$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
is an exact category. Also,
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
is a full exact subcategory of
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
which is closed under extensions. Moreover, for any left cell
$\omega\in\Omega$ and any $J\subseteq S$, there is a vanishing

$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}({\mathscr{L}}^{*})}(S_{\omega},x_{J}{\mathcal{H}})=0$
(2.3.5)

in both of these categories.

###### Proof.

The category
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
is the (contravariant) dual
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$. Hence,
it is exact, since
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$ is exact
by Proposition 2.1.3. Similarly,
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
is exact, by Corollary 2.1.8. Clearly,
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
is a full exact subcategory of
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
and is closed under extensions by Proposition 2.1.7. The displayed vanishing,
in the case of
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$,
follows from Theorem 2.1.6 and duality. However, we have already noted that
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
is an exact subcategory of
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$,
and ${\mathscr{A}}_{\flat}({\mathscr{L}}^{*})$ is full in
${\mathscr{A}}({\mathscr{L}}^{*})$, so the displayed vanishing also holds for
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$.
This completes the proof of the proposition. ∎

We conclude this section with the following proposition, a version of
Corollary 2.1.10 in our present dual setting. The proposition is essentially
equivalent to that corollary, and can be deduced from it using the duality
functor $(-)^{*}$. We omit the details. An alternative argument can be given
using the duality functors for part (a); then using the $3\times 3$-Lemma. See
the proof of Proposition 2.1.10(b).

######  Proposition 2.3.3.

(a) If $M\in{\mathscr{A}}_{\flat}({\mathscr{L}}^{*})$ has the
$\mathfrak{h}$-filtration (2.3.1), and $a\in\mathbb{Z}$ with $0\leq a\leq n$.
Then,

$(M^{a}\to M\to M/M^{a})\in{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}).$

In particular, each inclusion $M^{h-1}\subseteq M^{h}$ gives a conflation
$(M^{h-1}\to M^{h}\to M^{h}/M^{h-1})\in{\mathscr{E}}^{\flat}({\mathscr{L}})$.

(b) Suppose $(M\to N\to P)\in{\mathscr{E}}_{\flat}({\mathscr{L}}^{*})$. If
$h\in\mathbb{Z}$, then both $(M^{h}\to N^{h}\to P^{h})$ and $(M/M^{h}\to
N/N^{h}\to P/P^{h})$ belong to ${\mathscr{E}}_{\flat}({\mathscr{L}}^{*})$.

###  2.4. Another abstract setting

Our main goal in this section is to prove Theorem 2.4.10, which builds a kind
of injective envelope for certain exact categories, improving upon [DPS17, Th.
4.7]. This goal is achieved by refining the abstract framework used in the
earlier paper. At least some abstraction is desirable. The category
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
used in the main Theorem 3.3.1 is not easy to investigate directly, but it
inherits many properties from its “dual”
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$.
Those dual properties are often most easily checked in the abstract setting of
an exact category satisfying Conditions 2.4.2 below.

We follow the notation of [DPS17, Assump. 3.2 and Constr. 3.8].555In [DPS17,
Const.3.8] at the end of the first paragraph, it should be assumed that
$M^{j}/M^{j-1}$ is isomorphic to, not necessarily equal to, an object in
${\mathscr{S}}$ for all $j$. This is not a problem usually, since one often
requires that ${\mathscr{S}}$ be closed under isomorphism. Thus,
${\mathscr{K}}$ is a Noetherian integral domain with fraction field $K$. Also,
$H$ is a finite and torsion-free ${\mathscr{K}}$-algebra such that $H_{K}$ is
semisimple. Thus, $H_{K}$ has a finite set $\Lambda$ of isomorphism classes of
irreducible modules. In this section, “modules” for $H$ and $H_{K}$ are taken
as right modules, We are given a “height function”
$\mathfrak{h}:\Lambda\to\mathbb{Z}$, as defined in Appendix B. We will assume
$\mathfrak{h}$ takes positive values only on $\Lambda$, unless noted
otherwise.

Often ${\mathscr{K}}$ is the algebra ${\mathcal{Z}}=\mathbb{Z}[t,t^{-1}]$ of
Laurent polynomials, and $H=\mathcal{H}$ is a generic Hecke algebra (as
defined in Section 1.2 above and used in Sections 2.1 and 2.3). To pass from
the height function domain $\Omega$ used in Sections 2.1 and 2.3 to the set
$\Lambda$ above, see Notation 2.1.2 and Remark B.9. We will later apply the
abstract results of this section to Section 2.3, where the height function
$\mathfrak{h}=\mathfrak{h}_{r}$ also takes positive values.

Let $({\mathscr{A}},{\mathscr{E}})$ be the exact category given in Theorem
2.2.1. In particular, ${\mathscr{A}}$ is the full subcategory of mod-$H$
consisting of right $H$-modules which are finite and torsion-free over
${\mathscr{K}}$. For each integer $i$, let ${\mathscr{S}}_{i}$ be a set of
objects in a full additive subcategory of ${\mathscr{A}}$. For each
$S\in{\mathscr{S}}_{i}$, it is required that $S_{K}$ be a direct sum of
finitely many irreducible modules of the same height $i$. If $i$ is not in the
image of $\mathfrak{h}$, set ${\mathscr{S}}_{i}=[0]$, a full subcategory of
zero objects in ${\mathscr{A}}$. Let
${\mathscr{S}}:=\bigcup_{i}{\mathscr{S}}_{i}$. Finally, there is an exact
category $({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$ which
is constructed in [DPS17, Constr. 3.8]. (This construction with some details
has already been referenced in a left-module setting in (2.1.8). See also
(2.3.2) for a right-module setting.) The construction shows that
${\mathscr{A}}({\mathscr{S}})$ is a full subcategory of ${\mathscr{A}}$ (and
thus of mod-$H$), and it shows that ${\mathscr{E}}({\mathscr{S}})$ is
contained in ${\mathscr{E}}$.

######  Remark 2.4.1.

For convenience, we will assume that each category ${\mathscr{S}}_{i}$, and
thus ${\mathscr{A}}({\mathscr{S}})$, is closed under isomorphism in mod-$H$.

In particular,
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}({\mathscr{L}}^{*}))$, in the
Section 2.3 setting, is an example of an exact category
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$.

We will be interested in exact categories
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$, which satisfy the following

######  Conditions 2.4.2.

$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$ is a full exact subcategory of
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$, with
${\mathscr{A}}_{\flat}\hookrightarrow{\mathscr{A}}({\mathscr{S}})$ an
inclusion of ${\mathscr{K}}$-categories, such that the following hold:

  1. (1)

${\mathscr{A}}_{\flat}$ is object-closed under isomorphisms in
${\mathscr{A}}({\mathscr{S}})$ (and thus, by Remark 2.4.1 above, in mod-$H$).

  2. (2)

All objects of ${\mathscr{S}}$ are contained in ${\mathscr{A}}_{\flat}$.

  3. (3)

If $M\in{\mathscr{A}}_{\flat}$ and $h$ is an integer, then $M^{h}$, $M^{h-1}$
and $M^{h}/M^{h-1}$ belong to ${\mathscr{A}}_{\flat}$, and $(M^{h-1}\to
M^{h}\to M^{h}/M^{h-1})$ belongs to ${\mathscr{E}}_{\flat}$.

  4. (4)

For $M,N,P\in{\mathscr{A}}_{\flat}$, if $(M\to N\to
P)\in{\mathscr{E}}_{\flat}$, then $(M^{h}\to N^{h}\to
P^{h})\in{\mathscr{E}}_{\flat}$ for all integers $h$.

######  Example 2.4.3.

(a) The exact category
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$, a full exact
subcategory of itself, satisfies the conditions of Conditions 2.4.2. In
particular, in the setting of Section 2.3,
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}({\mathscr{L}}^{*}))$ provides
an example.

(b) Another example is
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$,
constructed in Section 2.3 and viewed as a full exact subcategory of
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}({\mathscr{L}}^{*}))$.
Conditions (1) and (2) are clear, while conditions (3) and (4) (for
$H={\mathcal{H}}$) are explicit in Proposition 2.3.3.

(c) The exact category
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
discussed in Proposition 2.3.2 does not satisfy the Conditions 2.4.2. In fact,
condition (3) fails to hold in general.

We now state several lemmas on
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}$-vanishing which hold for an
exact category $({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$ satisfying the
above conditions. Lemma 2.4.5 is difficult, while the remaining ones are more
routine generalizations of known results for the exact categories
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$ studied and
proved in [DPS17, §4]. Thus, we give a detailed proof of Lemma 2.4.5, and just
brief comments on the remaining ones. Everything is pulled together in Theorem
2.4.10.

Throughout the rest of this section,
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$ denotes a fixed exact category
which satisfies Conditions 2.4.2.

######  Lemma 2.4.4.

If $M,N\in{\mathscr{A}}_{\flat}$, then the natural maps
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(M,N)\rightarrow\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(M,N)$
and
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(M,N)\rightarrow\text{\rm{Ext}}^{1}_{H}(M,N)$,
as well as
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(M,N)\rightarrow\text{\rm{Ext}}^{1}_{H}(M,N)$,
are inclusions of ${\mathscr{K}}$-modules.

The above lemma follows from the discussion of Strategy 1.3.1.

######  Lemma 2.4.5.

Let $h$ be an integer. Let $S\in{\mathscr{S}}_{h}$ and
$M\in{{\mathscr{A}}_{\flat}}$, and suppose the map

$\text{\rm{Hom}}_{{{\mathscr{A}}_{\flat}}}(S,M^{h}/M^{h-1})\longrightarrow\text{\rm{Ext}}^{1}_{{{\mathscr{E}}_{\flat}}}(S,M^{h-1})$

defined by pull-back is surjective. Then,
$\text{\rm{Ext}}^{1}_{{{\mathscr{E}}_{\flat}}}(S,M)=0$.

###### Proof.

First, we claim it is sufficient to treat the case $M=M^{h}$: Given $(M\to
E\to S)\in{\mathscr{E}}_{\flat}$, $(M^{h}\to E^{h}\to
S)\in{\mathscr{E}}_{\flat}$ by Condition 2.4.2(4). The commutative diagram

$\begin{CD}M@>{}>{}>E@>{}>{}>S\\\ @A{\cup}A{}A@A{}A{}A\Big{\|}\\\
M^{h}@>{}>{}>E^{h}@>{}>{}>S\end{CD}$ (2.4.1)

identifies the original sequence $(M\to E\to S)$ as a push-out of $(M^{h}\to
E^{h}\to S)$ in mod-$H$. Since we have full subcategory inclusions
${\mathscr{A}}_{\flat}\subseteq{\mathscr{A}}({\mathscr{S}})\subseteq{\mathscr{A}}\subseteq{\text{mod-}}H$,
each map in the diagram (2.4.1) above belongs to ${\mathscr{A}}_{\flat}$.
Also, each row is a conflation in
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$. Thus, the diagram (2.4.1)
defines the original sequence $(M\to E\to S)$ as a push-out in
${\mathscr{A}}_{\flat}$. That is, the top row $(M\to E\to S)$ is isomorphic to
the push-out of the bottom row, regarded as a conflation in
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$. Obviously, this push-out is 0
in $\text{\rm{Ext}}^{1}_{{\mathscr{E}}^{\flat}}(S,M)$ if it is a push-out of 0
in $\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M^{h})$. Thus, from now on,
we can assume that $M=M^{h}$. (A somewhat shorter argument can be made using
[Bu10, Prop. 2.12], which relates push-outs to rectangles like (2.4.1) in
general, in an exact category.)

Let $\chi\in\text{\rm{Ext}}^{1}_{{{\mathscr{E}}_{\flat}}}(S,M)$ and let
$M\overset{\iota}{\to}E\overset{\delta}{\to}S$ be a representative conflation
sequence for $\chi$. Without loss of generality, we can assume $M$ is a
(right) $H$-submodule of $E$, with $i$ its inclusion. Also, $M^{h-1}$
identifies with $E^{h-1}$. Form the push-out diagram of the deflation
$M=M^{h}\to M^{h}/M^{h-1}$ and the sequence
$M\overset{\iota}{\to}E\overset{\delta}{\to}S$, to get a commutative diagram
in the abelian category mod-$H$:

$\begin{CD}M^{h}/M^{h-1}@>{}>{}>E/E^{h-1}@>{}>{}>S\\\
@A{}A{}A@A{}A{}A\Big{\|}\\\ M@>{\subseteq}>{}>E@>{}>{}>S\\\
@A{}A{}A@A{}A{}A\\\ M^{h-1}@>{=}>{}>E^{h-1}\end{CD}$ (2.4.2)

with exact rows and columns (after adding 0 maps at the beginning and end of
each row). Also, all objects in the diagram are in ${{\mathscr{A}}_{\flat}}$,
and each-three term row or column is in ${{\mathscr{E}}_{\flat}}$. (The only
non-obvious case, given our hypothesis on
$({{\mathscr{A}}_{\flat}},{{\mathscr{E}}_{\flat}})$, is the top row. However,
this row is also in ${{\mathscr{E}}_{\flat}}$, since the push-out of an
inflation is an inflation; see [DPS17, A.1(a)] which is based on [K90].)

In particular, the top row is defined by an element of

$\text{\rm{Ext}}^{1}_{{{\mathscr{E}}_{\flat}}}(S,M^{h}/M^{h-1})\subseteq\text{\rm{Ext}}^{1}_{{{\mathscr{E}}({\mathscr{S}})}}(S,M^{h}/M^{h-1})=0.$

It is, thus, split by an $H$-module map $\alpha:S\to E/E^{h-1}$. Let $j$
denote the map $M^{h}/M^{h-1}\to E/E^{h-1}$ in the top row, and let
$\beta:S\to M^{h}/M^{h-1}$ be any (right) $H$-module homomorphism. Fullness of
${{\mathscr{A}}_{\flat}}$ in ${{\mathscr{A}}({\mathscr{S}})}$, and of the
latter in mod-$H$, guarantees that $\beta$ is a map in the additive category
defined by ${{\mathscr{A}}_{\flat}}$. In particular, the pull-back through
$\beta$ of the element of
$\text{Ext}^{1}_{{\mathscr{E}}_{\flat}}(M^{h}/M^{h-1},M^{h-1})$ defining the
left-hand column is an element of
$\text{\rm{Ext}}^{1}_{{{\mathscr{E}}_{\flat}}}(S,M^{h-1})$. Indeed, this
process defines the natural homomorphism

$\text{\rm{Hom}}_{{{\mathscr{A}}_{\flat}}}(S,M^{h}/M^{h-1})\longrightarrow\text{\rm{Ext}}^{1}_{{{\mathscr{E}}_{\flat}}}(S,M^{h-1}),$

which our hypothesis in the lemma guarantees to be surjective. Notice that the
image of $\beta$ may also be described as the image of $j\circ\beta$ under a
similarly defined map

$\text{\rm{Hom}}_{{{\mathscr{A}}_{\flat}}}(S,E/E^{h-1})\longrightarrow\text{\rm{Ext}}^{1}_{{{\mathscr{E}}_{\flat}}}(S,E^{h-1})=\text{Ext}^{1}_{{\mathscr{E}}_{\flat}}(S,M^{h-1}).$

(The top left square in diagram (2.4.2) defines $M$ as a pull-back with top
row $j$. Extend it to the left by a pull-back square with top row $\beta$, and
consider the composite $j\circ\beta$.)

Observe that the map $\alpha+j\circ\beta:S\to E/E^{h-1}$ is still a splitting
map of the top row in (2.4.2). Now choose $\beta$ so that its image in
${\text{\rm{Ext}}}^{1}_{{\mathscr{E}}_{\flat}}(S,M^{h-1})$ is the negative of
the image of $\alpha$ under the display above, using its identification of
$E^{h-1}$ with $M^{h-1}$. With this choice, the image of $\alpha+j\circ\beta$
is 0. This implies that the map $\alpha+j\circ\beta:S\to E/E^{h-1}$ can be
lifted to a map $\alpha^{\prime}:S\to E$ of right $H$-modules. That is, the
composition of $\alpha^{\prime}$ with $E\to E/E^{h-1}$ is equal to
$\alpha+j\circ\beta$. Composing further with $E/E^{h-1}\to S$ gives the
identity map of $S$ to itself.

We have thus constructed a splitting map of the map $E\to S$ in mod-$H$. That
is, the short exact sequence defined by $\chi$ is 0 in
$\text{\rm{Ext}}^{1}_{H}(S,M)$. It is, therefore, also 0 in
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M)$.

This completes the proof of the lemma. ∎

The following result is a generalization of [DPS17, Lem. 4.1]. See [DPS17,
Prop. A.2] for the definition of Ext${}^{1}_{\mathscr{E}}$-groups.

######  Lemma 2.4.6.

Let $M,N\in{\mathscr{A}}_{\flat}$ and $h\in\mathbb{Z}$.

  * (a)

There is a natural isomorphism
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(N^{h},M)\cong\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(N^{h},M^{h})$.

  * (b)

In particular, if $S\in{\mathscr{S}}_{h},$ we have

$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M)\cong\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M^{h}).$

  * (c)

Assume that $S\in{\mathscr{S}}_{h}$. If $M=M^{h}$ and $M^{h-1}=0$, then
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M)=0$.

###### Proof.

Note that part (b) follows from part (a). Part (a) follows from [DPS17, Prop.
4.1] and its description of the isomorphism
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(N^{h},M)\cong\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}}}(N^{h},M^{h})$,
together with Condition 2.4.2(4) and Lemma 2.4.4. Since the proof will be
quoted later, we provide some of the details.

First, we give a map

$\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(N^{h},M)\overset{\phi}{\longrightarrow}\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(N^{h},M^{h})$

and show that it is an isomorphism by explicitly giving an inverse. The map
$\phi$ is obtained by applying the functor $(-)^{h}$ to a conflation. The
inverse of $\phi$ is defined by push-out through $M^{h}\rightarrow M$. The
restriction of $\phi$ to ${\mathscr{E}}_{\flat}$ makes sense using Condition
2.4.2(4) and Lemma 2.4.4.

This push-out also make sense in the exact category
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$ and agrees with the push-out
in $({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$. (This is an
elementary argument. The agreement of push-outs also follows from [Bu10, Prop.
2.12].) Thus, the isomorphism $\phi$ induces a corresponding isomorphism when
${\mathscr{E}}({\mathscr{S}})$ is replaced by ${\mathscr{E}}_{\flat}$ .

Finally, part (c) (which is independent of (a) and (b)) follows from [DPS17,
Lem. 4.1(c)] and Lemma 2.4.4 above. ∎

######  Lemma 2.4.7.

Let $M\in\mathscr{A}_{\flat}$. If $S\in{\mathscr{S}}_{h}$, then

$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M^{h-1})\hookrightarrow\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(S,M^{h-1})\cong\text{\rm{Ext}}^{1}_{H}(S,M^{h-1}).$

###### Proof.

The containment
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}\hookrightarrow\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}$
in the display is obtained from Lemma 2.4.4. The isomorphism in the display is
from [DPS17, Lem. 4.2]. ∎

######  Lemma 2.4.8.

Let $S\in\mathscr{S}_{h}$, and let $M\in\mathscr{A}_{\flat}$. Suppose that
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M^{h-1})$ is generated as a
$\mathscr{K}$-module by $\epsilon_{1},\ldots,\epsilon_{n}$ (regarded as
elements $\text{\rm{Ext}}^{1}_{H}(S,M^{h-1})$ using the inclusion above). Let
$M^{h-1}\to N\to S^{\oplus n}$ represent the element of
$\text{\rm{Ext}}^{1}_{H}(S^{\oplus n},M^{h-1})$ corresponding to
$\chi:=\epsilon_{1}\oplus\cdots\oplus\epsilon_{n}$ above. Suppose there is a
commutative diagram in mod-$H$

$\begin{CD}M^{h-1}@>{}>{}>N@>{}>{}>S^{\oplus n}\\\
\Big{\|}@V{}V{}V@V{f}V{}V\\\
M^{h-1}<EMAIL_ADDRESS>
Then, $\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M)=0$.

###### Proof.

By the construction of addition in $\text{\rm{Ext}}^{1}_{H}(S,M^{h-1})$, for
$1\leq j\leq n$, there is an inclusion $\iota_{j}:S\to S^{\oplus n}$, inducing
by pull-back a map
$\text{\rm{Ext}}^{1}_{H}(S,M)\longrightarrow\text{\rm{Ext}}^{1}_{H}(S^{\oplus
n},M^{h-1})$ taking $\chi$ to $\epsilon_{j}$. This gives the top rectangle in
the commutative diagram below, with top row representing $\epsilon_{j}$.

