Resolutions of locally analytic principal series representations of $GL_{2}$

Aranya Lahiri Department of Mathematics, Indiana University, Bloomington [email protected]

Abstract.

For a finite field extension $F/{{\mathbb{Q}}_{p}}$ we associate a coefficient system attached on the Bruhat-Tits tree of $G:={\rm GL}_{2}(F)$ to a locally analytic representation $V$ of $G$ . This is done in analogy to the work of Schneider and Stuhler for smooth representations. This coefficient system furnishes a chain-complex which is shown, in the case of locally analytic principal series representations $V$ , to be a resolution of $V$ .

1. Introduction

Let $F/{{\mathbb{Q}}_{p}}$ be a finite field extension and $G={\bf G}(F)$ the group of $F$ -valued points of a connected reductive group ${\bf G}$ over $F$ . In [5] Peter Schneider and Ulrich Stuhler associated to a smooth representation $V$ of $G$ (over the complex numbers) coefficient systems on the Bruhat-Tits building $BT=BT({\bf G})$ of ${\bf G}$ (and also defined sheaves associated to $V$ on $BT$ ). These coefficient systems were shown to furnish resolutions of $V$ . The purpose of this paper is to define analogous coefficient systems for locally analytic representations of ${\rm GL}_{2}(F)$ and to show that they too give rise to resolutions if the representation $V$ is a locally analytic principal series representation.

We quickly recall the coefficient system construction of Schneider and Stuhler for the general linear group, cf. [4].

For each vertex $v$ of $BT=BT({\rm GL}_{n,F})$ let $G_{v}(0):={\rm Stab}(v)\subset{\rm GL}_{n}(F)$ , and let $G_{v}(k)$ be the $k^{\rm th}$ congruence subgroup of $G_{v}(0)$ for some $k\geq 1$ . The coefficient system on $BT$ attached to a smooth representation $V$ of ${\rm GL}_{n}(F)$ is defined as follows

•

To each vertex $v\in BT$ we associate $V_{v}:=V^{G_{v}(k)}$ , the space of $G_{v}(k)$ -fixed vectors of $V$ . And to each simplex $\sigma:=\{v_{0},v_{1},\cdots,v_{d}\}$ we associate $V_{\sigma}=V^{G_{\sigma}(k)}$ where $G_{\sigma}(k):=\langle G_{v_{0}}(k),\cdots,G_{v_{d}}(k)\rangle$ , is the group generated by $G_{v_{i}}(k)$ ’s.
•

If $\sigma\subseteq\tau$ are two simplices, then $G_{\sigma}(k)\subset G_{\tau}(k)$ , and there is hence an inclusion $r_{\sigma}^{\tau}:V_{\tau}\rightarrow V_{\sigma}$ . These transition maps satisfy the conditions $r^{\sigma}_{\sigma}={\rm id}$ , and , $r_{\sigma}^{\tau}\circ r_{\tau}^{\xi}=r_{\sigma}^{\xi}$ for simplices $\sigma\subseteq\tau\subseteq\xi$ .

This coefficient system naturally gives rise to a chain-complex. One of the key results of [4, 5] is that this complex is exact.

In section 2 of this paper we associate an analogous coefficient system to a locally analytic representation $V$ of $G:={\rm GL}_{2}(F)$ , replacing $G_{v}(k)$ -fixed vectors by the rigid-analytic vectors $V_{{\mathbb{G}}_{v}(k)-{\rm an}}$ for the corresponding rigid analytic group ${\mathbb{G}}_{v}(k)$ and construct a chain complex [2.2.1] associated to it. In fact, the construction of the coefficient system and the associated chain-complex can be generalized to locally analytic representations of any (connected) $p$ -adic reductive group. It is natural to ask, when is this chain complex a resolution of $V$ ?

In section 3 we show that under some assumptions on $k$ ,

Theorem 1.1.1 (see 3.1.3).

For the locally analytic principal series $V:=Ind^{G}_{B}(\chi)$ , the chain-complex 2.2.1 is a resolution of $V$ .

Notation. We denote by $F/{{\mathbb{Q}}_{p}}$ a finite field extension of ${{\mathbb{Q}}_{p}}$ , by ${\mathcal{O}}$ its ring of integers, by ${\varpi}$ a uniformizer, and by ${{\mathbb{F}}_{q}}$ its residue field of cardinality $q$ . We will denote by $E/F$ a finite field extension (the ‘coefficient field’), and all locally analytic representations will be over $E$ . Throughout this paper $G$ will denote the group ${\rm GL}_{2}(F)$ . For a locally $F$ -analytic manifold $X$ , we denote by $C^{\rm la}(X,E)$ the the space of $E$ -valued locally $F$ -analytic functions on $X$ as defined in [6] (in the reference, the authors use $C^{\rm an}(X,E)$ for this space).

2. A coefficient system and complex

2.1. The Bruhat-Tits tree and associated subgroups

2.1.1.

The building. Recall that the semisimple Bruhat-Tits building $BT$ of ${\rm GL}_{2}$ over $F$ is a one-dimensional simplicial complex whose set of vertices $BT_{0}$ we can identify with the set of homothety classes of ${\mathcal{O}}$ -lattices $\Lambda\subset F^{2}$ . We write $[\Lambda]$ for the homothety class of $\Lambda$ . Two vertices $v$ and $v^{\prime}$ are adjacent (or form an edge) if and only if there are representing lattices $\Lambda$ of $v$ and $\Lambda^{\prime}$ of $v^{\prime}$ , such that ${\varpi}\Lambda\subsetneq\Lambda^{\prime}\subsetneq\Lambda$ . The set of edges of $BT$ will be denoted by $BT_{1}$ , and we write $e=\{v,v^{\prime}\}$ if the vertices $v$ and $v^{\prime}$ form an edge. We define the distance function $d:BT_{0}\times BT_{0}\rightarrow{\mathbb{Z}}_{\geq 0}$ as follows: $d(v,v^{\prime})=0$ , if $v=v^{\prime}$ , and $d(v,v^{\prime})=n$ , if there is a sequence of vertices $v=v_{0},v_{1},v_{2},\ldots,v_{n}=v^{\prime}$ such that $\{v_{i},v_{i+1}\}$ is an edge for all $i\in\{0,\ldots,n-1\}$ , and $v_{i+1}\neq v_{i-1}$ for $i\in\{1,\ldots,n-1\}$ . We further recall that $BT$ is a homogeneous tree of degree $q+1$ , in particular the distance function is well-defined.

An oriented edge is an ordered pair $(v,v^{\prime})$ of adjacent vertices. An orientation of $BT$ is a set $BT_{1}^{\rm or}\subset BT_{0}\times BT_{0}$ consisting of oriented edges, and such that the map $BT_{1}^{\rm or}\rightarrow BT_{1}$ , $(v,v^{\prime})\mapsto\{v,v^{\prime}\}$ , is bijective. For a vertex $v$ and an edge $e=\{v_{1},v_{2}\}$ , or an oriented edge $e=(v_{1},v_{2})$ , we set $d(e,v)=\max\{d(v_{1},v),d(v_{2},v)\}$ .

The group $G={\rm GL}_{2}(F)$ acts on $BT$ by $g.[\Lambda]=[g.\Lambda]$ , and by $g.\{v,v^{\prime}\}=\{g.v,g.v^{\prime}\}$ . The action of $G$ on $BT_{0}$ and $BT_{1}$ are well known to be transitive. In 3.4.7 we will need the following transitivity of $G$ -action.

Lemma 2.1.2.

Let $v,v^{\prime}$ and $w,w^{\prime}$ be two pairs of vertices with $d(v,v^{\prime})=d(w,w^{\prime})=n\geq 0$ , and let $v=v_{0},v_{1},v_{2},\ldots,v_{n}=v^{\prime}$ and $w=w_{0},w_{1},w_{2},\ldots,w_{n}=w^{\prime}$ be the unique paths connecting $v$ with $v^{\prime}$ and $w$ with $w^{\prime}$ , respectively. Then there exists $g\in G$ such that for all $i\in\{0,\ldots,n\}$ one has $g.v_{i}=w_{i}$ , i.e., the action of $G$ on paths of length $n\geq 0$ is transitive.

Proof.

The proof proceeds by induction on $n$ . The assertion is true in the case when $n\leq 1$ because of the transitivity of the $G$ -action on $BT_{0}$ and $BT_{1}$ , as mentioned above. We thus assume that $n\geq 2$ in the following. Furthermore, it is enough to prove the assertion under the assumption that $v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}]$ and $v_{i}=[({\varpi}^{i})\oplus{\mathcal{O}}]$ for $0\leq i\leq n$ . By induction, there this $h\in G$ such that $h.v_{i}=w_{i}$ for $0\leq i\leq n-1$ . Then $h^{-1}.w_{n}$ is adjacent to $v_{n-1}=h^{-1}.w_{n-1}$ and is different from $v_{n-2}$ . The vertices which have this property are of the form $v_{\alpha,n}:=[\langle({\varpi}^{n},0),([\alpha],1)\rangle]$ , the homothety class of ${\mathcal{O}}$ -lattice generated by $({\varpi}^{n},0)$ and $([\alpha],1)$ where $\alpha\in({\varpi})^{n-1}/({\varpi})^{n}$ . Then $h^{\prime}:=\begin{bmatrix}1&\alpha\\ 0&1\end{bmatrix}$ takes $v_{\alpha,n}$ to $v_{n}$ while fixing $v_{i}$ for $0\leq i\leq n-1$ . The element $g=h(h^{\prime})^{-1}$ has then the desired property. ∎

2.1.3.

The groups $G_{\sigma}(k)$ . Given a vertex $v=[\Lambda]$ we set $G_{v}(0)={\rm Stab}_{G}(\Lambda)=\{g\in G{\;|\;}g.\Lambda=\Lambda\}$ , and for a positive integer $k$ we put

G_{v}(k)=\{g\in G_{v}(0){\;|\;}\forall\,x\in\Lambda:\;g.x\equiv x\;{\rm mod}\;{\varpi}^{k}\Lambda\}

Given an edge $e=\{v,v^{\prime}\}$ (or an oriented edge $e=(v,v^{\prime})$ ) and $k\in{\mathbb{Z}}_{>0}$ we let $G_{e}(k)=\langle G_{v}(k),G_{v^{\prime}}(k)\rangle$ be the subgroup of $G$ generated by $G_{v}(k)$ and $G_{v^{\prime}}(k)$ . As $G$ acts transitively on $BT_{0}$ and $BT_{1}$ , $G$ acts transitively on $\{G_{v}(k){\;|\;}v\in BT_{0}\}$ as well as on $\{G_{e}(k){\;|\;}e\in BT_{1}\}$ by conjugation.

For later purposes it will be useful to describe some of the groups $G_{v}(k)$ and $G_{e}(k)$ explicitly.

Lemma 2.1.4.

Fix $k\in{\mathbb{Z}}_{>0}$ . If $v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}]$ , then

(2.1.5)

G_{v_{0}}(k)=\left\{\begin{bmatrix}1+{\varpi}^{k}a&{\varpi}^{k}b\\ {\varpi}^{k}c&1+{\varpi}^{k}d\end{bmatrix}\Big{|}\;a,b,c,d\in{\mathcal{O}}\right\}\;,

and if $e=\{v_{0},v_{1}\}$ with $v_{1}=[({\varpi})\oplus{\mathcal{O}}]$ , then

(2.1.6)

G_{e}(k)=\left\{\begin{bmatrix}1+{\varpi}^{k}a&{\varpi}^{k}b\\ {\varpi}^{k-1}c&1+{\varpi}^{k}d\end{bmatrix}\Big{|}\;a,b,c,d\in{\mathcal{O}}\right\}\;.

Proof.

The first assertion is well-known and straightforward to verify. For the second assertion we note that $v_{1}={\rm diag}({\varpi},1).v_{0}$ and therefore

G_{v_{1}}(k)={\rm diag}({\varpi},1)G_{v_{0}}(k){\rm diag}({\varpi},1)^{-1}=\left\{\begin{bmatrix}1+{\varpi}^{k}a&{\varpi}^{k+1}b\\ {\varpi}^{k-1}c&1+{\varpi}^{k}d\end{bmatrix}\Big{|}\;a,b,c,d\in{\mathcal{O}}\right\}\;,

By definition, $G_{e}(k)=\langle G_{v_{0}}(k),G_{v_{1}}(k)\rangle$ . Let $H$ be the group on the right hand side of 2.1.6. An easy computation shows that

(i) $G_{v_{0}}(k).G_{v_{1}}(k)=G_{v_{1}}(k).G_{v_{0}}(k)$ , and hence $G_{e}(k)=G_{v_{0}}(k).G_{v_{1}}(k)$ , and that

(ii) $G_{v_{0}}(k).G_{v_{1}}(k)\subset H$ .

Now we observe that for any $a,b,c,d\in{\mathcal{O}}_{F}$ we have

\begin{bmatrix}1+{\varpi}^{k}a&{\varpi}^{k}b\\ {\varpi}^{k-1}c&1+{\varpi}^{k}d\end{bmatrix}\;=\begin{bmatrix}1+{\varpi}^{k}a&0\\ {\varpi}^{k-1}c&1\end{bmatrix}.\begin{bmatrix}1&{\varpi}^{k}\frac{b}{1+{\varpi}^{k}a}\\ 0&1+{\varpi}^{k}\Big{(}d-{\varpi}^{k-1}\frac{cb}{1+{\varpi}^{k}a}\Big{)}\end{bmatrix}

This finishes the proof. ∎

2.1.7.

