# Parabolic induction in the homotopy category of pro-$p$ Iwahori-Hecke
modules

Nicolas Dupré  Universität Duisburg-Essen
Fakultät für Mathematik
Thea-Leymann-Straße 9
D–45127 Essen, Germany<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be the group of rational points of a split connected reductive group
over a non-archimedean local field of residue characteristic $p$, and let
${\mathcal{H}}$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over a field
of characteristic $p$. We study the parabolic induction functor for
${\mathcal{H}}$-modules in terms of the Gorenstein projective model structures
introduced by Hovey. Let ${\operatorname{Ho}}({\mathcal{H}})$ denote the
associated homotopy category of this model structure. We show that
${\operatorname{Ho}}({\mathcal{H}})$ and its thick subcategory generated by
the essential images of finitely many parabolic induction functors are related
via a recollement of triangulated categories. We then investigate the
isomorphism classes of simple ${\mathcal{H}}$-modules in
${\operatorname{Ho}}({\mathcal{H}})$ and give a complete classification when
$G=\mathrm{GL}_{n}$.

††footnotetext: 2020 Mathematics Subject Classification. Primary 20C08, 18N40,
18N55.

###### Contents

  1. 1 Introduction
  2. 2 Reminders on pro-$p$ Iwahori-Hecke algebras and model structures
  3. 3 Parabolic induction, recollements and supersingular modules
  4. 4 Bousfield localisations from recollements
  5. 5 Simple supersingular Hecke modules in the homotopy category

##  1\. Introduction

###  1.1. Background and motivation

Let $p$ be a prime and let $\mathfrak{F}$ be a non-archimedean local field of
residue characteristic $p$. Let $G=\mathbb{G}(\mathfrak{F})$ be the group of
$\mathfrak{F}$-rational points of a split connected reductive group
$\mathbb{G}$ over $\mathfrak{F}$. Let $k$ be a field of characteristic $p$ and
denote by ${\operatorname{Rep}}_{k}^{\infty}(G)$ the category of $k$-linear
smooth representations of $G$, i.e. those representations such that the
stabiliser of each vector is open. This category lies on one side of the
mod-$p$ local Langlands program.

Let $I$ be a pro-$p$ Iwahori subgroup of $G$ and let
${\mathcal{H}}=k[I\backslash G/I]$ be the corresponding _pro- $p$ Iwahori-
Hecke algebra_ of $G$, i.e. the convolution algebra of $I$-biinvariant
compactly supported functions on $G$. Write
${\operatorname{Mod}}({\mathcal{H}})$ for the category of left
${\mathcal{H}}$-modules. Then taking $I$-invariants defines a functor
$U:{\operatorname{Rep}}_{k}^{\infty}(G)\to{\operatorname{Mod}}({\mathcal{H}})$
which sends non-zero representations to non-zero modules. This functor is
expected to provide a strong relation between the two categories, however its
behaviour is not sufficiently understood beyond
$G=\mathrm{GL}_{2}(\mathbb{Q}_{p})$ and a few related cases (c.f. the work of
Ollivier [25]). In particular, one of the major open problems in the theory is
to determine the relationship between the isomorphism classes of irreducible
admissible smooth representations of $G$ and of simple
${\mathcal{H}}$-modules.

In previous work [10] with Kohlhaase, we showed that $U$ restricts to an
equivalence between an explicit full subcategory of
${\operatorname{Rep}}_{k}^{\infty}(G)$ and the full subcategory
$\mathrm{GProj}({\mathcal{H}})$ of _Gorenstein projective_
${\mathcal{H}}$-modules. These modules are a key component in the definition
of a model structure on ${\operatorname{Mod}}({\mathcal{H}})$ due to Hovey
[16]. We used this in [10] to construct and study a model structure on
${\operatorname{Rep}}_{k}^{\infty}(G)$ and the derived functor $\mathbf{R}U$
between the associated homotopy categories. We generally obtained that the
situation was improved at the homotopy level. This suggests that one should
instead investigate the relationship between the _homotopy classes_ of
irreducible admissible smooth representations of $G$ and of simple
${\mathcal{H}}$-modules.

This paper is concerned with the Hecke side of that question, i.e. we wish to
study the simple ${\mathcal{H}}$-modules viewed as objects of the homotopy
category ${\operatorname{Ho}}({\mathcal{H}})$ of
${\operatorname{Mod}}({\mathcal{H}})$. A first step towards this was given by
Koziol in [20] where the simple ${\mathcal{H}}$-modules of finite projective
dimension were classified. When rephrased in our language, this gives a
classification of the simple modules which are isomorphic to zero in
${\operatorname{Ho}}({\mathcal{H}})$. In particular, it was shown that the
simple supersingular ${\mathcal{H}}$-modules are generically never zero in
${\operatorname{Ho}}({\mathcal{H}})$.

Our main goal in this work is to treat the harder problem of determining
whether two nonisomorphic simple ${\mathcal{H}}$-modules of infinite
projective dimension can become isomorphic in
${\operatorname{Ho}}({\mathcal{H}})$. If $M$ denotes a standard Levi subgroup
of $G$ with associated Hecke algebra ${\mathcal{H}}_{M}$ then a key ingredient
in Abe’s construction of the simple modules in [1] is the _parabolic
induction_ functor
${\operatorname{Ind}}^{{\mathcal{H}}}_{{\mathcal{H}}_{M}}:{\operatorname{Mod}}({\mathcal{H}}_{M})\to{\operatorname{Mod}}({\mathcal{H}})$.
We therefore first aim to study the properties of
${\operatorname{Ind}}^{{\mathcal{H}}}_{{\mathcal{H}}_{M}}$ in the language of
Hovey’s Gorenstein projective model structure. This allows us to reduce our
question to the case of supersingular modules. In many situations of interest,
we are then able to obtain a rather complete understanding of the situation
(c.f. B & C below for details).

###  1.2. Main results

We first show that the parabolic induction functor induces decompositions of
${\operatorname{Ho}}({\mathcal{H}})$ via various recollements. To state this
more precisely, we note that ${\operatorname{Ho}}({\mathcal{H}})$ has the same
class of objects as ${\operatorname{Mod}}({\mathcal{H}})$, being a
localisation of it, and that it was proved by Hovey to be canonically a
triangulated category. Fixing a finite collection ${\mathcal{M}}$ of proper
standard Levi subgroups of $G$, we may therefore let
${\mathcal{S}}^{G}_{\mathcal{M}}$ be the thick subcategory of
${\operatorname{Ho}}({\mathcal{H}})$ generated by the essential images of
${\operatorname{Ind}}^{{\mathcal{H}}}_{{\mathcal{H}}_{M}}$ for all
$M\in{\mathcal{M}}$. When ${\mathcal{M}}$ equals the set of all proper
standard Levi subgroups of $G$, we simply write
${\mathcal{S}}^{G}_{\mathrm{all}}$ for this thick subcategory.

Also, associated to $G$ is a group $\Omega$ which is equal to the length zero
elements in the Iwahori-Weyl group $W=N_{G}(T)/(T\cap I^{\prime})$ (where
$I^{\prime}$ is the _Iwahori_ subgroup of $G$ with pro-$p$ radical $I$). This
group $\Omega$ is known to be a finitely generated abelian group, and we
denote by $\Omega_{\text{tor}}$ its (finite) torsion-subgroup. With this
notation established, our first main result then says:

######  Theorem A.

  1. (i)

The inclusion
$i^{G}_{\mathcal{M}}:{\mathcal{S}}^{G}_{\mathcal{M}}\hookrightarrow{\operatorname{Ho}}({\mathcal{H}})$
is part of a recollement

$\textstyle{{\mathcal{S}}^{G}_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{G}_{\mathcal{M}}}$$\textstyle{{\operatorname{Ho}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{G}_{\mathcal{M}}}$$\scriptstyle{\mathcal{L}^{G}_{\mathcal{M}}}$$\scriptstyle{\mathcal{R}^{G}_{\mathcal{M}}}$$\textstyle{{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{G}_{\mathcal{M}}}$$\scriptstyle{r^{G}_{\mathcal{M}}}$

of triangulated categories, where
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}={\operatorname{Ho}}({\mathcal{H}})/{\mathcal{S}}^{G}_{\mathcal{M}}$.

  2. (ii)

Assume that $k$ is algebraically closed, that the root system of $\mathbb{G}$
is irreducible and that $p\nmid\left|\Omega_{\text{tor}}\right|$. Suppose that
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}})$ has finite length. Then
the following are equivalent:

     * (a)

${\mathfrak{m}}$ lies in the essential image of $\ell^{G}_{\mathrm{all}}$

     * (b)

${\mathfrak{m}}$ lies in the essential image of $r^{G}_{\mathrm{all}}$

     * (c)

the inclusion ${\mathfrak{m}}_{\mathrm{ss}}\subseteq{\mathfrak{m}}$ is an
isomorphism in ${\operatorname{Ho}}({\mathcal{H}})$, where
${\mathfrak{m}}_{\mathrm{ss}}$ denotes the largest supersingular submodule of
${\mathfrak{m}}$.

This is proved in 3.13 and 3.31. Roughly speaking, part (i) of A says that the
triangulated category ${\operatorname{Ho}}({\mathcal{H}})$ can be decomposed
into two orthogonal thick subcategories, one equal to
${\mathcal{S}}^{G}_{\mathcal{M}}$ and the other one equal to
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$, where the latter category
is embedded fully faithfully into ${\operatorname{Ho}}({\mathcal{H}})$ via
either one of the functors $\ell^{G}_{\mathcal{M}}$ or $r^{G}_{\mathcal{M}}$
(c.f. 3.9 and the discussion following it for details). We point out that
while the proof of (i) is mostly formal, it however relies on the fact that
parabolic induction is fully faithful (c.f. 2.5), which appears to be a new
result.

We also point out that the recollement in A(i) can be entirely reconstructed
from various new model structures on ${\operatorname{Mod}}({\mathcal{H}})$.
This is proved in a more general context of a recollement in the homotopy
category of an abelian model category, see 4.2, 4.7, 4.8 and 4.9 for the
details.

The proof of part (ii) of A relies on Abe’s classification of simple
${\mathcal{H}}$-modules in [1], on Koziol’s results on the projective
dimension of simple modules from [20], and on a lift to the homotopy level of
a computation of Abe of the image of the simple modules under the adjoints of
the parabolic induction functor (c.f. 3.27). As a consequence of the latter
computation, we also obtain a new proof of a result of Koziol and reduce the
study of isomorphism classes of simple ${\mathcal{H}}$-modules in
${\operatorname{Ho}}({\mathcal{H}})$ to the case of simple supersingular
${\mathcal{H}}$-modules, c.f. 3.28.

Having established this, the next step is to determine when two simple
supersingular ${\mathcal{H}}$-modules can become isomorphic in
${\operatorname{Ho}}({\mathcal{H}})$. Here the main difficulty is that
computing the Hom spaces in ${\operatorname{Ho}}({\mathcal{H}})$ a priori
requires to compute explicit Gorenstein projective replacements in
${\operatorname{Mod}}({\mathcal{H}})$. Instead, inspired by the arguments of
Koziol in [20] and by the functorial resolutions of Ollivier-Schneider [26],
we use Bruhat-Tits theory. If $F$ denotes a face of the Bruhat-Tits building
contained in the closure of a chosen chamber $C$, then there is an associated
subalgebra ${\mathcal{H}}_{F}$ of ${\mathcal{H}}$. These subalgebras together
generate a Gorenstein subalgebra ${\mathcal{H}_{\operatorname{aff}}}$ of
${\mathcal{H}}$ called the _affine Hecke algebra_. The simple supersingular
Hecke modules were classified by Vignéras, c.f. [33], and a key input in this
classification are the so-called supersingular characters of
${\mathcal{H}_{\operatorname{aff}}}$. Our main insight is to consider the
‘diagonal’ functor

$\Delta=({\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F}})_{F}:{\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})\to\prod_{F\subseteq\overline{C}}{\operatorname{Mod}}({\mathcal{H}}_{F})$

which is right Quillen and in fact determines the model structure on
${\operatorname{Mod}}({\mathcal{H}_{\operatorname{aff}}})$ via a right
transfer property, c.f. 5.4. This indicates that we should in principle be
able to do computations in the product category
$\prod_{F\subseteq\overline{C}}{\operatorname{Ho}}({\mathcal{H}}_{F})$ instead
of ${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$. The advantage
being that since ${\mathcal{H}}_{F}$ is known to be self-injective, it follows
that every ${\mathcal{H}}_{F}$-module is Gorenstein projective and hence there
is no need to compute replacements in order to compute Hom spaces in
${\operatorname{Ho}}({\mathcal{H}}_{F})$.

More specifically, we have that $\Delta$ lifts to a functor
${\operatorname{Ho}}(\Delta):{\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})\to\prod_{F\subseteq\overline{C}}{\operatorname{Ho}}({\mathcal{H}}_{F})$.
One major technical input in this paper is the computation of the image of the
Hom space in ${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$ between
two supersingular characters under ${\operatorname{Ho}}(\Delta)$. Roughly, one
gets that this image is always zero unless the root system is of type
$\Phi_{1}\times\Phi_{2}$, where $\Phi_{1}$ is irreducible of rank 2 and
$\Phi_{2}$ is a product of $A_{1}$’s, and the two characters are of a very
explicit form which we will denote here by $\chi_{1}^{\text{special}}$ and
$\chi_{2}^{\text{special}}$, see 5.6 for the details. In order to go from
supersingular characters of ${\mathcal{H}_{\operatorname{aff}}}$ to simple
supersingular modules, we then analyse the actions of the group $\Omega$ on
these characters and on the Hom spaces between these characters. In
particular, given a supersingular character $\chi$ we denote by
$\Omega_{\chi}$ its stabiliser in $\Omega$. We also note that $\Omega$ acts on
the faces $F$ contained in $\overline{C}$, and we denote the stabiliser of a
face $F$ by $\Omega_{F}$. By combining this analysis of the $\Omega$-action on
Hom spaces in ${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$, c.f.
5.19, with the above computation of the image of Hom spaces under
${\operatorname{Ho}}(\Delta)$, we obtain the following:

######  Theorem B.

Let ${\mathfrak{m}}$ and ${\mathfrak{m}}^{\prime}$ be two simple supersingular
${\mathcal{H}}$-modules of infinite projective dimension.

  1. (i)

Suppose that $\mathbb{G}$ is simply-connected. Then
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}})$ if and only if
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Mod}}({\mathcal{H}})$.

  2. (ii)

Suppose that if one of
${\operatorname{Res}}^{\mathcal{H}}_{\mathcal{H}_{\operatorname{aff}}}({\mathfrak{m}})$
and
${\operatorname{Res}}^{\mathcal{H}}_{\mathcal{H}_{\operatorname{aff}}}({\mathfrak{m}}^{\prime})$
contains $\chi_{1}^{\text{special}}$ as a submodule then the other one does
not contain $\chi_{2}^{\text{special}}$. Furthermore, if we fix a
supersingular character $\chi$ occuring in
${\operatorname{Res}}^{\mathcal{H}}_{\mathcal{H}_{\operatorname{aff}}}({\mathfrak{m}})$,
assume that there exists a face $F\subseteq\overline{C}$ such that
$\Omega_{\chi}\subseteq\Omega_{F}$ and
${\operatorname{Res}}^{\mathcal{H}_{\operatorname{aff}}}_{{\mathcal{H}}_{F}}({\mathfrak{m}})$
is not projective. Then ${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}})$ if and only if
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Mod}}({\mathcal{H}})$.

The above is a combination of 5.14 and 5.23. We then use these results to give
a complete classification of the isomorphism classes of simple supersingular
modules in the situation where $G$ is a product of $\mathrm{GL}_{n}$’s.
Roughly, we then get that two nonisomorphic simple supersingular
${\mathcal{H}}$-modules almost never become isomorphic in
${\operatorname{Ho}}({\mathcal{H}})$ and when they do it is only in a
restricted small family of cases. In what follows, we denote by
$\widetilde{\Omega}$ the lift of $\Omega$ to $\widetilde{W}=N_{G}(T)/(T\cap
I)$. This group embeds canonically as a subgroup of ${\mathcal{H}}^{\times}$.

######  Theorem C (5.31).

Suppose that $G$ is a product of $\mathrm{GL}_{n}$’s and let ${\mathfrak{m}}$
and ${\mathfrak{m}}^{\prime}$ be two simple supersingular
${\mathcal{H}}$-modules of infinite projective dimension.

  1. (i)

If $G$ does not have root system of type $A_{2}\times A_{1}\times\cdots\times
A_{1}$ (possibly empty product of $A_{1}$’s), then
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}})$ if and only if
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Mod}}({\mathcal{H}})$.

  2. (ii)

If $G$ does have root system of type $A_{2}\times A_{1}\times\cdots A_{1}$
(possibly empty product of $A_{1}$’s), then
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Ho}}({\mathcal{H}})$ if and only if either
${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ in
${\operatorname{Mod}}({\mathcal{H}})$ or, up to swapping ${\mathfrak{m}}$ and
${\mathfrak{m}}^{\prime}$, we have that $\chi_{1}^{\text{special}}$ and
$\chi_{2}^{\text{special}}$ occur as submodules of
${\operatorname{Res}}^{\mathcal{H}}_{\mathcal{H}_{\operatorname{aff}}}({\mathfrak{m}})$
and
${\operatorname{Res}}^{\mathcal{H}}_{\mathcal{H}_{\operatorname{aff}}}({\mathfrak{m}}^{\prime})$
respectively and ${\mathfrak{m}}\cong{\mathfrak{m}}^{\prime}$ as
representations of $\widetilde{\Omega}$.

