# An exact category approach to Hecke endomorphism algebras∗

Jie Du  School of Mathematics and Statistics
University of New South Wales
UNSW Sydney 2052<EMAIL_ADDRESS>(Du) ,  Brian Parshall† †Department of
Mathematics
University of Virginia
Charlottesville, VA 22903  and  Leonard Scott  Department of Mathematics
University of Virginia
Charlottesville, VA 22903<EMAIL_ADDRESS>(Scott) *with an appendix by
Veronica Shalotenko

(Date: August 28, 2024)

###### Abstract.

Let $G$ be a finite group of Lie type. In studying the cross-characteristic
representation theory of $G$, the (specialized) Hecke algebra
$H=\text{\rm{End}}_{G}(\text{Ind}_{B}^{G}1_{B})$ has played a important role.
In particular, when $G=GL_{n}(\mathbb{F}_{q})$ is a finite general linear
group, this approach led to the Dipper-James theory of $q$-Schur algebras $A$.
These algebras can be constructed over ${\mathcal{Z}}:=\mathbb{Z}[t,t^{-1}]$
as the $q$-analog (with $q=t^{2}$) of an endomorphism algebra larger than $H$,
involving parabolic subgroups. The algebra $A$ is quasi-hereditary over
${\mathcal{Z}}$. An analogous algebra, still denoted $A$, can always be
constructed in other types. However, these algebras have so far been less
useful than in the $GL_{n}$ case, in part because they are not generally
quasi-hereditary.

Several years ago, reformulating a 1998 conjecture, the authors proposed (for
all types) the existence of a ${\mathcal{Z}}$-algebra $A^{+}$ having a
stratified derived module category, with strata constructed via Kazhdan-
Lusztig cell theory. The algebra $A$ is recovered as $A=eA^{+}e$ for an
idempotent $e\in A^{+}$. A main goal of this monograph is to prove this
conjecture completely. The proof involves several new homological techniques
using exact categories. Following the proof, we show that $A^{+}$ does become
quasi-hereditary after the inversion of the bad primes. Some first
applications of the result—e.g., to decomposition matrices—are presented,
together with several open problems.

This work was supported by grants from the ARC (DP120101436, Jie Du), and the
Simons Foundation (#359360, Brian Parshall; #359363, Leonard Scott). The last
author would like to thank UNSW for support from the Distinguished Research
Visitor Scheme.

August 28, 2024

###### Contents

  1. 0.1 Further results and applications
  2. 0.2 Outline and guide for the proof of Theorem 3.3.1
  3. 1 Preliminaries
    1. 1.1 Finite groups of Lie type
    2. 1.2 (Iwahori-)Hecke algebras
    3. 1.3 Exact categories and stratified algebras
  4. 2 Some exact categories
    1. 2.1 Three exact categories
    2. 2.2 Some generalities on duality
    3. 2.3 The dual category of $({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$
    4. 2.4 Another abstract setting
  5. 3 The conjecture and its proof
    1. 3.1 $q$-permutation modules and their filtrations
    2. 3.2 The conjecture
    3. 3.3 Proof of the conjecture
    4. 3.4 A uniqueness theory
    5. 3.5 The standard stratification of $A^{+}$
  6. 4 The quasi-heredity of $A^{+\natural}$
    1. 4.1 Maximal orders ${\mathcal{J}}^{\natural}:={\mathcal{Z}}^{\natural}\otimes_{\mathbb{Z}}{\mathsf{J}}$
    2. 4.2 Left cells and the order ${\mathcal{J}}:={\mathcal{Z}}\otimes_{\mathbb{Z}}{\mathsf{J}}$
    3. 4.3 The quasi-heredity of $A^{+\natural}$
  7. 5 Related topics
    1. 5.0 Introduction
    2. 5.1 Standardly based algebras
    3. 5.2 Decomposition matrices, I
    4. 5.3 Decomposition matrices, II
    5. 5.4 $i$-Quantum groups
    6. 5.5 Open problems.
  8. A Projective coverings for Steinberg modules
  9. B Height functions.
  10. C Decomposition numbers and Krull dimension $\leq 2$

## Introduction

Representations of the symmetric group $\mathfrak{S}_{r}$ are the first
examples of group representations systematically investigated by the pioneers
in representation theory such as F.G. Frobenius, A. Young, and W. Burnside.
When I. Schur used the symmetric group representations to determine polynomial
representations of the complex general linear group ${\text{\rm
GL}}_{n}(\mathbb{C})$, certain finite-dimensional algebras, known as Schur
algebras, played a bridging role between the two. The well-known Schur–Weyl
duality summarizes the relation between the representations of ${\text{\rm
GL}}_{n}(\mathbb{C})$ and $\mathfrak{S}_{r}$.

A Schur algebra is the endomorphism algebra of a direct sum of certain
permutation modules associated with Young (or parabolic) subgroups of
$\mathfrak{S}_{r}$. These permutation modules are filtered by Specht modules
(with multiplicities given by Young’s rule), and such filtrations result in a
standard stratification for the Schur algebra so that its representation
category is a so-called highest weight category, a notion naturally arising
from the study of representations of Lie groups/algebras and algebraic groups.
The Schur algebra itself is an example of a quasi-hereditary algebra, a notion
introduced by E. Cline together with the last two authors of this monograph
(CPS) in the late 1980s.

The endomorphism algebra of a permutation representation on a Borel subgroup
$B(q)$, consisting of all upper triangular matrices, of a finite general
linear group ${\text{\rm GL}}_{r}(q)$ is called an (Iwahori–)Hecke algebra
which is a $q$-deformation of the group algebra of $\mathfrak{S}_{r}$. Such
Hecke algebras and their generic versions are all defined for finite groups
$G$ of Lie type with a parameter $q$ [Iwa64, C85]. Often $q$ is a prime power
and the associated Hecke algebras are $q$-deformations of the group algebra of
the Weyl group $W$ of $G$. Now, using $q$-permutation modules associated with
parabolic subgroups of $W$, the notion of Schur algebras is generalized not
only to $q$-Schur algebras in the type $A$ case, but also to Hecke
endomorphism algebras (Hecke endo-algebras) in general.

More precisely, let ${\mathcal{H}}$ be a generic Hecke algebra over
${\mathcal{Z}}:=\mathbb{Z}[t,t^{-1}]$ associated to a finite group $G$ of Lie
type with Weyl group $W$. For each subset $J$ of the set $S$ of fundamental
reflections for $W$, form the associated $q$-permutation module
$x_{J}{\mathcal{H}}$. Put $T:=\bigoplus_{J}x_{J}{\mathcal{H}}$ and call the
endomorphism algebra $A:=\text{\rm{End}}_{\mathcal{H}}(T)$ a generic Hecke
endo-algebra. If $G$ is a finite general linear group and ${\mathscr{K}}$ is a
field, $A_{\mathscr{K}}={\mathscr{K}}\otimes A$ is (Morita equivalent to) a
$q$-Schur algebra, as introduced by Dipper and James [DJ89], and used
effectively in the study of cross-characteristic representation theory of $G$
(so that $q$ is nonzero in ${\mathscr{K}}$). The definition makes sense for
$G$ of any type, but $A$ itself, without modification, appears less useful
(compared to $q$-Schur algebras), partly for structural and homological
reasons. For simplicity, we will assume that $G$ is a fixed-point group of a
Frobenius map on a simple algebraic group ${\mathbb{G}}$ throughout the
monograph.

The structure of $q$-Schur algebras is a natural quantization from the
structure of Schur algebras, thanks to the existence of filtrations of
$q$-permutation modules by $q$-Specht modules. What is a replacement of Specht
or $q$-Specht modules in the general case?

In 1996, CPS developed a theory of stratifying endomorphism algebras. They
considered abstract Specht modules and a set of stratification hypotheses with
which the associated endomorphism algebra is a standardly stratified algebra,
a notion that is slightly weaker than that of a quasi-hereditary algebra. For
the aforementioned generic Hecke algebras, the authors used the modules
associated with Kazhdan–Lusztig left cells as “Specht modules” to filter these
$q$-permutation modules. This results in a standard stratification for
$q$-Schur algebras, but a coarser stratification for a generic Hecke endo-
algebra $A$ of other Lie types; see [DPS98b, DPS98a]. However, in the rank-two
case, if we enlarge $T$ to $T^{+}=T\oplus X$ where $X$ is a certain module
filtered by certain cell modules, we obtain a finer stratification in terms of
two-sided cells. In other words, if an enlargement $T^{+}$ of $T$ is used,
then $A^{+}=\text{\rm{End}}_{\mathcal{H}}(T^{+})$ could be standardly
stratified. This motivated the conjecture [DPS98a, Conj. 2.5.2], later
modified (slightly) in [DPS15, Conj. 1.2].

The conjecture is about the construction of an extended (generic) Hecke endo-
algebra $A^{+}$ with good homological properties. It proposes the existence of
a ${\mathcal{Z}}$-algebra $A^{+}$ with a nicely stratified derived module
category, with all strata constructed via Kazhdan-Lusztig cell theory. It is
required by the conjecture to extend the ${\mathcal{H}}$-module $T$ to
$T^{+}=T\oplus X$, for some right ${\mathcal{H}}$–module $X$ with a suitable
dual left-cell (module) filtration. Note, this implies that $A=eA^{+}e$, where
$e$ is the idempotent projection of $T^{+}$ onto $T$ along $X$. The conjecture
also specifies “standard modules,” parametrized by left cells, that filter a
projective generator for the category $A^{+}$–mod of finitely generated
$A^{+}$-modules.

More precisely, the $q$-permutation module $x_{J}{\mathcal{H}}$, filtered by
dual left-cell modules $S_{\omega}$, has a bottom section $S_{\omega^{0,J}}$,
where $\omega^{0,J}$ is the left cell that contains the longest element
$w_{0,J}$ of the parabolic subgroup $W_{J}$. For a left cell $\omega$ that
satisfies $S_{\omega}\not\cong S_{\omega^{0,J}}$ for all $J\subseteq S$, we
want to construct a right ${\mathcal{H}}$-module $X_{\omega}$ filtered by dual
left-cell modules and with bottom section $S_{\omega}$. We require
$X_{\omega}$ to satisfy a certain homological property, the Ext1 vanishing
property, which forms part of the Stratification Hypothesis [DPS98a, (1.2.9)].
Note that, for the symmetric group, every two-sided cell contains some
$w_{0,J}$ and every $S_{\omega}\cong S_{\omega^{0,J}}$ for some $J$. So, we
take $X=0$ in this case. In general, we may take $X$ to be a direct sum of
such $X_{\omega}$. Thus, the proof of the conjecture relies on the existence
of such $X_{\omega}$.

In [DPS15], we studied a local case of the conjecture, under the equal
parameter assumption for ${\mathcal{H}}$ in order to use a result from
[GGOR03]. In the DPS work, the module $X_{\omega}$ was constructed
inductively, using a new order on left cells, to satisfy the required Ext1
vanishing condition. This order is strictly compatible with the two-sided cell
order $\leq_{LR}$, and is defined by a height function on left cells, which is
called a sorting function in [GGOR03]. Note that the height function is
motivated by Lusztig’s $a$-function on $W$ that takes constant values on two-
sided cells. In this way, we proved a special case ([DPS15, Th. 5.6]) of the
conjecture [DPS15, Conj. 1.2].

To attack the conjecture in general, the authors developed in [DPS17] an exact
category approach to control the Ext1 vanishing property when constructing
certain modules (such as $X_{\omega}$) with some desired filtrations. Applying
this general construction to ${\mathcal{H}}$-modules yields the exact category
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}({\mathscr{L}}^{*}))$, where
${\mathscr{A}}({\mathscr{L}}^{*})$ is a full (additive) subcategory of
mod-${\mathcal{H}}$, and ${\mathscr{E}}({\mathscr{L}}^{*})$ is a subcategory
of the category of short exact sequences in the category mod-${\mathcal{H}}$
of finite right ${\mathcal{H}}$-modules. However, this general construction
does not decide if the $q$-permutation modules, though belonging to
${\mathscr{A}}({\mathscr{L}}^{*})$, satisfy the required Ext1 vanishing
property. To overcome the hurdle, two more exact subcategories are constructed
to obtain an exact category
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
in which $X_{\omega}$ may be constructed with a required filtration and both
$X_{\omega}$ and $x_{J}{\mathcal{H}}$ satisfy the Ext1 vanishing property.
Though the process is quite sophisticated, we eventually prove the conjecture
[DPS15, Conj. 1.2]; it is a consequence of Theorem 3.3.1 whose proof is
outlined in Section 0.2 below.

###  0.1. Further results and applications

After preliminaries in Chapters 1, we begin in Chapter 2 with a construction
of a sequence of exact categories, consisting of objects with filtrations of
left-cell modules, inside the category ${\mathcal{H}}$-mod of left
${\mathcal{H}}$-modules. By applying a standard duality theory to the
construction in Section 2.2, we obtain in Section 2.3 the exact categories
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$,
required in the proof of the conjecture, of right ${\mathcal{H}}$-modules with
filtrations of dual left-cell modules and the homological property satisfied
by $x_{J}{\mathcal{H}}$. Section 2.4 reviews and reformulates an abstract
setting developed in [DPS17] for the construction of these aforementioned
extra modules $X_{\omega}$.

The conjecture is stated and proved in Chapter 3; see Theorem 3.3.1. The proof
is technically quite difficult, so we provide at the end of this introduction
a conceptual framework for the way $A^{+}$ and $T^{+}$ evolve in these
sections.

Theorem 3.3.1 involves no mention of primes whatsoever, good, bad, or
otherwise. Nevertheless, the theorem proves that the category $A^{+}$-mod has
a natural (standard) stratifying system. Section 3.4 is devoted to a
“uniqueness theory.” (See similar uniqueness results for quasi-hereditray
covers, discussed by Rouquier in [Ro08, §4].) Its original motivation was to
sharpen Theorem 3.3.1: A main step in the proof of that theorem involved the
construction of certain ${\mathcal{H}}$-modules $X_{\omega}$ associated to
left cells $\omega$; see Construction 2.4.12. By Theorem 2.4.10, the
$X_{\omega}$ have remarkable homological properties, which are required in
realizing $A^{+}$ as an endomorphism algebra. Although no claim is made that
the modules $X_{\omega}$ are unique, in Theorem 3.4.5, the following
uniqueness result is proved: the Morita class of $A^{+}$ is independent of the
choice of the modules $X_{\omega}$. The uniqueness theory is formulated in
such a way (see Theorem 3.4.2) that it might be useful for other suitably
filtered endomorphism algebras. In Section 3.5, we turn the categorical
property — i.e., the stratifying system for $A^{+}$-mod, established in
Theorem 3.3.1 — into the structural property for $A^{+}$ by explicitly
displaying a defining sequence, called a standard stratification, for $A^{+}$
with stratifying ideals as sections; see Theoren 3.4.5.

######  Remark 0.1.1 (Good and bad primes).

The “bad primes” are the only integer primes that divide the denominators of
generic character degree formulas; see [DPS98a, (4.1.12)] or the discussion of
display (4.1.14) in the appendix to Section 4.1, where only the ${}^{2}F_{4}$
case is considered. They form a subset of $\\{2,3,5\\}$. We declare all such
prime divisors to be “bad” (for $W$, $G$, or the Lie series to which $G$
belongs) and all others to be “good.” Recall the classical definition that a
prime is good for ${\mathbb{G}}$ if it does not divide any coefficient of the
maximal root in the root system of ${\mathbb{G}}$. By [DPS98a, 4.1], all
primes that are good for ${\mathbb{G}}$ are good for $G$.

Suppose that
$A^{+\natural}:={\mathcal{Z}}^{\natural}\otimes_{\mathcal{Z}}A^{+}$ is
obtained from $A^{+}$ by inverting all the bad primes for $W$. Theorem 4.3.4
then proves that $A^{+\natural}$ is quasi-hereditary of separable type in all
cases (and split quasi-hereditary except in type ${}^{2}F_{4}$, becoming split
upon the adjunction of $\sqrt{2})$). Sections 4.1 and 4.2 prepare for the
above results in Section 4.3. They also provide an interesting study in
themselves of the base-changed version
${\mathcal{J}}:={\mathcal{Z}}\otimes{\mathsf{J}}$ of the Lusztig asymptotic
$\mathbb{Z}$–algebra ${\mathsf{J}}$. There is a natural inclusion of the Hecke
algebra ${\mathcal{H}}$ into (a copy of) ${\mathcal{J}}$, and the latter may
be regarded as an order in ${\mathcal{H}}_{\mathbb{Q}(t)}$ containing
${\mathcal{H}}$. One approach to understanding the representation theory of
the generic Hecke algebra ${\mathcal{H}}$ is to relate it to the simpler
algebra ${\mathcal{J}}$. Any ${\mathcal{J}}$-module $M$ can be restricted to
${\mathcal{H}}$, and our Theorem 4.2.1 says every left-cell module for
${\mathcal{H}}$ arises this way. Once again there is no mention of primes.
However, if bad primes are inverted in
$\mathbb{Z}^{\natural}\subseteq{\mathcal{Z}}^{\natural}$, we show the
resulting order ${\mathcal{J}}^{\natural}$ is maximal (in the sense of Reiner
[Reiner03]). We then prove that every stratifying ideal arising from the
defining sequence for $A^{+}$ in Theorem 3.4.5 is, after base change to
${\mathcal{Z}}^{\natural}$, a hereditary ideal of separable type (and of split
type, if ${}^{2}F_{4}$ is excluded), completing the proof of Theorem 4.3.4. In
the rest of Section 4.3, we focus on a refinement of the defining sequence for
$A^{+\natural}$ such that its sections are indecomposable split hereditary
ideals, a notion used in [Ro08]. In particular, standard modules and
Rouquier’s isomorphism are given in Corollary 4.3.5. Corollary 4.3.6 brings in
the idempotents that define these indecomposable split hereditary ideals.
(Note that the approach developed in [Ro08] is an idempotent-free approach.)
Finally, in Corollary 4.3.7, we prove that $A^{+\natural}$-mod is a highest
weight category in the sense of [DS94] and [Ro08].

Chapter 5 is devoted to applications to several related topics. In Section
5.1, we consider the algebras $A$ and $A^{+}$ from the point of view of
standardly based algebras, as introduced by Du and Rui [DR98]. For example,
Theorem 5.1.8 proves that the algebras $A^{\natural}$, $A^{+\natural}$ and
${\mathcal{H}}^{\natural}$ are all standardly based. The latter case was
previously known (see [G07]), though our proof is new.

Sections 5.2 focuses on a new study of decomposition numbers of specialized
Hecke endo-algebras relevant to those for finite groups of Lie type in cross-
characteristic representation theory. It is not a new theme, but our results
are quite general and often deeper than those found in the literature.
Theorems 5.2.2 and 5.2.8 identify a large collection of decomposition numbers
which remain constant for all groups in a given series, independent of their
$q$-parameter, the latter parameter, defined by [C85], is a general version of
the $q$ in $SL_{n}(q)$ for the $SL_{n}$-series. The uni-triangularity of the
associated decomposition matrix is an easy consequence of the split quasi-
hereditary structure.

