}{p-1}$ for all positive roots $\alpha$, and
$\frac{1}{e}+\frac{\text{ht}(\alpha)}{eh}\leq\frac{1}{e}-\frac{1}{eh}=\frac{h-1}{eh}<\frac{p}{p-1}$
also for negative roots $\alpha$. ∎

The observation that $(I,\omega)$ is saturated will be crucial in the next
section when we attach a rigid analytic group to this pair.

######  Remark 3.5.

For $G=\text{GL}_{n}(\mathbb{Q}_{p})$ the condition $p-1>eh$ is also necessary
for the existence of a $p$-valuation. In this example $e=1$ and $h=n$, and
when $p\leq n+1$ there is $p$-torsion in $I$. To see this, for $p\leq n$ just
consider a permutation matrix given by a $p$-cycle. A conjugate thereof lies
in $I$. For $p=n+1$ consider multiplication by $\zeta_{p}$ on the free rank
$n$ module $\mathbb{Z}_{p}[\zeta_{p}]$ over $\mathbb{Z}_{p}$. Again this gives
an element of order $p$ in $\text{GL}_{n}(\mathbb{Z}_{p})$ which can be
conjugated into $I$ because the latter is a Sylow pro-$p$ subgroup. (Since a
$p$-valuation satisfies $\omega(g^{p})=\omega(g)+1$ a $p$-valuable group is
torsion-free.)

For future reference we want to record the following consequence of the proof
of Proposition 3.4.

######  Corollary 3.6.

Assume $p-1>eh$ and pick a $\mathbb{Z}_{p}$-basis $(b_{1},\ldots,b_{\ell})$
for $\mathcal{O}$. If ${\bf{G}}$ is semisimple and simply connected the
following is an ordered basis for $(I,\omega)$.

  * •

$(u_{\alpha}(\varpi b_{1}),\ldots,u_{\alpha}(\varpi
b_{\ell}))_{\alpha\in\Phi^{-}}$,

  * •

$(\alpha^{\vee}(e^{\varpi b_{1}}),\ldots,\alpha^{\vee}(e^{\varpi
b_{\ell}}))_{\alpha\in\Delta}$,

  * •

$(u_{\alpha}(b_{1}),\ldots,u_{\alpha}(b_{\ell}))_{\alpha\in\Phi^{+}}$.

All basis elements have $p$-valuation at most $\frac{1}{e}$.

###### Proof.

This was shown above. When ${\bf{G}}$ is semisimple and simply connected one
can take $\mathcal{B}=\Delta^{\vee}$. ∎

##  4\. Rigid analytic groups

In this section $(H,\omega)$ denotes an arbitrary $p$-valued group equipped
with an ordered basis $(h_{1},\ldots,h_{d})$. Thus
$(x_{1},\ldots,x_{d})\mapsto h_{1}^{x_{1}}\cdots h_{d}^{x_{d}}$ gives a
homeomorphism $\mathbb{Z}_{p}^{d}\rightarrow H$ and
$\omega(h)=\min_{i=1,\ldots,d}\\{v(x_{i})+\omega(h_{i})\\}$ where $h\in H$ is
factored as $h_{1}^{x_{1}}\cdots h_{d}^{x_{d}}$.

We will review how to associate an affinoid rigid analytic group over
$\mathbb{Q}_{p}$ to a saturated pair $(H,\omega)$. The underlying rigid
analytic space is nothing but the $d$-dimensional closed ball
$\text{Sp}(\mathbb{Q}_{p}\langle Z_{1},\ldots Z_{d}\rangle)$. Here
$\mathbb{Q}_{p}\langle Z_{1},\ldots Z_{d}\rangle$ denotes the Tate algebra in
$d$ variables over $\mathbb{Q}_{p}$.

######  Proposition 4.1.

Let $(H,\omega)$ be a saturated $p$-valued group with basis
$(h_{1},\ldots,h_{d})$. Then there exists a unique rigid analytic group
$\mathbb{H}$ over $\mathbb{Q}_{p}$ such that

  * (a)

$\mathbb{H}=\text{Sp}(\mathbb{Q}_{p}\langle Z_{1},\ldots Z_{d}\rangle)$ as a
rigid analytic space;

  * (b)

the map $\varphi\mapsto h_{1}^{\varphi(Z_{1})}\cdots h_{d}^{\varphi(Z_{d})}$
gives an isomorphism of abstract groups

$\mathbb{H}(\mathbb{Q}_{p})=\text{Hom}_{\mathbb{Q}_{p}-\text{alg}}(\mathbb{Q}_{p}\langle
Z_{1},\ldots Z_{d}\rangle,\mathbb{Q}_{p})\overset{\sim}{\longrightarrow}H.$

###### Proof.

We construct $\mathbb{H}$. The multiplication map
$\mathbb{H}\times_{\text{Sp}(\mathbb{Q}_{p})}\mathbb{H}\longrightarrow\mathbb{H}$
arises from comultiplication on the Tate algebra $\mathbb{Q}_{p}\langle
Z_{1},\ldots Z_{d}\rangle$ defined as follows. By [Sch11, Prop. 29.2] and
[Sch11, Rem. 29.3] there exists power series $F_{i}\in\mathbb{Z}_{p}\langle
X_{1},\ldots X_{d},Y_{1},\ldots Y_{d}\rangle$ for $i=1,\ldots,d$ such that

$(h_{1}^{x_{1}}\cdots h_{d}^{x_{d}})\cdot(h_{1}^{y_{1}}\cdots
h_{d}^{y_{d}})=h_{1}^{F_{1}(x,y)}\cdots h_{d}^{F_{d}(x,y)}$

for all tuples $x=(x_{1},\ldots,x_{d})$ and $y=(y_{1},\ldots,y_{d})$ in
$\mathbb{Z}_{p}^{d}$. The fact that $F_{i}$ has coefficients in
$\mathbb{Z}_{p}$ (not just $\mathbb{Q}_{p}$) is due to the assumption that
$(H,\omega)$ is saturated, and this allows us to define the map

$\mathbb{Q}_{p}\langle Z_{1},\ldots
Z_{d}\rangle\longrightarrow\mathbb{Q}_{p}\langle X_{1},\ldots
X_{d}\rangle\widehat{\otimes}\mathbb{Q}_{p}\langle Y_{1},\ldots
Y_{d}\rangle=\mathbb{Q}_{p}\langle X_{1},\ldots X_{d},Y_{1},\ldots
Y_{d}\rangle$

by sending $Z_{i}\mapsto F_{i}(X,Y)$ where $X=(X_{1},\ldots,X_{d})$ and
similarly for $Y$. Since $(F_{1},\ldots,F_{d})$ is a formal group law one
quickly verifies this gives a rigid analytic group structure on
$\mathbb{H}=\text{Sp}(\mathbb{Q}_{p}\langle Z_{1},\ldots Z_{d}\rangle)$. (The
identity $\text{Sp}(\mathbb{Q}_{p})\rightarrow\mathbb{H}$ comes from the co-
unit on the Tate algebra and we leave it to the reader to construct the
inversion map $\mathbb{H}\rightarrow\mathbb{H}$ along the same lines.)

To show this $\mathbb{H}$ satisfies (b) let
$\varphi_{1},\varphi_{2}\in\mathbb{H}(\mathbb{Q}_{p})$. Unwinding definitions
it suffices to note that

(4.1)
$(\varphi_{1}\cdot\varphi_{2})(Z_{i})=F_{i}\big{(}\varphi_{1}(Z_{1}),\ldots,\varphi_{1}(Z_{d}),\varphi_{2}(Z_{1}),\ldots,\varphi_{2}(Z_{d})\big{)},$

but this is immediate in light of how multiplication on
$\mathbb{H}(\mathbb{Q}_{p})$ relates to comultiplication.

Finally, suppose $\text{Sp}(\mathbb{Q}_{p}\langle Z_{1},\ldots Z_{d}\rangle)$
carries some rigid analytic group structure for which (b) holds. This
structure is given by power series $G_{i}\in\mathbb{Z}_{p}\langle X_{1},\ldots
X_{d},Y_{1},\ldots Y_{d}\rangle$ for $i=1,\ldots,d$ satisfying the analogue of
(4.1) – still with $F_{i}$ on the right-hand side, but with
$\varphi_{1}\cdot\varphi_{2}$ defined relative to the $G_{i}$. This leads to
the equality

$G_{i}\big{(}\varphi_{1}(Z_{1}),\ldots,\varphi_{1}(Z_{d}),\varphi_{2}(Z_{1}),\ldots,\varphi_{2}(Z_{d})\big{)}=F_{i}\big{(}\varphi_{1}(Z_{1}),\ldots,\varphi_{1}(Z_{d}),\varphi_{2}(Z_{1}),\ldots,\varphi_{2}(Z_{d})\big{)}$

Since $\varphi_{1}$ and $\varphi_{2}$ are arbitrary we infer that $F_{i}$ and
$G_{i}$ define the same function
$\mathbb{Z}_{p}^{d}\times\mathbb{Z}_{p}^{d}\rightarrow\mathbb{Z}_{p}$, and
therefore $F_{i}=G_{i}$ for all $i$, cf. [Sch11, Cor. 5.8] with
$\varepsilon=1$ and $V=\mathbb{Q}_{p}$.∎

The next result shows how this construction behaves under a change of basis.