Combined with earlier results, the above gives a complete classification of
the isomorphism classes in ${\operatorname{Ho}}({\mathcal{H}})$ of the simple
${\mathcal{H}}$-modules when $G=\mathrm{GL}_{n}$. In that case, we note in
particular that C says that two simple supersingular ${\mathcal{H}}$-modules
are isomorphic in ${\operatorname{Ho}}({\mathcal{H}})$ if and only if they are
isomorphic in ${\operatorname{Mod}}({\mathcal{H}})$, except if $n=3$. As the
$n$-dimensional simple supersingular ${\mathcal{H}}$-modules are in bijection
with certain irreducible $n$-dimensional Galois representations by the work of
Grosse-Klönne [13], our results show that the modules appearing on the Hecke
side of this bijection are generically still non-isomorphic in
${\operatorname{Ho}}({\mathcal{H}})$.

###  1.3. Organisation of the paper

In §2.1-2.3 we recall background material on pro-$p$ Iwahori-Hecke algebras,
parabolic induction and simple ${\mathcal{H}}$-modules, and we prove that
parabolic induction is fully faithful. In §2.4-2.5 we recall the notion of an
abelian model category, left/right transfers of model structures and the
definition of Hovey’s Gorenstein projective model structure, and we prove some
general preparatory lemmas.

In §3.1-3.2, we study the properties of the functor of parabolic induction in
the language of model categories, culminating with the proof of A(i). We only
use the fact that parabolic induction is fully faithful, has both a left and a
right adjoint, and its left adjoint is exact. Consequently, §3.1 is written in
a much bigger level of generality c.f. 3.1. In §3.3, we study the relation
between the recollement of A(i) and supersingular modules and prove A(ii).

In §4.1, we investigate recollements in the homotopy category of an abelian
model category ${\mathcal{A}}$. We show that if ${\mathcal{A}}$ is equipped
with both a projective and injective model structure with the same class of
trivial objects, then any recollement in ${\operatorname{Ho}}({\mathcal{A}})$
can be reconstructed from new model structures on ${\mathcal{A}}$. In the case
of Hecke modules, we show that two of these new model structures can be right
transferred to ${\operatorname{Rep}}_{k}^{\infty}(G)$. In §4.2, we show that
these new model structures are Bousfield localisations of the original model
structures on ${\mathcal{A}}$.

Finally, in §5 we compute the isomorphism classes of simple supersingular
${\mathcal{H}}$-modules. After recalling the necessary background on the
affine Hecke algebra and supersingular characters in §5.1, we study the image
of Hom spaces between supersingular characters under the functor
${\operatorname{Ho}}(\Delta)$ in §5.2, proving B(i). In §5.3, we study the
action of $\Omega$ on the Hom spaces in
${\operatorname{Ho}}({\mathcal{H}_{\operatorname{aff}}})$, proving B(ii). We
finish in §5.4 by analysing the case where $G$ is a product of
$\mathrm{GL}_{n}$’s and prove C.

###  1.4. Acknowledgments

We would like to thank Jan Kohlhaase for helpful conversations and for giving
comments on an earlier draft of this paper. We have also further profited from
conversations with Claudius Heyer. The author was supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) within the project
Smooth modular representation theory of $p$-adic reductive groups, project
number 435414187. The author is also a member of the DFG Research Training
Group 2553 ‘Symmetries and Classifying Spaces – Analytic, Arithmetic and
Derived’. The financial support of the DFG is gratefully acknowledged.

###  1.5. Notation and conventions

A class of objects of a category ${\mathcal{C}}$ will usually be identified
with the corresponding full subcategory. If we denote an adjunction by
$F:{\mathcal{C}}\rightleftarrows{\mathcal{D}}:U$ then $F$ is always assumed to
be left adjoint to $U$. The unit (resp. the counit) of an adjunction will
always be denoted by $\eta$ (resp. $\varepsilon$). When such an adjunction is
a Quillen adjunction between model categories, we denote by
$\mathbf{L}F:{\operatorname{Ho}}({\mathcal{C}})\rightleftarrows{\operatorname{Ho}}({\mathcal{D}}):\mathbf{R}U$
the total derived adjunction between the homotopy categories. Given a functor
$F:{\mathcal{C}}\to{\mathcal{D}}$, we denote by ${\operatorname{Im}}(F)$ its
essential image. We also say that $F$ preserves (resp. reflects) a property
(P) if $F*$ (resp. $*$) has property (P) whenever $*$ (resp. $F*$) does.

For any unital ring $R$ we denote by ${\operatorname{Mod}}(R)$ the category of
left $R$-modules. Given
${\mathfrak{m}},\mathfrak{n}\in{\operatorname{Mod}}(R)$ we write
${\operatorname{Hom}}_{R}({\mathfrak{m}},\mathfrak{n})$ for the set of
$R$-linear maps from ${\mathfrak{m}}$ to $\mathfrak{n}$, and we denote by
${\operatorname{pd}}_{R}({\mathfrak{m}})$ the projective dimension of
${\mathfrak{m}}$. For any topological group $J$, we denote by
${\operatorname{Rep}}_{R}^{\infty}(J)$ the category of $R$-linear smooth
representation of $J$, i.e. the category of all $V\in{\operatorname{Mod}}(R)$
carrying an $R$-linear action of $J$ such that the stabiliser of any element
$v\in V$ is open in $J$. The $R$-module $R$ equipped with the trivial
$J$-action will be denoted by
$\mathbf{1}\in{\operatorname{Rep}}_{R}^{\infty}(J)$. Finally, for any
$V\in{\operatorname{Rep}}_{R}^{\infty}(J)$ and any subgroup $J^{\prime}\leq J$
we write $V^{J^{\prime}}$ for the $R$-submodule of $J^{\prime}$-invariant,
i.e. of elements $v\in V$ such that $j\cdot v=v$ for all $j\in J^{\prime}$.

##  2\. Reminders on pro-$p$ Iwahori-Hecke algebras and model structures

###  2.1. The Bruhat-Tits building and Weyl groups

Let $p$ be a prime and let $\mathfrak{F}$ be a nonarchimedean local field of
residue field $\mathbb{F}_{q}$ of characteristic $p$. Let
$G=\mathbb{G}(\mathfrak{F})$ denote the group of $\mathfrak{F}$-rational
points of a split connected reductive algebraic group defined over
$\mathfrak{F}$. We fix a maximal split $\mathfrak{F}$-torus $\mathbb{T}$ of
$\mathbb{G}$ and let $\mathbb{C}$ denote the connected component of the centre
of $\mathbb{G}$. We denote by $T=\mathbb{T}(\mathfrak{F})$ and
$Z=\mathbb{C}(\mathfrak{F})$ the corresponding groups of
$\mathfrak{F}$-rational points. Let $r_{ss}$, resp. $r_{Z}$, denote the
semisimple rank of $G$, resp. the rank of $Z$.

We let $\mathscr{X}$ denote the semisimple Bruhat-Tits building of $G$. In the
standard apartment
$X_{*}(\mathbb{T}/\mathbb{C})\otimes_{{\mathbb{Z}}}\mathbb{R}$ of
$\mathscr{X}$, we fix a chamber $C$ and a hyperspecial vertex $x_{0}$ such
that $x_{0}\in\overline{C}$. Given a facet $F$ in $\mathscr{X}$, we let
$P_{F}$ denote the parahoric subgroup of $G$ associated to $F$, and by
$P_{F}^{\dagger}$ the stabiliser in $G$ of the face $F$. The group $P_{F}$ has
a pro-$p$ radical which we denote by $I_{F}$. In particular we set $I=I_{C}$
and $I^{\prime}=P_{C}$, which are called a pro-$p$ Iwahori subgroup and
Iwahori subgroup of $G$, respectively. We note that whenever
$F^{\prime}\subseteq\overline{F}$, then we have inclusions

(2.1) $I_{F^{\prime}}\subseteq I_{F}\subseteq P_{F}\subseteq
P_{F^{\prime}}\subseteq P_{F^{\prime}}^{\dagger}$

in which $I_{F^{\prime}}$ and $P_{F^{\prime}}$ are normal in
$P_{F^{\prime}}^{\dagger}$. The chamber $C$ determines a set $\Phi^{+}$ of
positive roots of the root system $\Phi\subseteq X^{*}(\mathbb{T}/\mathbb{C})$
of $(\mathbb{G},\mathbb{T})$. We will view elements of $\Phi$ as characters of
$T$. We let $\Pi$ denote the basis of $\Phi$ defined by $\Phi^{+}$, and
$B=T\ltimes U$ be the Borel subgroup containing $T$ defined by $\Phi^{+}$,
where $U$ is the unipotent radical of $B$. A _standard parabolic subgroup_
$P=M\ltimes N$ is any parabolic subgroup of $G$ containing $B$. The
corresponding Levi subgroup $M$ will also be called _standard_. Given a
standard parabolic subgroup $P=MN$, we let $\Pi_{M}$ (resp. $\Phi_{M}$, resp.
$\Phi_{M}^{+}$) be the corresponding simple roots (resp. root system, resp.
positive roots) defined by $M$. We will sometimes also write $\Pi_{P}$ to
denote this same set of simple roots.

Now let $W_{0}=N_{G}(T)/T$ be the finite Weyl group of $G$, with length
function $\ell:W_{0}\to{\mathbb{Z}}_{\geq 0}$ defined with respect to $\Pi$.
Moreover we denote by

$W=N_{G}(T)/(T\cap I^{\prime})\cong\Lambda\rtimes W_{0}$

the extended Weyl group of $(G,T)$, where $\Lambda:=T/(T\cap I^{\prime})$. The
action of $G$ on $\mathscr{X}$ restricts to an action of $W$ on the standard
apartment by affine automorphisms. We let

$\langle-,-\rangle:X^{*}(\mathbb{T}/\mathbb{C})\times
X_{*}(\mathbb{T}/\mathbb{C})\to{\mathbb{Z}}$

denote the natural perfect pairing. The action of $T$ on the standard
apartment is then given by translation via $\nu$, where $\nu:T\to
X_{*}(\mathbb{T}/\mathbb{C})$ is the group homomorphism characterised by

$\langle\alpha,\nu(t)\rangle=-{\operatorname{val}}(\alpha(t))$

for all $\alpha\in\Phi$ and all $t\in T$. Here
${\operatorname{val}}:\mathfrak{F}^{\times}\to{\mathbb{Z}}$ denotes the
normalised valuation.

We now set

$\widetilde{\Lambda}:=T/(T\cap I),\quad\widetilde{W}:=N_{G}(T)/(T\cap I).$

Note that the map $\nu$ descends to $\Lambda$ and $\widetilde{\Lambda}$. The
group $(T\cap I^{\prime})/(T\cap I)$ identifies with the group of
$\mathbb{F}_{q}$-rational points of $\mathbb{T}$, which we denote by
$T(\mathbb{F}_{q})$. So we see that $\widetilde{\Lambda}$ is an extension of
$\Lambda$ by $T(\mathbb{F}_{q})$. For any standard Levi subgroup $M$, we let
$W_{0,M}\subset W_{0}$ be the corresponding finite Weyl group. We then denote
by $W_{M}$ (resp. $\widetilde{W}_{M}$) the preimage of $W_{0,M}$ under the
natural projection $W\twoheadrightarrow W_{0}$ (resp.
$\widetilde{W}\twoheadrightarrow W_{0}$). Given any $w\in W_{0,M}$, we fix a
lift $n_{w}\in\widetilde{W}_{M}$. Note that the length function $\ell$ extends
to both $W$ and $\widetilde{W}$.

By the identification $W\cong\Lambda\rtimes W_{0}$, we may (and indeed always
will) view $W_{0,M}$ as a subgroup of $W_{M}$. Moreover, any element
$\tilde{w}\in\widetilde{W}$ may be written as $\lambda n_{\overline{w}}$,
where $\lambda\in\widetilde{\Lambda}$ and $\overline{w}$ is the image of
$\tilde{w}$ in $W_{0}$ under the projection. We also define

$W_{0}^{M}:=\\{w\in W_{0}\mid w(\Phi^{+}_{M})\subset\Phi^{+}\\}.$

By the Bruhat decomposition,

(2.2) $G=\coprod_{\tilde{w}\in\widetilde{W}}I\tilde{w}I,\quad
M=\coprod_{\tilde{w}\in\widetilde{W}_{M}}I_{M}\tilde{w}I_{M}$

where $I_{M}:=I\cap M$ is a pro-$p$ Iwahori subgroup of $M$.

Let $\Omega=\\{w\in W\mid\ell(w)=0\\}$. Then we have a decomposition
$W=W_{\text{aff}}\rtimes\Omega$ where $W_{\text{aff}}$ denotes the affine Weyl
group, generated by the set $S$ of simple affine reflections fixing the walls
of $C$. Moreover, if we let
$\Pi_{\operatorname{aff}}=\Pi\cup\\{(\alpha_{1},1),\ldots,(\alpha_{r},1)\\}\subseteq\Phi^{+}_{\operatorname{aff}}$
where $\alpha_{1},\ldots,\alpha_{r}$ range over the longest roots in each
irreducible component of the root system $\Phi$, then the elements of $S$ are
all of the form $s_{\alpha}$ for $\alpha\in\Pi_{\operatorname{aff}}$ (cf. [26,
§4.3]). Given any affine root $\alpha$ of the form
$\alpha(t)=\beta(t)+\lambda$ with $\beta\in\Phi$, $\lambda\in{\mathbb{Z}}$,
and $t\in T$, we set $\alpha^{\vee}=\beta^{\vee}$. For any given
$s=s_{\alpha}\in S$, we let the subtorus $T_{s}\subseteq T$ denote the
$\mathfrak{F}$-rational points of the algebraic group image of the cocharacter
$\alpha^{\vee}$. Its group of $\mathbb{F}_{q}$-rational points
$T_{s}(\mathbb{F}_{q})$ is a subgroup of $T(\mathbb{F}_{q})$ isomorphic to
$\mathbb{F}_{q}^{\times}$. Now, given an $\overline{\mathbb{F}_{p}}$-valued
character $\xi$ of $T(\mathbb{F}_{q})$, we let

(2.3) $S_{\xi}:=\\{s\in S\mid\xi(t)=1\text{ for all }t\in
T_{s}(\mathbb{F}_{q})\\}.$

The group $\Omega$ is a finitely generated abelian group and we write
$\Omega_{\text{tor}}$ for its (finite) torsion subgroup.

To each face $F\subseteq\overline{C}$ we associate a subset $S_{F}\subseteq S$
of those reflections fixing $F$ pointwise. Note that $S_{C}=\emptyset$. If
$\Phi=\sqcup_{i=0}^{r}\Phi_{i}$ is the decomposition of $\Phi$ into
irreducible components and $\Pi=\sqcup_{i=0}^{r}\Pi_{i}$ is the corresponding
decomposition of the basis, we also have that $S=\sqcup_{i=0}^{r}S_{i}$ where,
in the above notation,
$S_{i}=\\{s_{\beta}\mid\beta\in\Pi_{i}\\}\cup\\{s_{(\alpha_{i},1)}\\}$. Then
the assignment $F\mapsto S_{F}$ is a bijection between the faces
$F\subseteq\overline{C}$ and the subsets $S^{\prime}\subseteq S$ with the
property that $S^{\prime}\cap S_{i}\neq S_{i}$ for every $1\leq i\leq r$ (c.f.
[9, §1.3.5]). Moreover, this bijection is order-reversing in the sense that
$F^{\prime}\subseteq\overline{F}$ if and only if $S_{F}\subseteq
S_{F^{\prime}}$.

###  2.2. Pro-$p$ Iwahori-Hecke algebras

Let $k$ denote a field of characteristic $p$, and set

${\mathbb{X}}:={\operatorname{ind}}_{I}^{G}(\mathbf{1})=\\{f:G/I\to k\mid
f\text{ has finite support}\\}\in{\operatorname{Rep}}_{k}^{\infty}(G).$

The representation ${\mathbb{X}}$ is a right module over its opposite
endomorphism ring

${\mathcal{H}}:={\operatorname{End}}_{G}({\mathbb{X}})^{{\operatorname{op}}},$

the so-called pro-$p$ Iwahori-Hecke algebra of $G$ with respect to $I$ over
$k$. By Frobenius reciprocity, ${\mathcal{H}}$ can be identified with the
convolution $k$-algebra

${\mathcal{H}}={\operatorname{Hom}}_{G}({\mathbb{X}},{\mathbb{X}})\cong{\mathbb{X}}^{I}={\operatorname{ind}}_{I}^{M}(\mathbf{1})^{I}=k[I\backslash
G/I]$

of $I$-biinvariant compactly supported maps $G\to k$. This construction has
analogues for various subgroups of $G$, which we briefly explain now.

For any face $F$ of $\mathscr{X}$, we set

${\mathbb{X}}_{F}={\operatorname{ind}}_{I_{C(F)}}^{P_{F}}(\mathbf{1})\quad\text{and}\quad{\mathbb{X}}_{F}^{\dagger}={\operatorname{ind}}_{I_{C(F)}}^{P_{F}^{\dagger}}(\mathbf{1}),$

where $C(F)$ is the chamber defined in [19, Lemma 1.3], and the corresponding
Hecke algebras are then defined analogously as

${\mathcal{H}}_{F}={\operatorname{End}}_{P_{F}}({\mathbb{X}}_{F})^{{\operatorname{op}}}\quad\text{and}\quad{\mathcal{H}}_{F}^{\dagger}={\operatorname{End}}_{P_{F}^{\dagger}}({\mathbb{X}}_{F}^{\dagger})^{{\operatorname{op}}}.$

When $F\subseteq\overline{C}$ then $C(F)=C$, $I_{C(F)}=I$ and so extending
functions by zero gives embeddings of ${\mathbb{X}}_{F}$ and
${\mathbb{X}}_{F}^{\dagger}$ into ${\mathbb{X}}$ which are $P_{F}$-equivariant
and $P_{F}^{\dagger}$-equivariant respectively. The algebras
${\mathcal{H}}_{F}$ and ${\mathcal{H}}_{F}^{\dagger}$ can then be naturally
identified with subalgebras of ${\mathcal{H}}$ by viewing them as
$I$-biinvariant functions on $P_{F}$ and $P_{F}^{\dagger}$ respectively and
extending functions by zero. We then have that ${\mathcal{H}}$ is free over
${\mathcal{H}}_{F}^{\dagger}$, which itself is also free over
${\mathcal{H}}_{F}$ (c.f.[26, Lemma 4.20 & Proposition 4.21]).

