The pro-$p$ Iwahori–Hecke $\ext$-algebra $E^\ast$ is a graded algebra that has been introduced and studied by Ollivier–Schneider, with the long-term goal of investigating the category of smooth mod-$p$ representations of $p$-adic reductive groups and its derived category. Its $0$th graded piece is the pro-$p$ Iwahori–Hecke algebra studied by Vignéras and others.

In the present article, we first show that the $\ext$-algebra $E^\ast$ associated with the group $\SL_2(\field)$, $\PGL_2(\field)$ or $\GL_2(\field)$, where $\field$ is an unramified extension of $\qq_p$ with $p \neq 2,3$, is finitely generated as a (non-commutative) algebra.

We then specialize to the case of the group $\SL_2(\qq_p)$, with $p \neq 2,3$, and we show that in this case the natural multiplication map from the tensor algebra $T^\ast_{E^0} E^1$ to $E^\ast$ is surjective and that its kernel is finitely generated as a two-sided ideal. Using this fact as main input, we then show that $E^\ast$ is finitely presented as an algebra. We actually compute an explicit presentation.

§ INTRODUCTION

§.§ Context

In the framework of the local Langlands program, a fundamental problem is to obtain an understanding of categories of representations of $p$-adic reductive groups $G$, such as $G = \GL_n(\qq_p)$ or $G = \SL_n(\qq_p)$. In particular, in the case of representations over fields of characteristic $p$, such an understating is still far from complete.

Let us introduce the setting: let $p$ be a prime number, let $\field$ be a finite extension of $\qq_p$ or the field of formal Laurent series $\ff_q((X))$, where $q$ is a power of $p$. Let $G$ be the group of $\field$-rational points of a connected split reductive group defined over $\field$ (for example, $G = \GL_n(\field)$ or $G = \SL_n(\field)$). It is naturally endowed with the structure of a locally profinite topological group. Let $k$ be a field of characteristic $p$, and let $\smooth{k}{G}$ be the category of smooth representations of $G$ on $k$-vector spaces.

Let $I$ be a pro-$p$ Iwahori subgroup of $G$. For example, in the case that $G = \GL_2(\qq_p)$ or $\SL_2(\qq_p)$ we may choose

\[
I =
\matr{1+p\zz_p}{\zz_p}{p\zz_p}{1+p\zz_p}.
\]

(with the understanding that matrices have determinant $1$ if we are considering $\SL_2(\qq_p)$).

A fundamental tool to try to study the category $\smooth{k}{G}$ is the pro-$p$ Iwahori–Hecke algebra defined by

\[
H \defeq \End_{\smooth{k}{G}}(k[G/I])^{\opp}.
\]

Here, $k[G/I]$ is the free $k$-vector space generated by the left cosets, endowed with the natural action of $G$ (in this way $k[G/I]$ is a smooth representation, that can be described more abstractly as the compact induction $\cind_I^G k$ of the trivial representation $k$). Furthermore, $\End_{\smooth{k}{G}}(k[G/I])$ denotes the ring of endomorphisms and $R^{\opp}$ denotes the opposite ring of a ring $R$. The ring $H$ is a (not necessarily commutative) $k$-algebra that has been studied in particular by Vignéras (see for example [26]).

The link between $H$ and the category $\smooth{k}{G}$ is the following functor:

\begin{equation}\label{eqInvariantFunctor}
\funcNN
{V^I \cong \homm_{\smooth{k}{G}}(k[G/I],V).}
\end{equation}

Here, $\modules{H}$ denotes the category of left modules over $H$ and $H$ acts on $ \homm_{\smooth{k}{G}}(k[G/I],V)$ by composition (defining a left action since we considered the opposite ring in the definition of $H$).

If $k$, instead of being of characteristic $p$, were the field of complex numbers, then the functor (<ref>) would be exact and, if furthermore $I$ were a Iwahori subgroup of $G$, then a fundamental result of Bernstein <cit.> would show that the functor induces an equivalence of categories when restricted to the full subcategory $\smoothh{k}{G}{I}$ of those smooth representations that are generated by their $I$-invariant vectors.

In our case instead, the functor (<ref>) is only left exact. The advantage of considering a pro-$p$ group when $k$ has characteristic $p$ is that a non-zero smooth representation $V$ is sent to a non-zero module $V^I$ by the functor (<ref>). In particular, the subcategory $\smoothh{k}{G}{I}$ contains all irreducible representations. The specific choice of a pro-$p$ Iwahori subgroup is done in order for the algebra $H$ to be as simple as possible.

In this setting, the functor (<ref>) restricted to $\smoothh{k}{G}{I}$ defines an equivalence of categories only in a few special cases (see [13, 14, 10] and, in particular, <cit.>). Schneider [19], however, proved that the situation does improve when one constructs a derived version

\begin{equation}\label{eqDerivedFunctor}
\funcInlineNN{D(\smooth{k}{G})}{D(\dgmod{\mathcal{H}^\bullet})}.
\end{equation}

of the functor (<ref>). Here, $D(\smooth{k}{G})$ denotes the derived category of $\smooth{k}{G}$, while $\mathcal{H}^\bullet$ denotes a certain differential-graded algebra that plays the role of $H$ (more precisely it is the $\homm$ complex $\homm^\bullet(\mathcal{J}^\bullet, \mathcal{J}^\bullet)^{\opp}$, where $\funcInlineNN{k[\quoz{G}{K}]}{\mathcal{J}^\bullet}$ is a fixed injective resolution) and $D(\dgmod{\mathcal{H}^\bullet})$ denotes the derived category of the category of differential-graded modules over $\mathcal{H}^\bullet$. The main result of [19] is that, under the assumption that $\field$ is a finite extension of $\qq_p$ and that $I$ is torsion-free, the functor (<ref>) defines an equivalence of triangulated categories (without the need to restrict to any subcategory).

The objects involved in (<ref>) are much more complicated than those in (<ref>), and an explicit description of the differential graded algebra $\mathcal{H}^\bullet$ is not known.

As an intermediate step, one may try to study instead the $\ext$-algebra

\[
\defeq
\ext^\ast(k[G/I],k[G/I])^{\opp}.
\]

This is the cohomology algebra of $\mathcal{H}^\bullet$, and its $0$th graded piece is of course $H$. Such idea is pursued in the work of Ollivier–Schneider [17, 18]. They describe $E^\ast$ via a certain isomorphism of $k$-vector spaces

\begin{equation}\label{eqDescrEastViaShapiro}
E^\ast \cong \bigoplus_{w \in \widetilde{W}} H^\ast(I \cap wIw^{-1},k).
\end{equation}

Here $\widetilde{W}$ is a certain subquotient of $G$ called the pro-$p$ Iwahori–Weyl group, and $H^\ast(I \cap wIw^{-1},k)$ denotes the cohomology of the profinite group $I \cap wIw^{-1}$ with respect to the trivial module $k$. This decomposition generalizes the Iwahori–Matsumoto basis of $H = E^0$ studied in [26].

Some of the main results by Ollivier-Schneider are the following: a description of the multiplication on $E^\ast$ via cohomological operations on the right hand side of (<ref>) <cit.>; a more detailed description of the (left) action of $E^0$ on $E^\ast$ <cit.>; the existence of an involutive anti-automorphism <cit.>; a duality theorem in the case that $\field$ is a finite extension of $\qq_p$ and $I$ is torsion-free <cit.>; a complete description of the structure of $E^\ast$ as an $E^0$-bimodule in the case that $G = \SL_2(\qq_p)$ with $p \neq 2,3$ <cit.>. As an interesting consequence, under the latter assumptions they show that an irreducible representation $V$ is supersingular if and only if its cohomology $H^\ast(I,V)$ is supersingular as a left $H$-module <cit.>.

§.§ Main results

The present paper is devoted to the study of finite generation properties of the $\ext$-algebra $E^\ast$, especially in the case $G = \SL_2(\qq_p)$ with $p \neq 2,3$.

In <ref>, we study the $1$st graded piece $E^1$ as an $E^0$-bimodule, and we prove the following result, under quite general assumptions.

[Proposition <ref>]
Assume that $\field$ is a finite extension of $\qq_p$. One has that $E^1$ is finitely generated as an $E^0$-bimodule.

To prove statements on the whole algebra $E^\ast$, in <ref> and <ref> we specialize to the case of small $G$ (as will be made precise in the statements below), and we study the multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$, where $T^\ast_{E^0} E^1$ denotes the tensor algebra of the $E^0$-bimodule $E^1$. We first prove the following result.
[Proposition <ref>, Remark <ref>]
Assume that $p \neq 2,3$, that $\field$ is a (finite) unramified extension of $\qq_p$ and that $G$ is either $\SL_2(\field)$ or $\PGL_2(\field)$. One has the image of the multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$ has finite codimension in $E^\ast$. If instead $G = \GL_2(\field)$, an analogous more technical result holds.

The two propositions above allow us to prove, with very little effort, that the $\ext$-algebra is finitely generated as a $k$-algebra in the above cases. This is the first main result of the paper.
[Theorem <ref>]
Assume that $p \neq 2,3$, that $\field$ is a (finite) unramified extension of $\qq_p$ and that $G$ is one of $\SL_2(\field)$, $\PGL_2(\field)$ or $\GL_2(\field)$. One has that the $\ext$-algebra $E^\ast$ is finitely generated as a $k$-algebra.

The longest part of the paper (namely, <ref>) deals with the case $G = \SL_2(\qq_p)$ with $p \neq 2,3$. Under this assumption, the explicit description of the $\ext$-algebra of [18] becomes available, and we exploit such description to obtain more precise results.

We first show that the multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$ is surjective, and then, with quite a long computation, we explicitly determine its kernel. We prove the following result.
[Theorem <ref>]
Assume that $G=\SL_2(\qq_p)$ with $p \neq 2,3$.
The multiplication map

\[
\funcInline{\mult}{T_{E^0}^\ast E^1}{E^\ast}
\]

is surjective and its kernel is generated by its $2$nd and $3$rd graded pieces as a two-sided ideal. Furthermore, the kernel is finitely generated as a two-sided ideal.

We will actually compute an explicit finite list of generators (see again Theorem <ref>).

Starting from the presentation of $E^\ast$ determined by the above theorem, we then compute a presentation of $E^\ast$ as a (non-commutative) $k$-algebra. This is achieved by computing a presentation of $E^0$ (the pro-$p$ Iwahori–Hecke algebra) as a $k$-algebra and a presentation of $E^1$ as an $E^0$-bimodule. Combining these two presentations with the above theorem, one then readily obtains a presentation of $E^\ast$ as a $k$-algebra. In this way, we obtain the following result.

[Theorem <ref>]
Assume that $G=\SL_2(\qq_p)$ with $p \neq 2,3$.
The $\ext$-algebra $E^\ast$ is finitely presented as a $k$-algebra.

We stress the fact that we obtain this result by explicitly computing a finite presentation  (see again Theorem <ref>).

§.§ Acknowledgements

A large portion of the material in this article was part of the author's PhD thesis [2] under the direction of Peter Schneider. I thank him for the invaluable help and support, his fruitful and precise comments, and for suggesting such a research topic.

I am indebted with Claudius Heyer for the extremely useful discussions and his great interest in my work. I thank Rachel Ollivier for very interesting discussions and her support during the write-up of this article.

I acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics – Geometry – Structure, and under the CRC 1442 Geometry: Deformations and Rigidity (Project Number 427320536).

§ SETTING AND BACKGROUND

§.§ Notation and assumptions

We fix the following data throughout the paper.

We consider a non-archimedean local field $\field$. We denote by $p > 0$ its residue characteristic and by $q = p^f$ its residue cardinality. We recall that $\field$ is either a finite extension of $\qq_p$ or the field of Laurent series $\ff_q((X))$. Almost everywhere in this paper we will exclude the second case, but we nevertheless allow such case in this section and in a few lemmas later on for consistency with [17], for sake of generality, and to point out a case in which our first result fails (Example <ref>). We denote by $\ooo$ the ring of integers of $\field$, and by $\mmm$ the maximal ideal of $\ooo$. We also fix a uniformizer $\pi$.

Let $\GG$ be a split connected reductive group over $\field$. Many results in this paper will only be valid for special choices of $\GG$, as will be explained later. We will always use the convention of denoting linear algebraic groups over $\field$ with boldface letters, and we will denote the associated group of $\field$-rational points with the corresponding non-boldface letter (e.g., in the case of $\GG$, we will have $G = \GG(\field)$). Such groups will be understood as locally profinite groups, with respect to the topology induced by $\field$.

Let us choose a split maximal torus $\TT$ of $\GG$. Let $T^0$ be the unique maximal compact subgroup of $T$ (isomorphic to $(\ooo^\times)^{\dim \TT}$), and let $T^1$ be the unique maximal pro-$p$ subgroup of $T$ (isomorphic to $(1+\mmm)^{\dim \TT}$).

Let $\mathbf{N}$ be the normalizer of $\TT$ in $\GG$. We define the (discrete) groups

\begin{align*}
&W_0 \defeq \quoz{N}{T},
&&W \defeq \quoz{N}{T^0},
&&\widetilde{W} \defeq \quoz{N}{T^1}.
\end{align*}

The group $W_0$ is the usual Weyl group, the group $W$ is sometimes called extended affine Weyl group or Iwahori–Weyl group, and the group $\widetilde{W}$ is sometimes called pro-$p$ Iwahori–Weyl group.

Let us consider the semisimple (or adjoint) building associated with $\GG$ (see <cit.>) and the apartment corresponding to the choice of $\TT$. Let us choose a chamber $\chamber$ in such apartment. Let $J$ be the Iwahori subgroup of $G$ corresponding to the choice of $\chamber$ (see [9] for some possible, and equivalent, definitions), and let $I$ be the corresponding pro-$p$ Iwahori subgroup, i.e., the pro-$p$ radical of $J$.

We fix a field $k$ of characteristic $p$ (i.e., the same characteristic as the residue field of $\field$), which will be the coefficient field of the algebras that we will consider.

Many results in the paper will be stated under stronger assumptions than those above. For <ref>, <ref> and <ref>, any further assumption will be made explicit in every single statement. In <ref>, instead, the assumptions will be made explicit only at the beginning and will hold throughout.

§.§ The apartment

In this section and the next ones, we will recall a few notions from Bruhat–Tits theory. We will only need a few facts from the treatise [3, 4]. For a more in-depth recollection of the what is needed in the context of the pro-$p$ Iwahori–Hecke algebra (or its $\ext$-version) see for example <cit.>. The present summary is partially inspired by those in [17] and [8].

The pair $(\GG,\TT)$ defines a root datum $\big( X^\ast(\TT), \Phi, X_\ast(\TT), \Phi^\vee \big)$, where $X^\ast(\TT)$ is the group of characters, $\Phi$ is the set of roots, $X_\ast(\TT)$ is the group of cocharacters, and $\Phi^\vee$ is the set of coroots.

Associated with every root $\alpha \in \Phi$ one has an additive subgroup $\UU_\alpha$ of $\GG$. We fix a Chevalley system $\big( \funcInline{x_\alpha}{\ga}{\UU_\alpha} \big)_{\alpha \in \Phi}$ according to the definition given in <cit.>. For all $\alpha \in \Phi$, we denote by $\funcInline{\varphi_\alpha}{\SL_2}{\GG}$ the associated morphism defined in <cit.>. In particular, we recall from loc. cit. that for all $\alpha \in \Phi$ and all $u \in \field$ one has

\begin{align*}
x_\alpha(u) &= \varphi_{\alpha} \matr{1}{u}{0}{1},
x_{-\alpha}(u) &= \varphi_{\alpha} \matr{1}{0}{-u}{1}.
\end{align*}

For all $\alpha \in \Phi$, the homomorphism of algebraic groups

\[
\func
{\varphi_{\alpha} \matr{x}{0}{0}{x^{-1}}}
\]

has values in $\TT$ (i.e., $\check{\alpha} \in X_\ast(\TT)$). It is called the coroot associated with $\alpha$, and seeing $\rr \otimes_\zz X_\ast(\TT)$ as the dual of $\rr \otimes_\zz X^\ast(\TT)$, the coroot $\check{\alpha}$ is indeed the coroot associated with $\alpha$ in the sense of abstract root systems (and hence we will write $\check{\alpha} \in \Phi^\vee$).

The apartment $\apart$ corresponding to $\TT$ of the semisimple building of $\GG$ is a real affine space whose underlying vector space is

\[
V_{\apart} \defeq (\quoz{X_\ast(\TT)}{X_\ast(\CC^\circ)}) \otimes_{\zz} \rr,
\]

where $\CC^\circ$ denotes the connected centre of $\GG$. We fix the valuation $x_0$ (in the sense of Bruhat–Tits) associated with the Chevalley system $(x_\alpha)_{\alpha \in \Phi}$ (see <cit.>). It is an element of $\apart$, and one actually has

\[
\apart = x_0 + V_{\apart}.
\]

We assume that the closure of the chamber $\chamber$ chosen in <ref> contains $x_0$. Then the choice of $\chamber$ is equivalent to the choice of a set of positive roots $\Phi^+ \subseteq \Phi$. We denote by $\Phi^-$ the corresponding set of negative roots.

Let us consider the perfect pairing $\funcInline{\left\langle {}_-, {}_- \right\rangle}{X_\ast(\TT) \times X^\ast(\TT)}{\zz}$. It is $W_0$-equivariant for the left actions of $W_0$ on $X_\ast(\TT)$ and $X^\ast(\TT)$ (defined via the conjugation action of $W_0$ on $T$). For all $t \in T$ there exists a unique cocharacter $\mu(t) \in X_\ast(\TT)$ satisfying the condition

\begin{align*}
&\langle \mu(t) , \chi \rangle = - \val_\field (\chi(t))
&&\text{for all $\chi \in X^\ast(\TT)$.}
\end{align*}

The map $\funcInline{\mu}{T}{X_\ast(\TT)}$ obtained in this way is thus a surjective group homomorphism, it is $W_0$-equivariant and it induces an isomorphism $\funcInlineNN{\quoz{T}{T^0}}{X_\ast(\TT)}$. We will actually use more often the induced surjective group homomorphism

\[
\funcInline{\nu}{\quoz{T}{T^0}}{\quoz{X_\ast(\TT)}{X_\ast(\CC^\circ)} \subseteq V_\apart.}
\]

We will often write $\nu(t)$ not only for $t \in \quoz{T}{T^0}$ but also for $t \in \quoz{T}{T^1}$.

We also consider the following perfect pairing, induced by the one above:

\[
\funcInline{\left\langle {}_-, {}_- \right\rangle}{(\quoz{X_\ast(\TT)}{X_\ast(\CC^\circ)}) \times \spann_{\zz} \Phi}{\zz.}
\]

In particular, with this notation we have that $\langle \nu(t) , \alpha \rangle = - \val_\field (\alpha(t))$ for all $t \in \quoz{T}{T^0}$ and all $\alpha \in \Phi$.

By <cit.>, the surjection $\funcInlineNN{N}{W_0}$ becomes split after one quotients out the subgroup of $T$ generated by $\check{\alpha}(-1)$ for $\alpha$ running over $\Phi$, and moreover an explicit splitting is obtained by sending, for all $\alpha \in \Phi$, the reflection defined by $\alpha$ to (the class of) $\varphi_\alpha \matr{0}{1}{-1}{0}$ (note that the result in loc. cit. is stated for split semisimple groups, but it might be applied to general split reductive groups by working with the derived subgroup). It thus follows that the exact sequence

\[\begin{tikzcd}
1 \ar[r]
\quoz{T}{T^0} \ar[r]
W \ar[r]
W_0 \ar[r]
\end{tikzcd}\]

is split: therefore we write $W = \quoz{T}{T^0} \rtimes W_0$ and we view $W_0$ as a subgroup of $W$, fixing once and for all the splitting determined by the above rule.

The group $W$ acts on the apartment (or equivalently, on its supporting vector space) by the following rule:

\[
\funcNN
{W \times V_{\apart}}
{(w_0 t , x)}
\\%[-0.5em]
\scriptstyle\text{(for $w_0 \in W_0$, $t \in \quoz{T}{T^0}$)}
\end{array}}
{w_0 \cdot (x + \nu(t)).}
\]

where the action of $w_0$ on $V_{\apart}$ is defined via the usual action of the Weyl group on cocharacters.