Section 5.3 links the decomposition matrix for a Hecke endo-algebra $A$ with
that of the corresponding finite groups of Lie type. We prove in Theorem 5.3.1
that, if $A$ is the Hecke endo-algebra associated with the standard finite
Coxeter system $(W,S,L)$ and $G$ is a finite group in the Lie type series
associated with $(W,S,L)$, then the decomposition matrix for the specialized
$A_{{\mathcal{O}}}$ is part of the decomposition matrix of ${\mathcal{O}}G$.
Part of the proof in the adjoint case is based on a note of Veronica
Shalotenko; see Appendix A. We thank her for allowing us to include this work
in our monograph. We also point out in a Remark 5.3.2(1) that her proof
continues to work if the hypothesis on the connected center of ${\mathbb{G}}$
is removed, due to a result of Dipper and Fleischmann [DF92, Lem. (3.8)].

Section 5.4 is motivated by Schur’s original work, the Schur–Weyl duality, as
mentioned at the beginning of this Introduction, and its quantum analog. In
their study of quantum symmetric pairs of type AIII, H. Bao and W. Wang [BW18]
established a new type of the Schur–Weyl duality between $i$-quantum groups
${\bf U}^{\jmath}$ and ${\bf U}^{\imath}$ and Hecke algebras of types $B$ and
$C$, respectively. Wu and the first author have partially lifted Bao–Wang’s
duality to the integral level (see [DW22, DW23]). In these works, certain
Hecke endomorphism algebras, which are centralizer subalgebras of
$A^{\ddagger}$, played the bridging role. In particular, the standard basis
theory in Section 5.1 and the discovery about the decomposition matrix for
$A^{+}$ or $A^{\ddagger}$ in Section 5.2 can be applied to representations of
$i$-quantum groups and their hyperalgebras. See Theorems 5.4.4 and 5.4.16.

In the final section of Chapter 5, we pose some problems, suggested by this
monograph, on decomposition numbers for finite groups of Lie type. Each of the
problems can be viewed as a step toward understanding the decomposition
numbers in reductions mod $p$ of ordinary irreducible representations in the
unipotent series. The problems are formulated at a “starter” level, assuming
the underlying field characteristic is a good prime. We hope this assumption
might be eventually removed in some future reformulation. The framework of
Theorem 3.3.1, which treats all primes alike, makes such a reworking of “good
prime” formulations possible, and provides a general starting point. Note that
Theorem 3.3.1 itself is a kind of reformulation of the “good prime” result
Theorem 4.3.4.

Finally, the monograph closes with several appendices which cover a number of
preparatory topics used in its main body.

###  0.2. Outline and guide for the proof of Theorem 3.3.1

Let $\Omega$ be the set of Kazhdan-Lusztig left cells in a finite standard
Coxeter system $(W,S,L)$ as defined in [Lus03], using parameters from an
ambient algebraic group and a fixed-point group of a Frobenius map (see [C85,
§1.17]).111See also footnote 1 in Section 1.1 below. In [DPS17], the authors
constructed an early version $A^{\dagger}$ of $A^{+}$ based on the existence
of suitable modules $X_{\omega}$, for all $\omega\in\Omega$. Each $X_{\omega}$
has a filtration with sections isomorphic to dual left-cell modules $S_{\nu}$,
$\nu\in\Omega$, and a bottom section $S_{\omega}$ (defined precisely in
display (1.2.10)), which, in addition, satisfy

$\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(S_{\tau},X_{\omega})=0,\quad\forall\tau,\omega\in\Omega.$
(0.2.1)

Here $S_{\tau}$ and $X_{\omega}$ are objects in a suitable exact category
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$. The category
${\mathscr{A}}({\mathscr{S}})$ is a full subcategory of the category of
$\mathbb{Z}[t,t^{-1}]$-free, finite right ${\mathcal{H}}$-modules. In
addition, it is required that each object in ${\mathscr{A}}({\mathscr{S}})$
have a certain filtration with sections direct sums of dual left-cell modules.
Also,
$A^{\dagger}:=\text{\rm{End}}_{{\mathscr{A}}({\mathscr{S}})}(T^{\dagger})$,
where $T^{\dagger}:=\bigoplus_{\omega\in\Omega}X_{\omega}^{\oplus m_{\omega}}$
with each $m_{\omega}$ a positive integer. This definition of $T^{\dagger}$,
together with the
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}$–vanishing (0.2.1),
implies that
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}({\mathscr{S}})}(S_{\tau},T^{\dagger})=0$,
for all $\tau\in\Omega$. This is a key property for managing filtrations of
$A^{\dagger}$ that arise from filtrations of $T^{\dagger}$.

Generally speaking, the pair $(A^{\dagger},T^{\dagger})$ has many desirable
properties, except we want $T^{\dagger}$ to contain, as an
${\mathscr{A}}({\mathscr{S}})$-summand, each of the $q$-permutation modules
$x_{J}{\mathcal{H}}$, as well as their direct sum (denoted $T$). This problem
presents a major difficulty in passing from [DPS17] to the set-up of this
monograph. It is solved by introducing a new exact category
$({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})=({\mathscr{A}}_{\flat}({\mathscr{S}}),{\mathscr{E}}_{\flat}({\mathscr{S}}))$
inside $({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$ with
$x_{J}{\mathcal{H}}\in{\mathscr{A}}_{\flat}$ and
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S_{\tau},x_{J}{\mathcal{H}})=0$,
for all $\tau$.

The construction of $({\mathscr{A}}_{\flat},{\mathscr{E}}_{\flat})$ is highly
nontrivial. It begins in Sections 1.3 and 2.1 which study a dual category
$({\mathscr{A}}^{\flat},{\mathscr{E}}^{\flat})$ in the category of
${\mathcal{Z}}$-free left ${\mathcal{H}}$-modules. Of course, it is possible
to pass between right and left ${\mathcal{Z}}$-free, finite
${\mathcal{H}}$-modules by simply applying the dual functor
$(-)^{*}=\text{\rm{Hom}}_{\mathcal{Z}}(-,{\mathcal{Z}})$. However, this
process interchanges left exact functors with right exact functors, while we
need to use the left-module versions to insure the various functors sending an
exact sequence $0\to M\to N\to P\to 0$ to the fixed-point sequences $0\to
M^{{\mathcal{H}}_{J}}\to N^{{\mathcal{H}}_{J}}\ \to P^{{\mathcal{H}}_{J}}$,
$J\subseteq S$, are left exact. This allows us to build an exact subcategory
by focusing on the short exact sequences $0\to M\to N\to P\to 0$ whose
associated fixed-point sequences are also all short exact sequences [DPS17,
Lem. 3.1]. A version of this approach is used in Proposition 2.1.3 to create
${\mathscr{E}}^{\flat}={\mathscr{E}}^{\flat}({\mathscr{L}})$ inside an exact
category $({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}({\mathscr{L}})$ dual to
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$. The category
${\mathscr{A}}^{\flat}$ is then defined to consist of objects with certain
“internal” candidate exact sequences belonging to ${\mathscr{E}}^{\flat}$.
Remarkably, the objects ${\mathcal{H}}x_{J}$ belong to
${\mathscr{A}}^{\flat}$; see Corollary 2.1.8. This implies that each dual
object $x_{J}{\mathcal{H}}$ belongs to ${\mathscr{A}}_{\flat}$. We want more
than that, however. We want $x_{J}{\mathcal{H}}$ to be a direct summand of a
new version of $T^{\dagger}$ with the key vanishing (0.2.1) preserved. This
forces us to prove that
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}_{\flat}}(S_{\omega},x_{J}{\mathcal{H}})=0$,
for each $\omega\in\Omega$ and $J\subseteq S$. This vanishing is proved in its
dual incarnation, Theorem 2.1.6. The short exact sequence (2.1.8) displayed in
the proof of Theorem 2.1.6 is the key ingredient. It is available from the
exactness of $(-)^{{\mathcal{H}}_{J}}$ there.

Section 2.3 can be viewed as dualizing the constructions of Section 2.1. Of
course, Section 2.1 was formulated with these duals in mind.

A main step in completing the proof of the conjecture, at this point, is
carried out in Section 2.4, namely, in Theorem 2.4.10, which imitates the
construction of $X_{\omega}$ in [DPS17, Th. 4.5].

Section 3.1 is also preparatory, providing more detail on dual left-cell
module filtrations of $q$-permutation modules. It refines the analysis of
$q$-permutation module filtrations begun above Corollary 2.1.5. The discussion
draws from earlier work in [DPS98a].

The main goal of Section 3.3 is to prove Theorem 3.3.1. This implies
Conjecture 3.2.1, equivalent to [DPS15, Conj. 1.2], a modification of [DPS98a,
Conj. 2.5.2]. A further consequence of the proof is the
$\text{\rm{Ext}}^{1}$-vanishing result Corollary 3.3.2. It is used later in
Section 3.4 in constructing a uniqueness Morita theory complementary to
Theorem 3.3.1, and discussed further below.

For further discussions in outline of this monograph, see the previous two
sections.

Sydney and Charlottesville, Jie Du

September 2022 Brian Parshall†

Leonard Scott

Acknowledgements

We would like to thank our colleague Weiqiang Wang for several discussions and
special assistances during the writing of this monograph. Section 5.4 was
heavily influenced by his work with H. Bao.

Part of the work was written while Jie Du was visiting the University of
Virginia in 2016 and 2022, using Simons grants there. Leonard Scott visited
the University of New South Wales in 2015 and 2019 with support there through
an ARC grant and the Distinguished Research Visitor Scheme. Both authors want
to thank the two universities for their support and hospitality.

Finally, we wish to thank Karen Parshall (University of Virginia) for reading
the entire manuscript and making many helpful comments.

A special note

Our lifelong collaborator and friend, Brian Parshall, passed away on January
17, 2022. This was a few days after a Zoom meeting we had held, generally
weekly, since mid 2020. This last meeting was very successful and lasted 2
hours. The last question, one Jie had for Brian, was about using “Hecke endo-
algebra” as a short form for “Hecke endomorphism algebra”. Brian answered “it
sounds good. I will think about it”. Unfortunately, these words from Brian
became the last words we heard from him. He passed away within a week of this
last meeting. It was a shock and surprise for everyone, certainly for Brian’s
collaborators. Most of the work on this monograph, however, had been done with
Brian’s participation. Its remaining co-authors, Jie Du and Leonard Scott,
wish to take this opportunity to acknowledge Brian Parshall as being a co-
author of this monograph. In a variation on the words above that Brian used in
our last meeting,“we didn’t have to think about it.”

Jie Du and Leonard Scott

##  Chapter 1 Preliminaries

In this chapter, we briefly review some notions that will be used throughout.
Topics include finite groups of Lie type, generic Hecke algebras, exact
categories and stratified algebras.

If $B$ is a ring, then $B$-mod (respectively, mod-$B$) will denote the
category of all finite (i.e., finitely generated) left (respectively, right)
$B$-modules. Usually, $B$ will be a finitely generated algebra over a
commutative Noetherian ring ${\mathscr{K}}$. In this case, and unless
otherwise noted, objects in $B$-mod will be assumed to be finite
${\mathscr{K}}$-modules (with a similar convention for mod-$B$).

###  1.1. Finite groups of Lie type

We review some basic notions concerning finite groups of Lie type. More
material on this topic will be introduced in Chapter 5. We closely follow the
treatment given in Carter’s book [C85]. In particular, we will assume the
reader is familiar with the notion [C85, pp.16–17] of a connected, reductive
algebraic group over an algebraically closed field of positive characteristic.
Such a group ${\mathbb{G}}$ may be characterized as the internal product
$\mathbf{Z}^{\circ}{\mathbb{G}}^{\prime}$, where $\mathbf{Z}^{\circ}$ is a
torus, the connected component of the identity in the center
$\mathbf{Z}=Z({\mathbb{G}})$ of ${\mathbb{G}}$, and ${\mathbb{G}}^{\prime}$ is
a semisimple algebraic group, defined by Carter to be nontrivial and an
“almost direct product” (a “central product” in traditional terminology) of
simple algebraic groups. The latter are defined as having a maximal normal
closed subgroup which is finite. Note that ${\mathbb{G}}^{\prime}$ is
necessarily the commutator subgroup of ${\mathbb{G}}$.

Now let ${\mathbb{G}}$ be a (connected) reductive algebraic group over an
algebraically closed field of positive characteristic $p$. Recall that a
Frobenius map $F:{\mathbb{G}}\to{\mathbb{G}}$ in Carter’s sense [C85, §1.17]
is a morphism, some power of which equals a standard Frobenius map. Fix a
Frobenius map $F$, and let $G:={\mathbb{G}}^{F}$ be the group of fixed points
of $F$. Necessarily, $G$ is finite. It is called a finite group of Lie type,
and all finite groups of Lie type arise this way.111Our notation differs
slightly from that in [C85], where ${\mathbb{G}}$ is sometimes written $G$.
Also, we sometimes call a Frobenius map a Frobenius endomorphism or a
Frobenius morphism. See [C85, p. 31] for a detailed definition. A Frobenius
endomorphism is sometimes called a Steinberg endomorphism in other sources
(see, e.g., [MT11]). We remark from [MT11, p.184, top] that Carter’s assertion
near the bottom of [C85, p. 31] (just above the definition) regarding
surjective endomorphisms is not correct. However, this does not affect the
validity of his definition.

The finite groups of Lie type most studied here arise in the context of the
previous paragraph when ${\mathbb{G}}$ is taken to be simple.222This
assumption is convenient for the description of the Lie type series and weight
function $L$ discussed below. Note also that sometimes it is useful to study
the more general case where ${\mathbb{G}}^{\prime}$ is simple, either to place
${\mathbb{G}}$ in a familiar context (e.g., ${\mathbb{G}}$ is a general linear
group), or, after beginning a study using a simple group ${\mathbb{G}}$,
viewing it as a commutator group of a larger reductive group
$\widetilde{{\mathbb{G}}}$ with favorable properties, such as a connected
center. See Lemma 5.2.4 in Chapter 5. In addition to the defining
(algebraically closed) field, the (simple) algebraic group ${\mathbb{G}}$ is
determined by its root datum and its isogeny class (see [C85, §§1.9, 1.11]),
and ${\mathbb{G}}^{F}$ is discussed in [C85, §1.19].

It is known that ${\mathbb{G}}$ has an $F$-stable Borel subgroup
${\mathbb{B}}$. In addition, ${\mathbb{B}}$ contains a maximal torus
$\mathbb{T}$ of ${\mathbb{G}}$ which is $F$-stable. The Weyl group
$\mathbb{W}:=N_{{\mathbb{G}}}(\mathbb{T})/{\mathbb{T}}$ is then part of a
finite Coxeter system $(\mathbb{W},\mathbb{S})$ for a (finite) subset
$\mathbb{S}$ of elements of order 2 and $\bf W$ is irreducible (i.e., the
Coxeter diagram is connected) under the assumption that ${\mathbb{G}}$ is
simple. In addition, the pair $(\mathbb{B},\mathbb{N})$, where
$\mathbb{N}=\mathbb{N}_{\mathbb{G}}(\mathbb{T})$, give $\mathbb{G}$ the
structure of a BN-pair. The group $G={\mathbb{G}}^{F}$ also has a natural BN-
pair structure $B,N$, where $B=\mathbb{B}^{F}$ and $N=\mathbb{N}^{F}$. Its
Weyl group is $W=\mathbb{W}^{F}$. In fact, $W$ is generated by the set $S$ of
all the long words in the various $\mathbb{W}_{J}$, as stated in [C85, p. 34]
and proved in [MT11, Lem. C.1]. Moreover, the proof in [MT11] shows that every
element $w\in W$ can be written as a product of elements of $S$ with lengths
additive. One can use these facts to complete Carter’s sketch [C85, p.34] that
$W$ is the Weyl group in a (split) BN-pair structure on $G$, so, in
particular, it is a Coxeter group (with $S$ as its set of fundamental
generators).

Let $\Phi$ be the root system of $\mathbb{G}$, and let $\Delta$ be the set of
simple roots associated to ${\mathbb{B}}$. The Frobenius map $F$ on
${\mathbb{G}}$ is closely related to a certain permutation $\rho$ of $\Delta$.
In fact, $\rho$ is a graph automorphism of the undirected Dynkin diagram
associated to $\Delta$. (See [C85, §1.19].) Two finite groups $G_{1}$ and
$G_{2}$ are said to be in the same Lie type series if there exist a connected
reductive group ${\mathbb{G}}$ and Frobenius morphisms $F_{1}$ and $F_{2}$,
inducing the same graph automorphism $\rho$, and such that
$G_{i}={\mathbb{G}}^{F_{i}}$, for $i=1,2$. By inspection, this amounts to
saying that $G_{1}$ and $G_{2}$ are in one of the groupings on [C85, pp.
39–41]; see also [MT11, p.193]. Further, if $\rho$ is the trivial (or
identity) graph automorphism, then $G$ is called a split (or untwisted) group
of Lie type; if $\rho$ is non-trivial, then $G$ is called a quasi-split (or
twisted) group of Lie type. Here, following Carter’s classification in [C85,
p.41], we exclude the Ree and Suzuki groups in the quasi-split case.

There is a natural action of $F$ on the character group $\mathbb{X}$
(respectively, cocharacter group $\mathbb{Y}$) of ${\mathbb{T}}$. The maximum
of the absolute values of the eigenvalues of $F$ as an operator on
$\mathbb{X}$ is denoted $q$. It is a (possibly fractional) power $q=p^{d}$,
where $p$ is the characteristic of the defining field of ${\mathbb{G}}$. For
example, if $G$ is a classical finite (split) Chevalley group over a finite
field $k$ of order $r$ (an integral power of the defining characteristic $p$),
then $q=r$. In this case, $\rho=1$, the identity map.

Returning to the general case, a useful construction of $W$ as a subgroup of
$\mathbb{W}$ using only the Dynkin diagram permutation $\rho$ associated to
$F$ (rather than $F$ itself) can be based on [C85, p. 34]: First, $\mathbb{W}$
may be viewed as constructed from the Dynkin diagram $\Gamma$, with the
fundamental reflections associated to the nodes of $\Gamma$. Thus, we can
associate to each orbit $J$ of $\rho$ on $\Gamma$ a parabolic subgroup
$\mathbb{W}_{J}$ of $\mathbb{W}$. Now, the fundamental reflections $s$ in $W$
may be defined as the longest words of the various subgroups $\mathbb{W}_{J}$.
Next, define the “weight function” $L$ on $W$ by taking it to be the
restriction of the length function $\boldsymbol{\ell}$ on $\mathbb{W}$ to $W$.
That is, $L(w):=\boldsymbol{\ell}(w)$, for $w\in W\subseteq\mathbb{W}$. Then,
$(W,S,L)$ is a finite (irreducible) Coxeter system with weight function
$L:W\to\mathbb{Z}$ in the sense of [Lus03, §3.1]. Thus, if $w=xy$ with
$\ell(w)=\ell(x)+\ell(y)$, then $L(xy)=L(x)+L(y)$, for $x,y,w\in W$.