$\begin{CD}M^{h-1}@>{}>{}>N_{j}@>{}>{}>S\\\ \Big{\|}@V{}V{}V@V{}V{}V\\\
M^{h-1}@>{}>{}>N@>{}>{}>S^{\oplus n}\\\ \Big{\|}@V{}V{}V@V{f}V{}V\\\
M^{h-1}<EMAIL_ADDRESS>
Thus, the conflation corresponding to $\epsilon_{j}$ may be obtained by a
pull-back of $M^{h-1}\to M^{h}\to M^{h}/M^{h-1}$ using the evident composite

$g_{j}:S\to S^{\oplus n}\overset{f}{\longrightarrow}M^{h}/M^{h-1}.$

Consequently, the image of
$g_{j}\in\text{\rm{Hom}}_{{\mathscr{A}}_{\flat}}(S,M^{h}/M^{h-1})\subseteq\text{\rm{Hom}}_{H}(S,M^{h}/M^{h+1})$
under the connecting homomorphism to
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M^{h-1})\subseteq\text{\rm{Ext}}^{1}_{H}(S,M^{h-1})$
is $\epsilon_{j}$. Since $j$ was arbitrary, the connecting homomorphism is
surjective, so $\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M)=0$ by Lemma
2.4.5. ∎

The next result will assume the following hypothesis for ${\mathscr{S}}$:

######  Finiteness Hypothesis 2.4.9.

For any integer $h$, ${\mathscr{S}}_{h}$ is strictly generated as an additive
category by a finite set $F(h)$, i.e., every object in ${\mathscr{S}}_{h}$ is
isomorphic to a finite direct sum of objects chosen from $F(h)$.

######  Theorem 2.4.10.

Suppose $({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$ is a full exact
subcategory of $({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$
satisfying conditions (1)–(4) in Conditions 2.4.2. Assume ${\mathscr{S}}$
satisfies Hypothesis 2.4.9. Then, given $M\in{\mathscr{A}}_{\flat}$, there
exists an object $X=X_{M}$ in $\mathscr{A}_{\flat}$ and an inflation
$M\overset{i}{\to}X$ in $({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$ such
that $\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,X)=0$, for all
$S\in\mathscr{S}$. In addition, if $h$ is chosen minimal such that
$M^{h-1}\neq 0$, it may be assumed that the inflation
$M\overset{i}{\longrightarrow}X$ induces an isomorphism $M^{h-1}\cong
X^{h-1}$.

###### Proof.

Without loss of generality, we can assume that $M\not=0$ and that
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,M)\neq 0$, for some
$S\in\mathscr{S}$. Choose an integer $h$ minimal with such a non-vanishing
occurring for some $S\in\mathscr{S}_{h}$. Note that $M^{h-1}\neq 0$ by Lemma
2.4.6(c). We will enlarge $M$ to an object $X$, closer to the $X$ required in
the theorem.

Let $S_{1},\ldots,S_{m}$ be generators for $\mathscr{S}_{h}$. For each index
$i$, let $\epsilon_{i,1},\cdots,\epsilon_{i,n_{i}}$ be a finite set of
generators for $\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S_{i},M^{h-1})$.
Form conflations $0\to M^{h-1}\to Y_{i}\to S_{i}^{\oplus n_{i}}\to 0$
corresponding to
$\chi_{i}:=\epsilon_{i,1}\oplus\cdots\oplus\epsilon_{i,n_{i}}\in\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S_{i}^{\oplus
n_{i}},M^{h-1})$. Put $\chi:=\chi_{1}\oplus\cdots\oplus\chi_{m}$, and let
$\chi^{\prime}\in\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(M^{h}/M^{h-1},M^{h-1})$
correspond to the conflation $M^{h-1}\to M^{h}\to M^{h}/M^{h-1}$. Put
$Z:=\oplus_{i}S_{i}^{\oplus n_{i}}\oplus M^{h}/M^{h-1}$, and let $M^{h-1}\to
X^{h}\to Z$ correspond to $\chi\oplus\chi^{\prime}$. Observe that there is a
commutative diagram

$\begin{CD}M^{h-1}@ >>>Y_{i}@>{}>{}>S_{i}^{\oplus n_{i}}\\\
\Big{\|}@V{}V{}V@V{}V{}V\\\ M^{h-1}@>{}>{}>X^{h}@>{}>{}>Z,\end{CD}$

in which the top row corresponds to $\chi_{i}$ and the bottom row to
$\chi\oplus\chi^{\prime}$ . Note that the bottom row belongs to
${\mathscr{E}}_{\flat}$. Comparison with Lemma 2.4.8, allowing for the
differences in notation, shows
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S_{i},X^{h})=0$, for all $i$.
Thus, $\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,X^{h})=0$, for all $S$ in
${\mathscr{S}}_{h}$. Note that we have the same vanishing for
$S\in{\mathscr{S}}_{j}$ with $j<h$, by our choice of $h$. In all cases, we can
replace $X^{h}$ with any $X^{\prime}$ containing it with
$(X^{\prime})^{h}=X^{h}$ by Lemma 2.4.6.

So far, we have not constructed an object $X$, only $X^{h}$. However, the
latter may be viewed as the middle term of an exact sequence of right
$H$-modules $0\to M^{h}\to X^{h}\to S^{\prime}\to 0$, where
$S^{\prime}:=\bigoplus S_{i}^{\oplus n_{i}}\in\mathscr{S}_{h}$. This sequence
clearly corresponds to a conflation in ${\mathscr{E}}_{\flat}$. (Notice that
$X^{h}\to S^{\prime}$ is the composition of the deflation $X^{h}\to Z$ and the
split deflation $Z\to S^{\prime}$, both for ${\mathscr{E}}_{\flat}$. It is
therefore a deflation for ${\mathscr{E}}_{\flat}$. The morphism $M^{h}\to
X^{h}$ belongs to ${\mathscr{A}}_{\flat}$ and is a kernel of $X^{h}\to
S^{\prime}$ in mod-$H$, hence is a kernel in ${\mathscr{A}}_{\flat}.$)
Applying a push-out construction using $M^{h}\to M$ (see Lemma 2.4.6(a,b) and
their proof), we obtain an object $X$ in $\mathscr{A}_{\flat}$ which contains
a copy of $M$ under an inflation and has our constructed $X^{h}$ as its image
under the functor $(-)^{h}$. In addition, $X^{j}=M^{j}$ for $j\leq h-1$.

As noted above, $\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(-,X)$ vanishes on
all objects in $\mathscr{S}_{j}$ with $j\leq h$. Now repeat the argument with
$X$ in the role of $M$. This requires a bigger $h$, unless
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S,X)$ already vanishes, for all
$S\in\mathscr{S}$. Eventually the process stops, at which point $X$ satisfies
all requirements of the theorem. ∎

######  Remark 2.4.11.

Note that [DPS17, Th. 4.7] is the special case of the theorem with
${\mathscr{A}}_{\flat}={\mathscr{A}}({\mathscr{S}})$ and
${\mathscr{E}}_{\flat}={\mathscr{E}}({\mathscr{S}})$. The result was applied
in the main theorem [DPS17, Th. 4.9] there to build, for each dual left-cell
module $S_{\omega}$, an object $X_{\omega}$ in ${\mathscr{A}}({\mathscr{S}})$
such that $S_{\omega}\subseteq X_{\omega}$ (part of the filtration structure
required for ${\mathscr{A}}({\mathscr{S}})$) and such that
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(S,X_{\omega})=0$, for all
$S\in{\mathscr{S}}$. We can apply the same construction here.

In preparation for the next section, we now assume that $H={\mathcal{H}}$ is
the generic Hecke algebra over ${\mathcal{Z}}=\mathbb{Z}[t,t^{-1}]$ as
discussed in Section 1.2. Thus, the Noetherian domain ${\mathscr{K}}$ will be
specialized to ${\mathcal{Z}}$.

######  Construction 2.4.12.

Let
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})=({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$,
using a height function $\mathfrak{h}=\mathfrak{h}_{r}$ as in Notation 2.3.1.
(Thus, $\mathfrak{h}$ is compatible with the preorder $\leq_{LR}^{\text{\rm
op}}$.) Let $\Omega$ be the set of (Kazhdan-Lusztig) left cells in $W$.

  * (a)

Let $\Omega^{\prime}$ be the set of left cells that do not contain the longest
element $w_{0,J}$ in a parabolic subgroup $W_{J}$, $J\subseteq S$. For
$\omega\in\Omega^{\prime}$, define (as given in Theorem 2.4.10)
$X_{\omega}=X_{M}$, for $M=S_{\omega}$, and let
$X:=\bigoplus_{\omega\in\Omega^{\prime}}X_{\omega}$.

  * (b)

Define right ${\mathcal{H}}$-modules

$\begin{cases}T:=\bigoplus_{J\subseteq S}\text{\bf\it
x}_{J}{\mathcal{H}}=\bigoplus_{\omega\in\Omega\backslash\Omega^{\prime}}T_{\omega},\\\
T^{+}:=T\bigoplus X,\end{cases}$ (2.4.3)

and form the generic Hecke endo-algebra and its extended version

$\begin{cases}A:=\text{\rm{End}}_{\mathcal{H}}(T),\\\
A^{+}:=\text{\rm{End}}_{{\mathcal{H}}}(T^{+}).\end{cases}$ (2.4.4)

  * (c)

Finally, for $\omega\in\Omega\backslash\Omega^{\prime}$, let
$T^{+}_{\omega}:=x_{J}{\mathcal{H}}$, where $J$ is the unique subset of $S$
with $w_{0,J}\in\omega$, and $T^{+}_{\omega}=X_{\omega}$, for
$\omega\in\Omega^{\prime}$. Define the following left $A^{+}$-modules

$\begin{cases}\Delta(\omega):=\text{\rm{Hom}}_{{\mathcal{H}}}(S_{\omega},T^{+}),\\\
P(\omega):=\text{\rm{Hom}}_{\mathcal{H}}(T^{+}_{\omega},T^{+}),\,\,{\text{\rm
for any}}\,\,\omega\in\Omega.\end{cases}$ (2.4.5)

As considered in (1.3.2) and also for later application in Section 5.4, we
define, for any poset ideal $\Theta$ of $(\Omega,\leq_{L})$,

$T^{\ddagger}_{\Theta}:=\bigoplus_{\omega\in\Theta}(T^{+}_{\omega})^{m_{\omega}}\;\text{
and
}\;A^{\ddagger}_{\Theta}=\text{\rm{End}}_{\mathcal{H}}(T^{\ddagger}_{\Theta}),$
(2.4.6)

and

$A^{\ddagger}=\text{\rm{End}}_{\mathcal{H}}(T^{\ddagger}_{\Omega}),$ (2.4.7)

where $\\{m_{\omega}\\}_{\omega\in\Theta}$ is any fixed set of positive
integers.666We view $T^{\ddagger}$ as a modification of $T^{+}$, rather than
$T$.

##  Chapter 3 The conjecture and its proof

The conjecture in the title of this chapter is Conjecture 3.2.1, which is
[DPS15, Conj.1.2] and also captures essentially the earlier conjecture
[DPS98a, Conj. 2.5.2], as discussed in Remarks 3.3.4. A special local case
[DPS15, Th. 3.6] of the [DPS15] conjecture was proved in a Cherednik algebra
context, using an Ext1 vanishing result [DPS15, Cor. 5.3] obtained in part
from [GGOR03]. The latter results seemed to be hard to generalize.
Nevertheless, in Chapter 2 here, we have used exact categories to formulate by
analogy a tractable global version of the Ext1 vanishing condition, not
mentioning localization or Cherednik algebras. Using it, we are now ready to
prove this new version of the conjecture.

For this chapter, we continue the notation in the previous chapter, but, as in
Construction 2.4.12, return to the special case where $H$ is the generic Hecke
algebra ${\mathcal{H}}$ (so that ${\mathscr{K}}={\mathcal{Z}}$) associated to
a standard finite Coxeter system $(W,S,L)$.

###  3.1. $q$-permutation modules and their filtrations

Let ${\mathcal{H}}$ be a generic Hecke algebra over
${\mathcal{Z}}=\mathbb{Z}[t,t^{-1}]$ associated to a standard finite Coxeter
system $(W,S,L)$. Let $\Omega$ be the set of left cells of $W$. The preorders
$\leq_{L}$, $\leq_{LR}$ on $W$ induce partial order $\leq_{L}$ and preorder
$\leq_{LR}$ on $\Omega$; see (1.2.11). We take this opportunity to repeat and
highlight the following result, mentioned after Proposition B.5.

######  Proposition 3.1.1.

Each two-sided cell module is a direct sum of left-cell modules.

###### Proof.

Let $\mathfrak{c}$ be a two-sided cell of $W$, and set
$M(\mathfrak{c}):={\mathcal{H}}_{\leq_{LR}\mathfrak{c}}/{\mathcal{H}}_{<_{LR}\mathfrak{c}}$,
the associated two-sided cell module, where
${\mathcal{H}}_{\leq_{LR}\mathfrak{c}}={\mathcal{H}}_{\leq_{LR}w}$ for
$w\in\mathfrak{c}$ is defined in (1.2.9). Write
$\mathfrak{c}=\cup_{i=1}^{m}\omega_{i}$ (a disjoint union) with
$\omega_{i}\in\Omega$. Then, the free ${\mathcal{Z}}$-module $M(\mathfrak{c})$
is the direct sum $\bigoplus_{j=1}^{m}S^{\prime}(\omega_{j})$, where each
submodule
$S^{\prime}(\omega_{j}):=\sum_{y\in\omega_{j}}({\mathcal{Z}}{\textsc{c}}_{y}+{\mathcal{H}}_{<_{LR}\mathfrak{c}})/{\mathcal{H}}_{<_{LR}\mathfrak{c}}$
is a free ${\mathcal{Z}}$-module. It remains to prove that each
$S^{\prime}(\omega_{j})$ is isomorphic to the left-cell module
$S(\omega_{j})={\mathcal{H}}_{\leq_{L}\omega_{j}}/{\mathcal{H}}_{<_{L}\omega_{j}}$.
This is clear, since the inclusion
${\mathcal{H}}_{\leq_{L}\omega_{j}}\subseteq{\mathcal{H}}_{\leq_{LR}\mathfrak{c}}$
induces by (1.2.11) an ${\mathcal{H}}$-module homomorphism
${\mathcal{H}}_{\leq_{L}\omega_{j}}\to M(\mathfrak{c})$ with image
$S^{\prime}(\omega_{j})$ and with kernel
${\mathcal{H}}_{<_{L}\omega_{j}}={\mathcal{H}}_{\leq_{L}\omega_{j}}\cap{\mathcal{H}}_{<_{LR}\mathfrak{c}}$.
(Observe that, for some $z\in\omega_{j}$, by (1.2.11), if $y<_{L}z$, then
$y<_{LR}z(\in\mathfrak{c})$, and if $y\leq_{L}z$ and $y<_{LR}w$ for some
$w\in\mathfrak{c}$, then $y<_{L}z$.) ∎

Observe that the argument for this result uses display (1.2.11). A similar
fact holds for right-cell modules or dual left-cell modules. Also, the
$q$-permutation modules ${\mathcal{H}}x_{J}$ in Proposition 2.1.4 have natural
sections arising from $\mathfrak{h}_{l}$ which are direct sums of left-cell
modules. Once more, this goes back to (1.2.11), as discussed at the end of the
proof of Proposition 2.1.4. The (right) $q$-permutation modules
$x_{J}{\mathcal{H}}$ have, similarly, filtrations by the direct sum of dual
left-cell modules $S_{\omega}:=S(\omega)^{*}$ defined as the
${\mathcal{Z}}$-linear dual of the left-cell module $S(\omega)$; see Notation
2.1.2.

We now define a new preorder $\preceq=\preceq_{\Omega}$ on $\Omega$ by
setting, for, $\omega,\omega^{\prime}\in\Omega$,

$\omega\preceq\omega^{\prime}\iff\mathfrak{h}(\omega)<\mathfrak{h}(\omega^{\prime}),\text{
or }\mathfrak{h}(\omega)=\mathfrak{h}(\omega^{\prime})\text{ and
}\omega\sim_{LR}\omega^{\prime}.$ (3.1.1)

Here, the height function $\mathfrak{h}=\mathfrak{h}_{r}=-\mathfrak{h}_{l}$ is
chosen so that it is compatible with $\leq^{\text{\rm op}}_{LR}$ on $\Omega$.
See Notation 2.3.1 and display (B.2) in Appendix B. In particular,
$\mathfrak{h}$ is constant on two-sided cells. Thus, $\preceq$ above is well-
defined on $\Omega$ and even induces a partial order on the set
$\overline{\Omega}$ of two-sided cells.

It also follows from the definition in display (3.1.1) that the equivalence
classes of the preorder $\preceq$ are precisely the two-sided cells,
satisfying a requirement in [DPS15, Conj. 1.2]. More is true along these
lines: the preorders $\leq^{\text{\rm op}}_{LR}$ and $\preceq$ share a
compatible height function $\mathfrak{h}$. In addition, $\leq^{\text{\rm
op}}_{LR}$ strictly dominates $\preceq$ (see Definition B.3 in Appendix
B)111To check this, suppose that $\omega,\omega^{\prime}\in\Omega$ and
$\omega<^{\text{\rm op}}_{LR}\omega^{\prime}$. Since $\mathfrak{h}$ is
compatible with $\leq^{\text{\rm op}}_{LR}$, we have
$\mathfrak{h}(\omega)<\mathfrak{h}(\omega^{\prime})$. Thus,
$\omega\preceq\omega^{\prime}$. However, it is not possible for
$\omega\sim_{LR}\omega^{\prime}$, since $\mathfrak{h}$ is constant on two-
sided cells. So $\omega\prec\omega^{\prime}$. Since we already know for
$\omega,\omega^{\prime}\in\Omega$, that $\omega\sim_{LR}\omega^{\prime}$
implies $\omega\sim_{\prec}\omega^{\prime}$, we have proved that
$\leq^{\text{\rm op}}_{LR}$ strictly dominates $\preceq$. so that any height
function compatible with $\preceq$ is also compatible with
$\leq_{LR}^{\text{\rm op}}$ (as well as with $\leq_{L}^{\text{\rm op}}$). See
Proposition B.6 (and Proposition B.5.) Again, the discussion traces back to
(1.2.11).

The right set of an element $w\in W$ is defined as the set
${\mathscr{R}}(x)=\\{s\in S\,|\,xs<x\\}.$ Then, if $x\leq_{L}y$, it is known
(and elementary) that ${\mathscr{R}}(x)\supseteq{\mathscr{R}}(y)$ [Lus03, Lem.
8.6]. For example, if $J\subseteq S$, let $w_{0,J}$ be the longest word in
$W_{J}$. Clearly, ${\mathscr{R}}(w_{0,J})=J$. Thus, if $\omega\in\Omega$
contains $w_{0,I}$ and $w_{0,J}$, for subsets $I,J$ of $S$, then, if
$w_{0,I}\leq_{L}w_{0,J}\leq_{L}w_{0,I}$, so $I=J$.

We have the following elementary result.

######  Lemma 3.1.2.

Let $J\subseteq S$, and let $w_{0,J}$ be the longest word in the parabolic
subgroup $W_{J}$. The following sets $A_{J}$, $B_{J}$, $C_{J}$ are equal:

  * (1)

$A_{J}=\\{w\in W\,|\,{\mathscr{R}}(w)\supseteq J\\}.$

  * (2)

$B_{J}=\\{w=uw_{0,J}\,|\,\ell(w)=\ell(u)+\ell(w_{0,J})\\}$.

  * (3)

$C_{J}=\\{w\in W\,|\,w\leq_{L}w_{0,J}\\}$.