§.§ Affine roots and the length function

We continue recalling a few notions from the literature, following in particular <cit.>.

We define the set of affine roots, the set of positive affine roots and the set of negative affine roots respectively as

\begin{align*}
&\Phi_{\aff} \defeq \Phi \times \zz,
&&\Phi_{\aff}^+ \defeq (\Phi \times \zz_{\geqslant 1}) \cup (\Phi^+ \times \{0\}),
&&\Phi_{\aff}^- \defeq \Phi_{\aff} \smallsetminus \Phi_{\aff}^+.
\end{align*}

The group $W$ acts on the set of affine roots via the following rule:

\[
\funcNN
{W \times \Phi_{\aff}}
{\big(w_0 t , (\alpha, \hhh)\big)}
\\%[-0.5em]
\scriptstyle\text{(for $w_0 \in W_0$, $t \in \quoz{T}{T^0}$)}
\end{array}}
{\big(w_0 \alpha, \hhh - \langle \nu(t), \alpha \rangle\big) = \big(w_0 \alpha, \hhh + \val_\field(\alpha(t)) \big).}
\]

The length function on $W$ is defined as

\[
\func
{\# \set{ (\alpha, \hhh) \in \Phi_{\aff}^+ }{ w(\alpha,\hhh) \in \Phi_{\aff}^- }}
\]

(see <cit.>). It satisfies the property $\ell(wv) \leqslant \ell(w) + \ell(v)$ for all $w,v \in W$. Further properties will be recalled in a moment.

Let $(\alpha, \hhh) \in \Phi_{\aff}$. The class of the element $s_{(\alpha, \hhh)} \defeq \varphi_\alpha \matr{0}{\pi^{\hhh}}{-\pi^{-\hhh}}{0}$ acts on the apartment as the reflection through the hyperplane

\[
\set{x \in V_{\apart}}{\langle x, \alpha \rangle + \hhh = 0}.
\]

We define $W_{\aff}$ as the subgroup of $W$ generated by the elements $\varphi_\alpha \matr{0}{\pi^{\hhh}}{-\pi^{-\hhh}}{0}$ for $(\alpha, \hhh)$ running over $\Phi_{\aff}$. This group can be identified with the group generated by the reflections through the above hyperplanes. It forms a Coxeter system $(W_{\aff}, S_{\aff})$, where

\[
S_{\aff} \defeq \set{ s_{(\alpha,\hhh)} }{(\alpha,\hhh) \in (\Pi \times \{0\}) \cup (\Pi_{\min} \times \{1\})},
\]

and where $\Pi$ denotes the basis of $\Phi$ corresponding to the choice of $\Phi^+$ and $\Pi_{\min}$ denotes the set of minimal roots with respect to the partial order $\leqslant$ on $\Phi$ defined by the rule $\alpha \leqslant \beta$ if and only if $\beta - \alpha$ is a linear combination with positive coefficients of elements of $\Pi$.

Recall from <cit.> that the group $W$ admits a semidirect product decomposition

\[
W = W_{\aff} \rtimes \Omega,
\]

where $\Omega$ is the subgroup of elements of length zero. Moreover, the length is constant on double cosets $(\Omega w \Omega)_{w \in W_{\aff}}$. When restricted to $W_{\aff}$, the length $\ell$ actually becomes the length function of the Coxeter system $(W_{\aff}, S_{\aff})$.

The explicit formulas to compute the length are the following:

\begin{align}
\label{eqLengthFormulaW0Right}
\ell(t w_0)
\sum_{\alpha \in \Phi^+ \cap w_0 \Phi^+}
\abs{\langle \nu(t), \alpha \rangle}
\sum_{\alpha \in \Phi^+ \cap w_0 \Phi^-}
\abs{\langle \nu(t), \alpha \rangle - 1},
\\
\label{eqLengthFormulaW0Left}
\ell(w_0 t)
\sum_{\alpha \in \Phi^+ \cap w_0^{-1} \Phi^+}
\abs{\langle \nu(t), \alpha \rangle}
\sum_{\alpha \in \Phi^+ \cap w_0^{-1} \Phi^-}
\abs{\langle \nu(t), \alpha \rangle + 1},
\end{align}

for all $w_0 \in W_0 \subseteq W$ and all $t \in \quoz{T}{T^0}$ (see <cit.>). In particular, for all $t \in \quoz{T}{T^0}$ one has

\begin{equation}\label{eqLengthFormulaTorus}
\ell(t)
\sum_{\alpha \in \Phi^+}
\abs{\langle \nu(t), \alpha \rangle}
\frac{1}{2}
\sum_{\alpha \in \Phi}
\abs{\langle \nu(t), \alpha \rangle}.
\end{equation}

For our purposes it will be fundamental to work with the already defined group $\widetilde{W} \defeq \quoz{N}{T^1}$ instead of $W$. We will slightly abuse notation and write $\ell(w)$ not only for $w \in W$ but also for $w \in \widetilde{W}$. For a subgroup $X \subseteq W$ we will write $\widetilde{X}$ for the preimage of $X$ in $\widetilde{W}$. In particular, we will write $\widetilde{\Omega}$ and $\widetilde{W_0}$ (recall that we view $W_0 \subseteq W$).

§.§ The pro-p Iwahori–Weyl group and the subgroups I_w

Recall the definition $\widetilde{W} \defeq \quoz{N}{T^1}$ of the pro-$p$ Iwahori–Weyl group from <ref>. Its importance stems from the following Bruhat decomposition:

\begin{equation}\label{eqBruhatDecomp}
G = \coprod_{w \in \widetilde{W}} IwI
\end{equation}

(see <cit.>), where we have not distinguished between elements of $\widetilde{W}$ and their representatives in $N$ since $T^1 \subseteq I$, as we will recall in <ref>.

For all $w \in \widetilde{W}$ we define the following subgroup of $I$:

\[
I_w \defeq I \cap wIw^{-1}.
\]

Here, again, we have not distinguished between elements of $\widetilde{W}$ and their representatives in $N$. Actually, since $T^0$ normalizes $I$, this definition makes sense also for $w \in W$. The group $I_w$ is useful in the context of the Bruhat decomposition (<ref>) because of the $I$-equivariant bijection

\begin{equation}\label{eqBijDoubleCoset}
\funcNN
\end{equation}

§.§ The Iwahori decomposition

Recall our fixed Chevalley system $\big( \funcInline{x_\alpha}{\ga}{\UU_\alpha} \big)_{\alpha \in \Phi}$. For all $\alpha \in \Phi$ let us consider the filtration $(\uuu_{(\alpha,m)})_{m \in \zz}$ of $U_\alpha$ defined by

\[
\uuu_{(\alpha,m)} \defeq x_\alpha(\pi^m \ooo).
\]

The groups $\uuu_{(\alpha,1)}$ for $\alpha \in \Phi^-$, the group $T^1$ and the groups $\uuu_{(\alpha,0)}$ for $\alpha \in \Phi^+$ are all contained in the pro-$p$ Iwahori subgroup $I$ and furthermore the multiplication map

\[
\funcInlineNN
\prod_{\alpha\in \Phi^-}
\uuu_{(\alpha, 1)}
\times
\times
\prod_{\alpha\in \Phi^+}
\uuu_{(\alpha, 0)}
\]

is bijective (see <cit.> and <cit.>; here the factors in the products are ordered in some arbitrarily chosen way). It is actually a homeomorphism, because it is a continuous bijection between compact Hausdorff spaces.

There is an analogous Iwahori decomposition for the subgroups $I_w$: for all $\alpha \in \Phi$ let us define

\begin{equation}\label{eqDefGWAlpha}
\defeq
\min \set{ m \in \zz }{ (\alpha, m) \in \Phi_{\aff}^+ \cap w \Phi_{\aff}^+ }.
\end{equation}

Then the multiplication map induce a bijection (actually, a homeomorphism)

\begin{equation}\label{eqIwaDecompIw}
\funcInlineNN
\prod_{\alpha\in \Phi^-}
\uuu_{(\alpha, g_w(\alpha))}
\times
\times
\prod_{\alpha\in \Phi^+}
\uuu_{(\alpha, g_w(\alpha))}
\end{equation}

where, again, the factors in the products are ordered in some arbitrarily chosen way (see <cit.>). Also note that the indices $g_w(\alpha)$ (and hence $I_w$) only depend on the image of $w$ in $W$ (and so we will sometimes use the notation $g_v(\alpha)$ for $v \in W$).

§.§ The pro-p Iwahori–Hecke algebra

As in the introduction, we consider the compact induction $\cind_I^G k$ from $I$ to $G$ of the trivial representation $k$. Concretely, the compact induction $\cind_I^G k$ can be described as the vector space $k[\quoz{G}{I}]$ endowed with the action defined by $g \cdot g'I \defeq gg'I$ (for all $g, g' \in G$).

As in <cit.>, we define the pro-$p$ Iwahori–Hecke algebra to be

\[
H \defeq \End_{\smooth{k}{G}}(k[\quoz{G}{I}])^{\opp},
\]

where $R^{\opp}$ denotes the opposite ring of a ring $R$. This definition can readily be seen to be equivalent to the one given in <cit.>, where $H$ is defined as $\End_{\smooth{k}{G}}(k[\quozright{I}{G}])$.

The Bruhat decomposition (<ref>) makes clear that one has a $k$-basis $(\tau_w)_{w \in \widetilde{W}}$ where $\tau_w$ is the characteristic function of the double coset $IwI$. This is called Iwahori–Matsumoto basis.

A fundamental property is the following braid relation (see <cit.>):

\begin{align*}
&\tau_{v} \cdot \tau_{w} = \tau_{vw}
&&\text{for $v,w \in \widetilde{W}$ such that $\ell(vw) = \ell(v) + \ell(w)$}.
\end{align*}

In the case that $\ell(vw) \neq \ell(v) + \ell(w)$, the formula is more complicated, but it is completely determined by the above braid relation and the quadratic relations (see <cit.>). We will only introduce the quadratic relations in the case $\GG = \SL_2$ (see (<ref>)), since we will not need them in the general case.

§.§ The Ext-algebra E*

We now introduce the pro-$p$ Iwahori–Hecke $\ext$-algebra $E^\ast$ following <cit.>. The definition, as anticipated in the introduction is:

\[
\defeq
\ext^\ast(k[G/I],k[G/I])^{\opp},
\]

where, this time, $R^{\opp}$ denotes the opposite a graded ring $R$.

There is a canonical identification of $k$-vector spaces

\[
E^\ast \cong H^\ast \left(I, k \left[ \quoz{G}{I} \right] \right),
\]

where $H^\ast$ denotes cohomology of profinite groups. Indeed, the abelian category $\smooth{k}{G}$ has enough injective objects (see <cit.>), and so we can fix an injective resolution $\funcInlineNN{k[\quoz{G}{I}]}{J^\bullet}$ in $\smooth{k}{G}$ and compute

\begin{align*}
\ext^\ast_{\smooth{k}{G}} \left( k \left[ \quoz{G}{I} \right], k \left[ \quoz{G}{I} \right] \right)
\\&\cong
H^\ast \big( \homm(k \left[ \quoz{G}{I} \right], J^\bullet) \big)
\\&\cong
H^\ast \big( (J^\bullet)^I \big)
\\&\cong
H^\ast \left(I, k \left[ \quoz{G}{I} \right] \right),
\end{align*}

where we have used Frobenius reciprocity for the compact induction and the fact that restriction functor from $\smooth{k}{G}$ to $\smooth{k}{I}$ preserves injective objects (see <cit.>).

Then, similarly to $H$, the Bruhat decomposition (<ref>) induces a $k$-vector space decomposition

\begin{align*}
\cong
\bigoplus_{w \in \widetilde{W}} H^\ast(I, \xx(w)),
&&\text{where $\xx(w) \defeq k[\quoz{IwI}{I}]$,}
\end{align*}

since the functor $H^\ast(I, {}_-)$ commutes with arbitrary direct sums (see <cit.>).

Let $w \in \widetilde{W}$. From (<ref>) we have a fixed isomorphism of $I$-representations $k[\quoz{IwI}{I}] \cong k[\quoz{I}{I_w}]$. Composing the induced isomorphism between cohomology spaces with the Shapiro isomorphism $H^j \left( I, k \left[ \quoz{I}{I_w} \right] \right) \cong H^j \left( I_w, k \right)$, we thus obtain an isomorphism

\[\begin{tikzcd}
\sh_w \colon
H^\ast \left( I, \xx(w) \right)
\ar[r, "\cong"]
H^\ast \left( I, k \left[ \quoz{I}{I_w} \right] \right)
\ar[r, "\cong"]
H^\ast \left( I_w, k \right),
\end{tikzcd}\]

which we will still call Shapiro isomorphism.

One of the main results of [17] consists in the description of the multiplication on $E^\ast$ in terms of cohomological operations on $\bigoplus_{w \in \widetilde{W}} H^\ast(I, \xx(w))$ (equivalently, on $\bigoplus_{w \in \widetilde{W}} H^\ast \left( I_w, k \right)$), see <cit.>. Except in the case $G = \SL_2(\qq_p)$, we will not use the full multiplicative structure of $E^\ast$, and so below, after a brief discussion on the needed cohomological operations, we will only state a partial (but interesting) result.

For all $w \in \widetilde{W}$ the cohomology space $H^\ast \left( I_w, k \right)$ has the structure of a graded-commutative $k$-algebra with respect to the cup product $\cupprod$. On the other side, the $G$-equivariant bilinear map
\funcInlineNN{k \left[ \quoz{G}{I} \right] \times k \left[ \quoz{G}{I} \right]}{k \left[ \quoz{G}{I} \right]}
gives rise to a cup product on the cohomology space $H^\ast(I,k[\quoz{G}{I}])$ (see <cit.>). This cup product is associative and graded-commutative but in general not unital. The two cup products are compatible in the following sense: if we transport the cup product from $H^\ast(I,k[\quoz{G}{I}])$ to $\bigoplus_{w \in \widetilde{W}} H^\ast \left( I_w, k \right)$, the resulting product is zero when we multiply two elements lying in distinct direct summands, and, when restricted to $H^\ast \left( I_w, k \right)$, coincides with the natural cup product (see <cit.>).

We also recall the conjugation on cohomology of profinite groups (with trivial coefficients). Let $H$ be a locally profinite group, let $K$ be a compact subgroup and let $h \in H$. The conjugation map by $h^{-1}$

\[
\funcNN{hKh^{-1}}{K}{x}{h^{-1}xh}
\]

defines a map

\[
\funcInline{h_\ast}{H^i(K,k)}{H^i(hKh^{-1},k)}
\]

for all $i \in \nn$.

We can now state the above-mentioned partial result about the multiplication in $E^\ast$ (see <cit.>): let us consider $v,w \in \widetilde{W}$ such that $\ell(vw) = \ell(v) + \ell(w)$, let $i,j \in \nn$, let $\alpha \in H^i(I,\xx(v))$ and let $\beta \in H^j(I,\xx(w))$. One has

\begin{align}\label{eqCupYoneda}
\alpha \cdot \beta = (\alpha \cdot \tau_w) \cupprod (\tau_v \cdot \beta)
\end{align}

Moreover, one has

\begin{equation}\label{eqActionRightLeftE0IfLengthsAddUp}
\begin{aligned}
\alpha \cdot \tau_w &\in H^i(I,\xx(vw)),
\\
\sh_{vw}(\alpha \cdot \tau_w) &= \res_{I_{vw}}^{I_v} \big( \sh_v(\alpha) \big),
\\
\tau_v \cdot \beta &\in H^i(I,\xx(vw)),
\\
\sh_{vw}(\tau_v \cdot \beta) &= \res_{I_{vw}}^{v I_w v^{-1}} \big( v_\ast \sh_w(\beta) \big).
\end{aligned}
\end{equation}

§.§ A few results on cohomology of pro-p groups

Let $K$ be a pro-$p$ group. Let us consider the Frattini subgroup $\overline{[K,K]K^p}$, i.e., the smallest closed subgroup of $K$ containing the commutator subgroup and the $p$-powers of the elements of $K$. We consider the Frattini quotient

\[
\fratt{K} \defeq \quoz{K}{\overline{[K,K]K^p}}.
\]

The first cohomology with trivial coefficients can then be computed as

\[
\cong
\homm_{\topgps}(K,k)
\cong
\homm_{\topgps}(\fratt{K},k),
\]

where $\homm_{\topgps}({}_-,{}_-)$ denotes continuous group homomorphisms.
If furthermore $\fratt{K}$ is finite, then we obtain

\[
\cong
\homm_{\vect{\ff_p}}(\fratt{K},k),
\]

where $\homm_{\vect{\ff_p}}({}_-,{}_-)$ denotes homomorphisms of $\ff_p$-vector spaces.

In our situation we consider the case $K = I_w$ for $w \in \widetilde{W}$. Then the Iwahori decomposition (<ref>) induces a surjective homomorphism of topological groups

\begin{equation}\label{eqPsiw}
\func
{\big( \quoz{T^1}{(T^1)^p} \big) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}
{\big( \overline{t}, (\overline{u_\alpha})_{\alpha \in \Phi} \big)}
{\overline{t} \cdot \prod\limits_{\alpha \in \Phi} \overline{x_\alpha(\pi^{g_w(\alpha)} u_\alpha)}.}
\end{equation}

If $\field$ is a finite extension of $\qq_p$ we have that both $\quoz{\ooo}{p\ooo}$ and $\quoz{1+\mmm}{(1+\mmm)^p}$ (and hence also $\quoz{T^1}{(T^1)^p}$) are finite. Therefore,

\begin{align}\label{eqFrattiniFinite}
\cong
\homm_{\vect{\ff_p}}(\fratt{I_w},k)
&&\text{if $\field$ is a finite extension of $\qq_p$.}
\end{align}

For the general structure of the cup-product algebra $H^\ast(K,k)$ of a pro-$p$ group $K$, we will use some results due to Lazard and Serre. Let $K$ be a torsion-free analytic pro-$p$ group of dimension $n$ as an analytic manifold over $\qq_p$. One has that $K$ is a Poincaré group of dimension $n$ (see <cit.>), meaning that the following conditions are satisfied: $H^n(K,k)$ is a one-dimensional $k$-vector space; for all $i > n$ the space $H^i(K,k)$ is zero; for all $i \in \{0,\dots,n\}$ the pairing

\[
\funcInline{\cupprod}{H^i(K,k) \times H^{n-i}(K,k)}{H^{n}(K,k)}
\]

is non-degenerate. In the literature the definition of Poincaré pro-$p$ group is usually given only in the case $k = \ff_p$, but for for a general $k$ of characteristic $p$, the same properties follow because of the natural isomorphism $H^\ast({}_-,k) \cong H^\ast({}_-,\ff_p) \otimes_{\ff_p} k$.

A pro-$p$ group $K$ is called uniform if it is topologically finitely generated, torsion-free and powerful (see <cit.>); by definition, powerful means that $\overline{[K,K]} \subseteq \overline{K^p}$ if $p$ is odd and $\overline{[K,K]} \subseteq \overline{K^4}$ if $p=2$. Under these assumptions, a result by Lazard (see <cit.>) states that one has natural isomorphisms of $k$-algebras

\[
H^\ast(K,k) \cong \exterior^\ast \big(H^1(K,k) \big) \cong \exterior^\ast \big(\homm_{\vect{\ff_p}} (\fratt{K},k) \big),
\]

where $\exterior^\ast({}_-)$ denotes the exterior algebra. Note that under the above assumptions $\fratt{K}$ is finite (see <cit.>).

§ THE FIRST GRADED PIECE Eˆ1 IS FINITELY GENERATED AS AN Eˆ0-BIMODULE

In this section we will prove that $E^1$ is finitely generated as an $E^0$-bimodule in the case that $\field$ is a finite extension of $\qq_p$ (Proposition <ref>). For the moment we do not make such assumption, and we start with a lemma on the generalized affine Weyl group $W$. The proof is inspired by the arguments in <cit.> (which the author already reused in <cit.>).