We also let $G_{\operatorname{aff}}$ be the subgroup of $G$ generated by all
the parahoric subgroups $G_{F}$ for $F$ varying among all the faces of
$\mathscr{X}$. We can then also construct a
$G_{\operatorname{aff}}$-representation
${\mathbb{X}}_{\operatorname{aff}}={\operatorname{ind}}_{I}^{G_{\operatorname{aff}}}(\mathbf{1})$
and an associated Hecke algebra
${\mathcal{H}}^{\operatorname{aff}}={\operatorname{End}}_{G_{\operatorname{aff}}}({\mathbb{X}}_{\operatorname{aff}})^{{\operatorname{op}}}$.
Via extension by zero, we can once again identify
${\mathbb{X}}_{\operatorname{aff}}$ with a
$G_{\operatorname{aff}}$-subrepresentation of ${\mathbb{X}}$ and
${\mathcal{H}}^{\operatorname{aff}}$ with a $k$-subalgebra of ${\mathcal{H}}$.
Viewing all the above Hecke algebras as subalgebras of ${\mathcal{H}}$, we
note that ${\mathcal{H}}^{\operatorname{aff}}$ contains all the
${\mathcal{H}}_{F}$ for $F\subseteq\overline{C}$ and is in fact generated by
them as an algebra (c.f. the proof of [19, Lemma 1.4]). Furthermore, for any
$F\subseteq\overline{C}$ we have that ${\mathcal{H}_{\operatorname{aff}}}$ is
free as both a left and right ${\mathcal{H}}_{F}$-module.

For any standard Levi subgroup $M$, we also let
${\mathbb{X}}_{M}:={\operatorname{ind}}_{I_{M}}^{M}(\mathbf{1})$ and
${\mathcal{H}}_{M}:={\operatorname{End}}_{M}({\mathbb{X}}_{M})^{{\operatorname{op}}}$.
By Frobenius reciprocity, ${\mathcal{H}}_{M}$ can again be identified with the
convolution $k$-algebra of $I_{M}$-biinvariant compactly supported maps $M\to
k$. Note however that ${\mathcal{H}}_{M}$ is _not_ a subalgebra of
${\mathcal{H}}$ in general. By Frobenius reciprocity again, we have a functor
$U_{M}:{\operatorname{Rep}}_{k}^{\infty}(M)\to{\operatorname{Mod}}({\mathcal{H}}_{M})$
defined by $U_{M}(V):={\operatorname{Hom}}_{M}({\mathbb{X}}_{M},V)\cong
V^{I_{M}}$ for any $V\in{\operatorname{Rep}}_{k}^{\infty}(M)$. When $M=G$, we
drop the subscript and write $U$ for the corresponding functor.

For any $\tilde{w}\in\widetilde{W}_{M}$, we set
$T^{M}_{\tilde{w}}:=I_{M}\tilde{w}I_{M}\in k[I_{M}\backslash
M/I_{M}]={\mathcal{H}}_{M}$, the characteristic function of the double coset
$I_{M}\tilde{w}I_{M}$. We again drop the superscript and write $T_{\tilde{w}}$
when $M=G$. The ring structure of ${\mathcal{H}}_{M}$ is determined by the
braid relation

$T^{M}_{\tilde{w}\tilde{w}^{\prime}}=T^{M}_{\tilde{w}}T^{M}_{\tilde{w}^{\prime}}\quad\text{if}\quad\ell_{M}(\tilde{w}\tilde{w}^{\prime})=\ell_{M}(\tilde{w})+\ell_{M}(\tilde{w}^{\prime}),$

and a certain quadratic relation, c.f. [26, §4.8, eq. (4.10)]. By the Bruhat
decomposition (2.2), the set
$\\{T^{M}_{\tilde{w}}\mid\tilde{w}\in\widetilde{W}\\}$ is a $k$-basis of
${\mathcal{H}}_{M}$. There are also certain dual elements
$T^{M,*}_{\tilde{w}}$ which form another $k$-basis of ${\mathcal{H}}_{M}$,
c.f. §4.8, eq. (4.11) in loc. cit. If $\widetilde{W}_{\operatorname{aff}}$
denotes the pre-image of $W_{\operatorname{aff}}$ under
$\widetilde{W}\twoheadrightarrow W$, then the set
$\\{T_{\tilde{w}}\mid\tilde{w}\in\widetilde{W}_{\operatorname{aff}}\\}$ is a
$k$-basis of ${\mathcal{H}}^{\operatorname{aff}}$. Furthermore, for any
$F\subseteq\overline{C}$ we let $W_{F}$ denote the subgroup of $W$ generated
by $S_{F}$ and $\widetilde{W}_{F}$ its pre-image in $\widetilde{W}$.

Recall from [1, §4.1] that for $w\in W_{0,M}$ and
$\lambda\in\widetilde{\Lambda}$, the element $\lambda
n_{w}\in\widetilde{W}_{M}$ is called _$M$ -positive_ (resp.  _$M$ -negative_)
if $\langle\alpha,\nu(\lambda)\rangle\leq 0$ (resp.
$\langle\alpha,\nu(\lambda)\rangle\geq 0$) for every
$\alpha\in\Phi^{+}\setminus\Phi_{M}^{+}$. The $k$-span of the
$T^{M}_{\tilde{w}}$ with $M$-positive (resp. $M$-negative)
$\tilde{w}\in\widetilde{W}$ forms a $k$-subalgebra of ${\mathcal{H}}_{M}$
which we denote by ${\mathcal{H}}_{M}^{+}$ (resp. ${\mathcal{H}}_{M}^{-}$). By
Lemma 4.6 of loc. cit. we have injective algebra homomorphisms

(2.4)
$j_{M}^{+}:{\mathcal{H}}_{M}^{+}\hookrightarrow{\mathcal{H}}\quad\text{and}\quad
j_{M}-:{\mathcal{H}}_{M}^{-}\hookrightarrow{\mathcal{H}}$

given respectively by $j_{M}^{+}(T^{M}_{\tilde{w}})=T_{\tilde{w}}$ and
$j_{M}^{+}(T^{M,*}_{\tilde{w}})=T^{*}_{\tilde{w}}$ for all $M$-positive (resp.
$M$-negative) $\tilde{w}\in\widetilde{W}_{M}$. Now fix
$\lambda_{0}^{+},\lambda_{0}^{-}\in\widetilde{\Lambda}$ such that they are in
the centre of $\widetilde{W}_{M}$ and
$\langle\alpha,\nu(\lambda_{0}^{+})\rangle<0$ (resp.
$\langle\alpha,\nu(\lambda_{0}^{-})\rangle>0$) for every
$\alpha\in\Phi^{+}\setminus\Phi_{M}^{+}$. We will need the following:

######  Proposition 2.1 ([30, Theorem 1.4]).

For any $\lambda_{0}^{+},\lambda_{0}^{-}\in\widetilde{\Lambda}$ chosen as
above, the elements $T^{M}_{\lambda_{0}^{\pm}}\in{\mathcal{H}}_{M}^{\pm}$ are
central and non-zero divisors, and ${\mathcal{H}}_{M}$ is the Ore localisation
of ${\mathcal{H}}_{M}^{\pm}$ at $T^{M}_{\lambda_{0}^{\pm}}$.

We finally need to introduce twists by the longest element $w_{0}\in W_{0}$ of
the finite Weyl group. Given a standard parabolic $P$ with Levi subgroup $M$,
let $w_{0}^{M}=w_{0}w_{0,M}$ where $w_{0,M}$ denotes the longest element in
$W_{0,M}$. Then we let $w_{0}(M)$ denote the standard Levi subgroup determined
by $W_{0,w_{0}(M)}=w_{0}W_{0,M}w_{0}$, and we let $w_{0}(P)$ be the standard
parabolic whose Levi subgroup is $w_{0}(M)$. Lastly, we fix a lift
$\hat{w}_{0}^{M}\in\widetilde{W}$ of $w_{0}^{M}$.

######  Proposition 2.2 ([30, Proposition 2.20]).

The $k$-linear map

${\mathcal{H}}_{M}\to{\mathcal{H}}_{w_{0}(M)},\quad\quad
T^{M}_{\tilde{w}}\mapsto
T^{w_{0}(M)}_{\tilde{w}_{0}^{M}\tilde{w}(\tilde{w}_{0}^{M})^{-1}}\quad\text{for
}\tilde{w}\in\widetilde{W}_{M},$

is an isomorphism of $k$-algebras. This induces an exact equivalence of
categories

$\iota_{0}^{M}:{\operatorname{Mod}}({\mathcal{H}}_{M})\to{\operatorname{Mod}}({\mathcal{H}}_{w_{0}(M)})$

such that $\iota_{M}({\mathcal{H}}_{M})\cong{\mathcal{H}}_{w_{0}(M)}$.

The above satisfies a transitivity property as follows. If $P\subseteq
P^{\prime}$ are standard parabolics with Levi subgroups $M\subseteq
M^{\prime}$, define $w_{M^{\prime}}^{M}:=w_{0,M^{\prime}}w_{0,M}$. Then as
above we have an equivalence of categories
$\iota_{M^{\prime}}^{M}:{\operatorname{Mod}}({\mathcal{H}}_{M})\to{\operatorname{Mod}}({\mathcal{H}}_{w_{0,M^{\prime}}(M)})$
induced by the isomorphism $T^{M}_{\tilde{w}}\mapsto
T^{w_{0,M^{\prime}}(M)}_{\tilde{w}_{M^{\prime}}^{M}\tilde{w}(\tilde{w}_{M^{\prime}}^{M})^{-1}}$.
Furthermore, the map $T^{w_{0,M^{\prime}}(M)}_{\tilde{w}}\mapsto
T^{w_{0}(M)}_{\tilde{w}_{0}^{M^{\prime}}\tilde{w}(\tilde{w}_{0}^{M^{\prime}})^{-1}}$
also defines an isomorphism
${\mathcal{H}}_{w_{0,M^{\prime}}(M)}\cong{\mathcal{H}}_{w_{0}(M)}$ which
induces an equivalence
$\iota_{0,M^{\prime}}^{M}:{\operatorname{Mod}}({\mathcal{H}}_{w_{0,M^{\prime}}(M)})\to{\operatorname{Mod}}({\mathcal{H}}_{w_{0}(M)})$.
Then the equality

(2.5) $\iota_{0}^{M}=\iota_{0,M^{\prime}}^{M}\circ\iota_{M^{\prime}}^{M}$

holds, c.f. [30, Proposition 2.21].

###  2.3. Parabolic induction and simple modules

We recall the construction of the parabolic induction and coinduction functors
from [30, §3]. We fix a standard Levi subgroup $M$ of $G$. By (2.4) we may
view the algebra ${\mathcal{H}}$ as a bimodule over the algebras
${\mathcal{H}}_{M}^{+}$ and ${\mathcal{H}}_{M}^{-}$ via the maps $j_{M}^{+}$
and $j_{M}^{-}$ respectively. Thus we may define two functors
${\operatorname{Mod}}({\mathcal{H}}_{M})\to{\operatorname{Mod}}({\mathcal{H}})$
by setting

${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}}):={\mathcal{H}}\otimes_{{\mathcal{H}}_{M}^{+}}{\mathfrak{m}}\quad\text{and}\quad{\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}}):={\operatorname{Hom}}_{{\mathcal{H}}_{M}^{-}}({\mathcal{H}},{\mathfrak{m}})$

for any ${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$. The
functors ${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$ and
${\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$ are called
_parabolic induction_ and _parabolic coinduction_ respectively.

By the tensor-hom adjunction, we see that
${\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$ has left adjoint
${\operatorname{Loc}}_{M}:={\mathcal{H}}_{M}\otimes_{{\mathcal{H}}_{M}^{-}}(-)$
and ${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$ has right
adjoint
${\mathcal{R}^{G}_{M}}:={\operatorname{Hom}}_{{\mathcal{H}}_{M}^{+}}({\mathcal{H}}_{M},-)$.
Note that since ${\mathcal{H}}_{M}$ is the Ore localisation of
${\mathcal{H}}_{M}^{-}$ at $T^{M}_{\lambda_{0}^{-}}$, the functor
${\operatorname{Loc}}_{M}$ is the corresponding localisation functor and hence
is exact. The counit of the adjoint pair
$({\operatorname{Loc}}_{M},{\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}})$
is given as follows. For
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$, we have a natural
map
${\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}})={\operatorname{Hom}}_{{\mathcal{H}}_{M}^{-}}({\mathcal{H}},{\mathfrak{m}})\to{\mathfrak{m}}$
by evaluating at $1$. Localising at $T^{M}_{\lambda_{0}^{-}}$ this gives the
counit

(2.6)
$\displaystyle\varepsilon_{{\mathfrak{m}}}:{\mathcal{H}}_{M}\otimes_{{\mathcal{H}}_{M}^{-}}{\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}})$
$\displaystyle\to{\mathfrak{m}}$
$\displaystyle(T^{M}_{\lambda_{0}^{-}})^{-n}\otimes\varphi$
$\displaystyle\mapsto(T^{M}_{\lambda_{0}^{-}})^{-n}\cdot\varphi(1)$

The main properties of these functors that we’ll need are summarised below:

######  Theorem 2.3 ([30, 1.6-1.10]).

Let $M$ be a standard Levi subgroup of $G$. Then:

  1. (i)

there are natural isomorphisms

${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}\cong{\operatorname{Coind}}_{{\mathcal{H}}_{w_{0}(M)}}^{{\mathcal{H}}}\circ\iota_{0}^{M}\quad\text{and}\quad{\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}\cong{\operatorname{Ind}}_{{\mathcal{H}}_{w_{0}(M)}}^{{\mathcal{H}}}\circ\iota_{0}^{M},$

where $\iota_{0}^{M}$ is the equivalence from 2.2; and

  2. (ii)

parabolic induction and coinduction are transitive and have both a left and a
right adjoint. In particular they are exact functors.

Note that the left (resp. right) adjoint of
${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$ (resp.
${\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$) can be described
explicitly, namely it is equal to the composite
$\mathcal{L}_{M}:=\iota_{0}^{w_{0}(M)}\circ{\operatorname{Loc}}_{w_{0}(M)}$
(resp. $\iota_{0}^{w_{0}(M)}\circ\mathcal{R}_{w_{0}(M)}$), c.f. [30, Theorem
1.9].

We will use the next general categorical fact several times, so we record it
here:

######  Lemma 2.4 ([18, Proposition 1.5.6]).

Suppose $F:{\mathcal{C}}\rightleftarrows{\mathcal{D}}:U$ is an adjunction, and
denote by $\eta$ and $\varepsilon$ the unit and counit respectively. Then $F$
(resp. $U$) is fully faithful if and only if $\eta$ (resp. $\varepsilon$) is
an isomorphism.

The next result is, to the best of our knowledge, not explicitly available
anywhere in the literature. Its proof was however inspired by the arguments
used in the proof of [20, Lemma 8.1]. We note that it also answers the
question asked in the last paragraph of [30, §1], in the case where the
coefficient ring is a field of characteristic $p$.

######  Proposition 2.5.

For any standard parabolic subgroup $M$ of $G$, the parabolic induction and
coinduction functors are fully faithful.

###### Proof.

It suffices to show the result for coinduction since they differ by a twist.
By 2.4 we must show that the counit $\varepsilon_{{\mathfrak{m}}}$ is an
isomorphism for all
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$. By [1, Lemma
4.10], the map $\varphi\mapsto(\varphi(T_{\hat{v}}))_{v\in W_{0}^{M}}$ gives a
$k$-linear isomorphism
${\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}})\cong\bigoplus_{v\in
W_{0}^{M}}{\mathfrak{m}}$, where $\hat{v}$ denotes a fixed lift of $v$ in
$\widetilde{W}$ (and by convention we impose $\hat{1}=1$). Under this
identification and from (2.6), we see that $\varepsilon_{{\mathfrak{m}}}$ is
the localisation of the projection onto the $v=1$ summand.

Now, by [1, Proposition 4.12] (whose statement assumes $k$ to be algebraically
closed but whose proof works in our level of generality) and [30, Lemma 2.18],
the action of $T_{\lambda_{0}^{-}}$ on the summand corresponding to $v\in
W_{0}^{M}$ is zero if $\hat{v}^{-1}\lambda_{0}^{-}$ is not $M$-negative, and
the action on the $v=1$ component agrees with the action of
$T^{M}_{\lambda_{0}^{-}}$ on ${\mathfrak{m}}$. Hence, from the definition of
$\varepsilon_{{\mathfrak{m}}}$, we’ll be done if we show that
$\hat{v}^{-1}\lambda_{0}^{-}$ is not $M$-negative for $v\neq 1$.