###### Proof.

Let $w\in A_{J}$, and write $w=uv$, where $v\in W_{J}$ and $u$ is a
distinguished left $W_{J}$-coset representative. For $s\in J$, it follows that
$ws<w$, so $vs<v$. Thus, $v=w_{0,J}$ and $w\in B_{J}$. Conversely, if
$w=uw_{0,J}\in B_{J}$, then it follows easily that $u$ is a distinguished left
$W_{J}$-coset representative. So $J\subseteq{\mathscr{R}}(uw_{0,J})$, and
$B_{J}\subseteq A_{J}$. Thus, $A_{J}=B_{J}$. Next, $w\leq_{L}w_{0,J}$ implies
${\mathscr{R}}(w)\supseteq{\mathscr{R}}(w_{0,J})$, so $C_{J}\subseteq A_{J}$.
Given $w=uw_{0,J}\in B_{J}$, let $s_{i_{1}}\cdots s_{i_{r}}$ be a reduced
expression for $u$. Because $\ell(w)=\ell(u)+\ell(w_{0,J})$, it follows that,
if $1<m\leq r$ and $u_{m}:=s_{i_{m}}\cdots s_{i_{r}}$, then
$s_{-_{m-1}}u_{m}w_{0,J}>u_{m}w_{0,J}$, so that
$s_{i_{m-1}}u_{m}w_{0,J}\leq_{L}u_{m}w_{0,J}$. Thus, $w\leq_{L}w_{0,J}$ and
$w\in C_{J}$.∎

######  Theorem 3.1.3.

Let $\omega\in\Omega$ and assume that, for some $J\subseteq S$,
$w_{0,J}\in\omega$, where $w_{0,J}$ is the longest word in $W_{J}$. Then
$x_{J}{\mathcal{H}}$ has an increasing filtration in the exact category
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
which satisfies Condition 2.4.2. The sections of the filtration are direct
sums of dual left-cell modules $S_{\omega^{\prime}}$, starting with
$S_{\omega}\subseteq x_{J}{\mathcal{H}}$. The other sections have summands
$S_{\omega^{\prime}}$ satisfying $\omega<_{L}\omega^{\prime}$ (equivalently,
$\omega^{\prime}>^{\text{\rm op}}_{L}\omega$).

###### Proof.

We consider the left module ${\mathcal{H}}x_{J}$ and then apply the
contravariant ${\mathcal{Z}}$-duality $(-)^{*}$ in Section 2.2. We use the
construction of the height filtration for ${\mathcal{H}}x_{J}$ in the proof of
Proposition 2.1.4. The quasi-poset ideal $V$ used in the construction from
[DPS98a, Lem. 2.3.5] is, by inspection of the latter lemma, the set $A_{J}$ in
Lemma 3.1.2, which is also the set $C_{J}$. In particular, $\omega$ generates
$V$ as a quasi-poset ideal with preorder $\leq_{L}$. The construction uses a
height function which we will take to be $\mathfrak{h}_{l}=-\mathfrak{h}$ for
a given height function $\mathfrak{h}=\mathfrak{h}_{r}$ compatible with
$\leq_{LR}^{\text{\rm op}}$. Thus, $\mathfrak{h}_{l}=-\mathfrak{h}$ is
compatible with $\leq_{LR}$ and, therefore, with $\leq_{L}$. Hence, $\omega$
has a unique maximal height among all left cells in the construction. The
left-cell module $S(\omega)$ is the top section in this height filtration of
${\mathcal{H}}x_{J}$, and the left-cell modules $S(\omega^{\prime})$, which
must satisfy $\omega^{\prime}<_{L}\omega$, must appear in lower sections. The
theorem now follows by duality. ∎

###  3.2. The conjecture

A main theorem in this monograph is Theorem 3.3.1, which, we will see, implies
Conjecture 3.2.1 below. As noted in the introduction to this chapter, this
latter conjecture was stated in [DPS15, Conj. 1.2], which revised
slightly222See Remarks 3.3.4. an earlier formulation [DPS98a, Conj. 2.5.2]. In
assertion (2), the “strict” stratifying system in [DPS15] is simply called a
stratifying system here (with the same meaning).

Recall that the set $\Omega$ of left cells (unequal parameter in the sense of
Lusztig [Lus03]) in the Weyl group $W$ has a natural preorder
$\leq_{LR}^{{\text{\rm op}}}$ which partitions $\Omega$ into its “two-sided
Kazhdan-Lusztig cells.” That is, $\omega$ and $\omega^{\prime}$ belong to the
same two-sided cell provided $\omega\leq_{LR}^{\text{\rm op}}\omega^{\prime}$
and $\omega^{\prime}\leq_{LR}^{\text{\rm op}}\omega$. Similarly, any preorder
$\leq$ on $\Omega$ also partitions $\Omega$ into “cells.” Let
$\widetilde{\Omega}$ be the corresponding set of cells. We say that $\leq$ is
strictly compatible with $\leq_{LR}^{\text{\rm op}}$ provided
$\overline{\Omega}=\widetilde{\Omega}$. The preorder $\preceq$ in display
(3.1.1) is clearly an example of such a preorder.

Recall also that given a left cell $\omega\in\Omega$, $S_{\omega}$ denotes the
corresponding dual left-cell module
$S(\omega)^{*}=\text{\rm{Hom}}_{\mathcal{Z}}(S(\omega),{\mathcal{Z}})$; see
Notation 2.1.2. For the definition of the element $x_{J}$, see (2.1.5).

######  Conjecture 3.2.1.

There exist a preorder $\leq$ on the set $\Omega$ of left cells in $W$,
strictly compatible with its partition $\overline{\Omega}$ into two-sided
cells, and a right ${\mathcal{H}}$-module $X$ such that the following
statements hold, where $T=\bigoplus_{J\subseteq S}x_{J}{\mathcal{H}}$:

  1. (1)

$X$ has a finite filtration with sections of the form $S_{\omega}$,
$\omega\in\Omega$.

  2. (2)

Let $T^{+}:=\\!T\oplus X$, $A^{+}:=\text{\rm{End}}_{{\mathcal{H}}}(T^{+})$
and, for any $\omega\in\Omega$, put

$\begin{cases}\Delta(\omega):=\text{\rm{Hom}}_{{\mathcal{H}}}(S_{\omega},T^{+}),\\\
P(\omega):=\text{\rm{Hom}}_{{\mathcal{H}}}(T^{+}_{\omega},T^{+}),\;\text{as in
\eqref{delta}.}\\\ \end{cases}$

Then, for any commutative, Noetherian ${\mathcal{Z}}$-algebra $R$, the set
$\\{\Delta(\omega)_{R},P(\omega)_{R}\\}_{\omega\in\Omega}$ is a stratifying
system for $A^{+}_{R}$-mod relative to the quasi-poset $(\Omega,\leq)$.

In [DPS15, p. 231], in the notion of a stratifying system for $A^{+}$-mod, it
is assumed, for each $\Delta(\omega)$, that there is a certain projective
$A^{+}$-module $P(\omega)$ satisfying certain properties. In the notation of
the present monograph, reviewed in Section 1.3, the projective modules
$P(\omega)$ are explicitly mentioned as part of the data for a stratifying
system. The $P(\omega)$’s are required to satisfy the same properties.
Finally, as already noted above Remark 1.3.2, the notion of a stratifying
system behaves well with respect to base change, so it suffices to prove the
conjecture for $R={\mathcal{Z}}$ and, thus, for $A^{+}_{R}=A^{+}$.

###  3.3. Proof of the conjecture

Before proceeding, we state and prove our main result, Theorem 3.3.1. The
preorder $\preceq$ in the statement of the theorem is defined explicitly in
display (3.1.1). It depends upon a suitable height function $\mathfrak{h}$.
Also, the exact category
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
depends on the same height function, as discussed, as does Construction
2.4.12.

######  Theorem 3.3.1.

Let
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})=({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$,
$T,X,T^{+}$, and $\mathfrak{h}=\mathfrak{h}_{r}$ be as in Construction 2.4.12.
For $\omega\in\Omega$, let

$T^{+}_{\omega}=\begin{cases}X_{\omega},&\text{if
}\omega\in\Omega^{\prime},\\\ x_{J}{\mathcal{H}},&\text{if
}w_{0,J}\in\omega.\end{cases}$

(Recall that $X_{\omega}$ is constructed from $S_{\omega}$, possibly non-
uniquely, as in Theorem 2.4.10. Here $\Omega^{\prime}$ is defined in
Construction 2.4.12(a).)

Form the endomorphism algebra

$A^{+}:=\text{\rm{End}}_{{\mathcal{H}}}(T^{+}).$ (3.3.1)

Similarly, for $\omega\in\Omega$,
$\Delta(\omega):=\text{\rm{Hom}}_{{\mathcal{H}}}(S_{\omega},T^{+})$ and
$P(\omega):=\text{\rm{Hom}}_{{\mathcal{H}}}(T_{\omega}^{+},T^{+})$.

Then $\\{\Delta(\omega),P(\omega)\\}_{\omega\in\Omega}$ is a stratifying
system for the category $A^{+}$-mod with respect to the quasi-poset
$(\Omega,\preceq)$.

###### Proof.

By Remarks 1.3.6(b)&(c), the left $A^{+}$-modules $\Delta(\omega)$,
$\omega\in\Omega$, are automatically ${\mathcal{Z}}$-projective, even
${\mathcal{Z}}$-free. If $\omega\in\Omega^{\prime}$, $X_{\omega}$ is, by
construction, a free ${\mathcal{Z}}$-module. Also, for
$\omega\in\Omega\backslash\Omega^{\prime}$, $T^{+}_{\omega}\cong
x_{J}{\mathcal{H}}$, for some $J\subseteq S$. By Theorem 3.1.3,
$T^{+}_{\omega}$ has a filtration by dual left-cell modules $S_{\tau}$,
$\tau\in\Omega$. (This follows already from (2.3.4).) Each $S_{\tau}$ is, by
construction, ${\mathcal{Z}}$-free, so $T^{+}_{\omega}$ is also
${\mathcal{Z}}$-free.

Now we must verify conditions (1), (2), and (3) in the Stratification
Hypothesis 1.3.3. Given $\omega\in\Omega^{\prime}$, Condition (1) follows from
Theorem 2.4.10. If $\omega\in\Omega\backslash\Omega^{\prime}$, then Condition
(1) follows from Theorem 3.1.3.

Suppose that $\omega,\tau\in\Omega$, and
$\text{\rm{Hom}}_{{\mathscr{A}}_{\flat}({\mathscr{L}}^{*})}(S_{\omega},T^{+}_{\tau})\not=0$.
Then, setting $K=\mathbb{Q}(t)$,
$\text{\rm{Hom}}_{{\mathcal{H}}_{K}}(S_{\omega,K},S_{\nu,K})\not=0$, so for
some $\tau\in\Omega$, $\tau\preceq\nu$. Since $S_{\omega,K}$ and $S_{\nu,K}$
share a composition factor, they belong to the same (dual) two-sided cell
module. Thus, $\omega$ and $\nu$ belong to the same two-sided cell for the
preorder $\leq_{LR}$, or, equivalently for $\leq_{LR}^{{\text{\rm op}}}$.
Since $\preceq$ is strictly compatible with $\leq_{LR}^{\text{\rm op}}$,
$\omega$ and $\nu$ belong to the same two-sided cell for $\preceq$. It follows
that $\tau\preceq\omega$. This proves (2).

Next, we prove condition (3). In case $\omega\in\Omega^{\prime}$, we can apply
the 4-term exact sequence in [DPS17, Prop. A.2] and the vanishing of
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}({\mathscr{L}}^{*})}(S,X)$ in
Theorem 2.4.10 to conclude that condition (3) holds in this case. If
$\omega\in\Omega\backslash\Omega^{\prime}$, we can argue in the same way, but
using Theorem 3.1.3 and Proposition 2.3.2.∎

The following is a corollary of the proof of the theorem, and Construction
2.4.12, using Theorem 2.4.10, Theorem 3.1.3, Proposition 2.3.2, displays
(2.3.4), and (3.1.1). The last assertion of the corollary follows from Theorem
2.4.10 ($\omega\in\Omega^{\prime}$) and the last assertion of Theorem 3.1.3
(when $\omega\not\in\Omega^{\prime}$).

######  Corollary 3.3.2.

Let
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})=({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
and keep the notation of the theorem above. Then, $S_{\omega}$ and
$T^{+}_{\tau}$ belong to ${\mathscr{A}}_{\flat}$, for all
$\omega,\tau\in\Omega$. Also,

$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S_{\omega},T^{+}_{\tau})=0.$

In addition, $S_{\omega}$ is a submodule (via an inflation in
${\mathscr{E}}_{\flat}$) of $T^{+}_{\omega}$ and is the lowest nonzero term in
its height filtration.

If $\Theta$ is a poset ideal of $(\Omega,\leq_{L})$, then $(\Theta,\preceq)$
is a quasi-poset co-ideal of $(\Omega,\preceq)$. Thus,
$T^{\ddagger}_{\Theta}:=\bigoplus_{\omega\in\Theta}(T^{+}_{\omega})^{m_{\omega}}\in{\mathscr{A}}_{\flat}$
with all $m_{\omega}>0$, as defined in (2.4.6), satisfies Stratification
Hypothesis 1.3.3 relative to the exact category
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$. Theorems 1.3.5 and 3.3.1
immediately imply the following.

######  Corollary 3.3.3.

Maintain the notation introduced above and form the endomorphism algebra
$A^{\ddagger}_{\Theta}=\text{\rm{End}}_{{\mathcal{H}}}(T^{\ddagger}_{\Theta})$
and $A^{\ddagger}_{\Theta}$-modules
$\Delta^{+}(\omega)=\text{\rm{Hom}}_{{\mathcal{H}}}(S_{\omega},T^{\ddagger}_{\Theta})$
and
$P^{+}(\omega)=\text{\rm{Hom}}_{{\mathcal{H}}}(T^{+}_{\omega},T^{\ddagger}_{\Theta})$.
Then, $\\{\Delta^{+}(\omega),P^{+}(\omega)\\}_{\omega\in\Theta}$ is a
stratifying system for the category $A^{\ddagger}_{\Theta}$-mod with respect
to the quasi-poset $(\Theta,\preceq)$.

We now complete the proof of Conjecture 3.2.1. The above theorem proves part
(1) of Conjecture 3.2.1. Part (1) of the conjecture deals with the filtration
of $X$ by dual left-cell modules and follows easily from the construction
given above. Our construction here is, in fact, much more informative than
that envisioned by part (1) of the conjecture. For example, $X_{\omega}$ is
built as an object in ${\mathscr{A}}_{\flat}({\mathscr{L}}^{*})$. As such, it
has a height filtration with sections that are direct sums of dual left-cell
modules. Also, left cells associated to distinct sections always lie in
distinct two-sided cells.

Part (2) of the conjecture is phrased in the language of base ring extensions
of ${\mathcal{Z}}$, but it is sufficient to check it over ${\mathcal{Z}}$, by
[DPS98a, Lem. 1.2.5(4)]. (Stratifying systems behave well under base change.)
Now taking $R={\mathcal{Z}}$, part (2) of the conjecture follows from Theorem
3.3.1, provided we know the preorder $\preceq$ has the same equivalence
classes as $\leq_{LR}^{\text{\rm op}}$. The actual preorder $\preceq$ in
Theorem 3.3.1 has this property (and is constructed from $\leq_{LR}^{\text{\rm
op}}$, sharing the height function $\mathfrak{h}$ with it; see display
(3.1.1)).

This completes the proof. ∎

The following remarks discuss further the difference between the Conjecture
proved in Theorem 3.3.1 and its original version given in [DPS98a]. It can be
skipped by readers interested only in the more recent version as stated in
Theorem 3.3.1 or in [DPS15, Conj. 1.2].

######  Remarks 3.3.4.

(a) When [DPS98a, Conj. 2.5.2] was written and checked for all rank 2
cases,333We correct some typos around the third display on [DPS98a, p. 203]:
…Each module ${\widetilde{S}}_{\\{u\\}}^{\mathscr{R}}=S_{0^{\prime}}\oplus
S_{0^{\prime\prime}}$ and ${\widetilde{S}}_{\\{v\\}}^{\mathscr{R}}=S_{0}\oplus
S_{0^{\prime\prime}}$ has a length two filtration with two dual left-cell
modules ${\widetilde{S}}_{\omega k}$ as sections, corresponding to left cells
$\omega=B,C_{1},C_{2}$, and $D$ (in the notation of (3.5) below). The
filtrations are of the form:
${\widetilde{S}}_{\\{u\\}}^{\mathscr{R}}=\boxed{\begin{matrix}C_{1}\\\
D\end{matrix}}\;\text{ and
}\;{\widetilde{S}}_{\\{v\\}}^{\mathscr{R}}=\boxed{\begin{matrix}B\\\
C_{2}\end{matrix}},$ and … the best choice of a preorder $\leq$ seemed to be
$\leq_{LR}^{\text{\rm op}}$. In the revised conjecture [DPS15, Conj.1.2]
(identical to Conjecture 3.2.1 above), we realized that this choice of a
preorder was too strong, and so it was weakened to require only a preorder
with the same equivalence classes as those of $\leq_{LR}^{\text{\rm op}}$. As
discussed in (a), the preorder $\preceq$ here in Theorem 3.3.1 satisfies this
condition and more, sharing also a height function $\mathfrak{h}$ with
$\leq_{LR}^{\text{\rm op}}$. Moreover, $\preceq$ is a preorder completely
determined by its equivalence classes and the height function $\mathfrak{h}$,
a specificity giving it some advantage over $\leq^{\text{\rm op}}_{LR}$. For
example, suppose $\mathfrak{h}$ is given by Lusztig’s $a$-function (see
(4.1.2) below), a choice of height function we can certainly make, provided
conjectures P4 and P11 hold; see [Lus03, 13.6, 14.1]. If
$\omega,\omega^{\prime}$ belong to distinct 2-sided cells, to decide if
$\omega\preceq\omega^{\prime}$, (equivalent here to
$\omega\prec\omega^{\prime}$), we need only check that
$a(\omega)<a(\omega^{\prime})$—which is a much more computable condition than
$\omega\leq^{\text{\rm op}}_{LR}\omega^{\prime}$ or $\omega<_{LR}^{\text{\rm
op}}\omega^{\prime}$.

(b) The other main difference between the conjecture in [DPS98a] and its
version in [DPS15] is that we generally worked over $\mathbb{Z}[t^{2},t^{-2}]$
in [DPS98a] rather than ${\mathcal{Z}}$. A case for using ${\mathcal{Z}}$
rather than $\mathbb{Z}[t^{2},t^{-2}]$ is that Lusztig’s explicit isomorphism
${\mathcal{H}}_{\mathbb{Q}(t)}\cong\mathbb{Q}(t)W$ holds without exception
(for the standard finite Coxeter system, even when $G$ has type
${}^{2}F_{4}$). These Hecke algebras are all semisimple and, in fact, split
semisimple except for the case ${}^{2}F_{4}$. We mention that the explicit
isomorphisms can be obtained by applying [Lus03, 18.12] to the cases
$R=\mathbb{Q}$ and $R=\mathbb{Q}(t)$. This works for any finite $W$ satisfying
P1–P15, and semisimplicity follows. The splitting, except for the
${}^{2}F_{4}$-case, follows from the isomorphisms
${\mathcal{H}}_{\mathbb{Q}(t)}\cong\mathbb{Q}(t)W$ above, and the fact that
$\mathbb{Q}(t)W$ (in fact, $\mathbb{Q}W$) is split semisimple; see [B71],
[BC71]. The latter reference discusses the Weyl groups arising in the “split”
(Chevalley group) cases, also including those $W$ for the “Steinberg twisted”
cases. The remaining groups, associated to the Ree and Suzuki groups, are
finite dihedral groups, and the only new group $W$ arising, that has not
already been considered, is the dihedral group $I_{2}(8)$ of order 16
associated to ${}^{2}F_{4}$. It has $\mathbb{Q}[\sqrt{2}]$ as its unique
minimal splitting field.

(c) [DPS98a, Conj. 2.5.2] left an ambiguity, removed in [DPS15, Conj. 1.2]. In
detail, the second sentence of part (2) of [DPS98a, Conj. 2.5.2] contains an
expression
“$\widetilde{A}_{{\mathcal{Z}}^{\prime}}=\text{\rm{End}}_{\widetilde{H}}({T}^{+}_{{\mathcal{Z}}^{\prime}})$”
which is a misprint, and should have been written
“${\widetilde{A}}_{{\mathcal{Z}}^{\prime}}=\text{\rm{End}}_{\widetilde{H}}(T^{+})_{{\mathcal{Z}}^{\prime}}$.”
However, it is shown in [DPS98a, Th. 2.4.4(b)] that
“$\text{\rm{End}}_{{\mathcal{H}}}(T)_{{\mathcal{Z}}^{\prime}}\cong\text{\rm{End}}_{{\mathcal{H}}}(T_{{\mathcal{Z}}^{\prime}})$,”
not using the “$+$” as a superscript on $T$. Possibly the same isomorphism
holds using $+$, but we only know this to be true if ${\mathcal{Z}}^{\prime}$
is flat over ${\mathcal{Z}}$.