Let $w_0 \in W_0 \subseteq W$, let $r \in \nn$, let $\overline{\mathfrak{c}}$ be a closed Weyl chamber in $V_{\apart} = \left( \quoz{X_\ast(\TT)}{X_\ast(\CC^\circ)} \right) \otimes_{\zz} \rr$, let $x_1, \dots, x_r \in \quoz{T}{T^0} \subseteq W$ be such that $\nu(x_i)$ lies in $\overline{\mathfrak{c}}$ for all $i \in \{1,\dots,r\}$, let $n_1, \dots, n_r \in \nn$ and let $m_1, \dots, m_r \in \nn$ with the property that if $m_i > 0$, then $n_i > 0$ (for all $i \in \{1,\dots,r\}$). One has that

\[
\ell \big( w_0 x_1^{n_1+m_1} \dots x_r^{n_r+m_r})
\ell \big( w_0 x_1^{n_1} \dots x_r^{n_r})
\ell \big(x_1^{m_1} \dots x_r^{m_r}).
\]

Recall the notations in <ref>. Let $x \defeq x_1^{n_1} \dots x_r^{n_r}$ and let $y \defeq x_1^{m_1} \dots x_r^{m_r}$. Looking at the length formula (<ref>), we see that

\begin{align*}
\ell(w_0 x y)
\sum_{\alpha \in \Phi^+ \cap w_0^{-1} \Phi^+}
\abs{\langle \nu(x y), \alpha \rangle}
\sum_{\alpha \in \Phi^+ \cap w_0^{-1} \Phi^-}
\abs{\langle \nu(x y), \alpha \rangle + 1}
\\&\leqslant
\sum_{\alpha \in \Phi^+ \cap w_0^{-1} \Phi^+}
\big( \abs{\langle \nu(x), \alpha \rangle} + \abs{\langle \nu(y), \alpha \rangle} \big)
\\&\qquad+
\sum_{\alpha \in \Phi^+ \cap w_0^{-1} \Phi^-}
\big( \abs{\langle \nu(x), \alpha \rangle + 1} + \abs{\langle \nu(y), \alpha \rangle} \big)
\\&=
\ell(w_0 x) + \ell(y),
\end{align*}

and that one has equality if and only if the following two conditions hold:

*  For all $\alpha \in \Phi^+ \cap w_0^{-1} \Phi^+$ the signs of $\langle \nu(x), \alpha \rangle$ and of $\langle \nu(y), \alpha \rangle$ are compatible, meaning that their product is bigger or equal than zero;

*  For all $\alpha \in \Phi^+ \cap w_0^{-1} \Phi^-$ the signs of $\langle \nu(x), \alpha \rangle + 1$ and of $\langle \nu(y), \alpha \rangle$ are compatible.

The condition <ref> is satisfied because $\nu(x)$ and $\nu(y)$ lie in a common closed Weyl chamber. To check the thesis of the lemma, it remains to show that condition <ref> is satisfied.

Let us consider $\alpha \in \Phi^+ \cap w_0^{-1} \Phi^-$. Since all the $\nu(x_i)$'s lie in a common closed Weyl chamber, we have either that $\langle \nu(x_i), \alpha \rangle \geqslant 0$ for all $i \in \{1,\dots,r\}$ or that $\langle \nu(x_i), \alpha \rangle \leqslant 0$ for all $i \in \{1,\dots,r\}$. In the first case both $\langle \nu(x), \alpha \rangle + 1$ and $\langle \nu(y), \alpha \rangle$ are bigger or equal than $0$, and so condition <ref> is satisfied. Now, let us assume that we are in the second case. If furthermore there exists $i_0 \in \{1,\dots,r\}$ such that $\langle \nu(x_{i_0}), \alpha \rangle < 0$ and $m_{i_0} > 0$ (and so also $n_{i_0} > 0$, by assumption), we obtain:

\begin{align*}
\langle \nu(x), \alpha \rangle + 1
n_{i_0} \cdot \langle \nu(x_{i_0}), \alpha \rangle + 1
\\&\leqslant 0,
\\
\langle \nu(y), \alpha \rangle
m_{i_0} \cdot \langle \nu(x_{i_0}), \alpha \rangle
\\&<
\end{align*}

Hence, in this case condition <ref> is satisfied. Therefore, it remains to consider the case that $\langle \nu(x_i), \alpha \rangle \leqslant 0$ for all $i \in \{1,\dots,r\}$ and that $m_i = 0$ for all $i \in \{1,\dots,r\}$ such that $\langle \nu(x_{i}), \alpha \rangle < 0$. These conditions force $\langle \nu(y), \alpha \rangle$ to be $0$, and so condition <ref> is satisfied.

Recall from (<ref>) the indices $g_w(\alpha)$ that appear in the Iwahori decomposition (<ref>). In the following lemma we will study a certain property of these indices that will be crucial to prove Proposition <ref>. The lack of symmetry in the two conditions below reflect the lack of symmetry in the formulas (<ref>) (where in one case only restriction appears and in the other case restriction and conjugation appear). The proof of the lemma is similar to <cit.>.

Let $w_0 \in W_0 \subseteq W$, let $r \in \nn$, let $\overline{\mathfrak{c}}$ be a closed Weyl chamber in $V_{\apart} = \left( \quoz{X_\ast(\TT)}{X_\ast(\CC^\circ)} \right) \otimes_{\zz} \rr$, let $x_1, \dots, x_r \in \quoz{T}{T^0} \subseteq W$ be such that $\nu(x_i)$ lies in $\overline{\mathfrak{c}}$ for all $i \in \{1,\dots,r\}$, let $n_1, \dots, n_r \in \nn$ and let $m_1, \dots, m_r \in \nn$ with the property that if $m_i > 0$, then $n_i > 0$ (for all $i \in \{1,\dots,r\}$). Finally, let $\alpha \in \Phi$.
One has that at least one of the following two conditions is true:

* $g_{w_0 x_1^{n_1+m_1} \dots x_r^{n_r+m_r}} (\alpha) = g_{w_0 x_1^{n_1} \dots x_r^{n_r}} (\alpha)$;

* $g_{w_0 x_1^{n_1+m_1} \dots x_r^{n_r+m_r}} (\alpha) = g_{w_0 x_1^{n_1} \dots x_r^{n_r}} (\alpha) - \langle \nu(x_1^{m_1} \dots x_r^{m_r}), w_0^{-1} \alpha \rangle$.

For brevity, let us define

\begin{align*}
a &\defeq g_{w_0 x_1^{n_1+m_1} \dots x_r^{n_r+m_r}} (\alpha),
\\
b &\defeq g_{w_0 x_1^{n_1} \dots x_r^{n_r}} (\alpha),
\\
c &\defeq g_{w_0 x_1^{n_1} \dots x_r^{n_r}} (\alpha) - \langle \nu(x_1^{m_1} \dots x_r^{m_r}), w_0^{-1} \alpha \rangle.
\end{align*}

We have to check that either $a = b$ or $a = c$.

For all $\beta \in \Phi$ let us set

\[
\varepsilon_\beta
\defeq
\begin{cases}
0 &\text{if $\beta \in \Phi^+$,}
\\
1 &\text{if $\beta \in \Phi^-$.}
\end{cases}
\]

Recalling the definition of $g_{({}_-)}(\alpha)$ from (<ref>), for all $x \in \quoz{T}{T^0}$ we compute:

\begin{align*}
\min \set{ m \in \zz }{ (\alpha, m) \in \Phi_{\aff}^+ \cap w_0x \Phi_{\aff}^+ }
\\&=
\min \set
{ m \in \zz }
{ \begin{matrix*}[l]
(\alpha, m) \in \Phi_{\aff}^+,
\\
\left( w_0^{-1} \alpha, m + \left\langle \nu(x) , w_0^{-1} \alpha \right\rangle \right) \in \Phi_{\aff}^+
\end{matrix*} }
\\&=
\max \left\{ \varepsilon_\alpha, \varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x) , w_0^{-1} \alpha \right\rangle \right\}.
\end{align*}

Plugging in, we obtain

\begin{align*}
a &= \max \left\{ \varepsilon_\alpha, \varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x_1^{n_1+m_1} \dots x_r^{n_r+m_r}) , w_0^{-1} \alpha \right\rangle \right\},
\\
b &= \max \left\{ \varepsilon_\alpha, \varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x_1^{n_1} \dots x_r^{n_r}) , w_0^{-1} \alpha \right\rangle \right\},
\\
c &= \max \left\{ \varepsilon_\alpha, \varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x_1^{n_1} \dots x_r^{n_r}) , w_0^{-1} \alpha \right\rangle \right\} - \langle \nu(x_1^{m_1} \dots x_r^{m_r}), w_0^{-1} \alpha \rangle.
\end{align*}

We distinguish three cases on the basis of the sign of $\left\langle \nu(x_1^{m_1} \dots x_r^{m_r}) , w_0^{-1} \alpha \right\rangle$.

* Let us consider the case $\left\langle \nu(x_1^{m_1} \dots x_r^{m_r}) , w_0^{-1} \alpha \right\rangle < 0$. Using the fact that the $\nu(x_i)$'s lie in a common closed Weyl chamber, we deduce that $\left\langle \nu(x_i) , w_0^{-1} \alpha \right\rangle \leqslant 0$ for all $i \in \{1,\dots,r\}$ and that $\left\langle \nu(x_{i_0}) , w_0^{-1} \alpha \right\rangle < 0$ and $m_{i_0} > 0$ for at least one $i_0 \in \{1,\dots,r\}$. By assumption, this implies that $n_{i_0} > 0$. We thus see that $\langle \nu(x_1^{n_1} \dots x_r^{n_r}), w_0^{-1} \alpha \rangle < 0$. Of course, we also have that $\left\langle \nu(x_1^{n_1+m_1} \dots x_r^{n_r+m_r}) , w_0^{-1} \alpha \right\rangle < 0$, and we conclude that

\begin{align*}
%		\max \left\{ \varepsilon_\alpha, \varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x_1^{n_1+m_1} \dots x_r^{n_r+m_r}) , w_0^{-1} \alpha \right\rangle \right\}
%		\\&=
\varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x_1^{n_1+m_1} \dots x_r^{n_r+m_r}) , w_0^{-1} \alpha \right\rangle,
\\
%		\max \left\{ \varepsilon_\alpha, \varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x_1^{n_1} \dots x_r^{n_r}) , w_0^{-1} \alpha \right\rangle \right\} - \langle \nu(x_1^{m_1} \dots x_r^{m_r}), w_0^{-1} \alpha \rangle
%		\\&=
\varepsilon_{w_0^{-1} \alpha} - \left\langle \nu(x_1^{n_1} \dots x_r^{n_r}) , w_0^{-1} \alpha \right\rangle - \langle \nu(x_1^{m_1} \dots x_r^{m_r}), w_0^{-1} \alpha \rangle,
\end{align*}

and hence that $a=c$.

* Let us consider the case $\left\langle \nu(x_1^{m_1} \dots x_r^{m_r}) , w_0^{-1} \alpha \right\rangle > 0$. With the same argument as before, we see that $\langle \nu(x_1^{n_1} \dots x_r^{n_r}), w_0^{-1} \alpha \rangle > 0$, and, of course, also that $\left\langle \nu(x_1^{n_1+m_1} \dots x_r^{n_r+m_r}) , w_0^{-1} \alpha \right\rangle > 0$. We thus conclude that

\[
a = \varepsilon_\alpha = b.
\]

* It remains to consider the case $\left\langle \nu(x_1^{m_1} \dots x_r^{m_r}) , w_0^{-1} \alpha \right\rangle = 0$, but under such assumption we clearly have $a=b=c$.

Before stating the main result of this section, we will need the following two elementary lemmas.

Let $r \in \nn$, let $F$ be a finite set, and let $\funcInline{\varphi}{\nn^r}{F}$ be a map of sets. One has that here exists a finite subset $N \subseteq \nn^r$ such that for all $(n_1, \dots n_r) \in \nn^n$ there exists $(m_1, \dots m_r) \in N$ with the property that:

* $\varphi(m_1, \dots m_r) = \varphi(n_1, \dots n_r)$,

* for all $i \in \{1,\dots,r\}$ one has $m_i \leqslant n_i$,

* for all $i \in \{1,\dots,r\}$ such that $n_i > 0$, one has $m_i > 0$.

Let $V$ be a finite-dimensional vector space over a field. Let $\rho$ and $\lambda$ be endomorphisms of $V$ such that there exists $n \in \zpiu$ satisfying the following properties:

\begin{align*}
\rho^{n+1} &= \rho^n,
\\
\ker(\rho^n) \cap \ker(\lambda) &= 0,
\\
\rho^n \circ \lambda &= \lambda \circ \rho^n,
\end{align*}

One has that the subspace $\image(\rho) + \image(\lambda)$ is the full $V$.

Let us choose a complement $V'$ of $\ker(\rho^n)$ in $V$. It suffices to prove that the endomorphism

\[
\funcNN
{V = \ker(\rho^n) \oplus V'}
{\lambda(x) + \rho(y)}
\]

is injective (hence surjective). So, let $(x,y)$ be in the kernel. Using the first and third assumptions of the lemma, we obtain

\begin{align*}
\rho^n(\lambda(x)) + \rho^{n+1}(y)
\\&=
\lambda(\rho^n(x)) + \rho^{n}(y)
\\&=
\rho^{n}(y).
\end{align*}

Therefore, $y \in \ker(\rho^n) \cap V'$ and so $y=0$. But now $x \in \ker(\rho^n) \cap \ker(\lambda) = 0$, and so we conclude that also $x=0$.

We are now able to prove the main result of <ref>.

Assume that $\field$ is a finite extension of $\qq_p$.
One has that $E^1$ is finitely generated as an $E^0$-bimodule.

We consider the Weyl chambers of $(\quoz{T}{T^0}) \otimes_{\zz} \rr \cong X_\ast(\TT) \otimes_{\zz} \rr$ (note that in Lemmas <ref> and <ref> we have instead considered the Weyl chambers of $V_{\apart}$).

For all closed Weyl chambers $\overline{\mathfrak{c}}$ in $(\quoz{T}{T^0}) \otimes_{\zz} \rr$ we fix a finite set of generators $x_{\overline{\mathfrak{c}}, 1}, \dots, x_{\overline{\mathfrak{c}}, r}$ of the monoid $(\quoz{T}{T^0}) \cap \overline{\mathfrak{c}}$ (as we can do by Gordan's Lemma; moreover, we choose $r$ to be independent of $\overline{\mathfrak{c}}$, as we can, just to simplify the notation). We also fix representatives $\dot{x}_{\overline{\mathfrak{c}}, 1}, \dots, \dot{x}_{\overline{\mathfrak{c}}, r} \in \widetilde{W}$ for $x_{\overline{\mathfrak{c}}, 1}, \dots, x_{\overline{\mathfrak{c}}, r}$.

Let $w \in \widetilde{W}$. Recall from (<ref>) the map

\[
\func
{\big( \quoz{T^1}{(T^1)^p} \big) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}
{\big( \overline{t}, (\overline{u_\alpha})_{\alpha \in \Phi} \big)}
{\overline{t} \cdot \prod\limits_{\alpha \in \Phi} \overline{x_\alpha(\pi^{g_w(\alpha)} u_\alpha)}.}
\]

We argued that, since $\field$ is a finite extension of $\qq_p$, the map $\Psi_{w}$ is a surjective homomorphism of finite-dimensional $\ff_p$-vector spaces.
We also consider the induced isomorphism

\[
\funcInline
{\frac{\big( \quoz{T^1}{(T^1)^p} \big) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{\ker(\Psi_{w})}}
\]

and the dual isomorphism

\[
\funcInline
{\homm_{\vect{\ff_p}} \left( \frac{\big( \quoz{T^1}{(T^1)^p} \big) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{\ker(\Psi_{w})}, k \right)}
\]

(compare (<ref>)).

Let $\widetilde{W_0}$ denote the preimage of $W_0$ via the quotient map $\funcInlineNN{\widetilde{W}}{W}$ (recall that we view $W_0$ as a subgroup of $W$). For all $w_0 \in \widetilde{W_0}$ and all closed Weyl chambers $\overline{\mathfrak{c}}$ in $(\quoz{T}{T^0}) \otimes_{\zz} \rr$ we consider the function

\[
\funcNN
{\left\{ \begin{matrix*}[c]
\text{sub-$\ff_p$-vector spaces of}
\\
\big( \quoz{T^1}{(T^1)^p} \big) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}
\end{matrix*} \right\}}
{\ker \big( \Psi_{w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{n_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{n_r}} \big).}
\]

By Lemma <ref>, there exists a finite set $N_{w_0, \overline{\mathfrak{c}}}$ with the property that, for all $(n_1, \dots, n_r) \in \nn^r$, there exists $(m_1, \dots, m_r) \in N_{w_0, \overline{\mathfrak{c}}}$ such that:

\begin{align}
&\text{$\ker \big( \Psi_{w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{m_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{m_r}} \big) = \ker \big( \Psi_{w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{n_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{n_r}} \big)$,}
\label{cond1}
\\
&\text{for all $i \in \{1,\dots,r\}$ one has $m_i \leqslant n_i$,}
\label{cond2}
\\
&\text{for all $i \in \{1,\dots,r\}$ such that $n_i > 0$, one has $m_i > 0$.}
\label{cond3}
\end{align}

We consider the following finite set:

\[
\mathcal{G}
\defeq
\set{
w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{m_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{m_r}
\begin{matrix*}[l]
w_0 \in \widetilde{W_0},
\\
\overline{\mathfrak{c}} \subseteq (\quoz{T}{T^0}) \otimes_{\zz} \rr \text{ closed Weyl chamber,}
\\
(m_1, \dots, m_r) \in N_{w_0, \overline{\mathfrak{c}}}
\end{matrix*}
\]

To prove the proposition, it suffices to show that the subspace

\[
\bigoplus_{v \in \mathcal{G}} H^1(I,\xx(v))
\]

generates $E^1$ as an $E^0$-bimodule, as we said that $\mathcal{G}$ is finite and that $H^1(I,\xx(v))$ is finite-dimensional over $k$.