A Coefficient System on $BT$ . We choose $k\in{\mathbb{Z}}_{>0}$ such that for any simplex (edge or vertex) $\sigma$ of $BT$ the group $G_{\sigma}(k)$ as defined above is a uniform pro- $p$ group [1]. Under this assumption we can associate to $G_{\sigma}(k)$ a rigid analytic subgroup ${\mathbb{G}}_{\sigma}(k)$ of the analytification ${\rm GL}_{2}^{\rm rig}$ of the algebraic group ${\rm GL}_{2}$ over $F$ . See [3, 3.5] for a description of this process.

Given a locally analytic representation $V$ of $G$ , we define a coefficient system ${\mathcal{C}}^{(k)}_{V}$ on $BT$ as follows.

•

To each simplex $\sigma\subset BT$ we associate $V_{\sigma}:=V_{{\mathbb{G}}_{\sigma}(k)-{\rm an}}$ , the space of ${\mathbb{G}}_{\sigma}(k)$ -analytic vectors¹¹1See [2, 3.3.13] for the definition of analytic vectors of a representation with respect to a good analytic subgroup. of $V$ .
•

If $\sigma\subseteq\tau$ are two simplices, then $G_{\sigma}(k)\subset G_{\tau}(k)$ , and there is hence an inclusion $r_{\sigma}^{\tau}:V_{\tau}\rightarrow V_{\sigma}$ . These transition maps satisfy the conditions $r^{\sigma}_{\sigma}={\rm id}$ , and therefore, $r_{\sigma}^{\tau}\circ r_{\tau}^{\xi}=r_{\sigma}^{\xi}$ for simplices $\sigma\subseteq\tau\subseteq\xi$ .

2.2. The chain complex associated to a coefficient system on $BT$

We associate to ${\mathcal{C}}^{(k)}_{V}$ the following complex

(2.2.1)

0\rightarrow\bigoplus_{e\in BT_{1}^{\rm or}}V_{e}\stackrel{{\scriptstyle\partial_{1}}}{{\longrightarrow}}\bigoplus_{v\in BT_{0}}V_{v}\stackrel{{\scriptstyle\partial_{0}}}{{\longrightarrow}}V\rightarrow 0

Where there maps are defined as follows: given an oriented edge $e=(v_{1},v_{2})\in BT_{1}^{\rm or}$ the map $\partial_{1}|_{V_{e}}$ is defined by

\begin{array}[]{ccc}\partial_{1}|_{V_{e}}:V_{e}&\longrightarrow&\bigoplus_{v\in BT_{0}}V_{v}\\ f&\longmapsto&(f_{v})_{v}\;,\end{array}

where $f_{v_{1}}=f$ , $f_{v_{2}}=-f$ , and $f_{v}=0$ for all $v\not\in\{v_{1},v_{2}\}$ . The map $\partial_{0}$ is defined by

	$\displaystyle\partial_{0}:\bigoplus_{v\in BT_{0}}V_{v}$	$\displaystyle\longrightarrow V$
	$\displaystyle(f_{v})_{v}$	$\displaystyle\rightarrow\sum_{v}f_{v}\;,$

where the sum is taken inside $V$ using the vector space embeddings $V_{v}\hookrightarrow V$ .

3. Locally analytic principal series representations

3.1. Locally analytic induction

3.1.1.

Let $T\subset G$ be the the maximal torus comprising of diagonal matrices and B with, $T\subset B\subset G$ be the standard Borel subgroup of upper triangular matrices. We also fix once and for all a finite extension $E/F$ . For a locally $F$ -analytic character $\chi:T\rightarrow E^{\times}$ we consider the locally analytic principal series representation

\begin{array}[]{rcl}V&:=&{\rm Ind}^{G}_{B}(\chi)\\ &&\\ &=&\Big{\{}f:G\rightarrow E\mbox{ locally $F$-analytic}\Big{|}\,\forall\,g\in G\;\forall\;b\in B:\;f(bg)=\chi(b)f(g)\Big{\}}\;.\end{array}

The action of $G$ on $V$ is given by $g.f(x)=f(x.g^{-1})$ . We will consider ${\rm Ind}^{G}_{B}(\chi)$ as a topological vector space as follows. Set $G_{0}={\rm GL}_{2}({\mathcal{O}})$ and $B_{0}=B\cap G_{0}$ . Then $G=B\cdot G_{0}$ and the canonical map of quotients $B_{0}\backslash G_{0}\rightarrow B\backslash G$ is an isomorphism of locally $F$ -analytic manifolds. Therefore, restricting locally $F$ -analytic functions from $G$ to $G_{0}$ gives an isomorphism of vector spaces ${\rm Ind}^{G}_{B}(\chi)\rightarrow{\rm Ind}^{G_{0}}_{B_{0}}(\chi)$ . Since $G_{0}$ is compact, the space $C^{\rm la}(G_{0},E)$ of $E$ -valued locally $F$ -analytic functions on $G_{0}$ naturally carries the structure of an $E$ -vector space of compact type [6, Lemma 2.1]. We equip ${\rm Ind}^{G}_{B}(\chi)$ with the structure of a locally convex $E$ -vector space so that the map ${\rm Ind}^{G}_{B}(\chi)\rightarrow{\rm Ind}^{G_{0}}_{B_{0}}(\chi)$ becomes an isomorphism of topological vectors spaces.

3.1.2.

For a simplex $\sigma\in BT$ (or an oriented edge $e\in BT_{1}^{{\rm or}}$ ), let

\Omega_{\sigma,k}=B\backslash G/G_{\sigma}(k)

be the set of $B$ - $G_{\sigma}(k)$ double cosets in $G$ , which is finite because $B\backslash G$ is compact and $G_{\sigma}(k)$ open in $G$ . As $G_{\sigma}(k)$ is an open subgroup, so is any double coset ${\Delta}\in\Omega_{\sigma,k}$ . Given ${\Delta}\in\Omega_{\sigma,k}$ , we set

I({\Delta},\chi)=\Big{\{}{\Delta}\stackrel{{\scriptstyle f}}{{\longrightarrow}}E\;\Big{|}\;f\mbox{ is loc. $F$-analytic},\,\forall\;b\in B,x\in{\Delta},f(bx)=\chi(b)f(x)\Big{\}}

Note that by extending functions in $I({\Delta},\chi)$ by zero outside ${\Delta}$ , we obtain an embedding $I({\Delta},\chi)\hookrightarrow V$ , and the image of this map, which we will henceforth identify with $I({\Delta},\chi)$ , is stable under the action of $G_{\sigma}(k)$ on $V$ . Because $G=\coprod_{{\Delta}\in\Omega_{\sigma,k}}{\Delta}$ , we have

V_{{\mathbb{G}}_{\sigma}(k)-{\rm an}}=\oplus_{{\Delta}\in\Omega_{\sigma,k}}I({\Delta},\chi)_{{\mathbb{G}}_{\sigma}(k)-{\rm an}}\;.

The main result of this article is

Theorem 3.1.3.

There exists an integer $k_{0}\geq 1$ such that for all $k\geq k_{0}$ the chain complex 2.2.1 is a resolution when applied to $V={\rm Ind}^{G}_{B}(\chi)$ .

The rest of the article is dedicated to the proof of 3.1.3. A crucial role in our proof will be played by the following sets of vertices and edges:

\begin{array}[]{lclclc}BT_{0,n}&=&\Big{\{}v\in BT_{0}\;\Big{|}\;d(v,v_{0})\leq n\Big{\}}\;,\\ &&\\ BT^{\rm or}_{1,n}&=&\Big{\{}e=(v,v^{\prime})\in BT^{\rm or}_{1}\;\Big{|}\;d(e,v_{0})\leq n\Big{\}}\end{array}

where $v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}]$ is as above.

3.2. Injectivity of $\partial_{1}$

Proposition 3.2.1.

The map $\partial_{1}:\bigoplus_{e\in BT_{1}^{\rm or}}V_{e}\longrightarrow\bigoplus_{v\in BT_{0}}V_{v}$ is injective.

Proof.

Let $f=(f_{e})_{e}\neq 0$ be supported on $BT_{1,n}^{\rm or}$ but not on $BT_{1,n-1}^{\rm or}$ for some $n\geq 1$ ( $BT_{1,0}^{\rm or}$ is the empty set), i.e., there is $e=(v_{1},v_{2})\in BT^{\rm or}_{1}$ such that $d(e,v_{0})=n$ and $f_{e}\neq 0$ , and $f_{e^{\prime}}=0$ for all edges $e^{\prime}$ with $d(e^{\prime},v_{0})>n$ . If then $i\in\{1,2\}$ is such that $d(v_{i},v_{0})=n$ , we have $\partial_{1}(f)_{v_{i}}=\pm f_{e}\neq 0$ . ∎

3.3. The action of $G$ on ${\mathbb{P}}^{1}(F)$

In the rest of the paper we will let $\infty$ be a symbol different from all elements in $F$ and put ${\mathbb{P}}^{1}(F)=F\cup\{\infty\}$ . We equip ${\mathbb{P}}^{1}(F)$ with an action from the right by $G$ via Möbius transformations. Explicitly, the action of $g=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in G$ on $z\in{\mathbb{P}}^{1}(F)$ is given by

(3.3.1)

z.g=\left\{\begin{array}[]{ccl}\frac{az+c}{bz+d}&,&z\neq\infty\\ \frac{a}{b}&,&z=\infty\end{array}\right.

where $\frac{az+c}{bz+d}$ (resp. $\frac{a}{b}$ ) is $\infty$ if the denominator vanishes. The stabilizer of the point $0\in{\mathbb{P}}^{1}(F)$ is $B$ , and the map

\iota:B\backslash G\longrightarrow{\mathbb{P}}^{1}(F)\;,\;\;Bg\mapsto 0.g\;,

is bijective and $G$ -equivariant, if we consider the right translation action of $G$ on $B\backslash G$ . The quotient $B\backslash G$ inherits the structure of a locally $F$ -analytic manifold from $G$ , and we equip ${\mathbb{P}}^{1}(F)$ with the structure of a locally $F$ -analytic manifold via $\iota$ (so that $\iota$ becomes an isomorphism of locally $F$ -analytic manifolds). Each $g\in G$ acts as a locally $F$ -analytic automorphism on ${\mathbb{P}}^{1}(F)$ .

We denote by ${\rm P\Omega}_{\sigma,k}$ the set of $G_{\sigma}(k)$ -orbits on ${\mathbb{P}}^{1}(F)$ . By the discussion in the preceding paragraph, there is a canonical bijection between $B$ - $G_{\sigma}(k)$ double cosets of $G$ and orbits of (right) action of $G_{\sigma}(k)$ on ${\mathbb{P}}^{1}(F)$ , given by

\Omega_{\sigma,k}\stackrel{{\scriptstyle 1:1}}{{\longrightarrow}}{\rm P\Omega}_{\sigma,k}\;,\;{\Delta}\mapsto{\rm P{\Delta}}:=B\backslash{\Delta}\;.

For $z_{0}\in F\subset{\mathbb{P}}^{1}(F)$ and $r\in{\mathbb{R}}_{>0}$ we then set ${\bf B}_{z}(z_{0},r)=\{x\in F{\;|\;}|x-z_{0}|\leq r\}$ , which is a closed disc of radius $r$ around $z_{0}$ . Similarly, for $r\in{\mathbb{R}}_{>0}$ and $w_{0}\in{\mathbb{P}}^{1}(F)\setminus\{0\}$ we set ${\bf B}_{w}(w_{0},r)=\{x\in F^{\times}\cup\{\infty\}{\;|\;}\left|\frac{1}{x}-\frac{1}{w_{0}}\right|\leq r\}$ , where we interpret $\frac{1}{\infty}$ as zero. In particular, ${\mathbb{P}}^{1}(F)={\bf B}_{z}(0,1)\sqcup{\bf B}_{w}(\infty,|{\varpi}|)$ .

Proposition 3.3.2.

For $v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}],v_{1}=[({\varpi})\oplus{\mathcal{O}}]$ and $e_{0}=\{v,v_{1}\}$ , the orbits of $G_{v_{0}}(k),G_{v_{1}}(k)$ and $G_{e_{0}}(k)$ can be described as follows

(i) The orbits of $G_{v_{0}}(k)$ on ${\mathbb{P}}^{1}(F)$ are discs of radius $|{\varpi}|^{k}$ of the form ${\bf B}_{z}(z_{0},|{\varpi}|^{k})$ on ${\bf B}_{z}(0,1)$ with $z_{0}\in{\bf B}_{z}(0,1)$ and are of the form ${\bf B}_{w}(w_{0},|{\varpi}|^{k})$ on ${\bf B}_{w}(\infty,|{\varpi}|)$ with $w_{0}\in{\bf B}_{w}(\infty,|{\varpi}|)$ .