(d) Finally, we observe that while the methods of [DPS98a] are useful in
studying low rank examples of the Conjecture, they do not provide much help in
the ultimate proof of the Conjecture!

###  3.4. A uniqueness theory

The main aim of this section is to provide a companion uniqueness theorem for
(the Morita classes of) the algebras $A^{+}$ resulting from Construction
2.4.12. If $A^{+}$ is quasi-hereditary, at least one such uniqueness result is
available in Rouquier’s theory of quasi-hereditary covers; see [Ro08, §4].

However, $A^{+}$ may not be quasi-hereditary. The method we use instead
applies in a quite general, but natural, context of endomorphism algebras of
suitably filtered injective–like objects in an exact category (e.g., the
object $T^{+}$ in Theorem 3.3.1). We begin with a description of this general
context (with a notation that is independent of previous sections, though
chosen to be suggestive).

Let ${\mathscr{K}}$ be a commutative Noetherian ring, and let $H$ be a finite
and projective ${\mathscr{K}}$-algebra. We view $H$ and its modules as
described by ${\mathscr{K}}$-module structure. Let
$({\mathscr{A}},{\mathscr{E}})$ be a strict and full exact subcategory of
mod–$H$. Recall this means that ${\mathscr{A}}$ is strict (closed under
isomorphisms) and full in mod–$H$ and that all conflations in ${\mathscr{E}}$
are also short exact sequences in mod$-H$. Given $M\in{\mathscr{A}}$, a
filtration $0=M^{-1}\subseteq M^{0}\subseteq\cdots\subseteq M^{\ell}=M$ in
mod–$H$ is called an ${\mathscr{E}}$-filtration provided each short exact
sequence $0\to M^{i-1}\to M^{i}\to M^{i}/M^{i-1}\to 0$ ($0\leq i\leq\ell$) in
mod-$H$ is a conflation in ${\mathscr{E}}$. (In particular, each $M^{i}$
belongs to ${\mathscr{A}}$.)

######  Lemma 3.4.1.

Suppose $M\in{\mathscr{A}}$ has an ${\mathscr{E}}$-filtration, indexed as
above, and that $T$ is an object in ${\mathscr{A}}$ such that
$\text{\rm{Ext}}^{1}_{\mathscr{E}}(N,T)=0$ for each section $N=M^{i}/M^{i-1}$,
$0\leq i\leq\ell$. Then,

$0\to\text{\rm{Hom}}_{H}(M^{j}/M^{i},T)\to\text{\rm{Hom}}_{H}(M^{j},T)\to\text{\rm{Hom}}_{H}(M^{i},T)\to
0$

is a short exact sequence, for all $0\leq i<j\leq\ell$. In particular, the map

$\text{\rm{Hom}}_{H}(M,T)\to\text{\rm{Hom}}_{H}(M^{0},T)$

is surjective.

For the proof, see the Claim in the proof of Theorem 1.3.5.

Now let ${\mathscr{S}}$ be a (strict) full additive subcategory of
${\mathscr{A}}$ generated by a finite set $\\{S_{\omega}\\}_{\omega\in\Omega}$
of nonzero objects indexed by a finite set $\Omega$. That is, every object in
${\mathscr{S}}$ is a finite direct sum of objects $S_{\omega}$,
$\omega\in\Omega$. Define ${\mathscr{A}}_{{\mathscr{E}}}({\mathscr{S}})$ to be
the subcategory of objects $M$ in ${\mathscr{A}}$ which have an
${\mathscr{E}}$-filtration with every section $M^{i}/M^{i-1}$ in
${\mathscr{S}}$.

Note that, by Corollary 3.3.2, the theorem below applies in particular to the
algebra $A^{+}$ in the conclusion of the main Theorem 3.3.1, using $A^{+}$ for
$A^{\dagger}$.

######  Theorem 3.4.2.

Assume that, for each $\omega\in\Omega$, there exists an object
$X_{\omega}\in{\mathscr{A}}_{\mathscr{E}}({\mathscr{S}})$ such that

  * (1)

$X_{\omega}\in{\mathscr{A}}_{\mathscr{E}}({\mathscr{S}})$ has an
${\mathscr{E}}$-filtration with lowest section $S_{\omega}$ (and all other
sections in ${\mathscr{S}}$), and

  * (2)

$\text{\rm{Ext}}^{1}_{{\mathscr{E}}}(S_{\omega^{\prime}},X_{\omega})=0$, for
all $\omega,\omega^{\prime}\in\Omega$.

If $A^{\dagger}=\text{\rm{End}}_{H}(T^{\dagger})$ where
$T^{\dagger}=\bigoplus_{\omega\in\Omega}X_{\omega}$, then the Morita class of
$A^{\dagger}$ is independent of the choices of the $X_{\omega}$ satisfying (1)
and (2).

As a corollary, we have the following result, whose statement will also be
used in the proof of the theorem.

######  Corollary 3.4.3.

Suppose that, for each $\omega\in\Omega$, there are given objects
$X_{\omega},Y_{\omega}\in{\mathscr{A}}_{\mathscr{E}}({\mathscr{S}})$ such that
conditions (1) and (2) in Theorem 3.4.2 hold for each $X_{\omega}$ and
maintain the notations $T^{\dagger}$ and $A^{\dagger}$. Assume also that each
$Y_{\omega}$ satisfies condition (2) in the sense that
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}}(S_{\omega^{\prime}},Y_{\omega})=0$, for
all $\omega^{\prime}\in\Omega$.

Put $T^{\dagger\prime}=\bigoplus_{\omega\in\Omega}(X_{\omega}\oplus
Y_{\omega})$ and $A^{\dagger\prime}:=\text{\rm{End}}_{H}(T^{\dagger\prime})$.
Then, $A^{\dagger}$ and $A^{\dagger\prime}$ are Morita equivalent.

Note that, as a special case of this corollary,
$A^{\ddagger}:=A^{\ddagger}_{\Omega}$ in (2.4.7) is Morita equivalent to
$A^{+}$.

###### Proof.

It is clear that the theorem implies the corollary by using $X_{\omega}\oplus
Y_{\omega}$ for $X_{\omega}$ satisfying (1) and (2). The filtration of the
direct sum starts with that of $X_{\omega}$, then uses that of $Y_{\omega}$.
However, it is also true that the corollary implies the theorem. In fact, if
$Y_{\omega}$ also satisfies condition (1), then the roles of $X_{\omega}$ and
$Y_{\omega}$ can be interchanged in the corollary, thus giving Morita
equivalences (designated here by $\sim$)

$\left\\{\begin{aligned}
&\text{\rm{End}}_{H}(\bigoplus_{\omega}Y_{\omega})\sim\text{\rm{End}}_{H}(\bigoplus_{\omega}(Y_{\omega}\oplus
X_{\omega}))\\\ \cong&\text{\rm{End}}_{H}(\bigoplus_{\omega}(X_{\omega}\oplus
Y_{\omega}))\sim\text{\rm{End}}_{H}(\bigoplus_{\omega}X_{\omega}).\end{aligned}\right.$
(3.4.1)

Thus, the corollary and the theorem are equivalent, so it remains to prove the
corollary directly. We follow the strategy in [DPS18, proof of Th. 4.1].

Let $e\in A^{\dagger\prime}=\text{\rm{End}}_{H}(T^{\dagger\prime})$ be the
idempotent projection of $T^{\dagger\prime}$ onto $T^{\dagger}\subseteq
T^{\dagger\prime}$ along $\bigoplus_{\omega}Y_{\omega}$. Since
$A^{\dagger}\cong eA^{\dagger\prime}e$, it suffices to show that the
projective module $A^{\dagger\prime}e$ is a progenerator, or, equivalently,
that every irreducible $A^{{+\prime}}$-module $L$ is a homomorphic image of
$A^{\dagger\prime}e$. (A better-known criterion is that
$A^{\dagger\prime}eA^{\dagger\prime}=A^{\dagger\prime}$. However, if this
fails, then the two-sided ideal generated by $e$ is contained in a maximal
left ideal $\mathfrak{m}$ of $A^{\dagger\prime}$. Take
$L=A^{\dagger\prime}/\mathfrak{m}$, and note that $eL=0$, so that $L$ cannot
be a homomorphic image of $A^{\dagger\prime}e$.)

However, $A^{\dagger\prime}$ is filtered by modules
$\text{\rm{Hom}}_{H}(S_{\omega},T^{\dagger\prime})$ by Lemma 3.4.1, and each
of these is a homomorphic image of
$\text{\rm{Hom}}_{H}(T^{\dagger},T^{\dagger\prime})$ again by Lemma 3.4.1. The
corollary now follows from the identification
$A^{\dagger\prime}e\cong\text{\rm{Hom}}_{H}(T^{\dagger},T^{\dagger\prime})$ of
$A^{\dagger\prime}$-modules. This completes the proof of the corollary and, as
noted, the theorem. ∎

######  Remark 3.4.4.

The proof of Corollary 3.4.3 can be used to give an explicit description of a
Morita (${\mathscr{K}}$-category) equivalence
$A^{\dagger}\text{-mod}\overset{\sim}{\to}A^{\dagger\prime}\text{-mod}$, where
$A^{\dagger}$ and $A^{\dagger\prime}$ are as stated in the corollary. First,
identify $A^{\dagger}$-mod with $eA^{\dagger\prime}e$-mod as above. Then, use
the well-known equivalence

$A^{\dagger\prime}\otimes_{eA^{\dagger\prime}e}(-):eA^{\dagger\prime}e\text{-mod}\longrightarrow
A^{\dagger\prime}\text{-mod}.$

######  Theorem 3.4.5.

Let $A^{+}$ be a ${\mathcal{Z}}$-algebra as constructed in the hypothesis of
Theorem 3.3.1 and let $A^{+\prime}$ be a ${\mathcal{Z}}$-algebra also
constructed to satisfy this hypothesis. (Here we use the same height function
for $A^{+}$ and $A^{+\prime},$ but allow the $X_{\omega}$ to vary.) Then
$A^{+}$ is Morita equivalent to $A^{+\prime}$.

The proof is immediate from Corollary 3.3.2 using Theorem 3.4.2. Further
details on the Morita equivalence can be obtained from the proof of the latter
theorem and its corollary.

######  Remarks 3.4.6.

We now mention several useful applications of Theorem 3.4.2 and its corollary:

(a) The identification
$A^{\dagger\prime}e\cong\text{Hom}_{H}(T^{\dagger},T^{\dagger\prime})$ in the
proof of the corollary, together with the corresponding identification for
$eA^{\dagger\prime}$ and the equality
$A^{\dagger\prime}eA^{\dagger\prime}=A^{\dagger\prime}=\text{End}_{H}(T^{\dagger\prime})$,
shows:

  * (i)

the trace of $T^{\dagger}$ is all of $T^{\dagger\prime}$. In particular,

  * (ii)

any $H$-projective direct summand of $T^{\dagger\prime}$ is isomorphic to a
direct summand of $(T^{\dagger})^{\oplus m}$, for a sufficiently large
$m\in\mathbb{N}$.

(b) Part (ii) of (a) may be used to prove the claim made in [DPS17, Remark
4.10(a)] that $H=\mathcal{H}$ is a direct summand of $T^{\dagger}$ there in
the conclusion of [DPS17, Th. 4.9]. (In the context of [DPS17, Remark
4.10(a)], we are allowed to replace $T^{\dagger}$ here by
$(T^{\dagger})^{\oplus m}$.) In fact, it can also be proved that $T^{\dagger}$
on [DPS17, p. 245], taking the multiplicities $m_{\omega}$ all equal to 1,
always has the regular representation $H=\mathcal{H}$ as a direct summand.

(c) Other Morita equivalences in the spirit of Corollary 3.4.3 are developed
in [DPS18, §4]. Conversely, Theorem 3.4.2 can be used to prove [DPS18, Th.
4.3]. The theorem gives a Morita equivalence between localizations of
stratified algebras in [DPS17] with algebras in [DPS15] which base change to
algebras arising in a Cherednik category ${\mathcal{O}}$ context (with equal
parameters).

###  3.5. The standard stratification of $A^{+}$

We now use the stratifying system
$\\{\Delta(\omega),P(\omega)\\}_{\omega\in\Omega}$ in the category $A^{+}$-mod
of finite $A^{+}$-modules given in Theorem 3.3.1 to construct a sequence of
stratifying ideals in $A^{+}$ and to show that $A^{+}$ is a standardly
stratified algebra.

Let ${\mathscr{K}}$ be a commutative Noetherian ring and let $B$ be a
${\mathscr{K}}$-algebra which is finite and projective as a
${\mathscr{K}}$-module.

######  Definition 3.5.1 ([DPS18, Defs. 3.1&3.2]).

An ideal $J$ in $B$ is called a standard stratifying ideal if the following
conditions hold:

  * (HI1)

$B/J$ is projective over ${\mathscr{K}}$;

  * (HI2)

${}_{B}J$ is $B$-projective as a left $B$-module;

  * (HI3)

$J^{2}=J$.

The algebra $B$ over ${\mathscr{K}}$ is called a standardly stratified algebra
(SSA) provided there exists a finite “defining sequence” of ideals in $B$:

$0=J_{0}\subseteq J_{1}\subseteq J_{2}\subseteq\cdots\subseteq J_{m}=B$
(3.5.1)

such that, for $0<i\leq m$, $J_{i}/J_{i-1}$ is a standard stratifying ideal in
$B/J_{i-1}$. We also call (3.5.1) a standard stratification.

We will see in Section 4.3 that stratifying ideals and standardly stratified
algebras are weakened (more general) versions of heredity ideals and quasi-
hereditary algebras.

We first observe the following nice properties.

######  Proposition 3.5.2.

(1) Every ideal in the standard stratification (3.5.1) is an idempotent ideal.

(2) Suppose that $J$ is a stratifying ideal in a ${\mathscr{K}}$-algebra $B$.
If $M,N$ are $B/J$-modules which are also regarded as $B$-modules by
inflation, then, for any integer $n\geq 0$, there is a ${\mathscr{K}}$-module
isomorphism

$\text{\rm{Ext}}^{n}_{B/J}(M,N)\longrightarrow\text{\rm{Ext}}^{n}_{B}(M,N).$

###### Proof.

A proof for (2) can be found in [DPS17, Appendix B]. We now prove (1). Since
$J_{1}^{2}=J_{1}$ and $(J_{2}/J_{1})^{2}=J_{2}/J_{1}$, it follows that

$J_{2}^{2}=J_{2}^{2}+J_{1}^{2}=J_{2}^{2}+J_{1}=J_{2}+J_{1}=J_{2}.$

Hence, $J_{2}$ is an idempotent ideal. An induction argument proves the first
assertion. ∎

We now prove that the Hecke endomorphism ${\mathcal{Z}}$-algebra $A^{+}$ is a
standardly stratified algebra. There is a general proof of such a fact for any
${\mathscr{K}}$-algebra $B$ which is finite projective over ${\mathscr{K}}$,
provided that the category $B$-mod has a stratifying system; see [DPS18, §3].
The proof for $A^{+}$ below is independent and more direct, laying down a
foundation for a further proof of the quasi-heredity of a base-changed
$A^{+\natural}$ in the next chapter.

######  Theorem 3.5.3.

If $A^{+}$ is the (extended) generic Hecke endo-algebra associated with a
standard finite Coxeter system $(W,S,L)$ as defined in (2.4.4), then the
${\mathcal{Z}}$-algebra $A^{+}$ is a standardly stratified algebra.

###### Proof.

We maintain the notation of Theorem 3.3.1, so, in particular,
$A^{+}=\text{\rm{End}}_{\mathcal{H}}(T^{+})$ is a certain endomorphism algebra
over ${\mathcal{Z}}$, where $T^{+}=\bigoplus_{\omega\in\Omega}T^{+}_{\omega}$,
and the category $A^{+}$-mod has a stratifying system
$\\{\Delta(\omega),P(\omega)\\}_{\omega\in\Omega}$. Recall also from Theorem
3.3.1 that, for each $\omega\in\Omega$, the left $A^{+}$-modules
$\Delta(\omega)$ and $P(\omega)$ are defined as
$\text{\rm{Hom}}_{\mathcal{H}}(S_{\omega},T^{+})$ and
$\text{\rm{Hom}}_{\mathcal{H}}(T_{\omega}^{+},T^{+})$, respectively, with the
action of $A^{+}:=\text{\rm{Hom}}_{\mathcal{H}}(T^{+},T^{+})$ through
compositions of morphisms (between right ${\mathcal{H}}$-modules).

Now, put $P:=\bigoplus_{\omega\in\Omega}P(\omega)$ and observe that $P\cong
A^{+}$ as left $A^{+}$-modules. (Use the equality
$\bigoplus_{\omega\in\Omega}T^{+}_{\omega}=T^{+}$ of right
${\mathcal{H}}$-modules.)

Let $\mathfrak{h}:\Omega\to\mathbb{Z}$ and $\preceq$ be as discussed above
Proposition 3.1.1, and used in Theorem 3.3.1. Here $\Omega$ is the set of
(Kazhdan–Lusztig) left cells of $W$, and $(\Omega,\preceq)$ is the quasi-poset
used in Theorem 3.3.1. For more details on our set-up here, see the paragraph
above the statement of Theorem 3.3.1 and the internal references it provides.

Our immediate strategy is to construct a tractable “defining sequence” of
ideals in $A^{+}$ (see Def. 3.5.1)

$0=J_{0}\subseteq J_{1}\subseteq\cdots\subseteq J_{m}=A^{+}$ (3.5.2)

such that all the conditions (HI1)–(HI3) in Definition 3.5.1, for every
$J=J_{i}/J_{i-1}$, hold as an ideal of $B=A^{+}/J_{i-1}$.

We construct the defining sequence for $A^{+}$ as follows. First, for any
$\omega\in\Omega$ and $j\in\mathbb{Z}$, we define a submodule
$P(\omega)_{j}\subseteq P(\omega)$ as the sum of all images in $P(\omega)$ of
all maps in $\text{\rm{Hom}}_{A^{+}}(P(\mu),P(\omega))$ with
$\mathfrak{h}(\mu)\geq j$. Using the ring-theoretic language of traces, we may
write this as

$P(\omega)_{j}=\sum_{\mathfrak{h}(\mu)\geq
j}\text{trace}_{P(\omega)}(P(\mu))=\text{trace}_{P(\omega)}\Big{(}\bigoplus_{\mathfrak{h}(\mu)\geq
j}P(\mu)\Big{)}:=\sum_{f\in\text{\rm{Hom}}_{A^{+}}(X,P(\omega))}\text{Im}(f),$

where $X=\bigoplus_{\mathfrak{h}(\mu)\geq j}P(\mu)$. Observe that
$P(\omega)_{j+1}\subseteq P(\omega)_{j}$. It is proved444The method of proof
reorders the sections $\Delta(\nu)$ in any given filtration of $P(\omega)$
that has sections given by various $\Delta(\nu)$’s with $\nu\in\Omega$. The
multiplicities of any given $\Delta(\nu)$ does change in this process. in
[DPS17, Prop. 2.2] and [DPS18, Prop. 3.9] that $P(\omega)_{j}/P(\omega)_{j+1}$
is a direct sum of modules $\Delta(\nu)$ with $\nu\in\Omega$ and
$\mathfrak{h}(\nu)=j$. Multiplicities are allowed. That is, isomorphic copies
of the same $\Delta(\nu)$ may appear in the direct sum.

Next, we may repeat the above discussion of $P(\omega)_{j}$ with $P$ replacing
$P(\omega)$. That is, we define

$P_{j}=\text{trace}_{P}\Big{(}\bigoplus_{\mathfrak{h}(\mu)\geq
j}P(\mu)\Big{)}.$ (3.5.3)

Clearly, $P_{j}=\bigoplus_{\omega\in\Omega}P(\omega)_{j}$, since
$P=\bigoplus_{\omega\in\Omega}P(\omega)$ as a left $A^{+}$-module. From this
we obtain $P_{j+1}\subseteq P_{j}$ and deduce also that $P_{j}/P_{j+1}$ is a
direct sum of $A^{+}$-modules isomorphic to $\Delta(\nu)$, for some
$\nu\in\Omega$, with $\mathfrak{h}(\nu)=j$. If the equality holds for a given
$\nu$, then some $\Delta(\nu)$ does occur.

We are now ready to return to $A^{+}$ and its ideals. We have observed near
the beginning of this proof that $P\cong A^{+}$ as left $A^{+}$-modules. For
each $j\in\mathbb{Z}$, let $A_{j}^{+}$ denote the left ideal in $A^{+}$
identifying with $P_{j}\subseteq P$.

