1. Introduction
1.1 History
Let 
$G$ be a connected reductive group over an algebraically closed field 
$k$ of characteristic 
$p>0$. A fundamental problem in the representation theory of 
$G$ is to understand the structure of Weyl modules or, equivalently, the structure of the induced modules 
$\nabla (\lambda )$ for 
$\lambda \in X^+$ a dominant integral weight. The determination of the composition factors of 
$\nabla (\lambda )$ for all 
$\lambda \in X^+$ is equivalent to knowing the characters of simple modules for 
$G$. This central problem still remains open even though there has been much work done to formulate new character formulas using tilting modules and 
$p$-Kazhdan–Lusztig polynomials. In order to enhance our understanding of the structure of 
$\nabla (\lambda )$, Jantzen asked the question in 1980 as to whether the induced modules 
$\nabla (\lambda )$ have good 
$p$-filtrations. Parshall and Scott [Reference Parshall and ScottPS15] proved that this holds as long as 
$p\geq 2(h-1)$ and the Lusztig character formula holds for 
$G$. Andersen [Reference AndersenAnd19] recently proved that 
$\nabla (\lambda )$ has a good 
$p$-filtration as long as 
$p\geq h(h-2)$.
Another important and still unresolved problem in the representation theory of 
$G$ is to determine whether a projective module for a Frobenius kernel of 
$G$ has a structure as a 
$G$-module. The expectation that this should be true dates back to the work of Humphreys and Verma [Reference Humphreys and VermaHV73]. Donkin later conjectured that such structures should arise from tilting modules for 
$G$ (cf. [Reference DonkinDon93]), which is now known to hold for 
$p \geq 2(h-2)$ (and some small rank cases). In some sense, this conjecture was implicit in [Reference Humphreys and VermaHV73], which predates the concept of tilting modules by many years. The general method involved tensoring by the Steinberg representation and locating an important 
$G$-summand that is known to be a tilting module by a result due to Pillen [Reference PillenPil93].
In [Reference Bendel, Nakano, Pillen and SobajeBNPS20b], the authors found counterexamples to (i) Jantzen's question (JQ) and (ii) the tilting module conjecture (TMC). For exact statements of JQ and the TMC, see Question 2.3.1 and Conjecture 2.3.2. The counterexamples occur for the simple algebraic group of type 
$\text {G}_2$ in characteristic 
$2$, where it was shown that the tilting module 
$T(2,2)$ is not indecomposable over the Frobenius kernel of 
$G$ and that 
$\nabla (2,1)$ does not have a good 
$p$-filtration. As the first surprising example of its kind, it provided some clarity regarding the TMC, explaining if nothing else why it had resisted proof for nearly 30 years. After that paper was written, a number of questions still remained to be answered. For instance, was this example an anomaly, or were there many others like it? There is also the basic question as to what features about the representation theory and cohomology give rise to this counterexample and what causes the tilting module 
$T(2,2)$ to split over the Frobenius kernel. It should be noted that Kildetoft and Nakano, and Sobaje made explicit connections between the TMC and good 
$p$-filtrations on 
$G$-modules (cf. [Reference Kildetoft and NakanoKN15, Reference SobajeSob18]).
1.2 Results
The main goal of this paper is to show that the type 
${\rm G}_2$ examples were not, in fact, anomalies. We show how to produce large families of counterexamples to JQ and the TMC. Our results are summarized in the following theorem.
Theorem 1.2.1 Let 
$G$ be a simple algebraic group over an algebraically closed field of characteristic 
$p>0$. Assume that the underlying root system 
$\Phi$ is not of type 
${\rm A}_{n}$ or 
${\rm B}_{2}$. Then there exists a prime 
$p$ for which 
$G$ produces a counterexample to JQ and the TMC.
In particular, we can exhibit counterexamples to JQ and the TMC for the following groups:
–
$\Phi ={\rm B}_{n}$, $p=2$, $n\geq 3$;–
$\Phi ={\rm C}_{n}$, $p=3$, $n\geq 3$;–
$\Phi ={\rm D}_{n}$, $p=2$, $n\geq 4$;–
$\Phi ={\rm E}_{n}$, $p=2$, $n=6,7,8$;–
$\Phi ={\rm F}_{4}$, $p=2,3$;–
$\Phi ={\rm G}_{2}$, $p=2$.
The counterexamples in type 
${\rm C}_n$ at 
$p = 3$ are particularly interesting as 3 is a good prime for the root system. We anticipate that counterexamples exist for all 
${\rm C}_n$ when 
$n = p$ is prime. In general, the question of when the TMC holds still seems quite subtle. In [Reference Bendel, Nakano, Pillen and SobajeBNPS22] and [Reference AndersenAnd19], it was shown that JQ and the TMC holds when 
$\Phi ={\rm B}_{2}$ for all 
$p$. Moreover, it was shown in [Reference Bendel, Nakano, Pillen and SobajeBNPS22] that the TMC holds in type 
${\rm G}_2$ for all 
$p > 2$ and for all primes for 
$\Phi ={\rm A}_{n}$ when 
$n\leq 3$. It is still possible that the TMC for 
$\Phi ={\rm A}_{n}$ holds for all 
$n$ and all 
$p$.
The paper is organized as follows. In § 2, we introduce the notation and conventions for the paper. Two reduction theorems (cf. Theorems 2.4.1 and 2.6.1) are established for JQ and the TMC that enable one to use counterexamples in low rank to establish counterexamples in large rank via Levi subgroups.
Section 3 focuses on new methods to use 
$\text {Ext}^{1}$ structural information to produce counterexamples to the TMC. Our new examples also give some indication as to why the failure of the TMC is occurring. Namely, in each case one finds that there are (restricted) dominant weights 
$\lambda$ and 
$\mu$ such that the 
$G/G_{1}$-module 
$\operatorname {Ext}_{G_1}^1(L(\lambda ),L(\mu ))$ is not a tilting module. Indeed, the counterexamples in this paper were actually discovered by first looking for such behavior in 
$\text {Ext}$-groups, having already observed its effect on the 
$\Phi ={\rm G}_{2}$, 
$p=2$, example in [Reference Bendel, Nakano, Pillen and SobajeBNPS20b].
In § 4 counterexamples to JQ and the TMC are worked out for 
$\Phi ={\rm B}_{3}$, 
${\rm C}_{3}$, and 
${\rm D}_{4}$ via the methods in § 3. In the following section (§ 5) the main theorem is proved. In § 6 we show how to produce other counterexamples using the Jantzen filtration. In particular, we exhibit other counterexamples to the TMC in characteristic 
$2$ for all groups of type 
$\text {B}_n$ with 
$n\geq 3$. We also construct additional counterexamples in type 
$\text {C}_n$ for 
$p=3$ and 
$n\geq 3$.
At the end of the paper, in § 7, we present some open problems involving the connections with the TMC and the question of whether 
$\operatorname {Ext}_{G_1}^1(L(\lambda ),L(\mu ))$ is a tilting 
$G/G_{1}$-module. We view these important observations as opening the door for future investigation.