######  Proposition 4.2.

Let $(H,\omega)$ be a saturated group with bases $(h_{1},\ldots,h_{d})$ and
$(h_{1}^{\prime},\ldots,h_{d}^{\prime})$. Let $\mathbb{H}$ and
$\mathbb{H}^{\prime}$ denote the rigid analytic groups associated with these
bases as in Proposition 4.1. There is a unique isomorphism of rigid analytic
groups $\mathbb{H}\overset{\sim}{\longrightarrow}\mathbb{H}^{\prime}$ such
that the induced map on $\mathbb{Q}_{p}$-points makes the following diagram
commute:

${\mathbb{H}(\mathbb{Q}_{p})}$${\mathbb{H}^{\prime}(\mathbb{Q}_{p})}$${H}$$\scriptstyle{\sim}$

###### Proof.

Uniqueness is a straightforward consequence of the fact that
$\mathbb{H}(\mathbb{Q}_{p})$ is Zariski dense in $\mathbb{H}$. (An
automorphism of $\mathbb{H}$ inducing the identity on
$\mathbb{H}(\mathbb{Q}_{p})$ must be the identity.)

To define the desired isomorphism we compare the formal group law
$\underline{F}=(F_{1},\ldots,F_{d})$ from the proof of Proposition 4.1 to the
formal group law
$\underline{F}^{\prime}=(F_{1}^{\prime},\ldots,F_{d}^{\prime})$ arising from
$(h_{1}^{\prime},\ldots,h_{d}^{\prime})$. Again by [Sch11, Prop. 29.2] and
[Sch11, Rem. 29.3] there are $d$ power series
$\Phi_{i}\in\mathbb{Z}_{p}\langle X_{1},\ldots X_{d}\rangle$ such that

$h_{1}^{x_{1}}\cdots
h_{d}^{x_{d}}=(h_{1}^{\prime})^{\Phi_{1}(x)}\cdots(h_{d}^{\prime})^{\Phi_{d}(x)}$

for all $x=(x_{1},\ldots,x_{d})\in\mathbb{Z}_{p}^{d}$. Similarly with the
bases interchanged. This gives an isomorphism of formal group laws
$\underline{F}\overset{\sim}{\longrightarrow}\underline{F}^{\prime}$, see
[Sch11, p. 143]. Mapping $Z_{i}^{\prime}\mapsto\Phi_{i}(Z_{1},\ldots,Z_{d})$
then yields an isomorphism

$\mathbb{Q}_{p}\langle Z_{1}^{\prime},\ldots
Z_{d}^{\prime}\rangle\overset{\sim}{\longrightarrow}\mathbb{Q}_{p}\langle
Z_{1},\ldots Z_{d}\rangle$

which intertwines the two comultiplications. Passing to spectra we finally get
an isomorphism of rigid analytic groups
$\mathbb{H}\overset{\sim}{\longrightarrow}\mathbb{H}^{\prime}$. The
commutativity of the diagram essentially just boils down to the definition of
the $\Phi_{i}$. ∎

Let us revisit the situation in Definition 3.3.

######  Remark 4.3.

Suppose every $(H_{i},\omega|_{H_{i}})$ is saturated, where $i=1,\ldots,s$.
Then so is $(H,\omega)$ and one can obtain a basis for the latter by stringing
together bases

$(h_{11},\ldots,h_{1d_{1}})\hskip 6.0pt\ldots\hskip
6.0pt(h_{s1},\ldots,h_{sd_{s}})$

for each subgroup. As in Proposition 4.1 this gives rigid analytic subgroups
$\mathbb{H}_{1},\ldots,\mathbb{H}_{s}$ of $\mathbb{H}$ over $\mathbb{Q}_{p}$,
and multiplication in $\mathbb{H}$ gives an isomorphism of rigid analytic
spaces

$\mathbb{H}_{1}\times_{\text{Sp}(\mathbb{Q}_{p})}\cdots\times_{\text{Sp}(\mathbb{Q}_{p})}\mathbb{H}_{s}\overset{\sim}{\longrightarrow}\mathbb{H}$

which on $\mathbb{Q}_{p}$-points corresponds to the bijection
$H_{1}\times\cdots\times H_{s}\longrightarrow H$. In this sense the
factorization of $H$ as a $p$-adic manifold can be promoted to an isomorphism
of rigid analytic spaces.

We end this section by introducing the space of rigid functions on a saturated
$p$-valued group $H$. These functions take values in a fixed finite extension
$K/\mathbb{Q}_{p}$. Choosing a basis $(h_{1},\ldots,h_{d})$ gives a rigid
analytic group $\mathbb{H}$, and a $K$-valued rigid analytic function on
$\mathbb{H}$ naturally defines a continuous function $f:H\rightarrow K$. In
more detail, an element $\sum_{I}c_{I}Z_{1}^{i_{1}}\cdots Z_{d}^{i_{d}}$ of
the global sections
$\Gamma(\mathbb{H},\mathcal{O}_{\mathbb{H}})\otimes_{\mathbb{Q}_{p}}K=K\langle
Z_{1},\ldots Z_{d}\rangle$ gives the function
$f(h)=\sum_{I}c_{I}x_{1}^{i_{1}}\cdots x_{d}^{i_{d}}$ where
$h=h_{1}^{x_{1}}\cdots h_{d}^{x_{d}}$. (Here and elsewhere111There should be
no risk of confusing the tuple $I$ with the pro-$p$ Iwahori. in this article
$I$ runs over tuples $I=(i_{1},\ldots,i_{d})\in\mathbb{N}_{0}^{d}$ and the
coefficients $c_{I}$ form a null-sequence in $K$ as
$|I|=i_{1}+\cdots+i_{d}\rightarrow\infty$.) By [Sch11, Cor. 5.8] for instance,
the power series is determined by the function $f$. The image of the map
$\Gamma(\mathbb{H},\mathcal{O}_{\mathbb{H}})\otimes_{\mathbb{Q}_{p}}K\hookrightarrow\mathcal{C}(H,K)$
is our space of rigid analytic functions $\mathcal{C}^{\text{rig}}(H,K)$. For
this to make sense we need to check that this space is independent of the
choice of basis, so suppose $(h_{1}^{\prime},\ldots,h_{d}^{\prime})$ is
another basis with corresponding rigid analytic group $\mathbb{H}^{\prime}$ as
above. The isomorphism
$\mathbb{H}\overset{\sim}{\longrightarrow}\mathbb{H}^{\prime}$ in Proposition
4.2 amounts to an isomorphism

${\Gamma(\mathbb{H}^{\prime},\mathcal{O}_{\mathbb{H}^{\prime}})\otimes_{\mathbb{Q}_{p}}K}$${\Gamma(\mathbb{H},\mathcal{O}_{\mathbb{H}})\otimes_{\mathbb{Q}_{p}}K}$${\mathcal{C}(H,K)}$$\scriptstyle{\sim}$

compatible with comultiplication. The diagram commutes so the images of the
two slanted arrows coincide. Furthermore, the isomorphism in the diagram
preserves the Gauss norms on the two Tate algebras (in the notation of the
proof of Proposition 4.2 one easily checks $\|\Phi_{i}\|=1$). Thus
$\mathcal{C}^{\text{rig}}(H,K)$ even carries a canonical norm. Let us
summarize this.

######  Definition 4.4.

The space $\mathcal{C}^{\text{rig}}(H,K)$ consists of all functions
$f:H\rightarrow K$ which can be expanded as

$f(h_{1}^{x_{1}}\cdots h_{d}^{x_{d}})=\sum_{I}c_{I}x_{1}^{i_{1}}\cdots
x_{d}^{i_{d}}$

relative to some or equivalently any basis $(h_{1},\ldots,h_{d})$ where
$c_{I}\rightarrow 0$ as $|I|\rightarrow\infty$. Under pointwise multiplication
$\mathcal{C}^{\text{rig}}(H,K)$ becomes a $K$-Banach algebra when equipped
with the Gauss norm

$\|f\|={\sup}_{I}|c_{I}|.$

Here $|p|=1/p$. The Gauss norm $\|\cdot\|$ is independent of the choice of
basis for $(H,\omega)$.

The group $H$ acts on $\mathcal{C}(H,K)$ by right (and left) translations.

######  Lemma 4.5.

The $H$-action preserves the subspace $\mathcal{C}^{\text{rig}}(H,K)$ and the
Gauss norm is $H$-invariant.