Within its Lie type series, the group $G={\mathbb{G}}^{F}$ is determined by
(and determines) a parameter $q$ [C85, p. 35] which is described as follows.
If $\delta$ is the smallest positive integer such that $F^{\delta}$ is a
standard Frobenius endomorphism (as per [C85, p. 31]), then $q^{\delta}$
(rather than $q$ itself) is defined to be the power of $p$ associated to the
latter endomorphism. The number $\delta$ is also the order of the Dynkin
diagram graph automorphism $\rho$. If $G$ is a (split) Chevalley group,
$\delta=1$ and $q$ can be any positive power of the defining characteristic
$p$. In the quasi-split cases which are not split, $\delta$ is the order of a
non-trivial automorphism of the Dynkin diagram, and $q$ is still any positive
power of $p$. However, in the Ree and Suzuki cases, $\delta=2$ and
$q^{2}=p^{2n+1}$ with $p=2$ for the ${}^{2}B_{2}$ and ${}^{2}F_{4}$ cases and
$p=3$ for the ${}^{2}G_{2}$ case. In particular, $q$ is a positive power of
$\sqrt{p}$. See [C85, p. 41]. Remarkably, in all cases, $q$ together with the
weight function $L$ gives

$q^{L(s)}=[B:B\cap B^{s}],$ (1.1.1)

where $B$ is a Borel subgroup of $G$ and $s$ is a fundamental reflection in
its Weyl group $W$. Here $B=\mathbb{B}^{F}$, where $\mathbb{B}$ is an
$F$-stable Borel subgroup of $\mathbb{G}$; see [C85, pp. 34-35, and p. 74] for
further details. This formula holds in the more general case where
${\mathbb{G}}$ is connected and reductive. (See the partial discussion in
[C85, p. 74] and also [St68, 11.8].)

Throughout this work, $W$ will be, unless otherwise noted, the Weyl group of a
finite group $G$ of Lie type (i.e., the Weyl group of the BN-pair associated
to $G$). We will use the natural positive weight function $L$, constructed
above and illustrated in [Lus03, Lem.16.2]. The resulting weighted Coxeter
system $(W,S,L)$ is irreducible and will be called standard finite.333The
Hecke algebras associated to these triples are especially important for the
modular representation theory of unipotent blocks for finite groups of Lie
type. According to a theory put forward by Bonafé and Rouquier [BR03], [BR17],
the case of general blocks (for finite groups of Lie type) could, eventually,
reduce to the unipotent case. In any event, all irreducible or indecomposable
modules with nontrivial cohomology belong to the unipotent principal block. We
also ignore the rank one case, where $W$ has order 2, unless otherwise
noted.444This connectedness assumption is a matter of convenience and is
unnecessary in most cases, e.g., if no component is dihedral of order 16, the
Weyl group of ${}^{2}F_{4}$. Note that standard finite Coxeter systems
$(W,S,L)$ can also be divided into the split case, where the weight function
$L$ equals the length function $\ell$ on $W$, and the quasi-split case, where
$L\neq\ell$. (The only non-trivial case for the Ree and Suzuki groups is the
${}^{2}F_{4}$ case, where $W$ is the dihedral group of order 16.) See [Lus03,
Chs. 15&16].

######  Example 1.1.1.

Let $\mathbb{F}_{q}$ be the finite field of $q$ elements, and let $n$ be a
positive integer. The finite general linear group ${\text{\rm
GL}}_{n}(\mathbb{F}_{q})$ is the fixed-point group of the following (standard)
Frobenius map

$F_{q}:{\text{\rm GL}}_{n}(k)\longrightarrow{\text{\rm
GL}}_{n}(k),\;(a_{i,j})\longmapsto(a_{i,j}^{q}),$ (1.1.2)

where $k$ is the algebraic closure of $\mathbb{F}_{q}$. Note that ${\text{\rm
GL}}_{n}(k)$ is a connected reductive group and the subgroup ${\text{\rm
SL}}_{n}(k)$ of determinant 1 matrices is simple with $F_{q}$-fixed-point
group ${\text{\rm SL}}_{n}(\mathbb{F}_{q})$.

For any field $k$, ${\text{\rm GL}}_{n}(k)$ has the following split BN-pair:

$\displaystyle B_{n}(k)$ $\displaystyle:=\\{\text{subgroup of all upper
triangular matrices in }{\text{\rm GL}}_{n}(k)\\}$ $\displaystyle N_{n}(k)$
$\displaystyle:=\\{\text{subgroup of all monomial matrices in }{\text{\rm
GL}}_{n}(k)\\}.$

Let $T_{n}(k)=B_{n}(k)\cap N_{n}(k)$ be the subgroup of diagonal matrices. If
$n_{i}$ denotes the permutation matrix obtained by interchanging the $i$-th
and $(i+1)$-th rows of $I_{n}$, then the Weyl group

$\widetilde{W}:=N_{n}(\mathbb{F})/T_{n}(\mathbb{F})\cong{\mathfrak{S}}_{n}\text{
under the correspondence }\;n_{i}\mapsto\tilde{s}_{i}=(i,i+1).$

See, e.g., [G03, 1.6.9] for more details.

The other finite classical groups can be constructed by fixed-point groups of
certain group automorphisms on ${\text{\rm GL}}_{n}(\mathbb{F}_{q})$. In
Example 1.1.2, we consider the symplectic and orthogonal groups. They are
relevant to our studies of $i$-quantum groups in Section 5.4. Unitary groups
are taken up in Example 1.1.3. These both help to complete the classical group
picture and to provide examples of “unequal parameter” weight functions.

######  Example 1.1.2.

For any field $k$ and $J\in{\text{\rm GL}}_{n}(k)$, consider the group
isomorphism

$\vartheta=\vartheta_{J}:{\text{\rm GL}}_{n}(k)\longrightarrow{\text{\rm
GL}}_{n}(k),\;x\longmapsto J^{-1}(x^{t})^{-1}J,$ (1.1.3)

and define orthogonal and symplectic groups as the fixed-point groups of some
$\vartheta$:

$\displaystyle{\text{\rm O}}_{n}(k)$ $\displaystyle:=\\{x\in{\text{\rm
GL}}_{n}(k)\mid J_{n}=x^{t}J_{n}x\\},$ $\displaystyle{\text{\rm Sp}}_{2m}(k)$
$\displaystyle:=\\{x\in{\text{\rm GL}}_{2m}(k)\mid
J_{2m}^{-}=x^{t}J_{2m}^{-}x\\},$

where

$J_{n}:=\begin{pmatrix}0&0&\cdots&0&1\\\ 0&0&\cdots&1&0\\\
&\cdots&\cdots&\cdots&\\\ 0&1&\cdots&0&0\\\ 1&0&\cdots&0&0\\\
\end{pmatrix},\quad J^{-}_{2m}:=\begin{pmatrix}0&J_{m}\\\ -J_{m}&0\\\
\end{pmatrix}.$ (1.1.4)

Let ${\text{\rm SO}}_{n}(k)={\text{\rm O}}_{n}(k)\cap{\text{\rm SL}}_{n}(k)$
and let ${\textbf{G}}\subseteq{\text{\rm GL}}_{n}(k)$ be one of the classical
groups:

${\textbf{G}}=\begin{cases}{\text{\rm
SO}}_{2m+1}(k),&n=2m+1,\text{char}(k)\neq 2;\\\ {\text{\rm
Sp}}_{2m}(k),&n=2m,\text{any characteristic};\\\ {\text{\rm
SO}}_{2m}(k),&n=2m,\text{char}(k)\neq 2.\end{cases}$

These are known as split twisted groups in [St67, Ch. 11].555See [MT11, p.8]
for the definition for char$(k)=2$.

If $k$ is the algebraic closure of $\mathbb{F}_{q}$, then G is a connected
simple group and has a split BN-pair ${\textbf{B}}={\textbf{G}}\cap B_{n}(k)$,
${\textbf{N}}={\textbf{G}}\cap N_{n}(k)$.

For the Frobenius map $F_{q}$ as defined in (1.1.2) and $J=J_{n}$ or
$J_{2m}^{-}$, since $J=x^{t}Jx$ implies $J=(F_{q}(x))^{t}JF_{q}(x)$, it
follows that $F_{q}$ induces the (standard) Frobenius maps

$F:{\textbf{G}}\longrightarrow{\textbf{G}},$

and ${\textbf{G}}^{F}={\text{\rm SO}}_{n}(\mathbb{F}_{q})\;(n=2m+1,2m)$ or
${\text{\rm Sp}}_{2m}(\mathbb{F}_{q})$. Note also that
${\textbf{G}}^{F}={\text{\rm GL}}_{n}(\mathbb{F}_{q})^{\vartheta}$ has BN-pair
${\textbf{B}}^{F}=B_{n}(\mathbb{F}_{q})^{\vartheta}$ and
${\textbf{N}}^{F}=N_{n}(\mathbb{F}_{q})^{\vartheta}$ since $\vartheta$
stabilizes the split BN-pair $(B_{n}(k),N_{n}(k))$. In particular, $\vartheta$
induced a group automorphism
$\sigma:{\widetilde{W}}\longrightarrow{\widetilde{W}}$, and there is a group
isomorphism

${\textbf{W}}:={\textbf{N}}/{\textbf{B}}\cap{\textbf{N}}\cong{\widetilde{W}}^{\sigma}=N_{n}(\mathbb{F}_{q})^{\vartheta}/T_{n}(\mathbb{F}_{q})^{\vartheta}.$

Note that the map induced by $F$ on W is the identity map. Note also that, for
${\textbf{G}}={\text{\rm SO}}_{2m+1}(k)$ or ${\text{\rm Sp}}_{2m}(k)$,
$W:={\textbf{W}}^{F}={\textbf{W}}$ is the Weyl group of type $B/C$ with
generators $S=\\{s_{1},\ldots,s_{m-1},t\\}$, where

$\displaystyle s_{i}$ $\displaystyle=\tilde{s}_{m-i}\tilde{s}_{m+1+i},\text{
and }t=\tilde{s}_{m}\tilde{s}_{m+1}\tilde{s}_{m},\text{ for }n=2m+1;$
$\displaystyle s_{i}$
$\displaystyle=\tilde{s}_{m-i}\tilde{s}_{m+i},\quad\text{ and
}t=\tilde{s}_{m},\text{ for }n=2m.$

Finally, $t$ is the homomorphic image of the elements in
$N_{n}(k)^{\vartheta}$:

$\begin{pmatrix}I_{m-1}&0&0\\\ 0&J&0\\\ 0&0&I_{m-1}\end{pmatrix},\text{ where
}J=\begin{pmatrix}0&0&1\\\ 0&-1&0\\\ 1&0&0\end{pmatrix}\text{ or
}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$

For more details, see, e.g., [G03, 1.7.3].

######  Example 1.1.3.

Now, following [DVV19, §4.1], we combine $F_{q}$ in (1.1.2) with $\vartheta$
in (1.1.3), using the choice $J=J_{n}$, to obtain a new Frobenius morphism

$F:=F_{q}\vartheta=\vartheta F_{q}:{\text{\rm
GL}}_{n}(k)\longrightarrow{\text{\rm GL}}_{n}(k).$

Since $\vartheta^{2}$ is the identity map, it follows that $F^{2}$ is the
standard Frobenius map $F_{q^{2}}$ and ${\text{\rm
GL}}_{n}(k)^{F}\subseteq{\text{\rm GL}}_{n}(\mathbb{F}_{q^{2}})$. We have

${\text{\rm GU}}_{n}(\mathbb{F}_{q}):={\text{\rm
GL}}_{n}(k)^{F}=\\{x\in{\text{\rm
GL}}_{n}(\mathbb{F}_{q^{2}})\mid(F_{q}(x))^{t}J_{n}x=J_{n}\\}.$

This is called the general unitary group associated with the hermitian form
defined by $J_{n}$. Replacing G by S defines the special unitary group. (See
also [MT11, Ex. 21.48(2)] or [G03, 4.1.10(c)].)

For ${\text{\rm SU}}_{4}(\mathbb{F}_{q^{2}})$, the $F$-orbits of the simple
roots $\alpha_{1,2},\alpha_{2,3},\alpha_{3,4}$ for ${\text{\rm SL}}_{4}(k)$,
with associated root subgroups $U_{i,j}=\\{I_{4}+aE_{i,j}\mid
a\in\mathbb{F}_{q^{2}}\\}$, are $\\{\alpha_{1,2},\alpha_{3,4}\\}$,
$\alpha_{2,3}$. The root subgroups ${\text{\rm
SU}}_{4}(\mathbb{F}_{q^{2}})={\text{\rm SL}}_{4}(k)^{F}$ are

$(U_{1,2}U_{3,4})^{F}=\\{I_{4}+aE_{1,2}-a^{q}E_{3,4}\mid
a\in\mathbb{F}_{q^{2}}\\},\quad U_{2,3}^{F}=\\{I_{4}+aE_{2,3}\mid
a+a^{q}=0\\}.$

The values $q^{L(s)}$ ($s\in S$) associated to the weight function $L$ on
${\textbf{W}}^{F}=W(B_{2})$ in this case are $q^{2},q$. In general, the values
$q^{L(s)}$ for ${\text{\rm SU}}_{2m}(\mathbb{F}_{q^{2}})$ are listed in the
Coxeter diagram:

$q^{2}$$q^{2}$$q^{2}$$q^{2}$$q$123m-1m

For ${\text{\rm SU}}_{3}(\mathbb{F}_{q^{2}})$, the $F$-orbits of the simple
roots $\alpha_{1,2},\alpha_{2,3}$ for ${\text{\rm SL}}_{4}(k)$ are
$\\{\alpha_{1,2},\alpha_{2,3}\\}$, and we have the root subgroup (cf. [G03,
4.5.12(b)])

$(U_{1,2}U_{2,3}U_{1,3})^{F}=\bigg{\\{}\begin{pmatrix}1&a&b\\\ 0&1&-a^{q}\\\
0&0&1\end{pmatrix}\;\bigg{|}\;a,b\in\mathbb{F}_{q},b^{q}+b+a^{q+1}=0\bigg{\\}}.$

So, $q^{L(s)}=q^{3}$. In general, the values $q^{L(s)}$ ($s\in S$) for
${\text{\rm SU}}_{2m+1}(\mathbb{F}_{q^{2}})$ take the form:

$q^{2}$$q^{2}$$q^{2}$$q^{2}$$q^{3}$123m-1m

###  1.2. (Iwahori-)Hecke algebras

We review the theory of Hecke algebras in [Lus03]. Each weighted Coxeter
system $(W,S,L)$ defines an algebra ${\mathcal{H}}$, called an Iwahori-Hecke
algebra in [Lus03] and called a generic Hecke algebra  in this monograph. See
(1.2.1) and (1.2.2) for more details.

Many results in [Lus03] are contingent on a series P1—P15 of conjectures
(involving $(W,S,L)$ and ${\mathcal{H}}$), all of which have been proved in
the standard finite case. See [Lus03, Chs. 15, 16], revisited in [GJ11, Ch. 2]
(especially [GJ11, 2.4.1(b)] for the case ${}^{2}F_{4}$). Thus, we can assume
Lusztig’s conjectures P1–P15 hold for the triples $(W,S,L)$ considered in this
monograph. Henceforth, we assume that $(W,S,L)$ is standard finite unless
otherwise noted. In particular, $L$ is positive and Lusztig’s conjectures
P1-P15 hold, as does condition (1.2.11) below. In fact, we mention that in
Chapters 1–3 which include the main theorem, we need only assume conjectures
P4 and P9, which, together with the rank 2 discussion in [DPS98a, p.
204],666See footnote 3 in Chapter 3 which corrects a typo. imply that property
(1.2.11) holds.

Let ${\mathcal{Z}}={\mathbb{Z}}[t,t^{-1}]$ be the ring of integer Laurent
polynomials in a variable $t$. Let ${\mathcal{H}}$ be the generic Hecke
algebra over ${\mathcal{Z}}$ associated with $(W,S,L)$. Recall that
${\mathcal{H}}$ has basis $\\{T_{w}\\}_{w\in W}$ and defining relations ($s\in
S,\,w\in W$):

$T_{s}T_{w}=\begin{cases}T_{sw},\quad sw>w;\\\
t_{s}^{2}T_{sw}+(t_{s}^{2}-1)T_{w},\quad sw<w,\end{cases}$ (1.2.1)

where $t_{s}:=t^{L(s)}$ (and, more generally, $t_{w}=t^{L(w)}$). Also, for
$x,y\in W$, $x<y$ denotes the Chevalley-Bruhat partial order. The algebra
${\mathcal{H}}$ is defined just using $t^{2}$, but, in practice, it is
convenient to have its square root $t$ available.777It is well known that the
algebra $\mathbb{Q}(t)\otimes_{{\mathcal{Z}}}{\mathcal{H}}$ is a semisimple
$\mathbb{Q}(t)$-algebra. See [CR81, Cor. 68.12] as well as Remark 3.3.4(b)
below.

We often use a different basis $\\{\widetilde{T}_{w}\\}$ also indexed by $W$
defined by putting

$\widetilde{T}_{w}:=t^{-L(w)}T_{w},\quad w\in W.$ (1.2.2)

(Our notation here differs slightly from that in [Lus03], where our
$\widetilde{T}_{w}$ is denoted $T_{w}$.)

The bar operator $\bar{\ }:{\mathcal{H}}\to{\mathcal{H}}$ sending $t$ to
$t^{-1}$ and $T_{w}$ to $(T_{w^{-1}})^{-1}$ is a ring involution on
${\mathcal{H}}$ (sending $t$ to $t^{-1})$, and ${\mathcal{H}}$ admits a
Kazhdan-Lusztig (or “canonical” in the sense of [Lus90]; see also [DDPW08,
§0.5]) basis $\\{{\textsc{c}}_{w}\\}_{w\in W}$. This basis is defined in
[Lus03, Chap.5] using the formula

${\textsc{c}}_{w}=\sum_{y\leq w}p_{y,w}{\widetilde{T}}_{y}.$

In addition to the condition $\overline{{\textsc{c}}}_{w}={\textsc{c}}_{w}$,
for all $w$, the Laurent polynomials $p_{y,w}\in{\mathcal{Z}}$ satisfy (1)
$p_{y,w}=0$ unless $y\leq w$, (2) $p_{w,w}=1$, and (3)
$p_{y,w}\in{\mathcal{Z}}_{<0}:=t^{-1}\mathbb{Z}[t^{-1}]$, if $y<w$. The
polynomials $p_{y,w}$ are determined by these properties. For example,

${\textsc{c}}_{s}=t_{s}^{-1}+\widetilde{T}_{s}=t_{s}^{-1}(1+T_{s}).$ (1.2.3)

(Note that the element ${\textsc{c}}_{x}$ is denoted $c_{x}$ in [Lus03].)