2. Preliminaries
2.1 Notation
In this paper we generally follow the standard conventions in [Reference JantzenJan03]. Throughout this paper 
$k$ is an algebraically closed field of characteristic 
$p>0$. Let 
$G$ be a connected semisimple algebraic group scheme defined over 
${\mathbb {F}}_{p}$. The Frobenius morphism is denoted by 
$F$ and the 
$r$th Frobenius kernel will be denoted by 
$G_{r}$. Given a maximal torus 
$T$, the root system 
$\Phi$ is associated to the pair 
$(G,T)$. Let 
$\Phi ^+$ be a set of positive roots and 
$\Phi ^{-}$ be the corresponding set of negative roots. The set of simple roots determined by 
$\Phi ^+$ is 
$\Delta =\{\alpha _1,\ldots,\alpha _{l}\}$. The ordering of simple roots is described in [Reference HumphreysHum72] following Bourbaki.
Let 
$B$ be the Borel subgroup given by the set of negative roots and let 
$U$ be the unipotent radical of 
$B$. More generally, if 
$J\subseteq \Delta$, let 
$P_{J}$ be the parabolic subgroup relative to 
$-J$, 
$L_{J}$ be the Levi factor of 
$P_{J}$ and 
$U_{J}$ be the unipotent radical. Let 
$\Phi _{J}$ be the root subsystem in 
$\Phi$ generated by the simple roots in 
$J$, with positive subset 
$\Phi _{J}^+ = \Phi _{J}\cap \Phi ^+$.
Let 
$\mathbb {E}$ be the Euclidean space associated with 
$\Phi$, and denote the inner product on 
${\mathbb {E}}$ by 
$\langle \, ,\, \rangle$. Set 
$\alpha _{0}$ to be the highest short root. Let 
$\rho$ be the half-sum of positive roots and 
$\alpha ^{\vee }$ be the coroot corresponding to 
$\alpha \in \Phi$. The Coxeter number associated to 
$\Phi$ is 
$h=\langle \rho,\alpha _{0}^{\vee } \rangle +1$. The Weyl group associated to 
$\Phi$ will be denoted by 
$W$, and, for any 
$J\subseteq \Delta$, let 
$W_{J}$ be the subgroup of 
$W$ generated by reflections corresponding to simple roots in 
$J$. Let 
$w_0$ (respectively, 
$w_{J,0}$) denote the longest word of 
$W$ (respectively, 
$W_{J}$, for 
$J \subseteq \Delta.$). Set 
$\rho _J$ to be the half-sum of all the roots in 
$\Phi _J^+$.
Let 
$X:=X(T)$ be the integral weight lattice spanned by the fundamental weights 
$\{\omega _1,\ldots,\omega _l\}$, 
$X^+$ be the dominant weights for 
$G$, and 
$X_{r}$ be the 
$p^{r}$-restricted weights. Moreover, for 
$J\subseteq \Delta$, let 
$X_{J}^+$ be the weights that are dominant on 
$J$. That is, 
$X_{J}^+ := \{\lambda \in X \,|\, \langle \lambda,\alpha ^{\vee }\rangle \geq 0\ \forall \ \alpha \in J\}$.
Let 
$\tau :G \rightarrow G$ be the Chevalley antiautomorphism of 
$G$ that is the identity morphism when restricted to 
$T$ (see [Reference JantzenJan03, II.1.16]). Given a finite-dimensional 
$G$-module 
$M$ over 
$k$, the module 
$^{\tau }M$ is 
$M^*$ (the ordinary 
$k$-linear dual of 
$M$) as a 
$k$-vector space, with action 
$g.f(m)=f(\tau (g).m)$. This defines a functor from 
$G$-mod to 
$G$-mod that preserves the character of 
$M$. In particular, it is the identity functor on all simple and tilting modules.
The following result is used throughout this section. Note that we follow the convention in [Reference JantzenJan03] that the set 
$\mathbb {N}$ includes 
$0$.
Lemma 2.1.1 Let 
$\mu \in X^+$ and 
$w \in W$ be such that 
$\mu - w \mu \in \mathbb {N}J$. Then there exists 
$w_J \in W_J$ with 
$w\mu =w_J\mu.$
Proof. There exists 
$u \in W_J$ such that 
$\langle u w \mu, \beta ^{\vee } \rangle \geq 0$ for all 
$\beta \in J,$ (i.e. 
$u w \mu$ is 
$J$-dominant). Since 
$u \in W_J$, 
$w\mu - u w \mu \in \mathbb {Z}J.$ With this and the hypothesis, 
$\mu - u w \mu = (\mu - w \mu ) + (w\mu - u w \mu ) \in \mathbb {Z}J.$ However, 
$\mu \geq u w \mu$, thus 
$\mu - u w\mu \in \mathbb {N}J$.
If 
$u w \mu \in X^+$, then 
$u w \mu = \mu$ and one may choose 
$w_J = u^{-1} \in W_J$. Thus, 
$w \mu = w_J \mu.$ If 
$u w \mu$ is not in 
$X^+$, then there exists a 
$\beta \in \Delta \setminus J$ such that 
$\langle u w \mu, \beta ^{\vee } \rangle < 0,$ which implies that 
$\langle \mu - u w \mu, \beta ^{\vee } \rangle > 0$. This contradicts the fact that 
$\mu - u w\mu \in \mathbb {N}J$.
2.2 Representations
For 
$\lambda \in X^+$, there are four fundamental families of 
$G$-modules (each having highest weight 
$\lambda$): 
$L(\lambda )$ (simple), 
$\nabla (\lambda )=H^{0}(\lambda )$ (costandard/induced), 
$\Delta (\lambda ) =V(\lambda )$ (standard/Weyl), and 
$T(\lambda )$ (indecomposable tilting). Let 
$\text {St}_r = L((p^r-1)\rho )$ be the 
$r$th Steinberg module. For 
$\lambda \in X_{r}$, let 
$Q_{r}(\lambda )$ denote the 
$G_{r}$-projective cover (equivalently, injective hull) of 
$L(\lambda )$ as a 
$G_{r}$-module. For 
$\lambda \in X$, if 
$\widehat {L}_{r}(\lambda )$ is the corresponding simple 
$G_{r}T$-module, let 
$\widehat {Q}_{r}(\lambda )$ denote the 
$G_{r}T$-projective cover (equivalently, injective hull) of 
$\widehat {L}_{r}(\lambda )$. For 
$\lambda \in X^+$, write 
$\lambda = \lambda _0 + p^r\lambda _1$ with 
$\lambda _0\in X_r$ and 
$\lambda _1\in X^+$. Define 
$\nabla ^{(p,r)}(\lambda ) = L(\lambda _0)\otimes \nabla (\lambda _1)^{(r)}$, where 
$(r)$ denotes the twisting of the module action by the 
$r$th Frobenius morphism.
A 
$G$-module 
$M$ has a good filtration (respectively, good 
$(p,r)$-filtration) if and only if 
$M$ has a filtration with factors of the form 
$\nabla (\mu )$ (respectively, 
$\nabla ^{(p,r)}(\mu )$) for suitable 
$\mu \in X^+$. In the case when 
$r=1$, good 
$(p,1)$-filtrations are often referred to as good 
$p$-filtrations.
For each 
$\lambda \in X_J^+$ there is a simple 
$L_J$-module 
$L_J(\lambda )$, a standard/Weyl module 
$\Delta _J(\lambda )$, a costandard/induced module 
$\nabla _J(\lambda ),$ and an indecomposable tilting module 
$T_J(\lambda )$. Specifically, for 
$\lambda \in X_{J}^+,$ 
$\nabla _J(\lambda )= \operatorname {ind}_{B}^{P_J}(\lambda )\cong \operatorname {ind}_{B\cap L_J}^{L_J}(\lambda )$. Furthermore, for 
$\lambda \in X$ and 
$r \geq 1$, set
\[ \widehat{Z}'_{J,r}(\lambda) = \operatorname{ind}_{B_rT}^{(P_J)_rT}(\lambda) \cong \operatorname{ind}_{(B_r \cap L_J)T}^{(L_J)_rT}(\lambda) \]
and 
$\widehat {Q}_{J,r}(\lambda )$ to be the 
$(L_J)_rT$-injective hull of the simple 
$(L_J)_rT$-module 
$\widehat {L}_{J,r}(\lambda )$. When 
$J = \Delta$, we simply denote 
$\widehat {Z}'_{J,r}(\lambda )$ by 
$\widehat {Z}'_r(\lambda )$.