###### Proof.

We right translate $f\in\mathcal{C}^{\text{rig}}(H,K)$ by an element
$h_{0}=h_{1}^{a_{1}}\cdots h_{d}^{a_{d}}$. The function $h_{0}f$ has the form

$(h_{0}f)(h_{1}^{x_{1}}\cdots h_{d}^{x_{d}})=f(h_{1}^{x_{1}}\cdots
h_{d}^{x_{d}}\cdot h_{1}^{a_{1}}\cdots
h_{d}^{a_{d}})=\sum_{I}c_{I}F_{1}(x,a)^{i_{1}}\cdots F_{d}(x,a)^{i_{d}}$

where $a=(a_{1},\ldots,a_{d})$. Recall the power series $F_{i}$ introduced in
the proof of Proposition 4.1 have coefficients in $\mathbb{Z}_{p}$. In
particular $\sum_{I}c_{I}F_{1}(X,a)^{i_{1}}\cdots F_{d}(X,a)^{i_{d}}$
converges in $K\langle X_{1},\ldots X_{d}\rangle$ and induces $h_{0}f$, which
is therefore in $\mathcal{C}^{\text{rig}}(H,K)$. In addition, this argument
shows $\|h_{0}f\|\leq\|f\|$. Since $h_{0}$ and $f$ are arbitrary we must have
equality. ∎

##  5\. Weyl conjugates

Let $W$ be the Weyl group of $(G,T)$. Recall, as noted in the proof of [OS19,
Lem. 2.3] for instance, that multiplication defines an injection

$\prod_{\alpha\in\Phi^{-}}U_{\alpha}\times
T\times\prod_{\alpha\in\Phi^{+}}U_{\alpha}\hookrightarrow G$

where the products are ordered arbitrarily. We conjugate it by $w\in W$ (or
rather a representative in the normalizer of $T$) which shows the
multiplication map

$\prod_{\alpha\in\Phi^{-}}U_{w\alpha}\times
T\times\prod_{\alpha\in\Phi^{+}}U_{w\alpha}\hookrightarrow G$

remains an injection. Here $U_{w\alpha}=wU_{\alpha}w^{-1}$ and
$(w\alpha)(t)=\alpha(w^{-1}tw)$.

######  Definition 5.1.

For $w\in W$ introduce the two subgroups

  * •

$U_{w}^{-}=\prod_{\alpha\in\Phi^{-}}(I\cap U_{w\alpha})$

  * •

$U_{w}^{+}=\prod_{\alpha\in\Phi^{+}}(I\cap U_{w\alpha})$

for some ordering of $\Phi^{-}$ and $\Phi^{+}$. Note that $I\cap
U_{w\alpha}=\begin{cases}\text{$U_{w\alpha,1}$ if $w\alpha\in\Phi^{-}$}\\\
\text{$U_{w\alpha,0}$ if $w\alpha\in\Phi^{+}$.}\end{cases}$

######  Lemma 5.2.

Assume the derived group ${\bf{G}}^{\text{der}}$ is simply connected. Then the
multiplication map gives a homeomorphism

$U_{w}^{-}\times T^{1}\times U_{w}^{+}\overset{\sim}{\longrightarrow}I$

for all $w\in W$. In particular $U_{w}^{\pm}=I\cap
w\big{(}\prod_{\alpha\in\Phi^{\pm}}U_{\alpha}\big{)}w^{-1}$. Furthermore, when
$p-1>eh$ the $p$-valuation $\omega$ on $I$ from Proposition 3.4 is compatible
with the above factorization for all $w$ (see Definition 3.3).

###### Proof.

The previous discussion shows the multiplication map

$\prod_{\alpha\in\Phi^{-}}(I\cap U_{w\alpha})\times
T^{1}\times\prod_{\alpha\in\Phi^{+}}(I\cap U_{w\alpha})\longrightarrow I$

is injective. It is also surjective, as we now explain. Since we are assuming
${\bf{G}}^{\text{der}}$ is simply connected, Bruhat-Tits theory shows that the
full Iwahori subgroup $J\supset I$ has a factorization

$T^{0}\times\prod_{\alpha\in\Phi}(J\cap
U_{\alpha})\overset{\sim}{\longrightarrow}J$

for any ordering of $\Phi$, see [OS10, Lem. 3.3.2] and the references given in
the proof thereof. For full disclosure those references to [Tit79, Sect.
3.1.1] require ${\bf{G}}$ to be semisimple and simply connected, but one can
easily reduce the general reductive case to the semisimple case by observing
that all root groups lie in the derived group.

Clearly $J\cap U_{\alpha}$ are pro-$p$ groups (subgroups of
$\mathfrak{m}^{-N}$ for large enough $N$) so $J\cap U_{\alpha}\subset I$ and
we therefore also have the analogous factorization of pro-$p$ Iwahori $I$
itself

$T^{1}\times\prod_{\alpha\in\Phi}(I\cap
U_{\alpha})\overset{\sim}{\longrightarrow}I.$

We order $\Phi$ as follows. The first batch of roots is $w\Phi^{-}$ (in some
order) and the second is $w\Phi^{+}$. Since $T^{1}$ normalizes each $I\cap
U_{\alpha}$ this gives the surjectivity.

The compatibility of $\omega$ follows from the case $w=1$ by noting that any
$p$-valuation is invariant under conjugation, see [Sch11, p. 169]. ∎

From now on we will assume ${\bf{G}}^{\text{der}}$ is simply connected and
$p-1>eh$. We do not know whether Lemma 5.2 continues to hold without the
simply connectedness assumption. The discussion in [Tit79] indicates that even
the definition of parahoric subgroups is more subtle in that case.

##  6\. Rigid induction and slopes

Start with a continuous character $\chi:T^{1}\rightarrow K^{\times}$ and
consider the conjugate $(w\chi)(t)=\chi(w^{-1}tw)$ for $w\in W$. Extend
$w\chi$ to a character of $U_{w}^{-}T^{1}$ by making it trivial on $U_{w}^{-}$
and introduce the rigid analytic principal series representation

(6.1)
$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)=\\{f\in\mathcal{C}^{\text{rig}}(I,K):f(bg)=(w\chi)(b)f(g)\hskip
6.0pt\forall b\in U_{w}^{-}T^{1}\\}.$

This is a closed subspace of $\mathcal{C}^{\text{rig}}(I,K)$ because
$|f(g)|\leq\|f\|_{\text{sup}}\leq\|f\|$ for all $g\in I$. Here
$\|f\|_{\text{sup}}$ is the supremum norm of $f$ on $I$. The group $I$ acts by
right translations. Of course, for the space in (6.1) to be non-trivial we
require that $\chi$ is rigid when viewed as a $K$-valued function
$T^{1}\rightarrow K^{\times}\hookrightarrow K$. In the sequel we will make
this assumption: $\chi\in\mathcal{C}^{\text{rig}}(T^{1},K)$.

######  Lemma 6.1.

Let $\chi:T^{1}\rightarrow K^{\times}$ be a rigid analytic character. Then the
restriction map $f\mapsto f|_{U_{w}^{+}}$ gives an isomorphism of
$U_{w}^{+}$-representations

$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)\overset{\sim}{\longrightarrow}\mathcal{C}^{\text{rig}}(U_{w}^{+},K)$

preserving the Gauss norms.

###### Proof.

By promoting the factorization in Lemma 5.2 to an isomorphism of rigid spaces
as in Remark 4.3 one can immediately extend a function in
$\mathcal{C}^{\text{rig}}(U_{w}^{+},K)$ to a function in
$\mathcal{C}^{\text{rig}}(I,K)$ with the required transformation property.

For the comment on norms, as Gauss norms are multiplicative it suffices to
note that $\|\chi\|=1$. Indeed, we certainly have
$\|\chi\|\geq\|\chi\|_{\text{sup}}=1$. Applying this to $\chi^{-1}$ we find
that $\|\chi\|=1$. ∎

Next we transfer the $T^{1}$-action on
$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ across the isomorphism in Lemma 6.1
to an action on $\mathcal{C}^{\text{rig}}(U_{w}^{+},K)$. It is given by the
formula

$(tf)(u)=(w\chi)(t)f(t^{-1}ut)$

for $t\in T^{1}$ and $u\in U_{w}^{+}$. In order to do explicit calculations we
choose coordinates on $U_{w}^{+}$, but first we introduce a bookkeeping
symbol.