If $J\subseteq S$ is non-empty, let $W_{J}:=\langle s\rangle_{s\in J}$, a
parabolic subgroup of $W$; if $J=\emptyset$, set $W_{J}:=\\{1\\}$. Then,
$W_{J}$ has a unique longest (reduced) word, denoted $w_{0,J}$, and

${\textsc{c}}_{w_{0,J}}=t^{-L(w_{0,J})}\sum_{y\in
W_{J}}T_{y}=:t^{-L(w_{0,J})}x_{J}$ (1.2.4)

(which defines $x_{J}$). See [Lus03, Cor. 12.2] (or [DDPW08, (7.2.5)] in the
split case). In the literature, the endomorphism algebra

$A:=\text{\rm{End}}_{{\mathcal{H}}}\left(\bigoplus_{J\subseteq
S}x_{J}{\mathcal{H}}\right)$ (1.2.5)

(or any algebra Morita equivalent to it) is called “the” generalized generic
$q$-Schur algebra or a generic Hecke endomorphism algebra (or generic Hecke
endo-algebra for short) by analogy with our naming of ${\mathcal{H}}$. Various
specializations of $A$ and its extensions will also be considered in later
chapters.

Let $h_{x,y,z}\in{\mathcal{Z}}$ denote the structure constants for
${\mathcal{H}}$, defined by the equations

${\textsc{c}}_{x}{\textsc{c}}_{y}=\sum_{z\in W}h_{x,y,z}{\textsc{c}}_{z}.$
(1.2.6)

In particular, we have [Lus03, Th. 6.6]

${\textsc{c}}_{s}{\textsc{c}}_{w}=\begin{cases}{\textsc{c}}_{sw}+\sum_{x:sx<x<w}\mu^{s}_{x,w}{\textsc{c}}_{x},&\text{
if }w<sw;\\\ (t_{s}+t_{s}^{-1}){\textsc{c}}_{w},&\text{ if }w>sw.\end{cases}$
(1.2.7)

Here $\mu^{s}_{x,w}\in{\mathcal{Z}}$ is fixed under the bar operator. In the
case of split groups, $\mu^{s}_{x,w}$ is constant (in $\mathbb{Z}$). Also,
(1.2.7) implies that the ${\textsc{c}}_{s}$, $s\in S$, generate the
${\mathcal{Z}}$-algebra ${\mathcal{H}}$.

If ${\textsc{c}}_{y}$ appears in ${\textsc{c}}_{s}{\textsc{c}}_{w}$ with a
nonzero coefficient, we set $y\leftarrow_{L}w$, following [Lus03, 8.1]. Define
a preorder (that is, a relation which is reflexive and transitive) on $W$ by
setting, for $w,w^{\prime}\in W$, $w\leq_{L}w^{\prime}$ if there exists
$w=w_{0},w_{1},\ldots,w_{n}=w^{\prime}$ in $W$ such that
$w_{i-1}\leftarrow_{R}w_{i}$, for all $i=1,2,\ldots,n$, and define
$w\leq_{R}w^{\prime}$ if $w^{-1}\leq_{L}(w^{\prime})^{-1}$.

We now consider the preorders $\leq_{L}$ and $\leq_{R}$ defined on $W$, and
the preorder $\leq_{LR}$ generated by $\leq_{L}$ and $\leq_{R}$ and their
associated quasi-poset $(W,\leq_{L})$ etc. See, e.g., [Lus03, Ch. 8]. Let
$\sim_{L},\sim_{R}$ and $\sim_{LR}$ be the equivalence relations associated
with $\leq_{L},\leq_{R}$, and $\leq_{LR}$, respectively. The equivalence
classes for $\sim_{L},\sim_{R}$, and $\sim_{LR}$ are called the left, right,
and two-sided cells, respectively.

For a quasi-poset, there is an associated poset (see Section B2 in Appendix
B). The poset associated with the quasi-poset $(W,\leq_{L})$ is the set
$\Omega$ of all left cells, while the poset associated with the quasi-poset
$(\Omega,\leq_{LR})$ is the set $\overline{\Omega}$ of all two-sided cells.
Note that if a left cell $\omega\in\Omega$ contains an element $w_{0,J}$, then
$J$ is a uniquely determined subset of $S$. See [Lus03, Lem. 8.6] and the
argument above Lemma 3.1.2 in this monograph.

We record that

$h_{x,y,z}\neq 0\implies z\leq_{L}y,z\leq_{R}x.$ (1.2.8)

See [Lus03, 13.1(d)]. In particular, for any $w\in W$,

${\mathcal{H}}_{\leq_{L}w}:=\sum_{y\in W\atop
y\leq_{L}w}{\mathcal{Z}}{\textsc{c}}_{y},\;\;{\mathcal{H}}_{\leq_{R}w}:=\sum_{y\in
W\atop
y\leq_{R}w}{\mathcal{Z}}{\textsc{c}}_{y},\;\;{\mathcal{H}}_{\leq_{LR}w}:=\sum_{y\in
W\atop y\leq_{LR}w}{\mathcal{Z}}{\textsc{c}}_{y}$ (1.2.9)

is a left, right, or two-sided ideal of ${\mathcal{H}}$, respectively (see
[Lus03, Lem. 8.2].). We may also define similar ideals by using
$<_{L},<_{R},<_{LR}$.

Each left cell $\omega\in\Omega$ defines a left-cell module
$S(\omega):={\mathcal{H}}_{\leq_{L}\omega}/{\mathcal{H}}_{<_{L}\omega}$—it is
explicitly defined in [Lus03, §8.3] (but with a different notation). Here
${\mathcal{H}}_{\leq_{L}\omega}={\mathcal{H}}_{\leq_{L}y}$ and
${\mathcal{H}}_{<_{L}\omega}={\mathcal{H}}_{<_{L}y}$, for any $y\in\omega$.
Similarly, we may define right- or two-sided-cell modules. Also, there are
base-change versions associated with the Hecke alegbra
${\mathscr{K}}\otimes_{\mathcal{Z}}{\mathcal{H}}$ and the basis
$\\{1\otimes{\textsc{c}}_{w}\\}_{w\in W}$ over any commutative ring
${\mathscr{K}}$ which is a ${\mathcal{Z}}$-module.

In addition, there is a corresponding

dual left-cell module
$S_{\omega}:=S(\omega)^{*}=\text{\rm{Hom}}_{\mathcal{Z}}(S(\omega),{\mathcal{Z}})$.
(1.2.10)

It is a right ${\mathcal{H}}$-module. Both $S(\omega)$ and $S_{\omega}$ are
free ${\mathcal{Z}}$-modules; see [Lus03, §8.3].

######  Remark 1.2.1.

If ${\mathcal{H}}$ is the generic Hecke algebra for a symmetric group
$W=\mathfrak{S}_{r}$, then it can be proved that the dual left-cell module
$S_{\omega}$ is a “classical” Specht module for ${\mathcal{H}}$, as defined
more generally by Dipper-James; see [DJ86, p 34], taking $R$ in the latter
reference to be the ring $\mathbb{Z}$ of integers. On the other hand, the
right cell $\gamma:=\omega^{-1}$, defines a right-cell module $K_{\gamma}$,
which is related to $S_{\omega}$ by the isomorphism
$\mathfrak{D}_{\mathcal{H}}S_{\omega}\cong K_{\gamma}$, $\gamma=\omega^{-1}$.
(Here, $\mathfrak{D}_{\mathcal{H}}$ is the contravariant duality functor on
mod-${\mathcal{H}}$ defined in [DPS98c, (1.2)].) We will not make use of any
of the theory of Specht modules in this monograph.

The preorders $\leq_{L},\leq_{LR}$ on $W$ induce partial orders
$\leq_{L},\leq_{R}$ on $\Omega$. Thus, for $\omega,\omega^{\prime}\in\Omega$
and $X\in\\{L,LR\\}$, $\omega\leq_{X}\omega^{\prime}\iff x\leq_{X}x^{\prime}$,
for some $x\in\omega,x^{\prime}\in\omega^{\prime}.$ Also, $\omega\in\Omega$
defines a unique two-sided cell $\bar{\omega}$ with the property that
$\omega\subseteq\bar{\omega}$. It is extremely important in our context here
(see the second paragraph of Section 1.2 or Proposition B.5 in Appendix B)
that, given $\omega_{1},\omega_{2}\in\Omega$,

$\omega_{1}<_{L}\omega_{2}\implies\overline{\omega}_{1}<_{LR}\overline{\omega}_{2}.$
(1.2.11)

Thus, $\leq_{L}$ strictly dominates $\leq_{LR}$ in the sense of Definition B.3
in Appendix B. As discussed at the beginning of the section, this property
holds when $(W,S,L)$ is standard finite, our standing assumption.

For each two-sided cell $\mathbf{c}$ of $W$, there is an
${\mathcal{H}}$-bimodule $M(\mathbf{c})$. The two-sided cell decomposition of
$W$ induces a decomposition of ${\mathcal{H}}_{\mathbb{Q}(t)}$ into two-sided
ideals $M(\mathbf{c})_{\mathbb{Q}(t)}$ and, hence, a decomposition of
$\Lambda=\text{Irr}({\mathcal{H}}_{\mathbb{Q}(t)})$. Further, the partial
order $\leq_{LR}$ on the set $\bar{\Omega}$ of two-sided cells induces a
preorder on $\Lambda$ by setting, for $E,E^{\prime}\in\Lambda$,

$E\leq_{LR}E^{\prime}\text{ if
}\mathbf{c}_{E}\leq_{LR}\mathbf{c}_{E^{\prime}}.$ (1.2.12)

Here, $\mathbf{c}_{E}$ denotes the two-sided cell $\mathbf{c}$ such that
$E|M(\mathbf{c})_{\mathbb{Q}(t)}$ in the sense that $E$ is a direct summand of
the left ${\mathcal{H}}_{\mathbb{Q}(t)}$-module defined by
$M(\mathbf{c})_{\mathbb{Q}(t)}$.

We end this section with a brief discussion about the origin of the generic
Hecke algebra ${\mathcal{H}}$ associated with a standard finite Coxeter system
$(W,S,L)$. Here $W=\mathbf{W}^{F}$ is the Weyl group of a finite group
$G={\mathbb{G}}^{F}$ of Lie type that has a Borel subgroup
$B={\mathbb{B}}^{F}$; see Section 1.1. The following result, which will not be
used until Section 5.2, was first discovered by Iwahori [Iwa64] for (split)
Chevalley groups.

######  Lemma 1.2.2.

Let ${\mathcal{H}}^{o}$ be the subalgebra over $\mathbb{Z}[t^{2}]$ generated
by all $T_{s}\in{\mathcal{H}}$, $s\in S$.888Thus, ${\mathcal{H}}^{o}$ is a
form for ${\mathcal{H}}$ over $\mathbb{Z}[t^{2}]$. Suppose ${\mathcal{O}}$ is
a commutative ring and $q\in{\mathcal{O}}$, and view ${\mathcal{O}}$ as a
$\mathbb{Z}[t^{2}]$-algebra by the homomorphism
$\mathbb{Z}[t^{2}]\to{\mathcal{O}}$ sending $t^{2}\mapsto q$. Then,

${\mathcal{H}}^{o}_{{\mathcal{O}}}\cong\text{\rm{End}}_{{\mathcal{O}}G}({\mathcal{O}}G/B),$

where ${\mathcal{O}}G/B$ is the permutation module for the action of $G$ on
the set of left cosets of $B$.

###### Proof.

A proof of display (1.1.1) is outlined in [C85, p.74], beginning with a
general root group cardinality formula

$|X_{\Psi}^{F}|=q^{|\Psi|},$

where $\Psi$ is a certain subset of the positive roots $\Phi^{+}$ of
${\mathbb{G}}$. For each fundamental reflection $s$ in the Weyl group $W$ of
$G$, one of the root groups $X^{F}_{\Psi}$ can be seen, using [C85, Prop.
2.5.11], to satisfy $|B|=|X^{F}_{\Psi}(B\cap B^{s})|=|X_{\Psi}^{F}|\cdot|B\cap
B^{s}|$. The set $\Psi$ here is the set of positive roots in $\Phi$ made
negative by the action of $s$. Thus, $|\Psi|=L(s)$ for this $\Psi$, where
$L(s)$ is the length of $s$ in the Weyl group $\mathbf{W}$ of ${\mathbb{G}}$;
see Section 1.1. Thus, we obtain the expression (1.1.1) above from the
expression

$q^{L(s)}=q^{|\Psi|}=|B/B\cap B^{s}|=[B:B\cap B^{s}].$

(Carter actually gives in [C85, p. 74] an analogous formula for $q^{L(w)}$,
for any $w$ in the Weyl group of ${\mathbb{G}}$.)

As in [CPS73, p. 188], the elements

$\frac{1}{|B|}{\underline{BwB}},\quad w\in W$ (1.2.13)

span a $\mathbb{Z}$-algebra $H$, well known to be the endomorphism algebra of
the permutation module $\mathbb{Z}G/B$ for the action of $G$ on coset space
$G/B$. This algebra base changes well, so that we can replace $\mathbb{Z}$ by
${\mathcal{O}}$, obtaining a basis for
$\text{\rm{End}}_{{\mathcal{O}}G}({\mathcal{O}}G/B)$. See, e.g., [DDPW08,
Exercise 9.3]. In [CPS73, fn. 31], a multiplication relation is showed to be
analogous to the right-hand version of the relation $T_{s}T_{w}=T_{sw}$ in
(1.2.1), when $\ell(sw)=1+\ell(w)$. Here, we think of
$\frac{1}{|B|}{\underline{BwB}}$ as analogous to $T_{w}$ in (1.2.1). There is
also the “quadratic relation”

$\left(\frac{1}{|B|}{\underline{BsB}}\right)^{2}=a\frac{1}{|B|}{\underline{BsB}}+b1$

with $b=[B:B\cap B^{s}]$ and $a=b-1$. Such a relation holds in any finite
doubly transitive group, such as $\langle B,s\rangle$ acting on the coset
space $\langle B,s\rangle/B$.

Hence $b=q^{L(s)}$ (and $a=b-1$). The two types of relations are analogous to
(1.2.1) with $w=s$, replacing the $t_{s}^{2}$ there with $q^{L(s)}$. The two
types of relations we have considered clearly form special cases of (1.2.1),
sufficient to provide generators and relations for
${\mathcal{H}}^{o}_{\mathcal{O}}$. Similarly, we have obtained from
$\text{\rm{End}}_{G}({\mathcal{O}}G/B)$ analogs with enough information to do
all multiplications of the basis elements $\frac{1}{|B|}{\underline{BwB}}$. ∎

###  1.3. Exact categories and stratified algebras

Beginning in this section, considerable use is made of exact categories
$({\mathscr{A}},{\mathscr{E}})$ as formulated in Keller [K90, Appendix A] and
revisited in Bühler’s exposition [Bu10]. The axioms given in these papers are
equivalent to (but shorter than) the original axioms of Quillen [Q73]. A
version999There are several different versions of exact categories in the
literature. In particular, following Quillen, we do not assume any kind of
idempotent splitting. is stated in [DPS17, Appendix] which follows [K90,
Appendix A] quite closely (as quoted in [DRSS99, Appendix A, Set 2] written by
Keller). Briefly, ${\mathscr{A}}$ is an additive category, while
${\mathscr{E}}$ consists of certain pairs $(i,d)$ of composable morphisms
$i:X\to Y$ and $d:Y\to Z$ in ${\mathscr{A}}$, such that $i$ is the kernel of
$d$ and $d$ is the cokernel of $i$. The pairs $(i,d)$ which belong to
${\mathscr{E}}$ are required to satisfy certain axioms; see explicitly [K90,
Appendix A], [DRSS99, Appendix A, Set 2], [Bu10, Def. 2.1], or [DPS17,
Appendix A]. As in [K90], we sometimes refer to $i$ as an inflation, $d$ as a
deflation, and the pair $(i,d)$ as a conflation. Conflations $(i,d)$ are
usually denoted by $(X\to Y\to Z)$.101010Such a sequence is denoted by
$X\rightarrowtail Y\twoheadrightarrow Z$ in [Bu10]. There are also other
notations in the literature used with the same meaning. As in [Bu10], we
sometimes call conflations “short exact sequences” in ${\mathscr{E}}$. When it
is convenient and clear from context, $(X\to Y\to Z)$ is denoted $0\to X\to
Y\to Z\to 0$ and called a short exact sequence (in ${\mathscr{E}}$).

Notice that ${\mathscr{E}}$ is itself an additive category, using the usual
mappings between the categorical diagrams formed by composable pairs of
morphisms in ${\mathscr{A}}$; see [Bu10, Cor. 2.10]. As such, ${\mathscr{E}}$
is required to be closed under isomorphisms as part of the exactness axioms.

One can consider a pair $({\mathscr{A}},{\mathscr{E}})$ consisting of an
additive category ${\mathscr{A}}$ together with a collection ${\mathscr{E}}$
of composable morphisms $(i,d)$ in ${\mathscr{A}}$ and ask if it defines an
exact category. Of course, one needs any $(i,d)$ in ${\mathscr{E}}$ to be a
kernel-cokernel pair (in the terminology of [Bu10]). Expanding on [DPS17],
Chapters 2 and 3 present some important examples of exact categories
$({\mathscr{A}},{\mathscr{E}})$ inspired by Hecke algebra representation
theory, especially involving the construction and control of endomorphism
rings, as in Theorem 3.3.1 below. Often ${\mathscr{A}}$ is a full subcategory
of an additive category ${\mathscr{B}}$, associated to an already introduced
exact category $({\mathscr{B}},{\mathscr{F}})$. The theme of constructing new
exact categories from old ones is continued in Sections 2.2 and 2.3 using
duality.

Finally, for an exact category $({\mathscr{A}},{\mathscr{E}})$, we can
consider the abelian group $\text{\rm{Ext}}^{1}_{{\mathscr{E}}}(X,Y)$ for any
pair of objects $X,Y\in{\mathscr{A}}$. In case ${\mathscr{A}}$ is an additive
${\mathscr{K}}$-category for a commutative ring ${\mathscr{K}}$,
$\text{\rm{Ext}}^{1}_{\mathscr{E}}(X,Y)$ carries a natural structure as a
${\mathscr{K}}$-module; see [DPS17, Prop. A.2]. The action of
$a\in{\mathscr{K}}$ on $\text{\rm{Ext}}^{1}_{{\mathscr{E}}}(X,Y)$ is given by
either pull-back of multiplication by $a$ on $X$ or (equally, as it turns out)
by push-out of multiplication of $a$ on $Y$.

1.3A. Exact subcategories: some terminology and a new strategy. A note on
exact category terminology: Let $({\mathscr{A}},{\mathscr{E}})$ and
$({\mathscr{B}},{\mathscr{F}})$ be exact categories. Suppose ${\mathscr{A}}$
is a subcategory of ${\mathscr{B}}$, and
${\mathscr{E}}\subseteq{\mathscr{F}}$, viewing ${\mathscr{E}}$ and
${\mathscr{F}}$ as classes of composable pairs of morphisms in ${\mathscr{A}}$
and ${\mathscr{B}}$, respectively. We say that $({\mathscr{A}},{\mathscr{E}})$
is an exact subcategory111111Some authors note that the set ${\mathscr{E}}$
has a natural category structure, using “morphisms” constructed from one short
exact sequence to another via commutative diagrams. This does not, however,
affect our definition of an exact subcategory. of
$({\mathscr{B}},{\mathscr{F}})$. It is a full exact subcategory if
${\mathscr{A}}$ is also a full subcategory of ${\mathscr{B}}$—that is,
$\text{\rm{Hom}}_{{\mathscr{A}}}(A_{1},A_{2})=\text{\rm{Hom}}_{{\mathscr{B}}}(A_{1},A_{2})$
whenever $A_{1},A_{2}$ are objects in ${\mathscr{A}}$.