Let $w \in \widetilde{W}$. We can write it in the form $w = w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{n_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{n_r}$ for some $w_0 \in \widetilde{W_0},$ some closed Weyl chamber $\overline{\mathfrak{c}} \subseteq (\quoz{T}{T^0}) \otimes_{\zz} \rr$ and some $r$-tuple $(n_1, \dots, n_r) \in \nn^r$. Let us pick an $r$-tuple $(m_1, \dots, m_r) \in N_{w_0, \overline{\mathfrak{c}}}$ satisfying properties (<ref>), (<ref>) and (<ref>). By (<ref>), for all $i \in \{1,\dots,r\}$ we have that $n_i-m_i \in \nn$, and from (<ref>) we see that if $n_i-m_i > 0$ then $m_i > 0$. Therefore, Lemma <ref> yields that

\[
\ell \big( w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{n_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{n_r} \big)
\ell \big( w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{m_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{m_r} \big)
\ell \big( \dot{x}_{\overline{\mathfrak{c}}, 1}^{n_1 - m_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{n_r - m_r} \big).
\]

To ease notation, let us set $v \defeq w_0 \dot{x}_{\overline{\mathfrak{c}}, 1}^{m_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{m_r}$ and $y \defeq \dot{x}_{\overline{\mathfrak{c}}, 1}^{n_1 - m_1} \cdots \dot{x}_{\overline{\mathfrak{c}}, r}^{n_r - m_r}$. Note that $v \in \mathcal{G}$. With the new notation, the above equation becomes

\begin{equation}\label{eqAddUp1}
\ell(w) = \ell(vy) = \ell(v) + \ell(y).
\end{equation}

Since $y \in \quoz{T}{T^1}$, we have that $\ell(y) = \ell(vyv^{-1})$ (as can be seen from (<ref>)), and so the above equation can also be rewritten as

\begin{equation}\label{eqAddUp2}
\ell(w) = \ell(vyv^{-1}v) = \ell(vyv^{-1}) + \ell(v).
\end{equation}

Although we will need this only at a much later point, we also apply Lemma <ref> (whose assumptions are satisfied, again by (<ref>) and (<ref>)). It tells us that for all $\alpha \in \Phi$ at least one of the following two equalities must be true:

\begin{align*}
g_{w_0 x_1^{n_1} \dots x_r^{n_r}} (\alpha) &= g_{w_0 x_1^{m_1} \dots x_r^{m_r}} (\alpha),
\\
g_{w_0 x_1^{n_1} \dots x_r^{n_r}} (\alpha) &= g_{w_0 x_1^{m_1} \dots x_r^{m_r}} (\alpha) - \langle \nu(x_1^{n_1-m_1} \dots x_r^{n_r-m_r}), w_0^{-1} \alpha \rangle.
\end{align*}

With the simplified notation we have introduced, these two equalities can be rewritten as

\begin{equation}\label{eqgw}\begin{aligned}
g_{w} (\alpha) &= g_{v} (\alpha),
\\
g_{w} (\alpha) &= g_{v} (\alpha) - \langle \nu(y), w_0^{-1} \alpha \rangle = g_{v} (\alpha) - \langle \nu(v y v^{-1}), \alpha \rangle.
\end{aligned}\end{equation}

Now let us consider an element $\beta_v \in H^1(I,\xx(v))$. By (<ref>) and (<ref>), we can apply formulas (<ref>) to compute the following products:

\begin{equation}\label{eqSimpleForm}\begin{aligned}
\beta_v \cdot \tau_y &\in H^1(I,\xx(vy)) = H^1(I,\xx(w))
\\
\sh_{w}(\beta_v \cdot \tau_y) &= \res_{I_{w}}^{I_v} \big( \sh_v(\beta_v) \big).
\\
\tau_{vyv^{-1}} \cdot \beta_v &\in H^1(I,\xx(vyv^{-1}v)) = H^1(I,\xx(w))
\\
\sh_{w}(\tau_{vyv^{-1}} \cdot \beta_v) &= \res_{I_{w}}^{vyv^{-1} I_v (vyv^{-1})^{-1}} \big( (vyv^{-1})_\ast \sh_v(\beta_v) \big).
\end{aligned}\end{equation}

If we prove that the elements of the form $\beta_v \cdot \tau_y$ and $\tau_{vyv^{-1}} \cdot \beta_v$, for $\beta_v$ running over $H^1(I,\xx(v))$, generate the full $k$-vector space $H^1(I,\xx(w))$, then the thesis of the proposition is proved.

From (<ref>) we see that for all $\beta_v \in H^1(I,\xx(v))$ and for all $\alpha \in \Phi$ we have that

\begin{equation}\label{eqbohh}\begin{aligned}
\restr{\sh_{w}(\beta_v \cdot \tau_y)}{T^1} &= \restr{\sh_v(\beta_v)}{T^1},
\\
\restr{\sh_{w}(\beta_v \cdot \tau_y)}{\uuu_{\alpha,g_w(\alpha)}} &= \restr{\sh_v(\beta_v)}{\uuu_{\alpha,g_w(\alpha)}},
%and that
\\
\restr{\sh_{w}(\tau_{vyv^{-1}} \cdot \beta_v)}{T^1}
\restr{\sh_v(\beta_v) \big( (vyv^{-1})^{-1} \cdot {}_- \cdot vyv^{-1} \big)}{T^1}
\\&=
\restr{\sh_v(\beta_v)}{T^1},
\\
\restr{\sh_{w}(\tau_{vyv^{-1}} \cdot \beta_v)}{\uuu_{\alpha,g_w(\alpha)}}
\restr{\sh_v(\beta_v) \big( (vyv^{-1})^{-1} \cdot {}_- \cdot vyv^{-1} \big)}{\uuu_{\alpha,g_w(\alpha)}}
\\&=
\restr{\sh_v(\beta_v) \big( \alpha(vyv^{-1})^{-1} \cdot {}_- \big)}{\uuu_{\alpha,g_w(\alpha)}}.
\end{aligned}\end{equation}

To ease notation, we set $({}_-)^\vee \defeq \homm_{\vect{\ff_p}}({}_-, k)$ and $K \defeq \ker(\Psi_v) = \ker(\Psi_w)$. We consider the following picture:

\[\begin{tikzcd}[column sep = 1.8em, row sep = 3em]
\ar[d, "{\overline{\Psi_{v}}^\vee}", "\cong"']
\ar[r, shift left, "{{}_- \cdot \tau_{y}}"]
\ar[r, shift right, "{\tau_{vyv^{-1}} \cdot {}_-}"']
\ar[d, "{\overline{\Psi_{w}}^\vee}", "\cong"']
\\
\left( \frac{( \quoz{T^1}{(T^1)^p} ) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{K} \right)^\vee
\ar[r, shift left, dashed, "{\rho}"]
\ar[r, shift right, dashed, "{\lambda}"']
\left( \frac{( \quoz{T^1}{(T^1)^p} ) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{K} \right)^\vee.
\end{tikzcd}\]

We want to describe explicitly the map $\rho$ (respectively, the map $\lambda$) making the square with the map ${}_- \cdot \tau_{y}$ (respectively, with the map $\tau_{vyv^{-1}} \cdot {}_-$) commute. Once we do this, to prove the proposition it remains to prove that the subspace $\image(\rho) + \image(\lambda)$ is the full space. Let us consider

\[
\gamma \in \left( \frac{( \quoz{T^1}{(T^1)^p} ) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{K} \right)^\vee,
\]

and let us compute $\rho(\gamma)$ and $\lambda(\gamma)$. To ease notation, we set $\beta_v \defeq \big( \overline{\Psi_{v}}^\vee \big)^{-1}(\gamma)$, getting that

\begin{equation}\label{eqMultExpli}\begin{aligned}
\gamma &= \overline{\Psi_{v}}^\vee (\beta_v) = \overline{\sh_v(\beta_v)} \circ \overline{\Psi_{v}},
\\
\rho(\gamma) &= \overline{\Psi_{w}}^\vee (\beta_v \cdot \tau_y) = \overline{\sh_w (\beta_v \cdot \tau_y)} \circ \overline{\Psi_{w}},
\\
\lambda(\gamma) &= \overline{\Psi_{w}}^\vee (\tau_{vyv^{-1}} \cdot \beta_v) = \overline{\sh_w (\tau_{vyv^{-1}} \cdot \beta_v)} \circ \overline{\Psi_{w}}.
\end{aligned}\end{equation}

Here, by $\overline{\sh_v(\beta_v)}$ we mean the element of $(\fratt{I_v})^\vee$ corresponding to the element $\sh_v(\beta_v) \in H^1(I_v,k)$ (and similarly in the other two lines).

Let $t \in T^1$ and let $\overline{t}$ be its image in $\frac{( \quoz{T^1}{(T^1)^p} ) \oplus \bigoplus_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{K}$. By (<ref>) we have

\begin{align*}
\gamma(\overline{t}) & = \sh_v(\beta_v) (t),
\\
\rho(\gamma)(\overline{t}) & = \sh_w (\beta_v \cdot \tau_y) (t),
\\
\lambda(\gamma)(\overline{t}) & = \sh_w (\tau_{vyv^{-1}} \cdot \beta_v)(t).
\end{align*}

From (<ref>) we then see that

\begin{equation}\begin{aligned}
\rho(\gamma)(\overline{t}) &= \gamma(\overline{t}),
\\
\lambda(\gamma)(\overline{t}) &= \gamma(\overline{t}).
\end{aligned}\end{equation}

Now let $\alpha \in \Phi$ and let $u_\alpha \in \ooo$. Let us denote by $\overline{u_\alpha}$ be the image of $u_\alpha$ in $\frac{( \quoz{T^1}{(T^1)^p} ) \oplus \bigoplus_{\alpha' \in \Phi} \quoz{\ooo}{p\ooo}}{K}$, when $\quoz{\ooo}{p\ooo}$ is embedded in $\bigoplus_{\alpha' \in \Phi} \quoz{\ooo}{p\ooo}$ via the inclusion on the summand indexed by $\alpha$. By (<ref>) we have

\begin{align*}
\gamma(\overline{u_\alpha}) & = \sh_v(\beta_v) \big( x_\alpha(\pi^{g_v(\alpha)} u_\alpha) \big),
\\
\rho(\gamma)(\overline{u_\alpha}) & = \sh_w (\beta_v \cdot \tau_y) \big( x_\alpha(\pi^{g_w(\alpha)} u_\alpha) \big),
\\
\lambda(\gamma)(\overline{u_\alpha}) & = \sh_w (\tau_{vyv^{-1}} \cdot \beta_v) \big( x_\alpha(\pi^{g_w(\alpha)} u_\alpha) \big).
\end{align*}

Combining this with (<ref>), we then see that

\begin{equation}\begin{aligned}
\rho(\gamma)(\overline{u_\alpha})
\sh_v (\beta_v) \big( x_\alpha(\pi^{g_w(\alpha)} u_\alpha) \big)
\\&=
\gamma \big( \pi^{g_w(\alpha)-g_v(\alpha)} \overline{u_\alpha} \big),
\\
\lambda(\gamma)(\overline{u_\alpha})
\sh_v(\beta_v) \big( \alpha(vyv^{-1})^{-1} \cdot x_\alpha(\pi^{g_w(\alpha)} u_\alpha) \big)
\\&=
\gamma \big( \alpha(vyv^{-1})^{-1} \pi^{g_w(\alpha)-g_v(\alpha)} \overline{u_\alpha} \big).
\end{aligned}\end{equation}

For all $\alpha \in \Phi$, we set $r_\alpha \defeq \pi^{g_w(\alpha)-g_v(\alpha)}$ and $l_\alpha \defeq \alpha(vyv^{-1})^{-1} \pi^{g_w(\alpha)-g_v(\alpha)}$. We have obtained that the following diagrams are commutative:

\[\begin{tikzcd}[column sep = 8em, row sep = 3em]
\left( \frac{( \quoz{T^1}{(T^1)^p} ) \times \prod_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{K} \right)^\vee
\ar[r, shift left, "{\rho}"]
\ar[r, shift right, "{\lambda}"']
\ar[d, hook]
\left( \frac{( \quoz{T^1}{(T^1)^p} ) \times \prod_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{K} \right)^\vee
\ar[d, hook]
\\
( \quoz{T^1}{(T^1)^p} )^\vee \oplus \bigoplus\limits_{\alpha \in \Phi} (\quoz{\ooo}{p\ooo})^\vee
\ar[r, shift left, "{\tilde{\rho} \defeq \id \oplus \bigoplus\limits_{\alpha \in \Phi} (r_\alpha \cdot {}_-)^\vee}"]
\ar[r, shift right, "{\tilde{\lambda} \defeq \id \oplus \bigoplus\limits_{\alpha \in \Phi} (l_\alpha \cdot {}_-)^\vee}"']
( \quoz{T^1}{(T^1)^p} )^\vee \oplus \bigoplus\limits_{\alpha \in \Phi} (\quoz{\ooo}{p\ooo})^\vee
\end{tikzcd}\]

If we check that there exists $n \in \zpiu$ such that

\begin{align*}
\rho^{n+1} &= \rho^n,
\\
\ker(\rho^n) \cap \ker(\lambda) &= 0,
\\
\rho^n \circ \lambda &= \lambda \circ \rho^n,
\end{align*}

then the assumptions of Lemma <ref> are satisfied and so $\image(\rho) + \image(\lambda)$ is the full space $\left( \frac{( \quoz{T^1}{(T^1)^p} ) \times \prod_{\alpha \in \Phi} \quoz{\ooo}{p\ooo}}{K} \right)^\vee$, as we wanted to show to conclude the proof of the proposition. But if we check that there exists $n \in \zpiu$ satisfying the above properties for $\tilde{\rho}$ in place of $\rho$ and for $\tilde{\lambda}$ in place of $\lambda$, then a fortiori the above properties remain true for $\rho$ and $\lambda$.

The property $\tilde{\rho}^n \circ \tilde{\lambda} = \tilde{\lambda} \circ \tilde{\rho}^n$ is completely trivial for all $n \in \zpiu$. The property $\tilde{\rho}^{n+1} = \tilde{\rho}^n$ is true for $n$ big enough because, for all $\alpha \in \Phi$, the constant $r_\alpha$ is either $1$ or a positive power of $\pi$, from which it follows that the sequence $(r_\alpha^n)_{n \in \zpiu}$ is either definitely $1$ or $0$ in $\quoz{\ooo}{p\ooo}$.

It remains to check the property $\ker(\tilde{\rho}^n) \cap \ker(\tilde{\lambda}) = 0$ (we will actually check it for all $n \in \zpiu$). Let us consider an arbitrary element

\[
(t^\vee, (u_\alpha^\vee)_{\alpha \in \Phi})
\in
( \quoz{T^1}{(T^1)^p} )^\vee \oplus \bigoplus\limits_{\alpha \in \Phi} (\quoz{\ooo}{p\ooo})^\vee;
\]

we have to check either $\tilde{\rho}^n(f)$ or $\tilde{\lambda}(f)$ is non-zero. If $t^\vee$ is non-zero, then actually both $\tilde{\rho}^n(f)$ and $\tilde{\lambda}(f)$ are non-zero. So we can now assume that $u_\alpha^\vee$ is nonzero for some $\alpha \in \Phi$. Recall from (<ref>) that at least one of the following equalities is true:

\begin{align*}
g_{w} (\alpha) &= g_{v} (\alpha),
\\
g_{w} (\alpha) &= g_{v} (\alpha) - \langle \nu(v y v^{-1}), \alpha \rangle.
\end{align*}

The first equality means that $r_\alpha = 1$ (and hence that $r_\alpha^n = 1$), while the second equality means that $l_\alpha \in \ooo^\times$, because

\begin{align*}
\val_\field(l_\alpha)
\val_\field \big( \alpha(vyv^{-1})^{-1} \big) + g_w(\alpha) - g_v(\alpha)
\\&=
\langle \nu(vyv^{-1}), \alpha \rangle + g_w(\alpha) - g_v(\alpha).
\end{align*}

In particular, we get that either $\tilde{\rho}^n(f)$ or $\tilde{\lambda}(f)$ is non-zero.

Assume that $\field = \ff_q((X))$ and that $\GG = \gm$. We show that $E^1$ is not finitely generated as an $E^0$-bimodule.

Since $\GG = \gm$, the length function on $\widetilde{W}$ is constantly zero and any conjugation by an element of $G$ is trivial. Therefore, looking at the formulas (<ref>) and (<ref>) for the multiplication on $E^\ast$ we see that we have an isomorphism of graded $k$-algebras

\[
E^\ast \cong E^0 \otimes_k H^\ast(I,k),
\]

where the multiplication on $H^\ast(I,k)$ is the cup product. Since left and right action of $E^0$ are the same, saying that $E^1$ is finitely generated as an $E^0$-bimodule is the same as saying that $H^1(I,k)$ is finitely generated as a $k$-vector space. This is not the case: indeed, since $\field = \ff_q((X))$, the pro-$p$ group $1+\mmm$ is isomorphic to $\prod_{i \in \nn} \zz_p$ (see <cit.>), and therefore

\[
\\\cong
\textstyle H^1 \big( \prod_{i \in \nn} \zz_p,k \big),
\]

which has infinite dimension as a $k$-vector space.

§ THE SUBALGEBRA GENERATED BY Eˆ0 AND Eˆ1 IN SOME SPECIAL CASES

The purpose of this section is to prove Proposition <ref>, i.e., to prove that, when $G$ is small, the multiplication map

\[
\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}
\]

is almost surjective (for the precise statement see the proposition as well as Remark <ref>).
To prove the proposition, we need to specialize the setting introduced in Section <ref> to the case that $\GG$ has semisimple rank $1$.

Let us assume that $\GG$ has semisimple rank $1$, or, equivalently, that the root system is of type $A_1$. Let $\alpha_0$ denote the positive root. Recalling the notation and facts on the Chevalley system reviewed in <ref> and on the structure of the group $W$ in <ref>, we define

\begin{align}\label{eqDefs0s1Gneneral}
&s_0 \defeq \overline{\varphi_{\alpha_0} \matr{0}{1}{-1}{0}},
&&s_1 \defeq \overline{\varphi_{-\alpha_0} \matr{0}{\pi}{-\pi^{-1}}{0}},
\end{align}

where $\overline{?}$ denotes the image of $?$ in $\widetilde{W}$. Strictly speaking this definition is not compatible with the one we will give at the beginning of <ref> (see there for details), but this will not cause problems.

With this definition, we have $S_{\aff} = \{\overline{s_0}, \overline{s_1}\}$, where $\overline{s_i}$ denotes the image of $s_i$ in $W$ and every element of $W_{\aff}$ can be expressed as a string of the elements $\overline{s_0}$ and $\overline{s_1}$, with no two identical letters one after the other. This representation is actually unique and we have

\begin{align*}
\ell((\overline{s_0} \cdot \overline{s_1})^i) &= i,
\ell((\overline{s_1} \cdot \overline{s_0})^i) &= i,
\\
\ell(\overline{s_0} \cdot (\overline{s_1} \cdot \overline{s_0})^i) &= i,
\ell(\overline{s_1} \cdot (\overline{s_0} \cdot \overline{s_1})^i) &= i,
\end{align*}

for all $i \in \nn$ (compare <cit.> in the case $\GG = \SL_2$). Note that in the above list the string representation of the elements of $W$ is not really unique, the trivial element being repeated twice.

We also deduce that every element of $W$ can be represented in a unique way as a product of an element of $\Omega$ and a string of the elements $\overline{s_0}$ and $\overline{s_1}$, without repetitions as above, and every element of $\widetilde{W}$ can be represented in a unique way as a product of an element of $\widetilde{\Omega}$ and a string of the elements ${s_0}$ and ${s_1}$, with no two identical letters one after the other.

Assume that $\GG$ has semisimple rank $1$. With notation as above, for all $w \in \widetilde{W}$ the following facts hold.

* If $\ell(s_0 w)= \ell(w) + 1$, then the multiplication map defines a homeomorphism

\[
\funcAbove{\cong}
x_{-\alpha_0} (\mmm^{\ell(w)+1})
\times
\times
x_{\alpha_0}  (\ooo)
\]

* If $\ell(s_1 w)= \ell(w) + 1$, then the multiplication map defines a homeomorphism

\[
\funcAbove{\cong}
x_{-\alpha_0} (\mmm)
\times
\times
x_{\alpha_0}  (\mmm^{\ell(w)})
\]

This is a variation of <cit.>, which only deals with the case $\GG = \SL_2$. Recall from <ref> that $I_w$ only depends on the image of $w$ in $W$, so we consider $w \in W$ rather than $w \in \widetilde{W}$. Moreover, for all $v \in W_{\aff}$ and all $\omega \in \Omega$ one has $I_{v\omega} = I_v$ (see <cit.>), and the condition $\ell(\overline{s_i}v\omega) = \ell(v\omega) + 1$ is satisfied if and only if the condition $\ell(\overline{s_i}v) = \ell(v) + 1$ is. Therefore, by the above argument, one only has to consider the cases $w = ({s_0} \cdot {s_1})^i)$, $w = ({s_1} \cdot {s_0})^i$, $w = {s_0} \cdot ({s_1} \cdot {s_0})^i$, $w = {s_1} \cdot ({s_0} \cdot {s_1})^i)$ for some $i \in \nn$, and those cases can be dealt with explicitly as in <cit.>.

Assume that $p \neq 2$ and that $\GG$ has semisimple rank $1$. With notation as above, for all $w \in \widetilde{W}$ the following facts hold.