(ii)The orbits of $G_{v_{1}}(k)$ are discs of radius $|{\varpi}|^{k-1}$ of the form ${\bf B}_{z}(z_{0},|{\varpi}|^{k-1}$ on ${\bf B}_{z}(0,1)$ with $z_{0}\in{\bf B}_{z}(0,1)$ and discs of radius $|{\varpi}|^{k+1}$ of the form ${\bf B}_{w}(w_{0},|{\varpi}|^{k+1})$ on ${\bf B}_{w}(\infty,|{\varpi}|)$ with $w_{0}\in{\bf B}_{w}(\infty,|{\varpi}|)$ .

(iii) The orbits $G_{e_{0}}(k)$ -orbits on ${\bf B}_{z}(0,1)$ are of the form ${\bf B}_{z}(z_{0},|{\varpi}|^{k-1})$ (i.e., same as orbits of of $G_{v_{1}}(k)$ ) with $z_{0}\in{\bf B}_{z}(0,1)$ and on ${\bf B}_{w}(\infty,|{\varpi}|)$ they are of the form ${\bf B}_{w}(w_{0},|{\varpi}|^{k})$ (i.e., same as orbits of of $G_{v_{0}}(k)$ ) with $w_{0}\in{\bf B}_{w}(\infty,|{\varpi}|)$ .

Proof.

(i) and (ii) To compute the orbits of $G_{v_{0}}(k)$ on ${\bf B}_{z}(0,1)$ we use the description of $G_{v_{0}}(k)$ from 2.1.5 and the description of the action in 3.3.1 to get for $z_{o}\in{\bf B}_{z}(0,1)$

\displaystyle\frac{az_{0}+c}{bz_{0}+d}=(z_{0}+O({\varpi}^{k}))(1+O({\varpi}^{k}))=z_{0}+O({\varpi}^{k})

for $g=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in G_{v_{0}}(k)$ . This shows that the orbit of $z_{0}$ is contained in ${\bf B}_{z}(z_{0},|{\varpi}|^{k})$ . Via the translation $z_{0}\rightarrow z_{0}.\begin{bmatrix}1&0\\ c&1\end{bmatrix}=z_{0}+c$ , we conclude that the orbit is indeed ${\bf B}_{z}(z_{0},|{\varpi}|^{k})$ . A similar computation shows that the orbit of $w_{0}\in{\bf B}_{w}(\infty,|{\varpi}|)$ is ${\bf B}_{w}(w_{0},|{\varpi}|^{k})$ .

For $g=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in G_{v_{1}}(k)$ we have,

\displaystyle\frac{az_{0}+c}{bz_{0}+d}=(z_{0}+O({\varpi}^{k-1}))(1+O({\varpi}^{k}))=z_{0}+O({\varpi}^{k-1})

And again by using matrices of the form $\begin{bmatrix}1&0\\ c&1\end{bmatrix}$ we obtain that the orbits on ${\bf B}_{z}(0,1)$ are ${\bf B}_{z}(z_{0},|{\varpi}|^{k-1})$ . On the other hand, on ${\bf B}_{w}(\infty,|{\varpi}|)$ we have,

\displaystyle\frac{1}{w_{0}.g}=\frac{bw_{0}+d}{aw_{0}+c}=\frac{b+d\frac{1}{w_{0}}}{a+c\frac{1}{w_{0}}}=(\frac{1}{w_{0}}+O({\varpi}^{k+1}))(1+O({\varpi}^{k}))=\frac{1}{w_{0}}+O({\varpi}^{k+1})

thus the orbits on ${\bf B}_{w}(\infty,|{\varpi}|)$ of $G_{v_{1}}(k)$ are disjoint discs ${\bf B}_{w}(w_{0},|{\varpi}|^{k+1})$ and ${\bf B}_{w}(\infty,|{\varpi}|^{k+1})$ . This shows assertion (ii).

(iii) Using 2.1.6, we can show via an analogous computation that the orbits of $G_{e_{0}}(k)$ on ${\bf B}_{z}(0,1)$ are of the form ${\bf B}_{z}(z_{0},|{\varpi}|^{k-1})$ (i.e., orbit of of $G_{v_{1}}(k)$ ) and on ${\bf B}_{w}(\infty,|{\varpi}|)$ they are of the form ${\bf B}_{w}(w_{0},|{\varpi}|^{k}),w_{0}\in{\bf B}_{w}(\infty,|{\varpi}|)$ (i.e., orbits of $G_{v_{0}}(k)$ ). ∎

We can make a more general statement based on the previous proposition.

Proposition 3.3.3.

Let $v,v^{\prime}\in BT_{0}$ and $e=\{v,v^{\prime}\}\in BT_{1}$ . We have,

(i) $\Omega_{v,k}\cap\Omega_{v^{\prime},k}=\emptyset$ .

(ii) For every ${\Delta}\in\Omega_{v,k}$ there is ${\Delta}^{\prime}\in\Omega_{v^{\prime},k}$ such that ${\Delta}\subset{\Delta}^{\prime}$ or ${\Delta}^{\prime}\subset{\Delta}$ . Furthermore, $\Omega_{v,k}^{v^{\prime}}:=\{{\Delta}\in\Omega_{v,k}{\;|\;}\exists{\Delta}^{\prime}\in\Omega_{v^{\prime},k}:{\Delta}^{\prime}\subset{\Delta}\}$ has cardinality $q^{k-1}$ and each ${\Delta}\in\Omega_{v,k}^{v^{\prime}}$ contains $q$ orbits of $\Omega_{v^{\prime},k}$ .

(iii) If ${\Delta}\in\Omega_{e,k}$ then ${\Delta}\in\Omega_{v,k}$ or ${\Delta}\in\Omega_{v^{\prime},k}$ .

(iv) If ${\Delta}\in\Omega_{v,k}\cap\Omega_{e,k}$ then there exists ${\Delta}^{\prime}\in\Omega_{v^{\prime},k}$ such that ${\Delta}^{\prime}\subset{\Delta}$ and similarly, if ${\Delta}^{\prime}\in\Omega_{v^{\prime},k}\cap\Omega_{e,k}$ then there exists ${\Delta}\in\Omega_{v,k}$ such that ${\Delta}\subset{\Delta}^{\prime}$ .

(v) If ${\Delta}^{\prime}\subset{\Delta}$ for double-cosets in $\Omega_{v^{\prime},k}$ and $\Omega_{v,k}$ , respectively, then ${\Delta}\in\Omega_{v,k}\cap\Omega_{e,k}.$

Proof.

Via the canonical bijection ${\Delta}\leftrightarrow{\rm P{\Delta}}$ it is enough to show the corresponding assertions for orbits on ${\mathbb{P}}^{1}(F)$ .

Since the action of $G$ on $BT_{0}$ and $BT_{1}$ is transitive it is enough to prove the statements for $v_{0},v_{1}$ and $e_{0}=\{v_{0},v_{1}\}$ , where $v_{0}$ and $v_{1}$ are as in 3.3.2.

(i), (ii) and (iii) Follows directly from 3.3.2.

(iv) If ${\rm P{\Delta}}={\bf B}_{w}(w_{0},|{\varpi}|^{k})\subset{\bf B}_{w}(\infty,|{\varpi}|)\in{\rm P\Omega}_{v_{0},k}\cap{\rm P\Omega}_{e_{0},k}$ then we can take ${\rm P{\Delta}}^{\prime}={\bf B}_{w}(w_{0},|{\varpi}|^{k+1})\in\Omega_{v_{1},k}$ . And if ${\rm P{\Delta}}={\bf B}_{z}(z_{0},|{\varpi}|^{k-1})\subset{\bf B}_{z}(0,1)\in{\rm P\Omega}_{v_{1},k}\cap{\rm P\Omega}_{e_{0},k}$ then we can take ${\rm P{\Delta}}^{\prime}={\bf B}_{z}(z_{0},|{\varpi}|^{k})$ .

(v) The only orbits for which the relation ${\rm P{\Delta}}^{\prime}\subset{\rm P{\Delta}}$ with ${\rm P{\Delta}}^{\prime}\in{\rm P\Omega}_{v_{1},k},{\rm P{\Delta}}\in{\rm P\Omega}_{v_{0},k}$ holds are of the form ${\bf B}_{w}(w_{0},|{\varpi}|^{k+1})\subset{\bf B}_{w}(w_{0},|{\varpi}|^{k})$ . And any such ${\bf B}_{w}(w_{0},|{\varpi}|^{k})\in{\rm P\Omega}_{v_{0},k}$ also belongs to ${\rm P\Omega}_{e_{0},k}$ . ∎

Remark 3.3.4.

Let $\sigma^{\prime},\sigma$ be two simplices in $BT$ with $\sigma^{\prime}=g.\sigma$ for $g\in G$ . The map

\begin{array}[]{cccccc}\Omega_{\sigma,k}&\longrightarrow&\Omega_{\sigma^{\prime},k},\;\;\;\;\;\;{\rm P\Omega}_{\sigma,k}&\longrightarrow&{\rm P\Omega}_{\sigma^{\prime},k}\\ {\Delta}&\longrightarrow&{\Delta}.g^{-1},\;\;\;\;{\rm P{\Delta}}&\longrightarrow&{\rm P{\Delta}}.g^{-1}\end{array}

defines a bijection between $\Omega_{\sigma,k}$ and $\Omega_{\sigma^{\prime},k}$ and between ${\rm P\Omega}_{\sigma,k}$ and ${\rm P\Omega}_{\sigma^{\prime},k}$ respectively.

Lemma 3.3.5.

Let $v_{\alpha,1}:=[\langle(1,\alpha),(0,{\varpi})\rangle]$ , $\alpha\in{\mathcal{O}}/({\varpi})$ , be the vertex corresponding to the homothety class of the ${\mathcal{O}}$ -lattice $\langle(1,[\alpha]),(0,{\varpi})\rangle\subset F^{\oplus 2}$ . Then the orbits of $G_{v_{\alpha,1}}(k)$ on ${\bf B}_{z}([\alpha],|{\varpi}|)\subset{\bf B}_{z}(0,1)$ are of the form ${\bf B}_{z}(\beta,|{\varpi}|^{k+1})$ , for $\beta\in{\bf B}_{z}([\alpha],|{\varpi}|)$ . Moreover each of the orbits of $G_{v_{\alpha,1},k}$ of the form ${\bf B}_{z}(\beta,|{\varpi}|^{k+1})$ on ${\bf B}_{z}([\alpha],|{\varpi}|)$ is contained in an orbit of the form ${\bf B}_{z}(z_{0},|{\varpi}|^{k})$ of $G_{v_{0},k}$ .

Each of the other orbits of $G_{v_{\alpha,1}}(k)$ contains $q$ orbits of $G_{v_{0}}(k)$ .

Proof.

Note that $v_{\alpha,1}=g_{\alpha}.v_{0}$ , where $g_{\alpha}=\begin{bmatrix}1&0\\ [\alpha]&{\varpi}\end{bmatrix}$ and $v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}]$ as in 3.3.3. Thus by 3.3.4 we have ${\rm P\Omega}_{v_{\alpha,1},k}={\rm P\Omega}_{v_{0},k}.g_{\alpha}^{-1}$ . Now

\beta.g_{\alpha}^{-1}=\beta.\begin{bmatrix}1&0\\ -\frac{[\alpha]}{{\varpi}}&{\varpi}^{-1}\end{bmatrix}=\frac{\beta-\frac{[\alpha]}{{\varpi}}}{{\varpi}^{-1}}={\varpi}.\beta-[\alpha].

This transformation takes ${\bf B}_{z}(0,|{\varpi}|^{k}).g_{\alpha}^{-1}={\bf B}_{z}([\alpha],|{\varpi}|^{k+1})$ and very similarly we can compute the other orbits. The containment relation is clear from the radius of the respective orbits. ∎

3.3.6.

We also introduce the following notation for the rest of the paper

\begin{array}[]{lclclclclclc}\Omega_{0,k}&=&\bigsqcup_{v\in BT_{0}}\Omega_{v,k}\;,&\;{\rm P\Omega}_{0,k}&=&\bigsqcup_{v\in BT_{0}}{\rm P\Omega}_{v,k}\\ &&\\ \Omega_{0,k,n}&=&\bigsqcup_{v\in BT_{0,n}}\Omega_{v,k}\;,&\;{\rm P\Omega}_{0,k,n}&=&\bigsqcup_{v\in BT_{0,n}}{\rm P\Omega}_{v,k}\\ &&\\ \Omega_{1,k}&=&\bigsqcup_{e\in BT_{1}^{{\rm or}}}\Omega_{e,k}\;,&\;{\rm P\Omega}_{1,k}&=&\bigsqcup_{e\in BT_{1}^{{\rm or}}}{\rm P\Omega}_{e,k}\\ &&\\ \Omega_{1,k,n}&=&\bigsqcup_{e\in BT^{{\rm or}}_{1,n}}\Omega_{e,k}\;,&\;{\rm P\Omega}_{1,k,n}&=&\bigsqcup_{e\in BT^{{\rm or}}_{1,n}}{\rm P\Omega}_{e,k}\\ &&\\ {\overline{\Omega}}_{0,k}&=&\bigcup_{v\in BT_{0}}\Omega_{v,k}\;,&\;{\overline{{\rm P\Omega}}}_{0,k}&=&\bigcup_{v\in BT_{0}}{\rm P\Omega}_{v,k}\\ &&\\ {\overline{\Omega}}_{0,k,n}&=&\bigcup_{v\in BT_{0,n}}\Omega_{v,k}\;,&\;{\overline{{\rm P\Omega}}}_{0,k,n}&=&\bigcup_{v\in BT_{0,n}}{\rm P\Omega}_{v,k}\\ &&\\ {\overline{\Omega}}_{1,k}&=&\bigcup_{e\in BT_{1}^{{\rm or}}}\Omega_{e,k}\;,&\;{\overline{{\rm P\Omega}}}_{1,k}&=&\bigcup_{e\in BT_{1}^{{\rm or}}}{\rm P\Omega}_{e,k}\\ &&\\ {\overline{\Omega}}_{1,k,n}&=&\bigcup_{e\in BT^{{\rm or}}_{1,n}}\Omega_{e,k}\;,&\;{\overline{{\rm P\Omega}}}_{1,k,n}&=&\bigcup_{e\in BT^{{\rm or}}_{1,n}}{\rm P\Omega}_{e,k}\end{array}

Remark 3.3.7.