With this terminology in hand, we can now describe our new approach and some
of its motivation.

######  Strategy 1.3.1.

If $({\mathscr{A}},{\mathscr{E}})$ is a full exact subcategory of
$({\mathscr{B}},{\mathscr{F}})$, then the fullness implies that the inclusion
${\mathscr{E}}\subseteq{\mathscr{F}}$ induces an inclusion, i.e., an injective
map,
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}}(A,A^{\prime})\hookrightarrow\text{\rm{Ext}}^{1}_{{\mathscr{F}}}(A,A^{\prime})$
for each pair $(A,A^{\prime})$ of objects in ${\mathscr{A}}$.121212The members
of exact category $\text{Ext}^{1}$ groups are defined as equivalence classes
of short exact sequences under a well-known equivalence relation. See [DPS17,
p.249]. In the presence of fullness, each equivalence class in ${\mathscr{F}}$
intersects at most one of the ${\mathscr{E}}$ equivalence classes, resulting
in the inclusions of this sentence. We take this opportunity to note that our
notion of “full exact subcategory” is quite different from the notion “fully
exact subcategory” in [Bu10, Def. 10.21], apparently aimed in a different
direction. The inclusion is even ${\mathscr{K}}$-linear if ${\mathscr{K}}$ is
a commutative Noetherian ring and the inclusion
${\mathscr{A}}\subseteq{\mathscr{B}}$ is also an inclusion of
${\mathscr{K}}$-categories (with ${\mathscr{A}}$ full in ${\mathscr{B}}$ as
before.) See [DPS17, Prop. A.2].

As in traditional homological algebra, the vanishing of a group
$\text{\rm{Ext}}^{1}_{{\mathscr{F}}}(A_{1},A_{2})$ leads to the exactness of
$\text{\rm{Hom}}_{{\mathscr{F}}}(A_{1},-)$ when applied to any given short
exact sequence $\gamma$ in ${\mathscr{F}}$ that begins with $A_{2}$. Assume
this exactness is desirable for some set $\Gamma$ of such $\gamma$’s, but the
proposed $\text{\rm{Ext}}^{1}$ vanishing is too much to ask for. In that case,
we can try to build a full exact subcategory $({\mathscr{A}},{\mathscr{E}})$
as above, but with $\Gamma\subseteq{\mathscr{E}}$ and
$\text{\rm{Ext}}^{1}_{\operatorname{{\mathscr{E}}}}(A_{1},A_{2})=0$. This
gives the desired exactness.

Though the discussion above is only an outline, we are able to use it when
constructing stratifying systems, as we now show.

1.3B. Stratifying Systems. Let ${\mathscr{K}}$ be a Noetherian commutative
ring. Let $B$ be an algebra which is finite (= finitely generated) and
projective as a ${\mathscr{K}}$-module. Let $\Lambda$ be a quasi-poset, i.e.,
a finite set $\Lambda$ with a preorder, denoted $\leq$ here. The associated
poset is $\bar{\Lambda},$ and $\bar{\lambda}$ is the element of
$\bar{\Lambda}$ associated to any given $\lambda\in\Lambda$.

For $\lambda\in\Lambda$, assume that there is given both a $B$-module
$\Delta(\lambda)$, which is finite and projective over ${\mathscr{K}}$, and a
finite and projective $B$-module $P(\lambda)$, together with an epimorphism
$P(\lambda)\twoheadrightarrow\Delta(\lambda)$. The following conditions are
assumed to hold:

  * (SS1)

For $\lambda,\mu\in\Lambda$,

$\text{\rm{Hom}}_{B}(P(\lambda),\Delta(\mu))\not=0\implies\lambda\leq\mu.$

  * (SS2)

Every irreducible $B$-module is a homomorphic image of some $\Delta(\lambda)$.

  * (SS3)

For $\lambda\in\Lambda$, the $B$-module $P(\lambda)$ has a finite filtration
by $B$-submodules with top section $\Delta(\lambda)$ and other sections of the
form $\Delta(\mu)$ with $\bar{\mu}>\bar{\lambda}$.

When these conditions hold, the data

$\\{\Delta(\lambda),P(\lambda)\\}_{\lambda\in\Lambda}$ (1.3.1)

form (by definition) a stratifying system131313Following [DPS17], we drop the
word “strict” in “strict stratifying system” as used in [DPS98a]. I.e., all
stratifying systems in this monograph are strict in the sense of [DPS98a],
thus, we insist on the strict comparison $\bar{\mu}>\bar{\lambda}$ in (SS3).
for the category $B\mbox{--mod}$ of finite $B$-modules. Note that if (1.3.1)
is a stratifying system for $B\mbox{--mod}$, then
$\\{\Delta(\lambda)_{\mathscr{K}^{\prime}},P(\lambda)_{\mathscr{K}^{\prime}}\\}_{\lambda\in\Lambda}$
is a stratifying system for $B_{\mathscr{K}^{\prime}}$-mod for any base change
$\mathscr{K}\to\mathscr{K}^{\prime}$, provided $\mathscr{K}^{\prime}$ is a
Noetherian commutative ring. (See [DPS15, p. 231]. For more details and
further results, see [DPS98a, Lem. 1.2.5].)

######  Remark 1.3.2.

If $B$ has a stratifying system as explained above, it has a “standard
stratification” by idempotent ideals $0\subseteq J_{0}\subseteq
J_{1}\subseteq\cdots\subseteq J_{n}=B$ much like a defining sequence in the
theory of quasi-hereditary algebras. Thus, $B$ is a kind of generalization of
a quasi-hereditary algebra, or, equivalently, if $B$-mod has a stratifying
system, then $B$-mod is a generalization of a highest weight category. We call
$B$ a standardly stratified algebra; see Section 3.5 for more details. We do
not pursue this further here, but refer to [DPS98a, §1.2] and [DPS18] for a
more complete discussion.

1.3C. Exact categories and the stratification hypothesis. Stratifying systems
can be constructed in an endomorphism-ring setting. Originally, the
construction was based on assuming a “stratification hypothesis” in [DPS98a,
Hyp. 1.2.9, Th. 1.2.10], which required a difficult
$\text{\rm{Ext}}^{1}$-vanishing condition (see [DPS98a, Th. 2.3.9, Th.
2.4.4]). This problem was partly circumvented in [DPS17, Hyp. 2.5 & Th. 2.6]
by introducing “designer” exact categories which had, a priori, smaller
$\text{\rm{Ext}}^{1}$-groups, and thus were more likely to vanish. However,
the new version was somewhat awkward to use in the Quillen axiom system. See
[DPS17, Rem. 2.7]. The updated version below, given in the Stratification
Hypothesis 1.3.3, removes this issue, by replacing all
$\text{\rm{Ext}}^{1}$-vanishings with existence assertions for needed short
exact sequences. As noted in Remark 1.3.4, these assertions are at least
implied by a natural exact category $\text{\rm{Ext}}^{1}$-vanishing condition.
This is the way they are verified in this monograph.

Let $R$ be a finite and projective ${\mathscr{K}}$-algebra. Our aim will be to
provide ingredients in mod-$R$, including a right $R$-module $T$ with suitable
filtrations, which enable the construction of a stratifying system for
$B\mbox{--mod}$, where $B:=\text{\rm{End}}_{R}(T)$. Clearly, $T$ is naturally
a left $B$-module.

Construction. We will construct $T$ as an object in a full subcategory
${\mathscr{A}}$ of mod-$R$, with the following assumptions. First, the
subcategory ${\mathscr{A}}$ is part of an exact category
$({\mathscr{A}},{\mathscr{E}})$. Second, the exact sequences $(X\to Y\to
Z)\in\mathscr{E}$ are among the short exact sequences $0\to X\to Y\to Z\to 0$
in mod-$R$. Thus, $({\mathscr{A}},{\mathscr{E}})$ is a full exact subcategory
of mod-$R$. (In our terminology, “full exact” is just an abbreviation for
“full and exact.”) Third, we have a collection of objects
$S_{\lambda},T_{\lambda}\in{\mathscr{A}}$ indexed by the elements $\lambda$ of
a finite quasi-poset $\Lambda$, where $T_{\lambda}$ is a finite and projective
${\mathscr{K}}$-module.141414This assumption is not often used directly, but
can be a step toward the required projectivity over ${\mathscr{K}}$ of the
modules $\Delta(\lambda)$; see Remark 1.3.6(b). The assumption for $T$ often
comes for free, since the $S_{\nu}$ are typically projective (or even free)
${\mathscr{K}}$-modules and each $T_{\lambda}$ is constructed to have a
filtration with each section a direct sum of various $S_{\nu}$. For each
$\lambda\in\Lambda$, $S_{\lambda}$ is a subobject of $T_{\lambda}$, with the
inclusion $S_{\lambda}\hookrightarrow T_{\lambda}$ an inflation in
${\mathscr{E}}$. For each $\lambda\in\Lambda$, fix a positive integer
$m_{\lambda}$. Define

$T:=\bigoplus_{\lambda\in\Lambda}T_{\lambda}^{\oplus m_{\lambda}}.$ (1.3.2)

With this notation, we can now state the following:

######  Stratification Hypothesis 1.3.3.

Assume the set-up introduced in the previous paragraph. In particular,
$({\mathscr{A}},{\mathscr{E}})$ is an exact category, a full exact subcategory
of mod-$R$, and $T\in{\mathscr{A}}$ with
$B=\text{\rm{End}}_{R}(T)=\text{\rm{End}}_{\mathscr{A}}(T)$. There are also
indexed objects $S_{\lambda},T_{\lambda}\,(\lambda\in\Lambda)$, and the
following statements are required as part of the hypothesis:

  * (1)

For $\lambda\in\Lambda$, there is an increasing filtration of $T_{\lambda}$:

$0=T^{-1}_{\lambda}\subseteq T^{0}_{\lambda}\subseteq\cdots\subseteq
T^{\ell(\lambda)}_{\lambda}=T_{\lambda}$

in which each inclusion $T^{i-1}_{\lambda}\subseteq T_{\lambda}^{i}$ defines a
conflation

$(T^{i-1}_{\lambda}\longrightarrow T_{\lambda}^{i}\longrightarrow
T_{\lambda}^{i}/T_{\lambda}^{i-1})\in{\mathscr{E}}.$ (1.3.3)

In addition, $T^{0}_{\lambda}\cong S_{\lambda}$, and, for $i>0$, the
“section”151515Observe that $T^{i}_{\lambda}\to
T^{i}_{\lambda}/T^{i-1}_{\lambda}$ is the unique (up to isomorphism) cokernel
in ${\mathscr{A}}$ of the inflation $T^{i-1}_{\lambda}\to T^{i}_{\lambda}$
and, thus, is a deflation. We generally leave such exact category exercises to
the reader.  $T^{i}_{\lambda}/T^{i-1}_{\lambda}$ is a direct sum of various
$S_{\mu}$, $\mu\in\Lambda$ and $\bar{\mu}>\bar{\lambda}$ (repetitions
allowed).

  * (2)

For $\lambda,\mu\in\Lambda$,
$\text{\rm{Hom}}_{\mathscr{A}}(S_{\mu},T_{\lambda})\not=0\implies\lambda\leq\mu.$

  * (3)

For all $\lambda\in\Lambda$ and integers $i\geq 0$, applying the contravariant
“diamond” functor161616This functor will also be used later in the book.
$(-)^{\diamond}:=\text{\rm{Hom}}_{R}(-,T)$ to (1.3.3) gives the exact sequence

$0\to(T_{\lambda}^{i}/T_{\lambda}^{i-1})^{\diamond}\longrightarrow(T^{i}_{\lambda})^{\diamond}\longrightarrow(T^{i-1}_{\lambda})^{\diamond}\to
0$

of $B$-modules.

######  Remark 1.3.4.

Notice that condition (3) is implied by the vanishing of
$\text{\rm{Ext}}^{1}_{\mathscr{E}}(S_{\lambda},T)$, for all
$\lambda\in\Lambda$; see [DPS17, Prop. A.2]. This is the way that we will use
the (updated) Stratification Hypothesis 1.3.3 in this monograph. We mention,
without proof, that it is implied by the less flexible earlier [DPS17, Hyp.
2.5], which was modeled closely on [DPS98a, Hyp. 1.2.9] and which is harder to
check, as written, in the Quillen exact category setting.

The following theorem updates [DPS98a, Th. 1.3.3] and [DPS17, Th. 2.6].

######  Theorem 1.3.5.

Assume that the Stratification Hypothesis 1.3.3 holds. For
$\lambda\in\Lambda$, put

$\begin{cases}\Delta(\lambda):=\text{\rm{Hom}}_{\mathscr{A}}(S_{\lambda},T)=\text{\rm{Hom}}_{R}(S_{\lambda},T)\in
B\mbox{--mod},\\\
P(\lambda):=\text{\rm{Hom}}_{\mathscr{A}}(T_{\lambda},T)=\text{\rm{Hom}}_{R}(T_{\lambda},T)\in
B\mbox{--mod}.\end{cases}$ (1.3.4)

If each $\Delta(\lambda)$ is projective over ${\mathscr{K}}$, then
$\\{\Delta(\lambda),P(\lambda)\\}_{\lambda\in\Lambda}$ is a stratifying system
for $B\mbox{--mod}$.

###### Proof.

First, we check (SS1). Observe that the contravariant functor $(-)^{\diamond}$
provides maps

$\text{\rm{Hom}}_{R}(S_{\mu},T_{\lambda})\longrightarrow\text{\rm{Hom}}_{B}(P(\lambda),\Delta(\mu)).$

Passing to $T=\bigoplus_{{\lambda}\in\Lambda}T_{\lambda}^{\oplus m_{\lambda}}$
on the left and ${}_{B}B=\bigoplus_{\lambda\in\Lambda}P(\lambda)^{\oplus
m_{\lambda}}$ on the right, we obtain an isomorphism

$\text{\rm{Hom}}_{R}(S_{\mu},T)\overset{\sim}{\longrightarrow}\text{\rm{Hom}}_{B}({{}_{B}B},\Delta(\mu))\cong\Delta(\mu).$

Consequently, the original maps must also be isomorphisms. Thus, (SS1) follows
from Condition (2) (of Hypothesis 1.3.3).

Next, we will prove (SS2) assuming (SS3) has been proved. By the construction
of $B=\text{\rm{End}}_{R}(T)$, we have that
${}_{B}B\cong\bigoplus_{\lambda\in\Lambda}P(\lambda)^{\oplus m_{\lambda}}$.
Using (SS3), ${}_{B}B$ has a filtration with sections various $\Delta(\nu)$,
$\nu\in\Lambda$. Any irreducible $B$-module is, of course, a homomorphic image
of ${}_{B}B$, and so (using irreducibility) a homomorphic image of some
$\Delta(\nu)$. This proves (SS2) in the presence of (SS3).

Now consider (SS3) itself for a given $\lambda$. Let $n$ be an integer such
that $T^{n}_{\lambda}=T_{\lambda}$. Let $j\geq i\geq-1$ be integers. Thus,
$T^{i}_{\lambda}$ is a submodule of $T^{j}_{\lambda}$.

Claim. The natural map
$(T^{j}_{\lambda})^{\diamond}\to(T^{i}_{\lambda})^{\diamond}$ is a surjection.

To see this, observe that the inclusion $T^{i}_{\lambda}\hookrightarrow
T^{j}_{\lambda}$ factors into a composition of inclusions
$T^{i}_{\lambda}\hookrightarrow T^{i+1}_{\lambda}\hookrightarrow
T^{i+2}_{\lambda}\hookrightarrow\cdots\hookrightarrow T^{j}_{\lambda}$. By
Condition (3), for $i\leq a<j$, the inclusion $T^{a}_{\lambda}\hookrightarrow
T^{a+1}_{\lambda}$ becomes a surjection
$(T^{a+1}_{\lambda})^{\diamond}\twoheadrightarrow(T^{a}_{\lambda})^{\diamond}$
upon applying $(-)^{\diamond}$. The Claim follows by taking compositions.

We return to the proof of (SS3) for a given $\lambda$. Let $n\geq j\geq 0$.
The maps $(T^{n}_{\lambda})^{\diamond}\to(T^{j}_{\lambda})^{\diamond}$ and
$(T^{n}_{\lambda})^{\diamond}\to(T^{j-1}_{\lambda})^{\diamond}$ are both
surjective by the Claim. This gives the two exact sequences forming the rows
of the natural commutative diagram

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>(T^{n}_{\lambda}/T^{j}_{\lambda})^{\diamond}@>{}>{}>(T^{n}_{\lambda})^{\diamond}@>{}>{}>(T^{j}_{\lambda})^{\diamond}@>{}>{}>0\\\
@V{\alpha}V{}V\Big{\|}@V{\beta}V{}V\\\
0@>{}>{}>(T^{n}_{\lambda}/T^{j-1}_{\lambda})^{\diamond}@>{}>{}>(T^{n}_{\lambda})^{\diamond}@>{}>{}>(T^{j-1}_{\lambda})^{\diamond}@>{}>{}>0\\\
\end{CD}$ (1.3.5)

Take a sufficiently large integer $n$, so that $T^{n}_{\lambda}=T_{\lambda}$.
Then, $P(\lambda)=T^{\diamond}_{\lambda}$ has a filtration by submodules of
the form $(T_{\lambda}/T^{j}_{\lambda})^{\diamond}$, and the map $\alpha$
above is an inclusion between adjacent filtration terms. The Snake Lemma shows
that the section ${\text{\rm Coker}}(\alpha)$ of these two filtration terms is
isomorphic to ${\text{\rm
Ker}}(\beta)=(T^{j}_{\lambda}/T^{j-1}_{\lambda})^{\diamond}$. Thus,
$P(\lambda)$ has a filtration with sections isomorphic to various
$(T^{j}_{\lambda}/T^{j-1}_{\lambda})^{\diamond}$. The module
$(T^{j}_{\lambda}/T^{j-1}_{\lambda})^{\diamond}$ is of the form
$S_{\nu}^{\diamond}=\text{\rm{Hom}}_{R}(S_{\nu},T)=\Delta(\nu)$, where
$\nu>\lambda$ if $j>0$, and $\nu=\lambda$ if $j=0$. In the $j=0$ case,
${\text{\rm Coker}}(\alpha)$ is the “top” section of the given filtration on
$P(\lambda)$, and it is isomorphic to ${\text{\rm
Ker}}(\beta)=(T^{0}_{\lambda})^{\diamond}=S_{\lambda}^{\diamond}=\Delta({\lambda})$.
This completes the proof of (SS3).