Pick $\alpha\in\Pi$ such that $s_{\alpha}v<v$. Note that
$\alpha\in\Phi^{+}\setminus\Phi_{M}^{+}$ because $v\in W_{0}^{M}$ (c.f. [8,
Ch. IV, Exercices du §1, (3)]). Then, using the fact that $v\in W_{0}^{M}$ and
that $v(-v^{-1}(\alpha))=-\alpha\in-\Phi^{+}$, we obtain that
$-v^{-1}(\alpha)\in\Phi^{+}\setminus\Phi_{M}^{+}$. Then, we get

$\langle-v^{-1}(\alpha),\nu(\hat{v}^{-1}\lambda_{0}^{-})\rangle=-\langle\alpha,\nu(\lambda_{0}^{-})\rangle<0$

by definition of $\lambda_{0}^{-}$, and so $\hat{v}^{-1}\lambda_{0}^{-}$ is
not $M$-negative as required. ∎

We now recall the classification of simple Hecke modules. Here we assume that
$k$ is algebraically closed. The classification was initially stated in terms
of parabolic coinduction by Abe [1, Theorem 4.22], and was later reworded in
terms of parabolic induction (cf. [4, Corollary 4.30]). We first need to
recall the definition of a supersingular ${\mathcal{H}}$-module as follows.
Recall from [31, §5] that for a choice of spherical orientation $o$, i.e. of a
closed Weyl chamber of $W_{0}$, we can associate a basis
$\\{E_{o}(\tilde{w})\\}_{\tilde{w}\in\widetilde{W}}$ of ${\mathcal{H}}$ with
the property that $E_{o}(\tilde{w})\in
T_{\tilde{w}}+\sum_{\tilde{v}<\tilde{w}}kT_{\tilde{v}}$. Now let $\mathcal{O}$
be a $\widetilde{W}$-conjugacy class in $\tilde{\Lambda}$, and let

$z_{\mathcal{O}}=\sum_{\tilde{\lambda}\in\mathcal{O}}E_{o}(\tilde{\lambda}).$

Then this element does not depend on the choice of orientation $o$, and
defines an element of the centre of ${\mathcal{H}}$, c.f. [33, Theorem 5.1].
The length of $\tilde{\lambda}\in\mathcal{O}$ is independent of the choice of
representative $\tilde{\lambda}$, and we denote it by $\ell(\mathcal{O})$.

######  Definition 2.6.

An ${\mathcal{H}}$-module ${\mathfrak{m}}$ is called supersingular if for each
$\widetilde{W}$-conjugacy class $\mathcal{O}$ in $\widetilde{\Lambda}$ with
$\ell(\mathcal{O})>0$ and for every $x\in{\mathfrak{m}}$, there exists some
$n\in\mathbb{N}$ such that $z_{\mathcal{O}}^{n}x=0$. If the integer $n$ can be
chosen independently of $x$ and $\mathcal{O}$, we call ${\mathfrak{m}}$
_uniformly supersingular_.

Note that the notions of supersingular and uniformly supersingular coincide
for ${\mathcal{H}}$-modules of finite length, and more generally finitely
generated modules, and hence in particular for simple modules.

Next, a _standard simple supersingular triple_ for ${\mathcal{H}}$ is a triple
$(P,\mathfrak{n},Q)$ such that

  * •

$P$ is a standard parabolic subgroup of $G$ with Levi decomposition $P=MN$;

  * •

$\mathfrak{n}$ is a simple supersingular module over ${\mathcal{H}}_{M}$; and

  * •

$Q$ is a parabolic subgroup of $G$ with $P\subseteq Q\subseteq
P(\mathfrak{n})$, where $P(\mathfrak{n})$ is the standard parabolic subgroup
of $G$ corresponding to
$\Pi(\mathfrak{n}):=\Pi_{M}\cup\Pi_{\mathfrak{n}}\subseteq\Pi$ and
$\alpha\in\Pi_{\mathfrak{n}}$ if and only if both
$\langle\Pi_{M},\alpha^{\vee}\rangle=0$ and $T_{\lambda}^{M}$ acts trivially
on $\mathfrak{n}$ for all $\lambda\in T^{\prime}_{\alpha}$.

In the above, for $\alpha\in\Pi$, we set $T^{\prime}_{\alpha}=T\cap
G^{\prime}_{\alpha}$ where $G^{\prime}_{\alpha}$ denotes the subgroup of $G$
generated by the root subgroups corresponding to $\pm\alpha$. Two
supersingular triples $(P,\mathfrak{n},Q)$ and
$(P^{\prime},\mathfrak{n}^{\prime},Q^{\prime})$ are called equivalent if
$P=P^{\prime}$, $Q=Q^{\prime}$ and $\mathfrak{n}\cong\mathfrak{n}^{\prime}$.

Given a supersingular triple $(P,\mathfrak{n},Q)$, we define
$I_{\mathcal{H}}(P,\mathfrak{n},Q),CI_{\mathcal{H}}(P,\mathfrak{n},Q)\in{\operatorname{Mod}}({\mathcal{H}})$
as follows. Suppose that $Q$ has Levi decomposition $Q=LV$. Now assume that
$Q^{\prime}$ is any parabolic subgroup of $G$ with $P\subseteq
Q^{\prime}\subseteq P(\mathfrak{n})$ and say it has Levi decomposition
$Q^{\prime}=L^{\prime}V^{\prime}$. Then $\mathfrak{n}$ can be extended
uniquely to an ${\mathcal{H}}_{L^{\prime}}$-module
$e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n})$ with certain properties, and
moreover whenever $P\subseteq Q\subseteq Q^{\prime}\subseteq P(\mathfrak{n})$
there is a natural injective ${\mathcal{H}}$-linear map
${\operatorname{Ind}}_{{\mathcal{H}}_{L^{\prime}}}^{\mathcal{H}}(e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n}))\to{\operatorname{Ind}}_{{\mathcal{H}}_{L}}^{\mathcal{H}}(e_{{\mathcal{H}}_{L}}(\mathfrak{n}))$
(cf. [4, Corollary 3.9 & Proposition 4.5]). We may then define the modules

$I_{\mathcal{H}}(P,\mathfrak{n},Q):={\operatorname{coker}}\left(\bigoplus_{Q\subseteq
Q^{\prime}\subseteq
P(\mathfrak{n})}{\operatorname{Ind}}_{{\mathcal{H}}_{L^{\prime}}}^{\mathcal{H}}(e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n}))\to{\operatorname{Ind}}_{{\mathcal{H}}_{L}}^{\mathcal{H}}(e_{{\mathcal{H}}_{L}}(\mathfrak{n}))\right)$

and

$CI_{\mathcal{H}}(P,\mathfrak{n},Q):={\operatorname{coker}}\left(\bigoplus_{Q\subseteq
Q^{\prime}\subseteq
P(\mathfrak{n})}{\operatorname{Coind}}_{{\mathcal{H}}_{L^{\prime}}}^{\mathcal{H}}(e_{{\mathcal{H}}_{L^{\prime}}}(\mathfrak{n}))\to{\operatorname{Coind}}_{{\mathcal{H}}_{L}}^{\mathcal{H}}(e_{{\mathcal{H}}_{L}}(\mathfrak{n}))\right).$

The classification of simple ${\mathcal{H}}$-modules then states that
$(P,\mathfrak{n},Q)\mapsto I_{\mathcal{H}}(P,\mathfrak{n},Q)$ and
$(P,\mathfrak{n},Q)\mapsto CI_{\mathcal{H}}(P,\mathfrak{n},Q)$ define two
bijections between the set of equivalence classes of standard simple
supersingular triples and the set of isomorphism classes of simple
${\mathcal{H}}$-modules, cf. [1, Theorem 4.22] and [4, Corollary 4.30].
Moreover, the two bijections are related by an ${\mathcal{H}}$-equivariant
isomorphism

(2.7) $CI_{\mathcal{H}}(P,\mathfrak{n},Q)\cong
I_{\mathcal{H}}(w_{0}(P),\iota_{0}^{M}(\mathfrak{n}),w_{0}(Q))$

cf. [4, Corollary 4.24]. Finally, a simple module
$I_{\mathcal{H}}(P,\mathfrak{n},Q)$ is supersingular if and only if $P=Q=G$
(cf. [1, Corollary 4.31]), and analogously for
$CI_{\mathcal{H}}(P,\mathfrak{n},Q)$.

###  2.4. Recollections on abelian model categories

We briefly recall some facts about model structures on abelian categories. For
details on the general theory of model categories, we refer the reader to the
book [15]. Fix a bicomplete abelian category ${\mathcal{A}}$. We will say that
a model structure on ${\mathcal{A}}$ is an _abelian model structure_ if the
class of (trivial) fibrations (resp. cofibrations) coincides with the class of
epimorphisms (resp. monomorphisms) with (trivially) fibrant (resp. cofibrant)
kernel (resp. cokernel). An abelian category equipped with such a model
structure will be called an _abelian model category_.

By work of Hovey, c.f. [16], constructing abelian model structures on
${\mathcal{A}}$ is equivalent to constructing certain cotorsion pairs. Given a
pair $({\mathcal{C}},{\mathcal{F}})$ of classes of objects of ${\mathcal{A}}$,
consider the pair $(^{\perp}{\mathcal{F}},{\mathcal{C}}^{\perp})$ defined by

${}^{\perp}{\mathcal{F}}=\\{X\in{\mathcal{A}}\mid{\operatorname{Ext}}^{1}_{\mathcal{A}}(X,Y)=0\text{
for all }Y\in{\mathcal{F}}\\}$

and

${\mathcal{C}}^{\perp}=\\{Y\in{\mathcal{A}}\mid{\operatorname{Ext}}^{1}_{\mathcal{A}}(X,Y)=0\text{
for all }X\in{\mathcal{C}}\\}.$

Then we say that $({\mathcal{C}},{\mathcal{F}})$ is a _cotorsion pair_ if
${\mathcal{C}}=^{\perp}{\mathcal{F}}$ and
${\mathcal{F}}={\mathcal{C}}^{\perp}$ (c.f. [16, Definition 2.3]). It is said
to have _enough functorial projectives_ if for all $X\in{\mathcal{A}}$, there
is a functorial epimorphism $f:Y\to X$ with $Y\in{\mathcal{C}}$ such that
$\ker(f)\in{\mathcal{F}}$. Dually, it is said to have _enough functorial
injectives_ if for all $X\in{\mathcal{A}}$, there is a functorial monomorphism
$g:X\to Z$ with $Z\in{\mathcal{F}}$ such that
${\operatorname{coker}}(g)\in{\mathcal{C}}$. When a cotorsion pair has both
enough functorial projectives and enough functorial injectives, then it is
said to be _functorially complete_.

Recall that a class of objects of ${\mathcal{A}}$ is called _thick_ if it is
closed under direct summands and satisfies the 2-out-of-3 property with
respect to short exact sequences. When ${\mathcal{A}}$ is equipped with a
model structure, we will say that an object $X\in{\mathcal{A}}$ is _trivial_
if the zero map $0\to X$ (or equivalently $X\to 0$) is a weak equivalence. The
fundamental result we need is the following.

######  Theorem 2.7 ([16, Theorem 2.2]).

Let ${\mathcal{A}}$ be a bicomplete abelian category.

  1. (i)

If ${\mathcal{A}}$ is an abelian model category, denote by ${\mathcal{C}}$
(resp. ${\mathcal{F}}$, resp. ${\mathcal{W}}$) the class of cofibrant (resp.
fibrant, resp. trivial) objects. Then ${\mathcal{W}}$ is thick, and
$({\mathcal{C}}\cap{\mathcal{W}},{\mathcal{F}})$ and
$({\mathcal{C}},{\mathcal{F}}\cap{\mathcal{W}})$ are functorially complete
cotorsion pairs.

  2. (ii)

Conversely, assume that $({\mathcal{C}},{\mathcal{F}},{\mathcal{W}})$ is a
triple of classes of objects of ${\mathcal{A}}$ such that ${\mathcal{W}}$ is
thick, and $({\mathcal{C}}\cap{\mathcal{W}},{\mathcal{F}})$ and
$({\mathcal{C}},{\mathcal{F}}\cap{\mathcal{W}})$ are functorially complete
cotorsion pairs. Then there is a unique abelian model structure on
${\mathcal{A}}$ such that ${\mathcal{C}}$, ${\mathcal{F}}$ and ${\mathcal{W}}$
coincide with the classes of cofibrant, fibrant and trivial objects,
respectively.

In fact, Hovey proved a more general version of the above theorem which
applies to a wider class of model structures on ${\mathcal{A}}$, using the
notion of a proper class of short exact sequences, and this has been further
generalised to the context of exact categories by Gillespie in [12]. In this
paper, we will however only need to work with abelian model structures. We
will sometimes denote an abelian model structure on ${\mathcal{A}}$ by the
triple $({\mathcal{C}},{\mathcal{F}},{\mathcal{W}})$ as in 2.7. Note that the
homotopy category ${\operatorname{Ho}}({\mathcal{A}})$ of an abelian model
category is canonically equivalent to the quotient
${\mathcal{C}}\cap{\mathcal{F}}/\Gamma$, where
$\Gamma={\mathcal{C}}\cap{\mathcal{F}}\cap{\mathcal{W}}$ (c.f. [12,
Proposition 4.4(5)]).

Given an abelian model structure $({\mathcal{C}},{\mathcal{F}},{\mathcal{W}})$
on ${\mathcal{A}}$, the completeness of the cotorsion pairs
$({\mathcal{C}}\cap{\mathcal{W}},{\mathcal{F}})$ and
$({\mathcal{C}},{\mathcal{F}}\cap{\mathcal{W}})$ ensures that for any
$X\in{\mathcal{A}}$ there are short exact sequences $0\to Y\to Q_{c}X\to X\to
0$ and $0\to X\to Q_{f}X\to Z\to 0$, with $Q_{c}X\in{\mathcal{C}}$,
$Q_{f}X\in{\mathcal{F}}$, $Y\in{\mathcal{F}}\cap{\mathcal{W}}$ and
$Z\in{\mathcal{C}}\cap{\mathcal{W}}$. This can be done functorially in our
setup, but by slight abuse of terminology we will refer to _any_ two elements
$Q_{c}X$ and $Q_{f}X$ fitting in short exact sequences as above as choices of
_cofibrant replacement_ and _fibrant replacement_ for $X$, respectively. We
will always denote such replacements by $Q_{c}X$ and $Q_{f}X$ as we did above.

As a final piece of notation, if a functor
${\mathsf{F}}:{\mathcal{A}}\to{\mathcal{B}}$ between the underlying categories
of two abelian model categories preserves all weak equivalences, it
immediately follows from the universal property of localisation that it lifts
to a functor between the homotopy categories which we will denote by
${\operatorname{Ho}}({\mathsf{F}}):{\operatorname{Ho}}({\mathcal{A}})\to{\operatorname{Ho}}({\mathcal{B}})$.
When $F$ is Quillen we can identify ${\operatorname{Ho}}({\mathsf{F}})$ with
the total derived functor of $F$ under mild assumptions as follows.

######  Lemma 2.8.

Suppose that ${\mathcal{A}}$ and ${\mathcal{B}}$ are abelian model categories,
and let ${\mathsf{F}}:{\mathcal{A}}\to{\mathcal{B}}$ be a (left or right)
Quillen functor which is exact. Then:

  1. (i)

if ${\mathsf{F}}$ preserves trivial objects then it preserves all weak
equivalences; and

  2. (ii)

if ${\mathsf{F}}$ reflects trivial objects then it reflects all weak
equivalences.

Furthermore, with the assumption of (i), the total derived functor of
${\mathsf{F}}$ is naturally isomorphic to ${\operatorname{Ho}}({\mathsf{F}})$.

###### Proof.

Suppose that $f$ is a morphism in ${\mathcal{A}}$. For (i), assume first that
$f$ is either a trivial fibration or trivial cofibration. Then in particular
it is either a monomorphism with trivial cokernel, or an epimorphism with
trivial kernel. Under our assumptions then ${\mathsf{F}}f$ is still a
monomorphism/epimorphism with trivial cokernel/kernel. It then follows from
[16, Lemma 5.8] that ${\mathsf{F}}f$ is a weak equivalence. The case of a
general $f$ follows immediately by factorising $f$ as the composite of a
trivial cofibration and a trivial fibration.

For (ii), we assume that ${\mathsf{F}}$ is left Quillen (the other case is
dual). Factorise $f=gh$ where $h$ is a trivial cofibration and $g$ is a
fibration. Then by the 2-out-of-3 property of weak equivalences it suffices to
show that if ${\mathsf{F}}g$ is a weak equivalence then so is $g$. Since
${\mathsf{F}}$ is exact, we may apply loc. cit. again to see that
${\mathsf{F}}g$ is a weak equivalence precisely when
${\mathsf{F}}({\operatorname{coker}}(g))$ is trivial. It then follows that $g$
is a trivial fibration since ${\mathsf{F}}$ reflects trivial objects.

Finally, the total derived functor of ${\mathsf{F}}$ is defined to be
${\operatorname{Ho}}({\mathsf{F}}\circ Q)$ where $Q$ is some fibrant or
cofibrant replacement functor, depending on whether ${\mathsf{F}}$ is right or
left Quillen. But by definition, any object is weakly equivalent to its
replacement and ${\mathsf{F}}$ preserves such weak equivalences under the
assumption of (i). Thus ${\operatorname{Ho}}({\mathsf{F}})$ is isomorphic the
total derived functor in that case, as required. ∎

An abelian model category ${\mathcal{A}}$ is called _hereditary_ if the class
${\mathcal{C}}$ of cofibrant objects is closed under taking kernels of
epimorphisms and if the class ${\mathcal{F}}$ of fibrant objects is closed
under taking cokernels of monomorphisms. Such an abelian model category is
automatically stable (c.f. [7, Corollary 1.1.15]), meaning that its associated
homotopy category ${\operatorname{Ho}}({\mathcal{A}})$ is canonically equipped
with the structure of a triangulated category. We quickly recall what this
triangulated structure is.

  * •

The shift functor is given by the suspension functor $\Sigma$ associated with
the model structure. For a cofibrant object $X\in{\mathcal{C}}$, choose a
cofibration $j_{X}:X\to W$ where
$W\in\Gamma={\mathcal{C}}\cap{\mathcal{F}}\cap{\mathcal{W}}$. Then we let
$\Sigma(X)$ be defined as ${\operatorname{coker}}(j_{X})$. Since
${\mathcal{A}}$ is hereditary, note that
$\Sigma(X)\in{\mathcal{C}}\cap{\mathcal{F}}$ whenever
$X\in{\mathcal{C}}\cap{\mathcal{F}}$. This construction doesn’t depend on the
choice of $j_{X}$ up to weak equivalence, and hence this defines an
endofunctor of
${\operatorname{Ho}}({\mathcal{A}})\equiv{\mathcal{C}}\cap{\mathcal{F}}/\Gamma$
which can be shown to be an auto-equivalence. There is also a dually defined
loop functor which is naturally isomorphic to $\Sigma^{-1}$. We note that the
loop and suspension functors can be defined more generally for any (not
necessarily stable) pointed model category, c.f. [6, Definition 3.1.2], and
are then adjoint functors.