* If $\ell(s_0 w)= \ell(w) + 1$, then the following equalities hold:

\begin{align*}
\overline{[I_w,I_w]}
\\&=
\varphi_{\alpha_0}
\left(
\matr{1+\mmm^{\ell(w)+1}}{\mmm}{\mmm^{\ell(w)+2}}{1+\mmm^{\ell(w)+1}}
\cap
\SL_2(\field)
\right)
\\&=
x_{-\alpha_0} (\mmm^{\ell(w)+2})
\cdot
\check{\alpha}_0(1+\mmm^{\ell(w)+1})
\cdot
x_{\alpha_0}  (\mmm).
\end{align*}

* If $\ell(s_1 w)= \ell(w) + 1$, then the following equalities hold:

\begin{align*}
\overline{[I_w,I_w]}
\\&=
\varphi_{\alpha_0}
\left(
\matr{1+\mmm^{\ell(w)+1}}{\mmm^{\ell(w)+1}}{\mmm^{2}}{1+\mmm^{\ell(w)+1}}
\cap
\SL_2(\field)
\right)
\\&=
x_{-\alpha_0} (\mmm^{2})
\cdot
\check{\alpha}_0(1+\mmm^{\ell(w)+1})
\cdot
x_{\alpha_0}  (\mmm^{\ell(w)+1}).
\end{align*}

This is a straightforward variation of <cit.>, which only deals with the case $\GG = \SL_2$.

Assume that $p \neq 2$ and that $\GG$ has semisimple rank $1$. With notation as above, for all $w \in \widetilde{W}$ the following facts hold.

* If $\ell(s_0 w)= \ell(w) + 1$, then one has the group isomorphism

\[
\begin{tikzcd}[row sep = 0pt, column sep = 0.5em]
\quoz{\ooo}{\mmm}
\times
\textstyle \frac{T^1}{\check{\alpha}_0(1+\mmm^{\ell(w)+1}) \cdot (T^1)^p}
\times
\quoz{\ooo}{\mmm}
\ar[r]
\\
{\left(\overline{c}, \overline{t}, \overline{b}\right)}
\ar[r, mapsto]
{\textstyle \overline{
x_{-\alpha_0} (-\pi^{\ell(w)+1} c)
\cdot
\cdot
x_{\alpha_0}  (b)
\end{tikzcd}
\]

* If $\ell(s_1 w)= \ell(w) + 1$, then one has the group isomorphism

\[
\begin{tikzcd}[row sep = 0pt, column sep = 0.5em]
\quoz{\ooo}{\mmm}
\times
\textstyle \frac{T^1}{\check{\alpha}_0(1+\mmm^{\ell(w)+1}) \cdot (T^1)^p}
\times
\quoz{\ooo}{\mmm}
\ar[r]
\\
{\left(\overline{c}, \overline{t}, \overline{b}\right)}
\ar[r, mapsto]
{\textstyle \overline{
x_{-\alpha_0} (-\pi c)
\cdot
\cdot
x_{\alpha_0} (\pi^{\ell(w)} b)
\end{tikzcd}
\]

This follows from the previous lemma. The minus signs in $-\pi^{\ell(w)+1} c$ and $-\pi c$ are of course completely inessential but we keep them as they seem more natural (for example, this choice is compatible with <cit.> and $x_{-\alpha_0} (-u) = \varphi_{\alpha_0} \matr{1}{0}{u}{1}$).

Assume that $p \neq 2$ and that $\GG$ has semisimple rank $1$. For all $w \in \widetilde{W}$ let us define

\[
\defeq
\frac{T^1}{\check{\alpha}_0(1+\mmm^{\ell(w)+1}) \cdot (T^1)^p}.
\]

Fixing the isomorphism

\[
\quoz{\ooo}{\mmm} \times T^1_w \times \quoz{\ooo}{\mmm} \cong \fratt{I_w}
\]

defined in the previous lemma, and using the notation $({}_-)^\vee \defeq \homm_{\vect{\ff_p}}({}_-, k)$, we have an induced isomorphism

\begin{equation}\label{eqIsoE1AsATriple}
\funcInline
{({}_-, {}_-, {}_-)_w}
{(\ooo/\mmm)^\vee \oplus (T^1_w)^\vee \oplus (\ooo/\mmm)^\vee}
\end{equation}

For the next lemma, recall from <ref> the definition of Poincaré pro-$p$ group and of uniform pro-$p$ group.

Assume that $p \neq 2,3$, that $\field$ is an unramified extension of $\qq_p$ and that $\GG$ has semisimple rank $1$. One has that $I$ is torsion-free (and so $I$, as well as all the subgroup $I_w$ with $w \in \widetilde{W}$ are Poincaré pro-$p$ groups) and that $I_w$ is a uniform pro-$p$ group for all $w \in \widetilde{W}$ such that $\ell(w) \geqslant 1$.

We claim that $G$ is $p$-torsion-free and that $I_w$ is a powerful pro-$p$ group for all $w \in \widetilde{W}$ such that $\ell(w) \geqslant 1$. Taking these two claims for granted, the two theses of the lemma follow: indeed $I$ is $p$-torsion-free and hence torsion-free, and furthermore, if $\ell(w) \geqslant 1$, saying that the pro-$p$ group $I_w$ is uniform amounts to saying that it is topologically finitely generated, powerful and torsion-free. We have the last two properties, and the fact that it is topologically finitely generated follows from the fact $\ooo$ and $1+\mmm$ are topologically finitely generated (since $\field$ is an extension of $\qq_p$) and from the Iwahori decomposition of $I_w$.

Let us consider the derived group $\GG'$ of $\GG$ (which must be isomorphic either to $\SL_2$ or to $\PGL_2$) and the semisimple cover $\SL_2$ of $\GG'$: we have central isogenies

\[\begin{tikzcd}
\gm^{\dim \CC^\circ} \times \SL_2
\cong
\CC^\circ \times \SL_2
\ar[r]
\CC^\circ \times \GG'
\ar[r]
\GG.
\end{tikzcd}\]

Let $\mmu$ be the kernel of the composite central isogeny; it must be isomorphic a central subgroup of $\SL_2$: i.e., either to the trivial group or to the group of $2$-units $\mmu_2$. Considering the exact sequence induced by Galois cohomology, we obtain an exact sequence

\[\begin{tikzcd}
\mmu(\field)
\ar[r]
(\field^\times)^{\dim \CC^\circ} \times \SL_2(\field)
\ar[r]
\ar[r]
H^1(\gal(\overline{\qq_p}/\field), \mmu(\overline{\qq_p})).
\end{tikzcd}\]

The group $(\field^\times)^{\dim \CC^\circ} \times \SL_2(\field)$ does not have $p$-torsion because $\field$ is an extension of $\qq_p$ without nontrivial $p$-th roots of unity and because $p \neq 2,3$ (an element of $\SL_2(\field)$ of order $p$ would have characteristic polynomial divisible by the $p$-th cyclotomic polynomial, which is absurd). The groups $\mmu(\field)$ and $H^1(\gal(\overline{\qq_p}/\field), \mmu(\overline{\qq_p}))$ are (finite) $2$-groups. It then follows formally that $G$ is $p$-torsion-free: indeed, assume by contradiction that we have a non-trivial $p$-torsion element $x \in G$: it lifts to an element of $y \in (\field^\times)^{\dim \CC^\circ} \times \SL_2(\field)$ since $H^1(\gal(\overline{\qq_p}/\field), \mmu(\overline{\qq_p}))$ is $p$-torsion free. The element $y^p$ must lift to an element $z \in \mmu(\field)$. Since $z^{2^r} = 1$ for some $r \in \zpiu$, it follows that $y^{2^r}$ is a $p$-torsion element in $(\field^\times)^{\dim \CC^\circ} \times \SL_2(\field)$. It cannot be trivial, because it maps to $x^{2^r}$.

Now it remains to prove that $I_w$ is a powerful pro-$p$ group for all $w \in \widetilde{W}$ such that $\ell(w) \geqslant 1$. We refer to the description of $I_w$ and of $\overline{[I_w,I_w]}$ we gave, respectively, in Lemmas <ref> <ref> and for brevity we only consider the case $\ell(s_0 w)= \ell(w) + 1$. It is enough to check that the elements of the form $\varphi_{\alpha_0} \matr{1}{0}{\mmm^{\ell(w)+2}}{1}$, of the form $\check{\alpha}_0(1+\mmm^{\ell(w)+1})$ and of the form $\varphi_{\alpha_0}  \matr{1}{\mmm}{0}{1}$ are contained in the subgroup topologically generated by the $p$-powers (we will actually show that all such elements are $p$-powers). Since $\field$ is an unramified extension of $\qq_p$, we have that

\[
\varphi_{\alpha_0} \matr{1}{0}{\mmm^{\ell(w)+2}}{1}
\left( \varphi_{\alpha_0} \matr{1}{0}{\mmm^{\ell(w)+1}}{1} \right)^p,
\]

and so we see that the elements in $\varphi_{\alpha_0} \matr{1}{0}{\mmm^{\ell(w)+2}}{1}$ are $p$-powers of elements in $I_w$. The same argument shows that the elements in $\varphi_{\alpha_0}  \matr{1}{\mmm}{0}{1}$ are $p$-powers of elements in $I_w$. It remains to work with $\check{\alpha}_0(1+\mmm^{\ell(w)+1})$. Since $\field$ is an unramified extension of $\qq_p$ with $p \neq 2$, the $p$-adic logarithm induces an isomorphism $1+\mmm \cong \mmm$, under which $1+\mmm^2$ corresponds to $\mmm^2$ (which is equal to $p \mmm$, using again that $\field$ is unramified). Therefore the elements in $1+\mmm^2$ are $p$-powers of elements in $1+\mmm$ and so, since $\ell(w) \geqslant 1$, the elements in $\check{\alpha}_0(1+\mmm^{\ell(w)+1})$ are $p$-powers of elements in $\check{\alpha}_0(1+\mmm) \subseteq I_w$. This concludes the proof that $I_w$ is a powerful pro-$p$ group.

Let $n \in \zpiu$ and let $w_1, \dots, w_n \in \widetilde{W}$ with the propriety that $\ell(w_1 \cdots w_n) = \ell(w_1) + \dots + \ell(w_n)$. For all $i \in \{1,\dots,n\}$ let $j_i \in \nn$ and let $\beta_i \in H^{j_i}(I,\xx(w_i))$. The following formula holds:

\[
\beta_1 \cdots \beta_n
(\beta_1 \cdot \tau_{w_2 \cdots w_n})
\cupprod
\dots
\cupprod
(\tau_{w_1 \cdots w_{i-1}} \cdot \beta_i \cdot \tau_{w_{i+1} \cdots w_n})
\cupprod
\dots
\cupprod
(\tau_{w_1 \cdots w_{n-1}} \cdot \beta_n).
\]

Applying the formula relating (the opposite of) the Yoneda product with the cup product (<ref>), we have

\[
\beta_1 \cdots \beta_n
(\beta_1 \cdots \beta_{n-1} \cdot \tau_{w_n})
\cupprod
(\tau_{w_1 \cdots w_{n-1}} \cdot \beta_n).
\]

So, by induction on $n$, it suffices to check that the map

\[
\funcInline
{{}_- \cdot \tau_{w_n}}
{H^\ast(I,\xx(w_1 \cdots w_{n-1}))}
{H^\ast(I,\xx(w_1 \cdots w_{n}))}
\]

commutes with cup products. Recall from (<ref>) that we have a commutative diagram

\[\begin{tikzcd}[column sep = large, row sep = large]
H^\ast(I,\xx(w_1 \cdots w_{n-1}))
\ar[r, "{{}_- \cdot \tau_{w_n}}"]
\ar[d, "\sh_{w_1 \cdots w_{n-1}}", "\cong"']
H^\ast(I,\xx(w_1 \cdots w_{n}))
\ar[d, "\sh_{w_1 \cdots w_{n}}", "\cong"']
\\
H^\ast(I_{w_1 \cdots w_{n-1}}, k)
\ar[r, "{\res^{I_{w_1 \cdots w_{n-1}}}_{I_{w_1 \cdots w_{n}}}}"]
H^\ast(I_{w_1 \cdots w_{n}}, k),
\end{tikzcd}\]

and the claim follows since restriction and Shapiro isomorphism commute with cup products.

Assume that $p \neq 2$ and that $\GG$ has semisimple rank $1$. Let $w \in \widetilde{W}$, let $c^-, c^+ \in \homm_{\vect{\ff_p}}(\quoz{\ooo}{\mmm}, k)$ and let $c^0 \in \homm_{\vect{\ff_p}}(T^1_w,k)$. With notation as at the beginning of this subsection and as in Remark <ref>, the following formulas hold.

If $v \in \widetilde{W}$ is such that $\ell(v) \geqslant 1$ and $\ell(s_0 w v) = 1 + \ell(w) + \ell(v)$, one has

\[
\big( c^-,c^0,c^+ \big)_w \cdot \tau_v
\big(0,c^0(\pr({}_-)),c^+\big)_{wv},
\]

where $\funcInline{\pr}{T^1_{wv}}{T^1_{w}}$ is the quotient map.

If $\ell(s_0w) = \ell(w) + 1$, one has

\[
\tau_{s_0} \cdot \big( c^-,c^0,c^+ \big)_w
\big( 0,c^0(w_0^{-1} \pr({}_-) w_0),-c^- \big)_{s_0w},
\]

where $\funcInline{\pr}{{T^1_{s_0w}}}{{T^1_w}}$ is the quotient map, and where $w_0$ is any element of $N \smallsetminus T$.

If $v \in \widetilde{W}$ is such that $\ell(v) \geqslant 1$ and $\ell(s_1 w v) = 1 + \ell(w) + \ell(v)$, one has

\[
\big(c^-,c^0,c^+\big)_w \cdot \tau_v
\big(c^-,c^0(\pr({}_-)),0\big)_{wv},
\]

where $\funcInline{\pr}{{T^1_{wv}}}{{T^1_{w}}}$ is the quotient map.

If $\ell(s_1w) = \ell(w) + 1$, one has

\[
\tau_{s_1} \cdot \big(c^-,c^0,c^+\big)_w
\big(-c^+,c^0(w_0^{-1} \pr({}_-) w_0),0\big)_{s_1w},
\]

where $\funcInline{\pr}{{T^1_{s_1w}}}{{T^1_w}}$ is the quotient map, and where $w_0$ is any element of $N \smallsetminus T$.

We prove the four statements using the description of the first cohomology of finitely generated pro-$p$ groups as

\[
H^1({}_-, k)
\cong
\homm_{\vect{\ff_p}} \big( \fratt{{}_-}, k \big).
\]

and the description of the relevant Frattini quotients given in Lemma <ref>.

Let us assume that $\ell(v) \geqslant 1$ and $\ell(s_0 w v) = 1 + \ell(w) + \ell(v)$ (in particular, if follows that $\ell(wv) = \ell(w) + \ell(v)$, that $\ell(s_0wv) = \ell(wv) + 1$ and that $\ell(s_0w) = \ell(w) + 1$) and let us prove the first formula.

We want to compute $\beta \cdot \tau_v$ where $\beta \defeq \big( c^-,c^0,c^+ \big)_w$. Since lengths add up, we can apply the formula (<ref>), obtaining

\begin{align*}
\beta \cdot \tau_v &\in H^1(I,\xx(wv))
&&\text{ and }
&\sh_{wv}(\alpha \cdot \tau_v) &= \res_{I_{wv}}^{I_w} \big( \sh_w(\beta) \big).
\end{align*}

Therefore, we have to compute the action of the restriction on cohomology, which corresponds to computing the map induced by the inclusion on the Frattini quotients. Such action can be computed via the following commutative diagram (where we use the description of the Frattini quotients given in Lemma <ref>: note that the conditions $\ell(s_0wv) = \ell(wv) + 1$ and $\ell(s_0w) = \ell(w) + 1$ tell us which of the descriptions in Lemma <ref> we have to consider; also note that the condition $\ell(v) \geqslant 1$ is used to say that $\left(\overline{c}, 0, 0\right)$ is mapped to zero by the map on the left):

\[\begin{tikzcd}[row sep = 3.5em, column sep = 16em]
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi^{\ell(wv)+1} c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{b}{0}{1}
\ar[d, "{\substack{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\\
\downmapsto
\\
\left(0, \overline{t}, \overline{b}\right)
\ar[d, "{\substack{\text{ind. by} \\ I_{wv} \hookrightarrow I_{w}}}"'  pos=0.6]
\\
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi^{\ell(w)+1} c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{b}{0}{1}
\end{tikzcd}\]

Dualizing (i.e., applying $\homm_{\vect{\ff_p}} \big( \fratt{{}_-}, k \big)$), we get the formula in the statement.

Now, let us assume that $\ell(s_0w) = \ell(w) + 1$ and let us prove the second formula.

We first remark that since lengths add up we can apply the formula (<ref>), obtaining

\begin{align*}
\tau_{s_0} \cdot \beta &\in H^i(I,\xx(wv)),
\\
\sh_{s_0w}(\tau_{s_0} \cdot \beta) &= \res_{I_{s_0w}}^{s_0 I_w s_0^{-1}} \big( (s_0)_\ast \sh_w(\beta) \big),
\end{align*}

where $\beta \defeq \big(c^-,c^0,c^+\big)_w$.

This time we have to compute the action of the inclusion map and of conjugation on the level of the Frattini quotients. We do this via the following commutative diagram (which uses again the description of the Frattini quotients given in Lemma <ref>: note that the description of $\fratt{s_0 I_{w} s_0^{-1}}$ follows from the description of $\fratt{I_w}$):

\[\begin{tikzcd}[row sep = 3.5em, column sep = 13em]
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{\pi^{\ell(s_0w)} b}{0}{1}
\ar[d, "{\substack{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\\
\downmapsto
\\
\left(0, \overline{t}, \overline{b}\right)
\ar[d, "{\substack{\text{ind. by} \\ I_{s_0w} \hookrightarrow s_0 I_{w} s_0^{-1}}}"' pos=0.6]
\\
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{\pi^{\ell(w)+1} b}{0}{1}
\ar[d, "{\substack{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\\
\downmapsto
\\
\left(-\overline{b}, \overline{w_0^{-1} t w_0}, -\overline{c}\right)
{\fratt{s_0 I_{w} s_0^{-1}}}
\ar[d, "{{\varphi_{\alpha_0}\matr{0}{1}{-1}{0}}^{-1} \cdot {}_- \cdot \varphi_{\alpha_0}\matr{0}{1}{-1}{0}}"' pos=0.6]
\\
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi^{\ell(w)+1} c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{b}{0}{1}
\end{tikzcd}\]

Dualizing, we get the formula in the statement.

The proof is analogous to <ref>. The commutative diagram in this case is

\[\begin{tikzcd}[row sep = 3.5em, column sep = 15em]
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{\pi^{\ell(wv)} b}{0}{1}
\ar[d, "{\substack{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\\
\downmapsto
\\
\left(\overline{c}, \overline{t}, 0\right)
\ar[d, "{\substack{\text{ind. by} \\ I_{wv} \hookrightarrow I_{w}}}"' pos=0.6]
\\
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{\pi^{\ell(w)} b}{0}{1}
\end{tikzcd}\]

The proof is analogous to <ref>. The commutative diagram in this case is

\[\begin{tikzcd}[row sep = 3.5em, column sep = 13.5em]
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi^{\ell(s_1w)+1} c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{b}{0}{1}
\ar[d, "{\substack{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\\
\downmapsto
\\
\left(\overline{c}, \overline{t}, 0\right)
\ar[d, "{\substack{\text{ind. by} \\ I_{s_1w} \hookrightarrow s_1 I_{w} s_1^{-1}}}"'  pos=0.6]
\\
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi^{\ell(w)+2} c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{\pi^{-1} b}{0}{1}
\ar[d, "{\substack{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\\
\downmapsto
\\
\left(-\overline{b}, \overline{w_0^{-1} t w_0}, -\overline{c}\right)
{\fratt{s_1 I_{w} s_1^{-1}}}
\ar[d, "{{\varphi_{\alpha_0}\matr{0}{-\pi^{-1}}{\pi}{0}}^{-1} \cdot {}_- \cdot \varphi_{\alpha_0}\matr{0}{-\pi^{-1}}{\pi}{0}}"' pos=0.6]
\\
\quoz{\ooo}{\mmm}
\times
\times
\quoz{\ooo}{\mmm}
\ar[r, "{
\left(\overline{c}, \overline{t}, \overline{b}\right)
\mapsto
\overline{
\varphi_{\alpha_0} \matr{1}{0}{\pi c}{1}
\cdot
\cdot
\varphi_{\alpha_0}  \matr{1}{\pi^{\ell(w)} b}{0}{1}
\end{tikzcd}\]

Assume that $p \neq 2,3$, that $\field$ is an unramified extension of $\qq_p$ and that $\GG$ has semisimple rank $1$. Let us denote by $f$ the residue degree of $\field$ over $\qq_p$. Let us consider the tensor algebra $T^\ast_{E^0} E^1$ associated with the $E^0$-bimodule $E^1$ and the natural multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$. One has that

\[
\image \big( \funcInlineNN{T^\ast_{E^0} E^1}{E^\ast} \big)
\supseteq
\bigoplus_{\substack{w \in \widetilde{W} \\ \text{s.t. $\ell(w) \geqslant f \cdot \dim \TT$}}}
\]

We first note that $f \cdot \dim \TT = \dim_{\ff_p} \quoz{T^1}{(T^1)^p}$. Indeed, we have

\[
\quoz{T^1}{(T^1)^p}
\cong
\big( \quoz{1+\mmm}{(1+\mmm)^p} \big)^{\oplus \dim \TT}
\cong
\big( \quoz{\mmm}{p\mmm} \big)^{\oplus \dim \TT}
\cong
\big( \quoz{\ooo}{\mmm} \big)^{\oplus \dim \TT},
\]

where in the second step we have used the $p$-adic logarithm/exponential (which we may apply since $\field$ is an unramified extension of $\qq_p$ and $p \neq 2$) and in the third step we have further used that $\field$ is an unramified extension of $\qq_p$. The claimed formula $f \cdot \dim \TT = \dim_{\ff_p} \quoz{T^1}{(T^1)^p}$ now follows.