######  Definition 6.2.

For all roots $\beta$ we let $\varpi_{\beta}=\begin{cases}\text{$\varpi$ if
$\beta\in\Phi^{-}$}\\\ \text{$1$ if $\beta\in\Phi^{+}$.}\end{cases}$

######  Notation 6.3.

With this symbol
$(u_{w\alpha}(\varpi_{w\alpha}b_{1}),\ldots,u_{w\alpha}(\varpi_{w\alpha}b_{\ell}))$
is an ordered basis for $I\cap U_{w\alpha}$ for all $\alpha\in\Phi^{+}$.
Recall that $(b_{1},\ldots,b_{\ell})$ is a choice of $\mathbb{Z}_{p}$-basis
for $\mathcal{O}$, cf. the proof of Proposition 3.4. Letting $\alpha$ vary
gives a basis for $U_{w}^{+}$. Once and for all we label the positive roots as
$\Phi^{+}=\\{\alpha_{1},\ldots,\alpha_{N}\\}$. Then our basis for $U_{w}^{+}$
is the string

$(u_{w\alpha_{1}}(\varpi_{w\alpha_{1}}b_{1}),\ldots,u_{w\alpha_{1}}(\varpi_{w\alpha_{1}}b_{\ell}))\hskip
6.0pt\ldots\hskip
6.0pt(u_{w\alpha_{N}}(\varpi_{w\alpha_{N}}b_{1}),\ldots,u_{w\alpha_{N}}(\varpi_{w\alpha_{N}}b_{\ell})).$

This lets us identify $\mathcal{C}^{\text{rig}}(U_{w}^{+},K)$ with the space
of rigid functions on the ball $\mathbb{Z}_{p}^{N\ell}$ in terms of the
corresponding $\ell$-tuples of variables

$z_{1}=(z_{11},\ldots,z_{1\ell})\hskip 6.0pt\ldots\hskip
6.0ptz_{N}=(z_{N1},\ldots,z_{N\ell}).$

Our power series expansions $f(z_{1},\ldots,z_{N})=\sum_{I}c_{I}z^{I}$ are
indexed by tuples $I=(i_{1},\ldots,i_{N})$ where

$i_{1}=(i_{11},\ldots,i_{1\ell})\hskip 6.0pt\ldots\hskip
6.0pti_{N}=(i_{N1},\ldots,i_{N\ell}),$

and we adopt the following notation:

  * •

$z^{I}=z_{1}^{i_{1}}\cdots z_{N}^{i_{N}}=z_{11}^{i_{11}}\cdots
z_{N\ell}^{i_{N\ell}}$,

  * •

$|I|=|i_{1}|+\cdots+|i_{N}|=i_{11}+\cdots+i_{N\ell}$.

By choosing a Chevalley basis we get a model for ${\bf{G}}$ over $\mathbb{Z}$.
In particular ${\bf{G}}={\bf{G}}_{0}\times_{\mathbb{Q}_{p}}L$ for some
connected reductive group ${\bf{G}}_{0}$ split over $\mathbb{Q}_{p}$.
Similarly ${\bf{T}}={\bf{T}}_{0}\times_{\mathbb{Q}_{p}}L$ for a split
$\mathbb{Q}_{p}$-torus ${\bf{T}}_{0}\subset{\bf{G}}_{0}$. We let
$T_{0}={\bf{T}}_{0}(\mathbb{Q}_{p})$. The maximal compact subgroup of $T_{0}$
will be denoted by $T_{0}^{0}$, and $T_{0}^{1}$ is the Sylow pro-$p$-subgroup
of $T_{0}^{0}$. The main reason for passing to the $\mathbb{Q}_{p}$-groups
${\bf{G}}_{0}$ and ${\bf{T}}_{0}$ is that all roots $\alpha\in\Phi$ are
$\mathbb{Z}_{p}^{\times}$-valued on $T_{0}^{0}$, and
$\alpha:T_{0}^{1}\rightarrow 1+p\mathbb{Z}_{p}$. In particular
$tu_{\alpha}(x)t^{-1}=u_{\alpha}(x)^{\alpha(t)}$ for $t\in T_{0}^{0}$. Note
that the right-hand side only makes sense when $\alpha(t)\in\mathbb{Z}_{p}$,
which is why we have to put restrictions on $t$.

We recommend the reader initially considers the case $L=\mathbb{Q}_{p}$ and
$w=1$ at a first reading, which simplifies the notation tremendously.

In these coordinates the $T_{0}^{1}$-action on
$\mathcal{C}^{\text{rig}}(U_{w}^{+},K)$ is given by

$(tf)(z_{1},\ldots,z_{N})=(w\chi)(t)\cdot
f\big{(}(w\alpha_{1})(t^{-1})z_{1},\ldots,(w\alpha_{N})(t^{-1})z_{N}\big{)}.$

We will differentiate this and get a $\text{Lie}(T_{0}^{1})$-module structure.
Pick an $H\in\text{Lie}(T_{0}^{1})$ (there is no risk of confusion with the
use of the letter $H$ in Section 4). The exponential map $\gamma:s\mapsto
e^{sH}$ is defined on a small enough open ball $\Omega\subset\mathbb{Z}_{p}$
around $0$. This gives a locally $\mathbb{Q}_{p}$-analytic homomorphism
$\gamma:\Omega\rightarrow T_{0}^{1}$ such that $\gamma(0)=1$ is the identity
and $\gamma^{\prime}(0)=\text{Lie}(\gamma)(1)=H$. Thus $H$ sends $f$ to222We
apologize for the double meaning of $s$ in this calculation.

$\displaystyle(Hf)(z_{1},\ldots,z_{N})$
$\displaystyle=\frac{d}{ds}(e^{sH}f)(z_{1},\ldots,z_{N})|_{s=0}$
$\displaystyle=\frac{d}{ds}\big{[}(w\chi)(e^{sH})\cdot
f\big{(}(w\alpha_{1})(e^{-sH})z_{1},\ldots,(w\alpha_{N})(e^{-sH})z_{N}\big{)}\big{]}|_{s=0}$
$\displaystyle=d(w\chi)(H)f(z_{1},\ldots,z_{N})-\sum_{r=1}^{N}\sum_{s=1}^{\ell}d(w\alpha_{r})(H)z_{rs}\frac{\partial
f}{\partial z_{rs}}(z_{1},\ldots,z_{N}),$

by the product rule and the chain rule; see [Sch11, Lem. 9.5, Lem. 9.12]. Let
us also emphasize that $d(w\chi)$ means the derivative at the identity of
$w\chi$ viewed as a function $T_{0}^{1}\rightarrow K$; see the definition in
[Sch11, p. 59]. Similarly $d(w\alpha_{r})$ is the derivative (at the identity)
of the conjugate root $w\alpha_{r}$ viewed as a function
$T_{0}^{1}\rightarrow\mathbb{Q}_{p}^{\times}$.

######  Lemma 6.4.

The monomial $f_{I}(z_{1},\ldots,z_{N})=z^{I}$ is an eigenfunction for the
$\text{Lie}(T_{0}^{1})$-action with eigensystem

$H\longmapsto d(w\chi)(H)-\sum_{r=1}^{N}d(w\alpha_{r})(H)|i_{r}|.$

Any vector in $\text{Lie}(T_{0}^{1})$ acts continuously on
$\mathcal{C}^{\text{rig}}(U_{w}^{+},K)$.

###### Proof.

This is immediate from the above calculation. Indeed
$z_{rs}\frac{\partial}{\partial z_{rs}}f_{I}=i_{rs}f_{I}$. Continuity follows
since the eigenvalues of any $H$ are bounded uniformly in $I$ as $|i_{r}|$ is
an integer. ∎

We think of $w\in W$ as being fixed in the entire discussion, and choose a
cocharacter $\mu\in X_{*}({\bf{T}})$ adapted to $w$ in the sense that $\langle
w\alpha,\mu\rangle>0$ for all $\alpha\in\Phi^{+}$. We view $\mu$ as a
homomorphism $\mu:1+p\mathbb{Z}_{p}\rightarrow T_{0}^{1}$ and introduce its
derivative $H_{\mu}=\text{Lie}(\mu)(1)$. A simple calculation shows
$d(w\alpha)(H_{\mu})=\langle w\alpha,\mu\rangle$. In particular the operator

$d(w\chi)(H_{\mu})\text{Id}-H_{\mu}$

has eigenvalue $\lambda_{I}=\sum_{r=1}^{N}\langle
w\alpha_{r},\mu\rangle|i_{r}|$ on $f_{I}$ by Lemma 6.4. (Obviously
$\lambda_{I}$ depends on $w$ and $\mu$, but we think of those as being fixed,
and we omit them to lighten the notation.) Note that
$\lambda_{I}\in\mathbb{N}_{0}$, so it makes sense to consider its $p$-adic
valuation $v(\lambda_{I})$.

We will explore the spectral theory of the operator
$d(w\chi)(H_{\mu})\text{Id}-H_{\mu}$ in more detail. First
$C^{\text{rig}}(U_{w}^{+},K)$ has a slope decomposition

(6.2) $C^{\text{rig}}(U_{w}^{+},K)=C^{\text{rig}}(U_{w}^{+},K)^{<s}\oplus
C^{\text{rig}}(U_{w}^{+},K)^{\geq s}$

for all $s\in\mathbb{N}_{0}$. Here $C^{\text{rig}}(U_{w}^{+},K)^{\geq s}$ is
the subspace of functions $f(z_{1},\ldots,z_{N})=\sum_{I:v(\lambda_{I})\geq
s}c_{I}z^{I}$. The space $C^{\text{rig}}(U_{w}^{+},K)^{<s}$ is defined
similarly. We also consider the space $C^{\text{rig}}(U_{w}^{+},K)^{=s}$
spanned (topologically) by those monomials $z^{I}$ with $v(\lambda_{I})=s$.

In accordance with (6.2) we decompose a $K$-valued rigid function on
$U_{w}^{+}$ as $f=f^{<s}+f^{\geq s}$ where $f^{<s}$ is the projection of $f$
onto $C^{\text{rig}}(U_{w}^{+},K)^{<s}$, and similarly for $f^{\geq s}$.