Note that, (iii) of 3.3.3 says, If ${\Delta}\in\Omega_{e,k}$ for $e=\{v,v^{\prime}\}$ then ${\Delta}\in\Omega_{v,k}$ or ${\Delta}\in\Omega_{v^{\prime},k}$ . From this it is clear that we can think of $\Omega_{1,k}$ as a subset of $\Omega_{0,k}$ . And we can similarly conclude ${\rm P\Omega}_{1,k}\subset{\rm P\Omega}_{0,k},{\overline{\Omega}}_{1,k}\subset{\overline{\Omega}}_{1,k}$ and ${\overline{{\rm P\Omega}}}_{1,k}\subset{\overline{{\rm P\Omega}}}_{0,k}$ .

We equip ${\overline{\Omega}}_{0,k}$ (and ${\overline{{\rm P\Omega}}}_{0,k}$ ) with the partial order via inclusion.

Remark 3.3.8.

Note that $\Omega_{0,k,n}$ and ${\overline{\Omega}}_{0,k,n}$ (and by extension $\Omega_{0,k}$ and ${\overline{\Omega}}_{0,k}$ ) are not the same. For example, by 3.3.5, ${\bf B}_{w}(\infty,|{\varpi}|)$ is in both $\Omega_{v_{1,1},2}$ and $\Omega_{v_{0,1},2}$ , where $v_{0,1}$ and $v_{1,1}$ are as in 3.3.5. There is an obvious map ${\rm pr}:\Omega_{0,k}\rightarrow{\overline{\Omega}}_{0,k}$ which restricts to a map from ${\rm pr}:\Omega_{0,k,n}\rightarrow{\overline{\Omega}}_{0,k,n}$ , taking every orbit in ${\overline{\Omega}}_{0,k}$ that are same as sets to the same set in ${\overline{\Omega}}_{0,k}$ . We will use the same notation for the map ${\rm pr}:{\rm P\Omega}_{0,k}\rightarrow{\overline{{\rm P\Omega}}}_{0,k}$ .

For sake of clarity, from now on let $\overline{{\Delta}}={\rm pr}({\Delta})$ and $\overline{{\rm P{\Delta}}}={\rm pr}({\rm P{\Delta}})$ . When we want to emphasize that the double coset ${\Delta}\in\Omega_{v,k}$ (resp. the orbit ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ ) is just considered as a subset of $G$ (resp. a subset of ${\mathbb{P}}^{1}(F)$ ), then we write $\overline{{\Delta}}$ (resp. $\overline{{\rm P{\Delta}}}$ ) for it.

Proposition 3.3.9.

For $G_{v}(k)$ -orbits on ${\mathbb{P}}^{1}(F)$ we have the following:

(i) Given any $z\in{\mathbb{P}}^{1}(F)$ and $k\in{\mathbb{Z}}_{>0}$ , the set

\{\overline{{\rm P{\Delta}}}{\;|\;}v\in BT_{0}\,,\;{\rm P{\Delta}}\in{\rm P\Omega}_{v,k}\,,\;z\in\overline{{\rm P{\Delta}}}\}

is a fundamental system of open compact neighborhoods of $z$ .

(ii) Given any covering ${\mathcal{U}}$ of ${\mathbb{P}}^{1}(F)$ there is a finite subset of ${\overline{{\rm P\Omega}}}_{0,k}$ that refines ${\mathcal{U}}$ .

Proof.

(i) Without loss of generality we may assume that $z=0$ . Consider the vertices $v_{n}:=[{\mathcal{O}}\oplus({\varpi}^{n})],n\geq 1$ . We have $g_{n}.v_{0}=v_{n}$ , where $g_{n}=\begin{bmatrix}1&0\\ 0&{\varpi}^{n}\end{bmatrix}$ . Let ${\rm P{\Delta}}_{0}:={\bf B}_{z}(0,|{\varpi}|^{k})\in{\rm P\Omega}_{v_{0},k}$ , then we can compute that ${\rm P{\Delta}}_{n}:={\rm P{\Delta}}_{0}.g_{n}^{-1}={\bf B}_{z}(0,|{\varpi}|^{n+k})\in{\rm P\Omega}_{v_{n},k}$ . And, $\{\overline{{\rm P{\Delta}}}_{n}\}_{n\geq 1}\subset\{\overline{{\rm P{\Delta}}}\in{\rm P\Omega}_{v,k}{\;|\;}v\in BT_{0},z\in\overline{{\rm P{\Delta}}}\}$ forms a fundamental neighborhood around $z=0$ of compact open subsets of ${\mathbb{P}}^{1}(F)$ .

(ii) This is an immediate corollary of (i). ∎

3.4. Containment relations between the orbits

3.4.1.

Another description of ${\rm Ind}^{G}_{B}(\chi)$ . We define an locally $F$ -analytic embedding $s:{\mathbb{P}}^{1}(F)\rightarrow G$ by

s(z)=\left(\begin{array}[]{cc}1&0\\ z&1\end{array}\right),\;\mbox{if }|z|\leq 1\,,\;s(z)=\left(\begin{array}[]{cc}0&-1\\ 1&\frac{1}{z}\end{array}\right),\;\mbox{if }|z|>1\;.

Then, for every $z\in{\mathbb{P}}^{1}(F)$ and $g\in G$ , one has $\xi(z,g):=s(z)gs(z.g)^{-1}\in B$ and

(3.4.2)

g=\xi(0,g)s(0.g)\,,\mbox{ and }\xi(z,gg^{\prime})=\xi(z,g)\xi(z.g,g^{\prime})\;.

Lemma 3.4.3.

The map

\zeta:{\rm Ind}^{G}_{B}(\chi)\longrightarrow C^{\rm la}({\mathbb{P}}^{1}(F),E),\;\zeta(f)(z)=f(s(z))\;

is an isomorphism of topological vector spaces.

Proof.

Because $s$ and $f$ are locally $F$ -analytic, so is $\zeta(f)$ . We define another map $\tilde{\zeta}:C^{\rm la}({\mathbb{P}}^{1}(F),E))\rightarrow{\rm Ind}^{G}_{B}(\chi)$ by $\tilde{\zeta}(f_{1})(g)=\chi(\xi(0,g))f_{1}(s(0.g))$ . We leave it to the reader to check that these maps are continuous when ${\rm Ind}^{G}_{B}(\chi)$ is equipped with the structure of a compact inductive limit, as explained in 3.1.1, and $C^{\rm la}({\mathbb{P}}^{1}(F),E)$ is also considered as a vector space of compact type, cf. [6, Lemma 2.1]. And it is easy to see that $\zeta$ and $\tilde{\zeta}$ are inverses of each other. ∎

Using $\zeta$ we equip $C^{\rm la}({\mathbb{P}}^{1}(F),E)$ with a $G$ -action. Explicitly, on $f\in C^{\rm la}({\mathbb{P}}^{1}(F),E)$ , the group action is given by

(3.4.4)

(g._{\chi}f)(z)=\chi(\xi(z,g))f(z.g)\;.

If we write $\chi({\rm diag}(a,d))=\chi_{1}(ad)\chi_{2}(d)$ , and if $g=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)$ , then 3.4.4 becomes

(3.4.5)

(g._{\chi}f)(z)=\chi_{1}(ad-bc)\chi_{2}(bz+d)f\Big{(}\frac{az+c}{bz+d}\Big{)}\;.

Lemma 3.4.6.

Let $v$ be a vertex in $BT$ , then for every orbit ${\rm P{\Delta}}$ of $G_{v}(k)$ there is an edge $e=\{v,v^{\prime}\}$ and an orbit ${\rm P{\Delta}}^{\prime}$ of $G_{v^{\prime}}(k)$ which is contained properly in ${\Delta}$ .

Proof.

Without loss of generality we can assume that $v=v_{0}:=[{\mathcal{O}}\oplus{\mathcal{O}}]$ . Then let ${\rm P{\Delta}}={\bf B}_{z}(z_{0},|{\varpi}|^{k})\in{\rm P\Omega}_{v_{0},k}$ , if $z_{0}\in{\bf B}_{z}([\alpha],|{\varpi}|)$ for some $\alpha\in{\bf F}_{q}$ , then $G_{v_{\alpha,1}}(k)$ has as an orbit ${\bf B}_{z}(z_{0},|{\varpi}|^{k+1})$ where $v_{\alpha,1}$ is as in 3.3.5. If ${\rm P{\Delta}}={\bf B}_{w}(w_{0},|{\varpi}|^{k})\subset{\bf B}(\infty,|{\varpi}|)$ then by 3.3.3 we have ${\rm P{\Delta}}^{\prime}={\bf B}_{w}(w_{0},|{\varpi}|^{k+1})$ as an orbit of $G_{v_{1}}(k)$ .

∎

Lemma 3.4.7.

We have the following relations between $G_{v}(k)$ -orbits for varying $v\in BT_{0}$ .

(i) Let $\{v,v^{\prime}\}\in BT_{1}$ be an edge. Suppose there are ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ and ${\rm P{\Delta}}^{\prime}\in{\rm P\Omega}_{v^{\prime},k}$ such that $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}^{\prime}$ . Then for any edge $\{v^{\prime\prime},v^{\prime}\}\in BT_{1}$ with $d(v,v^{\prime\prime})=2$ there is ${\rm P{\Delta}}^{\prime\prime}\in{\rm P\Omega}_{v^{\prime\prime},k}$ with $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ .

(ii) Let $\{v^{\prime},v^{\prime\prime}\}\in BT_{1}$ be an edge. Suppose there are ${\rm P{\Delta}}^{\prime}\in{\rm P\Omega}_{v^{\prime},k}$ and ${\rm P{\Delta}}^{\prime\prime}\in{\rm P\Omega}_{v^{\prime\prime},k}$ such that $\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ . Then there exists some $\{v,v^{\prime}\}\in BT_{1}$ with $d(v,v^{\prime\prime})=2$ and ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ such that $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ .

(iii) Let $\{v^{\prime},v^{\prime\prime}\}\in BT_{1}$ be an edge. Suppose that for some ${\rm P{\Delta}}^{\prime}\in{\rm P\Omega}_{v^{\prime},k}$ and ${\rm P{\Delta}}^{\prime\prime}\in{\rm P\Omega}_{v^{\prime\prime},k}$ we have $\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ . Then

\overline{{\rm P{\Delta}}}^{\prime}=\bigcup_{\begin{subarray}{c}\{v^{\prime},v\}\in BT_{1},v\neq v^{\prime\prime}\\ {\rm P{\Delta}}\in{\rm P\Omega}_{v,k}\\ \overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}^{\prime}\end{subarray}}\overline{{\rm P{\Delta}}}

Proof.

(i) By 2.1.2 we can assume that $v^{\prime\prime}=v_{0},v^{\prime}=v_{1}$ and $v=v_{2}$ where $v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}]$ and $v_{1}=[({\varpi})\oplus{\mathcal{O}}]$ as before and $v_{2}$ is defined as $v_{2}:=[({\varpi}^{2})\oplus{\mathcal{O}}]$ .

We have already shown that the orbits of $G_{v_{0}}(k)$ on ${\mathbb{P}}^{1}(F)$ are discs of radius $|{\varpi}|^{k}$ . On the other hand, the orbits of $G_{v_{1}}(k)$ are discs of radius $|{\varpi}|^{k-1}$ on ${\bf B}_{z}(0,1)$ and discs of radius $|{\varpi}|^{k+1}$ on ${\bf B}_{w}(\infty,|{\varpi}|)$ . Similarly, orbits of $G_{v_{2}}(k)$ on ${\bf B}_{z}(0,|{\varpi}|^{-1})$ are discs of radius $|{\varpi}|^{k-2}$ and on ${\bf B}_{w}(\infty,|{\varpi}|^{2})$ they are discs of radius $|{\varpi}|^{k+2}$ .