  * Claim.

Each $A_{j}^{+}$ is a two-sided ideal of $A^{+}$. It is idempotent (allowing
zero), and $A^{+}/A^{+}_{j}$ is free as a $\mathcal{Z}$-module.

To prove the claim, we first show that each $A^{+}_{j}$ is a two-sided ideal
in $A^{+}$. By definition, it is the left ideal of $A^{+}$ corresponding to
the left $A^{+}$-submodule $P_{j}$ of $P$ under the natural identification
$P\cong A^{+}$ of left $A^{+}$-modules discussed above. Similarly,
$A^{+}/A^{+}_{j}$ is isomorphic to $P/P_{j}$ as left $A^{+}$-modules. As such,
it is filtered by copies of various modules $\Delta(\nu)$ with
$\mathfrak{h}(\nu)<j$, and $A^{+}_{j}$ is similarly filtered by modules
$\Delta(\tau)$ with $\mathfrak{h}(\tau)\geq j$. (In both cases, “filtered by”
refers informally to a filtration by left $A^{+}$-modules, with the given
$\Delta$’s as the only nonzero sections.)

If $A^{+}_{j}$ were not a two-sided ideal in $A^{+}$, it would not be a right
ideal, and there would be an element $a\in A^{+}$ with
$A^{+}_{j}a\not\subseteq A^{+}_{j}$. This implies that the composition

$A^{+}_{j}\longrightarrow
A^{+}_{j}a\longrightarrow(A^{+}_{j}a+A^{+}_{j})/A^{+}_{j}\subseteq
A^{+}/A^{+}_{j}$

of left $A^{+}$-module homomorphisms would be non-zero. Combining this with
the above paragraph, we find from this that there is a nonzero map
$\Delta(\tau)\to\Delta(\nu)$ with $\mathfrak{h}(\tau)>\mathfrak{h}(\nu)$.
However, this is impossible. It leads to a nonzero composite
$P(\tau)\twoheadrightarrow\Delta(\tau)\to\Delta(\nu)$, implying
$\tau\preceq\nu$, by the axiom (SS2) for stratifying systems. But
$\tau\preceq\nu$ and $\mathfrak{h}(\tau)>\mathfrak{h}(\nu)$ cannot both hold,
by the compatibility of $\mathfrak{h}$ with $\preceq$; see (3.1.1). This
contradiction shows that each $A^{+}_{j}$ is a two-sided ideal.

Next, we show each ideal $A_{j}^{+}$ is idempotent. In fact, we will show it
is generated by an idempotent element $e_{j}\in A^{+}_{j}$ (if $A^{+}_{j}\neq
0$).

First, we need some refinements of our notation. Define

$P_{j,0}=\bigoplus_{\mathfrak{h}(\omega)\geq j}P(\omega)_{j}\;\;\text{ and
}\;\;P_{j,1}=\bigoplus_{\mathfrak{h}(\omega)<j}P(\omega)_{j}.$

Note that the trace definition
$P(\omega)_{j}=\text{trace}_{P(\omega)}\Big{(}\bigoplus_{\mathfrak{h}(\mu)\geq
j}P(\mu)\Big{)}$ implies $P(\omega)_{j}\subseteq P(\omega)$, for any
$\omega\in\Omega$, and $P(\omega)_{j}=P(\omega)$ if $\mathfrak{h}(\omega)\geq
j$. (In particular, $P_{j,0}=\bigoplus_{\mathfrak{h}(\omega)\geq
j}P(\omega)$.) Recalling the identity
$P_{j}=\bigoplus_{\omega\in\Omega}P(\omega)_{j}$, we have $P_{j}=P_{j,0}\oplus
P_{j,1}$. It is also useful to set
$\widehat{P}_{j,1}=\bigoplus_{\mathfrak{h}(\omega)<j}P(\omega)$; so that
$P_{j,1}\subseteq\widehat{P}_{j,1}$ and $P=P_{j,0}\oplus\widehat{P}_{j,1}$.

We are almost ready to define $e_{j}$. First, let
$A^{+},A^{+}_{j},A^{+}_{j,0},A^{+}_{j,1},\widehat{A}^{+}_{j,1},\ldots$
correspond to $P,P_{j},P_{j,0},P_{j,1},\widehat{P}_{j,1},\ldots$ under our
standard isomorphism $A^{+}\cong P$ of left $A^{+}$-modules. Next, let
$f_{j}:P\twoheadrightarrow P_{j,0}\subseteq P_{j}\subseteq P$ be the obvious
idempotent endomorphism of $P$ passing through $P_{j}$. Then, there is a
unique idempotent $e_{j}\in A^{+}$ such that right multiplication by $e_{j}$
gives the map $A^{+}\twoheadrightarrow A^{+}_{j,0}\subseteq A^{+}_{j}\subseteq
A^{+}$ corresponding to $f_{j}$. Thus, $e_{j}$ is idempotent (since
$e_{j}^{2}=1\cdot e_{j}^{2}=f_{j}^{2}(1)=f_{j}(1)=1\cdot e_{j}=e_{j}$), and
$e_{j}\in A^{+}_{j}$ (since $e_{j}=1\cdot e_{j}\in A^{+}_{j,0}\subseteq
A_{j}^{+}$).

To complete the proof that $A^{+}_{j}$ is an idempotent ideal in $A^{+}$, it
is enough now (and even necessary and sufficient) to show
$A^{+}e_{j}A^{+}=A^{+}_{j}$. This is actually fairly easy. The submodule
$A^{+}e_{j}$ of $A^{+}_{j}$ corresponds, by construction, to the submodule
$P_{j,0}$ of $P$. Also, we have the equality $P_{j}=\text{trace}_{P}(P_{j,0})$
by definition. It follows, passing to $A^{+}$, that

$A^{+}_{j}=\text{trace}_{A^{+}}(A^{+}_{j,0})=\text{trace}_{A^{+}}(A^{+}e_{j})=\sum_{f\in\text{\rm{Hom}}_{A^{+}}(A^{+}e_{j},A^{+})}\text{Im}(f)=A^{+}e_{j}A^{+}.$
(3.5.4)

Here the last equality follows from a well-known identification of
$\text{\rm{Hom}}_{A^{+}}(A^{+}e_{j},A^{+})$ with $e_{j}A^{+}$, which leads to
an identification of trace${}_{A^{+}}(A^{+}e_{j})$ with $A^{+}e_{j}A^{+}$.

To prove the last remaining assertion of the claim that $A^{+}/A^{+}_{j}$ is
free as a ${\mathcal{Z}}$-module, it is enough to show $A^{+}_{i}/A^{+}_{i+1}$
is free for every $i<j$. However, we have already observed that the isomorphic
module $P_{i}/P_{i+1}$ is a direct sum of $A^{+}$-modules isomorphic to
$\Delta(\nu)$ for some $\nu\in\Omega$. Thus, it is projective (of finite rank)
over ${\mathcal{Z}}$, hence free, by Swan’s theorem. See Remark 1.3.6(c). The
claim is now proved.

We are now able to construct a candidate for the defining sequence (3.5.2)
previewed earlier in this proof.

Observe that $A^{+}_{j}=0$, for all sufficiently large integers $j$. In fact,
it suffices to take $j>\mathfrak{h}(\omega)$, for all $\omega\in\Omega$, using
the trace formula for $P_{j}$. Put $J_{0}=0$, and define $J_{1}=A^{+}_{j(1)}$,
where $j(1)$ denotes the largest value of $\mathfrak{h}$ on $\Omega$.
Continuing, define $J_{2}=A^{+}_{j(2)}$, where $j(2)$ denotes the largest
value of $\mathfrak{h}$ on $\Omega\backslash\Omega_{j(1)}$ with
$\Omega_{j(1)}=\\{\omega\in\Omega\mid\mathfrak{h}(\omega)=j(1)\\}$. This
process continues until $j(1),j(2),\ldots,j(m)$ have been defined as all the
elements of $\mathfrak{h}(\Omega)$, listed in decreasing order. For each
integer $i$ with $1\leq i\leq m$, we set $J_{i}=A^{+}_{j(i)}$, generalizing
our definitions for $J_{1}$ and $J_{2}$. We note that $A^{+}_{j(m)}=A^{+}$,
since $P_{j(m)}=P$ by the trace formula. Thus, $J_{m}=A^{+}_{j(m)}=A^{+}$, the
terminal member of our ascending sequence of $J_{i}$’s:

$0=J_{0}\subseteq J_{1}\subseteq\cdots\subseteq J_{m}=A^{+}\;\text{ with
}\;j(1)>j(2)>\cdots>j(m).$ (3.5.5)

(The general inclusion $P_{j}\subseteq P_{j-1}$, which holds for any
$j\in\mathbb{Z}$, also implies that $P_{j}\subseteq P_{j-a}$, for any positive
$a\in\mathbb{Z}$. Consequently, $A^{+}_{j}\subseteq A^{+}_{j-a}$. So
$A^{+}_{j(i)}\subseteq A^{+}_{j(i+1)}$, using $j=j(i)$ and $a=j(i)-j(i+1)$.
Here, $i$ is an integer with $1\leq i\leq m-1$. Since we have defined
$J_{i}=A^{+}_{j(i)}$ for each $i$, as well as $i=m$, we have the desired
sequence $0=J_{0}\subseteq J_{1}\subseteq\cdots\subseteq J_{m}=A^{+}$.)

Continuing the discussion above, we now wish to analyze more closely the left
$A^{+}$-modules $J_{1}=J_{1}/J_{0}$ and $J_{i}/J_{i-1}$ for $2\leq i\leq m$.
We start with $J_{1}=A^{+}_{j(1)}$. Recall that $j(1)$ is the largest value of
the height function so that $A^{+}_{j}=0$, for $j>j(1)$. (Use the trace
formula for $P_{j}$.) Thus, $A^{+}_{j(1)}=A^{+}_{j(1)}/A^{+}_{j(1)+1}$ is a
direct sum of various left $A^{+}$-modules $\Delta(\omega)$ with
$\mathfrak{h}(\omega)=j(1)$. These modules are projective for $A^{+}$, by the
maximality of $\mathfrak{h}(\omega)$. (Just observe that
$P(\omega)=\Delta(\omega)$ by axiom (SS3).)

We now consider $J_{i}/J_{i-1}$ for $2\leq i\leq m$ (the case $i=1$ having
been just discussed). We note

$J_{i}/J_{i-1}=A^{+}_{j(i)}/A^{+}_{j(i-1)}=A^{+}_{j(i)}/A^{+}_{j(i)+1}.$
(3.5.6)

The right-hand equality holds because $j(i)<j(i-1)$ are adjacent values of the
height function. Consequently, for any integer $j$ with $j(i)+1\leq j<j(i-1)$,
the $A^{+}$-module $A^{+}_{j}/A^{+}_{j+1}$ must be 0. (Use our description,
earlier in this proof, of $P_{j}/P_{j+1}$ as a direct sum of modules
$\Delta(\omega)$ with $j=\mathfrak{h}(\omega)$.) But this gives
$A^{+}_{j}=A^{+}_{j+1}$, with repeated application giving
$A^{+}_{j(i)+1}=A^{+}_{j(i-1)}$, as desired.

We are now in a position to prove condition (HI2), the projectivity of
$J:=J_{i}/J_{i-1}$ as a left $B$-module, for $B:=A^{+}/J_{i-1}$, $1\leq i\leq
m$. (The case $i=1$ has already been treated.) We have
$J=A^{+}_{j(i)}/A^{+}_{j(i)+1}$ by (3.5.6). Since $j(i)$ is a height function
value, we know that $J$ in this latter quotient form is the direct sum of
various $A^{+}/A^{+}_{j(i)+1}$-modules isomorphic to $\Delta(\omega)$ with
$\mathfrak{h}(\omega)=j(i)$. (Such a module may be regarded as a left
$A^{+}$\- or $A^{+}/A^{+}_{j(i)+1}$-module.) Recall that we have direct sum
formulas $P=\bigoplus_{\omega\in\Omega}P(\omega)$ and
$P_{j}=\bigoplus_{\omega\in\Omega}P(\omega)_{j}$ for each integer $j$, and
variations. In particular, note that
$P/P_{j+1}=\bigoplus_{\omega\in\Omega}P(\omega)/P(\omega)_{j+1}$. When
interpreted with $A^{+}$ replacing $P$ and $j=j(i)$, we find that the left-
hand side is $B=A^{+}/A^{+}_{j(i)+1}$, and the right-hand side, viewed a left
$A^{+}$-module, has $\Delta(\omega)$ as a direct summand whenever
$\mathfrak{h}(\omega)=j(i)$, as is the case for $\Delta(\omega)$ above. Thus,
$\Delta(\omega)$, as a left $B$-module, is projective. Since the
$\Delta(\omega)$ also represents the general direct summand of $J$, we have
proved $J$ is projective as a left $B$-module, as desired. (Note also the
$A^{+}$ analog of the formula
$P_{j}/P_{j+1}=\bigoplus_{\omega\in\Omega}P(\omega)_{j}/P(\omega)_{j+1}$. For
$j=j(i)$, the left-hand side becomes
$A^{+}_{j(i)}/A^{+}_{j(i)+1}=A^{+}_{j(i)}/A^{+}_{j(i-1)}=J_{i}/J_{i-1}$, which
is $J$ in the current context. On the right, we obtain a direct sum of modules
$\Delta(\omega)$ with $\mathfrak{h}(\omega)=j(i)$, all now known to be
projective as left $B$-modules.)

This completes the proof of (HI2) for $B$ and $J$ and all choices of $i$ with
$1\leq i\leq m$. Properties (HI1) and (HI3) hold as immediate consequence of
our claim studied several paragraphs above. This completes the proof of the
theorem. ∎

In Chapter 4, we will consider the localization ${\mathcal{Z}}^{\natural}$ of
${\mathcal{Z}}$ at the multiplicative set generated by all bad primes for $W$;
see Remark 0.1.1. By base change from ${\mathcal{Z}}$ to
${\mathcal{Z}}^{\natural}$, (3.5.2) gives rise to a sequence of ideals of
$A^{+\natural}:={\mathcal{Z}}^{\natural}\otimes A^{+}$:

$0=J_{0}^{\natural}\subseteq J_{1}^{\natural}\subseteq\cdots\subseteq
J_{m}^{\natural}=A^{+\natural}.$ (3.5.7)

We will prove in Section 4.3 that every ideal
$J^{\natural}:=J_{i}^{\natural}/J_{i-1}^{\natural}$ ($1\leq i\leq m$) is a
“heredity ideal of split type” in
$B^{\natural}:=A^{+\natural}/J_{i-1}^{\natural}$. More precisely, every
$\text{\rm{End}}_{B^{\natural}}(J^{\natural})$ is a direct product of matrix
algebras $M_{n}({\mathcal{Z}}^{\natural})$. Consequently, if $W$ is not of
type ${}^{2}F_{4}$, $A^{+\natural}$ is a (split) quasi-hereditary algebra.

######  Remark 3.5.4.

Replacing $\Omega$ by a poset ideal $\Theta$ of $(\Omega,\leq_{L})$, $T^{+}$
by the ${\mathcal{H}}$-module $T^{\ddagger}_{\Theta}$, $A^{+}$ by
$A^{\ddagger}_{\Theta}$ as defined in (2.4.6), and using the stratifying
system in Corollary 3.3.3 (thus, replacing $\Delta(\omega),P(\omega),P$, etc.
by $\Delta^{+}(\omega),P^{+}(\omega),P^{+}$, etc.) throughout the proof of
Theorem 3.5.3, we may construct a standard stratification

$0=J_{0}^{+}\subseteq J_{1}^{+}\subseteq\cdots\subseteq
J_{m}^{+}=A^{\ddagger}_{\Theta}.$ (3.5.8)

for the standardly stratified algebra $A^{\ddagger}_{\Theta}$.

##  Chapter 4 The quasi-heredity of $A^{+\natural}$

We proved in the last chapter that the enlarged Hecke endomorphism algebra
$A^{+}$ is standardly stratified, a property reminiscent of quasi-hereditary
algebras [CPS88]. In this chapter, we will prove that its base change
$A^{+\natural}:={\mathcal{Z}}^{\natural}\otimes_{{\mathcal{Z}}}A^{+}$ over
${\mathcal{Z}}^{\natural}$ (in which all bad primes are invertible) is
integral quasi-hereditary; see, e.g., [CPS90] (also [DPS18]) for integral
quasi-hereditary algebras.

###  4.1. Maximal orders
${\mathcal{J}}^{\natural}:={\mathcal{Z}}^{\natural}\otimes_{\mathbb{Z}}{\mathsf{J}}$
arising from Lusztig’s asymptotic form ${\mathsf{J}}$

Let $(W,S,L)$ be a standard finite Coxeter system associated to a finite group
$G$ of Lie type, and let ${\mathcal{H}}$ be its generic Hecke algebra over
${\mathcal{Z}}:=\mathbb{Z}[t,t^{-1}]$. As noted in Section 1.2, Lusztig’s
conjectures P1–P15 all hold for this situation. Also, $W$ is, of course,
finite, so the standing hypotheses of Lusztig used in [Lus03, §18] hold. In
this section, we temporarily exclude the case that $G$ is of type
${}^{2}F_{4}$, i.e., we assume $W$ is not the dihedral group $I_{2}(8)$ of
order 16. We will treat this case in a short appendix at the end of this
section. (However, everything up through (4.1.6) holds for ${}^{2}F_{4}$.
Recall that the smallest splitting field of $I_{2}(8)$ is
$\mathbb{Q}(\sqrt{2})$, the field generated by its irreducible character
values. The other Weyl groups have splitting field $\mathbb{Q}$.)

Let ${\mathcal{Z}}^{\natural}:={\mathcal{S}}^{-1}{\mathcal{Z}}$, the
localization of ${\mathcal{Z}}$ at the multiplicative set ${\mathcal{S}}$
generated by the bad primes of $W$.111As noted with more detail in Remark
0.1.1, one could use “bad prime” in the traditional sense here, or, for our
purposes, simply define it to be a prime divisor of the generic degree
denominator $m$ for a principal series character. Set
${\mathcal{H}}^{\natural}:={\mathcal{Z}}^{\natural}\otimes_{\mathcal{Z}}{\mathcal{H}}$.
Of course, ${\mathcal{Z}}^{\natural}$ is a unique factorization domain of
Krull dimension 2 (which is regular). In particular, its height one prime
ideals $\mathfrak{p}$ are all principal and are given (in this case) as
follows: (i) $\mathfrak{p}=p{\mathcal{Z}}^{\natural}$, where $p\in\mathbb{Z}$
is a good prime for $W$; and (ii) $\mathfrak{p}=f(t){\mathcal{Z}}^{\natural}$,
where $f(t)$ is an irreducible polynomial in $\mathbb{Z}[t]$, of positive
degree and not equal to $\pm t$. This description follows easily from [Mum88,
Example H, p. 74], noting that the prime ideals of ${\mathcal{Z}}$ correspond
naturally to those of $\mathbb{Z}[t]$ which do not contain $t$.

Before going further we introduce some general terminology on lattices and
orders over a Noetherian integral domain ${\mathscr{K}}$. We have
${\mathscr{K}}={\mathcal{Z}}$ or ${\mathcal{Z}}^{\natural}$ in mind. Often it
will be convenient to let $K$ be the fraction field of ${\mathscr{K}}$, e.g.,
$K=\mathbb{Q}(t)$, if ${\mathscr{K}}={\mathcal{Z}}$ or
${\mathcal{Z}}^{\natural}$.

######  Definitions 4.1.1.

(a) For any finite-dimensional $K$-space $V$, a (full) ${\mathscr{K}}$-lattice
in $V$ is a finite ${\mathscr{K}}$-submodule $M$ in $V$ such that $KM=V$,
where

$KM:=\Big{\\{}\sum r_{i}m_{i}(\text{finite sum})\mid r_{i}\in K,m_{i}\in
M\Big{\\}}.$

Following [Reiner03, p. 108], a ${\mathscr{K}}$-order is a
${\mathscr{K}}$-subalgebra ${\mathcal{A}}$ of a finite-dimensional $K$-algebra
$\mathfrak{A}$, having the same identity as $\mathfrak{A}$, such that
${\mathcal{A}}$ is a full ${\mathscr{K}}$-lattice in $\mathfrak{A}$. Unless
otherwise noted, from now on, the term “lattice” refers to a “full lattice.”