Now, for the moment let us consider $a, b, c \in \nn$ and arbitrary elements

\begin{align*}
\mathfrak{a}_1, \dots \mathfrak{a}_{a} &\in \homm_{\vect{\ff_p}}(\quoz{\ooo}{\mmm}, k),
\\
\mathfrak{b}_1, \dots \mathfrak{b}_{b} &\in \homm_{\vect{\ff_p}}(\quoz{T^1}{(T^1)^p}, k),
\\
\mathfrak{c}_1, \dots \mathfrak{c}_{c} &\in \homm_{\vect{\ff_p}}(\quoz{\ooo}{\mmm}, k).
\end{align*}

In the next computations it will be handy to work with the following notation: for all $n \in \nn$ we define $s_{\overline{n}}$ to be $s_0$ if $n$ is even and $s_1$ if $n$ is odd.

Let $m \in \nn$ be an odd number with $m \geqslant b-1$. With the notation of Remark <ref>, we make the following computation, whose steps will be explained below.

\begin{equation}\label{eqFirstBigComputation}\begin{aligned}
&\prod_{i=1}^{a} (\mathfrak{a}_i,0,0)_1
\cdot
\prod_{i=1}^{b} (0,\mathfrak{b}_i,0)_{s_{\overline{i-1}}}
\cdot
\tau_{s_{\overline{b}} \cdots s_{\overline{m}}}
\cdot
\prod_{i=1}^{c} (0,0,\mathfrak{c}_i)_1
\\&\qquad=
\bigcupprod_{i=1}^{a} \big( (\mathfrak{a}_i,0,0)_1 \cdot \tau_{s_{\overline{0}} \cdots s_{\overline{m}}} \big)
\cupprod
\bigcupprod_{i=1}^{b}
\big(
\tau_{s_{\overline{0}} \cdots s_{\overline{i-2}}}
\cdot
\cdot
\tau_{s_{\overline{i}} \cdots s_{\overline{m}}}
\big)
\\&\qquad\qquad\cupprod
\tau_{s_{\overline{0}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{c} \big( \tau_{s_{\overline{0}} \cdots s_{\overline{m}}} \cdot (0,0,\mathfrak{c}_i)_1 \big)
\\&\qquad=
\bigcupprod_{i=1}^{a} (\mathfrak{a}_i,0,0)_{s_{\overline{0}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{b}
\big( 0,\mathfrak{b}_i(w_0^{-(i-1)} {}_- w_0^{i-1}),0 \big)_{s_{\overline{0}} \cdots s_{\overline{m}}}
\\&\qquad\qquad\cupprod
\bigcupprod_{i=1}^{c} (0,0,\mathfrak{c}_i)_{s_{\overline{0}} \cdots s_{\overline{m}}}.
\end{aligned}\end{equation}

Here, $s_{\overline{b}} \cdots s_{\overline{m}}$ means the empty product in the case $m = b-1$. To prove the first equality it suffices to apply the formula of Remark <ref>. Also note that cup product by $\tau_{s_{\overline{0}} \cdots s_{\overline{m}}}$ is the identity on $H^\ast(I,\xx(s_{\overline{0}} \cdots s_{\overline{m}}))$, and so it can be ignored. Regarding the second equality, the fact that
(\mathfrak{a}_i,0,0)_1 \cdot \tau_{s_{\overline{0}} \cdots s_{\overline{m}}}
(\mathfrak{a}_i,0,0)_{s_{\overline{0}} \cdots s_{\overline{m}}}
follows from Lemma <ref> part <ref>; the fact that

\[
\tau_{s_{\overline{0}} \cdots s_{\overline{i-2}}}
\cdot
\cdot
\tau_{s_{\overline{i}} \cdots s_{\overline{m}}}
\big( 0,\mathfrak{b}_i(w_0^{-(i-1)} {}_- w_0^{i-1}),0 \big)_{s_{\overline{0}} \cdots s_{\overline{m}}}
\]

follows from Lemma <ref> (applying iteratively parts <ref> and <ref> to compute the action of the left multiplication by $\tau_{s_{\overline{0}} \cdots s_{\overline{i-2}}}$ and applying a single time parts <ref> or <ref>, depending on the parity of $i$, to compute the action of the right multiplication by $\tau_{s_{\overline{i}} \cdots s_{\overline{m}}}$.
Finally, the fact that
\tau_{s_{\overline{0}} \cdots s_{\overline{m}}} \cdot (0,0,\mathfrak{c}_i)_1
(0,0,\mathfrak{c}_i)_{s_{\overline{0}} \cdots s_{\overline{m}}}
follows from Lemma <ref> (applying iteratively parts <ref> and <ref>): here is the only point where we use that $m$ is odd.

We have thus checked the computation (<ref>). Before deducing some consequences, let us make some completely similar computations.

Let $m \in \nn$ be an even number with $m \geqslant b-1$. The following computation is exactly as (<ref>) except that the factors on the right are different because now $m$ is even (recall that in (<ref>) we have used that $m$ was odd only in the very final step).

\begin{equation}\label{eqSecondBigComputation}\begin{aligned}
&\prod_{i=1}^{a} (\mathfrak{a}_i,0,0)_1
\cdot
\prod_{i=1}^{b} (0,\mathfrak{b}_i,0)_{s_{\overline{i-1}}}
\cdot
\tau_{s_{\overline{b}} \cdots s_{\overline{m}}}
\cdot
\prod_{i=1}^{c} (\mathfrak{c}_i,0,0)_1
\\&\qquad=
\bigcupprod_{i=1}^{a} (\mathfrak{a}_i,0,0)_{s_{\overline{0}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{b}
\big( 0,\mathfrak{b}_i(w_0^{-(i-1)} {}_- w_0^{i-1}),0 \big)_{s_{\overline{0}} \cdots s_{\overline{m}}}
\\&\qquad\qquad\cupprod
\bigcupprod_{i=1}^{c} (0,0,-\mathfrak{c}_i)_{s_{\overline{0}} \cdots s_{\overline{m}}}.
\end{aligned}\end{equation}

Now, let $m \in \zpiu$ be an odd number with $m \geqslant b$. We make the following computation

\begin{equation}\label{eqThirdBigComputation}\begin{aligned}
&\prod_{i=1}^{a} (0,0,\mathfrak{a}_i)_1
\cdot
\prod_{i=1}^{b} (0,\mathfrak{b}_i,0)_{s_{\overline{i}}}
\cdot
\tau_{s_{\overline{b+1}} \cdots s_{\overline{m}}}
\cdot
\prod_{i=1}^{c} (0,0,\mathfrak{c}_i)_1
\\&\qquad=
\bigcupprod_{i=1}^{a} \big( (0,0,\mathfrak{a}_i)_1 \cdot \tau_{s_{\overline{1}} \cdots s_{\overline{m}}} \big)
\cupprod
\bigcupprod_{i=1}^{b}
\big(
\tau_{s_{\overline{1}} \cdots s_{\overline{i-1}}}
\cdot
\cdot
\tau_{s_{\overline{i+1}} \cdots s_{\overline{m}}}
\big)
\\&\qquad\qquad\cupprod
\tau_{s_{\overline{1}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{c} \big( \tau_{s_{\overline{1}} \cdots s_{\overline{m}}} \cdot (0,0,\mathfrak{c}_i)_1 \big)
\\&\qquad=
\bigcupprod_{i=1}^{a} (0,0,\mathfrak{a}_i)_{s_{\overline{1}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{b}
\big( 0,\mathfrak{b}_i(w_0^{-(i-1)} {}_- w_0^{i-1}),0 \big)_{s_{\overline{1}} \cdots s_{\overline{m}}}
\\&\qquad\qquad\cupprod
\bigcupprod_{i=1}^{c} (-\mathfrak{c}_i,0,0)_{s_{\overline{1}} \cdots s_{\overline{m}}}.
\end{aligned}\end{equation}

Now, let $m \in \zpiu$ be an even number with $m \geqslant b$. We make the following computation

\begin{equation}\label{eqFourthBigComputation}\begin{aligned}
&\prod_{i=1}^{a} (0,0,\mathfrak{a}_i)_1
\cdot
\prod_{i=1}^{b} (0,\mathfrak{b}_i,0)_{s_{\overline{i}}}
\cdot
\tau_{s_{\overline{b+1}} \cdots s_{\overline{m}}}
\cdot
\prod_{i=1}^{c} (\mathfrak{c}_i,0,0)_1
\\&\qquad=
\bigcupprod_{i=1}^{a} (0,0,\mathfrak{a}_i)_{s_{\overline{1}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{b}
\big( 0,\mathfrak{b}_i(w_0^{-(i-1)} {}_- w_0^{i-1}),0 \big)_{s_{\overline{1}} \cdots s_{\overline{m}}}
\\&\qquad\qquad\cupprod
\bigcupprod_{i=1}^{c} (\mathfrak{c}_i,0,0)_{s_{\overline{1}} \cdots s_{\overline{m}}}.
\end{aligned}\end{equation}

Now let us go back to (<ref>) and let us draw some consequences from it. Recall that all of $a$, $b$, $c$ and all of $\mathfrak{a}_i$, $\mathfrak{b}_i$, $\mathfrak{c}_i$ were arbitrary, and $m \in \nn$ was an odd integer with $m \geqslant b-1$.

Let $m \in \nn$ be an arbitrary odd natural number. Recall that (since $m+1 \geqslant 1$) the pro-$p$ group $I_{s_{\overline{0}} \cdots s_{\overline{m}}}$ is uniform (Lemma <ref>). Therefore, the cup product algebra on $H^\ast(I, \xx(s_{\overline{0}} \cdots s_{\overline{m}})) \cong H^\ast(I_{s_{\overline{0}} \cdots s_{\overline{m}}},k)$ is the exterior algebra on $H^1$, as recalled in <ref>. The computation (<ref>) tells us that the elements of the following form lie in the image of the multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$:

\begin{equation}\label{eqCupObtained}
\bigcupprod_{i=1}^{a} (\mathfrak{a}_i,0,0)_{s_{\overline{0}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{b}
\big( 0,\mathfrak{b}_i',0 \big)_{s_{\overline{0}} \cdots s_{\overline{m}}}
\cupprod
\bigcupprod_{i=1}^{c} (0,0,\mathfrak{c}_i)_{s_{\overline{0}} \cdots s_{\overline{m}}},
\end{equation}

for $a, c \in \nn$, for $b \in \nn$ with $b \leqslant m+1$, for $\mathfrak{a}_i, \mathfrak{c}_{i} \in \homm_{\vect{\ff_p}}(\quoz{\ooo}{\mmm}, k)$ and for $\mathfrak{b}_i' \in \homm_{\vect{\ff_p}}(\quoz{T^1}{(T^1)^p}, k)$. Now let $m \geqslant f \cdot \dim \TT - 1$: in this way, since $\quoz{T^1}{(T^1)^p}$ has dimension $f \cdot \dim \TT$ as an $\ff_p$-vector space, a $k$-basis of $\homm_{\vect{\ff_p}}(\quoz{T^1}{(T^1)^p}, k)$ has cardinality less or equal than $m+1$. Then the elements of the form (<ref>) (with $b \leqslant m+1$) span the cup product algebra/exterior algebra $H^\ast(I, \xx(s_{\overline{0}} \cdots s_{\overline{m}}))$.

All in all, this shows that the image of the multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$ contains $H^\ast(I, \xx(s_{\overline{0}} \cdots s_{\overline{m}}))$ for $m \in \nn$ odd with $m \geqslant f \cdot \dim \TT - 1$. Reasoning exactly in the same way for the other computations we made, we obtain that the image of the multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$ contains the following vector spaces:

\begin{align*}
&H^\ast(I, \xx(s_{\overline{0}} \cdots s_{\overline{m}}))
&&\text{for $m \in \nn$ odd with $m \geqslant f \cdot \dim \TT - 1$,}
\\
&H^\ast(I, \xx(s_{\overline{0}} \cdots s_{\overline{m}}))
&&\text{for $m \in \nn$ even with $m \geqslant f \cdot \dim \TT - 1$,}
\\
&H^\ast(I, \xx(s_{\overline{1}} \cdots s_{\overline{m}}))
&&\text{for $m \in \zpiu$ odd with $m \geqslant f \cdot \dim \TT$,}
\\
&H^\ast(I, \xx(s_{\overline{1}} \cdots s_{\overline{m}}))
&&\text{for $m \in \zpiu$ even with $m \geqslant f \cdot \dim \TT$.}
\end{align*}

By the discussion at the beginning of this subsection, every element of $\widetilde{W}$ of length greater or equal than $f \cdot \dim \TT$ can be represented as a product $\omega w$, where $\omega \in \widetilde{\Omega}$ (i.e., $\omega$ has length zero) and $w$ is of the form $s_{\overline{0}} \cdots s_{\overline{m}}$ or $s_{\overline{1}} \cdots s_{\overline{m}}$ as above.

With this notation, we thus know that the image of the multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$ contains $H^\ast(I,\xx(w))$, and to show that it also contains $H^\ast(I,\xx(\omega w))$ it suffices to recall that, since $\omega$ has length zero, the formula (<ref>) implies that multiplication on the left by $\tau_\omega$ induces an isomorphism between $H^\ast(I,\xx(w))$ and $H^\ast(I,\xx(\omega w))$.

Assume that $p \neq 2,3$, that $\GG$ is $\SL_2$ or $\PGL_2$ and that $\field$ is an unramified extension of $\qq_p$. One has that the image of the natural multiplication map $\funcInlineNN{T^\ast_{E^0} E^1}{E^\ast}$ has finite codimension in $E^\ast$.

For all $w \in \widetilde{W}$ the $k$-vector space $H^\ast(I,\xx(w)) \cong H^\ast(I_w,k)$ is finite-dimensional since $I_w$ is a Poincaré group (Lemma <ref>). If one shows that, since $\GG$ is semisimple, there are only finitely many elements of $\widetilde{W}$ of any given length, then by Proposition <ref> we conclude.

It suffices to show the claim for $W$ in place of $\widetilde{W}$ by the definition of length and since $W$ is a quotient of $\widetilde{W}$ of finite index. Recall from <ref> that the length $\ell$ is constant on double cosets $(\Omega w \Omega)_{w \in W_{\aff}}$. Therefore, it suffices to show that $\Omega$ is finite. But $\Omega$ is isomorphic to $X_\ast(\TT)/\spann_{\zz} \Phi^\vee$ (see <cit.>, with the difference that in the construction of $W$ in loc.cit. roots are swapped with coroots and characters with cocaracters with respect to our conventions). Since $\GG$ is semisimple, $X_\ast(\TT)/\spann_{\zz} \Phi^\vee$ is indeed finite (and isomorphic to the fundamental group).

§ THE EXT-ALGEBRA E* IS FINITELY GENERATED IN SOME SPECIAL CASES

Putting together the results obtained in Proposition <ref> and Proposition <ref>, we are able to show that, under the assumptions under which those results have been proven, the $\ext$-algebra $E^\ast$ is finitely generated as a (non-commutative) $k$-algebra.

Assume that $p \neq 2,3$, that $\field$ is an unramified extension of $\qq_p$ and that $\GG$ has semisimple rank $1$. One has that $E^\ast$ is finitely generated as a $k$-algebra.

Proposition <ref> shows that

\[
\image \big( \funcInlineNN{T^\ast_{E^0} E^1}{E^\ast} \big)
\supseteq
\bigoplus_{\substack{w \in \widetilde{W} \\ \text{s.t. $\ell(w) \geqslant f \cdot \dim \TT$}}}
\]

Therefore $E^\ast$ is generated as a $k$-algebra by $E^0$, $E^1$ and

\[
\bigoplus_{\substack{w \in \widetilde{W} \\ \text{s.t. $\ell(w) < f \cdot \dim \TT$}}} H^\ast(I,\xx(w)).
\]

By (<ref>), for all $w \in \widetilde{W}$ and all $\omega \in \widetilde{\Omega}$, multiplication on the left by $\tau_\omega$ induces an isomorphism between $H^\ast(I,\xx(w))$ and $H^\ast(I,\xx(\omega w))$. Furthermore, $E^1$ is finitely generated as an $E^0$-bimodule by Proposition <ref> (in this setting a more explicit proof could also be provided starting from the formulas in Lemma <ref>).

This shows that $E^\ast$ is generated as a $k$-algebra by $E^0$, a finite subset of $E^1$ and

\[
\bigoplus_{\substack{w \in \widetilde{W_{\aff}} \\ \text{s.t. $\ell(w) < f \cdot \dim \TT$}}} H^\ast(I,\xx(w)).
\]

The last set is finite-dimensional as a $k$-vector space because there are only finitely many elements of bounded length in $W_{\aff}$ (and so in $\widetilde{W_{\aff}}$) and because for all $w \in \widetilde{W}$ the $k$-vector space $H^\ast(I,\xx(w)) \cong H^\ast(I_w,k)$ is finite-dimensional since $I_w$ is a Poincaré group (Lemma <ref>).

Therefore, to show the thesis of the proposition it suffices to notice that $E^0$ is finitely generated as a $k$-algebra. This is clear from the fact that $E^0$ is a finitely generated module over its centre, which is a finitely generated (commutative) $k$-algebra (see <cit.>), but it can also be shown more directly from the braid relations, the fact that $\widetilde{\Omega}$ is a finitely generated group, and the fact that $S_{\aff}$ is finite.

§ THE EXT-ALGEBRA E* IS FINITELY PRESENTED WHEN G = SL_2(Q_P) WITH P ≠ 2,3

For the whole section we make the following assumptions. We consider $\field \defeq \qq_p$ and $G \defeq \SL_2(\qq_p)$, and we assume that $p \neq 2,3$. Furthermore, we fix the uniformizer $\pi$ to be $p$. We consider the ($\qq_p$-split maximal) torus $\TT$ of diagonal matrices, and we fix the following pro-$p$ Iwahori subgroup $I$:

\begin{equation}\label{eqIwaSL2}
I = \matr{1+p\zz_p}{\zz_p}{p\zz_p}{1+p\zz_p} \cap \SL_2(\qq_p).
\end{equation}

(the corresponding choices of the positive root and the Chevalley system will be made explicit in <ref>).

§.§ The Ext-algebra in the case G = SL_2(Q_p): notation and recollection of  and results

Under the current assumptions, the $\ext$-algebra $E^\ast$ has been studied extensively in [18]. We need to recall quite a few facts from loc. cit. and from [17].