Finally, because it is assumed that each $\Delta(\nu)$, $\nu\in\Lambda$, is a
projective ${\mathscr{K}}$-module, it follows that each $P(\lambda)$,
$\lambda\in\Lambda$, is also a projective ${\mathscr{K}}$-module. Since $B$ is
a direct sum of the $P(\lambda)$ (repetitions allowed), $B$ is a projective
${\mathscr{K}}$-module. This completes the proof. ∎

######  Remarks 1.3.6.

(a) Correspondingly, ${\mathscr{A}}$ can be one of the categories
${\mathscr{A}}({\mathscr{L}}^{*})$ in Section 2.3 below (or a full
subcategory, such as ${\mathscr{A}}_{\flat}({\mathscr{L}}^{*})$ there.) A
similar remark holds for the corresponding exact category in Section 2.2;
however, the notation above is chosen to be compatible with the applications
in Section 2.4 as developed in Section 2.3.

(b) If ${\mathscr{K}}$ is a regular ring of Krull dimension at most 2, then,
in the statement of Theorem 1.3.5, the sentence “Assume that each
$\Delta(\lambda)$ is projective over ${\mathscr{K}}$” can be omitted, since
$\Delta(\lambda)$ is automatically projective over ${\mathcal{Z}}$. This is
shown in the proof of [DPS98a, Cor. 1.2.12)], using earlier work of Auslander-
Goldman together with the projectivity of $T$ as a ${\mathscr{K}}$-module
(assumed in the preamble to the Stratification Hypothesis 1.3.3). For a
detailed account, see [DDPW08, §C.3]. A similar result holds if the
stratifying system is “defined over $Z$” where $Z$ is a regular ring of Krull
dimension $\leq 2$. This means that the stratifying system over
${\mathscr{K}}$ is obtained by base change from the regular ring $Z$.

(c) As an example in (b), we will be mainly interested in the special case in
which $R={\mathcal{H}}$ is the Hecke algebra defined over
${\mathcal{Z}}=\mathbb{Z}[t,t^{-1}]$ (and its localizations
${\mathcal{Z}}^{\natural}$ at multiplicative subsets, not containing 0, of
$\mathbb{Z}$). Since ${\mathcal{Z}}$ is a regular ring of dimension 2, (b)
above applies to show that each $\Delta(\lambda)$ is projective over
${\mathcal{Z}}$. In fact, by a result of Swan [Sw78, p.111, first paragraph],
$\Delta(\lambda)$ is then ${\mathcal{Z}}$-free. (Swan’s paragraph also implies
that any finite, projective ${\mathcal{Z}}^{\natural}$-module is free. In
addition, if ${\mathcal{Z}}^{\natural\oplus
m}\cong{\mathcal{Z}}^{\natural\oplus n}$ for integers $m,n$, then $m=n$.)

##  Chapter 2 Some exact categories

Section 1.3 in Chapter 1 provides a general theory of stratifying endomorphism
algebras using exact categories, reformulating that developed in [DPS17]. To
apply the theory in the context of the conjecture [DPS98a, Conj. 2.5.2] by the
authors, we need to construct a nested sequence of exact categories involving
Hecke algebra modules with cell module filtrations. We eventually prove the
Ext1 vanishing condition (see Remark 1.3.4) in one of the exact categories
and, then, prove the (slightly modified) conjecture in the next chapter.

###  2.1. Three exact categories

In this section, we work exclusively with left modules for an algebra. Let
${\mathcal{H}}$ be the generic Hecke algebra over ${\mathcal{Z}}$, associated
to a standard finite Coxeter system $(W,S,L)$ (as defined in Section 1.1).

We begin with the following variation on [DPS15, Lem. 4.3]. If $V$ is a left
${\mathcal{H}}$-module, the “fixed-point module” for the action of
${\mathcal{H}}$ on $V$ is defined to be

$V^{\mathcal{H}}:=\\{v\in
V\,|\,{\textsc{c}}_{s}v=(t_{s}+t_{s}^{-1})v,\quad\forall s\in
S\\}\in{\mathcal{Z}}{\text{\rm--mod}},$

where $t_{s},{\textsc{c}}_{s}$ are defined in (1.2.1) and (1.2.3),
respectively. Equivalently,

$v\in V^{\mathcal{H}}\iff T_{s}v=t_{s}^{2}v,\text{ for all }s\in S.$

The assignment $V\mapsto V^{\mathcal{H}}$ is part of an evident left-exact
endo-functor. Similarly, $V^{{\mathcal{H}}_{J}}$ is defined for any parabolic
subalgebra ${\mathcal{H}}_{J}$ of ${\mathcal{H}}$ ($J\subseteq S$). The next
result follows from [DPS15, Lem.4.3] and the above definition of fixed points,
using (1.2.1) and (1.2.3).

######  Lemma 2.1.1.

Let ${\mathfrak{I}\subseteq\mathfrak{K}}$ be left ideals in the Hecke algebra
${\mathcal{H}}$ over ${\mathcal{Z}}$. Assume that both $\mathfrak{I}$ and
$\mathfrak{K}$ are spanned by the Kazhdan-Lusztig basis elements
${\textsc{c}}_{x}$, $x\in W$, that they contain. Then, for any $J\subseteq S$,
the natural map

${\mathfrak{K}}^{{\mathcal{H}}_{J}}\longrightarrow({\mathfrak{K}}/{\mathfrak{J}})^{{\mathcal{H}}_{J}}$
(2.1.1)

is surjective.

######  Notation 2.1.2.

At this point, the reader should review the notion of height functions
$\mathfrak{h}$ discussed in Appendix B below. In this section, we fix

$\mathfrak{h}=\mathfrak{h}_{l}:\Omega\to\mathbb{Z},$

a function defined on the set $\Omega$ of left cells in $W,$ and compatible
with the preorder $\leq_{LR}$; see display (B.2) in Appendix B for the meaning
of “compatible.” For another example, see the order defined in display
(3.1.1). The subscript $l$ is there to remind the reader of our left-module
context. Such a height function $\mathfrak{h}$ is also compatible with
$\leq_{L}$, as formalized in Corollary B.8 (which uses property (1.2.11) via
Proposition B.5). In particular, $\mathfrak{h}$ is constant on left cells
which belong to the same two-sided cell. It is also observed in Remark B.9
that $\mathfrak{h}$ may be viewed as assigning heights to left-cell modules
$S(\omega)$. Thus, if $\omega\in\Omega$, $\mathfrak{h}$ assigns the left-cell
module $S(\omega)$ height $\mathfrak{h}(\omega)$. Also, the value
$\mathfrak{h}(\omega)=\mathfrak{h}(\bar{\omega})$, where $\bar{\omega}$ is the
two-sided cell containing $\omega$, may be assigned without ambiguity as a
“height” to the irreducible constituents of the base-changed module
$S(\omega)_{\mathbb{Q}(t)}$. In particular, this means that $\mathfrak{h}$
uniquely determines a height function on the set $\Lambda$ of irreducible
modules for the semisimple algebra ${\mathcal{H}}_{\mathbb{Q}(t)}$. Since this
height function also determines the original
$\mathfrak{h}:\Lambda\to\mathbb{Z}$ (see Remark B.9), we continue to denote it
by $\mathfrak{h}$. As explained in Example B.1, $\mathfrak{h}$ induces a
canonical decreasing filtration

$M=M_{0}\supseteq M_{1}\supseteq M_{2}\supseteq\cdots\supseteq
M_{n-1}\supseteq M_{n}=0$ (2.1.2)

on each ${\mathcal{H}}$-module $M$. We call this filtration the
$\mathfrak{h}$-filtration or the height filtration, when $\mathfrak{h}$ is
understood on $M$. Similar terminology can be used for right modules in the
next three sections, although the height filtrations are increasing and the
subscripts here become superscripts there.

Finally, as a convenience, we assume in this section (unless noted otherwise)
that $\mathfrak{h}=\mathfrak{h}_{l}:\Omega\to\mathbb{Z}$ only takes negative
values. See Appendix B. Later, we make use of
$\mathfrak{h}_{r}:=-\mathfrak{h}_{l}:\Omega\to\mathbb{Z}$, which is, in
particular, used to assign positive height values to dual left-cell modules
$S_{\omega}=S(\omega)^{*}=\text{\rm{Hom}}_{\mathcal{Z}}(S(\omega),{\mathcal{Z}}),\;\;\omega\in\Omega$
(see (1.2.10)) which are naturally right ${\mathcal{H}}$-modules. See Remark
B.8.

Relative to such a height function $\mathfrak{h}:\Omega\to\mathbb{Z}$ that
induces $\mathfrak{h}:\Lambda\to\mathbb{Z}$ as above, there is an exact
category $({\mathscr{A}},{\mathscr{E}})$, as constructed in Example B.1. Thus,
$({\mathscr{A}},{\mathscr{E}})$ consists of the full subcategory
${\mathscr{A}}={\mathscr{A}}_{l}$ of ${\mathcal{H}}$-mod whose objects are
left ${\mathcal{H}}$-modules, finite and torsion-free over ${\mathcal{Z}}$,
and the (full) “subcategory” ${\mathscr{E}}={\mathscr{E}}_{l}$ (in the sense
of footnote 11 in Chapter 1) of short exact sequences $0\to M^{\prime}\to M\to
M^{\prime\prime}\to 0$ in ${\mathscr{A}}$ satisfying the condition that $0\to
M^{\prime}_{h}\to M_{h}\to M^{\prime\prime}_{h}\to 0$ is exact, for all
heights $h$. (Compare the right-module version given in [DPS17, Constr. 3.5]
or Theorem 2.2.1 below.)

Below, building on
$({\mathscr{A}},{\mathscr{E}})=({\mathscr{A}}_{l},{\mathscr{E}}_{l})$, we
present three more examples of exact categories:

${\mathcal{H}}\text{-mod}\geq({\mathscr{A}},{\mathscr{E}})\geq({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}({\mathscr{L}}))\geq({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))\geq({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}})),$

which will be important in the sequel. Here “$\geq$” indicates “a full
subcategory”. In particular,
$\text{\rm{Hom}}_{\mathscr{C}}(M,N)=\text{\rm{Hom}}_{\mathcal{H}}(M,N)$ for
objects $M,N$ in $\mathscr{C}$ and every
$\mathscr{C}={\mathscr{A}},{\mathscr{A}}({\mathscr{L}})$, etc.

2.1A. The exact category
$\boldsymbol{({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}({\mathscr{L}}))}$.
Its construction involves the category ${\mathcal{H}}$-mod of (finite) left
${\mathcal{H}}$-modules, but it closely follows the more abstract discussion
in [DPS17, Assump. 3.2 and Constr. 3.8] which worked with the category mod-$H$
of right modules for a suitable algebra $H$ more general than the generic
Hecke algebra ${\mathcal{H}}$.

For any integer $h$, let ${\mathscr{L}}_{h}$ be the full subcategory of
${\mathscr{A}}$ consisting of modules $N$ which are a finite direct sum of
modules isomorphic to left-cell modules $S(\omega)$ of height $h$. Let
${\mathscr{L}}$ be the set-theoretic union of the ${\mathscr{L}}_{h}$. Next,
let ${\mathscr{A}}({\mathscr{L}})$ be the full additive subcategory of
${\mathcal{H}}$-mod with objects $M$ (necessarily in ${\mathscr{A}}_{l}$)
whose $\mathfrak{h}$-filtration (2.1.2) has the property that, for any integer
$h$, $M_{h-1}/M_{h}$ is an object in ${\mathscr{L}}_{-h}$.111The minus sign is
chosen to be compatible with the duality theory developed later in Sections
2.2 and 2.3. In particular, we will consider dual left-cell modules
$S_{\omega}$, for each $\omega\in\Omega$, and assign to each $S_{\omega}$ a
positive height, negative to that of $S(\omega)$. See the beginning paragraphs
of Section 2.3 and also (2.2.2). Note that, with Notation 2.1.2, this height
filtration coincides with the one defined in Example B.1. We will also
informally call such an $\mathfrak{h}$-filtration a left-cell
filtration.222Clearly, every $\mathfrak{h}$-filtration here may be refined to
a filtration with sections isomorphic to left-cell modules. This refined
filtration is a left-cell filtration in the sense of [DPS98a, §2.2].

By analogy with [DPS17, §3], let ${\mathscr{E}}({\mathscr{L}})$ consist of
those sequences $(A\to B\to C)$ of objects in ${\mathscr{A}}({\mathscr{L}})$
such that, for every $h$,

$0\longrightarrow A_{h-1}/A_{h}\longrightarrow B_{h-1}/B_{h}\longrightarrow
C_{h-1}/C_{h}\longrightarrow 0$

is a split short exact sequence in the additive category ${\mathscr{L}}_{-h}$
(or, equivalently, in the category ${\mathcal{H}}$–mod). Then,

$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}({\mathscr{L}}))$ is an exact
category. (2.1.3)

This was proved in the more abstract setting of right $H$-modules in [DPS17,
Th. 3.9] for an analogous category
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$ associated to
$H$. The proof in the present situation is entirely similar. The result can
also be deduced from the special case $H={\mathcal{H}}^{\text{op}}$ of the
cited theorem.

Note that left-cell modules are, by definition, finite and free over
${\mathcal{Z}}$. This implies that any object in
${\mathscr{A}}({\mathscr{L}})$ is also finite and free over ${\mathcal{Z}}$.
Looking ahead to Section 2.2, this implies that
${\mathscr{A}}({\mathscr{L}})\subseteq{\mathscr{A}}^{\prime}_{l}$, the left-
hand version of the category ${\mathscr{A}}^{\prime}$ introduced in Theorem
2.2.2. Thus, ${\mathscr{A}}^{\prime}_{l}$ denotes the full additive
subcategory of objects $M\in{\mathscr{A}}_{l}$ for which $M_{h}/M_{h+1}$ is
projective as a ${\mathscr{K}}$-module, for each $h\in\mathbb{Z}$, and
${\mathscr{E}}^{\prime}_{l}={\mathscr{E}}_{l}|_{{\mathscr{A}}_{l}^{\prime}}$.
Also, ${\mathscr{E}}({\mathscr{L}})\subseteq{\mathscr{E}}^{\prime}_{l}$, so
that $({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}({\mathscr{L}}))$ is a (full)
exact subcategory of
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$; see the definition
at the beginning of Subsection 1.3A.

Note also that, for each object $M$ in ${\mathscr{A}}({\mathscr{L}})$ with the
$\mathfrak{h}$-filtration given in display (2.1.2), each inclusion
$M_{h}\subseteq M_{h-1}$ gives a conflation $(M_{h}\to M_{h-1}\to
M_{h-1}/M_{h})\in{\mathscr{E}}({\mathscr{L}})$, for all $1\leq h\leq n$, so
that its ${\mathcal{Z}}$-dual $M^{*}$ satisfies the condition (1) in the
Stratification Hypothesis 1.3.3.

2.1B. The exact category
$\boldsymbol{({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))}$.
With the notation above, let ${\mathscr{E}}^{\flat}({\mathscr{L}})$ denote the
full subcategory of ${\mathscr{E}}({\mathscr{L}})$ with objects $(M\to N\to
P)\in{\mathscr{E}}({\mathscr{L}})$ such that, for any integer $h$ and any
subset $J$ of $S$, the sequence

$0\longrightarrow(M_{h})^{{\mathcal{H}}_{J}}\longrightarrow(N_{h})^{{\mathcal{H}}_{J}}\longrightarrow(P_{h})^{{\mathcal{H}}_{J}}\longrightarrow
0$ (2.1.4)

of fixed points is exact. Note that, for any $h$, each $0\to M_{h}/M_{h-1}\to
N_{h}/N_{h-1}\to P_{h}/P_{h-1}\to 0$ is split in ${\mathcal{H}}$-mod, since
$(M\to N\to P)\in{\mathscr{E}}({\mathscr{L}})$, but this does not mean that
the short exact sequence $0\to M_{h}\to N_{h}\to P_{h}\to 0$ is split. In
particular, the exactness of (2.1.4) is a nontrivial property.

We have:

######  Proposition 2.1.3.

$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$ is an
exact category.

###### Proof.

Let $\mathscr{C}$ be the abelian category of all finite
${\mathcal{Z}}$-modules. Put

$F(-):=\bigoplus_{J,h}((-)_{h})^{{\mathcal{H}}_{J}}:{\mathscr{A}}({\mathscr{L}})\to{\mathscr{C}}.$

Then, $F$ is an ${\mathscr{E}}({\mathscr{L}})$-left exact functor. Now we
apply [DPS17, Lem. 3.1], with ${\mathscr{E}}$ there taken to be
${\mathscr{E}}^{\flat}({\mathscr{L}})$ as defined above. ∎

For $J{\subseteq}S$, if $W_{J}$ is the parabolic subgroup of $W$ defined by
$J$, then ${\mathcal{H}}x_{J}$ is the usual (left) $q$-permutation module
determined by any $J\subseteq S$, where

$x_{J}:=\sum_{w\in W_{J}}T_{w}$ (2.1.5)

was defined in display (1.2.4). For future use in Section 5.2, we also define

$y_{J}:=\sum_{w\in W_{J}}\text{{\rm sgn}}(w)(t^{2})^{-L(w)}T_{w},$ (2.1.6)

where $\text{{\rm sgn}}(w):=(-1)^{\ell(w)}$ is the sign of $w$. See [DPS98c,
pp. 324–326], taking $q=t^{2}$ in our current notation.

######  Proposition 2.1.4.

The ${\mathcal{H}}$-module ${\mathcal{H}}x_{J}$ belongs to
${\mathscr{A}}({\mathscr{L}})$.

###### Proof.

Let $M$ denote ${\mathcal{H}}x_{J}$. Using display (1.2.8), we find $M$ has a
basis of elements ${\textsc{c}}_{v}$, with $v\in W$, in a certain quasi-poset
ideal $V$ for $\leq_{L}$, necessarily closed under $\sim_{L}$. In fact, we
have $V=\\{w\in W\mid ws<w,\forall s\in J\\}=\\{w\in W\mid
w\leq_{L}w_{0,J}\\}$; see [DPS98a, (2.3.3)] or Lemma 3.1.2 in this monograph.
The quasi-poset ideal $V$ is, thus, a union of left cells. For any integer
$h$, let $V_{h-1}$ be the set of all $v\in V$ belonging to a left cell
$\omega$ with $\mathfrak{h}(\omega)\leq-h$. Then, each $V_{h-1}$ is, by
construction, a union of left cells, and is a quasi-poset ideal, since
$\mathfrak{h}$ is compatible with $\leq_{L}$ on $\Omega$. (This was observed
in Notation 2.1.2 as a consequence of (1.2.11).) The ${\mathcal{Z}}$-span
$M_{h-1}$ of $V_{h-1}$ is, thus, a left ${\mathcal{H}}$-module, as follows
from the definition of $\leq_{L}$ [Lus03, p. 43]. Similar considerations
identify $M_{h-1}/M_{h}$ as a direct sum of left-cell modules of height $-h$,
one summand for each left cell in $V$ of height $-h$. Thus,
$M\in{\mathscr{A}}({\mathscr{L}})$, proving the assertion. ∎

We remark that improvements of Proposition 2.1.4 occur in several places in
this monograph. For example, Corollary 2.1.8 gives the sharper result that,
for $J\subseteq S$,
${\mathcal{H}}x_{J}\in{\mathscr{A}}^{\flat}({\mathscr{L}})$. See Theorem 3.1.3
for a right-module version involving the exact category
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$.