Let 
$r \ge 1$ and write 
$\lambda = \lambda _0 + p^r\lambda _1$, where 
$\lambda _0 \in X_r(T)$ and 
$\lambda _1 \in X_J^+$. By 
$\nabla _J^{(p,r)}(\lambda )$ denote the module
\[ \nabla_J^{(p,r)}(\lambda) = L_J(\lambda_0) \otimes \nabla_J(\lambda_1)^{(r)}. \]
An 
$L_J$-module with a filtration whose factors have this form is said to have a good 
$(p,r)$-filtration. In [Reference JantzenJan03, Propositions II.2.11 and II.5.21], it is observed that one has the following facts with 
$\lambda, \mu \in X^+$:
(i)
$\nabla _J(\lambda )= \bigoplus _{ \nu \in \mathbb {N}J} \nabla (\lambda )_{\lambda -\nu }$;(ii)
$L_J(\lambda )= \bigoplus _{ \nu \in \mathbb {N}J} L(\lambda )_{\lambda -\nu }$;(iii)
$[\nabla (\lambda ) : L(\mu )] = [\nabla _J(\lambda ): L_J(\mu )],$ whenever $\lambda - \mu \in \mathbb {N}J.$
Furthermore, the arguments in [Reference DonkinDon83] can be adapted to yield for 
$\lambda, \mu, \lambda ', \mu ' \in X$:
(iv)
$\widehat {Z}'_{J,r}(\lambda ) = \bigoplus _{ \nu \in \mathbb {N}J} \widehat {Z}_r'(\lambda )_{\lambda -\nu }$;(v) [
$\widehat {Z}'_r(\lambda ) : \widehat {L}_r(\mu )] = [\widehat {Z}'_{J,r}(\lambda ): \widehat {L}_{J,r}(\mu )],$ whenever $\lambda - \mu \in \mathbb {N}J$;(vi) If
$\lambda -\mu \in \mathbb {N}J$ and $\langle \lambda - \lambda ', \beta ^{\vee } \rangle = \langle \mu - \mu ', \beta ^{\vee } \rangle = 0$ for all $\beta \in J,$ then
\[ [\widehat{Z}'_{J,r}(\lambda): \widehat{L}_{J,r}(\mu)] =[\widehat{Z}'_{J,r}(\lambda'): \widehat{L}_{J,r}(\mu')]. \]
The following proposition discusses the restriction of tensor products of 
$G$-modules with unique highest weights to Levi subgroups, with an immediate application involving induced modules. Note that a module with ‘unique highest weight’ is not assumed to be generated by a highest weight vector. This allows the proposition to apply, for example, to indecomposable tilting modules.
Proposition 2.2.1 Let 
$\lambda, \mu \in X^+$.
(a) If
$Y(\lambda )$ and $Y(\mu )$ are $G$-modules with unique highest weights $\lambda$ and $\mu$, respectively, then we have an equality of $L_J$-modules
\[ \bigg(\bigoplus_{\nu \in \mathbb{N}J} (Y(\lambda) )_{\lambda-\nu}\bigg) \otimes \bigg(\bigoplus_{\nu \in \mathbb{N}J} (Y(\mu) )_{\mu-\nu}\bigg) = \bigoplus_{\nu \in \mathbb{N}J} (Y(\lambda) \otimes Y(\mu))_{\lambda+ \mu-\nu}. \]
(b) There exists an equality of
$L_J$-modules
\[ \nabla_J^{(p,r)}(\lambda) = \bigoplus_{\nu \in \mathbb{N}J} \nabla^{(p,r)}(\lambda)_{\lambda-\nu}. \]
Proof. (a) Clearly we have that
\[ \bigg(\bigoplus_{\nu \in \mathbb{N}J} Y(\lambda)_{\lambda -\nu}\bigg) \otimes \bigg(\bigoplus_{\nu \in \mathbb{N}J} Y(\mu)_{\mu -\nu}\bigg) \subseteq \bigoplus_{\nu \in \mathbb{N}J} (Y(\lambda) \otimes Y(\mu))_{\lambda+\mu-\nu}. \]
On the other hand, if 
$Y(\lambda )_{\gamma }$ is a nonzero weight space and 
$\gamma$ is not of the form 
$\lambda - \nu$ for some 
$\nu \in \mathbb {N}J$, then there is some simple root 
$\alpha \in \Delta \backslash J$ such that when 
$\lambda -\gamma$ is expressed in the basis of simple roots, the coefficient 
$n_{\alpha }$ of 
$\alpha$ has 
$n_{\alpha }>0$. For there to be a weight 
$\sigma$ of 
$Y(\mu )$ such that 
$\gamma +\sigma =\lambda +\mu -\nu$ with 
$\nu \in \mathbb {N}J$, we need that
\begin{align*} \lambda+\mu - (\gamma+\sigma) \in \mathbb{N}J \end{align*}
or, equivalently, that
\[ (\lambda - \gamma) +(\mu - \sigma) \in \mathbb{N}J. \]
Thus, we have that in writing 
$\mu - \sigma$ in the basis 
$\Delta$, the coefficient of 
$\alpha$ is 
$-n_{\alpha }$, and 
$-n_{\alpha }<0$. Since 
$\mu \ge \sigma$, this cannot happen. A similar argument, reversing the roles of 
$\lambda$ and 
$\mu$, shows then that we have an equality
\[ \bigg(\bigoplus_{\nu \in \mathbb{N}J} (Y(\lambda) )_{\lambda-\nu}\bigg) \otimes \bigg(\bigoplus_{\nu \in \mathbb{N}J} (Y(\mu) )_{\mu-\nu}\bigg) = \bigoplus_{\nu \in \mathbb{N}J} (Y(\lambda) \otimes Y(\mu))_{\lambda+ \mu-\nu}. \]
(b) With facts (i) and (ii), applying part (a) to 
$Y(\lambda _0)=L(\lambda _0),$ with 
$\lambda _0 \in X_r,$ and 
$Y(p^r\lambda _1)= \nabla (\lambda _1)^{(r)},$ with 
$\lambda _1 \in X^+,$ immediately yields the result.
2.3 JQ and the TMC
For a complete description of JQ, Donkin's 
$(p,r)$-filtration conjecture, and Donkin's TMC, we refer the reader to [Reference Bendel, Nakano, Pillen and SobajeBNPS20b, § 2.2]. In 1980, Jantzen [Reference JantzenJan80] asked the following question which lead to the formulation of the 
$(p,r)$-filtration conjecture.
Question 2.3.1 For 
$\lambda \in X^+$, does 
$\nabla (\lambda )$ admit a good 
$(p,r)$-filtration?
The TMC, introduced by Donkin in 1990 [Reference DonkinDon93], states that a projective indecomposable module for 
$G_r$ can be realized as an indecomposable tilting 
$G$-module.