First of all, it follows from <cit.> that $E^\ast$ is supported in the following degrees:

\[
E^\ast = E^0 \oplus E^1 \oplus E^2 \oplus E^3.
\]

With reference to <ref>, we have a Chevalley system such that, denoting the positive root by $\alpha_0$, the map $\varphi_{\alpha_0}$ is the identity of $\SL_2$ and $\varphi_{-\alpha_0}$ is the inverse transpose. With reference to <ref>, in this setting the group $\Omega$ is trivial. Following <cit.>, we define

\begin{align}\label{eqDefs0s1Special}
&s_0 \defeq \overline{\matr{0}{1}{-1}{0}},
&&s_1 \defeq \overline{\matr{0}{-\pi^{-1}}{\pi}{0}},
\end{align}

where $\overline{?}$ denotes the image of $?$ in $\widetilde{W}$. This is almost compatible with the definition we gave in (<ref>), the only difference being that the two definitions of $s_1$ differ by multiplication by the central element $\overline{\matr{-1}{0}{0}{-1}}$. However, this will not cause any problems. For example, in <cit.> some general formulas are stated only for the specific choices (<ref>) (see the comment at the end of the statement), but the only difference being multiplication by a central element, the formulas remain valid for the lifts (<ref>). We make the choice (<ref>) to preserve compatibility of notation with <cit.>.

We also define

\begin{equation}\label{eqcminus1}
c_{-1} \defeq \overline{\matr{-1}{0}{0}{-1}} = s_0^2 = s_1^2.
\end{equation}

Since the group $\Omega$ is trivial, every element of $\widetilde{W}$ can be represented in a unique way as a product of an element of $T^0/T^1$ and a string of the elements ${s_0}$ and ${s_1}$, with no two identical letters one after the other (compare the beginning of <ref>). In other words, the following is a basis of $E^0$:

\begin{align*}
&\tau_{\omega} &&\text{for $\omega \in T^0/T^1$,}
\\
&\tau_{\omega s_0(s_1s_0)^i} &&\text{for $\omega \in T^0/T^1$ and $i \in \nn$,}
\\
&\tau_{\omega s_1(s_0s_1)^i} &&\text{for $\omega \in T^0/T^1$ and $i \in \nn$,}
\\
&\tau_{\omega(s_0s_1)^i} &&\text{for $\omega \in T^0/T^1$ and $i \in \zpiu$,}
\\
&\tau_{\omega(s_1s_0)^i} &&\text{for $\omega \in T^0/T^1$ and $i \in \zpiu$.}
\end{align*}

As in <cit.>, we parametrize the group $T^0/T^1 \cong \ff_p^\times$ as follows: for all $u \in \ff_p^\times$ we define

\begin{equation}\label{eqOmegaU}
\omega_u \defeq \overline{\matr{[u]^{-1}}{0}{0}{[u]}},
\end{equation}

where $[?]$ denotes the Teichmüller lift of $?$ and $\overline{?}$ denotes the class of $?$ in $T^0/T^1$.

In <cit.>, it is shown that the map $\invol_w$ defined by the commutativity of the diagram

\begin{equation}\label{eqqDefOfTheAntiInvol}
\begin{tikzcd}
\arrow[r, "{\invol_w}", "{\cong}"']
\arrow[d, "{\sh_w}"', "{\cong}"]
\arrow[d, "{\sh_{w^{-1}}}", "{\cong}"']
\\
\arrow[r, "{(w^{-1})_\ast}"', "{\cong}"]
\end{tikzcd}
\end{equation}

gives rise to a map $\funcInline{\invol \defeq \bigoplus_{w \in \widetilde{W}} \invol_w}{E^\ast}{E^\ast}$ that is an involutive anti-automorphism of $E^\ast$ (anti-automorphism meaning that it is an automorphism of $k$-vector spaces and that it satisfies the equation $\invol(\alpha \cdot \beta) = (-1)^{\abs{\alpha} \cdot \abs{\beta}} \invol(\beta) \cdot \invol(\alpha)$ for all homogeneous $\alpha \in E^{\abs{\alpha}}$ and $\beta \in E^{\abs{\beta}}$).

In <cit.> a certain involutive automorphism $\Gamma_\varpi$ of $E^\ast$ is constructed as follows. One defines $\varpi \defeq \matr{0}{1}{p}{0} \in \GL_2(\qq_p)$ and one considers the map $\Gamma_{\varpi,w}$ defined by the commutativity of the diagram

\[
\begin{tikzcd}
\ar[d, "{\sh_w}"', "{\cong}"]
\ar[r, "{\Gamma_{\varpi,w}}"]
H^\ast(I,\xx(\varpi w \varpi^{-1}))
\ar[d, "{\sh_{\varpi w \varpi^{-1}}}", "{\cong}"']
\\
\ar[r, "{\varpi_\ast}"]
H^\ast(I_{\varpi w \varpi^{-1}},k).
\end{tikzcd}
\]

Defining $\funcInline{\Gamma_\varpi \defeq \bigoplus_{w \in \widetilde{W}} \Gamma_{\varpi,w}}{E^\ast}{E^\ast}$, one has that $\Gamma_\varpi$ is an involutive automorphism of $E^\ast$ (as a graded $k$-algebra).
Moreover, the involutions $\Gamma_\varpi$ and $\invol$ commute (see again <cit.>):

\begin{equation}\label{eqCCCNInvolCommute}
\Gamma_\varpi \circ \invol = \invol \circ \Gamma_\varpi.
\end{equation}

The Poincaré duality of <ref> for the groups $I_w$ implies that there is a duality between $H^i(I,\xx(w))$ and $H^{3-i}(I,\xx(w))$ for all $i \in \{0,1,2,3\}$ and all $w \in \widetilde{W}$. In <cit.>, such duality is extended to a suitably defined duality of $E^0$-bimodules involving $E^i$ and $E^{3-i}$. We recall a few details. One fixes (and we will make a specific choice later) an isomorphism of $1$-dimensional vector spaces

\begin{equation}\label{eqEta}
\funcInline{\eta}{H^3(I,k)}{k}
\end{equation}

as well as  the $G$-equivariant map

\begin{equation}\label{eqDefSSS}
\func
{k \left[ \quoz{G}{I} \right]}
{\sum_{\overline{g} \in \quoz{G}{I}} f(g)}
\end{equation}

and its induced map

\[
\sss^i \defeq \funcInline{H^i(I, \sss)}{E^i = H^i\left(I, k \left[ \quoz{G}{I} \right]\right)}{H^i(I, k)}.
\]

Furthermore, one defines the finite twisted dual $\biinvol{(E^{i})^{\vee, \finite}}$ of $E^{i}$ as follows: as a $k$-vector space it is defined as

\[
\bigoplus_{w \in \widetilde{W}} H^{i}(I,\xx(w))^\vee \subseteq (E^{i})^{\vee},
\]

where $({}_-)^\vee \defeq \homm_{\vect{k}}({}_-, k)$. One may check that this vector space is actually a sub-$E^0$-bimodule of the full dual $(E^{i})^{\vee}$, but we endow $\biinvol{(E^{i})^{\vee, \finite}}$ with the following twisted action instead:

\begin{align*}
h \cdot \varphi &\defeq \varphi(\invol(h) \cdot {}_-),
\varphi \cdot h &\defeq \varphi( {}_- \cdot \invol(h))
\end{align*}

for all $h \in E^0$ and all $\varphi \in \biinvol{(E^{i})^{\vee, \finite}}$, thus justifying the complicated notation.
Having introduced these notations, we can recall the statement of the above-mentioned duality theorem; namely, the map

\[
\func
{\biinvol{(E^{3-i})^{\vee, \finite}}}
{\Big( \beta \mapsto \big( \eta \circ \sss^d \big) (\alpha \cupprod \beta) \Big)}
\]

is a well-defined isomorphism of $E^0$-bimodules. The link with Poincaré duality of the groups $I_w$ is as follows: The Poincaré duality of <ref> defines an isomorphism of $k$-vector spaces

\begin{equation}\label{eqDualityIso}
\func
{\big( H^{3-i}(I_w,k) \big)^\vee}
{\big( \beta \mapsto \eta_w (\alpha \cupprod \beta) \big),}
\end{equation}

once one fixes an isomorphism of $1$-dimensional $k$-vector spaces

\[
\funcInline{\eta_w}{H^3(I_w,k)}{k}
\]

and we make the choice $\eta_w \defeq \eta \circ \cores^{I_w}_I$ (recalling from <cit.> that the corestriction is an isomorphism on the top graded piece). With these choices the following diagram of solid arrows commutes (see <cit.>):

\begin{equation}\label{eqDiagramDualities}\begin{tikzcd}
\ar[d, "\sh_w", "\cong"']
\ar[r, "{\Delta^i_w}"]
\ar[d, leftarrow, "\sh_w^\vee"', "\cong"]
\\
\ar[d, hook]
\ar[r, dashed]
\ar[d, hook]
\\
\ar[r, "\Delta^i"]
\biinvol{(E^{3-i})^{\vee, \finite}}
\end{tikzcd}\end{equation}

One can thus define a dashed map making the whole diagram commute either as the unique map making the upper square commute or as the restriction of $\Delta^i$.

Following <cit.>, we want to fix $k$-bases for the graded pieces of $E^\ast$ in a compatible way.
Denoting $({}_-)^\vee \defeq \homm_{\vect{\ff_p}}({}_-,k)$ let us recall from (<ref>) that for all $w \in \widetilde{W}$ we have fixed a specific isomorphism

\begin{equation}\label{eqIsoE1AsATripleSL2Qp}
\funcInline
{({}_-, {}_-, {}_-)_w}
{(\ff_p)^\vee \oplus (T^1_w)^\vee \oplus \ff_p^\vee}
\end{equation}

Using the $p$-adic logarithm and exponential, one sees that $(1+\mmm)^p = 1+\mmm^2$, and therefore

\begin{align*}
\quoz{(1+\mmm)}{\big( (1+\mmm^{\ell(w)+1})\cdot (1+\mmm)^p \big)}
\\&\cong
\begin{cases}
\ff_p &\text{if $\ell(w) \geqslant 1$,}
\\
0 &\text{if $\ell(w) = 0$.}
\end{cases}
\end{align*}

Henceforth, in the case that $\ell(w) \geqslant 1$ we fix the following isomorphism, induced by the $p$-adic logarithm:

\begin{equation}
\label{eqDefIota}
\func
{T^1_w \cong \quoz{(1+\mmm)}{(1+\mmm^2)}}
{\quoz{\ooo}{\mmm} = \ff_p}
\end{equation}

Dualizing the isomorphism (<ref>) we obtain an isomorphism

\[
\funcAbove{\cong}{H^1(I,\xx(w))^\vee}{\ff_p \oplus T^1_w \oplus \ff_p}
\]

We combine the inverse of this isomorphism with the inverse of the isomorphism $\funcInlineNN{H^2(I,\xx(w))}{H^1(I,\xx(w))^\vee}$ defined in the diagram (<ref>), obtaining a composite isomorphism

\[\begin{tikzcd}
\ff_p \oplus T^1_w \oplus \ff_p
\ar[r, "\cong"]
\ar[r, "\cong"]
\end{tikzcd}\]

for which we will use the notation

\[
\funcInline{({}_-,{}_-,{}_-)_w}{\ff_p \oplus T^1_w \oplus \ff_p}{H^2(I,\xx(w))}
\]

similarly to the isomorphism describing $H^1(I,\xx(w))$.

Now, let us fix an element $\mathbf{c} \in \homm_{\vect{\ff_p}} \left( \ff_p, k \right)$ and an element $\mathbfSpecial{\alpha} \in \ff_p$ with the property that

\[
\mathbf{c}(\mathbfSpecial{\alpha}) = 1.
\]

For all $w \in \widetilde{W}$ we define the following $k$-basis of $H^1(I,\xx(w))$:

\begin{align*}
\bm{w} &\defeq (\mathbf{c},0,0)_w,
\\
\bp{w} &\defeq (0,0,\mathbf{c})_w,
\\
\bz{w} &\defeq (0,\mathbf{c}\iota,0)_w \qquad\qquad\text{if $\ell(w) \geqslant 1$,}
\end{align*}

and for all $w \in \widetilde{W}$ we define the following $k$-basis of $H^2(I,\xx(w))$:

\begin{align*}
\am{w} &\defeq (\mathbfSpecial{\alpha},0,0)_w,
\\
\ap{w} &\defeq (0,0,\mathbfSpecial{\alpha})_w,
\\
\az{w} &\defeq (0,\iota^{-1}(\mathbfSpecial{\alpha}),0)_w \qquad\qquad\text{if $\ell(w) \geqslant 1$.}
\end{align*}

Furthermore, we define the basis $(\phi_v)_{v \in \widetilde{W}}$ of $E^3$ as the dual basis of $(\tau_v)_{v \in \widetilde{W}}$ (dual with respect to the duality isomorphism (<ref>) with $i = 3$), meaning that for all $w \in \widetilde{W}$ the element $\phi_w$ is characterized by the properties

\begin{align*}
\left( \eta \circ \sss^d \right) (\phi_w \cupprod \tau_w) &= 1,
\\
\left( \eta \circ \sss^d \right) (\phi_w \cupprod \tau_v) &= 0 &&\text{for all $v \in \widetilde{W} \smallsetminus \{w\}$.}
\end{align*}

As anticipated above, we make a specific choice for $\funcInline{\eta}{H^3(I,k)}{k}$. In <cit.>, it is shown that there exists a (necessarily unique) choice of $\eta$ such that the following property holds:

\begin{align}\label{eqCond2}
&\bm{w} \cupprod \bz{w} \cupprod \bp{w} = \phi_w &&\text{for all $w \in \widetilde{W}$ such that $\ell(w) \geqslant 1$.}
\end{align}

Henceforth, we will always work with this fixed choice of $\eta$.

It is possible to show that the following relations hold (see <cit.>):

\begin{align}
\az{w} &= \bp{w} \cupprod \bm{w}
&&\text{for all $w \in \widetilde{W}$ such that $\ell(w) \geqslant 1$,}
\notag
\\
\am{w} &= \bz{w} \cupprod \bp{w}
&&\text{for all $w \in \widetilde{W}$ such that $\ell(w) \geqslant 1$,}
\label{eqFormulaCupUniform}
\\
\ap{w} &= \bm{w} \cupprod \bz{w}
&&\text{for all $w \in \widetilde{W}$ such that $\ell(w) \geqslant 1$.}
\notag
\end{align}

We have thus fixed the following $k$-basis of $H^\ast(I,\xx(w))$:

\begin{equation}\label{eqBasis}
\begin{aligned}
\\
&\bm{w}, &&\qquad\qquad\bz{w} \quad\text{(if $\ell(w) \geqslant 1$),} &&\qquad\qquad\bp{w},
\\
&\am{w}, &&\qquad\qquad\az{w} \quad\text{(if $\ell(w) \geqslant 1$),} &&\qquad\qquad\ap{w},
\\
\end{aligned}
\end{equation}

It is useful to introduce some idempotent elements of $E^0$. We denote by $\widehat{T^0/T^1}$ the group of characters $\funcInlineNN{T^0/T^1}{k^\times}$. To each $\lambda \in \widehat{T^0/T^1}$ we attach

\begin{equation}\label{eqDefELambda}
e_\lambda \defeq - \sum_{t \in T^0/T^1} \lambda(t)^{-1} \tau_{t}.
\end{equation}

Using the braid relations and that $T^0/T^1 \cong \ff_p^\times$, one sees that $(e_\lambda)_{\lambda \in \widehat{T^0/T^1}}$ is a system of orthogonal idempotents, that they form a basis of the subalgebra $\spann_k \set{\tau_t}{t \in \quoz{T^0}{T^1}} \cong k[T^0/T^1]$ of $E^0$, and that the formula

\begin{equation}\label{eqELambdaTauT}
e_\lambda \cdot \tau_t = \tau_t \cdot e_\lambda = \lambda(t) e_\lambda
\end{equation}

holds for all $\lambda \in \widehat{T^0/T^1}$ and all $t \in T^0/T^1$ (compare <cit.>). With this notation, $e_1$ denotes the idempotent associated with the trivial character. Furthermore, we define

\begin{equation}\label{eqDefIdd}
\func
\end{equation}

where the notation $\omega_u$ has been introduced in (<ref>). We may thus write $e_{\idd}$ for the corresponding idempotent. More generally, every idempotent of the form (<ref>) can be written as $e_{\idd^m}$ for some uniquely determined $m \in \{0,\dots,p-2\}$.

Using the basis (<ref>), we will now recall the formulas for the multiplication on $E^0$, for the left action of $E^0$ on $E^\ast$ (and partially, also on the right) as well as the action of the anti-involution $\invol$ and of the involutive automorphism $\Gamma_\varpi$.

* Braid and quadratic relations on $E^0$ (see <cit.>):

One has:

\begin{align}\label{eqQuadrRelSL2}
&\tau_{v} \cdot \tau_w = \tau_{vw} &&\text{for $v,w \in \widetilde{W}$ such that $\ell(vw)=\ell(v)+\ell(w)$,}
\\
&\tau_{s_i}^2 = - e_1 \cdot \tau_{s_i} = - \tau_{s_i} \cdot e_1,
&&\text{for $i \in \{0,1\}$.}
\end{align}

* Action of the anti-involution $\invol$ on $E^0$ (see <cit.>):

For all $w \in \widetilde{W}$ one has:

\begin{equation}\label{eqInvolTauw}
\invol(\tau_w) = \tau_{w^{-1}}.
\end{equation}

* Action of the involutive automorphism $\Gamma_\varpi$ on $E^0$:

For all $w \in \widetilde{W}$ one has that

\[
\Gamma_\varpi(\tau_w) = \tau_{\varpi w \varpi^{-1}}
\]

(immediate from the definition of $\Gamma_\varpi$). With an explicit computation, this can be made explicit as follows:

\begin{equation}\label{eqCCCNOnH}\begin{aligned}
\Gamma_\varpi(\tau_\omega) &= \tau_{\omega^{-1}} \qquad\qquad\qquad\qquad\qquad \text{for all $\omega \in \quoz{T^0}{T^1}$,}
\\
\Gamma_\varpi(\tau_{s_0}) &= \tau_{s_1},
\\
\Gamma_\varpi(\tau_{s_1}) &= \tau_{s_0}.
\end{aligned}\end{equation}

* Left and right action of $\tau_{\omega}$ on $E^1$ for $\omega \in \quoz{T^0}{T^1}$ (see <cit.>):

Let $u \in \ff_p^\times$ and let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\bz{w}$). Recall the notation $\omega_u$ from (<ref>). One has:

\begin{equation}\label{eqOmegaOnE1Left}\begin{aligned}
\tau_{\omega_{u}} \cdot \bm{w} &= u^{-2} \bm{\omega_u w},
\\
\tau_{\omega_{u}} \cdot \bz{w} &= \bz{\omega_u w},
\\
\tau_{\omega_{u}} \cdot \bp{w} &= u^2 \bp{\omega_u w},
\end{aligned}\end{equation}

\begin{equation}\label{eqOmegaOnE1Right}\begin{aligned}
\bm{w} \cdot \tau_{\omega_{u}} &=\bm{w \omega_{u}},
\\
\bz{w} \cdot \tau_{\omega_{u}} &= \bz{w \omega_{u}},
\\
\bp{w} \cdot \tau_{\omega_{u}} &= \bp{w \omega_{u}}.
\end{aligned}\end{equation}

* Action of the idempotents on $E^1$:

Let $\lambda \in \widehat{\quoz{T^0}{T^1}}$ and let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\bz{w}$). Recalling the notation (<ref>) and (<ref>), one has:

\begin{equation}\label{eqIdempotentsE1}\begin{aligned}
\bm{w} \cdot e_\lambda &= e_{\lambda^{(-1)^{\ell(w)}} \idd^{-2}} \cdot \bm{w},
\\
\bz{w} \cdot e_\lambda &= e_{\lambda^{(-1)^{\ell(w)}}} \cdot \bz{w},
\\
\bp{w} \cdot e_\lambda &= e_{\lambda^{(-1)^{\ell(w)}} \idd^{2}} \cdot \bp{w}.
\end{aligned}\end{equation}

These formulas can be easily computed from formulas (<ref>) and (<ref>), and they are also proven in <cit.> (for $\lambda = \idd^m$ for some $m \in \zz$, i.e., for every $\lambda$).