(b) Also, by some abuse of notation, we shall refer to a module $M$ for a
${\mathscr{K}}$-order ${\mathcal{A}}$ as an ${\mathcal{A}}$-lattice if it
inherits a ${\mathscr{K}}$-lattice structure from the action of
${\mathcal{A}}$. A ${\mathscr{K}}$-order ${\mathcal{A}}$ in $\mathfrak{A}$ is
a maximal ${\mathscr{K}}$-order if it is not properly contained in a larger
${\mathscr{K}}$-order in $\mathfrak{A}$; see [Reiner03, p. 110].

To give a common example of a maximal order in a finite-dimensional
$K$-algebra $\mathfrak{A}$, assume that ${\mathscr{K}}$ is integrally closed
and $\mathfrak{A}$ is a direct product $\prod_{i}M_{n_{i}}(K)$ of a finite
number of matrix algebras $M_{n_{i}}(K)$. Then
${\mathcal{A}}:=\prod_{i}M_{n_{i}}({\mathscr{K}})$ is a maximal
${\mathscr{K}}$-order in $\mathfrak{A}$. (See [Reiner03, Th. 8.7].)

We need some further notation, mostly taken from [Lus03]. For $w\in W$, let
$\text{\text{{\rm sgn}}}(w)=(-1)^{\ell(w)}$. Recall the notation
$\widetilde{T}_{w}$ from display (1.2.2). The map

$\widetilde{T}_{w}\longmapsto\widetilde{T}_{w}^{\dagger}:=\text{sgn}(w)\widetilde{T}_{w^{-1}}^{-1}$
(4.1.1)

defines a ${\mathcal{Z}}$-algebra involution $h\mapsto h^{\dagger}$ of
${\mathcal{H}}$. See [Lus03, 3.5]. Next, for $z\in W$, let $a(z)$ be the
smallest non-negative integer such that

$t^{a(z)}h_{x,y,z}\in\mathbb{Z}[t],\;\;\text{for all }x,y\in W.$ (4.1.2)

Here $h_{x,y,z}$ is the structure constant defined in (1.2.6). Then let
$\gamma_{x,y,z}\in\mathbb{Z}$ be the coefficient of $t^{a(z)}$ in $h_{x,y,z}$.

Obviously, ${\mathcal{H}}$ is a ${\mathscr{K}}$-order in
${\mathcal{H}}_{\mathbb{Q}(t)},$ using ${\mathscr{K}}={\mathcal{Z}}$. Next, we
define two further orders in (isomorphic copies of)
${\mathcal{H}}_{\mathbb{Q}(t)}$ in the sense of Definition 4.1.1. The first
one ${\mathcal{J}}:={\mathcal{Z}}\otimes_{\mathbb{Z}}{\mathsf{J}}$ is the base
change of a $\mathbb{Z}$-algebra ${\mathsf{J}}$; the latter, defined below, is
called Lusztig’s asymptotic form. As we shall see, the order ${\mathcal{J}}$
contains a copy of ${\mathcal{H}}$, as discussed below. The second order
${\mathcal{J}}^{\natural}:={\mathcal{Z}}^{\natural}\otimes_{\mathbb{Z}}{\mathsf{J}}\cong{\mathcal{Z}}^{\natural}\otimes_{\mathcal{Z}}{\mathcal{J}}$,
a ${\mathcal{Z}}^{\natural}$-order, is obtained from the first by inverting
all bad primes for $W$.

The algebra ${\mathsf{J}}$ is a free $\mathbb{Z}$-algebra with
$\mathbb{Z}$-basis $\\{{\mathsf{j}}_{x}\,|\,x\in W\\}$ and multiplication
given by

${\mathsf{j}}_{x}{\mathsf{j}}_{y}=\sum_{z\in
W}\gamma_{x,y,z^{-1}}{\mathsf{j}}_{z},\quad x,y\in W.$ (4.1.3)

Then ${\mathsf{J}}$ is associative and unital [Lus03, 18.3]. (Note that
rank$({\mathsf{J}})=|W|$.)

Recall that $W$ has a subset $\mathcal{D}$ of distinguished involutions
[Lus03, P6, 14.1]. Each left cell $\omega$ contains a unique element
$d\in{\mathcal{D}}$. Also, for $d\in\mathcal{D}$, an integer $n_{d}=\pm 1$ is
defined in [Lus03, 14.1, P5 & P13]. Given $z\in W$, let $d\in\mathcal{D}$ be
the unique distinguished involution satisfying $z^{-1}\sim_{L}d$, and set
$\hat{n}_{z}:=n_{d}$. By [Lus03, Th. 18.9], there is a ${\mathcal{Z}}$-algebra
homomorphism $\phi:{\mathcal{H}}\longrightarrow{\mathcal{J}}$ defined by

$\phi({\textsc{c}}_{w}^{\dagger})=\sum h_{w,d,z}\hat{n}_{z}{\mathsf{j}}_{z}.$
(4.1.4)

In this expression, $w\in W$ and the sum is over all pairs
$(d,z)\in\mathcal{D}\times W$ such that $a(z)=a(d)$. Then, for any commutative
${\mathcal{Z}}$-algebra $R$, $\phi$ induces an $R$-algebra homomorphism
$\phi_{R}:{\mathcal{H}}_{R}\to{\mathcal{J}}_{R}$. By [Lus03, Prop. 18.12a],
the kernel of $\phi_{R}$ is a nilpotent ideal. Thus, when $R$ is a field and
${\mathcal{H}}_{R}$ is a semisimple algebra, $\phi_{R}$ is injective and,
thus, an isomorphism by dimension considerations. In particular,
$\phi_{\mathbb{Q}(t)}$ is an isomorphism. Of course, this means that $\phi$ is
an injection, so that ${\mathcal{J}}$ and ${{\mathcal{J}}}^{\natural}$ both
contain subalgebras isomorphic to ${\mathcal{H}}$. The case $R=\mathbb{Q}$
leads to a similar isomorphism
${\mathcal{H}}_{\mathbb{Q}}\overset{\sim}{\rightarrow}{\mathcal{J}}_{\mathbb{Q}}$,
regarding $\mathbb{Q}$ as a ${\mathcal{Z}}$-algebra by means of the
specialization $t\mapsto 1$. Note
${\mathcal{J}}_{\mathbb{Q}}\cong{\mathsf{J}}_{\mathbb{Q}}$ by construction.
For later use in the proof of the theorem below, note also that
${\mathcal{H}}_{\mathbb{Q}}\cong{\mathbb{Q}}W$, and so,
${\mathsf{J}}_{\mathbb{Q}}\cong{\mathcal{J}}_{\mathbb{Q}}\cong{\mathcal{H}}_{\mathbb{Q}}\cong\mathbb{Q}W$.

######  Remark 4.1.2.

All the assertions of the above paragraph (below display (4.1.4)) hold as
written if the homomorphism $\phi$ is replaced by the composition

$\varpi:=\phi\circ\dagger:{\mathcal{H}}\longrightarrow{\mathcal{J}},$ (4.1.5)

where $\dagger$ is defined in display (4.1.1). This is not needed right away,
but will be useful in showing in the next section that all left-cell modules
for ${\mathcal{H}}$ extend to ${\mathcal{J}}$-modules. For the convenience of
the reader, we write down explicitly the expression for the composite
$\varpi=\phi\circ\dagger$:

$\varpi({\textsc{c}}_{w})=\sum_{z\in W}\sum_{d\in\mathcal{D}\atop
a(d)=a(z)}h_{w,d,z}\hat{n}_{z}{\mathsf{j}}_{z}\quad(w\in W).$ (4.1.6)

We will also use $\varpi$ in the context of ${\mathcal{H}}^{\natural}$ and
${{\mathcal{J}}}^{\natural}$, in which case we define $\varpi^{\natural}$ to
be the base-changed map

$\varpi^{\natural}:\,{\mathcal{H}}^{\natural}={\mathcal{H}}_{{\mathcal{Z}}^{\natural}}\overset{\varpi_{{\mathcal{Z}}^{\natural}}}{\longrightarrow}{\mathcal{J}}_{{\mathcal{Z}}^{\natural}}={\mathcal{J}}^{\natural}.$
(4.1.7)

Moreover, the isomorphisms
${\mathsf{J}}_{\mathbb{Q}}\cong{\mathcal{H}}_{\mathbb{Q}}\cong\mathbb{Q}W$ and
${\mathsf{J}}_{\mathbb{Q}(t)}={\mathcal{J}}_{\mathbb{Q}(t)}\cong{\mathcal{H}}_{\mathbb{Q}(t)}$
induce an isomorphism $\mathbb{Q}(t)W\cong{\mathcal{H}}_{\mathbb{Q}(t)}$ and,
hence, a bijection

$\text{Irr}(\mathbb{Q}W)\longrightarrow\text{Irr}({\mathcal{H}}_{\mathbb{Q}(t)}),\;E\longmapsto
E_{\mathbb{Q}(t)}.$ (4.1.8)

If $G$ is of type ${}^{2}F_{4}$, $\mathbb{Q}$ should be replaced by
$\mathbb{Q}(\sqrt{2})$.

Next, we pursue some of the remarkable properties of the
${\mathcal{Z}}^{\natural}$-order ${{\mathcal{J}}}^{\natural}$. Here,
${\mathscr{K}}={\mathcal{Z}}^{\natural}$ and $K=\mathbb{Q}(t)$.

Note that ${{\mathcal{J}}}^{\natural}_{{\mathbb{Q}}(t)}$ identifies with
${\mathcal{J}}_{\mathbb{Q}(t)}$ which is isomorphic (via the inverse of
$\phi_{\mathbb{Q}(t)}$) to ${\mathcal{H}}_{{\mathbb{Q}}(t)}$. The latter is
semisimple by Remark 3.3.4(b) and even split, since $\mathbb{Q}W$ is split
semisimple. (Recall that we have postponed the case ${}^{2}F_{4}$ to an
appendix at the end of this section.) Hence, there is a “natural” isomorphism

${\mathcal{J}}^{\natural}_{{\mathbb{Q}}(t)}\overset{\sim}{\longrightarrow}\prod_{i}\text{\rm{End}}_{{\mathbb{Q}}(t)}(E_{i}),$
(4.1.9)

where $E_{i}$ ranges over the distinct irreducible
${\mathcal{J}}_{{\mathbb{Q}}(t)}$-modules. The map is obtained from the action
of ${{\mathcal{J}}}^{\natural}_{\mathbb{Q}(t)}={\mathcal{J}}_{\mathbb{Q}(t)}$
on the various $E_{i}$. We can assume that $E_{i}=KL_{i}$ where $L_{i}$ is a
${{\mathcal{J}}}^{\natural}$-stable lattice in $E_{i}$. In fact, using
Auslander-Goldman theory (see [DDPW08, Cor. C.18]), it can also be assumed
(after replacing $L_{i}$ by its double dual $L_{i}^{**}$) that all the $L_{i}$
are projective over ${\mathcal{Z}}^{\natural}$ . Then, by the result of Swan
mentioned in Remark 1.3.6(c)), the $L_{i}$ are free over
${\mathcal{Z}}^{\natural}$.

Thus, (4.1.9) may be obtained by base change from ${\mathcal{Z}}^{\natural}$
to $\mathbb{Q}(t)$ of an injection

$\psi:{{\mathcal{J}}}^{\natural}\to\prod_{i}\text{\rm{End}}_{{\mathcal{Z}}^{\natural}}(L_{i}),$
(4.1.10)

where each $L_{i}$ is ${\mathcal{Z}}^{\natural}$-free. Note that each
${{\mathcal{J}}}^{\natural}$-module $L_{i}$ is an
${\mathcal{H}}^{\natural}$-module by restriction through the map
$\varpi^{\natural}$ in (4.1.7).

We can now prove:

######  Theorem 4.1.3.

Maintain the assumptions on $W$, ${\mathcal{H}}$, etc. above. The map $\psi$
in (4.1.10) is an isomorphism of ${\mathcal{Z}}^{\natural}$-algebras.

###### Proof.

We have already noted that the map $\psi$ is an injection (since its base
change to ${\mathbb{Q}}(t)$ is obviously an injection). Clearly,
${{\mathcal{J}}}^{\natural}$ is a ${\mathcal{Z}}^{\natural}$-order (and, in
particular, a full lattice) in ${{\mathcal{J}}}^{\natural}_{\mathbb{Q}(t)}$.
To prove the theorem, it suffices to prove that ${{\mathcal{J}}}^{\natural}$
is a maximal ${\mathcal{Z}}^{\natural}$-order in
${{\mathcal{J}}}^{\natural}_{\mathbb{Q}(t)}$. This will show that $\psi$ is an
isomorphism, since it is an inclusion of orders. To do this, it is necessary
and sufficient, by the Auslander-Goldman theorem (see [Reiner03, Th. 11.4]),
to check two conditions:

  * (a)

the ${\mathcal{Z}}^{\natural}$-order $\Xi:={{\mathcal{J}}}^{\natural}$ is
reflexive in the sense that $\Xi^{**}\cong\Xi$ as
${\mathcal{Z}}^{\natural}$-modules;

  * (b)

for any height one prime ideal $\mathfrak{p}$ in ${\mathcal{Z}}^{\natural}$,
${{\mathcal{J}}}^{\natural}_{\mathfrak{p}}$ is a maximal
${\mathcal{Z}}^{\natural}_{\mathfrak{p}}$-order in
${{\mathcal{J}}}^{\natural}_{\mathbb{Q}(t)}$.

But ${{\mathcal{J}}}^{\natural}$ is ${\mathcal{Z}}^{\natural}$-free, so it is
certainly reflexive. Thus, (a) holds.

For part (b), it suffices to show, for any height one prime ideal
$\mathfrak{p}$ in ${\mathcal{Z}}^{\natural}$, that there is an abstract
algebra isomorphism

${{\mathcal{J}}}^{\natural}_{\mathfrak{p}}\cong\prod_{i}M_{n_{i}}({\mathcal{Z}}^{\natural}_{\mathfrak{p}})$
(4.1.11)

for some finite list of positive integers $n_{i}$. See [Reiner03, Ths. 8.7,
10.5].

Consider first the case in which ${\mathfrak{p}}=(p)$, where $p$ is a good
prime for $W$. Then [DPS98a, Ths. 4.2.1, 4.2.2] and their proofs show that

${\mathcal{H}}_{\mathbb{F}_{p}(t)}\cong\prod_{i}\text{\rm{End}}_{\mathbb{F}_{p}(t)}(L_{i{\mathbb{F}_{p}(t)}})$
(4.1.12)

with $L_{i}$ as above, and
$L_{i{\mathbb{F}_{p}(t)}}=(L_{i})_{\mathbb{F}_{p}(t)}$ denoting the base-
changed module $\mathbb{F}_{p}(t)\otimes_{{\mathcal{Z}}^{\natural}}L_{i}$.
Note that $\mathbb{F}_{p}(t)$ is the residue field of
${\mathcal{Z}}^{\natural}_{\mathfrak{p}}$. (Essentially, the argument in
[DPS98a] uses Schur matrix relations and the fact that good primes avoid
generic degree denominators [DPS98a, (4.1.2)] to show that all matrix units on
the right have preimages on the left.) Since the action of
${\mathcal{H}}_{{\mathcal{Z}}^{\natural}}$ on $L_{i}$ is by restriction
through the algebra ${{\mathcal{J}}}^{\natural}$, we have a surjective
composite map

$({\mathcal{H}}_{{\mathcal{Z}}^{\natural}})_{\mathfrak{p}}\hookrightarrow{{\mathcal{J}}}^{\natural}_{\mathfrak{p}}\longrightarrow
M_{n_{i}}({\mathcal{Z}}^{\natural}_{\mathfrak{p}})\longrightarrow
M_{n_{i}}({\mathbb{F}}_{p}(t)),$

where the two matrix algebras are the full endomorphism rings of
$(L_{i})_{{\mathcal{Z}}^{\natural}_{\mathfrak{p}}}$ and
$(L_{i})_{\mathbb{F}_{p}(t)}$, respectively. Consequently, the map

${{\mathcal{J}}}^{\natural}_{\mathfrak{p}}\longrightarrow\prod_{i}M_{n_{i}}({\mathbb{F}}_{p}(t))$

is surjective, as is the map

${{\mathcal{J}}}^{\natural}_{\mathfrak{p}}\longrightarrow\prod_{i}M_{n_{i}}({\mathcal{Z}}^{\natural}_{\mathfrak{p}})\qquad(\text{by
Nakayama's Lemma}).$

Since the latter map is an injection, it is also an isomorphism.

Finally, it is enough to show that an isomorphism (4.1.11) holds whenever
${\mathfrak{p}}=(f(t))$ with $f(t)\in{\mathbb{Z}}[t]$ an irreducible
polynomial of positive degree and $\not=\pm t$. (These are the remaining
height one primes in ${\mathcal{Z}}^{\natural}$.) In this case, no nonzero
integer $z\not=0$ is divisible by $f(t)$ in the unique factorization domain
${\mathcal{Z}}^{\natural}$. Thus,
$\mathbb{Q}\subseteq{\mathcal{Z}}^{\natural}_{\mathfrak{p}}$. Since

${\mathsf{J}}_{\mathbb{Q}}\cong{\mathbb{Q}}W\cong\prod_{j}M_{n_{i}}({\mathbb{Q}})$

for some positive integers $n_{i}$,222The first isomorphism is discussed above
Remark 4.1.2. It may be regarded as one of the many well-known results of
Lusztig generalized in [Lus03] to an unequal parameter setting. we have

$\displaystyle{\mathcal{J}}^{\natural}_{\mathfrak{p}}:={\mathcal{Z}}^{\natural}_{\mathfrak{p}}\otimes_{{\mathcal{Z}}^{\natural}}{\mathcal{J}}^{\natural}$
$\displaystyle\cong{\mathcal{Z}}^{\natural}_{\mathfrak{p}}\otimes_{{\mathcal{Z}}^{\natural}}{\mathcal{Z}}^{\natural}\otimes_{{\mathcal{Z}}}{\mathcal{J}}\cong{\mathcal{Z}}^{\natural}_{\mathfrak{p}}\otimes_{{\mathcal{Z}}^{\natural}}{\mathcal{Z}}^{\natural}\otimes_{{\mathcal{Z}}}{\mathcal{Z}}\otimes_{\mathbb{Z}}{\mathsf{J}}$
$\displaystyle\cong{\mathcal{Z}}^{\natural}_{\mathfrak{p}}\otimes_{\mathbb{Z}}{\mathsf{J}}\cong{\mathcal{Z}}^{\natural}_{\mathfrak{p}}\otimes_{\mathbb{Q}}\mathbb{Q}\otimes_{\mathbb{Z}}{\mathsf{J}}\cong{\mathcal{Z}}^{\natural}_{\mathfrak{p}}\otimes_{\mathbb{Q}}{\mathsf{J}}_{\mathbb{Q}}\cong\prod_{i}M_{n_{i}}({\mathcal{Z}}^{\natural}_{\mathfrak{p}}),$

completing part (b) of the proof. ∎

This has several consequences. The first corollary, which we call a theorem,
follows either from the proof of Theorem 4.1.3 or from its statement.
(Remember that $K=\mathbb{Q}(t)$ and $G$ is not of type ${}^{2}F_{4}$.)

######  Theorem 4.1.4.

(a) The ${\mathcal{Z}}^{\natural}$-algebra
${{\mathcal{J}}}^{\natural}={\mathcal{J}}_{{\mathcal{Z}}^{\natural}}$ is
isomorphic to a maximal order in ${\mathcal{H}}_{K}$.

(b) Furthermore,

${{\mathcal{J}}}^{\natural}={\mathcal{J}}_{{\mathcal{Z}}^{\natural}}\cong\prod_{i}M_{n_{i}}({\mathcal{Z}}^{\natural}),$
(4.1.13)

for a finite sequence $n_{1},n_{2},\cdots$ of positive integers.

######  Remarks 4.1.5.

(1) The above theorem has an analogous version in which the
${\mathcal{Z}}^{\natural}$-algebra ${\mathcal{J}}^{\natural}$ is replaced by
the $\mathbb{Z}^{\natural}$-algebra ${\mathsf{J}}^{\natural}$. In this
context, another consequence of Theorem 4.1.4 is that
${\mathsf{J}}^{\natural}$ is a maximal order in
${\mathcal{H}}_{\mathbb{Q}}=\mathbb{Q}W.$ Explicitly,
${\mathsf{J}}^{\natural}$ is isomorphic to a direct product of full matrix
algebras over $\mathbb{Z}^{\natural}$. This follows by a base-change argument
using the specialization
${\mathcal{Z}}^{\natural}\rightarrow\mathbb{Z}^{\natural}$ given by $t\mapsto
1$. Finally, all corollaries below have an analogous version for
${\mathsf{J}}^{\natural}$. They are easily derived through a similar proof, or
through a base-change argument together with the statements for
${\mathcal{Z}}$.