Therefore, if for ${\rm P{\Delta}}\in{\rm P\Omega}_{v_{2},k},\overline{{\rm P{\Delta}}}$ is contained in $\overline{{\rm P{\Delta}}}^{\prime}$ for ${\rm P{\Delta}}^{\prime}\in{\rm P\Omega}_{v_{1},k}$ , then $\overline{{\rm P{\Delta}}}$ is a disc of radius $|{\varpi}|^{k+2}$ and $\overline{{\rm P{\Delta}}}^{\prime}$ is a disc of $|{\varpi}|^{k+1}$ containing it. But any such disc of radius $|{\varpi}|^{k+1}$ is contained in a disc of radius $|{\varpi}|^{k}$ which is an orbit of $G_{v_{0}}(k)$ . This proves our claim.

(ii) For computational ease we pick $v^{\prime\prime}=v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}]$ and $v^{\prime}=v_{1}=[({\varpi})\oplus{\mathcal{O}}]$ . From 3.3.2 we see that if $\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ with ${\rm P{\Delta}}^{\prime}\in{\rm P\Omega}_{v^{\prime},k}$ and ${\rm P{\Delta}}^{\prime\prime}\in{\rm P\Omega}_{v^{\prime\prime},k}$ then $\overline{{\rm P{\Delta}}}^{\prime}$ is a disc of radius $|{\varpi}|^{k+1}$ inside ${\bf B}_{w}(\infty,|{\varpi}|)$ . Let $a\in F$ belong to $\overline{{\rm P{\Delta}}}^{\prime}$ , then choosing $v=[({\varpi}^{2},0),(a,1)]$ and noting $v=\begin{bmatrix}{\varpi}^{2}&a\\ 0&1\end{bmatrix}.v_{0}$ , we can see that $\begin{bmatrix}{\varpi}^{2}&a\\ 0&1\end{bmatrix}^{-1}$ transforms ${\bf B}_{w}(\infty,|{\varpi}|^{k})$ to ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ , a disc of radius $|{\varpi}|^{k+2}$ such that $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ . (iii) This is an extension of calculations of (ii), and it can be seen that the any disc of radius $|{\varpi}|^{k+1}$ that is an orbit of $G_{v_{1}}(k)$ is covered by a subset of discs of radius $|{\varpi}|^{k+2}$ which are orbits of various $G_{v}(K)$ with $\{v_{1},v\}\in BT_{1,2}$ .

∎

3.4.8.

Minimal Orbits. Recall that we partially order ${\overline{\Omega}}_{0,k,n}$ and ${\overline{{\rm P\Omega}}}_{0,k,n}$ via inclusion. Then we define

{\overline{\Omega}}_{0,k,n}^{\min}:=\big{\{}\overline{{\Delta}}\in{\overline{\Omega}}_{0,k,n}\big{|}\;\overline{{\Delta}}\;\mbox{ minimal w.r.t. the partial ordering of }{\overline{\Omega}}_{0,k,n}\big{\}}

and

{\overline{{\rm P\Omega}}}_{0,k,n}^{\min}:=\big{\{}\overline{{\rm P{\Delta}}}\in{\overline{{\rm P\Omega}}}_{0,k,n}\big{|}\;\overline{{\rm P{\Delta}}}\;\mbox{ minimal w.r.t. the partial ordering of }{\overline{{\rm P\Omega}}}_{0,k,n}\big{\}}\;.

Lemma 3.4.9.

Fix ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ with $v\in BT_{0,n}$ , let $n\geq 1$ .

(a) $\overline{{\rm P{\Delta}}}\in{\overline{{\rm P\Omega}}}_{0,k,n}$ belongs to ${\overline{{\rm P\Omega}}}_{0,k,n}^{\min}$ iff both of the following conditions hold

(i) $d(v,v_{0})=n$

(ii) Let $v_{1}$ be the unique vertex of $BT$ such that $\{v,v_{1}\}\in BT_{1,n}$ . Then there exists ${\rm P{\Delta}}_{1}\in{\rm P\Omega}_{v_{1},k}$ such $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}_{1}$ .

(b) Condition (ii) above is equivalent to the condition

(ii)’ For every vertex $v^{\prime}\in BT_{0,n}$ if $v=v_{1},\cdots,v_{m}=v^{\prime}$ is the path connecting $v$ and $v^{\prime}$ there exists ${\rm P{\Delta}}_{i}\in{\rm P\Omega}_{v_{i},k}$ such that $\overline{{\rm P{\Delta}}}=\overline{{\rm P{\Delta}}}_{1}\subset\cdots\subset\overline{{\rm P{\Delta}}}_{m}$ .

Proof.

(b) To see the equivalence of (ii)’ and (ii) we first note that (ii)’ definitely implies (ii). Now assume (ii) holds, i.e., there are orbits ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ and ${\rm P{\Delta}}_{1}\in{\rm P\Omega}_{v_{1},k}$ such that $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}_{1}$ , where $v_{1}$ is the unique neighboring vertex of $v$ in $BT_{0,n}$ . For any vertex $v^{\prime}\in BT_{0,n}$ let the path connecting $v,v^{\prime}$ be $v,v_{1},\cdots,v_{m}=v^{\prime}$ . Then by 3.4.7 we can find an orbit ${\rm P{\Delta}}_{2}$ in ${\rm P\Omega}_{v_{2},k}$ such that $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}_{1}\subset\overline{{\rm P{\Delta}}}_{2}$ . Continuing .this process we get a chain of nested orbits as in (ii)’

(a) We give a proof of (i) and (ii)’ below.

First, suppose $\overline{{\rm P{\Delta}}}$ is minimal in ${\overline{{\rm P\Omega}}}_{0,k,n}$ . Then, $d(v_{0},v)=n$ , because otherwise all adjacent vertices $v^{\prime}$ with $d(v,v^{\prime})=1$ are also in $BT_{0,n}$ , and hence, by 3.4.6, any orbit of $G_{v}(k)$ properly contains an orbit of $G_{v^{\prime}}(k)$ for some such $v^{\prime}$ . And thus is not minimal in $\Omega_{0,k,n}$ . We thus have $d(v_{0},v)=n$ .

Let $v_{2}$ be the unique vertex adjacent to $v=v_{1}$ and contained in $BT_{0,n}$ . Then every orbit of $G_{v}(k)$ is contained in an orbit of $G_{v_{2}}(k)$ , or contains an orbit of $G_{v_{2}}(k)$ . Since $\overline{{\rm P{\Delta}}}=\overline{{\rm P{\Delta}}}_{1}$ is minimal, $\overline{{\rm P{\Delta}}}$ is contained in $\overline{{\rm P{\Delta}}}_{2}$ for an orbit $\overline{{\rm P{\Delta}}}_{2}$ of $G_{v_{2}}(k)$ .

Now let $v^{\prime}\neq v$ be any vertex in $BT_{0,n}$ (the assertion is trivial for $v^{\prime}=v$ ). Let $v=v_{1},v_{2},...,v_{m}=v^{\prime}$ be the path from $v$ to $v^{\prime}$ . As we have seen, $\overline{{\rm P{\Delta}}}_{1}$ is contained in $\overline{{\rm P{\Delta}}}_{2}$ for an orbit ${\rm P{\Delta}}_{2}$ of $v_{2}$ . By 3.4.7 we then find an orbit ${\rm P{\Delta}}_{3}$ of $G_{v_{3}}(k)$ such that $\overline{{\rm P{\Delta}}}_{2}\subset\overline{{\rm P{\Delta}}}_{3}$ , and repeatedly applying 3.4.7 in the same way we conclude that (ii)’ holds true.

Conversely, assume that ${\rm P{\Delta}}$ satisfies assumption (ii)’. Suppose $\overline{{\rm P{\Delta}}}$ properly contains $\overline{{\rm P{\Delta}}}^{\prime}$ for an orbit ${\rm P{\Delta}}^{\prime}$ of some $G_{v^{\prime}}(k)$ with $v^{\prime}$ in $BT_{0,n}$ . Let $v=v_{1},...,v_{m}=v^{\prime}$ and $\overline{{\rm P{\Delta}}}=\overline{{\rm P{\Delta}}}_{1}\subset\overline{{\rm P{\Delta}}}_{2}\subset...\subset\overline{{\rm P{\Delta}}}_{m}$ be as in (ii). Then $\overline{{\rm P{\Delta}}}^{\prime}\subsetneq\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}_{m}$ is a contradiction, because ${\rm P{\Delta}}^{\prime}$ and ${\rm P{\Delta}}_{m}$ are both orbits of $G_{v^{\prime}}(k)$ . Hence ${\rm P{\Delta}}$ is minimal.

∎

Proposition 3.4.10.

If $v,v^{\prime}\in BT_{0,n}$ , and if ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ and ${\rm P{\Delta}}^{\prime}\in{\rm P}\Omega_{v^{\prime},k}$ are such that their images in $\overline{{\rm P}\Omega}_{0,k,n}$ are minimal and (whose underlying sets) are equal, then $v=v^{\prime}$ .

Proof.

Let ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}$ and ${\rm P{\Delta}}^{\prime}\in{\rm P}\Omega_{v^{\prime},k}$ belong to $\overline{{\rm P}\Omega}_{0,k,n}$ and assume $\overline{{\rm P{\Delta}}}=\overline{{\rm P{\Delta}}}^{\prime}$ . Let $v=v_{1},v_{2},...,v_{m}=v^{\prime}$ be the path from $v$ to $v^{\prime}$ . By 3.4.9 there exists ${\rm P{\Delta}}_{i}\in{\rm P\Omega}_{v_{i},k}$ such that $\overline{{\rm P{\Delta}}}=\overline{{\rm P{\Delta}}}_{1}\subset\cdots\subset\overline{{\rm P{\Delta}}}_{m}$ . Note that $\overline{{\rm P{\Delta}}}_{1}\neq\overline{{\rm P{\Delta}}}$ since by 3.3.3 ${\rm P\Omega}_{v,k}\cap{\rm P\Omega}_{v_{1},k}=\emptyset$ . Hence, in particular $\overline{{\rm P{\Delta}}}^{\prime}\subsetneq\overline{{\rm P{\Delta}}}_{m}\in{\rm P\Omega}_{v^{\prime},k}$ . Which is a contradiction. ∎

Lemma 3.4.11.

${\overline{{\rm P\Omega}}}_{0,k,n}^{\min}$ forms a disjoint covering of ${\mathbb{P}}^{1}(F)$ for all $n\geq 1$ .

Proof.

This is evidently true for $n=0$ . Assuming it is true for $n-1$ , $n\geq 1,$ observe that it is enough to prove ${\overline{{\rm P\Omega}}}_{0,k,n}^{\min}$ forms a covering of ${\overline{{\rm P\Omega}}}_{0,k,n-1}^{\min}$ . Let $\overline{{\rm P{\Delta}}}^{\prime}\in{\overline{{\rm P\Omega}}}_{0,k,n-1}^{\min}$ then there exists a ${\rm P{\Delta}}^{\prime\prime}\in{\rm P\Omega}_{v^{\prime\prime},k}$ with $\{v^{\prime},v^{\prime\prime}\}\in BT_{1}$ such that $\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ (this is true for $n=1$ by 3.3.3 and 3.3.5). Then by 3.4.7 $\overline{{\rm P{\Delta}}}^{\prime}$ is covered by disjoint sets $\overline{{\rm P{\Delta}}}\in{\rm P\Omega}_{v,k}$ with $\overline{{\rm P{\Delta}}}\subset\overline{{\rm P{\Delta}}}^{\prime}\subset\overline{{\rm P{\Delta}}}^{\prime\prime}$ where $v$ ranges over all vertices of $BT$ such that $\{v,v^{\prime}\}\in BT_{1},v\neq v^{\prime\prime}$ . But such a $\overline{{\rm P{\Delta}}}\in{\overline{{\rm P\Omega}}}_{0,k,n}^{\min}$ by 3.4.9. This proves our claim.

∎

Remark 3.4.12.

The bijection between $\Omega_{v,k}$ and ${\rm P\Omega}_{v,k}$ naturally extends to $\Omega_{0,k,n}$ and ${\rm P\Omega}_{0,k,n}$ , and the containment relations stated in this section about elements of ${\rm P\Omega}_{0,k,n}$ can be applied exactly in the same way to elements of $\Omega_{0,k,n}$ . With this remark we will shift to working with $\Omega_{0,k,n}$ in the subsequent sections. Further, let

\Omega_{0,k,n}^{\min}={\rm pr}^{-1}({\overline{\Omega}}_{0,k,n}^{\min})

Note that 3.4.10 applies to $\Omega_{0,k,n}$ and ${\overline{\Omega}}_{0,k,n}$ as well and ${\rm pr}:\Omega_{0,k,n}^{\min}\rightarrow{\overline{\Omega}}_{0,k,n}^{\min}$ is a bijection.

Remark 3.4.13.

We record the following containment relations

(i) From 3.4.9 we see that for any $v\in BT_{0,n-1}$ we have

\Omega_{v,k}\cap\Omega_{0,k,n}^{\min}=\emptyset

(ii) From (iv) of 3.3.3 and 3.4.9 it follows that for any ${\Delta}\in\Omega_{e,k}$ with $e=\{v,v^{\prime}\}$ and $d(e,v_{0})=n$ ,

\Omega_{e,k}\cap\Omega_{0,k,n}^{\min}=\emptyset

3.5. Detailed description of rigid analytic vectors

We will need the following lemma to prove an important detail in the subsequent lemma.