######  Corollary 2.1.5.

Let $M={\mathcal{H}}x_{J}$, and consider the decreasing filtration

$M=M_{0}\supseteq M_{1}\supseteq\cdots\supseteq M_{l_{J}}=0$

defined by the height function $\mathfrak{h}$. For any $h$, the short exact
sequence $0\to M_{h}\to M_{h-1}\to M_{h-1}/M_{h}\to 0$ belongs of
${\mathscr{E}}^{\flat}({\mathscr{L}})$.

###### Proof.

By the discussion at the end of the proof of Proposition 2.1.4, each inclusion
$M_{h}\subseteq M_{h-1}$ is an inflation. Thus, each $(M_{h}\to M_{h-1}\to
M_{h-1}/M_{h})\in{\mathscr{E}}({\mathscr{L}})$. By Lemma 2.1.1, the sequence
$0\to(M_{h})^{{\mathcal{H}}_{J}}\to(M_{h-1})^{{\mathcal{H}}_{J}}\to(M_{h-1}/M_{h})^{{\mathcal{H}}_{J}}\to
0$ is exact for all $J\subseteq S$. Hence, by definition (2.1.4), the short
exact sequence $0\to M_{h}\to M_{h-1}\to M_{h-1}/M_{h}\to 0$ belongs of
${\mathscr{E}}^{\flat}({\mathscr{L}})$. ∎

In the following theorem, the
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}^{\flat}({\mathscr{L}})}$-group is
calculated in the exact category
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$.
Subsection 2.1C below introduces a second exact category
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$,
possibly setting up a conflict of notation. However, as pointed out in
Proposition 2.1.7 (or the proof of its corollary), the
$\text{\rm{Ext}}^{1}_{{\mathscr{E}}^{\flat}({\mathscr{L}})}(M,N)$-groups can
be calculated in either exact category, as long as $M,N$ both belong to
${\mathscr{A}}^{\flat}({\mathscr{L}})$. We also remark that the condition
(2.1.1) plays an essential role in the proof of the theorem.

######  Theorem 2.1.6.

For any $J\subseteq S$,
$\text{\rm{Ext}}_{{\mathscr{E}}^{\flat}({\mathscr{L}})}^{1}({\mathcal{H}}x_{J},S(\omega))=0$,
for all $\omega\in\Omega$.

###### Proof.

Consider an exact sequence

$0\longrightarrow S(\omega)\longrightarrow E\
\overset{g}{\longrightarrow}{\mathcal{H}}x_{J}\longrightarrow 0$ (2.1.7)

in ${\mathscr{E}}^{\flat}({\mathscr{L}})$. We want to prove that the sequence
is split. By definition, we obtain an exact sequence

$0\longrightarrow S(\omega)^{{\mathcal{H}}_{J}}\longrightarrow
E^{{\mathcal{H}}_{J}}\overset{g_{J}}{\longrightarrow}({\mathcal{H}}x_{J})^{{\mathcal{H}}_{J}}\longrightarrow
0.$ (2.1.8)

Consider the ${\mathcal{H}}_{J}$-submodule ${\mathcal{Z}}x_{J}$ of
$({\mathcal{H}}x_{J})^{{\mathcal{H}}_{J}}$. The exactness of (2.1.8) implies
that there exists $e\in E^{{\mathcal{H}}_{J}}$ such that $g_{J}(e)=x_{J}$.
Thus, ${\mathcal{Z}}x_{J}\cong{\mathcal{Z}}e$ as ${\mathcal{H}}_{J}$-modules.
Since ${\mathcal{Z}}e\subseteq E$, this induces an ${\mathcal{H}}_{J}$-module
homomorphism $f_{J}:{\mathcal{Z}}x_{J}\to E$. By Frobenius reciprocity, there
is an ${\mathcal{H}}$-module homomorphism $f:{\mathcal{H}}x_{J}\to E$ such
that $f(x_{J})=e$. Hence, $gf(x_{J})=x_{J}$, and, consequently, $gf$ is the
identity map on ${\mathcal{H}}x_{J}$. In other words, the sequence (2.1.7) is
split. ∎

Again, we note that
${\mathscr{E}}^{\flat}({\mathscr{L}})\subseteq{\mathscr{E}}({\mathscr{L}})\subseteq{\mathscr{E}}^{\prime}_{l}$,
the latter defined in Section 2.2 (see Theorem 2.2.2). Thus,
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$ is a
full exact subcategory of
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$.

Note that, in the exact category
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$, not
every object satisfies the property in Corollary 2.1.5. This motivates the
definition of the following full subcategory.

2.1C. The exact category
$\boldsymbol{({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))}$.
Let ${\mathscr{A}}^{\flat}({\mathscr{L}})$ be the full subcategory of
${\mathscr{A}}({\mathscr{L}})$ with objects $M$ having the property that, for
any integer $h$ and any $J\subseteq S$, the map

$(M_{h-1})^{{\mathcal{H}}_{J}}\longrightarrow(M_{h-1}/M_{h})^{{\mathcal{H}}_{J}}$
(2.1.9)

is surjective. In other words, each conflation $(M_{h}\to M_{h-1}\to
M_{h-1}/M_{h})$ in ${\mathscr{E}}({\mathscr{L}})$ gives rise, for each
$J\subseteq S$, to a short exact seqeunce
$0\to(M_{h})^{{\mathcal{H}}_{J}}\to(M_{h-1})^{{\mathcal{H}}_{J}}\to(M_{h-1}/M_{h})^{{\mathcal{H}}_{J}}\to
0$. Hence, $(M_{h}\to M_{h-1}\to
M_{h-1}/M_{h})\in{\mathscr{E}}^{\flat}({\mathscr{L}})$.

######  Proposition 2.1.7.

Let $M,P\in{\mathscr{A}}^{\flat}({\mathscr{L}})$, and assume a given $(M\to
N\to P)\in{\mathscr{E}}({\mathscr{L}})$. Then,
$N\in{\mathscr{A}}^{\flat}({\mathscr{L}})$ if and only if $(M\to N\to
P)\in{\mathscr{E}}^{\flat}({\mathscr{L}})$. In particular, the full
subcategory ${\mathscr{A}}^{\flat}({\mathscr{L}})$ of
${\mathscr{A}}({\mathscr{L}})$ is closed under extensions in
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$.

###### Proof.

First, assume that $(M\to N\to P)\in{\mathscr{E}}^{\flat}({\mathscr{L}})$. For
an integer $h$ and a subset $J\subseteq S$, form the commutative diagram

$\begin{CD}(M_{h-1}/M_{h})^{{\mathcal{H}}_{J}}@>{}>{}>(N_{h-1}/N_{h})^{{\mathcal{H}}_{J}}@>{}>{}>(P_{h-1}/P_{h})^{{\mathcal{H}}_{J}}\\\
@A{}A{}A@A{}A{}A@A{}A{}A\\\
(M_{h-1})^{{\mathcal{H}}_{J}}@>{}>{}>(N_{h-1})^{{\mathcal{H}}_{J}}@>{}>{}>(P_{h-1})^{{\mathcal{H}}_{J}}\\\
@A{}A{}A@A{}A{}A@A{}A{}A\\\
(M_{h})^{{\mathcal{H}}_{J}}@>{}>{}>(N_{h})^{{\mathcal{H}}_{J}}@>{}>{}>(P_{h})^{{\mathcal{H}}_{J}.}\end{CD}$
(2.1.10)

Since $M,P\in{\mathscr{A}}^{\flat}({\mathscr{L}})$, columns 1 and 3 are short
exact sequences. Since $(M\to N\to P)\in{\mathscr{E}}^{\flat}({\mathscr{L}})$,
the three rows are all short exact sequences as well. Row 1 is already, before
taking fixed points, a split short exact sequence of ${\mathcal{H}}$-modules,
because $(M\to N\to P)\in{\mathscr{E}}({\mathscr{L}})$. Thus, it remains exact
upon taking ${\mathcal{H}}_{J}$-fixed points. Hence, column 2 is also exact,
using [Mac94, Ex. 2, p. 51], so that
$N\in{\mathscr{A}}^{\flat}({\mathscr{L}})$, as required.

Conversely, if $N\in{\mathscr{A}}^{\flat}({\mathscr{L}})$, then all columns in
(2.1.10) are short exact sequences. Row 1 is exact as in the previous
paragraph. By induction on $h$, the bottom row is also a short exact sequence.
By [Mac94, Ex. 2, p. 51] again, row 2 is exact, proving that $(M\to N\to
P)\in{\mathscr{E}}^{\flat}({\mathscr{L}})$, as required. ∎

If $\mathscr{F}$ is a category of composible morphism pairs from an additive
category $\mathscr{B}$, and $\mathscr{C}$ is an additive subcategory of
$\mathscr{B}$, let $\mathscr{F}|_{\mathscr{C}}$ denote the category of pairs
from $\mathscr{F}$ whose objects belong to $\mathscr{C}$. In the corollary
below and elsewhere, we abuse notation by writing
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$
for
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}})|_{\mathscr{A}^{\flat}(\mathscr{L})})$.

######  Corollary 2.1.8.

With the above notation,
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$
is an exact category, and is a full exact subcategory of
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$.
Moreover, for every $J\subseteq S$,
${\mathcal{H}}x_{J}\in{\mathscr{A}}^{\flat}({\mathscr{L}})$. (In particular,
Theorem 2.1.6 holds in the full exact subcategory
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$.)

###### Proof.

Observe that the category ${\mathscr{A}}^{\flat}({\mathscr{L}})$ is closed
under extensions in the exact category
$({\mathscr{A}}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$ by
Proposition 2.1.7. So
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$
is itself exact by [Bu10, Lem. 10.20]. (The proof in [Bu10] is largely left as
an exercise, in part using the diagram of [Bu10, Lem. 3.5].) The assertion of
the corollary regarding ${\mathcal{H}}x_{J}$ follows from Lemma 2.1.1.
Alternately, it follows from a downward induction on $h$ by applying
Proposition 2.1.7 to Corollary 2.1.5. ∎

######  Lemma 2.1.9.

Let $M\in{\mathscr{A}}^{\flat}({\mathscr{L}})$, $h<a$ be integers, and
$J\subseteq S$. Then, $M_{h}$ and $M_{h}/M_{a}$ belong to
${\mathscr{A}}^{\flat}({\mathscr{L}})$, and the natural map

$(M_{h})^{{\mathcal{H}}_{J}}\longrightarrow(M_{h}/M_{a})^{{\mathcal{H}}_{J}}$
(2.1.11)

is surjective. The same result is true if $M_{h}$ is replaced by $M$.

###### Proof.

Obviously, $M_{h}\in{\mathscr{A}}^{\flat}({\mathscr{L}})$ by definition. Once
the surjectivity of (2.1.11) is proven, it follows that
$M_{h}/M_{a}\in{\mathscr{A}}^{\flat}({\mathscr{L}})$, by writing the map
$M_{h}\longrightarrow M_{h}/M_{h+1}$ as the composite $M_{h}\to M_{h}/M_{a}\to
M_{h}/M_{h+1}$ and taking ${\mathcal{H}}_{J}$-fixed points. It remains to
prove the surjectivity of (2.1.11). Of course, it holds for $a=h+1$ by the
definition of ${\mathscr{A}}^{\flat}({\mathscr{L}})$. By induction on $a-h$,
it holds if $h$ is replaced by $h+1$. The required surjectivity follows from,
say, a diagram chase in ${\mathcal{Z}}$-mod, using the commutative diagram

$\begin{CD}M^{{\mathcal{H}}_{J}}_{h+1}@>{}>{}>M_{h}^{{\mathcal{H}}_{J}}@>{}>{}>(M_{h}/M_{h+1})^{{\mathcal{H}}_{J}}\\\
@V{f}V{}V@V{g}V{}V\Big{\|}\\\
(M_{h+1}/M_{a})^{{\mathcal{H}}_{J}}@>{}>{}>(M_{h}/M_{a})^{{\mathcal{H}}_{J}}@>{}>{}>(M_{h}/M_{h+1})^{{\mathcal{H}}_{J}},\end{CD}$

whose rows are short exact sequences. ∎

######  Corollary 2.1.10.

  * (a)

If $M\in{\mathscr{A}}^{\flat}({\mathscr{L}})$ has the
$\mathfrak{h}$-filtration (2.1.2), and $a\in\mathbb{Z}$ with $0\leq a\leq n$,
then

$(M_{a}\longrightarrow M\longrightarrow
M/M_{a})\in{\mathscr{E}}^{\flat}({\mathscr{L}}).$

In particular, $(M_{h}\to M_{h-1}\to
M_{h-1}/M_{h})\in{\mathscr{E}}^{\flat}({\mathscr{L}})$, for all $1\leq h\leq
n$.

  * (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 ${\mathscr{E}}^{\flat}({\mathscr{L}})$.

###### Proof.

(a) follows from Lemma 2.1.9, using $M$ for $M_{h}$.

(b) Clearly, $(M_{h}\to N_{h}\to
P_{h})\in{\mathscr{E}}^{\flat}({\mathscr{L}})$. The corresponding statement
for the quotients follows from the $3\times 3$-lemma for exact categories
[Bu10, Cor. 3.6], using (a) to obtain the columns.∎

Note that ${\mathscr{A}}^{\flat}({\mathscr{L}})$ is, by definition, a full
subcategory of ${\mathscr{A}}({\mathscr{L}})$ and, thus, of
${\mathscr{A}}^{\prime}_{l}$. The latter is defined, along with
${\mathscr{E}}^{\prime}_{l}$, in Section 2.2. Also,
${\mathscr{E}}^{\flat}({\mathscr{L}})\subseteq{\mathscr{E}}^{\prime}_{l}$ so
that
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$
is a full exact subcategory of
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$.333See the
discussion above (1.3.2) for the terminology. By the discussion in Section 2.2
below Proposition 2.2.3 (including Remark 2.2.4), the contravariant dual
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))^{{\text{\rm
op}}}$ is realized as
$({\mathscr{A}}^{\flat}({\mathscr{L}})^{*},{\mathscr{E}}^{\flat}({\mathscr{L}})^{*})$
inside
$({\mathscr{A}}^{\prime*}_{l},{\mathscr{E}}^{\prime*}_{l})=({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r})=({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$.

Remarkably, it is this dual of the exact category in the corollary that
interests us most. We pursue this in Section 2.3.

###  2.2. Some generalities on duality

This section uses the (more abstract) right-module setup of [DPS17, Assump.
3.2]; see Example B.1 in Appendix B for the left-module setup. Thus,
${\mathscr{K}}$ is a Noetherian integral domain, $K$ is its fraction field,
and $H$ is a finite and torsion-free algebra over ${\mathscr{K}}$ such that
$H_{K}$ is semisimple. The (isomorphism classes of) irreducible right
$H_{K}$-modules are indexed by a given set $\Lambda$, and
$\mathfrak{h}:\Lambda\to\mathbb{Z}$ is a height function.

It was also assumed “for convenience” in [DPS17, Assump. 3.2] that
$\mathfrak{h}$ takes non-negative values. We do not make that assumption here.
The “convenience” was in being able to use 0 as a lower bound for indexing
elements of $\Lambda$ via $\mathfrak{h}$. In no case is the validity of a
stated result in [DPS17] affected by dropping the requirement that
$\mathfrak{h}$ takes non-negative values. One such general result is the
following.

######  Theorem 2.2.1 ([DPS17, Th. 3.6]).

Let ${\mathscr{A}}$ denote the category of right $H$-modules which are finite
and torsion-free over ${\mathscr{K}}$. For each $M$ in ${\mathscr{A}}$, and
$h\in\mathbb{Z}$, let $M^{h}$ denote $M\cap(M_{K})^{\leq h}$, where
$(M_{K})^{\leq h}$ is the sum in $M_{K}$ of all irreducible right
$H_{K}$-submodules indexed by $\lambda\in\Lambda$ with
$\mathfrak{h}(\lambda)\leq h$. Let ${\mathscr{E}}$ denote the set of short
exact sequences $0\to X\overset{i}{\to}Y\overset{d}{\to}Z\to 0$ in mod-$H$ of
objects $X,Y,Z\in{\mathscr{A}}$, such that, for each $h\in\mathbb{Z}$, the
induced sequence

$0\longrightarrow X^{h}/X^{h-1}\longrightarrow Y^{h}/Y^{h-1}\longrightarrow
Z^{h}/Z^{h-1}\longrightarrow 0$ (2.2.1)

is also exact in mod-$H$. Then, $({\mathscr{A}},{\mathscr{E}})$ is an exact
category.

We also have

######  Theorem 2.2.2.

Keep the notation above, but let ${\mathscr{A}}^{\prime}$ denote the full
additive subcategory of objects $M\in{\mathscr{A}}$ for which $M^{h}/M^{h-1}$
is projective as a ${\mathscr{K}}$-module, for each $h\in\mathbb{Z}$. If
${\mathscr{E}}^{\prime}$ denotes ${\mathscr{E}}|_{{\mathscr{A}}^{\prime}}$,
the short exact sequences in ${\mathscr{E}}$ whose objects all belong to
${\mathscr{A}}^{\prime}$, then
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$ is an exact category.

###### Proof.

The category ${\mathscr{A}}^{\prime}$ is precisely the category
${\mathscr{A}}({\mathscr{S}})$ in [DPS17, Constr. 3.8] with
${\mathscr{S}}=\bigcup_{i\in\mathbb{Z}},{\mathscr{S}}_{i}$ in the special case
that each ${\mathscr{S}}_{i}$ is the full additive category of ${\mathscr{A}}$
consisting of objects $S$ projective over ${\mathscr{K}}$ with $S_{K}$ a sum
of irreducible right $H_{K}$-modules of height $i$, for all $i\in\mathbb{Z}$.

Note that ${\mathscr{A}}^{\prime}$ is closed under extensions in
${\mathscr{A}}$, since the middle term of the exact sequence (2.2.1) is
projective over ${\mathscr{K}}$ when the other terms are projective over
${\mathscr{K}}$. Consequently, [Bu10, Lem. 10.20] implies that
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$ is an exact category. ∎

Observe that the category ${\mathscr{E}}({\mathscr{S}})$, as defined in
[DPS17, Constr. 3.8], is not used in the proof of the above result.
Nevertheless, $({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$ is
a (full) exact subcategory of
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$, while
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$ provides us with a general
place to park duals, arising from the construction below. Those duals of most
interest to us will actually belong to
$({\mathscr{A}}({\mathscr{S}}),{\mathscr{E}}({\mathscr{S}}))$ (with
${\mathscr{S}}$ restricted to suitable direct sums of dual left-cell modules).

2.2A. Duality. In Section 2.1, we emphasized the differences between left and
right modules, because of the historical use of left-cell modules and their
duals. Here, we want to do the opposite and try to view left and right modules
on a more equal footing.