(2) By the theorem, we have
${\mathcal{J}}_{K}={\mathcal{J}}^{\natural}_{K}\cong\prod_{i=1}^{s}M_{n_{i}}(K)$,
for some $s$, when $K=\mathbb{Q}(t)$. Thus, analyzing projections, there are a
total of $s$ non-isomorphic irreducible ${\mathcal{J}}_{K}$-modules
$L_{1},\ldots,L_{s}$ such that, for each $i$ with $1\leq i\leq s$, $L_{i}\cong
K^{n_{i}}$ as ${\mathcal{J}}_{K}$-modules, where $K^{n_{i}}$ is the
irreducible left $M_{n_{i}}(K)$-module via matrix multiplication. The
discussion of the map $\psi$, in and above Theorem 4.1.3, implies that $L_{i}$
has a ${\mathcal{J}}^{\natural}$-lattice $L_{i,{\mathcal{Z}}^{\natural}}$
isomorphic to $({\mathcal{Z}}^{\natural})^{n_{i}}$. In particular,
$L_{i,{\mathcal{Z}}^{\natural}}$ is ${\mathcal{Z}}^{\natural}$-free.

(3) In the equal parameter case, Lusztig first proved in [Lus87d,
(3.1j),(1.2a)] that, for a large enough field $F$, ${\mathsf{J}}_{F}$ is split
semisimple. Further, in [G98, Remark 2.5], Geck observed that this result
applies to the case where the characteristic of $F$ is a good prime. See also
[GJ11, Props. 1.5.7&2.3.16] for a similar result in general over a
sufficiently large field $\mathbb{K}$ of characteristic 0.

We note that a maximal order copy of ${{\mathcal{J}}}^{\natural}$ in
${\mathcal{H}}_{K}$ is $\phi_{K}^{-1}({{\mathcal{J}}}^{\natural})$, in the
notation of (4.1.4). Another useful copy is obtained by applying to this copy
the involution on ${\mathcal{H}}_{K}$ induced by $\dagger$. The resulting copy
may also be written $\varpi_{K}^{-1}({{\mathcal{J}}}^{\natural})$, with
$\varpi$ as in Remark 4.1.2.

Recall that if $R$ is a commutative ring, an algebra $B$ over $R$ is separable
if $B$ is a projective $B\otimes_{R}B^{{\text{\rm op}}}$-module; see
Auslander-Goldman [AG60b, p. 1].

######  Corollary 4.1.6.

(a) The algebra ${{\mathcal{J}}}^{\natural}$ is separable over
${\mathcal{Z}}^{\natural}$.

(b) Every finite module for ${{\mathcal{J}}}^{\natural}$, which is projective
over ${\mathcal{Z}}^{\natural}$, is projective over
${{\mathcal{J}}}^{\natural}$.

###### Proof.

The reader may directly check that the direct product of a finite set of
separable algebras is a separable algebra. Then, in view of Theorem 4.1.3,
part (a) follows from [AG60b, Prop. 5.1]. Finally, part (b) follows from
[Hat63, Th. 2.11]. (See also [Reiner03, Ex. 9, p. 106].) ∎

######  Corollary 4.1.7.

Let $M$ and $M^{\prime}$ be finite ${{\mathcal{J}}}^{\natural}$-modules,
projective over ${\mathcal{Z}}^{\natural}$. If $M_{K}\cong M^{\prime}_{K}$ as
${{\mathcal{J}}}^{\natural}_{K}$-modules, then $M\cong M^{\prime}$ as
${{\mathcal{J}}}^{\natural}$-modules.

###### Proof.

Note that $M$ and $M^{\prime}$ are projective over
${{\mathcal{J}}}^{\natural}$ by Corollary 4.1.6(b). By Theorem 4.1.4, we can
reduce to the case where $M$ and $M^{\prime}$ are finite projective modules
for a matrix algebra $M_{n}({\mathcal{Z}}^{\natural})$ which become isomorphic
over the fraction field $K$. By Morita theory, we can assume $n=1$. (Let
$F:\text{mod-}M_{m}({\mathcal{Z}}^{\natural})\rightarrow{\text{
mod-}}{\mathcal{Z}}^{\natural}$ be an equivalence of
${\mathcal{Z}}^{\natural}$-categories, e.g., a suitable idempotent projection.
Note that $F$ induces by base change an equivalence
$F_{K}:{\text{mod-}}M_{m}(K)\rightarrow\text{mod-}K$. For
$X,Y\in{\text{mod-}}M_{m}({\mathcal{Z}}^{\natural})$, we clearly have $X\cong
Y$ if and only if $F(X)\cong F(Y)$, and $X_{K}\cong Y_{K}$ in mod-$M_{m}(K)$
if and only if $F(X)_{K}\cong F(Y)_{K}$ in mod-$K$.) By Swan’s theorem (see
Remark 1.3.6(c)), we can assume that $M$ and $M^{\prime}$ are free
${\mathcal{Z}}^{\natural}$-modules which are isomorphic over $K$. Obviously,
this means $M\cong M^{\prime}$. ∎

Finally, we have:

######  Corollary 4.1.8.

If $M$ is a ${{\mathcal{J}}}^{\natural}$-module which is finite and projective
over ${\mathcal{Z}}^{\natural}$, then

  * (a)

$M$ is a direct sum of ${{\mathcal{J}}}^{\natural}$-modules which are
projective over ${{\mathcal{J}}}^{\natural}$ and which have irreducible base
change to ${{\mathcal{J}}}^{\natural}_{K}$-modules.

  * (b)

Moreover, $M$ is uniquely determined (up to isomorphism) by the multiplicity
of each isomorphism type of the irreducible summands of the
${{\mathcal{J}}}^{\natural}_{K}$-module $M_{K}$.

###### Proof.

We first prove part (a). By Theorem 4.1.4, we may replace
${{\mathcal{J}}}^{\natural}$ by one of its direct factors
$M_{m}({\mathcal{Z}}^{\natural})$. Using Morita theory, we can even replace
$M_{m}({\mathcal{Z}}^{\natural})$ by ${\mathcal{Z}}^{\natural}$. Once again,
applying Swan’s theorem, the ${\mathcal{Z}}^{\natural}$-module $M$ must be
free, and so clearly is a direct sum of ${\mathcal{Z}}^{\natural}$-modules
with irreducible base change to $K$. This proves (a).

For the proof of (b), the reduction is simply reversed. Further details are
left to the reader.∎

#### Appendix to Section 4.1: the case of the Ree groups of type
${}^{2}\\!F_{4}$

The Weyl group $W$ is the dihedral group $I_{2}(8)$ of order 16, generated by
reflections $s_{1}$ and $s_{2}$ with $s_{1}s_{2}$ having order 8. The weights
are determined from the longest words in each individual orbit of a graph
“symmetry”; see [C85, pp. 36-38] or by analogy with the construction [Lus03,
16.1–16.2] formulated in the quasi-split case. We can assume $L(s_{1})=2$ and
$L(s_{2})=4$ for the values of the weight function $L$ on fundamental
reflections $s_{1}$, $s_{2}$. See [C85, pp. 489-490] for the character degrees
of the unipotent characters in the principal series. (Note that the degree
associated to $\rho_{2}$ should be corrected to
$(1/2)q^{4}\Phi_{8}^{2}\Phi_{24}$.)333The character degree given in [C85] does
not appear in Lusztig’s list of unipotent character degrees. The value we have
given does appear in [Lus84, p. 374] but just as a unipotent character degree.
It can be seen to be a generic/principal series degree by direct calculation
using [GP00, pp. 270-272]. There are 7 degrees in all, labeled
$1,\epsilon^{\prime},\epsilon^{\prime\prime},\epsilon$ (for the degree 1
representations of ${\mathbb{C}}W$) and
$\rho^{\prime}_{2},\rho_{2}^{\prime\prime},\rho_{2}$ (for the degree 2
representations of ${\mathbb{C}}W$). The degrees of all the unipotent
characters are in terms of polynomial functions, elements of
$\mathbb{Q}(\sqrt{2})[q]$, evaluated with $q$ an odd power of $\sqrt{2}$. The
generic degrees are obtained by replacing $q$ by the variable $t$. From the
corrected list, we see that the generic degrees have the form

${\frac{1}{m}}t^{a}\Phi^{\prime}_{b_{1}}(t)\cdots\Phi^{\prime}_{b_{r}}(t),$
(4.1.14)

where $a$ is a non-negative integer. Also, $\Phi^{\prime}_{b_{i}}(t)$ is a
factor in $\mathbb{Z}[\sqrt{2}][t]$ of the corresponding cyclotomic polynomial
$\Phi_{b_{i}}(t)$ in $\mathbb{Z}[t]$. The integer $m$ is equal to 1, 2, or 4.

The expression (4.1.14) is sufficiently similar to [DPS98a, (4.1.2)] that
analogs of [DPS98a, Th. 4.2.1] and [DPS98a, Th. 4.2.2] may be proved with
parallel arguments. It is only necessary to replace $\mathbb{Z}$ by
$\mathbb{Z}^{\prime}=\mathbb{Z}[\sqrt{2}]$, $\mathbb{Q}$ by
$\mathbb{Q}^{\prime}=\mathbb{Q}[\sqrt{2}]$, ${\mathcal{H}}$ by
${\mathcal{H}}^{\prime}={\mathcal{H}}_{\mathbb{Z}^{\prime}[t,t^{-1}]}$, $K$ by
$K^{\prime}=\mathbb{Q}^{\prime}(t)=K[\sqrt{2}]$, etc.

It will also be useful to define
${\mathcal{Z}}^{\prime}=\mathbb{Z}^{\prime}[t,t^{-1}]=\mathbb{Z}[\sqrt{2},t,t^{-1}]$,
${\mathcal{Z}}^{\prime\natural}={\mathcal{Z}}^{\prime}[1/\sqrt{2},t,t^{-1}]$,
$\mathbb{Z}^{\natural}=\mathbb{Z}[1/2],$ and
$\mathbb{Z}^{\natural\prime}=\mathbb{Z}[1/2][1/\sqrt{2}]=\mathbb{Z}[1/\sqrt{2}]$.
Finally, defining
${\mathcal{Z}}^{\natural\prime}=\mathbb{Z}^{\natural\prime}[t,t^{-1}]$, we
have ${\mathcal{Z}}^{\natural\prime}={\mathcal{Z}}^{\prime\natural}.$

The only “bad prime” is $p=2$ in $\mathbb{Z}$ (or $p=\sqrt{2}$ in
$\mathbb{Z}^{\prime}$) in the sense of dividing $m$ in (4.1.14). Fortunately,
$\mathbb{Z}^{\prime}=\mathbb{Z}[\sqrt{2}]$ is a principal ideal domain. Using
these facts, the argument in the proof of Theorem 4.1.3 establishing the
intermediate isomorphism (4.1.12) determines an analogous isomorphism. In the
analogy, ${\mathcal{H}}^{\prime}$ replaces ${\mathcal{H}}$, and
${\mathcal{Z}}^{\prime\natural}$-free $\mathcal{J}^{\prime\natural}$-modules
$L_{i}^{\prime}$ replaces $L_{i}$, for each $i$. Also, $\mathbb{F}_{p}(t)$ is
replaced by $\mathbb{F}^{\prime}_{p}(t)$, the residue field of
${\mathcal{Z}}^{\prime}_{\mathfrak{p}}$ (which equals the residue field of
${\mathcal{Z}}^{\prime\natural}_{\mathfrak{p}}.$) Here, $\mathfrak{p}$ is the
ideal generated by $p\in{\mathcal{Z}}^{\prime}$, where $p$ is a prime not
dividing $\sqrt{2}$. With this analog of the isomorphism (4.1.12) in hand, an
evident version of the isomorphism (4.1.11) follows for each $\mathfrak{p}$,
using the argument in the proof of Theorem 4.1.3. Consequently, each
$\mathcal{J}_{\mathfrak{p}}^{\prime\natural}$ is a maximal order. Thus,
$\mathcal{J}^{\prime\natural}$, which is clearly reflexive, is also a maximal
order, and the map

$\psi:\mathcal{J}^{\prime\natural}\to\prod_{i}\text{\rm{End}}_{{\mathcal{Z}}^{\prime\natural}}(L^{\prime}_{i}),$
(4.1.15)

analogous to (4.1.10), is an isomorphism of
${\mathcal{Z}}^{\prime\natural}$-algebras. This proves the following theorem,
an analog of Theorem 4.1.3 above, in the Ree group setting of this appendix.

######  Theorem 4.1.9.

The map $\psi$ in (4.1.15) is an isomorphism of
${\mathcal{Z}}^{\prime\natural}$-algebras.

Each of the endomorphism (“matrix”) components of (4.1.15) is a maximal order
in some $M_{n}(K^{\prime})$ by [Reiner03, Th. 8.7], since
${\mathcal{Z}}^{\prime\natural}$ is integrally closed in
$K^{\prime}=\mathbb{Q}^{\prime}(t)$. The result below is a corollary of the
above theorem, but given its importance, we upgrade it to a theorem.

######  Theorem 4.1.10.

The ${\mathcal{Z}}^{\prime\natural}$-algebra
$\mathcal{J}^{\prime\natural}=\mathcal{J}_{{\mathcal{Z}}^{\prime\natural}}$ is
a maximal order in ${\mathcal{H}}_{\mathbb{Q}^{\prime}(t)}$. Moreover,

${\mathcal{J}}^{\prime\natural}={\mathcal{J}}_{{\mathcal{Z}}^{\prime\natural}}\cong\prod_{i}M_{m_{1}}({\mathcal{Z}}^{\prime\natural})$
(4.1.16)

for a finite sequence $m_{1},m_{2},\cdots$ of positive integers.

######  Remarks 4.1.11.

(a) Similarly, Corollaries 4.1.6–4.1.8 above have analogs for ${}^{2}F_{4}$ in
the ${\mathcal{Z}}^{\natural\prime}={\mathcal{Z}}^{\prime\natural}$-setting.
We leave the details and proofs, which follow the original arguments, to the
reader. Theorem 4.1.10 above is similarly analogous to Theorem 4.1.4.

(b) Also, Corollaries 4.1.6 to 4.1.8 and Theorem 4.1.4(a) remain true in the
${}^{2}F_{4}$-case, as stated, that is, using
${\mathcal{Z}}^{\natural}:=\mathbb{Z}[{{1}\over{2}},t,t^{-1}]$, and
${\mathcal{J}}^{\natural}:={\mathcal{J}}_{{\mathcal{Z}}^{\natural}}$.
(However, Theorem 4.1.4(b) is not true without further modification; see
display (4.1.17) below.) The proofs below, in this new setting for the
${}^{2}F_{4}$ case, are somewhat more involved.

To prove Theorem 4.1.4(a) in this new setting, note that if
${{\mathcal{J}}}^{\natural}$ were properly contained in a
${\mathcal{Z}}^{\natural}$-order—call it ${\mathcal{J}}^{\clubsuit}$—then the
${\mathcal{Z}}^{\natural\prime}$-order
${\mathcal{J}}^{\clubsuit}_{{\mathcal{Z}}^{\natural\prime}}$ would properly
contain ${\mathcal{J}}^{\natural\prime}$, contradicting Theorem 4.1.10 above.

Next, to prove Corollary 4.1.6 in the new setting, observe first that part (b)
follows from part (a), using the argument already given in the original
context. Part (a) follows from the following faithfully flat descent theorem:

######  Theorem 4.1.12 ([F17, Th. 8.1.20(1)]).

Let $R$ be a commutative ring, $B$ a finite $R$-algebra. Suppose $S$ is a
commutative faithfully flat $R$-algebra such that $B\otimes_{R}S$ is separable
over $S$. Then, $B$ is separable over $R$.

For a more elementary proof of Corollary 4.1.6(a), first show, using the
maximality of ${{\mathcal{J}}}^{\natural}$ (Theorem 4.1.4(a)), that
${\mathcal{J}}^{\natural}$ is a direct product of $1\times 1$ and $2\times 2$
full matrix algebras over ${\mathcal{Z}}^{\natural}$, and a $2\times 2$ full
matrix algebra over ${\mathcal{Z}}^{\natural\prime}$:

${\mathcal{J}}^{\natural}\cong{\mathcal{Z}}^{\natural}\oplus{\mathcal{Z}}^{\natural}\oplus{\mathcal{Z}}^{\natural}\oplus{\mathcal{Z}}^{\natural}\oplus
M_{2}({\mathcal{Z}}^{\natural})\oplus M_{2}({\mathcal{Z}}^{\natural\prime}).$
(4.1.17)

Here, the factor $M_{2}({\mathcal{Z}}^{\natural\prime})$ arises, using the
field automorphism $\sqrt{2}\mapsto-\sqrt{2}$, as a “diagonal”
${\mathcal{Z}}$-subalgebra of $M_{2}({\mathcal{Z}}^{\natural\prime})\oplus
M_{2}({\mathcal{Z}}^{\natural\prime})$. It suffices to prove the separability
of this factor over ${\mathcal{Z}}^{\natural}$, which reduces to the
separability of ${\mathcal{Z}}^{\natural\prime}$ over
${\mathcal{Z}}^{\natural}$, and then to the separability of
$\mathbb{Z}^{\natural\prime}$ over $\mathbb{Z}^{\natural}$. A splitting of the
(bimodule) multiplication surjection
$\mathbb{Z}^{\natural\prime}\otimes_{\mathbb{Z}^{\natural}}\mathbb{Z}^{\natural^{\prime}}\longrightarrow\mathbb{Z}^{\natural\prime}$
is given by sending $1\in\mathbb{Z}^{\natural\prime}$ to
$(1/4)(\sqrt{2}\otimes\sqrt{2})+(1/2)(1\otimes 1)$. This proves the needed
separability and completes the elementary proof.

To prove Corollary 4.1.7 in the new setting, note that proving a similar
statement for each component algebra of ${{\mathcal{J}}}^{\natural}$ in
(4.1.17) is equivalent. The original proof obviously works for all components
except for the ${\mathcal{Z}}^{\natural}$-algebra
$M_{n}({\mathcal{Z}}^{\natural\prime})$. Here, it is necessary and sufficient
to show (call it a claim) that projective modules $M,M^{\prime}$ for this
component, which are isomorphic after base change to $K$, are themselves
isomorphic. If we view $M_{n}({\mathcal{Z}}^{\natural\prime})$ as a
${\mathcal{Z}}^{\natural\prime}$-algebra in this claim, and replace $K$ by
$K^{\prime}$, its validity follows from the proof of the original Corollary
4.1.7, in the spirit of (a) above. However, the module categories for this
matrix component, whether viewed as a ${\mathcal{Z}}^{\natural}$\- or
${\mathcal{Z}}^{\natural\prime}$-algebra, are the same. Moreover, the
respective base-change functors, $K\otimes_{{\mathcal{Z}}^{\natural}}(-)$ and
$K^{\prime}\otimes_{{\mathcal{Z}}^{\natural\prime}}(-)$ are naturally
isomorphic, even in the larger category of
${\mathcal{Z}}^{\natural\prime}$-modules. In view of these identifications,
the cited argument proves the claim, and the version of Corollary 4.1.6 in new
setting follows.

Corollary 4.1.8 follows in the new setting from its current proof together
with the decomposition of ${\mathcal{J}}^{\natural}$ given in (4.1.17) above.
The above identification of $K-$ and $K^{\prime}-$base changes for
$M_{n}({\mathcal{Z}}^{\natural\prime})$-modules is also useful here.

Note, too, that specializing $t\mapsto 1$ in the expression (4.1.17) gives an
analogous and very precise version of Remark 4.1.5 in the ${}^{2}F_{4}$ case.

Finally, we alert the reader that Theorem 4.1.12 will be needed in Section
4.3.