Lemma 3.5.1.

Let ${\mathbb{H}}$ be an affinoid rigid analytic group over $F$ , and assume that ${\mathbb{H}}$ is isomorphic as a rigid analytic space to ${\mathbb{B}}^{d}$ via a chart $x:{\mathbb{H}}\rightarrow{\mathbb{B}}^{d}$ . Let ${\mathbb{U}}$ be an affinoid rigid analytic space which is equipped with a rigid analytic action of ${\mathbb{H}}$ from the right. Furthermore, we assume that there is $z_{0}\in U={\mathbb{U}}(F)$ and a closed rigid analytic subgroup ${\mathbb{H}}^{\prime}\subset{\mathbb{H}}$ such that the map ${\mathbb{H}}^{\prime}\rightarrow{\mathbb{U}}$ , $h\mapsto z_{0}.h$ , is an isomorphism of rigid analytic spaces.

We let $H$ act on $V=C^{\rm la}(U,E)$ by $(g.f)(z)=f(zg)$ . Then $f\in V$ is ${\mathbb{H}}$ -analytic if and only if $f\in{\mathcal{O}}({\mathbb{U}})$ .

Proof.

(i) If $f\in{\mathcal{O}}({\mathbb{U}})$ , then we consider $f$ as a rigid analytic morphism $f:{\mathbb{U}}\rightarrow{\mathbb{A}}^{1,{\rm rig}}$ . Because the group action $\mu:{\mathbb{U}}\times{\mathbb{H}}\rightarrow{\mathbb{U}}$ is a rigid analytic morphism, so is $f\circ\mu$ , and hence $f\circ\mu\in{\mathcal{O}}({\mathbb{U}}\times{\mathbb{H}})={\mathcal{O}}({\mathbb{U}})\widehat{\otimes}_{F}{\mathcal{O}}({\mathbb{H}})={\mathcal{O}}({\mathbb{U}})\langle x_{1},\ldots,x_{d}\rangle$ , where the latter is the ring of strictly convergent power series over the Banach algebra ${\mathcal{O}}({\mathbb{U}})$ . Then, for $h\in{\mathbb{H}}$ and $z\in{\mathbb{U}}$ we have

(h.f)(z)=f(z.h)=(f\circ\mu)(z,h)=\sum_{\nu\in{\mathbb{N}}^{d}}f_{\nu}(z)x(h)^{\nu}\;.

And hence $f$ is ${\mathbb{H}}$ -analytic.

(ii) Conversely, if $f$ is ${\mathbb{H}}$ -analytic, then we can write

h.f=\sum_{\nu\in{\mathbb{N}}^{d}}f_{\nu}x(h)^{\nu}

where all functions $f_{\nu}$ are in a single BH-subspace of $C^{\rm la}(U,E)$ . This means that there is a finite covering $(U_{i})_{i=1}^{n}$ of $U$ consisting of disjoint compact open subsets. After refining this covering, we may assume that each $U_{i}$ is the set of $F$ -valued points of an afinoid subspace ${\mathbb{U}}_{i}\subset{\mathbb{U}}$ . Put ${\mathbb{U}}^{\prime}=\coprod_{i=1}^{n}{\mathbb{U}}_{i}$ , which is again an affinoid subdomain of ${\mathbb{U}}$ , and which has the property that ${\mathbb{U}}^{\prime}(F)={\mathbb{U}}(F)\,(=U)$ . The functions $f_{\nu}$ are then all in ${\mathcal{O}}({\mathbb{U}}^{\prime})$ , and we have

\|f_{\nu}\|_{{\mathbb{U}}^{\prime}}\,\cdot\,\sup\{|x(h)^{\nu}|{\;|\;}h\in{\mathbb{H}}\}\;\longrightarrow\;0\;\;\mbox{as}\;\;|\nu|\rightarrow\infty\;.

If $z_{0}\in U={\mathbb{U}}^{\prime}(F)$ is as in the statement of the lemma, we find for all $h\in H$ :

(3.5.1)

f(z_{0}h)=\sum_{\nu}f_{\nu}(z_{0})x(h)^{\nu}

The right hand side of 3.5.1 makes sense and converges for all $h\in{\mathbb{H}}$ , and is thus a rigid analytic function on ${\mathbb{H}}$ . When we restrict the right hand side of 3.5.1 to ${\mathbb{H}}^{\prime}$ it is hence rigid analytic on ${\mathbb{H}}^{\prime}$ . Let $\alpha:{\mathbb{H}}^{\prime}\rightarrow{\mathbb{U}}$ , be defined by $h\mapsto z_{0}.h$ . Then we find that $f\circ\alpha$ is a rigid analytic map on ${\mathbb{H}}^{\prime}$ , and $f=(f\circ\alpha)\circ\alpha^{-1}$ is rigid analytic on ${\mathbb{U}}$ . ∎

Lemma 3.5.2.

There exists $k_{0}=k_{0}(\chi)\in{\bf Z}_{\geq 1}$ such that for all $k\geq k_{0}$ the following statements holds. If $e=\{v,v^{\prime}\}$ is a vertex and ${\Delta}\in\Omega_{e,k}\cap\Omega_{v,k}$ , then

I({\Delta},\chi)_{{\mathbb{G}}_{e}(k)-{\rm an}}=I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}

Proof.

Let ${\Delta}\leftrightarrow{\rm P{\Delta}}$ be the bijection $\Omega_{v,k}\leftrightarrow{\rm P\Omega}_{v,k}$ , as in 3.3.6. We set $I({\rm P{\Delta}})=C^{\rm la}({\rm P{\Delta}},E)$ . Under the bijection $\zeta$ of 3.4.3, the space $I({\Delta},\chi)$ gets mapped isomorphically to $I({\rm P{\Delta}})$ . The latter carries the action of $G_{e}(k)$ (and hence of $G_{v}(k)\subset G_{e}(k))$ defined by 3.4.5. With respect to this group action, it suffices to show that

(3.5.3)

I({\rm P{\Delta}})_{{\mathbb{G}}_{e}(k)-{\rm an}}=I({\rm P{\Delta}})_{{\mathbb{G}}_{v}(k)-{\rm an}}

From the transitivity of the $G$ -action on $BT_{0}$ and $BT_{1}$ it is easy to see given any two pairs $(v_{1},e_{1}),(v_{2},e_{2})$ with $v_{i}\in BT_{0}$ and $e_{i}\in BT_{1}$ there is a $g\in G$ such that $(g.v_{2},g.e_{2})=(v_{1},e_{1})$ . It follows that $G_{e_{1}}(k)=gG_{e_{2}}(k)g^{-1}$ and $G_{v_{1}}(k)=gG_{v_{2}}(k)g^{-1}$ . From [2, 3.5.1] it follows that for ${\rm P{\Delta}}\in\Omega_{e_{1},k}\cap\Omega_{v_{1},k}$ one has

(3.5.4)

g.\Big{(}I({\rm P{\Delta}})_{{\mathbb{G}}_{e_{1}}(k)-{\rm an}}\Big{)}=I({\rm P{\Delta}}.g)_{{\mathbb{G}}_{e_{2}}(k)-{\rm an}}

and

(3.5.5)

g.\Big{(}I({\rm P{\Delta}})_{{\mathbb{G}}_{v_{1}}(k)-{\rm an}}\Big{)}=I({\rm P{\Delta}}.g)_{{\mathbb{G}}_{v_{2}}(k)-{\rm an}}\;.

In particular, it suffices to show 3.5.3 for a single pair $(v,e)$ which we are free to choose. The following choice is particularly convenient: $e=\{v,v^{\prime}\}$ with $v=[{\mathcal{O}}\oplus{\mathcal{O}}]$ and $v^{\prime}=[({\varpi})\oplus{\mathcal{O}}]$ . By computations in 3.3.3 any ${\rm P{\Delta}}\in{\rm P\Omega}_{v,k}\cap{\rm P\Omega}_{e,k}$ is of the form ${\bf B}_{z}(a,|{\varpi}|^{k})$ with $|a|\leq 1$ . Now action of $G_{e}(k)$ and $G_{v}(k)$ on $I({\rm P{\Delta}})$ is given by

(g._{\chi}f)(x)=\chi_{1}(\det g)\chi_{2}(bx+d)f(z.g)

Note that for all $g\in G_{e}(k)$ one has $|\det g-1|\leq|{\varpi}|^{k}$ , (and automatically for $g\in G_{v}(k)$ , since $G_{v}(k)\subset G_{e}(k)$ ) and for all $|x|\leq 1$ and for all $g\in G_{e}(k)$ one has $|bx+d-1|\leq|{\varpi}|^{k}$ . Thus for large enough $k$ by local analyticity of $\chi_{1}$ and $\chi_{2}$ we can make sure that both $\chi_{1}(\det g)$ and $\chi_{2}(bx+d)$ are expressible as a power series for $x\in{\bf B}_{z}(a,|{\varpi}|^{k})$ and $g\in G_{e}(k)$ . Now we see from 3.5.1 that both $I({\rm P{\Delta}},\chi)_{{\mathbb{G}}_{e}(k)-{\rm an}}$ and $I({\rm P{\Delta}},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}$ are same as ${\mathcal{O}}({\rm P{\Delta}})$ . Here we have used 3.5.1 twice with ${\mathbb{H}}={\mathbb{G}}_{e}(k)$ and ${\mathbb{H}}={\mathbb{G}}_{v}(k)$ , respectively. This finishes the proof. ∎

3.6. Surjectivity of $\partial_{0}$

Proposition 3.6.1.

There exists $k_{0}=k_{0}(\chi)\in{\bf Z}_{\geq 1}$ such that for all $k\geq k_{0}$ , the map $\partial_{0}:\bigoplus_{v\in BT_{0}}V_{v}\longrightarrow V$ is surjective.

Proof.

Let $f\in C^{\rm la}({\mathbb{P}}^{1}(F),E)$ . Let us partition ${\mathbb{P}}^{1}(F)$ into finitely many discs such that $f$ restricted to each of these discs is a rigid analytic function on that disc. Let $\alpha\in{\mathbb{P}}^{1}(F)$ and ${\bf B}_{\alpha}$ be the disc containing $\alpha$ such that $f|_{{\bf B}_{\alpha}}\in{\mathcal{O}}({\bf B}_{\alpha})$ . Let $g\in G$ be such that $\alpha=0.g$ and let $0\in U$ be a neighborhood around $0$ such that $U.g={\bf B}_{\alpha}$ . For $v_{n}:=[{\mathcal{O}}\oplus({\varpi}^{n})]$ we can show that ${\bf B}_{z}(0,|{\varpi}|^{k+n})\in{\rm P\Omega}_{v_{n},k}$ . Thus for any $k\geq 0$ there exists $n\geq 0$ such that ${\bf B}_{z}(0,|{\varpi}|^{k+n})\subset U$ . By 3.4.5 the group action is given by the formula

(g._{\chi}f)(z)=\chi_{1}(\det g)\chi_{2}(bz+d)f(z.g)

Note that for all $g\in G_{v_{n}}(k)$ and for all $z\in{\bf B}(0,|{\varpi}|^{k+n})$ one has $|\det g-1|\leq|{\varpi}|^{k}$ , and $|bz+d-1|\leq|{\varpi}|^{k}$ .

Since $f(z.g)$ is rigid analytic in ${\bf B}(0,|{\varpi}|^{k+n})$ , by 3.5.1 we see that for large enough $k=k_{0}(\chi)$ we have $(g._{\chi}f)|_{{\bf B}(0,|{\varpi}|^{k+n})}\in V_{v_{n}}$ , note that $k_{0}(\chi)$ depends only on $\chi$ and not on the choice of $f$ . Let $U_{\alpha}={\bf B}_{z}(0,|{\varpi}|^{k+n}).g$ , then it is clear that $U_{\alpha}\in{\rm P\Omega}_{g.v_{n},k}$ and by 3.5.5 it follows that $f|_{U_{\alpha}}\in V_{g.v_{n}}$ . For each $\alpha\in{\mathbb{P}}^{1}(F)$ we find such a neighborhood $U_{\alpha}$ and assign a unique vertex $v_{\alpha}$ using the process above with $f|_{U_{\alpha}}\in V_{v_{\alpha}}$ . With this setup, for the covering ${\mathbb{P}}^{1}(F)=\cup_{\alpha}U_{\alpha}$ , there is a disjoint finite sub-covering ${\mathcal{D}}$ such that if we define

(3.6.2)

f_{v}=\left\{\begin{array}[]{ccl}f|_{U_{\alpha}}&,&v=v_{\alpha},U_{\alpha}\in{\mathcal{D}}\\ 0&,&\mbox{otherwise.}\end{array}\right.

Then we have

\partial_{0}((f_{v})_{v})=f

∎

3.7. Counting Arguments

Lemma 3.7.1.

For any $v\in BT_{0}$ and $e\in BT_{1}$ (or $BT_{1}^{{\rm or}}$ ) we have

|\Omega_{v,k}|=q^{k-1}+q^{k}=(q+1)q^{k-1},\;\;|\Omega_{e,k}|=2q^{k-1}.