The discussion and results above in this section are formulated in terms of
right $H$-modules. They all have analogs for left $H$-modules as may be seen
from the right $H$-module case by substituting the opposite algebra
$H^{\text{op}}$ for $H$. In a context where both left and right $H$-modules
are used, it is useful to use a subscript $l$ or $r$ to distinguish the two
cases. Thus, ${\mathscr{A}}_{l}$, $({\mathscr{A}}_{l},{\mathscr{E}}_{l})$,
$({\mathscr{A}}_{l}^{\prime},{\mathscr{E}}_{l}^{\prime})$ are ${\mathscr{A}}$,
$({\mathscr{A}},{\mathscr{E}})$,
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$ analogs in the left
$H$-module case, and a subscript $r$ could be used similarly to give a
balanced treatment of the (original) right $H$-module case.

Next, let $(-)^{*}=\text{\rm{Hom}}_{{\mathscr{K}}}(-,{\mathscr{K}})$ be the
usual duality functor—actually, there are two functors,
${\mathscr{A}}_{l}\rightarrow{\mathscr{A}}_{r}$ and
${\mathscr{A}}_{r}\rightarrow{\mathscr{A}}_{l}$. Both are contravariant and
are denoted by the same symbol. The functors apply also to
${\mathscr{A}}_{l}^{\prime}$ and ${\mathscr{A}}_{r}^{\prime}$ and induce
equivalences of additive categories

${\mathscr{A}}_{l}^{\prime}\overset{\sim}{\longrightarrow}{\mathscr{A}}_{r}^{\prime\text{op}}\quad{\text{\rm
and}}\quad{\mathscr{A}}_{r}^{\prime}{\overset{\sim}{\longrightarrow}{\mathscr{A}}_{l}^{\prime\text{{\text{\rm
op}}}}}.$

Similarly, we can apply $(-)^{*}$ to ${\mathscr{E}}^{\prime}_{l}$ and
${\mathscr{E}}^{\prime}_{r}$. Finally, $(-)^{*}$ can be applied to the pairs
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$ and
$({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r})$ as well as the
opposite categories.

Our notation hides the height function $\mathfrak{h}$, which affects the
definition of ${\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime},\cdots$. If we
wish, we can always add a constant integer to any given $\mathfrak{h}$
function without affecting
${\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime}\ldots$. We assume in the
proposition below, and unless otherwise noted, that the dual of any
irreducible right $H$-module indexed by ${\lambda}\in\Lambda$ is also indexed
by ${\lambda}$ and that

$\mathfrak{h}_{l}({\lambda})=-\mathfrak{h}({\lambda})=-\mathfrak{h}_{r}(\lambda),$
(2.2.2)

for all ${\lambda}\in\Lambda$.

######  Proposition 2.2.3.

The duality functor $(-)^{*}$ induces a contravariant equivalence between
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$ and
$({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r})$.

We leave the proof to the reader, except for noting that the projectivity
(over ${\mathscr{K}}$) property of objects in ${\mathscr{A}}_{l}^{\prime}$ is
preserved by $(-)^{*}$ and also guarantees that $(-)^{*}$ sends
${\mathscr{E}}_{l}^{\prime}$ to ${\mathscr{E}}_{r}^{\prime}$. Similar
properties hold with $r$ and $l$ interchanged. Also, $(-)^{**}=((-)^{*})^{*}$
is naturally isomorphic to the identity functor.

######  Remark 2.2.4.

The functor $(-)^{*}$ above is more than a contravariant equivalence. It
defines an isomorphism of one category with a full representative subcategory
of another. (Recall that a representative subcategory contains a member of
every isomorphism class of objects in the larger category. If it is also full,
its inclusion is an equivalence.)

In view of the proposition, we may think of
$({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r})$ as a dual of
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$, via the
contravariant functor $(-)^{*}$. More interesting to note is the possibility
of using $(-)^{*}$ to construct new exact categories inside
$({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r})$ as duals (i.e.,
strict images under $(-)^{*}$) of exact subcategories of
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$. More precisely, if
$({\mathscr{B}},{\mathscr{F}})$ is a full exact subcategory of
$({\mathscr{A}}_{l}^{\prime},{\mathscr{E}}_{l}^{\prime})$, let
${\mathscr{B}}^{*}$ denote the (strict) image of ${\mathscr{B}}$ under
$(-)^{*}$ in ${\mathscr{A}}^{\prime}_{r}$, and define
${\mathscr{F}}^{*}\subseteq{\mathscr{E}}^{\prime}_{r}$ similarly. Then,
$(-)^{*}$ induces a contravariant equivalence
$({\mathscr{B}},{\mathscr{F}})\overset{\sim}{\to}({\mathscr{B}}^{*},{\mathscr{F}}^{*})$.
That is, the subcategory
$({\mathscr{B}}^{*},{\mathscr{F}}^{*})\subseteq({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$
is equivalent to the contravariant dual
$({\mathscr{B}},{\mathscr{F}})^{\text{\rm op}}$ of
$({\mathscr{B}},{\mathscr{F}})$.

In general, the notation $M^{h}$ applies to both the left- and right-module
case. If it is really needed to distinguish the left case from the right,
subscripts may be used, with the convention $M_{h}=M^{-h}$. This notation is
implicit in Section 2.1, and we use it below.

2.2B. Balanced subcategories. A property we are looking for in exact
subcategories of $({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$
$(=({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r}))$ is balance. More
generally, if $({\mathscr{B}},{\mathscr{F}})$ is a full444We could use a
weaker condition than“full” in this definition. All we really need is that,
for any $X,Y\in{\mathscr{B}}$, if
$f\in\text{\rm{Hom}}_{{\mathscr{A}}^{\prime}}(X,Y)$ is an isomorphism in
${\mathscr{A}}^{\prime}$, then $f\in\text{\rm{Hom}}_{{\mathscr{B}}}(X,Y)$. One
could call the embedding ${\mathscr{B}}\to{\mathscr{A}}^{\prime}$ isomorphism-
full under such a condition. Note that the strict closure
$\overline{{\mathscr{B}}}$ of ${\mathscr{B}}$ in ${\mathscr{A}}^{\prime}$
(throwing in all objects in ${\mathscr{A}}^{\prime}$ isomorphic to an object
in ${\mathscr{B}}$) makes sense as a category in the presence of this
condition, using for morphisms $f:X^{\prime}\to Y^{\prime}$ between its
objects those morphisms $f$ in ${\mathscr{A}}^{\prime}$ for which there are in
${\mathscr{A}}^{\prime}$ isomorphisms $e:X\to X^{\prime}$ and $g:Y^{\prime}\to
Y$ with objects $X,Y\in{\mathscr{B}}$ and composite $efg$ a morphism in
${\mathscr{B}}$. As such, the inclusion of ${\mathscr{B}}$ into
$\overline{{\mathscr{B}}}$ is full and an equivalence of categories, and
${\mathscr{B}}$ is a subcategory of ${\mathscr{A}}^{\prime}$ closed under
object isomorphisms. exact subcategory of
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$, we say
$({\mathscr{B}},{\mathscr{F}})$ is balanced provided, for each
$M\in{\mathscr{B}}$ and $h\in\mathbb{Z}$, the objects $M^{h}$ and $M/M^{h}$
are both isomorphic to objects in ${\mathscr{B}}$, and the sequence $(M^{h}\to
M\to M/M^{h})$ (which is obviously in ${\mathscr{E}}^{\prime}$) is isomorphic
to a member of ${\mathscr{F}}$. Here, we are using the defining properties of
${\mathscr{A}}^{\prime}$ and ${\mathscr{E}}^{\prime}$ introduced in Theorems
2.2.1 and 2.2.2 above. Of course,
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$ itself is balanced.

We have not assumed in the definition above that ${\mathscr{B}}$ is closed
under isomorphism in ${\mathscr{A}}^{\prime}$. However,
$({\mathscr{B}},{\mathscr{F}})$ is balanced if and only if its strict closure
in $({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$ is balanced. See footnote
4 above for further discussion.

A similar discussion applies for
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$.

######  Proposition 2.2.5.

Suppose $({\mathscr{B}},{\mathscr{F}})$ is a balanced exact subcategory of
$({\mathscr{A}}^{\prime}_{l},{\mathscr{E}}^{\prime}_{l})$. If
$({\mathscr{B}}^{*},{\mathscr{F}}^{*})$ denotes its strict image in
$({\mathscr{A}}_{r}^{\prime},{\mathscr{E}}_{r}^{\prime})$ under $(-)^{*}$,
then $({\mathscr{B}}^{*},{\mathscr{F}}^{*})$ is balanced.

We leave the proof of this result to the reader.

We have already observed in Corollary 2.1.10 and Proposition 2.3.3 below that
the exact categories
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$
and
$({\mathscr{A}}_{\flat}({\mathscr{L}}^{*}),{\mathscr{E}}_{\flat}({\mathscr{L}}^{*}))$
are balanced. As a consequence, the following proposition applies to both.
Note that ${\mathscr{A}}^{\flat}({\mathscr{L}})$ and
${\mathscr{A}}_{\flat}({\mathscr{L}}^{*})$ are, respectively, closed under
isomorphism of objects in ${\mathscr{A}}^{\prime}_{l}$ and
${\mathscr{A}}^{\prime}_{r}$.

######  Proposition 2.2.6.

Let $({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$ denote either
$({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r})$ or its left-hand
counterpart. Suppose $({\mathscr{B}},{\mathscr{F}})$ is a balanced exact
subcategory of $({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})$. If
$M\in{\mathscr{B}}$, $h\in\mathbb{Z}$, then the sequence

$M^{h-1}\longrightarrow M^{h}\longrightarrow M^{h}/M^{h-1}$ (2.2.3)

belongs to ${\mathscr{E}}^{\prime}$ and is isomorphic to a sequence in
${\mathscr{F}}$. In fact, the sequence belongs to ${\mathscr{F}}$, if
${\mathscr{B}}$ is closed under isomorphisms in ${\mathscr{A}}^{\prime}$.

###### Proof.

We treat the right-hand case
$({\mathscr{A}}^{\prime},{\mathscr{E}}^{\prime})=({\mathscr{A}}^{\prime}_{r},{\mathscr{E}}^{\prime}_{r})$,
which also implies the left-hand version of the proposition.

Fix $M$ in ${\mathscr{B}}$ and let $h$ be an integer. Applying the definition
of balanced twice, we obtain that $M^{h},(M^{h})^{h-1},$ and
$M^{h}/(M^{h})^{h-1}$ all belong to ${\mathscr{B}}$. Also, $((M^{h})^{h-1}\to
M^{h}\to M^{h}/M^{h-1})\in{\mathscr{F}}$. Using the definition of
${\mathscr{A}}$ in Theorem 2.2.1 we obtain the following commutative diagram
in ${\mathscr{A}}$ with all vertical arrows isomorphisms.

$\begin{CD}(M^{h})^{h-1}@>{}>{}>M^{h}@>{}>{}>M^{h}/(M^{h})^{h-1}\\\
@V{\wr}V{}V\Big{\|}@V{\wr}V{}V\\\
M^{h-1}<EMAIL_ADDRESS>
Since ${\mathscr{A}}^{\prime}$ is closed under isomorphism (and full) in
${\mathscr{A}}$, this diagram and its isomorphic images are all in
${\mathscr{A}}^{\prime}$. The top row is in ${\mathscr{B}}$ and, in fact,
belongs to ${\mathscr{F}}$. The bottom rows belongs to
${\mathscr{E}}^{\prime}$ and are isomorphic to the top row (as a diagram).
This proves the first assertion of the proposition.

To prove the second, assume ${\mathscr{B}}$ is closed under isomorphism. Then,
the objects and morphisms in the bottom row all belong to ${\mathscr{B}}$, so
the bottom row belongs to ${\mathscr{F}}$. (Recall that the conflations in any
exact category are required to be closed under isomorphism as part of their
definition.) This completes the proof of the proposition.∎

The last part of the proof above implicitly used the fact that ${\mathscr{B}}$
is full, assumed in the definition of balanced. It would have been enough to
assume the weaker property in the footnote 4 of this chapter.

###  2.3. The dual category of
$({\mathscr{A}}^{\flat}({\mathscr{L}}),{\mathscr{E}}^{\flat}({\mathscr{L}}))$

We now turn back to the generic Hecke algebra ${\mathcal{H}}$ associated with
a standard finite Coxeter system $(W,S,L)$. We first consider the right
${\mathcal{H}}$-module analog
$({\mathscr{A}}_{r}({\mathscr{L}}^{*}),{\mathscr{E}}_{r}({\mathscr{L}}^{*}))$
of $({\mathscr{A}}_{l}({\mathscr{L}}),{\mathscr{E}}_{l}({\mathscr{L}}))$
defined in display (2.1.3) (see (2.3.2) below).

Thus, ${\mathscr{A}}_{r}$ is the full subcategory of right
${\mathcal{H}}$-modules which are finite and torsion-free over
${\mathscr{K}}={\mathcal{Z}}$. We often denote the category
${\mathscr{A}}_{r}$ simply as ${\mathscr{A}}$ when no confusion can arise.

######  Notation 2.3.1.

It will be convenient in this section to use the height function
$\mathfrak{h}=\mathfrak{h}_{r}$, which is the negative of the height function
$\mathfrak{h}_{l}$ introduced in Notation 2.1.2. Thus, the symbol
$\mathfrak{h}$ in this section, without further comment, denotes
$\mathfrak{h}_{r}=-\mathfrak{h}_{l}$; it takes positive values. It is also
compatible with the preorder $\leq_{LR}^{{\text{\rm op}}}$ and partial order
$\leq_{L}^{{\text{\rm op}}}$ on $\Omega$.

For an integer $h$, let ${\mathscr{L}}^{*}_{h}$ be the full subcategory of
${\mathscr{A}}$ consisting of right ${\mathcal{H}}$-modules $N$ such that $N$
is a finite direct sum of modules which are isomorphic to dual left-cell
modules $S_{\omega}:=S(\omega)^{*}$ of height $h$. Then let
${\mathscr{L}}^{*}:=\bigcup_{h}{\mathscr{L}}^{*}_{h}$. We remind the reader
that, by a slight abuse of notation, a left-cell module $S(\omega)$ has a
negative height $h=\mathfrak{h}_{l}(\omega)<0$, and the dual left-cell module
$S_{\omega}$ has height $-h=\mathfrak{h}_{r}(\omega)>0$.

Let ${\mathscr{A}}({\mathscr{L}}^{*})$ be the full additive subcategory of
mod-${\mathcal{H}}$ with objects $M$ whose $\mathfrak{h}_{r}$-filtration

$0=M^{0}\subseteq M^{1}\subseteq M^{2}\subseteq\cdots\subseteq
M^{n-1}\subseteq M^{n}=M$ (2.3.1)

has the property that, for any integer $h>0$, $M^{h}/M^{h-1}$ is an object in
${\mathscr{L}}^{*}_{h}$. See [DPS17, §3], which also allowed $h=0$ (but did
not use this case). Like the refinement in footnote 2, such a filtration will
be informally called a dual left-cell filtration.

By analogy with [DPS17, §3], let ${\mathscr{E}}({\mathscr{L}}^{*})$ consist of
those sequences $(M\to N\to P)$ in ${\mathscr{A}}$ such that, for every $h$,

$0\to M^{h}/M^{h-1}\to N^{h}/N^{h-1}\to P^{h}/P^{h-1}\to 0$

is a split short exact sequence in the additive category
${\mathscr{L}}^{*}_{h}$. Then, as a special case of [DPS17, Th. 3.9], we have

$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}({\mathscr{L}}^{*}))$ is an
exact category. (2.3.2)

Recall from Section 2.2 the standard ${\mathcal{Z}}$-duality functor $(-)^{*}$
discussed in the context of the exact categories used here. Thus, if
$N\in{\mathscr{A}}_{l}({\mathscr{L}})$, then
$N^{*}\in{\mathscr{A}}_{r}({\mathscr{L}}^{*})$ and
$(N_{h-1}/N_{h})^{*}\cong(N^{*})^{h}/(N^{*})^{h-1}$, for each
$h\in\mathbb{Z}$. More precisely, if $N$ has a (left-cell)
$\mathfrak{h}_{l}$-filtration as in (2.1.2), then $N^{*}$ has a (dual left-
cell) $\mathfrak{h}_{r}$-filtration

$0=(N^{*})^{0}\subseteq(N^{*})^{1}\subseteq(N^{*})^{2}\subseteq\cdots\subseteq(N^{n-1})^{*}\subseteq(N^{*})^{n}=N$

as given in (2.3.1), where $(N^{*})^{i}=(N/N_{i})^{*}$ so that
$(N^{*})^{h}/(N^{*})^{h-1}\cong(N_{h-1}/N_{h})^{*}\in({\mathscr{L}}_{-h})^{*}=({\mathscr{L}}^{*})_{h}$,
for all $1\leq h\leq n$. Hence, $N^{*}$ is an object of
${\mathscr{A}}_{r}({\mathscr{L}}^{*})$. Similarly, for a conflation $(M\to
N\to P)\in{\mathscr{E}}_{l}({\mathscr{L}})$, we have $(P^{*}\to N^{*}\to
M^{*})\in{\mathscr{E}}_{r}({\mathscr{L}}^{*})$, since
$\big{(}(P^{*})^{h}/(P^{*})^{h-1}\to(N^{*})^{h}/(N^{*})^{h-1}\to(M^{*})^{h}/(M^{*})^{h-1}\big{)}$,
which is isomorphic to
$\big{(}(P_{h-1}/P_{h})^{*}\to(N_{h-1}/N_{h})^{*}\to(M_{h-1}/M_{h})^{*}\big{)}$,
is split exact, for all $1\leq h\leq n$.

Hence, the exact category
$({\mathscr{A}}({\mathscr{L}}^{*}),{\mathscr{E}}({\mathscr{L}}^{*}))$ is
naturally isomorphic to the dual category
$({\mathscr{A}}_{l}({\mathscr{L}})^{*},{\mathscr{E}}_{l}({\mathscr{L}})^{*})$
of $({\mathscr{A}}_{l}({\mathscr{L}}),{\mathscr{E}}_{l}({\mathscr{L}}))$. As
in Section 2.2, $\mathscr{A}_{l}(\mathscr{L})^{*}$ and
$\mathscr{E}_{l}(\mathscr{L})^{*}$ both denote strict images (closed under
isomorphism).

Let
${\mathscr{E}}_{\flat}({\mathscr{L}}^{*})={\mathscr{E}}^{\flat}({\mathscr{L}})^{*}$
and let
${\mathscr{A}}_{\flat}({\mathscr{L}}^{*})={\mathscr{A}}^{\flat}({\mathscr{L}})^{*}$.
We easily prove the following key proposition, motivated by the weaker [DPS15,
Cor. 4.5] and [DPS98a, Th. 2.3.9]. For $J\subseteq S$,

$x_{J}{\mathcal{H}}\cong({\mathcal{H}}x_{J})^{*}$ (2.3.3)

by [DPS98a, Lem. 2.3.9]. In particular,

$x_{J}{\mathcal{H}}\,\,\,{\text{\rm belongs to
${\mathscr{A}}_{\flat}({\mathscr{L}}^{*})$ }}$ (2.3.4)

by Corollary 2.1.8.

######  Proposition 2.3.2.

With the notation above,