The next observation is inspired by the Hida idempotent in the theory of
$p$-adic modular forms. This point of view is one of the main novelties of
this paper.

######  Lemma 6.5.

For a given $s\in\mathbb{N}_{0}$ consider the decomposition

$C^{\text{rig}}(U_{w}^{+},K)^{\geq s}=C^{\text{rig}}(U_{w}^{+},K)^{=s}\oplus
C^{\text{rig}}(U_{w}^{+},K)^{\geq s+1}.$

Introduce the operator
$\mathcal{U}_{s}=\bigg{(}p^{-s}\big{(}d(w\chi)(H_{\mu})\text{Id}-H_{\mu}\big{)}\bigg{)}^{p-1}$
on $C^{\text{rig}}(U_{w}^{+},K)$. Then

$\lim_{n\rightarrow\infty}\mathcal{U}_{s}^{n!}$

defines the projection map $C^{\text{rig}}(U_{w}^{+},K)^{\geq
s}\longrightarrow C^{\text{rig}}(U_{w}^{+},K)^{=s}$.

###### Proof.

Take some $f(z_{1},\ldots,z_{N})=\sum_{I:v(\lambda_{I})\geq s}c_{I}z^{I}$
belonging to $C^{\text{rig}}(U_{w}^{+},K)^{\geq s}$ and apply the
$\mathcal{U}_{s}$-operator,

$(\mathcal{U}_{s}f)(z_{1},\ldots,z_{N})={\sum}_{I:v(\lambda_{I})\geq
s}c_{I}(p^{-s}\lambda_{I})^{p-1}z^{I}.$

Doing this $n!$ times multiplies the coefficients $c_{I}$ by
$(p^{-s}\lambda_{I})^{(p-1)n!}$. If $v(\lambda_{I})>s$ this clearly goes to
$0$ as $n\rightarrow\infty$. On the other hand, if $v(\lambda_{I})=s$ we have
$(p^{-s}\lambda_{I})^{p-1}\equiv 1$ modulo $p$ by Fermat. Therefore
$(p^{-s}\lambda_{I})^{(p-1)l}\rightarrow 1$ as $v(l)\rightarrow\infty$. For
$l=n!$ this amounts to letting $n\rightarrow\infty$. Consequently

$(\mathcal{U}_{s}^{n!}f)(z_{1},\ldots,z_{N})\longrightarrow{\sum}_{I:v(\lambda_{I})=s}c_{I}z^{I}$

for $n\rightarrow\infty$ as desired. ∎

This yields a slope decomposition for closed $T_{0}^{1}$-invariant subspaces.

######  Lemma 6.6.

Let $V\subset C^{\text{rig}}(U_{w}^{+},K)$ be a closed $T_{0}^{1}$-invariant
subspace. Then $V=V^{<s}\oplus V^{\geq s}$ for all $s\in\mathbb{N}_{0}$ where

$V^{<s}=V\cap C^{\text{rig}}(U_{w}^{+},K)^{<s}\hskip 6.0pt\hskip 6.0pt\hskip
6.0pt\hskip 6.0pt\hskip 6.0ptV^{\geq s}=V\cap
C^{\text{rig}}(U_{w}^{+},K)^{\geq s}.$

###### Proof.

It suffices to show $f\in V\Rightarrow f^{\geq s}\in V$ for all
$s\in\mathbb{N}_{0}$. We use induction on $s$. The base case $s=0$ is trivial
since $V=V^{\geq 0}$. Suppose $s\geq 0$ and the implication holds for $s$.
Decompose $f^{\geq s}=f^{=s}+f^{\geq s+1}$. Here

$f^{=s}=\lim_{n\rightarrow\infty}\mathcal{U}_{s}^{n!}f^{\geq s}$

by the previous Lemma. By the induction hypothesis $f^{\geq s}\in V$. Thus
each $\mathcal{U}_{s}^{n!}f^{\geq s}\in V$ by $T_{0}^{1}$-invariance. Since
$V$ is closed $f^{=s}\in V$. We conclude that also $f^{\geq s+1}\in V$. ∎

We now consider closed $T_{0}^{1}U_{w}^{+}$-invariant subspaces.

######  Proposition 6.7.

Let $V\subset C^{\text{rig}}(U_{w}^{+},K)$ be a nonzero closed
$T_{0}^{1}U_{w}^{+}$-invariant subspace. Then $V$ contains all the constant
functions.

###### Proof.

Pick a nonzero $h\in V$. Then $h(u_{0})\neq 0$ for some $u_{0}\in U_{w}^{+}$.
In other words the right $u_{0}$-translate $f=u_{0}h\in V$ does not vanish at
the identity. Therefore, when expressed in coordinates we have

$f(z_{1},\ldots,z_{N})=\sum_{I}c_{I}z^{I}$

with $c_{0}\neq 0$. (The subscript $0$ denotes the zero $N\ell$-tuple.) By the
previous Lemma we known $f^{\geq s}\in V$ for all $s\in\mathbb{N}_{0}$. We
claim $f^{\geq s}\rightarrow c_{0}$ as $s\rightarrow\infty$. Given
$\epsilon>0$ we must find an $S$ such that $\forall I$:

$s\leq v(\lambda_{I})<\infty\Longrightarrow|c_{I}|<\epsilon$

provided $s>S$. First pick an $S^{\prime}$ such that
$|I|>S^{\prime}\Rightarrow|c_{I}|<\epsilon$, which is possible by rigidity. We
will check that any $S$ satisfying

$p^{S}>S^{\prime}\cdot{\max}_{\alpha\in\Phi^{+}}\langle w\alpha,\mu\rangle$

works. Indeed, suppose $s>S$ and $I$ is a nonzero tuple for which
$p^{s}|\lambda_{I}$. Then

$p^{S}<p^{s}\leq\lambda_{I}=\sum_{r=1}^{N}\langle
w\alpha_{r},\mu\rangle|i_{r}|\leq|I|\cdot{\max}_{\alpha\in\Phi^{+}}\langle
w\alpha,\mu\rangle$

and thus $|I|>S^{\prime}$ by choice of $S$. This finishes the proof. ∎

To summarize, every nonzero closed $T_{0}^{1}U_{w}^{+}$-invariant subspace of
the rigid induction $\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ contains all the
functions which are constant on $U_{w}^{+}$. In other words every such
subspace must contain the function $\tilde{f}_{0}$ which extends the constant
function $f_{0}=1$ on $U_{w}^{+}$. Note that
$H\tilde{f}_{0}=d(w\chi)(H)\tilde{f}_{0}$ for all $H\in\text{Lie}(T_{0}^{1})$.
In fact this holds for all $H\in\text{Lie}(T^{1})$ since any $t\in T^{1}$ acts
by $(t\tilde{f}_{0})(u)=(w\chi)(t)$. This observation will be used shortly in
the next section where we address the $U_{w}^{-}$-action.

##  7\. Verma modules and irreducibility

We now take into account the $U_{w}^{-}$-action which does not have a nice
explicit description on $C^{\text{rig}}(U_{w}^{+},K)$. First we introduce some
more notation. All our groups will be viewed as locally
$\mathbb{Q}_{p}$-analytic groups:

  * •

$\mathfrak{g}=\text{Lie}(I)\otimes_{\mathbb{Q}_{p}}K$

  * •

$\mathfrak{t}=\text{Lie}(T^{1})\otimes_{\mathbb{Q}_{p}}K$

  * •

$\mathfrak{t}_{0}=\text{Lie}(T_{0}^{1})\otimes_{\mathbb{Q}_{p}}K$

  * •

$\mathfrak{u}_{w}^{\pm}=\text{Lie}(U_{w}^{\pm})\otimes_{\mathbb{Q}_{p}}K$

  * •

$\mathfrak{b}_{w}^{\pm}=\mathfrak{t}\oplus\mathfrak{u}_{w}^{\pm}$

We assume $K\supset L$. In particular this implies $\mathfrak{g}_{\text{der}}$
is split in the sense of [Dix96, Sect. 1.10.1].

We view $d(w\chi)$ as a linear form $\mathfrak{t}\rightarrow K$ and extend it
to $\mathfrak{b}_{w}^{+}$ by declaring it to be $0$ on $\mathfrak{u}_{w}^{+}$.
We denote the resulting $U(\mathfrak{b}_{w}^{+})$-module by $K_{d(w\chi)}$ and
consider the Verma module

$\mathcal{V}_{d(w\chi)}=U(\mathfrak{g})\otimes_{U(\mathfrak{b}_{w}^{+})}K_{d(w\chi)}.$

The rigid functions $C^{\text{rig}}(I,K)$ form a $\mathfrak{g}$-module. The
map $X\mapsto X\tilde{f}_{0}$ factors through the Verma module and gives a
nonzero $U(\mathfrak{g})$-linear map

$\psi:\mathcal{V}_{d(w\chi)}\longrightarrow\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi).$

Indeed $H\tilde{f}_{0}=d(w\chi)(H)\tilde{f}_{0}$ for all $H\in\mathfrak{t}$,
and $X\tilde{f}_{0}=0$ for $X\in\mathfrak{u}_{w}^{+}$ (as we are
differentiating a constant function on $U_{w}^{+}$). This is a major contrast
between this argument and the one in [OS10] say. In the locally analytic setup
one maps the Verma module to the dual of the induction, whereas $\psi$ maps
$\mathcal{V}_{d(w\chi)}$ to the actual induction.