###  4.2. Left cells and the order
${\mathcal{J}}:={\mathcal{Z}}\otimes_{\mathbb{Z}}{\mathsf{J}}$

Recall from (4.1.5) that $\varpi:{\mathcal{H}}\rightarrow{\mathcal{J}}$ is the
${\mathcal{Z}}$-algebra homomorphism $\varpi=\phi\circ\dagger$, where
$\phi:{\mathcal{H}}\to{\mathcal{J}}$ is defined in (4.1.4) and
$\dagger:{\mathcal{H}}\to{\mathcal{H}}$ is defined in (4.1.1). Given a
${\mathcal{J}}$-module $M$, there is a unique ${\mathcal{H}}$-module
$M^{\varpi}=\\{m^{\varpi}\,|\,m\in M\\}$ obtained by letting
${\mathcal{H}}$-act on $M$ through $\varpi$, i.e.,
$hm^{\varpi}=(\varpi(h)m)^{\varpi}$, for each $h\in{\mathcal{H}}$, $m\in M$.
In other words, $M^{\varpi}$ is the restriction of the ${\mathcal{J}}$-module
$M$ to ${\mathcal{H}}$ through $\varpi$. (Throughout this section, modules are
taken to be left modules, unless otherwise indicated.)

One approach to understanding the representation theory of a generic Hecke
algebra ${\mathcal{H}}$ is to relate it to the representation theory of the
“simpler” asymptotic algebra ${\mathcal{J}}$, by means of the inclusion
$\varpi:{\mathcal{H}}\rightarrow{\mathcal{J}}$. For example, as just
explained, any ${\mathcal{J}}$-module $M$ can be restricted to
${\mathcal{H}}$, giving rise to a functor $M\mapsto M^{\varpi}$. Conversely,
Theorem 4.2.1 below shows that (under the standing hypotheses on
${\mathcal{H}}$ of Section 1.2) for certain ${\mathcal{H}}$-modules $S$, there
exists a unique (up to isomorphism) ${\mathcal{J}}$-module $V$ such that
$V^{\varpi}\cong S$. In fact, we prove that this is true whenever $S$ is a
finite direct sum of left-cell modules for ${\mathcal{H}}$.

If bad primes are inverted as in Section 4.1, then ${\mathcal{Z}}$ becomes
${\mathcal{Z}}^{\natural}$, and ${\mathcal{J}}$ becomes a maximal (and
separable) order ${\mathcal{J}}^{\natural}$ over ${\mathcal{Z}}^{\natural}$.
See Theorem 4.1.4, and Remark 4.1.11(b) in the case ${}^{2}F_{4}$. This
implies that every finite and ${\mathcal{Z}}^{\natural}$-projective
${\mathcal{J}}^{\natural}$-module is a direct sum of finite projective
${\mathcal{J}}^{\natural}$-modules, each of which becomes irreducible upon
base change to ${\mathcal{J}}_{\mathbb{Q}(t)}$. Such a decomposition is unique
up to isomorphism. Also, because ${\mathcal{Z}}^{\natural}$ is regular of
Krull dimension $\leq 2$ (it is dimension 2, in fact), every irreducible
${\mathcal{J}}_{\mathbb{Q}(t)}$-module is the base change of a
${\mathcal{J}}^{\natural}$-module which is projective over
${\mathcal{Z}}^{\natural}$. In fact, applying Swan’s theorem, any such module
is free over ${\mathcal{Z}}^{\natural}$. In effect, this provides a new
refinement of the set of left-cell modules.

######  Theorem 4.2.1.

Let $S$ be a left-cell module (or even a finite direct sum of left-cell
modules) for ${\mathcal{H}}$. Then, there exists a ${\mathcal{J}}$-module $M$,
unique up to isomorphism, such that $S\cong M^{\varpi}$.

###### Proof.

We use several results from [Lus03, Ch. 18]. Let $*$ be as defined in [Lus03,
18.10]; it is a left-module action of ${\mathcal{J}}$ on ${\mathcal{H}}$. Now
define a new action $\boxdot$ of ${\mathcal{J}}$ on ${\mathcal{H}}$ by putting
$j\boxdot h:=(j*h^{\dagger})^{\dagger}$, for $j\in{\mathcal{J}}$ and
$h\in{\mathcal{H}}$. For each integer $a$, let

${\mathcal{H}}_{a}=\oplus_{w;a(w)=a}{\mathcal{Z}}{\textsc{c}}^{\dagger}_{w}\quad{\text{and}}\quad{\mathcal{H}}_{\geq
a}=\oplus_{w;a(w)\geq a}{\mathcal{Z}}{\textsc{c}}_{w}^{\dagger}$ (4.2.1)

as in [Lus03, 18.10], where $a(-)$ is the Lusztig $a$-function defined in
(4.1.2); see [Lus03, 13.6]. Lusztig observes that
${\mathsf{j}}_{x}*c^{\dagger}_{w}\in{\mathcal{H}}_{a(w)}$, for all $x,w\in W$.
In particular, ${\mathcal{H}}_{\geq a}$ is stable under the $*$ action of
${\mathcal{J}}$. He proves [Lus03, 18.10(a)] that

$h{\textsc{c}}_{w}^{\dagger}\equiv\phi(h)*{\textsc{c}}_{w}^{\dagger}\,\,{\rm
mod}\,\,{\mathcal{H}}_{\geq a(w)+1},\quad\forall w\in W.$

Replacing $h$ by $h^{\dagger}$, and then applying $\dagger$ to both sides, we
get

$h{\textsc{c}}_{w}=(\phi(h^{\dagger})*{\textsc{c}}^{\dagger}_{w})^{\dagger}=\varpi(h)\boxdot{\textsc{c}}_{w}\quad{\rm
mod}({\mathcal{H}}_{\geq a(w)+1})^{\dagger}.$ (4.2.2)

It follows from Lusztig’s observation above that $({\mathcal{H}}_{\geq
a})^{\dagger}$ is stable under the $\boxdot$-action of ${\mathcal{J}}$, as is
$({\mathcal{H}}_{\geq q})^{\dagger}/({\mathcal{H}}_{\geq a+1})^{\dagger}$.
Observe that $-a(-)$ is a height function in the sense of Appendix B
compatible with $\leq_{LR}$ by [Lus03, Ch. 14, P4]. By Corollary 2.1.8 above,
the left ${\mathcal{H}}$-module
${\mathcal{H}}={{}_{\mathcal{H}}{\mathcal{H}}}$ has a height filtration with
respect to the (compatible and negatively-valued) height function $-a(-)-1$.
Note that $({\mathcal{H}}_{\geq a})^{\dagger}/({\mathcal{H}}_{\geq
a+1})^{\dagger}$ is a section of such a filtration. The equation (4.2.2) shows
that this natural ${\mathcal{H}}$-module is also the restriction through
$\varpi$ of a module for ${\mathcal{J}}$, the latter acting via $\boxdot$ on
the same ${\mathcal{Z}}$-module. Since the ${\mathcal{H}}$-module
${}_{{\mathcal{H}}}{\mathcal{H}}$ belongs to ${\mathscr{A}}({\mathscr{L}})$
using Corollary 2.1.8, its filtration sections are direct sums of left-cell
modules. Also, it is clear that every left-cell module for ${\mathcal{H}}$
appears as a direct summand of some such section. The theorem now follows from
the elementary Lemma 4.2.2 below, which proves uniqueness (see part (c)) as
well as completing the proof of existence (see part (d)—the notation differs
slightly).∎

Temporarily, let ${\mathscr{K}}$ be any Noetherian integral domain, and let
$K$ be its fraction field. The following lemma is inspired by the arguments in
[Reiner03, below (8.2), p. 109]. Note the definition of $\mathfrak{B}$-modf is
given in part (a). A similar definition applies for $\mathfrak{A}$-modf. Note
also that part (b) will be used in the proof of Theorem 4.3.4.

######  Lemma 4.2.2.

Let $\mathfrak{A}$ be a finite and torsion-free ${\mathscr{K}}$-algebra and
let $\mathfrak{B}$ be an order in the $K$-algebra $\mathfrak{A}_{K}$
containing $\mathfrak{A}$. Then,

  * (a)

The restriction functor
$\mathfrak{B}$-mod$\,{}^{f}\rightarrow\mathfrak{A}$-mod defines a fully
faithful ${\mathscr{K}}$-linear functor from the ${\mathscr{K}}$-category
$\mathfrak{B}$–mod$\,{}^{f}$ of ${\mathscr{K}}$-finite and
${\mathscr{K}}$-torsion-free $\mathfrak{B}$-modules to the
${\mathscr{K}}$-category of $\mathfrak{A}$-modules.

  * (b)

In particular, for any $M\in\mathfrak{B}$–mod$\,{}^{f}$, we have

$\text{\rm{End}}_{\mathfrak{B}}(M)\cong\text{\rm{End}}_{\mathfrak{A}}(M|_{\mathfrak{A}})$

as ${\mathscr{K}}$-algebras. Also,
$M|_{\mathfrak{A}}\in\mathfrak{A}$–mod$\,{}^{f}$.

  * (c)

Suppose that $M,M^{\prime}\in\mathfrak{B}$-mod$\,{}^{f}$, and let
$N,N^{\prime}$ denote the $\mathfrak{A}$-modules which are their respective
restrictions to $\mathfrak{A}$. Then, $M\cong M^{\prime}$ as
$\mathfrak{B}$-modules if and only if $N\cong N^{\prime}$ as
$\mathfrak{A}$-modules.

  * (d)

Suppose $M\in\mathfrak{B}$-mod$\,{}^{f}$. Assume that $N:=M|_{\mathfrak{A}}$
can be written as a direct sum $N=N_{1}\oplus N_{2}$ in $\mathfrak{A}$-mod.
Then, $M=M_{1}\oplus M_{2}$ in $\mathfrak{B}$-modf, where
$M_{i}|_{\mathfrak{A}}\cong N_{i}$, $i=1,2$. (Also, $N_{1},N_{2}$ are in
$\mathfrak{A}$–modf.)

###### Proof.

We first prove part (a). Obviously, restriction is a ${\mathscr{K}}$-linear
functor. We have to prove it induces an isomorphism on Hom-spaces between
objects. For $M,M^{\prime}\in\mathfrak{B}$–mod$\,{}^{f}$, note that
$\text{\rm{Hom}}_{\mathfrak{B}}(M,M^{\prime})$ identifies with the
${\mathscr{K}}$-submodule of
$\text{\rm{Hom}}_{\mathfrak{B}_{K}}(M_{K},M^{\prime}_{K})$ consisting of the
elements of the latter that send $M\subseteq M_{K}$ to $M^{\prime}\subseteq
M^{\prime}_{K}$. A similar identification holds for
$\text{\rm{Hom}}_{\mathfrak{A}}(M|_{\mathfrak{A}},M^{\prime}|_{\mathfrak{A}})$
in $\text{\rm{Hom}}_{\mathfrak{A}_{K}}(M_{K},M^{\prime}_{K})$. Since
$\mathfrak{B}_{K}=K\mathfrak{B}=K\mathfrak{A}=\mathfrak{A}_{K}$, the
isomorphism required in part (a) holds.

Part (b) follows from the argument above with $M=M^{\prime}$ (using the fact
that restriction is a functor, hence it induces an algebra homomorphism). The
fact that $M|_{\mathfrak{A}}$ belongs to $\mathfrak{A}$–modf is obvious. Part
(c) also follows from a functorial argument using (a). Finally, (d) follows
from the functorial properties of (a) and (b). It is also necessary to observe
for part (d) that idempotents split in $\mathfrak{B}$–modf. More precisely, in
terms of the argument for (a), if $e\in\text{\rm{End}}_{\mathfrak{B}}(M)$ is
an idempotent, with $M\in\mathfrak{B}$–mod$\,{}^{f}$, then $eM$ also belongs
to $\mathfrak{B}$–modf. ∎

Now return to the setting of Theorem 4.2.1, but consider modules over
${\mathcal{Z}}^{\natural}$, rather than ${\mathcal{Z}}$. For each
$E\in\text{Irr}(\mathbb{Q}W)$, let $E_{{\mathbb{Q}}(t)}$ be the corresponding
irreducible $\mathbb{Q}(t)W$-module, regarded as a
${\mathcal{J}}_{\mathbb{Q}(t)}$-module through the isomorphism
$\mathbb{Q}(t)W\cong{\mathcal{J}}_{\mathbb{Q}(t)}$. (In the ${}^{2}F_{4}$
case, $E$ and $E_{\mathbb{Q}(t)}$ may not be absolutely irreducible.) Thus, by
Remark 4.1.5(2), $E_{\mathbb{Q}(t)}\cong L_{i}$, for some $i$. Choose a
${\mathcal{J}}^{\natural}$-lattice $E_{\natural}$ in $E_{\mathbb{Q}(t)}$ that
is a projective as a ${\mathcal{Z}}^{\natural}$-module (thus free by Swan’s
theorem [Sw78, first paragraph, p. 111]). For example, for a type other than
${}^{2}F_{4}$, we may choose $E_{\natural}\cong
L_{i,{\mathcal{Z}}^{\natural}}$, for some $i$, which is
${\mathcal{Z}}^{\natural}$-free in the notation of Remark 4.1.5(2). Since
${\mathcal{J}}^{\natural}$ is projective over ${\mathcal{Z}}^{\natural}$, the
${\mathcal{J}}^{\natural}$-module $E_{\natural}$ is projective over
${\mathcal{J}}^{\natural}$; see Corollary 4.1.6(b) and its analog in Remark
4.1.11(b).

We remark that $E_{\natural}$ is uniquely determined (up to isomorphism) by
Corollary 4.1.7 and its twisted ${}^{2}F_{4}$ analog in Remark 4.1.11(b).
Define

$S^{\natural}(E):=(E_{\natural})^{\varpi^{\natural}},\quad
S^{\natural}_{E}:=\text{\rm{Hom}}_{{\mathcal{Z}}^{\natural}}(S^{\natural}(E),{\mathcal{Z}}^{\natural}),$
(4.2.3)

which belong to ${\mathcal{H}}^{\natural}$-mod and
mod-${\mathcal{H}}^{\natural}$, respectively. Note that $S^{\natural}(E)$ and
$S^{\natural}_{E}$ are natural ${\mathcal{H}}^{\natural}$-forms for the
${\mathcal{H}}_{{\mathbb{Q}}[t,t^{-1}]}$-modules $S(E)$ and $S_{E}$ defined in
[DPS15, §3]. Note also that $S^{\natural}(E)_{\mathbb{Q}(t)}\cong
E_{\mathbb{Q}(t)}\cong L_{i}$.

######  Remark 4.2.3.

Since $\\{S^{\natural}(E)_{K}\mid E\in\text{Irr}(\mathbb{Q}W)\\}$, where
$K=\mathbb{Q}(t)$, forms a complete set of irreducible left
${\mathcal{H}}_{K}$-modules (see Remark 4.1.5(2)), it follows that
$\\{(S^{\natural}_{E})_{K}\mid E\in\text{Irr}(\mathbb{Q}W)\\}$ is a complete
set of irreducible right ${\mathcal{H}}_{K}$-modules. Thus, since every
$(S_{E}^{\natural})_{K}$ is an irreducible constituent of $T^{+}_{K}$, the set

$\\{\text{\rm{Hom}}_{{\mathcal{H}}^{\natural}_{K}}((S_{E}^{\natural})_{K},T^{+\natural}_{K})\mid
E\in\text{Irr}(\mathbb{Q}W)\\}$

forms a complete set of irreducible left $A^{+}_{K}$-modules. (Note that
$A^{+}_{K}=\text{\rm{End}}_{{\mathcal{H}}^{\natural}}(T^{+\natural})_{K}$. So,
$A^{+}_{K}$ is split semisimple if ${}^{2}F_{4}$ is excluded.) Now, if
$\Delta(E):=\text{\rm{Hom}}_{{\mathcal{H}}^{\natural}}(S_{E}^{\natural},T^{+\natural})$,
then
$\Delta(E)\subseteq\text{\rm{Hom}}_{{\mathcal{H}}^{\natural}_{K}}((S_{E}^{\natural})_{K},T^{+\natural}_{K})$
is an $A^{+\natural}$-submodule, which is a torsion-free
${\mathcal{Z}}^{\natural}$-module. Hence, $\Delta(E)_{K}=K\Delta(E)$ is a
(nonzero) $A^{+}_{K}$-submodule of
$\text{\rm{Hom}}_{{\mathcal{H}}^{\natural}_{K}}((S_{E}^{\natural})_{K},T^{+\natural}_{K})$.
Irreducibility (treated above) forces
$\Delta(E)_{K}=\text{\rm{Hom}}_{{\mathcal{H}}^{\natural}_{K}}((S_{E}^{\natural})_{K},T^{+\natural}_{K})$.

Therefore, $\\{\Delta(E)_{K}\mid E\in\text{Irr}(\mathbb{Q}W)\\}$ forms a
complete set of irreducible left $A^{+}_{K}$-modules. To include the
${}^{2}F_{4}$ case, we may simply replace ${\mathcal{Z}}^{\natural}$ by
${\mathcal{Z}}^{\natural\prime}={\mathcal{Z}}^{\natural}[\sqrt{2}]$, $K$ by
$K^{\prime}$, and $\Delta(E)$ by
$\Delta^{\prime}(E):=\Delta(E)_{{\mathcal{Z}}^{\natural\prime}}$ to get a
complete set $\\{\Delta^{\prime}(E)_{K^{\prime}}\mid
E\in\text{Irr}(\mathbb{Q}^{\prime}W)\\}$ of irreducible left
$A^{+}_{K^{\prime}}$-modules. Here, the fields $K^{\prime}$ ,
$\mathbb{Q}^{\prime}$ are obtained by adjoining $\sqrt{2}$ to $K$,
$\mathbb{Q}$, respectively.

######  Theorem 4.2.4.

(a) Let $N$ be a finite direct sum of left-cell modules for ${\mathcal{H}}$,
and let $N^{\natural}:=N_{{\mathcal{Z}}^{\natural}}$ be the corresponding
(base-changed) module for ${\mathcal{H}}^{\natural}$. If $M$ is a direct
summand of the ${\mathcal{H}}^{\natural}$-module $N^{\natural}$, then $M$
admits a direct sum decomposition $M=\oplus_{i=1}^{r}M_{i}$ in
${\mathcal{H}}^{\natural}$-mod such that each $M_{i}\cong
S^{\natural}(E_{i})$, for some $E_{i}\in\text{\rm Irr}({\mathbb{Q}}W)$. The
modules $M_{i}$ and their multiplicities in $M$ are determined up to
isomorphism by $M_{K}$, where $K=\mathbb{Q}(t)$.

(b) Assume that $(W,S,L)$ is of type ${}^{2}F_{4}$ (so the base-change
notations $(-)^{\natural\prime}$ and $(-)^{\prime}$ are defined). Then, (a)
also holds after substituting the top row for the bottom row in the display
(see the appendix to Section 4.1):

$\begin{matrix}{\mathcal{H}}^{\prime}\,&{\mathcal{Z}}^{\natural\prime}\,&{\mathcal{H}}^{\natural\prime}\,&\mathbb{Q}^{\prime}\,&K^{\prime}\,&N^{\natural\prime}\\\
{\mathcal{H}}\,&{\mathcal{Z}}^{\natural}\,&{\mathcal{H}}^{\natural}\,&\mathbb{Q}\,&K\,&N^{\natural}\end{matrix}$
(4.2.4)

###### Proof.

(a) First, we establish some projectivity properties. Note that all left-cell
modules for ${\mathcal{H}}$ are projective (even free) over ${\mathcal{Z}}$.
Thus, $N$ is projective over ${\mathcal{Z}}$, and both $N^{\natural}$ and its
direct summand $M$ are projective over ${\mathcal{Z}}^{\natural}$.
Consequently, if $M\cong X^{\varpi^{\natural}}$ (the restriction of $X$ to
${\mathcal{H}}^{\natural}$ through $\varpi_{{\mathcal{Z}}^{\natural}}$) for a
given ${\mathcal{J}}^{\natural}$-module $X$, then $X$ is projective over
${\mathcal{Z}}^{\natural}$. Consequently, $X$ is also projective over
${\mathcal{J}}^{\natural}$, by Corollary 4.1.6(b).

By Theorem 4.2.1, $N$ is (isomorphic to) the restriction (through $\varpi$) to
${\mathcal{H}}$ of a module for ${\mathcal{J}}$. Thus, $N^{\natural}$ is the
restriction to ${\mathcal{H}}^{\natural}$ of a module $Y$ for the order
${{\mathcal{J}}}^{\natural}$, and the analogous statement holds for $M$ by
Lemma 4.2.2(d). (Use $\mathfrak{A}={\mathcal{H}}^{\natural}$ and let
$\mathfrak{B}=\varpi_{K}^{-1}({\mathcal{J}}^{\natural})$.) Accordingly, let