Conjecture 2.3.2 For all 
$\lambda \in X_{r}$,
\[ T((p^{r}-1)\rho+\lambda)|_{G_{r}T}\cong\widehat{Q}_{r}((p^{r}-1)\rho+w_{0}\lambda). \]
An alternative and equivalent formulation of the conjecture is that
\[ T(2(p^{r}-1)\rho+w_{0}\lambda)|_{G_{r}T}\cong\widehat{Q}_{r}(\lambda) \]
for all 
$\lambda \in X_{r}$.
One direction of Donkin's 
$(p,r)$-filtration conjecture is equivalent to the following.
Conjecture 2.3.3 For 
$\lambda \in X_r$, 
$\operatorname {St}_r\otimes L(\lambda )$ is a tilting module.
In [Reference SobajeSob18], it was shown that an affirmative answer to Question 2.3.1 and Conjecture 2.3.3 implies that Conjecture 2.3.2 holds.
2.4 JQ and Levi subgroups
The following result demonstrates that JQ is compatible with restricting to Levi subgroups. This theorem will be used later to produce counterexamples to JQ by analyzing examples in low rank.
Theorem 2.4.1 Let 
$L_J$ be a Levi subgroup of 
$G$, and let 
$\lambda \in X^+$. If 
$\nabla (\lambda )$ has a good 
$(p,r)$-filtration as a 
$G$-module, then 
$\nabla _J(\lambda )$ has a good 
$(p,r)$-filtration as an 
$L_J$-module. Moreover, one obtains that 
$[\nabla (\lambda ): \nabla ^{(p,r)}(\mu )] = [\nabla _J(\lambda ): \nabla _J^{(p,r)}(\mu )]$, whenever 
$\lambda - \mu \in \mathbb {N}J$.
Proof. As an 
$L_J$-module, 
$\nabla _J(\lambda )$ is a direct summand of 
$\nabla (\lambda )$ and, from fact (i) in § 2.2, is given explicitly as the sum of weight spaces
\[ \bigoplus_{\nu \in \mathbb{N}J} \nabla(\lambda)_{\lambda-\nu}. \]
In particular, there is a projection map 
$\pi :\nabla (\lambda ) \rightarrow \nabla _J(\lambda )$, which is a homomorphism of 
$L_J$-modules. In addition, it follows that on each 
$T$-weight space, 
$\pi$ is the zero map if the weight is not of the form 
$\lambda -\nu$ for 
$\nu \in \mathbb {N}J$, otherwise 
$\pi$ is an isomorphism onto the corresponding weight space in the image.
If 
$\nabla (\lambda )$ has a good 
$(p,r)$-filtration, then there is a filtration of 
$G$-submodules
\[ \{0\}=F_0 \subseteq F_1 \subseteq \cdots \subseteq F_m = \nabla(\lambda) \]
and dominant weights 
$\mu _1,\ldots,\mu _m$ such that each 
$F_i/F_{i-1} \cong \nabla ^{(p,r)}(\mu _i)$. We can apply 
$\pi$ to this filtration, obtaining a filtration on 
$\nabla _J(\lambda )$:
\[ \{0\}=\pi(F_0) \subseteq \pi(F_1) \subseteq \cdots \subseteq \pi(F_m) = \nabla_J(\lambda). \]
For each 
$i$, we have (by restricting 
$\pi$ and composing with a quotient map) an 
$L_J$-module homomorphism
\[ \pi_i: F_i \rightarrow \pi(F_i)/\pi(F_{i-1}), \]
and clearly 
$F_{i-1}$ is contained in the kernel of 
$\pi _i$. Thus, we obtain a homomorphism of 
$L_J$-modules
\[ \nabla^{(p,r)}(\mu_i) \rightarrow \pi(F_i)/\pi(F_{i-1}). \]
As noted previously, this homomorphism is nonzero if and only if 
$\nabla ^{(p,r)}(\mu _i)$ has nonzero weight vectors of the form 
$\lambda -\nu$ for 
$\nu \in \mathbb {N}J$. Since 
$\lambda \ge \mu _i$, it follows (by an argument as the proof of Proposition 2.2.1) that it is necessary (and clearly sufficient) that 
$\lambda - \mu _i \in \mathbb {N}J$.
Thus, we have that
\[ \pi(F_i)/\pi(F_{i-1}) \cong \nabla_J^{(p,r)}(\mu_i) \]
if 
$\lambda -\mu _i \in \mathbb {N}J$, and
\[ \pi(F_i)/\pi(F_{i-1}) = \{0\} \]
otherwise, so that 
$\nabla _J(\lambda )$ has a good 
$(p,r)$-filtration.
2.5 Injective hulls and Levi subgroups
It was first observed by Donkin [Reference DonkinDon93] that the restriction of a projective indecomposable 
$G_{r}T$ module to 
$(L_{J})_{r}T$ yields a decomposition that is similar to those listed in § 2.2. As a proof was only outlined in [Reference DonkinDon93], for the reader's convenience, we include a self-contained proof that will also be used in [Reference Bendel, Nakano, Pillen and SobajeBNPS23].
Theorem 2.5.1 [Reference DonkinDon93, Proposition 2.7]
Let 
$L_J$ be a Levi subgroup of G and 
$\lambda \in X_r$. Then we have an equality of 
$(L_J)_rT$-modules:
\[ \widehat{Q}_{J,r}((p^r-1)\rho + w_{J,0}\lambda) = \bigoplus_{\nu \in \mathbb{N}J} \widehat{Q}_r((p^r-1)\rho+w_0 \lambda)_{(p^r-1)\rho+\lambda-\nu}. \]
Proof. To prove the theorem, we define the 
$(L_J)_rT$-summand 
$M$ of 
$\widehat {Q}_r((p^r-1)\rho +w_0 \lambda )$ via
\[ M=\bigoplus_{\nu \in \mathbb{N}J} \widehat{Q}_r((p^r-1)\rho+w_0 \lambda)_{(p^r-1)\rho+\lambda-\nu}. \]
The restriction of 
$\widehat {Q}_r((p^r-1)\rho +w_0 \lambda )$ to 
$(L_J)_rT$ is still injective and projective. Therefore, 
$M$ is also injective and projective as an 
$(L_J)_rT$-module. It suffices to show that
\[ \text{ch } M = \text{ch } \widehat{Q}_{J,r}((p^r-1)\rho + w_{J,0}\lambda). \]
Verification of this latter claim will be carried out over the next several steps.