Proof.

By 3.3.4, the sets $\Omega_{v,k}$ and $\Omega_{v^{\prime},k}$ , for any two vertices $v,v^{\prime}\in BT_{0}$ , have the same cardinality. By 3.3.6, $\Omega_{v,k}$ and ${\rm P\Omega}_{v,k}$ have the same cardinality, hence it is enough to find $|{\rm P\Omega}_{v_{0},k}|$ where $v_{0}=[{\mathcal{O}}\oplus{\mathcal{O}}]$ as above. From the descriptions as given in 3.3.3 we see that the orbits of $G_{v_{0}}(k)$ in both ${\bf B}_{z}(0,1)$ and ${\bf B}_{w}(\infty,|{\varpi}|)$ are balls of radius $|{\varpi}|^{k}$ . Thus $|{\rm P\Omega}_{v_{0},k}|=\frac{1}{|{\varpi}|^{k}}+\frac{|{\varpi}|}{|{\varpi}|^{k}}=q^{k}+q^{k-1}$ . Similarly, we compute $|{\rm P\Omega}_{e,k}|$ , where $e=\{v_{0},v_{1}\}$ with $v_{1}=[({\varpi})\oplus{\mathcal{O}}]$ by the explicit description in 3.3.3. ∎

Lemma 3.7.2.

Given a vertex $v\in BT_{0}$ with $d(v,v_{0})=n$ ,

|\Omega_{v,k}\cap\Omega_{0,k,n}^{\min}|=q^{k}

Proof.

Given such a vertex $v$ , let $v^{\prime}$ be the unique vertex in $BT_{0,n-1}$ such that $\{v,v^{\prime}\}\in BT_{1,n}$ . Then by 3.4.9 ${\Delta}\in\Omega_{v,k}$ belongs to $\Omega_{0,k,n}^{\min}$ iff $\overline{{\Delta}}\subset\overline{{\Delta}}^{\prime}$ for some ${\Delta}^{\prime}\in\Omega_{v^{\prime},k}$ . By transitivity of $G$ -action on $BT_{1}$ , we can look at the vertices $v_{0},v_{1}$ constituting the edge $e=\{v_{0},v_{1}\}$ where $v_{0},v_{1}$ are as in 2.1.4. By 3.3.3 (ii), $\Omega_{v^{\prime},k}^{v}:=\{{\Delta}^{\prime}\in\Omega_{v^{\prime},k}{\;|\;}\exists{\Delta}\in\Omega_{v,k}:{\Delta}\subset{\Delta}^{\prime}\}$ has cardinality $q^{k-1}$ and each ${\Delta}^{\prime}\in\Omega_{v^{\prime},k}^{v}$ contains $q$ orbits of $\Omega_{v,k}$ . ∎

Recall that we have remarked in 3.3.7 that $\Omega_{1,k}\subset\Omega_{0,k}$ and in fact it is easy to see that this induces an inclusion $\Omega_{1,k,n}\subset\Omega_{0,k,n}$ on these subsets. With this identification in place we claim,

Proposition 3.7.3.

\displaystyle\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}=\Omega_{1,k,n}.

Proof.

Recall that from 3.4.13 it follows that $\Omega_{1,k,n}\cap\Omega_{0,k,n}^{\min}=\phi$ . Thus, from the remark just before the proposition, it is enough to prove $\big{|}\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}\big{|}=\big{|}\Omega_{1,k,n}\big{|}$ . First we compute $|\Omega_{1,k,n}|$ .

\begin{array}[]{lllll}|\{e\in BT_{1}^{{\rm or}}|d(v_{0},e)=i\}|&=&(q+1).q^{i-1}\\ |BT_{1,n}^{{\rm or}}|&=&\sum_{i=1}^{n}|\{e\in BT_{1}^{{\rm or}}|d(v_{0},e)=i\}|\\ &=&(q+1).q^{0}+\cdots+(q+1)q^{n-1}\\ &=&(q+1)\frac{q^{n}-1}{q-1}\\ |\Omega_{e,k}|&=&2q^{k-1}\\ \big{|}\Omega_{1,k,n}\big{|}&=&|\Omega_{e,k}|\times|BT_{1,n}^{{\rm or}}|=2q^{k-1}(q+1)\frac{q^{n}-1}{q-1}\end{array}

where the last but one formula is 3.7.1. To find $|\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}|$ first we collect the following (where here $i\geq 1$ in the first formula)

\begin{array}[]{lclc}|\{v\in BT_{0}|d(v_{0},v)=i\}|&=&(q+1).q^{i-1}\\ |BT_{0,n-1}|&=&1+(q+1)q^{0}+\cdots+(q+1)q^{n-2}\\ |\Omega_{0,k,n}^{\min}\cap\Omega_{v,k}|&=&0\;\;\;\;\;\;\text{if}\;\;\;d(v_{0},v)\leq n-1\;\;(\ref{edgenotminimal}\mbox{ (i)})\\ |\Omega_{0,k,n}^{\min}\cap\Omega_{v,k}|&=&q^{k}\;\;\;\;\text{if}\;\;\;d(v_{0},v)=n\;\;(\ref{cardmin})\end{array}

Putting all these together we get

\begin{array}[]{lll}&&\big{|}\Omega_{0,k,n}{\;\setminus\;}\Omega_{0,k,n}^{\min}\big{|}=(q+1)q^{k-1}|BT_{0,n-1}|+q^{k-1}|\{v\in BT_{0}|d(v_{0},v)=n\}|\\ &=&(q+1)q^{k-1}(1+(q+1)q^{0}+\cdots+(q+1)q^{n-2})+q^{k-1}.(q+1).q^{n-1}\\ &=&(q+1)q^{k-1}(1+q+1+q^{2}+q+\cdots+q^{n-1}+q^{n-2}+q^{n-1})\\ &=&2q^{k-1}(q+1)\frac{q^{n}-1}{q-1}\end{array}

∎

3.8. Exactness in the middle

Let,

	$\displaystyle C_{0,n,k}$	$\displaystyle=\bigoplus_{\begin{subarray}{c}{\Delta}\in\Omega_{v,k}\\ v\in BT_{0,n}\end{subarray}}I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}},$
	$\displaystyle C_{1,n,k}^{\rm or}$	$\displaystyle=\bigoplus_{\begin{subarray}{c}{\Delta}\in\Omega_{e,k}\\ e\in BT_{1,n}^{\rm or}\end{subarray}}I({\Delta},\chi)_{{\mathbb{G}}_{e}(k)-{\rm an}}$

In order to show that the sequence 2.2.1 is exact in the middle, it is enough to show that

0\rightarrow C_{1,n,k}^{\rm or}\xrightarrow{\partial_{1}}C_{0,n,k}\xrightarrow{\partial_{0}}V\rightarrow 0

is exact in the middle for every $n\geq 1$ , where $\partial_{1}$ and $\partial_{0}$ denote what technically are restrictions of those maps to $C_{1,n,k}^{\rm or}$ and $C_{0,n,k}$ respectively.

3.8.1.

Some sub-spaces of $C_{0,n,k}$ . We put

C_{0,n,k}^{\min}=\bigoplus_{\begin{subarray}{c}{\Delta}\in\Omega_{0,k,n}^{\min}\end{subarray}}I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}

and

C_{0,n,k}^{\rm\mathop{non\mathchar 45\relax min}}=\bigoplus_{\begin{subarray}{c}{\Delta}\in\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}\end{subarray}}I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}\;.

In the following, we write an element

f_{v}\in V_{v}=\bigoplus_{{\Delta}^{\prime}\in\Omega_{v,k}}I({\Delta}^{\prime},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}

as $f_{v}=(f_{{\Delta}^{\prime},v})_{{\Delta}^{\prime}\in\Omega_{v,k}}$ .

Lemma 3.8.2.

Given $k>0$ let $v,v^{\prime}$ be vertices in $BT_{0,n}$ and let ${\Delta}\in\Omega_{v,k}$ and ${\Delta}^{\prime}\in\Omega_{v^{\prime},k}$ be such that $\overline{{\Delta}}\subset\overline{{\Delta}}^{\prime}$ and ${\Delta}\in\Omega_{0,k,n}^{\min}$ . Then for $f\in I({\Delta}^{\prime},\chi)_{{\mathbb{G}}_{v}^{\prime}(k)-{\rm an}}$ we have $f|_{{\Delta}}\in I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}$

Proof.

Let us first note that for an edge $e=\{v,v^{\prime}\}\in BT_{1}$ such that $\overline{{\Delta}}\subset\overline{{\Delta}}^{\prime}$ with ${\Delta}^{\prime}\in\Omega_{v^{\prime},k}$ and ${\Delta}\in\Omega_{v,k}$ we have ${\Delta}^{\prime}\in\Omega_{e,k}$ (by 3.3.3 (v)). Now by 3.5.2 we have $I({\Delta}^{\prime},\chi)_{{\mathbb{G}}_{v}^{\prime}(k)-{\rm an}}=I({\Delta}^{\prime},\chi)_{{\mathbb{G}}_{e}(k)-{\rm an}}$ . For $f\in I({\Delta}^{\prime},\chi)_{{\mathbb{G}}_{v^{\prime}}(k)-{\rm an}}=I({\Delta},\chi)_{{\mathbb{G}}_{e}(k)-{\rm an}}\subset V_{e}$ we have $f|_{{\Delta}}\in I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}$ , since $V_{e}\hookrightarrow V_{v}$ .

Now let $d(v,v^{\prime})>1$ and let $v=v_{0},\cdots,v_{m}=v^{\prime}$ be the path from $v$ to $v^{\prime}$ . By 3.4.9 we have, for ${\Delta}\in\Omega_{0,k,n}^{\min}$ a nested sequence of double cosets (seen as a subset of $G$ ) $\overline{{\Delta}}=\overline{{\Delta}}_{0}\subset\overline{{\Delta}}_{1}\subset\overline{{\Delta}}_{n-1}\subset\cdots\subset\overline{{\Delta}}_{m}=\overline{{\Delta}}^{\prime}$ with ${\Delta}_{i}\in\Omega_{v_{i},k}$ . Let $e_{i}:=\{v_{i},v_{i+1}\}\in BT_{1}$ . Since $\overline{{\Delta}}_{i}\subset\overline{{\Delta}}_{i+1}$ by 3.3.3 (v) we have that ${\Delta}_{i+1}\in\Omega_{e_{i},k}$ . Thus $I({\Delta}_{i+1},\chi)_{{\mathbb{G}}_{e_{i}(k)}-{\rm an}}=I({\Delta}_{i+1},\chi)_{{\mathbb{G}}_{v_{i+1}(k)}-{\rm an}}$ and we successively get that $f|_{{\Delta}_{i}}\in I({\Delta}_{i},\chi)_{{\mathbb{G}}_{v_{i}}(k)-{\rm an}}$ for each $i$ . Proving our claim. ∎

Lemma 3.8.3.

The projection of

(3.8.4)

C_{0,n,k}=C_{0,n,k}^{\min}\oplus C_{0,n,k}^{\rm\mathop{non\mathchar 45\relax min}}

onto $C_{0,n,k}^{\rm\mathop{non\mathchar 45\relax min}}$ maps ${\rm Ker}(C_{0,n,k}\xrightarrow{\partial_{0}}V)$ isomorphically onto $C_{0,n,k}^{\rm\mathop{non\mathchar 45\relax min}}$ .

Proof.

By 3.4.11 the union of the minimal orbit in ${\overline{{\rm P\Omega}}}_{0,k,n}$ is equal to ${\mathbb{P}}^{1}(F)$ . Hence, the union of the minimal double cosets in $\Omega_{0,k,n}$ is equal to $G$ . Now, if $(f_{v})_{v\in BT_{0,n}}$ is in $\ker(\partial_{0})$ , then $\sum_{v\in BT_{0,n}}\sum_{{\Delta}^{\prime}\in\Omega_{v,k}}f_{{\Delta}^{\prime},v}=0$ . Now we restrict both sides of this equation to a minimal double coset ${\Delta}\in\Omega_{0,k,n}^{\min}$ and obtain

\sum_{{\Delta}^{\prime}\in\Omega_{0,k,n}}\sum_{{\Delta}^{\prime}\supset{\Delta}}f_{{\Delta}^{\prime},v}|_{{\Delta}}=0\;,

equivalently

f_{{\Delta},v}=-\sum_{{\Delta}^{\prime}\in\Omega_{0,k,n}}\sum_{{\Delta}^{\prime}\supsetneq{\Delta}}f_{{\Delta}^{\prime},v^{\prime}}|_{{\Delta}}\;.