  * •

The distinguished triangles are obtained from the so-called cofiber sequences
as in [6, Definition 3.4.1 & Theorem 4.2.1]. In the context of hereditary
abelian model categories, these are given as follows. Suppose that $0\to
X\stackrel{{\scriptstyle f}}{{\to}}Y\stackrel{{\scriptstyle g}}{{\to}}Z\to 0$
is a short exact sequence in ${\mathcal{A}}$ with $X,Y,Z\in{\mathcal{C}}$. Fix
$j_{X}:X\to W$ as above, so that $W\in{\mathcal{F}}\cap{\mathcal{W}}$ and
$\Sigma X\cong{\operatorname{coker}}(j_{X})$. Since
${\operatorname{Ext}}^{1}_{\mathcal{A}}(Z,W)=0$ we have that $j_{X}$ extends
along $f$ to a map $Y\to W$. Postcomposing with the projection $W\to\Sigma X$,
we obtain a map $l:Y\to\Sigma X$ such that $l\circ f=0$. Thus $l$ descends to
a map $h:Z\to\Sigma X$. The distinguished triangles in
${\operatorname{Ho}}({\mathcal{A}})$ are then, up to isomorphism, the
triangles of the form

(2.8) $X\stackrel{{\scriptstyle f}}{{\to}}Y\stackrel{{\scriptstyle
g}}{{\to}}Z\stackrel{{\scriptstyle h}}{{\to}}\Sigma X$

as above.

######  Remark 2.9.

There is a dual notion of fiber sequences which may be used to construct the
distinguished triangles as well, c.f. [6, Definition 3.4.2, Corollaries 4.4.1
& 4.4.2]. In our context, those are obtained similarly to the above by using
short exact sequences of fibrant objects and the loop functor. That these two
constructions give the same triangulated structure on the homotopoy category
holds in the general context of stable model categories (c.f. loc. cit.) but
for hereditary abelian model categories this can be also seen directly by
applying [7, Lemma 1.4.4] and its dual.

Recall that a ring $R$ is called Gorenstein if it is both left and right
noetherian and if it has finite injective dimension both as a left and as a
right $R$-module. In this case the left and right selfinjective dimensions of
$R$ coincide (cf. [11], Proposition 9.1.8). If $n$ denotes their common value
then $R$ is called an $n$-Gorenstein ring. We will frequently make use of the
fact that if $R$ is an $n$-Gorenstein ring then an $R$-module ${\mathfrak{m}}$
has finite projective dimension if and only if it has finite injective
dimension. In this case both dimensions are bounded above by $n$ (cf. [11],
Theorem 9.1.10). The relevance of this notion for us comes from the following
result of Ollivier-Schneider.

######  Theorem 2.10 ([26, Theorem 0.1]).

The pro-$p$ Iwahori-Hecke algebra ${\mathcal{H}}$ is $n$-Gorenstein where
$n\leq r_{\text{ss}}+r_{Z}$.

The bound on the selfinjective dimension of ${\mathcal{H}}$ is actually sharp
(c.f. [20, Remark 8.2]). The theorem applies generally to the algebras
${\mathcal{H}}_{M}$ for any standard Levi subgroup $M$ of $G$.

We now recall the construction, due to Hovey, of two abelian model structures
on the category ${\operatorname{Mod}}(R)$ for any Gorenstein ring $R$. Let
${\mathcal{A}}$ be an abelian category with enough projectives. An object
$A\in{\mathcal{A}}$ is called Gorenstein projective if there is an acyclic
complex

$Y=\cdots\longrightarrow Y_{1}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}Y_{0}\stackrel{{\scriptstyle
d_{0}}}{{\longrightarrow}}Y_{-1}\longrightarrow\cdots$

of projective objects of ${\mathcal{A}}$ for which $A=Z_{0}Y=B_{0}Y$ and which
remains exact upon applying ${\operatorname{Hom}}_{{\mathcal{A}}}(\cdot,P)$
for any projective object $P$ of ${\mathcal{A}}$. When
${\mathcal{A}}={\operatorname{Mod}}(R)$ for a Gorenstein ring $R$, then a
module ${\mathfrak{m}}$ is Gorenstein projective as an object of
${\operatorname{Mod}}(R)$ if and only if
${\operatorname{Ext}}^{i}_{R}({\mathfrak{m}},\mathfrak{p})=0$ for all $i\geq
1$ and all projective $R$-modules $\mathfrak{p}$ (cf. [11, Corollary 11.5.3]).
This is true if and only if
${\operatorname{Ext}}^{i}_{R}({\mathfrak{m}},\mathfrak{p})=0$ for all $i\geq
1$ and all modules $\mathfrak{p}$ of finite projective dimension. Then
${\mathfrak{m}}$ is Gorenstein projective if and only if there is an acyclic
complex $Y$ of projectives with ${\mathfrak{m}}=Z_{0}Y$, c.f. [10, Remark
3.1]. There is also dually a notion of _Gorenstein injective_ object of
${\mathcal{A}}$, and for ${\mathcal{A}}={\operatorname{Mod}}(R)$ these have
dual properties to the above (c.f. [11, Definition 10.1.1 & Corollary
11.2.2])). We denote the classes of Gorenstein projective and Gorenstein
injective modules over a ring $R$ by $\mathrm{GProj}(R)$ and
$\mathrm{GInj}(R)$, respectively.

The following fundamental theorem is due to Hovey.

######  Theorem 2.11 ([16, Theorem 8.6]).

Let $R$ be a Gorenstein ring and let ${\mathcal{W}}$ denote the class of
$R$-modules which have finite projective/injective dimension. Then there is a
cofibrantly generated abelian model structure on ${\operatorname{Mod}}(R)$
given by the triple
$(\mathrm{GProj}(R),{\operatorname{Mod}}(R),{\mathcal{W}})$. There is dually a
cofibrantly generated abelian model structure on ${\operatorname{Mod}}(R)$
given by the triple
$({\operatorname{Mod}}(R),\mathrm{GInj}(R),{\mathcal{W}})$.

We call these the Gorenstein projective and Gorenstein injective model
structures on ${\operatorname{Mod}}(R)$, respectively. We will write
${\operatorname{Mod}}(R)^{\text{GP}}$, resp.
${\operatorname{Mod}}(R)^{\text{GI}}$, for the category
${\operatorname{Mod}}(R)$ viewed as an abelian model category with the
Gorenstein projective, resp. Gorenstein injective, model structure.

######  Remark 2.12.

The identity functor
${\operatorname{id}}:{\operatorname{Mod}}(R)^{\text{GP}}\to{\operatorname{Mod}}(R)^{\text{GI}}$
is left Quillen and in fact a Quillen equivalence (cf. [16, bottom of p.583]).
The latter means that if
${\mathfrak{m}},\mathfrak{n}\in{\operatorname{Mod}}(R)$ are such that
${\mathfrak{m}}$ is Gorenstein projective and $\mathfrak{n}$ is Gorenstein
injective, then a map between them is a weak equivalence in one model
structure if and only if it is a weak equivalence in the other. But this holds
because, more generally, the weak equivalences in both model structures are
the same. Indeed, that follows from 2.8 as the trivial objects are the same in
either model structure. Even without using the notion of Quillen equivalences,
we therefore see that the two model structures have equivalent homotopy
categories as these are localisations of ${\operatorname{Mod}}(R)$ with
respect to the same collection of weak equivalences. We will always denote
this homotopy category by ${\operatorname{Ho}}(R)$.

We record here a particular application of 2.8 which will be useful in order
to compare the Gorenstein projective model structures over various Gorenstein
rings.

######  Lemma 2.13.

Suppose that $A$ and $B$ are Gorenstein rings, with $A$ isomorphic to a
subring of $B$. Further, assume that $B$ is free as both a left and a right
$A$-module. Write
$F^{B}_{A}:{\operatorname{Mod}}(A)\to{\operatorname{Mod}}(B)$ for the functor
${\mathfrak{m}}\mapsto B\otimes_{A}{\mathfrak{m}}$ and
${\operatorname{Res}}^{B}_{A}$ for its right adjoint given by restriction of
scalars. Then the adjunction $F^{B}_{A}\dashv{\operatorname{Res}}^{B}_{A}$ is
Quillen with respect to the Gorenstein projective model structure and there
are natural isomorphisms
$\mathbf{L}F^{B}_{A}\cong{\operatorname{Ho}}(F^{B}_{A})$ and
$\mathbf{R}{\operatorname{Res}}^{B}_{A}\cong{\operatorname{Ho}}({\operatorname{Res}}^{B}_{A})$.

###### Proof.

Under our assumptions, both $F^{B}_{A}$ and ${\operatorname{Res}}^{B}_{A}$ are
exact. We claim that they preserve trivial objects, i.e. modules of finite
projective/injective dimension. Indeed, $F^{B}_{A}$ and
${\operatorname{Res}}^{B}_{A}$ preserve projective and injective modules,
respectively, since they are exact and ajoint. It therefore follows by
exactness that $F^{B}_{A}$, resp. ${\operatorname{Res}}^{B}_{A}$, preserves
finite projective resolutions, resp. finite injective resolutions, and thus we
get our claim because both $A$ and $B$ are Gorenstein. From all of this we
also obtain that ${\operatorname{Res}}^{B}_{A}$ preserves epimorphisms, i.e.
fibrations, and epimorphisms with kernels of finite projective dimension, i.e.
trivial fibrations, and hence is right Quillen as required. Finally, the
isomorphism on the total derived functors is immediate from 2.8. ∎

The following characterisation of the morphisms in the homotopy category of
any abelian model category is due to Gillespie.

######  Proposition 2.14 ([12, Proposition 4.4]).

Let ${\mathcal{A}}$ be an abelian model category with model structure
$({\mathcal{C}},{\mathcal{F}},{\mathcal{W}})$. Suppose $X\in{\mathcal{C}}$ and
$Y\in{\mathcal{F}}$. Then the abelian group
${\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{A}})}(X,Y)$ is isomorphic
to the quotient of ${\operatorname{Hom}}_{{\mathcal{A}}}(X,Y)$ by the
equivalence relation given by $f\sim g$ if and only if $f-g$ factors through
an object of $\Gamma={\mathcal{C}}\cap{\mathcal{F}}\cap{\mathcal{W}}$, if and
only if $f-g$ factors through an object of ${\mathcal{C}}\cap{\mathcal{W}}$,
if and only if $f-g$ factors through an object of
${\mathcal{F}}\cap{\mathcal{W}}$.

For a Gorenstein ring $R$ and writing ${\mathcal{W}}$ for the class of
$R$-modules of finite projective dimension, it is shown in [16, Corollary 8.5]
that the classes $\mathrm{GProj}(R)\cap{\mathcal{W}}$ and
$\mathrm{GInj}(R)\cap{\mathcal{W}}$ are that of projective and injective
$R$-modules respectively. We thus see from 2.14 that for ${\mathfrak{m}}$ a
Gorenstein projective module and $\mathfrak{n}$ any module, two maps
$f,g:{\mathfrak{m}}\to\mathfrak{n}$ are equal in ${\operatorname{Ho}}(R)$ if
and only if $f-g$ factors through a projective module. Similarly, if
${\mathfrak{m}}$ is any module and if $\mathfrak{n}$ is Gorenstein injective,
then two maps $f,g:{\mathfrak{m}}\to\mathfrak{n}$ are equal in
${\operatorname{Ho}}(R)$ if and only if $f-g$ factors through an injective
module.

To make our notation simpler, for any two
${\mathfrak{m}},\mathfrak{n}\in{\operatorname{Mod}}(R)$ we will denote the
$k$-vector space
${\operatorname{Hom}}_{{\operatorname{Ho}}(R)}({\mathfrak{m}},\mathfrak{n})$
by $[{\mathfrak{m}},\mathfrak{n}]_{R}$. More generally, if $R$ and $S$ are two
Gorenstein rings with $R$ isomorphic to a subring of $S$ and if
${\mathfrak{m}},\mathfrak{n}\in{\operatorname{Mod}}(S)$, we will write
$[{\mathfrak{m}},\mathfrak{n}]_{R}$ for short instead of
$[{\operatorname{Res}}^{S}_{R}({\mathfrak{m}}),{\operatorname{Res}}^{S}_{R}(\mathfrak{n})]_{R}$.

######  Remark 2.15.

Suppose that $R$ is a Gorenstein ring and $f:{\mathfrak{m}}\to\mathfrak{n}$ is
a map in ${\operatorname{Mod}}(R)$. Fix a surjection
$\theta:\mathfrak{p}\to\mathfrak{n}$ where $\mathfrak{p}$ is a projective
module. Assume that $f$ factors as a composite
${\mathfrak{m}}\to\mathfrak{q}\stackrel{{\scriptstyle g}}{{\to}}\mathfrak{n}$
where $\mathfrak{q}$ is a projective module. Since $\mathfrak{q}$ is
projective, it follows that $g$, and hence $f$, factor through $\theta$.
Consequently, we see that the maps
$f\in{\operatorname{Hom}}_{R}({\mathfrak{m}},\mathfrak{n})$ which factor
through a projective module are precisely those in the image of
$\theta\circ(-):{\operatorname{Hom}}_{R}({\mathfrak{m}},\mathfrak{p})\to{\operatorname{Hom}}_{R}({\mathfrak{m}},\mathfrak{n})$.
Dually, if $\iota:{\mathfrak{m}}\to\mathfrak{e}$ is a monomorphism into an
injective module $\mathfrak{e}$ then the maps
$f\in{\operatorname{Hom}}_{R}({\mathfrak{m}},\mathfrak{n})$ which factor
through a injective module are precisely those in the image of
$(-)\circ\iota:{\operatorname{Hom}}_{R}(\mathfrak{e},\mathfrak{n})\to{\operatorname{Hom}}_{R}({\mathfrak{m}},\mathfrak{n})$.

By combining 2.14 and 2.15, we see that for
${\mathfrak{m}},\mathfrak{n}\in{\operatorname{Mod}}(R)$ with ${\mathfrak{m}}$
Gorenstein projective and for $\theta:\mathfrak{p}\to\mathfrak{n}$ as above,
$[{\mathfrak{m}},\mathfrak{n}]_{R}\cong{\operatorname{coker}}(\theta\circ(-))\cong{\operatorname{Ext}}^{1}_{S}({\mathfrak{m}},\ker(\theta))$.
This generalises as follows. We say that an abelian model structure
$({\mathcal{C}},{\mathcal{F}},{\mathcal{W}})$ on an abelian category
${\mathcal{A}}$ is _projective_ , resp. _injective_ , if
${\mathcal{F}}={\mathcal{A}}$, resp. ${\mathcal{C}}={\mathcal{A}}$. We note
that any projective or injective abelian model structure on ${\mathcal{A}}$ is
hereditary, c.f. [7, Corollary 1.1.12].

######  Proposition 2.16.

Let ${\mathcal{A}}$ be an abelian model category with model structure
$({\mathcal{C}},{\mathcal{F}},{\mathcal{W}})$, and assume that ${\mathcal{A}}$
is either injective or projective. Let $X\in{\mathcal{C}}$ and
$Y\in{\mathcal{F}}$. Then we have isomorphisms

${\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{A}})}(\Sigma^{-k}X,Y)\cong{\operatorname{Hom}}_{{\operatorname{Ho}}({\mathcal{A}})}(X,\Sigma^{k}Y)\cong{\operatorname{Ext}}^{k}_{\mathcal{A}}(X,Y)$

of abelian groups for all $k>0$.

###### Proof.

The arguments given in [16, Theorem 9.3] for Gorenstein projective/injective
model structures carry over mutatis mutandis to this more general situation. ∎

Since the Gorenstein projective, resp. Gorenstein injective, model structure
on ${\operatorname{Mod}}(R)$ is projective, resp. injective, we see from the
above that it is hereditary. In particular, it is stable which is also proved
directly in [16, Theorem 9.3]. Hence we see that the homotopy category
${\operatorname{Ho}}(R)$ has the structure of a triangulated category as
described earlier. We note that since every object is cofibrant in the
Gorenstein injective model structure the triangles (2.8) are obtained for
_any_ short exact sequence in ${\operatorname{Mod}}(R)$.

###  2.5. Left and right transfers

We now recall the notion of left/right transfer of a model structure along an
adjunction. If $F:\mathcal{E}\rightleftarrows{\mathcal{D}}:U$ is an adjunction
where $\mathcal{E}$ is a model category and ${\mathcal{D}}$ is a locally
small, bicomplete category, there is then a natural candidate for a model
structure on ${\mathcal{D}}$ making the adjunction Quillen. Namely, one
defines fibrations (resp. weak equivalences) in ${\mathcal{D}}$ to be the
morphisms $f$ such that $U(f)$ is a fibration (resp. weak equivalence). The
cofibrations are then defined as those morphisms which satisfy the left
lifting property with respect to trivial fibrations. This, however, may not in
general give a well-defined model structure on ${\mathcal{D}}$. When it does,
we call it the _right transfer_ of the model structure on $\mathcal{E}$ along
$U$, and we say that the right transfer exists. In [10, Proposition 4.10], it
was shown that the right transfer of Hovey’s Gorenstein projective model
structure on ${\operatorname{Mod}}({\mathcal{H}})$ along the functor $U$ of
pro-$p$ Iwahori invariants exists. This gives a model structure on
${\operatorname{Rep}}_{k}^{\infty}(G)$, called the $I$-Gorenstein projective
model structure.