(1) Formal character of 
$\widehat {Q}_{J,r}((p^r-1)\rho +w_{J,0}\lambda )$. The 
$(L_J)_rT$-module
\[ \nabla_J((p^r-1)\rho)= L_J((p^r-1) \rho)=\widehat{Z}'_{J,r}((p^r-1) \rho) \]
is irreducible, injective and projective. In addition,
\begin{align*} & {\operatorname{Hom}}_{(L_J)_rT}(L_J((p^r-1)\rho+w_{J,0}\lambda),\widehat{Z}'_{J,r}((p^r-1) \rho)\otimes L_J(\lambda))\\ &\quad \cong {\operatorname{Hom}}_{(L_J)_rT}(L_J((p^r-1)\rho+w_{J,0}\lambda) \otimes L_J(-w_{J,0}\lambda), L_J((p^r-1) \rho)) \\ &\quad \cong k. \end{align*}
Therefore, 
$\widehat {Q}_{J,r}((p^r-1)\rho +w_{J,0}\lambda )$ is an 
$(L_J)_rT$-summand of 
$\widehat {Z}'_{J,r}((p^r-1) \rho )\otimes L_J(\lambda )$. The latter has a 
$\widehat {Z}'_{J,r}$-filtration with factors of the form 
$\widehat {Z}'_{J,r}((p^r-1) \rho +\gamma )$, with 
$\gamma$ being a weight of 
$L_J(\lambda )$. The weights of 
$L_J(\lambda )$ are all of the form 
$w\mu$ with 
$\mu \in X_J^+,$ 
$w \in W_J,$ and 
$\lambda -\mu \in \mathbb {N}J$. Note that 
$\lambda \in X_r,$ 
$\mu \in X_J^+,$ and 
$\lambda -\mu \in \mathbb {N}J$ implies that 
$\mu \in X^+$. Hence, 
$\widehat {Q}_{J,r}((p^r-1)\rho +w_{J,0}\lambda )$ has an 
$(L_J)_rT$-filtration with factors of the form 
$\widehat {Z}_{J,r}'((p^r-1)\rho + w \mu )$ with 
$\mu \in \{\gamma \in X^+ \,|\, \lambda - \gamma \in \mathbb {N}J\}$ and 
$w \in W_J$. By making use of Brauer–Humphreys reciprocity [Reference JantzenJan03, II.11.4] and 
$W_J$-invariance [Reference JantzenJan03, II.9.16(5)], one obtains
\begin{align*} &
\text{ch}\,\widehat{Q}_{J,r}((p^r-1)\rho +w_{J,0}\lambda)\\
&\quad = \sum_{\{\mu \in X^+ \,|\, \lambda - \mu \in
\mathbb{N}J\}} \frac{1}{|\text{Stab}_{W_J}(\mu)|} \sum_{w
\in W_J} [\widehat{Q}_{J,r}((p^r-1)\rho + w_{J,0}\lambda):
\widehat{Z}_{J,r}'((p^r-1)\rho +w\mu)] \\ &\qquad \cdot
\text{ch } \widehat{Z}_{J,r}'((p^r-1)\rho +w\mu)\\ &\quad =
\sum_{\{\mu \in X^+ \,|\, \lambda - \mu \in \mathbb{N}J\}}
\frac{1}{|\text{Stab}_{W_J}(\mu)|} \sum_{w \in W_J}
[\widehat{Z}_{J,r}'((p^r-1)\rho +w\mu): L_J((p^r-1)\rho +
w_{J,0}\lambda)]\\ &\qquad \cdot \text{ch }
\widehat{Z}_{J,r}'((p^r-1)\rho +w\mu) \\ &\quad =
\sum_{\{\mu \in X^+ \,|\, \lambda - \mu \in \mathbb{N}J\}}
\frac{1}{|\text{Stab}_{W_J}(\mu)|} \cdot
[\widehat{Z}_{J,r}'((p^r-1)\rho +\mu): L_J((p^r-1)\rho +
w_{J,0}\lambda)]\\ &\qquad \cdot \sum_{w \in W_J}\text{ch }
\widehat{Z}_{J,r}'((p^r-1)\rho +w\mu).
\end{align*}
Using 
$(L_J)_rT$-duality [Reference JantzenJan03, II.9.2], we obtain
\begin{align*} & {}[\widehat{Z}_{J,r}'((p^r-1)\rho +\mu): L_J((p^r-1)\rho + w_{J,0}\lambda)]\\ &\quad = [ \widehat{Z}_{J,r}'(2(p^r-1)\rho_J-(p^r-1)\rho -\mu): L_J(-w_{J,0}(p^r-1)\rho -\lambda)]\\ &\quad = [ \widehat{Z}_{J,r}'(-w_{J,0}(p^r-1)\rho -\mu): L_J(-w_{J,0}(p^r-1)\rho -\lambda)]. \end{align*}
Hence,
\begin{align*} & \text{ch}\,\widehat{Q}_{J,r}((p^r-1)\rho +w_{J,0}\lambda)\\ &\quad = \sum_{\{\mu \in X^+ | \lambda - \mu \in \mathbb{N}J\}} \frac{1}{|\text{Stab}_{W_J}(\mu)|} \cdot [\widehat{Z}_{J,r}'(-w_{J,0}(p^r-1)\rho -\mu): L_J(-w_{J,0}(p^r-1)\rho -\lambda)]\\ &\qquad \cdot \sum_{w \in W_J}\text{ch } \widehat{Z}_{J,r}'((p^r-1)\rho +w\mu). \end{align*}
Observe also that the highest weight of 
$\widehat {Q}_{J,r}((p^r-1)\rho +w_{J,0}\lambda )$ is 
$(p^r-1)\rho + \lambda.$
(2) Formal character of 
$\widehat {Q}_r((p^r-1)\rho +w_0 \lambda )$. If one applies the calculation of step (1) to the case 
$J=\Delta,$ one obtains a filtration of 
$\widehat {Q}_r((p^r-1)\rho +w_0 \lambda )$ with factors of the form 
$\widehat {Z}_r'((p^r-1)\rho +w\mu )$ where 
$\mu \in X^+$ and 
$w \in W.$ The equivalent of the last equation in step (1) is
\begin{align*} \text{ch } \widehat{Q}_r((p^r-1)\rho+w_0 \lambda) &= \sum_{\{\mu \in X^+\,|\, \mu \leq \lambda\}} \frac{1}{|\text{Stab}_{W}(\mu)|} \cdot [ \widehat{Z}_r'((p^r-1)\rho -\mu): L((p^r-1)\rho -\lambda)]\\ &\quad \cdot \sum_{w \in W}\text{ch } \widehat{Z}_r'((p^r-1)\rho +w\mu). \end{align*}
(3) Formal character of 
$M$. The aforementioned 
$\widehat {Z}_r'$-filtration of 
$\widehat {Q}_r((p^r-1)\rho +w_0 \lambda )$ induces a filtration on the 
$(L_J)_rT$-summand 
$M$. Analogous to the argument in the proof of Theorem 2.4.1, the resulting filtration will have factors of the form 
$\widehat {Z}'_{J,r}(\sigma )$. One observes that a factor of the form 
$\widehat {Z}_r'((p^r-1)\rho + w \mu )$ can only contribute to the filtration of 
$M$ if
\[ (p^r-1)\rho +\lambda - (p^r-1)\rho -w \mu= (\lambda - \mu)+(\mu - w\mu) \in \mathbb{N}J. \]
Since both 
$\lambda - \mu \geq 0$ and 
$\mu - w \mu \geq 0$, it follows that 
$\lambda - \mu \in \mathbb {N}J$ and that 
$\mu - w \mu \in \mathbb {N}J$. From Lemma 2.1.1, the latter implies that 
$w\mu = w_J\mu$ for some 
$w_J \in W_J$. If these conditions are satisfied, then it follows from fact (iv) in § 2.2 that the resulting 
$(L_J)_rT$-factor of 
$M$ is isomorphic to 
$\bigoplus _{ \nu \in \mathbb {N}J} \widehat {Z}'_{r}((p^r-1)\rho +w_J\mu )_{(p^r-1)\rho +\lambda - \nu } =\widehat {Z}'_{J,r}((p^r-1)\rho +w_J\mu ).$ One obtains from the above discussion and step (2) that
\begin{align*} \text{ch } M &= \sum_{\{\mu \in X^+ | \lambda - \mu \in \mathbb{N}J\}} \frac{1}{|\text{Stab}_{W_J}(\mu)|} \cdot [\widehat{Z}_r'((p^r-1)\rho -\mu):L((p^r-1)\rho -\lambda)] \\ &\quad \cdot \sum_{w \in W_J} \text{ch } \widehat{Z}_{J,r}'((p^r-1)\rho +w\mu). \end{align*}
Since 
$(p^r-1)\rho -\mu -((p^r-1)\rho -\lambda ) =\lambda - \mu \in \mathbb {N}J$, by fact (v) in § 2.2, it follows that
\[ [\widehat{Z}_r'((p^r-1)\rho -\mu):L((p^r-1)\rho -\lambda)] =[\widehat{Z}_{J,r}'((p^r-1)\rho -\mu):L_{J}((p^r-1)\rho -\lambda)] \]
and
\begin{align*} \text{ch } M &= \sum_{\{\mu \in X^+ | \lambda - \mu \in \mathbb{N}J\}} \frac{1}{|\text{Stab}_{W_J}(\mu)|} \cdot [\widehat{Z}_{J,r}'((p^r-1)\rho -\mu):L_J((p^r-1)\rho -\lambda)] \\ &\quad \cdot \sum_{w \in W_J} \text{ch } \widehat{Z}_{J,r}'((p^r-1)\rho +w\mu). \end{align*}
(4) Comparison and completion of the proof. We claim that for 
$\lambda \in X_r$ and 
$\mu \in X^+$ with 
$\lambda - \mu \in \mathbb {N}J$,
\[ [\widehat{Z}_{J,r}'((p^r-1)\rho -\mu):L_J((p^r-1)\rho -\lambda)]= [\widehat{Z}_{J,r}'(-w_{J,0}(p^r-1)\rho -\mu): L_J(-w_{J,0}(p^r-1)\rho -\lambda)]. \]
Note the pairs of weights on each side of the equation both differ by 
$\lambda - \mu$. In addition, 
$\langle (p^r-1)\rho + w_{J,0}(p^r-1)\rho, \beta ^{\vee } \rangle =0,$ for all 
$\beta \in J$. The claim follows now from fact (vi) in § 2.2. Therefore, comparing the final equations of steps (1) and (3) yields the assertion of Theorem 2.5.1.