Since
$\mathcal{V}_{d(w\chi)}=\bigoplus_{\lambda\in\mathfrak{t}_{0}^{*}}\mathcal{V}_{d(w\chi)}[\lambda]$
is a sum of weight spaces the map $\psi$ clearly takes values in the subspace
$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}$ of
$\mathfrak{t}_{0}$-finite vectors (those $f$ for which
$\text{span}_{K}\\{Hf:H\in\mathfrak{t}_{0}\\}$ is finite-dimensional).

######  Lemma 7.1.

The restriction map $f\mapsto f|_{U_{w}^{+}}$ gives an isomorphism of vector
spaces

$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}\overset{\sim}{\longrightarrow}C^{\text{alg}}(U_{w}^{+},K).$

(The right-hand side denotes the dense subspace of polynomial functions:
$f(z_{1},\ldots,z_{N})=\sum_{I}c_{I}z^{I}$ with $c_{I}=0$ for $|I|>\\!\\!>0$.)

###### Proof.

Let $f\in\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}$
and consider the finite-dimensional space
$W=\text{span}_{K}\\{Hf:H\in\mathfrak{t}_{0}\\}$. Decompose it into weight
spaces $W=\bigoplus_{\lambda\in\mathfrak{t}_{0}^{*}}W[\lambda]$, after
possibly enlarging $K$. In order to show $f|_{U_{w}^{+}}$ is a polynomial
function we may assume $f\in W[\lambda]$ for some
$\lambda\in\mathfrak{t}_{0}^{*}$. Expanding $f$ as $\sum_{I}c_{I}f_{I}$ and
applying $H\in\text{Lie}(T_{0}^{1})$ to it gives

$\lambda(H)f=Hf=\sum_{I}c_{I}\big{(}d(w\chi)(H)-\sum_{r=1}^{N}d(w\alpha_{r})(H)|i_{r}|\big{)}f_{I}.$

Comparing coefficients shows $c_{I}\neq
0\Rightarrow\lambda(H)=d(w\chi)(H)-\sum_{r=1}^{N}d(w\alpha_{r})(H)|i_{r}|$.
For $H=H_{\mu}$ this last equation reads

$\lambda(H_{\mu})-d(w\chi)(H_{\mu})=\lambda_{I}=\sum_{r=1}^{N}\langle
w\alpha_{r},\mu\rangle|i_{r}|.$

There are only finitely many tuples $I$ with this property.

The map is obviously surjective: A polynomial function belongs to
$C^{\text{rig}}(U_{w}^{+},K)_{\mathfrak{t}_{0}-\text{fin}}$ since the $f_{I}$
are eigenfunctions for $\text{Lie}(T_{0}^{1})$. ∎

As a byproduct
$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}=\bigoplus_{\lambda\in\mathfrak{t}_{0}^{*}}\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}[\lambda]$
decomposes as a direct sum of weight spaces, and

$\dim_{K}\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}[\lambda]=\\#\\{I:d(w\chi)-\lambda=\sum_{r=1}^{N}d(w\alpha_{r})|i_{r}|\\}=\dim_{K}\mathcal{V}_{d(w\chi)}[\lambda].$

The last equality is part (ii) of [Dix96, Prop. 7.1.6, p. 233] applied to
$w\Phi^{+}$ in place of $\Phi^{+}$, or rather a variant of the proof: As a
vector space $\mathfrak{g}=\mathfrak{u}_{w}^{-}\oplus\mathfrak{b}_{w}^{+}$
where $\mathfrak{u}_{w}^{-}=\bigoplus_{r=1}^{N}\text{Lie}(I\cap
U_{-w\alpha_{r}})\otimes_{\mathbb{Q}_{p}}K$. In this decomposition each space
$\text{Lie}(I\cap U_{-w\alpha_{r}})$ is $\ell$-dimensional over
$\mathbb{Q}_{p}$ and $\mathfrak{t}_{0}$ acts on it via $-d(w\alpha_{r})$.
Inspecting the $\mathfrak{t}_{0}$-action on a PBW basis for
$\mathcal{V}_{d(w\chi)}\simeq U(\mathfrak{u}_{w}^{-})$ gives the result as in
[Dix96].

######  Proposition 7.2.

If $\mathcal{V}_{d(w\chi)}$ is a simple $U(\mathfrak{g})$-module then $\psi$
is an isomorphism of $U(\mathfrak{g})$-modules

$\mathcal{V}_{d(w\chi)}\overset{\sim}{\longrightarrow}\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}$

and $\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ is topologically irreducible.
Conversely, if $\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ is topologically
irreducible the Verma module $\mathcal{V}_{d(w\chi)}$ is necessarily simple.

###### Proof.

The map $\psi$ is nonzero and $\mathfrak{g}$-linear, therefore injective by
the simplicity of $\mathcal{V}_{d(w\chi)}$. Since the dimensions match up the
inclusions
$\mathcal{V}_{d(w\chi)}[\lambda]\hookrightarrow\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}[\lambda]$
are bijective for all $\lambda\in\mathfrak{t}_{0}^{*}$. Now take the direct
sum.

The irreducibility follows: Suppose
$V\subset\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ is a nonzero closed
$I$-invariant subspace. By the $T^{1}U_{w}^{+}$-invariance we know $V$ must
contain the vector $\tilde{f}_{0}$, and therefore also all elements
$X\tilde{f}_{0}$ for $X\in U(\mathfrak{u}_{w}^{-})$ by the
$U_{w}^{-}$-invariance. From what we have just shown these $X\tilde{f}_{0}$
constitute all of
$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}$, which
is a dense subspace (the polynomial functions on $U_{w}^{+}$ are dense in the
space of rigid functions). Since $V$ is closed it must be all of
$\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$.

To show the converse we start by making the following observation. Let
$E\subset C^{\text{alg}}(U_{w}^{+},K)$ be an arbitrary
$U(\mathfrak{t})$-submodule with closure $\bar{E}\subset
C^{\text{rig}}(U_{w}^{+},K)$. Then we assert that

(7.1) $E=\bar{E}\cap C^{\text{alg}}(U_{w}^{+},K).$

To see this suppose $f\in C^{\text{alg}}(U_{w}^{+},K)$ is a limit
$f=\lim_{i\rightarrow\infty}f_{i}$ of functions $f_{i}\in E$. Let
$\lambda\in\mathfrak{t}_{0}^{*}$ and consider the projection map onto the
$\lambda$-weight space
$\text{pr}_{\lambda}:C^{\text{alg}}(U_{w}^{+},K)\rightarrow
C^{\text{alg}}(U_{w}^{+},K)[\lambda]$. The target
$C^{\text{alg}}(U_{w}^{+},K)[\lambda]$ is finite-dimensional, as we have seen
above. Since $C^{\text{alg}}(U_{w}^{+},K)$ is topologized by the Gauss norm,
and the $f_{I}$ are eigenfunctions, it is easy to see $\text{pr}_{\lambda}$ is
continuous. Therefore
$\text{pr}_{\lambda}(f)=\lim_{i\rightarrow\infty}\text{pr}_{\lambda}(f_{i})$.
Observe that $E$ is a semisimple $U(\mathfrak{t}_{0})$-module since
$C^{\text{alg}}(U_{w}^{+},K)$ is, and thus $E$ has a weight space
decomposition $E=\bigoplus_{\lambda\in\mathfrak{t}_{0}^{*}}E[\lambda]$. Hence
all the terms $\text{pr}_{\lambda}(f_{i})$ lie in $E[\lambda]$. Since
$\dim_{K}E[\lambda]<\infty$ so does the limit $\text{pr}_{\lambda}(f)$
($E[\lambda]$ is complete and therefore closed). Consequently
$f=\sum_{\lambda}\text{pr}_{\lambda}(f)\in E$. This shows our assertion. In
particular

(7.2) $E\subsetneq C^{\text{alg}}(U_{w}^{+},K)\Longrightarrow\bar{E}\subsetneq
C^{\text{rig}}(U_{w}^{+},K).$

Now suppose $\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ is topologically
irreducible. Apply the above observation to $\text{im}(\psi)$. Since
$\overline{\text{im}(\psi)}=\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ we must
have
$\text{im}(\psi)=\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}$.
In other words
$\psi:\mathcal{V}_{d(w\chi)}\rightarrow\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}$
is surjective. Again, since
$\mathcal{V}_{d(w\chi)}[\lambda]\twoheadrightarrow\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)_{\mathfrak{t}_{0}-\text{fin}}[\lambda]$
must be bijective $\forall\lambda$ as the two spaces have the same finite
dimension, $\psi$ is bijective as well. Finally, to verify
$\mathcal{V}_{d(w\chi)}$ is simple, suppose
$\mathcal{E}\subsetneq\mathcal{V}_{d(w\chi)}$ is a
$U(\mathfrak{g})$-submodule. Then, by the above implication applied to
$E=\psi(\mathcal{E})$, we find that
$\overline{\psi(\mathcal{E})}\subsetneq\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$
and so $\overline{\psi(\mathcal{E})}=\\{0\\}$. Equivalently
$\mathcal{E}=\\{0\\}$ by the bijectivity of $\psi$. ∎

######  Remark 7.3.

Assertion (7.1) and the implication (7.2) in the above proof are analogous to
[Ray20b, Lem. 3.13, Cor. 3.14] respectively. Our proof is more or less just a
rewording of the argument given there.