Dually, when ${\mathcal{D}}$ is a model category and $\mathcal{E}$ is a
locally small, bicomplete category, the _left transfer_ of the model structure
on ${\mathcal{D}}$ along $F$, if it exists, is the model structure on
$\mathcal{E}$ whose cofibrations (resp. weak equivalences) are the morphisms
$g$ such that $F(g)$ is a cofibration (resp. weak equivalence), and the
fibrations are defined as those morphisms which satisfy the right lifting
property with respect to the trivial cofibrations.

##  3\. Parabolic induction, recollements and supersingular modules

###  3.1. Parabolic induction and recollement: one parabolic

We now wish to study how the parabolic (co)induction functor behaves with
respect to Hovey’s Gorenstein projective and Gorenstein injective model
structures on ${\operatorname{Mod}}({\mathcal{H}}_{M})$ and
${\operatorname{Mod}}({\mathcal{H}})$. Since we only know that the Gorenstein
projective model structure right transfers to the category
${\operatorname{Rep}}_{k}^{\infty}(G)$, we a priori favour that model
structure over the injective one. But as we will see, when it comes to
parabolic induction it is more fruitful to consider the Gorenstein injective
model structure as well. Since working specifically with Hecke algebras does
not simplify the approach, we work in this subsection in the more general
situation described below.

######  Setup 3.1.

We let $R,S$ be two Gorenstein rings. Moreover, we assume the following:

  * •

there is an adjoint triple of additive functors

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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{{\mathsf{R}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
7.9628pt\raise-5.49513pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces,$

meaning that ${\mathsf{L}}$ (resp. ${\mathsf{R}}$) is the left (resp. right)
adjoint of ${\mathsf{G}}$;

  * •

the functor ${\mathsf{G}}$ is fully faithful, and ${\mathsf{L}}$ is exact.

######  Example 3.2.

The above setup applies to $S={\mathcal{H}}_{M}$ ($M$ some standard Levi
subgroup), $R={\mathcal{H}}$ and
${\mathsf{G}}\in\\{{\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}},{\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}\\}$
along with its two adjoints. The left adjoint is exact in either case because
${\operatorname{Loc}}_{M}$ is a localisation functor and the equivalence 2.2
is also exact. We’ve also seen that parabolic (co)induction is fully faithful
in 2.5.

Note that the functor ${\mathsf{G}}$ is automatically exact as it is both a
left and a right adjoint. But ${\mathsf{R}}$ is only known to be left exact,
being a right adjoint. Part (i) of the Lemma below is a generalisation of [20,
Lemma 8.1] to our setup:

######  Lemma 3.3.

Let ${\mathfrak{m}}\in{\operatorname{Mod}}(S)$ and
$\mathfrak{n}\in{\operatorname{Mod}}(R)$. Then:

  1. (i)

${\operatorname{pd}}_{S}({\mathfrak{m}})<\infty$ if and only if
${\operatorname{pd}}_{R}({\mathsf{G}}({\mathfrak{m}}))<\infty$;

  2. (ii)

if ${\operatorname{pd}}_{S}(\mathfrak{n})<\infty$ then
${\operatorname{pd}}_{R}({\mathsf{L}}(\mathfrak{n}))<\infty$; and

  3. (iii)

${\mathfrak{m}}$ is Gorenstein injective if and only if
${\mathsf{G}}({\mathfrak{m}})$ is Gorenstein injective.

###### Proof.

(i)-(ii) Since ${\mathsf{L}}$ and ${\mathsf{G}}$ are exact adjoint functors,
we have an isomorphism of abelian groups

(3.1)
${\operatorname{Ext}}^{i}_{S}({\mathsf{L}}(\mathfrak{n}),{\mathfrak{m}})\cong{\operatorname{Ext}}^{i}_{R}(\mathfrak{n},{\mathsf{G}}({\mathfrak{m}}))$

for all ${\mathfrak{m}}\in{\operatorname{Mod}}(S)$, all
$\mathfrak{n}\in{\operatorname{Mod}}(R)$ and all $i\geq 0$. This immediately
implies that if ${\operatorname{pd}}_{S}({\mathfrak{m}})<\infty$ then
${\operatorname{pd}}_{R}({\mathsf{G}}({\mathfrak{m}}))<\infty$ by considering
their injective dimensions, and it also immediately implies (ii). For the
converse of (i), given any
${\mathfrak{m}}^{\prime}\in{\operatorname{Mod}}(S)$, 2.4 and (3.1) together
give isomorphisms

(3.2)
${\operatorname{Ext}}^{i}_{S}({\mathfrak{m}},{\mathfrak{m}}^{\prime})\cong{\operatorname{Ext}}^{i}_{S}({\mathsf{L}}({\mathsf{G}}({\mathfrak{m}})),{\mathfrak{m}}^{\prime})\cong{\operatorname{Ext}}^{i}_{R}({\mathsf{G}}({\mathfrak{m}}),{\mathsf{G}}({\mathfrak{m}}^{\prime})).$

This now gives that
${\operatorname{pd}}_{R}({\mathsf{G}}({\mathfrak{m}}))<\infty$ implies
${\operatorname{pd}}_{S}({\mathfrak{m}})<\infty$.

(iii) Recall that ${\mathfrak{m}}$ is Gorenstein injective if and only if
${\operatorname{Ext}}^{i}_{S}(\mathfrak{e},{\mathfrak{m}})=0$ for all
$\mathfrak{e}\in{\operatorname{Mod}}(S)$ of finite injective dimension (and
similarly for ${\mathsf{G}}({\mathfrak{m}})$ and $R$-modules of finite
injective dimension). Therefore, (ii) and (3.1) give that if ${\mathfrak{m}}$
is Gorenstein injective then so is ${\mathsf{G}}({\mathfrak{m}})$. Conversely,
(i) and (3.2) give that if ${\mathsf{G}}({\mathfrak{m}})$ is Gorenstein
injective then so is ${\mathfrak{m}}$. ∎

We next show that the Gorenstein projective/injective model structures on
${\operatorname{Mod}}(R)$ are in fact determined by those on
${\operatorname{Mod}}(S)$. Note that since the notion of weak equivalences
doesn’t depend on which model structure we choose to work with (c.f. 2.12),
there is no ambiguity in the statement of part (iii) of the Proposition below.

######  Proposition 3.4.

With $S$, $R$, ${\mathsf{G}}$ and ${\mathsf{L}}$ as in our setup, we have:

  1. (i)

the Gorenstein projective model structure on ${\operatorname{Mod}}(S)$ is the
right transfer of the Gorenstein projective model structure on
${\operatorname{Mod}}(R)$ along ${\mathsf{G}}$. In particular, ${\mathsf{G}}$
is a right Quillen functor for the Gorenstein projective model structures;

  2. (ii)

the Gorenstein injective model structure on ${\operatorname{Mod}}(S)$ is both
the left transfer and the right transfer of the Gorenstein injective model
structure on ${\operatorname{Mod}}(R)$ along ${\mathsf{G}}$. In particular,
${\mathsf{G}}$ is both a left and a right Quillen functor for the Gorenstein
injective model structures; and

  3. (iii)

the functor ${\mathsf{L}}$ preserves weak equivalences.

###### Proof.

We first show that ${\mathsf{G}}$ both preserves and reflects weak
equivalences. Since ${\mathsf{G}}$ is exact, it follows from 2.8(i)-(ii) that
we only need to show that ${\mathfrak{m}}\in{\operatorname{Mod}}(S)$ is a
trivial object, i.e. has finite projective dimension, if and only if
${\mathsf{G}}({\mathfrak{m}})$ is a trivial object in
${\operatorname{Mod}}(R)$. This holds by 3.3(i) and thus proves our claim. To
show (i), we are therefore left to prove that for a morphism
$f:{\mathfrak{m}}\to{\mathfrak{m}}^{\prime}$ in ${\operatorname{Mod}}(S)$, $f$
is a fibration if and only if ${\mathsf{G}}(f)$ is a fibration, in the
Gorenstein projective model structures on ${\operatorname{Mod}}(S)$ and
${\operatorname{Mod}}(R)$ respectively. But in this case the fibrations are
the epimorphisms and the result follows because ${\mathsf{G}}$ is exact and
faithful.

To show (ii), we instead need to show that $f$ as above is a fibration, resp.
cofibration, if and only if ${\mathsf{G}}(f)$ is a fibration, resp.
cofibration, in the Gorenstein injective model structures on
${\operatorname{Mod}}(S)$ and ${\operatorname{Mod}}(R)$ respectively. The case
of cofibrations is dual to the above: they are the monomorphisms, so that the
claim holds by exactness and faithfulness of ${\mathsf{G}}$. The fibrations on
the other hand are precisely the epimorphisms with Gorenstein injective
kernels. Now the claim holds instead by exactness of ${\mathsf{G}}$ and
3.3(iii).

Finally, (iii) follows from 2.8(i) and 3.3(ii), given that (i) above implies
that ${\mathsf{L}}$ is left Quillen. ∎

######  Corollary 3.5.

Continuing with our setup, we have:

  1. (i)

the two total right derived functors of ${\mathsf{G}}$ (with respect to the
Gorenstein projective and Gorenstein injective model structures respectively)
are naturally isomorphic to ${\operatorname{Ho}}({\mathsf{G}})$. Similarly,
the two total left derived functor of ${\mathsf{L}}$ are naturally isomorphic
to ${\operatorname{Ho}}({\mathsf{L}})$, and the total left derived functor of
${\mathsf{G}}$ with respect to the Gorenstein injective model structure is
also naturally isomorphic to ${\operatorname{Ho}}({\mathsf{G}})$;

  2. (ii)

the functor ${\operatorname{Ho}}({\mathsf{G}})$ is fully faithful; and

  3. (iii)

${\mathsf{L}}$ reflects trivial objects, i.e. the converse of 3.3(ii) holds,
if and only if $({\mathsf{L}},{\mathsf{G}})$ is a Quillen equivalence (for
either/both model structures).

###### Proof.

Part (i) is immediate from 2.8. For the rest of this proof, we work with the
Gorenstein injective model structure. This suffices to show (iii) because the
derived functors of ${\mathsf{G}}$ (resp. ${\mathsf{L}}$) for both model
structures can be identified with ${\operatorname{Ho}}({\mathsf{G}})$ (resp.
${\operatorname{Ho}}({\mathsf{L}})$), and moreover an adjoint pair is a
Quillen equivalence precisely when the derived functors form an adjoint
equivalence of categories (c.f. [15, Proposition 1.3.13]).

(ii) By 2.4 we need to show that the counit of the adjunction
$({\operatorname{Ho}}({\mathsf{L}}),{\operatorname{Ho}}({\mathsf{G}}))$ is an
isomorphism. By the proof of [15, Proposition 1.3.13], this holds if and only
if, for any ${\mathfrak{m}}\in{\operatorname{Mod}}(S)$, the composite

${\mathsf{L}}Q_{c}{\mathsf{G}}{\mathfrak{m}}\stackrel{{\scriptstyle{\mathsf{L}}q_{c}}}{{\longrightarrow}}{\mathsf{L}}{\mathsf{G}}{\mathfrak{m}}\stackrel{{\scriptstyle\varepsilon_{{\mathfrak{m}}}}}{{\longrightarrow}}{\mathfrak{m}}$

is a weak equivalence. Here $Q_{c}{\mathsf{G}}{\mathfrak{m}}$ denotes a
cofibrant replacement of ${\mathsf{G}}{\mathfrak{m}}$ in
${\operatorname{Mod}}(R)$, which comes equipped with a trivial fibration
$q_{c}:Q_{c}{\mathsf{G}}{\mathfrak{m}}\to{\mathsf{G}}{\mathfrak{m}}$. But by
3.4(ii) we have that ${\mathsf{L}}q_{c}$ is a weak equivalence and
$\varepsilon_{{\mathfrak{m}}}$ is an isomorphism (c.f. 2.4 again), so that our
claim holds.

(iii) By the above $\varepsilon_{{\mathfrak{m}}}{\mathsf{L}}q_{c}$ is always a
weak equivalence. It therefore follows from [15, Corollary 1.3.16] that
$({\mathsf{L}},{\mathsf{G}})$ is a Quillen equivalence if and only if
${\mathsf{L}}$ reflects weak equivalences between cofibrant objects. But in
the Gorenstein injective model structure all objects are cofibrant, so the
latter condition says that ${\mathsf{L}}$ reflects all weak equivalences. The
result now follows from 2.8(ii). ∎

Note that from 3.4(iii) and exactness we get that ${\mathsf{L}}$ preserves
both fibrations and trivial fibrations in the Gorenstein projective model
structure. This is somewhat unusual for a _left_ Quillen functor. Also, we
will see later that the Quillen equivalence as in 3.5(iii) essentially never
holds in the case of parabolic (co)induction from a standard Levi subgroup $M$
unless $M=G$, c.f. 3.20. For now we only illustrate that this is clear when
$M$ is the maximal split torus.

######  Example 3.6.

If $M=T$ is the split maximal torus, then
${\mathcal{H}}_{T}=k[\widetilde{\Lambda}]$ is the group algebra of
$\widetilde{\Lambda}=T/(T\cap I)\cong{\mathbb{Z}}^{r_{ss}+r_{Z}}\times
T(\mathbb{F}_{q})$. Hence ${\mathcal{H}}_{T}$ is free of finite rank, coprime
with $p$, over the Laurent polynomial ring $k[x_{1}^{\pm
1},\ldots,x_{r_{ss}+r_{Z}}^{\pm 1}]$. The latter ring has finite global
dimension, and therefore so does ${\mathcal{H}}_{T}$ (c.f. [22, Theorem
7.5.6(iii)]). In particular the homotopy category
${\operatorname{Ho}}({\operatorname{Mod}}({\mathcal{H}}_{T}))=0$. On the other
hand, ${\mathcal{H}}$ typically has infinite global dimension. Indeed, when
$G$ does not have root system of type $A_{1}\times\cdots\times A_{1}$ or when
$r_{ss}>0$ and $q>2$, then ${\mathcal{H}}$ has infinite global dimension and
in fact has a simple module ${\mathfrak{m}}$ of infinite projective dimension
(c.f. [26, 7.1-7.3]). So under these hypotheses,
${\operatorname{Ho}}({\mathcal{H}})\neq 0$ and
${\operatorname{Ind}}_{{\mathcal{H}}_{T}}^{{\mathcal{H}}}$ is not a Quillen
equivalence. And indeed, for this module ${\mathfrak{m}}$ one automatically
has
${\operatorname{pd}}_{{\mathcal{H}}_{T}}({\operatorname{Loc}}_{M}(\mathfrak{n}))<\infty$
even though ${\operatorname{pd}}_{{\mathcal{H}}}(\mathfrak{n})=\infty$.

Suppose that ${\mathfrak{m}}\in{\operatorname{Mod}}(R)$ and
$\mathfrak{n}\in{\operatorname{Mod}}(S)$. We have an isomorphism
${\operatorname{Hom}}_{R}({\mathsf{G}}(\mathfrak{n}),{\mathfrak{m}})\cong{\operatorname{Hom}}_{S}(\mathfrak{n},{\mathsf{R}}({\mathfrak{m}}))$.
Since ${\mathsf{G}}$ is exact we have that ${\mathsf{R}}$ preserves
injectives, and hence we obtain a first quadrant Grothendieck spectral
sequence

(3.3)
$E_{2}^{ij}={\operatorname{Ext}}_{S}^{i}(\mathfrak{n},R^{j}{\mathsf{R}}({\mathfrak{m}}))\Rightarrow{\operatorname{Ext}}_{R}^{i+j}({\mathsf{G}}(\mathfrak{n}),{\mathfrak{m}}).$

Note that 3.4(ii) implies that ${\mathsf{R}}$ preserves fibrations in the
Gorenstein injective model structures, i.e. surjections with Gorenstein
injective kernels. Given that the functor ${\mathsf{R}}$ is not necessarily
right exact, this may seem surprising but it is actually fairly
straightforward to show this directly using the above spectral sequence.

######  Lemma 3.7.

Suppose that $R$ is an $n$-Gorenstein ring. Then:

  1. (i)

the functor ${\mathsf{R}}$ has cohomological dimension bounded above by $n$;

  2. (ii)

the Gorenstein injective modules are all acyclic for ${\mathsf{R}}$;

  3. (iii)

an $R$-module ${\mathfrak{m}}$ is ${\mathsf{R}}$-acyclic if and only if there
is an isomorphism of abelian groups

${\operatorname{Ext}}_{R}^{j}({\mathsf{G}}(\mathfrak{n}),{\mathfrak{m}})\cong{\operatorname{Ext}}^{j}_{S}(\mathfrak{n},{\mathsf{R}}({\mathfrak{m}}))$

for all $j\geq 0$ and all $\mathfrak{n}\in{\operatorname{Mod}}(S)$; and

  4. (iv)

${\mathsf{R}}$ preserves Gorenstein injectives.

###### Proof.

Let ${\mathfrak{m}}\in{\operatorname{Mod}}(R)$. By plugging $\mathfrak{n}=S$
in the spectral sequence (3.3), the $E_{2}^{ij}$ term becomes zero for $i>0$,
and we obtain an isomorphism

(3.4) ${\operatorname{Ext}}_{R}^{j}({\mathsf{G}}(S),{\mathfrak{m}})\cong
R^{j}{\mathsf{R}}({\mathfrak{m}})$

for all $j\geq 0$. But by 3.3(i), ${\mathsf{G}}(S)$ has finite projective
dimension and hence ${\operatorname{pd}}_{R}({\mathsf{G}}(S))\leq n$ by
assumption on $R$. Thus the left hand side of (3.4) vanishes for $i>n$,
proving (i).

For (ii), assume now that ${\mathfrak{m}}$ is Gorenstein injective. Then for
any module of finite injective dimension
$\mathfrak{e}\in{\operatorname{Mod}}(R)$ and any $j>0$, we have
${\operatorname{Ext}}_{R}^{j}(\mathfrak{e},{\mathfrak{m}})=0$. Applying this
to $\mathfrak{e}={\mathsf{G}}(S)$, we immediately get from (3.4) that
$R^{j}{\mathsf{R}}({\mathfrak{m}})=0$ for all $j>0$ as claimed.