2.6 The TMC and Levi subgroups
Using Theorem 2.5.1, we can show that the validity of the TMC is compatible with restriction to Levi subgroups.
Theorem 2.6.1 Let 
$L_J$ be a Levi subgroup of 
$G$. If the TMC holds for 
$G$, then it also holds for 
$L_J$.
Proof. Verifying the TMC for 
$L_J$ is equivalent to showing the following: given a weight 
$\lambda$ that is 
$p^r$-restricted on 
$L_J,$ the unique indecomposable 
$L_J$-summand containing the highest weight 
$(p^r-1)\rho _J + \lambda$ in the tensor product 
$L_J((p^r-1)\rho _J) \otimes L_J(\lambda )$ remains indecomposable as an 
$(L_J)_rT$-module.
Note that 
$\langle (p^r-1)\rho -(p^r-1)\rho _J, \beta ^{\vee } \rangle =0$ for all 
$\beta \in J$ and that, given any simple 
$p^r$-restricted 
$L_J$-module 
$V$, one can always find 
$\lambda \in X_r$ such that the highest weights of 
$V$ and 
$L_J(\lambda )$ agree on all weight components corresponding to 
$J$.
It is therefore sufficient to show that, for all 
$\lambda \in X_r,$ the unique indecomposable 
$L_J$-summand containing the highest weight 
$(p^r-1)\rho + \lambda$ in the tensor product 
$L_J((p^r-1)\rho ) \otimes L_J(\lambda )$ is indecomposable as an 
$(L_J)_rT$-module. We denote this 
$L_J$-summand, which is tilting, by 
$T_J((p^r-1)\rho + \lambda )$.
The indecomposable 
$G$-tilting module 
$T((p^r-1)\rho + \lambda )$ appears as the unique 
$G$-summand containing the weight 
$(p^r-1)\rho + \lambda$ in 
$L((p^r-1) \rho ) \otimes L(\lambda ).$ From fact (ii) in § 2.2 and Lemma 2.2.1,
\[ L_J((p^r-1)\rho) \otimes L_J(\lambda) = \bigoplus_{\nu \in \mathbb{N}J}(L((p^r-1) \rho) \otimes L(\lambda))_{(p^r-1)\rho+\lambda-\nu}. \]
It follows that 
$T_J((p^r-1)\rho + \lambda )$ appears as an 
$L_J$-summand of 
$T((p^r-1)\rho + \lambda )$. More precisely, it is a summand of the 
$L_J$-module 
$N=\bigoplus _{\nu \in \mathbb {N}J} T((p^r-1)\rho + \lambda )_{(p^r-1)\rho +\lambda -\nu }$.
The TMC for 
$G$ implies that 
$T((p^r-1)\rho + \lambda )=\widehat {Q}_r((p^r-1)\rho +w_0 \lambda )$, as 
$G_rT$-modules. We obtain from Theorem 2.5.1 that
\[ N= \bigoplus_{\nu \in \mathbb{N}J}\widehat{Q}_r((p^r-1)\rho+w_0 \lambda)_{(p^r-1)\rho+\lambda-\nu}= \widehat{Q}_{J,r}((p^r-1)\rho + w_{J,0}\lambda), \]
as 
$(L_J)_rT$-modules. Hence, 
$N$ and 
$T_J((p^r-1)\rho + \lambda )$ are indecomposable as 
$(L_J)_rT$-modules.
Remark 2.6.2 Donkin observed in [Reference DonkinDon93, Proposition 1.5(ii)] that indecomposable tilting modules behave nicely when restricted to Levi subgroups. More precisely, he showed that for any 
$\lambda \in X^+$
\begin{equation} T_J( \lambda)=\bigoplus_{\nu \in \mathbb{N}J} T(\lambda)_{\lambda-\nu}. \end{equation}
Now (2.6.1) together with Theorem 2.5.1 may also be used to prove Theorem 2.6.1. In particular, in the preceding proof of Theorem 2.6.1, it would immediately follow that 
$T_J((p^r-1)\rho + \lambda )$ was, in fact, equal to the module 
$N$, rather than simply being a summand; a conclusion that is also reached at the end of the proof.
In subsequent sections, the contrapositives of Theorems 2.4.1 and 2.6.1 will be used to obtain infinite families of counterexamples to JQ and the TMC from low rank counterexamples. To conclude this section, we record the relationship between the validity of the TMC and of the character of 
$\nabla _{J}(\lambda )$ admitting the character of a module with a 
$p$-filtration.
Theorem 2.6.3 If 
$G$ satisfies the TMC, then 
$\nabla _{J}(\lambda )$ has the character of a module admitting a good 
$(p,r)$-filtration for all 
$\lambda \in X^+$, 
$J\subseteq \Delta$ and 
$r\geq 1$.
Proof. Let 
$r\geq 1$. First recall that if the TMC holds for 
$G$, then 
$\text {Hom}_{G_{r}}(\widehat {Q}_{r}(\sigma ),\nabla (\lambda ))$ has a good filtration for all 
$\sigma \in X_{r}$ and 
$\lambda \in X^+$ (cf. [Reference Kildetoft and NakanoKN15, Theorem 9.2.3]).