As explained in [OS10, Cor. 3.4.11] for example, $\mathcal{V}_{d(w\chi)}$ is
simple for all $w\in W$ if (and only if) $\mathcal{V}_{d\chi}$ is simple. This
is a straightforward argument; conjugation by $w$ identifies the Verma
modules.

The BGG criterion [Dix96, Thm. 7.6.24] tells us $\mathcal{V}_{d\chi}$ is
simple if and only if $(d\chi+\delta)(H_{\alpha})\notin\\{1,2,3,\ldots\\}$ for
all positive roots $\alpha$ relative to the pair
$(\mathfrak{g},\mathfrak{t})$. (To aid comparison, in the notation of [Dix96]
we have $\mathcal{V}_{d\chi}=M(d\chi+\delta)$.)

##  8\. Multiplicity one

Suppose $\mathcal{V}_{d\chi}$ is simple. In this section we note that as $w\in
W$ varies the representations $\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ are
non-isomorphic. This relies on the non-existence of a Haar measure on the
space of rigid analytic functions on $\mathbb{Z}_{p}$. We include the standard
argument for convenience.

######  Lemma 8.1.

There is no nonzero translation-invariant continuous linear form
$C^{\text{rig}}(\mathbb{Z}_{p},K)\rightarrow K$.

###### Proof.

We let $T:C^{\text{rig}}(\mathbb{Z}_{p},K)\rightarrow
C^{\text{rig}}(\mathbb{Z}_{p},K)$ denote the translation operator
$(Tf)(x)=f(x+1)$. Suppose $\ell:C^{\text{rig}}(\mathbb{Z}_{p},K)\rightarrow K$
is a translation-invariant form; meaning $\ell(Tf)=\ell(f)$. To show $\ell=0$
we consider the values of $\ell$ on the monomials $f_{N}(x)=x^{N}$. We will
verify that $\ell(f_{N})=0$ by strong induction on $N$. The case $N=0$ follows
by observing $Tf_{1}=f_{0}+f_{1}$ and applying $\ell$. Now suppose
$\ell(f_{i})=0$ for all $i=0,\ldots,N-1$. Note that

$Tf_{N+1}=\sum_{i=0}^{N+1}\binom{N+1}{i}f_{i}=f_{N+1}+(N+1)f_{N}+\sum_{i=0}^{N-1}\binom{N+1}{i}f_{i}.$

Applying $\ell$ yields $\ell(Tf_{N+1})=\ell(f_{N+1})+(N+1)\ell(f_{N})$ by the
induction hypothesis. Since the left-hand side equals $\ell(f_{N+1})$ we
conclude $\ell(f_{N})=0$. By continuity $\ell=0$ on all of
$C^{\text{rig}}(\mathbb{Z}_{p},K)$. ∎

Without any simplicity assumption on $\mathcal{V}_{d\chi}$ we have the
following result.

######  Proposition 8.2.

$\text{Hom}_{I}^{\text{cont}}\big{(}\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi),\text{Ind}_{U_{w^{\prime}}^{-}T^{1}}^{I}(w^{\prime}\chi)\big{)}=0$
for $w\neq w^{\prime}$.

###### Proof.

Rigid induction is easily checked to satisfy Frobenius reciprocity for
globally analytic representations (one of the adjunction maps is given by the
orbit map which is rigid analytic). So we have to show that

$\text{Hom}_{U_{w^{\prime}}^{-}T^{1}}^{\text{cont}}\big{(}\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi),K_{w^{\prime}\chi}\big{)}=0.$

We identify $\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$ with
$C^{\text{rig}}(U_{w}^{+},K)$ as a $U_{w}^{+}$-representation. It suffices to
show there is no nonzero $K$-linear form
$\ell:C^{\text{rig}}(U_{w}^{+},K)\rightarrow K$ such that $\ell(uf)=\ell(f)$
for all $u\in U_{w}^{+}\cap U_{w^{\prime}}^{-}$. Note that this intersection
is non-trivial since $w\Phi^{+}\cap w^{\prime}\Phi^{-}\neq\varnothing$. Indeed
$w^{-1}w^{\prime}\neq 1$ has length $l(w^{-1}w^{\prime})=|\Phi^{+}\cap
w^{-1}w^{\prime}\Phi^{-}|>0$. Pick a $\beta\in w\Phi^{+}\cap
w^{\prime}\Phi^{-}$ and consider the subgroup $I\cap U_{\beta}\subset
U_{w}^{+}\cap U_{w^{\prime}}^{-}$. By ordering $\Phi^{+}$ appropriately we may
factor $U_{w}^{+}$ as $U_{w}^{+}=Y\times(I\cap U_{\beta})$ and promote this to
the level of rigid spaces as in remark 4.3. Thus
$C^{\text{rig}}(U_{w}^{+},K)\simeq
C^{\text{rig}}(Y,K)\hat{\otimes}_{K}C^{\text{rig}}(I\cap U_{\beta},K)$ as
representations of $I\cap U_{\beta}$. (The action on $C^{\text{rig}}(Y,K)$ is
trivial.) This reduces the problem to checking there is no nonzero $I\cap
U_{\beta}$-invariant linear form $C^{\text{rig}}(I\cap U_{\beta},K)\rightarrow
K$. Since $I\cap U_{\beta}\simeq\mathbb{Z}_{p}^{\ell}$ we just have to check
there is no nonzero translation-invariant form
$C^{\text{rig}}(\mathbb{Z}_{p}^{\ell},K)\rightarrow K$. This easily reduces to
the case $\ell=1$ which is Lemma 8.1. ∎

The previous argument is an adaptation of the proof of [OS10, Prop. 3.5.1]
which they in turn partially attribute to Frommer.

##  9\. Principal series

To add some finishing strokes to the proof of our main theorem, suppose
$\chi:T\rightarrow K^{\times}$ is a continuous character and inflate it to
$B^{-}=\prod_{\alpha\in\Phi^{-}}U_{\alpha}\times T$. Consider the continuous
induction

$\text{Ind}_{B^{-}}^{G}(\chi)_{\text{cont}}=\\{f\in
C(G,K):f(bg)=\chi(b)f(g)\hskip 6.0pt\forall b\in B^{-}\\}.$

By the Bruhat-Tits decomposition $G=\bigsqcup_{w\in W}B^{-}wJ$ it decomposes
as a $J$-representation into

$\text{Ind}_{B^{-}}^{G}(\chi)_{\text{cont}}\overset{\sim}{\longrightarrow}\bigoplus_{w\in
W}\text{Ind}_{J\cap wB^{-}w^{-1}}^{J}(w\chi)_{\text{cont}}$

by sending $f\mapsto(f_{w})_{w\in W}$ where $f_{w}(g)=f(w^{-1}g)$. Since
$J=T^{0}I$ and $T^{1}=T^{0}\cap I$ further restriction gives an isomorphism of
$I$-representations

$\text{Ind}_{J\cap
wB^{-}w^{-1}}^{J}(w\chi)_{\text{cont}}\overset{\sim}{\longrightarrow}\text{Ind}_{I\cap
wB^{-}w^{-1}}^{I}(w\chi)_{\text{cont}}.$

The space of rigid vectors (or as in Emerton’s terminology
$\mathbb{I}$-analytic vectors, cf. [Eme17, Df. 3.3.1]) of the right-hand side
coincides with the rigid induction $\text{Ind}_{U_{w}^{-}T^{1}}^{I}(w\chi)$
from above. This latter observation is a consequence of [Eme17, Prop. 3.3.7].
Our space $\mathcal{C}^{\text{rig}}(I,K)$ corresponds to
$\mathcal{C}^{\text{an}}(\mathbb{I},K)$ in his notation.

Altogether this proves the main theorem 1.1 in the introduction.