One direction of (iii) is immediate from the spectral sequence (3.3). So
assume now that
${\operatorname{Ext}}_{R}^{j}({\mathsf{G}}(\mathfrak{n}),{\mathfrak{m}})\cong{\operatorname{Ext}}^{j}_{S}(\mathfrak{n},{\mathsf{R}}({\mathfrak{m}}))$
for all $\mathfrak{n}\in{\operatorname{Mod}}(S)$ and all $j\geq 0$. Then
combining this isomorphism with (3.4) we obtain

$R^{j}{\mathsf{R}}({\mathfrak{m}})\cong{\operatorname{Ext}}^{j}_{S}(S,{\mathsf{R}}({\mathfrak{m}}))=0$

for all $j>0$, as required.

Finally, for (iv) we can apply (ii), (iii), and 3.3(i) again to see that if
${\mathfrak{m}}\in{\operatorname{Mod}}(R)$ is Gorenstein injective and if
$\mathfrak{e}\in{\operatorname{Mod}}(S)$ has finite injective dimension then
${\operatorname{Ext}}_{S}^{j}(\mathfrak{e},{\mathsf{R}}({\mathfrak{m}}))\cong{\operatorname{Ext}}_{R}^{j}({\mathsf{G}}(\mathfrak{e}),{\mathfrak{m}})=0$
for $j>0$, and thus ${\mathsf{R}}({\mathfrak{m}})$ is Gorenstein injective. ∎

Next, we investigate properties of ${\operatorname{Mod}}(S)$ (resp.
${\operatorname{Ho}}(S)$) viewed as a full subcategory of
${\operatorname{Mod}}(R)$ (resp. ${\operatorname{Ho}}(R)$) via the functor
${\mathsf{G}}$ (resp. ${\operatorname{Ho}}({\mathsf{G}}))$. We first note that
${\operatorname{Mod}}(S)$ is not in general closed under subquotients in
${\operatorname{Mod}}(R)$, and so in particular is not a Serre subcategory, as
seen in the following example.

######  Example 3.8.

When ${\mathsf{G}}={\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$
or ${\operatorname{Coind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$ for some
proper Levi subgroup $M\neq G$, the essential image of ${\mathsf{G}}$ is not
closed under subobjects. Indeed, first let
$V={\operatorname{Ind}}_{P}^{G}(\mathbf{1}_{M})\in{\operatorname{Rep}}_{k}^{\infty}(G)$.
By standard properties of induction for smooth representations, the trivial
representation $\mathbf{1}_{G}$ embeds as a subrepresentation of $V$. By left
exactness of the functor $U$ and applying [28, Theorem 1], we see that the
trivial character $\mathbf{1}_{{\mathcal{H}}}$ of ${\mathcal{H}}$ embeds as a
submodule of
${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}(\mathbf{1}_{{\mathcal{H}}_{M}})$.
However, there does not exist
${\mathfrak{m}}\in{\operatorname{Mod}}({\mathcal{H}}_{M})$ such that
${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}})\cong\mathbf{1}_{{\mathcal{H}}}$.
This is because, by [30, Lemmas 3.5 & 3.6], we have an isomorphism
${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}})\cong{\mathfrak{m}}^{\oplus
n}$ of $k$-vector spaces, where $n=\left|W_{0}\right|/\left|W_{0,M}\right|>1$.
So in particular
$\dim_{k}{\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}({\mathfrak{m}})>1$
whenever ${\mathfrak{m}}\neq 0$.

Hence we cannot in cases of interest form an abelian quotient of
${\operatorname{Mod}}(R)$ by ${\operatorname{Mod}}(S)$ via ${\mathsf{G}}$. We
now show how the situation is improved by working at the homotopy level, by
making use of the triangulated structures of ${\operatorname{Ho}}(S)$ and
${\operatorname{Ho}}(R)$.

Let ${\mathcal{T}}$ be a triangulated category with shift functor
$\Sigma:{\mathcal{T}}\to{\mathcal{T}}$. Recall that a subcategory
${\mathcal{S}}$ of ${\mathcal{T}}$ is called a _triangulated subcategory_ if
$\Sigma$ restricts to an autoequivalence of ${\mathcal{S}}$, if
${\mathcal{S}}$ is strictly full (meaning any object of ${\mathcal{T}}$
isomorphic to an object of ${\mathcal{S}}$ lies in ${\mathcal{S}}$) and if
whenever $X\to Y\to Z\to\Sigma X$ is a distinguished triangle in
${\mathcal{T}}$ with two of $X,Y,Z$ lying in ${\mathcal{S}}$, then so does the
third one. A triangulated subcategory ${\mathcal{S}}$ is called _thick_ if,
furthermore, it is closed under direct summands. If ${\mathcal{T}}^{\prime}$
is another triangulated category with shift functor $\Sigma^{\prime}$, an
additive functor ${\mathsf{F}}:{\mathcal{T}}\to{\mathcal{T}}^{\prime}$ is
called _triangulated_ if ${\mathsf{F}}\Sigma\cong\Sigma^{\prime}{\mathsf{F}}$
and ${\mathsf{F}}$ sends distinguished triangles to distinguished triangles.
Note that the essential image of a full triangulated functor is a triangulated
subcategory. Moreover, when ${\mathcal{S}}$ is a triangulated subcategory then
it inherits from ${\mathcal{T}}$ a triangulated structure and the inclusion
${\mathcal{S}}\to{\mathcal{T}}$ is a triangulated functor.

Suppose now that ${\mathcal{T}}$ contains all small coproducts. For example,
the homotopy category of any Gorenstein ring has all small coproducts since
Gorenstein projective modules are closed under arbitrary direct sums. Then
recall that a triangulated subcategory ${\mathcal{S}}$ is called _localising_
if it is closed under arbitrary coproducts. All such subcategories are
automatically thick, c.f. [23, Proposition 1.6.8].

Recall that in this general setup, Verdier defined a quotient category
${\mathcal{T}}/{\mathcal{S}}$ whenever ${\mathcal{T}}$ is triangulated and
${\mathcal{S}}$ is a triangulated subcategory, which is itself a triangulated
category equipped with a triangulated quotient functor
$\pi:{\mathcal{T}}\to{\mathcal{T}}/{\mathcal{S}}$, and this construction
satisfies a suitable universal property (c.f. [29, §II.2.1-II.2.3]).

We will now see how our setup lifts to the homotopy level. For this we need
the notion of a recollement of triangulated categories, which we now recall.

######  Definition 3.9.

A _recollement_ of triangulated categories is a diagram of triangulated
functors

$\textstyle{{\mathcal{T}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{G}}$$\textstyle{{\mathcal{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{I}}$$\scriptstyle{\mathsf{L}}$$\scriptstyle{\mathsf{R}}$$\textstyle{{\mathcal{T}}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{Q}}$$\scriptstyle{\mathsf{P}}$

such that the following conditions are satisfied:

  * •

$\mathsf{Q}$ (resp. $\mathsf{P}$) is the left (resp. right) adjoint of
$\mathsf{I}$;

  * •

$\mathsf{L}$ (resp. $\mathsf{R}$) is the left (resp. right) adjoint of
$\mathsf{G}$;

  * •

the functors $\mathsf{G}$, $\mathsf{Q}$ and $\mathsf{P}$ are fully faithful;
and

  * •

the essential image of $\mathsf{G}$ is equal to the kernel
${\operatorname{Ker}}(\mathsf{I})$.

We briefly describe some properties implied by this definition (c.f. [21,
§4.9-4.13]). First, ${\mathcal{T}}^{\prime}$ identifies with a thick
subcategory of ${\mathcal{T}}$ via ${\mathsf{G}}$ such that
${\mathcal{T}}^{\prime\prime}\simeq{\mathcal{T}}/{\mathcal{T}}^{\prime}$.
Moreover, the adjunction morphisms give for any $X\in{\mathcal{T}}$ several
functorial distinguished triangles, for example

$\mathsf{G}\mathsf{R}(X)\to
X\to\mathsf{P}\mathsf{I}(X)\to\Sigma(\mathsf{G}\mathsf{R}(X))$

or

$\mathsf{Q}\mathsf{I}(X)\to
X\to\mathsf{G}\mathsf{L}(X)\to\Sigma(\mathsf{Q}\mathsf{I}(X)).$

In particular, we see that the smallest triangulated subcategory of
${\mathcal{T}}$ containing ${\mathcal{T}}^{\prime}$ and
${\mathcal{T}}^{\prime\prime}$ is ${\mathcal{T}}$ itself, where
${\mathcal{T}}^{\prime\prime}$ is identified as a subcategory of
${\mathcal{T}}$ via either one of the fully faithful functors $\mathsf{Q}$ and
$\mathsf{P}$.

For any category ${\mathcal{C}}$ and any full subcategory
${\mathcal{D}}\subseteq{\mathcal{C}}$, we define two full subcategories
${\mathcal{D}}^{\ddagger}$ and ${}^{\ddagger}{\mathcal{D}}$ (as the symbol
$\perp$ was already taken) of ${\mathcal{C}}$ by

${\mathcal{D}}^{\ddagger}:=\\{Y\in{\mathcal{C}}\mid{\operatorname{Hom}}_{\mathcal{C}}(X,Y)=0\text{
for all }X\in{\mathcal{D}}\\}$

and

${}^{\ddagger}{\mathcal{D}}:=\\{X\in{\mathcal{C}}\mid{\operatorname{Hom}}_{\mathcal{C}}(X,Y)=0\text{
for all }Y\in{\mathcal{D}}\\}.$

Then, in the above setting of a recollement, the category
${\mathcal{T}}^{\prime\prime}$ is equivalent to both
$({\mathcal{T}}^{\prime})^{\ddagger}$ and
${}^{\ddagger}({\mathcal{T}}^{\prime})$ via the functors $\mathsf{P}$ and
$\mathsf{Q}$ respectively, where we view ${\mathcal{T}}^{\prime}$ as a full
subcategory of ${\mathcal{T}}$ via the functor $\mathsf{G}$. Note that by the
adjunctions ${\mathsf{L}}\dashv{\mathsf{G}}\dashv{\mathsf{R}}$, we have
$({\mathcal{T}}^{\prime})^{\ddagger}={\operatorname{Ker}}({\mathsf{R}})$ and
${}^{\ddagger}({\mathcal{T}}^{\prime})={\operatorname{Ker}}({\mathsf{L}})$.

######  Proposition 3.10.

  1. (i)

There is an adjoint triple

$\textstyle{{\operatorname{Ho}}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{Ho}}({\mathsf{G}})}$$\textstyle{{\operatorname{Ho}}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{Ho}}({\mathsf{L}})}$$\scriptstyle{\mathbf{R}{\mathsf{R}}}$

and ${\operatorname{Ho}}({\mathsf{G}})$ is fully faithful.

  2. (ii)

All the above functors are triangulated.

  3. (iii)

The essential image of ${\operatorname{Ho}}({\mathsf{G}})$ is a localising
subcategory of ${\operatorname{Ho}}(R)$.

###### Proof.

(i) We view our homotopy categories as coming from the Gorenstein injective
model structures. Then the existence of such an adjoint triple is immediate
from 3.4(ii) and 3.5(i), and 3.5(ii) says that
${\operatorname{Ho}}({\mathsf{G}})$ is fully faithful.

(ii) Recall that for a Gorenstein projective module
${\mathfrak{m}}\in{\operatorname{Mod}}(S)$, its suspension in
${\operatorname{Ho}}(S)$ is defined to be the cokernel of a cofibration to a
projective module. Being an exact left Quillen functor with exact right
adjoint, the functor ${\mathsf{L}}$ preserves this construction so that
${\operatorname{Ho}}({\mathsf{L}})\circ\Sigma\cong\Sigma\circ{\operatorname{Ho}}({\mathsf{L}})$.
Taking successive right adjoints of this identity gives us that all three
functors commute with the shift functor. Furthermore, since ${\mathsf{L}}$ is
exact and preserves the construction of $\Sigma$, we see that it in fact
preserves the distinguished triangles as given in (2.8). Thus
${\operatorname{Ho}}({\mathsf{L}})$ is triangulated. A completely dual
argument using fiber sequences and the loop functor shows that
${\operatorname{Ho}}({\mathsf{G}})$ and $\mathbf{R}{\mathsf{R}}$ preserve
distinguished triangles, recalling that ${\mathsf{R}}$ preserves short exact
sequences of Gorenstein injective modules by 3.7.

(iii) This is immediate from the fact that ${\operatorname{Ho}}({\mathsf{G}})$
commutes with arbitrary direct sums as it is a left adjoint. ∎

######  Remark 3.11.

Part (ii) of the above proposition also follows immediately from general
properties of stable (pointed) model categories. Indeed, if
$({\mathsf{F}},{\mathsf{G}})$ is a Quillen adjunction between stable pointed
model categories, then the derived functors $\mathbf{L}{\mathsf{F}}$ and
$\mathbf{R}{\mathsf{G}}$ commute with the suspension by [6, Corollary 3.1.4],
noting that the loop functor is the inverse of suspension. Moreover, in any
such model category the distinguished triangles in the homotopy category are
given by cofiber sequences, and left/right derived functors preserve those
(c.f. [15, Proposition 6.4.1 & Theorem 7.1.11]).

######  Corollary 3.12.

The adjoint triple from 3.10(i) is part of a recollement

$\textstyle{{\operatorname{Ho}}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{Ho}}({\mathsf{G}})}$$\textstyle{{\operatorname{Ho}}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{\pi}}$$\scriptstyle{{\operatorname{Ho}}({\mathsf{L}})}$$\scriptstyle{\mathbf{R}{\mathsf{R}}}$$\textstyle{{\operatorname{Ho}}(R)/{\operatorname{Ho}}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{Q}}$$\scriptstyle{\mathsf{P}}$

of triangulated categories, where we identify ${\operatorname{Ho}}(S)$ with
${\operatorname{Im}}({\operatorname{Ho}}({\mathsf{G}}))$ and
$\pi:{\operatorname{Ho}}(R)\to{\operatorname{Ho}}(R)/{\operatorname{Im}}({\operatorname{Ho}}({\mathsf{G}}))$
is the quotient functor.

###### Proof.

By 3.10, the inclusion
${\operatorname{Im}}({\operatorname{Ho}}({\mathsf{G}}))\to{\operatorname{Ho}}(R)$
has both a left and a right adjoint and
${\operatorname{Im}}({\operatorname{Ho}}({\mathsf{G}}))$ is a thick
subcategory. Then [21, Propositions 4.9.1 & 4.13.1] show that the quotient
functor $\pi$ also has a left and a right adjoint and that these two adjoint
triples form a recollement as required. ∎

Note that both
${\operatorname{Im}}(\mathsf{Q})={{}^{\ddagger}}{\operatorname{Ho}}(S)$ and
${\operatorname{Ho}}(S)$ are in fact localising subcategories of
${\operatorname{Ho}}(R)$ since they are kernels of triangulated functors which
are left adjoints.

###  3.2. Extension to several parabolics

We now extend the constructions from the previous subsection in the case where
$R={\mathcal{H}}$, $S={\mathcal{H}}_{M}$ for some standard Levi subgroup $M$,
and ${\mathsf{G}}={\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$
with left adjoint ${\mathsf{L}}={\mathcal{L}^{G}_{M}}$ and right adjoint
${\mathsf{R}}={\mathcal{R}^{G}_{M}}$. To simplify the notation, we will write
$I^{G}_{M}$ for ${\operatorname{Ind}}_{{\mathcal{H}}_{M}}^{{\mathcal{H}}}$. We
will denote by ${\operatorname{Ho}}({\mathcal{H}})_{M}$ the quotient
triangulated category of
${\operatorname{Ho}}({\mathcal{H}})/{\operatorname{Ho}}({\mathcal{H}}_{M})$,
and we have an adjoint triple

$\textstyle{{\operatorname{Ho}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{G}_{M}}$$\textstyle{{\operatorname{Ho}}({\mathcal{H}})_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{G}_{M}}$$\scriptstyle{r^{G}_{M}}$

with $\ell^{G}_{M}$ and $r^{G}_{M}$ both fully faithful, as given by the
recollement in 3.12. The following result will be our main aim in this
subsection.

######  Theorem 3.13.

Let ${\mathcal{M}}=\\{M_{1},\ldots,M_{n}\\}$ be a collection of standard Levi
subgroups of $G$, and let ${\mathcal{S}}^{G}_{\mathcal{M}}$ be the smallest
thick subcategory of ${\operatorname{Ho}}({\mathcal{H}})$ containing
${\operatorname{Im}}({\operatorname{Ho}}(I_{M_{1}}^{G})),\ldots,{\operatorname{Im}}({\operatorname{Ho}}(I_{M_{n}}^{G}))$.

  1. (i)

${\mathcal{S}}^{G}_{\mathcal{M}}$ is a localising subcategory and the
inclusion
$i^{G}_{\mathcal{M}}:{\mathcal{S}}^{G}_{\mathcal{M}}\hookrightarrow{\operatorname{Ho}}({\mathcal{H}})$
has both a left and a right adjoint, which form part of a recollement

$\textstyle{{\mathcal{S}}^{G}_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{G}_{\mathcal{M}}}$$\textstyle{{\operatorname{Ho}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{G}_{\mathcal{M}}}$$\scriptstyle{\mathcal{L}^{G}_{\mathcal{M}}}$$\scriptstyle{\mathcal{R}^{G}_{\mathcal{M}}}$$\textstyle{{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{G}_{\mathcal{M}}}$$\scriptstyle{r^{G}_{\mathcal{M}}}$

of triangulated categories, where
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}={\operatorname{Ho}}({\mathcal{H}})/{\mathcal{S}}^{G}_{\mathcal{M}}$.