Next observe that if 
$M$ is a finite-dimensional 
$G$-module, then
\[ \text{ch }M=\sum_{\sigma\in X_{r}}\text{ch }L(\sigma)\otimes \text{ch }\text{Hom}_{G_{r}}(\widehat{Q}_r(\sigma),M). \]
This can be proved via induction on the composition length of 
$M$ and using the fact that for 
$\lambda \in X^+$ with 
$L(\lambda ) \cong L(\lambda _0) \otimes L(\lambda _1)^{(r)}$, 
$\lambda _0 \in X_r$, and 
$\lambda _1 \in X^+$, the expression 
$ {\operatorname {Hom}}_{G_r}(\widehat {Q}_r(\sigma ), L(\lambda )) \cong {\operatorname {Hom}}_{G_r}(\widehat {Q}_r(\sigma ), L(\lambda _0))\otimes L(\lambda )^{(r)}$ vanishes unless 
$\sigma = \lambda _0$, in which case it is 
$L(\lambda _1)^{(r)}.$
Therefore, by using these facts, it follows that 
$M=\nabla (\lambda )$ has the character of a module with a good 
$(p,r)$-filtration. The statement for 
$\nabla _{J}(\lambda )$ where 
$J\subseteq \Delta$ follows by Proposition 2.2.1.
The statement of the prior theorem for 
$J=\Delta$ was also observed by Kildetoft and conveyed to the third author via a private correspondence.
3. Using extensions between simple modules to generate counterexamples
In this section, we show that the structure of the 
$\text {Ext}^{1}$ between two simple 
$G_{1}$-modules is a key ingredient to the validity of the TMC. In the process of our analysis we present several related methods for constructing counterexamples to the TMC.
3.1 First method
We begin by making the following observation.
Proposition 3.1.1 Let 
$\lambda, \mu \in X_1$ with 
$\lambda \neq \mu$. If the TMC holds, then 
$\operatorname {Ext}_{G_1}^1(L(\lambda ), L(\mu ))$ is a 
$G$-submodule of 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ), Q_1(\mu ))$.
Proof. The TMC implies that 
$Q_1(\sigma )$ can be lifted to 
$G$-modules for all 
$\sigma \in X_{1}$ and are tilting modules. In particular, 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ), Q_1(\mu ))$ has a 
$G$-module structure. Consider
\[ 0 \to L(\mu) \to Q_1(\mu) \to Q_1(\mu)/L(\mu) \to 0. \]
Using 
$\lambda \neq \mu$ one immediately obtains
\[ {\operatorname{Hom}}_{G_1}(L(\lambda), Q_1(\mu)/L(\mu)) \cong \operatorname{Ext}_{G_1}^1(L(\lambda), L(\mu)). \]
Next we use
\[ 0 \to \mbox{rad}(Q_1(\lambda)) \to Q_1(\lambda) \to L(\lambda) \to 0, \]
to obtain that
\[ \operatorname{Ext}_{G_1}^1(L(\lambda), L(\mu)) \cong {\operatorname{Hom}}_{G_1}(L(\lambda), Q_1(\mu)/L(\mu)) \hookrightarrow {\operatorname{Hom}}_{G_1}(Q_1(\lambda), Q_1(\mu)/L(\mu)). \]
Finally, using the first short exact sequence and the operator 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ), - )$ yields
\[ \operatorname{Ext}_{G_1}^1(L(\lambda), L(\mu)) \hookrightarrow {\operatorname{Hom}}_{G_1}(Q_1(\lambda), Q_1(\mu)/L(\mu))\cong {\operatorname{Hom}}_{G_1}(Q_1(\lambda), Q_1(\mu)).\]
Note that the TMC implies that 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ), Q_1(\mu ))^{(-1)}$ is a tilting module [Reference Kildetoft and NakanoKN15]. Moreover, the weights appearing in 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ), Q_1(\mu ))$ are less than or equal to 
$2(p-1)\rho - \lambda + \omega _0\mu.$ One obtains immediately the following theorem.
Theorem 3.1.2 Let 
$\lambda, \mu \in X_1$ with 
$\lambda \neq \mu$. Assume the TMC holds.
(a) Then
$\operatorname {Ext}_{G_1}^1(L(\lambda ), L(\mu ))^{(-1)}$ is a $G$-submodule of some tilting module whose weights $\gamma$ satisfy $p \gamma \leq 2(p-1)\rho - \lambda + w_0\mu$.(b) If
$L(\nu ) \hookrightarrow \operatorname {Ext}_{G_1}^1(L(\lambda ), L(\mu ))^{(-1)},$ then $L(\nu )$ has to be a submodule of a Weyl module $\Delta (\gamma )$, with $p \gamma \leq 2(p-1)\rho - \lambda + w_0\mu$.
Proof. (a) This follows by the observation stated before the theorem and Proposition 3.1.1.
(b) From part (a), we know that if the TMC holds, then 
$L(\nu )$ has to appear in the socle of some indecomposable tilting module 
$T(\gamma )$, with 
$p \gamma \leq 2(p-1)\rho - \lambda + w_0\mu$. The assertion follows from the fact that a tilting module has a Weyl filtration.
3.2 Counterexample revisited for $\Phi ={\rm G}_{2}$
Let 
$G$ be of type 
${\rm G}_2$ and 
$p=2.$ It was shown in [Reference Bendel, Nakano, Pillen and SobajeBNPS20b] that the TMC fails in this case. We will make use of the above set-up and give a new argument.
Let 
$\lambda = 0$ and 
$\mu = \omega _2$. According to [Reference JantzenJan91, Proposition 5.2],
\[ \operatorname{Ext}_{G_1}^1(k, L(\omega_2))^{(-1)} \cong \operatorname{Ext}_{G_1}^1(k, \nabla(\omega_2))^{(-1)} \cong \nabla(\omega_1). \]
Moreover, 
$2\rho - \lambda +w_0\mu = 2\rho - \omega _2 = 2\omega _1 + \omega _2$. The only weights 
$\gamma$ with 
$2 \gamma \leq 2\omega _1 + \omega _2$ are 
$0$, 
$\omega _1$, and 
$\omega _2$. The corresponding Weyl modules 
$\Delta (0)$, 
$\Delta (\omega _1)$, and 
$\Delta (\omega _2)$ have simple socles 
$k$, 
$k$, and 
$L(\omega _2)$, respectively. If the TMC held, this would contradict Theorem 3.1.2(b). Thus, the TMC fails for 
${\rm G}_2$ and 
$p=2$.
3.3 Second method
If we assume that 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ),\nabla (\mu ))$ vanishes, one obtains the following modifications of Proposition 3.1.1 and Theorem 3.1.2.
Proposition 3.3.1 Let 
$\lambda, \mu \in X_1$ with 
$\lambda \neq \mu$. If the TMC holds and 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ),\nabla (\mu ))= 0$, then 
$\operatorname {Ext}_{G_1}^1(L(\lambda ), \nabla (\mu ))$ is a 
$G$-submodule of 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ), Q_1(\mu ))$.
Theorem 3.3.2 Let 
$\lambda, \mu \in X_1$ with 
$\lambda \neq \mu$. Assume the TMC holds and 
$ {\operatorname {Hom}}_{G_1}(Q_1(\lambda ), \nabla (\mu ))= 0$.
(a) Then
$\operatorname {Ext}_{G_1}^1(L(\lambda ), \nabla (\mu ))^{(-1)}$ is a $G$-submodule of some tilting module $T(\gamma )$, with $p \gamma \leq 2(p-1)\rho - \lambda + w_0\mu$.(b) If
$L(\nu ) \hookrightarrow \operatorname {Ext}_{G_1}^1(L(\lambda ), \nabla (\mu ))^{(-1)},$ then $L(\nu )$ has to be a submodule of a Weyl module $\Delta (\gamma )$, with $p \gamma \leq 2(p-1)\rho - \lambda + w_0\mu$.