This shows that the components $f_{{\Delta},v}$ with ${\Delta}\in\Omega_{v,k}$ and in $\Omega_{0,k,n}^{\min}$ are uniquely determined by the other components. Note that by 3.8.2 we see that $f_{{\Delta}^{\prime},v^{\prime}}|_{{\Delta}}\in I({\Delta},\chi)_{{\bf G}_{v}(k)-{\rm an}}$ for any ${\Delta}\in\Omega_{v,k}$ such that ${\Delta}\in\Omega_{0,k,n}^{\min}$ and $\overline{{\Delta}}\subset\overline{{\Delta}}^{\prime}$ . So the equation is well-defined in particular. Conversely, for any element

(f_{{\Delta}^{\prime},v})_{{\Delta}^{\prime}\in\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}}\in C_{0,n,k}^{\rm\mathop{non\mathchar 45\relax min}}

we obtain an element $(f_{v})_{v}$ of $\ker(\partial_{0})$ by defining the components $f_{{\Delta},v}$ for ${\Delta}\in\Omega_{v,k}$ and in $\Omega_{0,k,n}^{\min}$ by the previous equation, the equation being well defined by 3.8.2. This proves our claim. ∎

Let $p:C_{0,n,k}\rightarrow\ker(\partial_{0})$ be the projection on to the second summand as in 3.8.4. With these identifications we write the composition map as

C_{1,n,k}^{\rm or}\xrightarrow{\partial_{1}}C_{0,n,k}\xrightarrow{p}\ker(\partial_{0})

and call it, ${\overline{\partial}}_{1}:=p\circ\partial_{1}$ .

Proposition 3.8.5.

There exists some $k_{0}(\chi)\in{\mathbb{Z}}_{>0}$ such that for all $k\geq k_{0}(\chi)$ the composite map ${\overline{\partial}}_{1}=p\circ\partial_{1}$ is an isomorphism from $C_{1,k,n}^{{\rm or}}$ onto $\ker(\partial_{0})$ .

Proof.

We refine the partial ordering on ${\overline{\Omega}}_{0,k,n}$ to a total ordering on $\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}$ . By 3.5.2 and 3.7.3 $C_{1,k,n}$ maps isomorphically onto $C_{0,k,n}^{\rm\mathop{non\mathchar 45\relax min}}$ for all $k\geq k_{0}(\chi)$ for some $k_{0}(\chi)\in{\mathbb{Z}}_{>0}$ . Thus we can view (for $k\geq k_{0}(\chi)$ ) ${\overline{\partial}}_{1}$ explicitly as a map between

\left[\bigoplus_{\begin{subarray}{c}{\Delta}\in\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}\end{subarray}}I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}\right]\stackrel{{\scriptstyle{\overline{\partial}}_{1}}}{{\longrightarrow}}\left[\bigoplus_{\begin{subarray}{c}{\Delta}\in\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}\end{subarray}}I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}\right]

Let us define for $e=\{v_{1},v_{2}\}\in BT_{1}$ , ${\rm sgn}^{e}_{v_{i}}=\begin{cases}1\;\;\;\;\;\mbox{if}\;(v_{i},v_{j})\in BT_{1}^{{\rm or}}\\ -1\;\;\;\mbox{if}\;(v_{j},v_{i})\in BT_{1}^{{\rm or}}\end{cases}$ .

Given $f_{{\Delta},v}\in I({\Delta},\chi)_{{\mathbb{G}}_{e}-{\rm an}}=I({\Delta},\chi)_{{\mathbb{G}}_{v}-{\rm an}}$ for some $e=\{v,v^{\prime}\}\in BT_{1,n}$ and ${\Delta}\in\Omega_{e,k}\cap\Omega_{v,k}$ indexing elements of $\bigoplus_{\begin{subarray}{c}{\Delta}\in\Omega_{0,k,n}\setminus\Omega_{0,k,n}^{\min}\end{subarray}}I({\Delta},\chi)_{{\mathbb{G}}_{v}(k)-{\rm an}}$ as $(f_{{\Delta},v})_{{\Delta},v}$ we see that,

(3.8.6)

\begin{array}[]{cl}&{\overline{\partial}}_{1}(0,\cdots,f_{{\Delta},v},\cdots,0)\\ &\\ =&(g_{{\Delta}_{\alpha},v_{\alpha}})_{{\Delta}_{\alpha},v_{\alpha}}\\ &\\ =&\left\{\begin{array}[]{ccl}{\rm sgn}^{e}_{v}.f_{{\Delta},v}&,&\mbox{if}\;\;{\Delta}_{\alpha}={\Delta},v_{\alpha}=v,\\ {\rm sgn}^{e}_{v}.f_{{\Delta},v}|_{{\Delta}^{\prime}}&,&\mbox{if}\;\;{\Delta}_{\alpha}={\Delta}^{\prime},v_{\alpha}=v^{\prime},\overline{{\Delta}}^{\prime}\subset\overline{{\Delta}}\\ 0&,&\mbox{otherwise}\end{array}\right.\end{array}

This shows that ${\overline{\partial}}_{1}$ can be expressed as a lower triangular $r\times r$ matrix with $\pm Id$ on the diagonal and $\pm{\rm res}^{{\Delta}}_{{\Delta}^{\prime}}$ maps on the off-diagonal elements which corresponds to $(({\Delta},v),({\Delta}^{\prime},v^{\prime}))$ such that ${\Delta}^{\prime}\subset{\Delta}$ and $\{v,v^{\prime}\}\in BT_{1}$ and $0$ elsewhere, where $r=2q^{k-1}(q+1)\frac{q^{n}-1}{q-1}$ and ${\rm res}^{{\Delta}}_{{\Delta}^{\prime}}(f_{{\Delta},v}):=g_{{\Delta}^{\prime},v^{\prime}}=f_{\Delta}|_{{\Delta}^{\prime}}$ . Thus ${\overline{\partial}}_{1}$ is an isomorphism as claimed. ∎

The exactness of the chain complex 2.2.1 for $V:={\rm Ind}^{G}_{B}(\chi)$ now follows from 3.2.1, 3.6.1 and 3.8.5.

3.9. An example of the matrix of ${\overline{\partial}}_{1}$

We give an example of the matrix of ${\overline{\partial}}_{1}$ in the case of $n=1,k=1$ . Let $v^{0},v^{1},v^{\infty}$ be the vertices adjacent to $v_{0}$ . Let ${\Delta}_{0}$ , ${\Delta}_{1}$ , ${\Delta}_{\infty}$ be the $B$ - $G_{v_{0}}(1)$ double cosets. Let $e_{0},e_{1},e_{\infty}$ be the oriented edges connecting $v_{0}$ (origin) with $v^{0},v^{1},v^{\infty}$ , respectively. We may assume that ${\Delta}_{i}$ and ${\Delta}_{j\cup k}:={\Delta}_{j}\cup{\Delta}_{k}$ are the two $B$ - $G_{e_{i}}(1)$ double cosets of $G_{e_{i}}(1)$ , where $\{i,j,k\}=\{0,1,\infty\}$ . The group $G_{v^{i}}(1)$ has then the $B$ - $G_{v^{i}}(1)$ double cosets ${\Delta}^{i}_{0},{\Delta}^{i}_{1}$ , whose union is ${\Delta}_{i}$ , and the double coset ${\Delta}_{j\cup k}$ for $j\neq i$ , $k\neq i$ (and $j\neq k$ ). We then have

\Omega_{0,1,1}^{\min}=\{{\Delta}^{0}_{0},{\Delta}^{0}_{1},{\Delta}^{1}_{0},{\Delta}^{1}_{1},{\Delta}^{\infty}_{0},{\Delta}^{\infty}_{1}\}

and

\Omega_{0,1,1}^{{\rm\mathop{non\mathchar 45\relax min}}}=\{{\Delta}_{0},{\Delta}_{1},{\Delta}_{\infty},{\Delta}_{0\cup 1},{\Delta}_{0\cup\infty},{\Delta}_{1\cup\infty}\}

Let $p$ be the projection from $C_{0,n,k}$ to $C_{0,n,k}^{\rm\mathop{non\mathchar 45\relax min}}$ , and set ${\overline{\partial}}_{1}=p\circ\partial_{1}$ . We write

\begin{array}[]{rcl}C_{1,1,1}^{\rm or}&=&I({\Delta}_{1\cup\infty},\chi)_{G_{e_{0}}-{\rm an}}\oplus I({\Delta}_{0\cup\infty},\chi)_{G_{e_{1}}-{\rm an}}\oplus I({\Delta}_{0\cup 1},\chi)_{G_{e_{\infty}}-{\rm an}}\\ &&\oplus I({\Delta}_{0},\chi)_{G_{e_{0}}-{\rm an}}\oplus I({\Delta}_{1},\chi)_{G_{e_{1}}-{\rm an}}\oplus I({\Delta}_{\infty},\chi)_{G_{e_{\infty}}-{\rm an}}\end{array}

\begin{array}[]{rcl}C_{0,1,1}^{\rm\mathop{non\mathchar 45\relax min}}&=&I({\Delta}_{1\cup\infty},\chi)_{G_{v_{0,0}}-{\rm an}}\oplus I({\Delta}_{0\cup\infty},\chi)_{G_{v_{0,1}}-{\rm an}}\oplus I({\Delta}_{0\cup 1},\chi)_{G_{v_{0,\infty}}-{\rm an}}\\ &&\oplus I({\Delta}_{0},\chi)_{G_{v_{0}}-{\rm an}}\oplus I({\Delta}_{1},\chi)_{G_{v_{0}}-{\rm an}}\oplus I({\Delta}_{\infty},\chi)_{G_{v_{0}}-{\rm an}}\end{array}

The the matrix of ${\overline{\partial}}_{1}$ is given by

\left(\begin{array}[]{cccccc}-1&0&0&0&0&0\\ 0&-1&0&0&0&0\\ 0&0&-1&0&0&0\\ 0&{\rm res}^{0\cup\infty}_{0}&{\rm res}^{0\cup 1}_{0}&1&0&0\\ {\rm res}^{1\cup\infty}_{1}&0&{\rm res}^{0\cup 1}_{1}&0&1&0\\ {\rm res}^{1\cup\infty}_{\infty}&{\rm res}^{0\cup\infty}_{\infty}&0&0&0&1\\ \end{array}\right)

where ${\rm res}^{i\cup j}_{i}(f_{{\Delta}_{i\cup j,v^{k}}}):=g_{{\Delta}_{i},v_{0}}=f_{{\Delta}_{i\cup j}}|_{{\Delta}_{i}}$ .

References

[1] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal. Analytic pro- $p$ -groups, volume 157 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1991.
[2] M. Emerton. Locally analytic vectors in representations of locally $p$ -adic analytic groups. Mem. Amer. Math. Soc., 248(1175):iv+158, 2017.
[3] A. Lahiri. Rigid analytic vectors in locally analytic representations. Annales mathématiques du Québec. Available at https://arxiv.org/pdf/1907.12220.pdf, 2020.
[4] P. Schneider and U. Stuhler. Resolutions for smooth representations of the general linear group over a local field. J. Reine Angew. Math., 436:19–32, 1993.
[5] P. Schneider and U. Stuhler. Representation theory and sheaves on the Bruhat-Tits building. Inst. Hautes Études Sci. Publ. Math., (85):97–191, 1997.
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Resolutions of locally analytic principal series representations of G​L2GL_{2}

Abstract.

1. Introduction

Theorem 1.1.1 (see 3.1.3).

2. A coefficient system and complex

2.1. The Bruhat-Tits tree and associated subgroups

2.1.1.

Lemma 2.1.2.

Proof.

2.1.3.

Lemma 2.1.4.

Proof.

2.1.7.

2.2. The chain complex associated to a coefficient system on B​TBT

3. Locally analytic principal series representations

3.1. Locally analytic induction

3.1.1.

3.1.2.

Theorem 3.1.3.

3.2. Injectivity of ∂1\partial_{1}

Proposition 3.2.1.

Proof.

3.3. The action of GG on ℙ1​(F){\mathbb{P}}^{1}(F)

Proposition 3.3.2.

Proof.

Proposition 3.3.3.

Proof.

Remark 3.3.4.

Lemma 3.3.5.

Proof.

3.3.6.

Remark 3.3.7.

Remark 3.3.8.

Proposition 3.3.9.

Proof.

3.4. Containment relations between the orbits

3.4.1.

Lemma 3.4.3.

Proof.

Lemma 3.4.6.

Proof.

Lemma 3.4.7.

Proof.

3.4.8.

Lemma 3.4.9.

Proof.

Proposition 3.4.10.

Proof.

Lemma 3.4.11.

Proof.

Remark 3.4.12.

Remark 3.4.13.

3.5. Detailed description of rigid analytic vectors

Lemma 3.5.1.

Proof.

Lemma 3.5.2.

Proof.

3.6. Surjectivity of ∂0\partial_{0}

Proposition 3.6.1.

Proof.

3.7. Counting Arguments

Lemma 3.7.1.

Proof.

Lemma 3.7.2.

Proof.

Proposition 3.7.3.

Proof.

3.8. Exactness in the middle

3.8.1.

Lemma 3.8.2.

Proof.

Lemma 3.8.3.

Proof.

Proposition 3.8.5.

Proof.

3.9. An example of the matrix of ∂¯1{\overline{\partial}}_{1}

References

Resolutions of locally analytic principal series representations of $GL_{2}$

2.2. The chain complex associated to a coefficient system on $BT$

3.2. Injectivity of $\partial_{1}$

3.3. The action of $G$ on ${\mathbb{P}}^{1}(F)$

3.6. Surjectivity of $\partial_{0}$

3.9. An example of the matrix of ${\overline{\partial}}_{1}$