  2. (ii)

Suppose $n>1$. Let $1\leq i\leq n$ and
${\mathfrak{m}}\in{\operatorname{Ho}}({\mathcal{H}})$, and write
${\mathcal{M}}_{i}={\mathcal{M}}\setminus\\{M_{i}\\}$. Then there is a
distinguished triangle

$i^{G}_{{\mathcal{M}}_{i}}\mathcal{R}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})\longrightarrow
i^{G}_{\mathcal{M}}\mathcal{R}^{G}_{\mathcal{M}}({\mathfrak{m}})\longrightarrow{\mathfrak{m}}_{i}\longrightarrow\Sigma
i^{G}_{{\mathcal{M}}_{i}}\mathcal{R}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})$

where
${\mathfrak{m}}_{i}=r^{G}_{{\mathcal{M}}_{i}}\pi^{G}_{{\mathcal{M}}_{i}}{\operatorname{Ho}}(I^{G}_{M_{i}})\mathbf{R}\mathcal{R}^{G}_{M_{i}}({\mathfrak{m}})$,
and similarly there is a distinguished triangle

$i^{G}_{{\mathcal{M}}_{i}}\mathcal{L}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})\longrightarrow
i^{G}_{\mathcal{M}}\mathcal{L}^{G}_{\mathcal{M}}({\mathfrak{m}})\longrightarrow{\mathfrak{m}}_{i}^{\prime}\longrightarrow\Sigma
i^{G}_{{\mathcal{M}}_{i}}\mathcal{L}^{G}_{{\mathcal{M}}_{i}}({\mathfrak{m}})$

where
${\mathfrak{m}}_{i}^{\prime}=\ell^{G}_{{\mathcal{M}}_{i}}\pi^{G}_{{\mathcal{M}}_{i}}{\operatorname{Ho}}(I^{G}_{M_{i}}){\operatorname{Ho}}(\mathcal{L}^{G}_{M_{i}})({\mathfrak{m}})$.

  3. (iii)

We have
${\operatorname{Im}}(\ell^{G}_{\mathcal{M}})=\bigcap_{i=1}^{n}{\operatorname{Ker}}({\operatorname{Ho}}(\mathcal{L}^{G}_{M_{i}}))$
and
${\operatorname{Im}}(r^{G}_{\mathcal{M}})=\bigcap_{i=1}^{n}{\operatorname{Ker}}(\mathbf{R}\mathcal{R}^{G}_{M_{i}})$.

As a first step, we lift a result of Abe to the homotopy level.

######  Lemma 3.14.

Suppose that $M$ and $L$ are two standard Levi subgroups. Then we have:

  1. (i)

${\operatorname{Ho}}({\mathcal{L}^{G}_{M}})\circ{\operatorname{Ho}}(I^{G}_{L})\cong{\operatorname{Ho}}(I_{M\cap
L}^{M})\circ{\operatorname{Ho}}(\mathcal{L}_{M\cap L}^{L})$; and

  2. (ii)

$\mathbf{R}{\mathcal{R}^{G}_{M}}\circ{\operatorname{Ho}}(I^{G}_{L})\cong{\operatorname{Ho}}(I_{M\cap
L}^{M})\circ R\mathcal{R}_{M\cap L}^{L}$.

###### Proof.

It was proved in [2, Lemma 5.2] that ${\mathcal{R}^{G}_{M}}\circ
I^{G}_{L}\cong I_{M\cap L}^{M}\circ\mathcal{R}_{M\cap L}^{L}$. By taking left
adjoints, we deduce that ${\mathcal{L}^{G}_{M}}\circ I_{L}^{G}\cong I_{M\cap
L}^{M}\circ\mathcal{L}_{M\cap L}^{L}$. Part (i) is now immediate, and part
(ii) then follows by taking right adjoints in (i). ∎

To prove 3.13 we will argue by induction on $n:=\left|{\mathcal{M}}\right|$,
the case $n=1$ being 3.12. We will need to establish some preparatory results
first, under the assumption that the statement of 3.13(i) holds for some fixed
$n\geq 1$ and all split reductive groups. Assuming this from now on, we let
${\mathcal{M}}=\\{M_{1},\ldots,M_{n}\\}$ be a collection of $n$ distinct
standard Levi subgroups and we fix another standard Levi subgroup $L$. Then
$L_{i}:=M_{i}\cap L$ is a standard Levi subgroup of $L$, and we may embed
fully faithfully each ${\operatorname{Ho}}({\mathcal{H}}_{L_{i}})$ inside
${\operatorname{Ho}}({\mathcal{H}}_{L})$ via
${\operatorname{Ho}}(I_{L_{i}}^{L})$. Thus we may consider the smallest thick
subcategory ${\mathcal{S}}_{{\mathcal{M}}\cap L}^{L}$ of
${\operatorname{Ho}}({\mathcal{H}}_{L})$ which contains these essential
images, and by assumption we also have a recollement

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With this notation and assumptions established, we have the following:

######  Lemma 3.15.

  1. (i)

The functors ${\operatorname{Ho}}(I^{G}_{L})$, $\mathbf{R}\mathcal{R}_{L}^{G}$
and ${\operatorname{Ho}}(\mathcal{L}_{L}^{G})$ induce a triple of triangulated
functors

$\textstyle{{\operatorname{Ho}}({\mathcal{H}}_{L})_{{\mathcal{M}}\cap
L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(I^{G}_{L})_{\mathcal{M}}}$$\textstyle{{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathcal{L}^{G}_{L})_{\mathcal{M}}}$$\scriptstyle{(\mathcal{R}_{L}^{G})_{\mathcal{M}}}$

such that

$\displaystyle\pi_{\mathcal{M}}^{G}\circ{\operatorname{Ho}}(I^{G}_{L})$
$\displaystyle=(I^{G}_{L})_{\mathcal{M}}\circ\pi_{{\mathcal{M}}\cap L}^{L},$
$\displaystyle\pi_{{\mathcal{M}}\cap
L}^{L}\circ{\operatorname{Ho}}(\mathcal{L}_{L}^{G})$
$\displaystyle=(\mathcal{L}^{G}_{L})_{\mathcal{M}}\circ\pi_{\mathcal{M}}^{G},$
$\displaystyle\text{and}\quad\quad\pi_{{\mathcal{M}}\cap
L}^{L}\circ\mathbf{R}\mathcal{R}_{L}^{G}$
$\displaystyle=(\mathcal{R}^{G}_{L})_{\mathcal{M}}\circ\pi_{\mathcal{M}}^{G}.$

  2. (ii)

The functor $(I^{G}_{L})_{\mathcal{M}}$ is naturally isomorphic to both
$\pi_{\mathcal{M}}^{G}\circ{\operatorname{Ho}}(I^{G}_{L})\circ\ell^{L}_{{\mathcal{M}}\cap
L}$ and $\pi_{\mathcal{M}}^{G}\circ{\operatorname{Ho}}(I^{G}_{L})\circ
r^{L}_{{\mathcal{M}}\cap L}$. Similarly, we have natural isomorphisms

$(\mathcal{L}^{G}_{L})_{\mathcal{M}}\cong\pi_{{\mathcal{M}}\cap
L}^{L}\circ{\operatorname{Ho}}(\mathcal{L}_{L}^{G})\circ\ell_{\mathcal{M}}^{G}$

and

$(\mathcal{R}^{G}_{L})_{\mathcal{M}}\cong\pi_{{\mathcal{M}}\cap
L}^{L}\circ\mathbf{R}\mathcal{R}_{L}^{G}\circ r_{\mathcal{M}}^{G}.$

###### Proof.

(i) By transitivity of parabolic induction (c.f. [30, Proposition 4.3]), for
each $1\leq i\leq n$ we have

$I^{G}_{L}\circ I^{L}_{L_{i}}\cong I^{G}_{L_{i}}\cong I^{G}_{M_{i}}\circ
I^{M_{i}}_{L_{i}}$

and so in particular $I^{G}_{L}\circ I^{L}_{L_{i}}$ factors through
$I^{G}_{M_{i}}$. Hence the same holds for the corresponding homotopy functors.
As a result it follows that the composite
$\pi_{\mathcal{M}}^{G}\circ{\operatorname{Ho}}(I^{G}_{L}):{\operatorname{Ho}}({\mathcal{H}}_{L})\to{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$
kills the essential image of each ${\operatorname{Ho}}(I^{L}_{L_{i}})$ and
hence kills ${\mathcal{S}}_{{\mathcal{M}}\cap L}^{L}$. By the universal
property of quotients, this implies that there is an induced triangulated
functor

$(I^{G}_{L})_{\mathcal{M}}:{\operatorname{Ho}}({\mathcal{H}}_{L})_{{\mathcal{M}}\cap
L}\to{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$

with the property that
$\pi_{\mathcal{M}}^{G}\circ{\operatorname{Ho}}(I^{G}_{L})=(I^{G}_{L})_{\mathcal{M}}\circ\pi_{{\mathcal{M}}\cap
L}^{L}$ as required.

The construction of $(\mathcal{R}^{G}_{L})_{\mathcal{M}}$ is very similar. By
3.14(ii), the composite
$\mathbf{R}\mathcal{R}_{L}^{G}\circ{\operatorname{Ho}}(I^{G}_{M_{i}})$ factors
through ${\operatorname{Ho}}(I^{L}_{L_{i}})$ for each $1\leq i\leq n$. Hence
it follows that the composite $\pi_{{\mathcal{M}}\cap
L}^{L}\circ\mathbf{R}\mathcal{R}_{L}^{G}$ kills the essential image of
${\operatorname{Ho}}(I^{G}_{M_{i}})$ for each $i$, and hence it kills
${\mathcal{S}}^{G}_{\mathcal{M}}$. Again, we then have that
$\pi_{{\mathcal{M}}\cap L}^{L}\circ\mathbf{R}\mathcal{R}_{L}^{G}$ factorises
as $(\mathcal{R}^{G}_{L})_{\mathcal{M}}\circ\pi_{\mathcal{M}}^{G}$ for some
triangulated functor

$(\mathcal{R}^{G}_{L})_{\mathcal{M}}:{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}\to{\operatorname{Ho}}({\mathcal{H}}_{L})_{{\mathcal{M}}\cap
L}.$

The construction of $(\mathcal{L}^{G}_{L})_{\mathcal{M}}$ is completely
analogous, using 3.14(i).

(ii) The two functors $r_{\mathcal{M}}^{G}$ and $\ell_{\mathcal{M}}^{G}$ are
both fully faithful and hence we have natural isomorphisms

$\pi_{\mathcal{M}}^{G}\circ\ell_{\mathcal{M}}^{G}\cong{\operatorname{id}}_{{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}}\cong\pi_{\mathcal{M}}^{G}\circ
r_{\mathcal{M}}^{G}$

by 2.4. So we get the identity for $(\mathcal{R}^{G}_{L})_{\mathcal{M}}$ by
pre-composing the functor $\pi_{{\mathcal{M}}\cap
L}^{L}\circ\mathbf{R}\mathcal{R}_{L}^{G}=(\mathcal{R}^{G}_{L})_{\mathcal{M}}\circ\pi_{\mathcal{M}}^{G}$
by $r_{\mathcal{M}}^{G}$ and applying the above natural isomorphism. The
identities for $(\mathcal{L}^{G}_{L})_{\mathcal{M}}$ and
$(I^{G}_{L})_{\mathcal{M}}$ are obtained similarly. ∎

######  Proposition 3.16.

Let ${\mathcal{M}}^{\prime}={\mathcal{M}}\cup\\{L\\}$. The triple of functors
from 3.15 is an adjoint triple, and it forms part of a recollement of the form

(3.5) $\textstyle{{\operatorname{Ho}}({\mathcal{H}}_{L})_{{\mathcal{M}}\cap
L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(I^{G}_{L})_{\mathcal{M}}}$$\textstyle{{\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\mathcal{M}}_{{\mathcal{M}}^{\prime}}}$$\scriptstyle{(\mathcal{L}^{G}_{L})_{\mathcal{M}}}$$\scriptstyle{(\mathcal{R}_{L}^{G})_{\mathcal{M}}}$$\textstyle{{\mathcal{T}}_{{\mathcal{M}}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{\mathcal{M}}_{{\mathcal{M}}^{\prime}}}$$\scriptstyle{r^{\mathcal{M}}_{{\mathcal{M}}^{\prime}}}$

for some triangulated category ${\mathcal{T}}_{{\mathcal{M}}^{\prime}}$.

###### Proof.

First, we show that the functors are adjoints and that
$(I^{G}_{L})_{\mathcal{M}}$ is fully faithful. This follows from the
identities in 3.15(ii). Indeed, we see that
$(\mathcal{R}_{L}^{G})_{\mathcal{M}}\cong\pi_{{\mathcal{M}}\cap
L}^{L}\circ\mathbf{R}\mathcal{R}_{L}^{G}\circ r_{\mathcal{M}}^{G}$ is a
composite of right adjoints so is a right adjoint, with left adjoint
$\pi_{\mathcal{M}}^{G}\circ{\operatorname{Ho}}(I^{G}_{L})\circ\ell^{L}_{{\mathcal{M}}\cap
L}\cong(I^{G}_{L})_{\mathcal{M}}$. Similarly,
$(\mathcal{L}_{L}^{G})_{\mathcal{M}}\cong\pi_{{\mathcal{M}}\cap
L}^{L}\circ{\operatorname{Ho}}(\mathcal{L}_{L}^{G})\circ\ell_{\mathcal{M}}^{G}$
is a composite of left adjoints, so is left adjoint to
$\pi_{\mathcal{M}}^{G}\circ{\operatorname{Ho}}(I^{G}_{L})\circ
r^{L}_{{\mathcal{M}}\cap L}\cong(I^{G}_{L})_{\mathcal{M}}$ as required.
Finally,
$(I^{G}_{L})_{\mathcal{M}}\cong\pi_{L}^{G}\circ{\operatorname{Ho}}(I^{G}_{L})\circ
r^{L}_{{\mathcal{M}}\cap L}$ is a composite of fully faithful functors and so
is fully faithful.

Next, we show that the essential image of $(I^{G}_{L})_{\mathcal{M}}$ is
localising (and in particular thick). Indeed, the quotient categories
${\operatorname{Ho}}({\mathcal{H}}_{L})_{{\mathcal{M}}\cap L}$ and
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}$ have arbitrary direct sums,
because the same is true of ${\operatorname{Ho}}({\mathcal{H}}_{M})$ and
${\operatorname{Ho}}({\mathcal{H}})$, and because the quotient functors are
essentially surjective and preserve direct sums (as they are left adjoints).
Then $(I^{G}_{L})_{\mathcal{M}}$ preserve these direct sums since it is a left
adjoint.

Therefore, we are in the same situation as the one obtained in 3.10, and we
may again apply [21, Propositions 4.9.1 & 4.13.1] to obtain a recollement. ∎

###### Proof of 3.13.

(i) As stated earlier, we work by induction on $n$. Assuming the statements
hold for $n\geq 1$, let $M_{1},\ldots,M_{n+1}$ be distinct standard Levi
subgroups and write ${\mathcal{M}}=\\{M_{1},\ldots,M_{n}\\}$ and
${\mathcal{M}}^{\prime}={\mathcal{M}}\cup\\{M_{n+1}\\}$ as before. By setting
$L=M_{n+1}$ in 3.16, we see that there is a recollement as in (3.5) and we
write again ${\mathcal{T}}_{{\mathcal{M}}^{\prime}}$ for the quotient
${\operatorname{Ho}}({\mathcal{H}})_{\mathcal{M}}/{\operatorname{Ho}}({\mathcal{H}}_{L})_{{\mathcal{M}}\cap
L}$.

Then define the functor $\pi^{G}_{{\mathcal{M}}^{\prime}}$ to be the composite
$\pi^{\mathcal{M}}_{{\mathcal{M}}^{\prime}}\circ\pi^{G}_{\mathcal{M}}:{\operatorname{Ho}}({\mathcal{H}})\to{\mathcal{T}}_{{\mathcal{M}}^{\prime}}$
and let
${\mathcal{S}}_{{\mathcal{M}}^{\prime}}:={\operatorname{Ker}}(\pi^{G}_{{\mathcal{M}}^{\prime}})$.
Note that $\pi^{G}_{{\mathcal{M}}^{\prime}}$ has fully faithful left and right
adjoints, given by
$\ell^{G}_{{\mathcal{M}}^{\prime}}:=\ell^{G}_{\mathcal{M}}\circ\ell^{\mathcal{M}}_{{\mathcal{M}}^{\prime}}$
and $r^{G}_{{\mathcal{M}}^{\prime}}:=r^{G}_{\mathcal{M}}\circ
r^{\mathcal{M}}_{{\mathcal{M}}^{\prime}}$ respectively. Therefore it follows
from [21, Proposition 4.9.1] and its dual that the inclusion
${\mathcal{S}}_{{\mathcal{M}}^{\prime}}\to{\operatorname{Ho}}({\mathcal{H}})$
also has both a left and a right adjoint, and and so these adjunctions form a
recollement. Note also that ${\mathcal{S}}_{{\mathcal{M}}^{\prime}}$ is
automatically localising, because the functor
$\pi^{G}_{{\mathcal{M}}^{\prime}}$ commutes with arbitrary direct sums as it
is a left adjoint. We are thus only left to check that
${\mathcal{S}}^{G}_{{\mathcal{M}}^{\prime}}={\mathcal{S}}_{{\mathcal{M}}^{\prime}}$.

For this, we need to show that if ${\mathcal{S}}$ is an arbitrary thick
subcategory of ${\operatorname{Ho}}({\mathcal{H}})$ containing the essential
images of
${\operatorname{Ho}}(I_{M_{1}}^{G}),\ldots,{\operatorname{Ho}}(I_{M_{n+1}}^{G})$,
then ${\mathcal{S}}$ contains ${\mathcal{S}}_{{\mathcal{M}}^{\prime}}$. It
suffices to show that the quotient functor