##  10\. The example $\text{Sp}_{4}(\mathbb{Q}_{p})$

To highlight a concrete new example of our main result, we work out the case
of $\text{Sp}_{4}(\mathbb{Q}_{p})$ explicitly. We follow the conventions in
[RS07]. Thus

$G=\text{Sp}_{4}(\mathbb{Q}_{p})=\\{g\in\text{GL}_{4}(\mathbb{Q}_{p}):{{}^{t}g}Jg=J\\},\hskip
6.0pt\hskip 6.0pt\hskip 6.0ptJ=\begin{pmatrix}&&&1\\\ &&1&\\\ &-1&&\\\
-1&&&\end{pmatrix}.$

We take our Borel subgroup $P$ to consist of all upper triangular matrices in
$\text{Sp}_{4}(\mathbb{Q}_{p})$. The maximal torus is the subgroup of all
diagonal matrices

$T=\\{t_{a,b}=\begin{pmatrix}a&&&\\\ &b&&\\\ &&b^{-1}&\\\
&&&a^{-1}\end{pmatrix}:a,b\in\mathbb{Q}_{p}^{\times}\\}.$

The Weyl group $W$ is dihedral of order $8$. There are two simple roots;
$\Delta=\\{\alpha,\beta\\}$ where $\alpha(t_{a,b})=\frac{a}{b}$ and
$\beta(t_{a,b})=b^{2}$. Then
$\Phi^{+}=\\{\alpha,\beta,\alpha+\beta,2\alpha+\beta\\}$. In particular
$\theta=2\alpha+\beta$ is the highest root and the Coxeter number is $h=4$. We
therefore require that $p>5$ in order to apply our theorem. Moreover, one
immediately verifies that $\delta(t_{a,b})=a^{2}b$. See [RS07, Ch. 2] for a
discussion of the root system.

Let $\chi:T\rightarrow K^{\times}$ be a character. Write it as
$\chi(t_{a,b})=\chi_{1}(a)\chi_{2}(b)$ for a pair of characters
$\chi_{i}:\mathbb{Q}_{p}^{\times}\rightarrow K^{\times}$. We require that
their restrictions $1+p\mathbb{Z}_{p}\rightarrow K^{\times}$ are rigid
analytic. This means they can be expanded as $\chi_{i}(\exp(px))=\sum_{r\geq
0}\gamma_{i,r}x^{r}$ for all $x\in\mathbb{Z}_{p}$. (Here $\gamma_{i,r}\in K$
converge to $0$ as $r\rightarrow\infty$.) If we write $\chi_{i}(t)=t^{c_{i}}$
for $t$ close to $1$, where $c_{i}=(d\chi_{i})(1)\in K$, then
$\gamma_{i,r}=\frac{1}{r!}p^{r}c_{i}^{r}\rightarrow 0$ precisely when

$v_{p}(c_{i})>\frac{1}{p-1}-1=-\frac{p-2}{p-1}.$

Here $v_{p}$ denotes the valuation on $K$ with $v_{p}(p)=1$. We will assume
this in the sequel.

The BGG criterion for the simplicity of $\mathcal{V}_{d\chi}$ involves
$H_{\alpha}\in\mathfrak{t}$ for all positive roots $\alpha$. One can read
those off from the coroots given in [RS07, p. 42]. We list them here for
convenience.

$H_{\alpha}=\begin{pmatrix}1&&&\\\ &-1&&\\\ &&1&\\\
&&&-1\end{pmatrix}H_{\beta}=\begin{pmatrix}0&&&\\\ &1&&\\\ &&-1&\\\
&&&0\end{pmatrix}H_{\alpha+\beta}=\begin{pmatrix}1&&&\\\ &1&&\\\ &&-1&\\\
&&&-1\end{pmatrix}H_{2\alpha+\beta}=\begin{pmatrix}1&&&\\\ &0&&\\\ &&0&\\\
&&&-1\end{pmatrix}.$

This allows us to make the BGG criterion explicit in terms of $c_{1}$ and
$c_{2}$.

  * •

$(d\chi+\delta)(H_{\alpha})=c_{1}-c_{2}+1\notin\mathbb{Z}_{>0}$;

  * •

$(d\chi+\delta)(H_{\beta})=c_{2}+1\notin\mathbb{Z}_{>0}$;

  * •

$(d\chi+\delta)(H_{\alpha+\beta})=c_{1}+c_{2}+3\notin\mathbb{Z}_{>0}$;

  * •

$(d\chi+\delta)(H_{2\alpha+\beta})=2c_{1}+2\notin\mathbb{Z}_{>0}$.

All four conditions are satisfied precisely when the Verma module
$\mathcal{V}_{d\chi}$ is simple. If so, our main result shows that
$\text{Ind}_{P}^{G}(\chi)_{\mathbb{I}-\text{an}}$ breaks up as a direct sum of
eight mutually non-isomorphic topologically irreducible representations
$\text{Ind}_{I\cap wPw^{-1}}^{I}(w\chi)_{\mathbb{I}-\text{an}}$ parametrized
by the Weyl elements $w$. Here the pro-$p$ Iwahori subgroup $I$ consist of
symplectic matrices which are lower triangular modulo $p$,

$I=\\{g\in\text{Sp}_{4}(\mathbb{Z}_{p}):g\equiv\begin{pmatrix}1&&&\\\ *&1&&\\\
*&*&1&\\\ *&*&*&1\end{pmatrix}\text{mod $p$}\\}.$

When $w=1$ the subgroup $I\cap wPw^{-1}$ consists of upper triangular matrices
$\begin{pmatrix}a&*&*&*\\\ &b&*&*\\\ &&b^{-1}&*\\\ &&&a^{-1}\end{pmatrix}$ in
$\text{Sp}_{4}(\mathbb{Z}_{p})$ with $a,b\in 1+p\mathbb{Z}_{p}$ and all
$\ast$-entries in $p\mathbb{Z}_{p}$.

### Acknowledgments

We are grateful to Matthias Strauch for sharing his expertise with us at
various stages of this project. Thanks are also due to Jishnu Ray for some
initial guidance and encouragement. Thanks to both for extensive feedback on a
preliminary draft.

## References

  * [BH16] D. Ban and J. Hundley, On reducibility of p -adic principal series representations of p -adic groups. Represent. Theory 20 (2016), 249–262.
  * [Clo17] L. Clozel, Globally analytic p-adic representations of the pro–p–Iwahori subgroup of GL(2) and base change, I : Iwasawa algebras and a base change map. Special Issue of the Bulletin of the Iranian Mathematical Society in Honor of Professor Freydoon Shahidi’s 70th birthday. Vol. 43 (2017), No. 4, pp. 55–76.
  * [Clo18] L. Clozel, Globally Analytic p-adic Representations of the Pro–p Iwahori Subgroup of GL(2) and Base Change, II: A Steinberg Tensor Product Theorem. On the Occasion of Joachim Schwermer’s 66th Birthday, Bonn, Germany, June 2016. Springer Proceedings in Mathematics and Statistics, 2018.
  * [CR18] C. Cornut and J. Ray, Generators of the pro-p Iwahori and Galois representations. International Journal of Number Theory Vol. 14, No. 1 (2018) 37–53.
  * [Dix96] J. Dixmier, Enveloping algebras. Graduate Studies in Mathematics, Vol. 11, AMS, 1996.
  * [Eme17] M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups. Memoirs of the American Mathematical Society, 2017; Volume 248, Number 1175.
  * [OS10] S. Orlik and M. Strauch, On the irreducibility of locally analytic principal series representations. Representation Theory, Volume 14, Pages 713–746, 2010.
  * [OS19] R. Ollivier and P. Schneider, The modular pro-p Iwahori-Hecke Ext-algebra. Proc. Symp. Pure Math. 101, 255–308, 2019.
  * [Ray20a] J. Ray, Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of $\text{SL}_{n}(\mathbb{Z}_{p})$. Forum Mathematicum. March 2020, Vol. 32 Issue 2, p319–338.
  * [Ray20b] J. Ray, Globally analytic principal series representation and Langlands base change. Pacific Journal of Mathematics, Volume 307, Issue 2, Pages 455–490, 2020.
  * [Rob84] A. Robert, Representations p-adiques irreductibles de sous-groupes ouverts de $\text{SL}_{2}(\mathbb{Z}_{p})$, C. R. Acad. Sci. Paris Ser. I Math. 298:11 (1984), 237–240.
  * [Rob85] A. Robert, Representations p-adiques de dimension infinie de sous-groupes ouverts de $\text{SL}_{2}(\mathbb{Q}_{p})$, exposé 23, pp. 1–14 in Groupe d’étude d’analyse ultramétrique. 12e année: 1984/85. Fasc. 2, edited by Y. Amice et al., Secrétariat Math., Paris, 1985.
  * [RS07] B. Roberts and R. Schmidt, Local newforms for $\text{GSp}(4)$. Springer, LNM Vol. 1918, 2007.
  * [Sch06] P. Schneider, Continuous representation theory of p-adic Lie groups. Proc. ICM Madrid 2006, vol. II, 1261–1282.
  * [Sch11] P. Schneider, p-Adic Lie groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 344. Springer, Heidelberg, 2011.
  * [ST02] P. Schneider and J. Teitelbaum, Locally analytic distributions and p-adic representation theory, with applications to $\text{GL}_{2}$. J. AMS 15, 443–468 (2002).
  * [ST03] P. Schneider and J. Teitelbaum, Algebras of p-adic distributions and admissible representations. Invent. math. 153, 145–196 (2003).
  * [Tit79] J. Tits, Reductive groups over local fields. Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., vol. XXXIII, Providence, R.I.: American Mathematical Society, pp. 29–69.

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Department of Mathematics, UCSD, La Jolla, CA, USA.