* Left action of $\tau_{s_0}$ and $\tau_{s_1}$ on $E^1$ when lengths add up:

Let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\bz{w}$). One has:

\begin{equation}\label{eqFormulasTauQp}\begin{aligned}
\tau_{s_0} \cdot \bm{w}
\\
\tau_{s_0} \cdot \bz{w}
\\
\tau_{s_0} \cdot \bp{w}
\end{array}
\quad\right\}\qquad\text{if $\ell(s_0 w) = \ell(w) + 1$,}
\\
\tau_{s_1} \cdot \bm{w}
\\
\tau_{s_1} \cdot \bz{w}
-\bz{s_1 w}
\\
\tau_{s_1} \cdot \bp{w}
-\bm{s_1 w}
\end{array}
\quad\right\}\qquad\text{if $\ell(s_1 w) = \ell(w) + 1$.}
\end{aligned}\end{equation}

These formulas are proved in <cit.>, and they also follow from Lemma <ref> parts <ref> and <ref>.

* Right action of $\tau_v$ on $E^1$ when lengths add up (for $v \in \widetilde{W}$):

Let $w, v \in \widetilde{W}$ such that $\ell(wv) = \ell(w) + \ell(v)$ and such that $\ell(v) \geqslant 1$ (and furthermore let us assume that $\ell(w) \geqslant 1$ when we speak about $\bz{w}$). One has:

\begin{equation}\label{eqFormulaRighteasyQp}\begin{aligned}
\bm{w} \cdot \tau_{v}
\bm{wv}
\\
\bz{w} \cdot \tau_{v}
\bz{wv}
\\
\bp{w} \cdot \tau_{v}
\end{array}
\quad\right\}\qquad\text{if $\ell(s_1 w v) = \ell(w v) + 1$,}
\\
\bm{w} \cdot \tau_{v}
\\
\bz{w} \cdot \tau_{v}
\bz{wv}
\\
\bp{w} \cdot \tau_{v}
\bp{wv}
\end{array}
\quad\right\}\qquad\text{if $\ell(s_0 w v) = \ell(w v) + 1$.}
\end{aligned}\end{equation}

These formulas are proved in <cit.>, and they also follow from Lemma <ref> parts <ref> and <ref>.

* Left action of $\tau_{s_0}$ and $\tau_{s_1}$ on $E^1$ when lengths do not add up:

Let $w \in \widetilde{W}$ be such that $\ell(s_0w) = \ell(w) - 1$. One has:

\begin{equation}\label{eqTauS0OnE1LeftNotAddUp}\begin{aligned}
\tau_{s_0} \cdot \bm{w}
\begin{cases}
-e_1\bm{w} -2e_{\idd} \bz{w} -\bp{s_0w}
&\text{if $\ell(w) \geqslant 2$,}
\\
-e_1\bm{w} -2e_{\idd} \bz{w} + e_{\idd^2} \bp{w} - \bp{s_0w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\\
\tau_{s_0} \cdot \bz{w}
\begin{cases}
&\text{if $\ell(w) \geqslant 2$,}
\\
-e_1\bz{w} + e_{\idd} \bp{w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\\
\tau_{s_0} \cdot \bp{w}
\end{aligned}\end{equation}

Let $w \in \widetilde{W}$ be such that $\ell(s_1w) = \ell(w) - 1$. One has:

\begin{equation}\label{eqTauS1OnE1LeftNotAddUp}\begin{aligned}
\tau_{s_1} \cdot \bm{w}
-e_1 \bm{w}
\\
\tau_{s_1} \cdot \bz{w}
\begin{cases}
-e_1 \bz{w}
&\text{if $\ell(w) \geqslant 2$,}
\\
-e_1 \bz{w} - e_{\idd^{-1}}\bm{w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\\
\tau_{s_1} \cdot \bp{w}
\begin{cases}
-e_1 \bp{w} + 2e_{\idd^{-1}}\bz{w} - \bm{s_1w}
&\text{if $\ell(w) \geqslant 2$,}
\\
-e_1 \bp{w} + 2e_{\idd^{-1}} \bz{w} + e_{\idd^{-2}} \bm{w} - \bm{s_1w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\end{aligned}\end{equation}

This is proved in <cit.>.

* Right action of $\tau_{s_0}$ and $\tau_{s_1}$ on $E^1$ when lengths do not add up (some cases):

Let $v \in \widetilde{W}$ such that $\ell(s_0v) = \ell(v) - 1$. One has:

\begin{equation}\label{eqRightBadLengthS0}
\bz{s_0} \cdot \tau_v = -e_1\bz{v} - e_{\idd^{-1}}\bm{v}.
\end{equation}

Let $v \in \widetilde{W}$ such that $\ell(s_1v) = \ell(v) - 1$. One has:

\begin{equation}\label{eqRightBadLengthS1}
\bz{s_1} \cdot \tau_v = -e_1\bz{v} + e_{\idd}\bp{v}.
\end{equation}

This is proved in <cit.>.

* Action of the anti-involution $\invol$ on $E^1$ (see <cit.>):

Let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\bz{w}$) and let $u_w \in \ff_p^\times$ be such that $\omega_{u_w}^{-1}w$ lies in the subgroup of $\widetilde{W}$ generated by $s_0$ and $s_1$ (it is easy to see that such $u_w$ exists and, although it is not unique, it has the property that $u_w^2$ is uniquely determined by $w$; for the notation $\omega_?$ see (<ref>)). One has:

\begin{equation}
\label{eqInvolOnE1}
\begin{aligned}
\invol (\bm{w})
\begin{cases}
u_w^2 \bm{w^{-1}} &\text{if $\ell(w)$ is even,}
\\
- u_w^{2} \bp{w^{-1}} &\text{if $\ell(w)$ is odd,}
\end{cases}
\\
\invol (\bz{w})
\begin{cases}
\bz{w^{-1}} &\text{if $\ell(w)$ is even,}
\\
-\bz{w^{-1}} &\text{if $\ell(w)$ is odd,}
\end{cases}
\\	\invol (\bp{w})
\begin{cases}
u_w^{-2} \bp{w^{-1}} &\text{if $\ell(w)$ is even,}
\\
- u_w^{-2} \bm{w^{-1}} &\text{if $\ell(w)$ is odd.}
\end{cases}
\end{aligned}
\end{equation}

* Action of the involutive automorphism $\Gamma_\varpi$ on $E^1$ (see <cit.>):

Let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\bz{w}$). One has:

\begin{equation}\label{eqCCCNOnE1}\begin{aligned}
\Gamma_\varpi (\bm{w})
\bp{\varpi w \varpi^{-1}},
\\
\Gamma_\varpi (\bz{w})
-\bz{\varpi w \varpi^{-1}},
\\
\Gamma_\varpi (\bp{w})
\bm{\varpi w \varpi^{-1}}.
\end{aligned}\end{equation}

* Left and right action of $\tau_{\omega}$ on $E^2$ for $\omega \in \quoz{T^0}{T^1}$:

Let $u \in \ff_p^\times$, let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\az{w}$). Recall the notation $\omega_u$ from (<ref>). One has:

\begin{equation}\label{eqTauOmegaE2Left}\begin{aligned}
\tau_{\omega_u} \cdot \am{w}
u^2 \am{\omega_u w},
\\
\tau_{\omega_u} \cdot \az{w}
\az{\omega_u w},
\\
\tau_{\omega_u} \cdot \ap{w}
u^{-2} \ap{\omega_u w},
\end{aligned}\end{equation}

\begin{equation}\label{eqTauOmegaE2Right}\begin{aligned}
\am{w} \cdot \tau_{\omega_u}
\am{w \omega_u},
\\
\az{w} \cdot \tau_{\omega_u}
\az{w \omega_u},
\\
\ap{w} \cdot \tau_{\omega_u}
\ap{w \omega_u}.
\end{aligned}\end{equation}

For the proof of the first formula see <cit.>. The second formula can be proved exactly in the same way using the corresponding formula for $E^1$, or from the first formula by using the anti-involution (see (<ref>) later).

* Action of the idempotents on $E^2$:

Let $\lambda \in \widehat{\quoz{T^0}{T^1}}$ and let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\az{w}$). One has:

\begin{equation}\label{eqFormulasIdempotentsLeftRight}
\begin{aligned}
\am{w} \cdot e_\lambda
e_{\lambda^{(-1)^{\ell(w)}} \cdot \idd^2} \cdot \am{w},
\\
\az{w} \cdot e_\lambda
e_{\lambda^{(-1)^{\ell(w)}}} \cdot \az{w}
\qquad\qquad\qquad\text{(if $\ell(w) \geqslant 1$)},
\\
\ap{w} \cdot e_\lambda
e_{\lambda^{(-1)^{\ell(w)}} \cdot \idd^{-2}} \cdot \ap{w}.
\end{aligned}
\end{equation}

These formulas can be proved using (<ref>) and (<ref>).

* Left action of $\tau_{s_0}$ and $\tau_{s_1}$ when lengths add up (see <cit.>):

Let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\az{w}$). One has:

\begin{equation}\label{eqExplicitDeg2AddUp}\begin{aligned}
\tau_{s_0} \cdot \am{z}
\\
\tau_{s_0} \cdot \az{z}
\\
\tau_{s_0} \cdot \ap{z}
-\am{s_0 w}
\end{array}
\quad\right\}\qquad\text{if $\ell(s_0 w) = \ell(w) + 1$,}
\\
\tau_{s_1} \cdot \am{z}
-\ap{s_1 w}
\\
\tau_{s_1} \cdot \az{z}
\\
\tau_{s_1} \cdot \ap{z}
\end{array}
\quad\right\}\qquad\text{if $\ell(s_1 w) = \ell(w) + 1$.}
\end{aligned}\end{equation}

* Left action of $\tau_{s_0}$ and $\tau_{s_1}$ when lengths do not add up (see <cit.>):

Let $w \in \widetilde{W}$ be such that $\ell(s_0w) = \ell(w) - 1$. One has:

\begin{equation}\label{eqTauS0OnE2LeftNotAddUp}\begin{aligned}
\tau_{s_0} \cdot \am{w}
- e_1 \am{w}
\\
\tau_{s_0} \cdot \az{w}
\begin{cases}
-e_1 \az{w} + 2 e_{\idd} \am{w} - \az{s_0 w}
&\text{if $\ell(w) \geqslant 2$,}
\\
-e_1 \az{w} + 2 e_{\idd} \am{w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\\
\tau_{s_0} \cdot \ap{w}
\begin{cases}
-e_1 \ap{w} - \am{s_0 w}
&\text{if $\ell(w) \geqslant 2$,}
\\
- e_1 \ap{w} - e_{\idd} \az{w} + e_{\idd^2} \am{w} - \am{s_0 w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\end{aligned}\end{equation}

Let $w \in \widetilde{W}$ be such that $\ell(s_1w) = \ell(w) - 1$. One has:

\begin{equation}\label{eqTauS1OnE2LeftNotAddUp}\begin{aligned}
\tau_{s_1} \cdot \am{w}
\begin{cases}
- e_1 \am{w} - \ap{s_1 w}
&\text{if $\ell(w) \geqslant 2$,}
\\
- e_1 \am{w} + e_{\idd^{-1}} \az{w} + e_{\idd^{-2}} \ap{w} - \ap{s_1 w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\\
\tau_{s_1} \cdot \az{w}
\begin{cases}
- e_1 \az{w} -2 e_{\idd^{-1}} \ap{w} - \az{s_1 w}
&\text{if $\ell(w) \geqslant 2$,}
\\
- e_1 \az{w} -2 e_{\idd^{-1}} \ap{w}
&\text{if $\ell(w) = 1$,}
\end{cases}
\\
\tau_{s_1} \cdot \ap{w}
- e_1 \ap{w}
\end{aligned}\end{equation}

* Action of the anti-involution $\invol$ on $E^2$ (see <cit.>):

Let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\az{w}$) and let $u_w \in \ff_p^\times$ as in (<ref>). One has:

\begin{equation}\label{eqInvolOnE2}\begin{aligned}
\invol (\am{w})
\begin{cases}
u_w^{-2} \am{w^{-1}} &\text{if $\ell(w)$ is even,}
\\
- u_w^{-2} \ap{w^{-1}} &\text{if $\ell(w)$ is odd,}
\end{cases}
\\
\invol (\az{w})
\begin{cases}
\az{w^{-1}} &\text{if $\ell(w)$ is even,}
\\
-\az{w^{-1}} &\text{if $\ell(w)$ is odd,}
\end{cases}
\\
\invol (\ap{w})
\begin{cases}
u_w^{2} \ap{w^{-1}} &\text{if $\ell(w)$ is even,}
\\
- u_w^{2} \am{w^{-1}} &\text{if $\ell(w)$ is odd.}
\end{cases}
\end{aligned}\end{equation}

* Action of the involutive automorphism $\Gamma_\varpi$ on $E^2$ (see <cit.>):

Let $w \in \widetilde{W}$ (with the understanding that $\ell(w) \geqslant 1$ when we speak about $\az{w}$). One has:

\begin{equation}\label{eqCCCNOnE2}\begin{aligned}
\Gamma_\varpi (\am{w})
\ap{\varpi w \varpi^{-1}},
\\
\Gamma_\varpi (\az{w})
-\az{\varpi w \varpi^{-1}},
\\
\Gamma_\varpi (\ap{w})
\am{\varpi w \varpi^{-1}}.
\end{aligned}\end{equation}

* Left and right action of $E^0$ on $E^3$:

For all $\omega \in \quoz{T^0}{T^1}$, for all $j \in \{0,1\}$ and for all $w \in \widetilde{W}$ one has

\begin{equation}\label{eqFormulasTopDegSL2}\begin{aligned}
\tau_\omega \cdot \phi_w &= \phi_{\omega w},
\\
\phi_w \cdot \tau_\omega &= \phi_{w \omega},
\\
\tau_{s_j} \cdot \phi_w
\begin{cases}
\phi_{s_j w} - e_1 \cdot \phi_w &\text{if } \ell(s_j w) = \ell(w) - 1,\\
0 &\text{if } \ell(s_j w) = \ell(w) + 1,
\end{cases}
\\
\phi_w \cdot \tau_{s_j}
\begin{cases}
\phi_{w s_j} - \phi_w \cdot e_1 &\text{if } \ell(w s_j) = \ell(w) - 1,\\
0 &\text{if } \ell(w s_j) = \ell(w) + 1.
\end{cases}
\end{aligned}\end{equation}

These formulas follow from the general ones in <cit.>.

* Action of the anti-involution $\invol$ on $E^3$ (see <cit.>):

For all $w \in \widetilde{W}$ one has

\begin{equation}\label{eqInvolOnE3}
\invol(\phi_w) = \phi_{w^{-1}}.
\end{equation}

* Action of the involutive automorphism $\Gamma_\varpi$ on $E^3$ (see <cit.>):

For all $w \in \widetilde{W}$ one has:

\begin{equation}\label{eqCCCNOnE3}
\Gamma_\varpi(\phi_w) = \phi_{\varpi w \varpi^{-1}}.
\end{equation}

Recall that $E^1$ is finitely-generated as an $E^0$-bimodule (Proposition <ref>). In the next lemma we compute a (nice) finite set of generators, along with some useful formulas.

One has the following facts.

* The following formulas hold:

\begin{align}
\bm{1} \cdot \tau_w &= \bm{w}
\label{eqA}
&&\substack{\text{for $w \in \widetilde{W}$ with $\ell(s_1w) = \ell(w) + 1$,}}
\\
\bp{1} \cdot \tau_w &= \bp{w}
\label{eqB}
&&\substack{\text{for $w \in \widetilde{W}$ with $\ell(s_0w) = \ell(w) + 1$,}}
\\
\tau_{(s_1s_0)^i} \cdot \bm{1} &= \bm{(s_1s_0)^i}
\label{eqC}
&&\substack{\text{for all $i \in \nn$,}}
\\
\tau_{s_0(s_1s_0)^i} \cdot \bm{1} &= - \bp{s_0(s_1s_0)^i}
\label{eqD}
&&\substack{\text{for all $i \in \nn$,}}
\\
\tau_{(s_0s_1)^i} \cdot \bp{1} &= \bp{(s_0s_1)^i}
\label{eqE}
&&\substack{\text{for all $i \in \nn$,}}
\\
\tau_{s_1(s_0s_1)^i} \cdot \bp{1} &= -\bm{s_1(s_0s_1)^i}
\label{eqF}
&&\substack{\text{for all $i \in \nn$,}}
\\
\bz{s_i} \cdot \tau_{w} &= \bz{s_iw}
\label{eqG}
&&\substack{\text{for $i \in \{0,1\}$ and $w \in \widetilde{W}$}\\\text{with $\ell(s_iw) = \ell(w) + 1$,}}
\\
\tau_w \cdot \bz{s_i} &= (-1)^{\ell(w)} \bz{ws_i}
\label{eqH}
&&\substack{\text{for $i \in \{0,1\}$ and $w \in \widetilde{W}$}\\\text{with $\ell(ws_i) = \ell(w) + 1$.}}
\end{align}

* The following elements generate $E^1$ as an $E^0$-bimodule:

\begin{align*}
\end{align*}

Let us prove the two statements.

* Formulas (<ref>), (<ref>) and (<ref>) are immediate consequence of the formulas describing the right action of $E^0$ when lengths add up (namely, formulas (<ref>) in the case that $\ell(w) \geqslant 1$ and formulas (<ref>) in the case that $\ell(w)=0$). Formulas (<ref>), (<ref>), (<ref>) and (<ref>) can be shown using the formulas describing the left action of $E^0$ when lengths add up (namely, formulas (<ref>)). The same is true for formula (<ref>), also recalling the left action of $\tau_\omega$ for $\omega \in \quoz{T^0}{T^1}$ of formula (<ref>).

* This is a consequence of part (i). More precisely, for all $v \in \widetilde{W}$ we want to show that the elements $\bm{v}$, $\bp{v}$ and $\bz{v}$ (the last one if $\ell(v) \geqslant 1$) lie in the sub-$E^0$ bimodule generated by the four elements in the statement. For $\bz{v}$ this is clear form formula (<ref>) (or from formula (<ref>)). For $\bm{v}$ and $\bp{v}$, using (<ref>) (or (<ref>)), we might assume that $v$ is of the form $(s_1s_0)^i$, $s_0(s_1s_0)^i$, $(s_0s_1)^i$ or $s_1(s_0s_1)^i$ for some $i \in \nn$. Then we can apply the formulas in part (i) to conclude.

§.§ Outline of the strategy for the first main result

Let us consider the multiplication map

\[
\funcInline
{T_{E^0}^\ast E^1}
\]

where the left hand side is the tensor algebra generated by the $E^0$-bimodule $E^1$. Proposition <ref> can be used to prove surjectivity of $\mult$, even if it does not directly imply such property (see Lemmas <ref> and <ref>).
The first main result we want to prove is the computation of the kernel of $\mult$ (Theorem <ref>), hence obtaining a presentation of $E^\ast$.

This computation will be quite long and will require many intermediate statements, and we now outline the strategy of the proof, working in a general abstract setting to simplify the notation.

Let $A$ be an associative $k$-algebra with $1$ (in our case $A=E^0$); let $M$ and $N$ be $A$-bimodules (in our case, at first, $M = E^1 \otimes_{E^0} E^1$ and $N=E^2$) and let $\funcInline{\mult}{M}{N}$ be a surjective homomorphism of $A$-bimodules (in our case $\mult$ will be the above multiplication map, at first restricted to $E^1 \otimes_{E^0} E^1$). To compute the kernel of $\mult$ we fix the following:

* generators $(a_i)_{i \in \mathbf{I}}$ of $A$ as a $k$-algebra (for a suitable index set $\mathbf{I}$);

* generators $(m_j)_{j \in \mathbf{J}}$ of $M$ as an $A$-bimodule (for a suitable index set $\mathbf{J}$);

* a basis $(n_l)_{l \in \mathbf{L}}$ of $N$ as a $k$-vector space (for a suitable index set $\mathbf{L}$).