The next result enables the employment of Theorem 3.3.2. This strategy will be used later to produce a counterexample to the TMC in type 
${\rm B}_n$.
Proposition 3.3.3 Let 
$\lambda, \mu \in X_1$ satisfy the equation 
$\lambda + p \omega _i = \mu + \alpha _i,$ where 
$\alpha _i$ denotes a simple root and 
$\omega _i$ the corresponding fundamental weight. In addition, assume that 
$\langle \lambda, \alpha _i^{\vee }\rangle =0$. Then there exists a 
$G$-module monomorphism
\[ \nabla(\omega_i) \hookrightarrow \operatorname{Ext}_{G_1}^1(L(\lambda), \nabla(\mu))^{(-1)}. \]
Proof. In computing 
$G_1$-extensions, one has (cf. [Reference JantzenJan03, Lemma II.12.8])
\[ \operatorname{Ext}_{G_1}^1(L(\lambda), \nabla(\mu))^{(-1)} \cong \operatorname{ind}_{B}^G[\operatorname{Ext}_{B_1}^1(L(\lambda), \mu)^{(-1)}]. \]
There exists a short exact sequence of 
$B$-modules
\[ 0 \to \mu \to k[U_1] \otimes \mu \to k[U_1]/k \otimes \mu \to 0, \]
which yields the exact sequence
\[ 0 \to {\operatorname{Hom}}_{B_1}(L(\lambda),k[U_1] \otimes \mu) \to {\operatorname{Hom}}_{B_1}(L(\lambda),k[U_1]/k \otimes \mu) \to \operatorname{Ext}_{B_1}^1(L(\lambda), \mu) \to 0. \]
From weight considerations one obtains a 
$B$-module injection
\[ p \omega_i \cong {\operatorname{Hom}}_{B_1}(L(\lambda),\mu + \alpha_i)\hookrightarrow {\operatorname{Hom}}_{B_1}(L(\lambda),k[U_1]/k \otimes \mu). \]
Note that 
$\langle \lambda, \alpha _i^{\vee }\rangle =0$ implies that 
$\lambda - \alpha _i$ is not a weight of 
$L(\lambda )$. This implies that 
$p\omega _i$ is not a weight of 
$ {\operatorname {Hom}}_{B_1}(L(\lambda ),k[U_1] \otimes \mu )$. Consequently, there is an injection
\[ p \omega_i \hookrightarrow \operatorname{Ext}_{B_1}^1(L(\lambda), \mu). \]
Since induction is left-exact, one obtains a 
$G$-module monomorphism:
\[ \nabla(\omega_i) \hookrightarrow \operatorname{Ext}_{G_1}^1(L(\lambda), \nabla(\mu))^{(-1)}. \]
3.4 Third method
For a restricted weight 
$\lambda \in X_1$ and a dominant weight 
$\mu$, we will look at 
$ {\operatorname {Hom}}_{G_1}({Q}_1(\lambda ), \nabla (\mu ))$. If the TMC holds, then it follows from [Reference Kildetoft and NakanoKN15, Theorem 9.2.3] that 
$ {\operatorname {Hom}}_{G_1}({Q}_1(\lambda ), \nabla (\mu ))^{(-1)}$ has a good filtration. The idea now is to find appropriate weights 
$\lambda$ and 
$\mu$ that violate this last statement. We will use this technique to produce counterexamples for groups of type 
${\rm B}_3$ with 
$p = 2$ and type 
${\rm C}_3$ with 
$p=3$.
4. Low-rank counterexamples
4.1 Summary of results
In this section, we show that the TMC fails and JQ has a negative answer for 
$G$ when 
$\Phi ={\rm B}_{3}$, 
${\rm C}_{3}$, and 
${\rm D}_{4}$. In particular, the following theorem is proved.
Theorem 4.1.1 Let 
$G$ be a simple algebraic group. Then 
$T(2(p-1)\rho )$ is not isomorphic to 
$\widehat {Q}_1(0)$ as a 
$G_1T$-module in the following cases:
(a)
$\Phi ={\rm B}_{3}$, $p=2;$(b)
$\Phi ={\rm C}_{3}$, $p=3;$(c)
$\Phi ={\rm D}_{4}$, $p=2.$
In the cases (a)–(c), there exists an induced module 
$\nabla (\lambda )$, 
$\lambda \in X^+$, that does not admit a good 
$p$-filtration.
4.2 Jantzen filtration and Jantzen sum formula
Several of the important calculations that we will use are derived via the Jantzen filtration (cf. [Reference JantzenJan03, Proposition II.8.19]) as described in the following.
For each 
$\lambda \in X^+$, there is a filtration of 
$G$-modules
\[ \Delta(\lambda) = \Delta(\lambda)^0 \supseteq \Delta(\lambda)^1 \supseteq \Delta(\lambda)^2 \supseteq \cdots \]
such that 
$\text {rad}_{G}\Delta (\lambda )=\Delta (\lambda )^{1}$ and 
$\Delta (\lambda )/\Delta (\lambda )^1 \cong L(\lambda )$. Moreover,
\[ \sum_{i > 0} \operatorname{ch} \Delta(\lambda)^i = \sum_{\alpha \in \Phi^+} \sum_{0 < mp < \langle\lambda + \rho,\alpha^{\vee}\rangle} \nu_p(mp)\chi(s_{\alpha,mp}\cdot \lambda). \]
4.3 Some character data
For 
$\Phi ={\rm B}_{n}$ when 
$p=2$, we record the following information about the structure of various representations.
Lemma 4.3.1 Let 
$G$ be of type 
${\rm B}_n$, 
$n \geq 3,$ and 
$p=2$.
(a) The composition factors of
$\nabla (\omega _1)$ are $L(\omega _1)$ and the trivial module, each appearing once. In particular, the head of $\nabla (\omega _1)$ consists of the trivial module.(b) For
$n$ odd, the module $\nabla (\omega _2)$ is uniserial with the three composition factors of $L(\omega _2)$, $L(\omega _1)$ and the trivial module, listed from bottom to top. Its head consists of the trivial module.(c) For
$n$ even, the module $\nabla (\omega _2)$ has four composition factors: $L(\omega _2)$, $L(\omega _1)$, and the trivial module with multiplicity two. Its head consists of just one copy of the trivial module.(d) For
$n = 3$, the dominant weights appearing in the simple module $L(\omega _1+\omega _2)$ are $\omega _1 + \omega _2$ with multiplicity one, $2 \omega _3$ with multiplicity two, and $\omega _1$ with multiplicity four.
Proof. Note that 
$\nabla (\omega _2)$ is the dual of the adjoint representation. Claims (a) through (c) can be found in [Reference JantzenJan91, Proposition 6.9(a)] and its proof.
For part (d), one can make use of the special isogenies between types 
${\rm B}_n$ and 
${\rm C}_n$ that exist for 
$p=2$ (see [Reference Dowd and SinDS96, I.3]). As a result, one obtains a sharpened version of the Steinberg tensor product theorem [Reference Dowd and SinDS96, I.4.2(3)]. The character of the 
${\rm B}_3$-module 
$L(\omega _1+ \omega _2)$ can be obtained via the character of the 
${\rm C}_3$-module 
$L(\omega _1 + \omega _2),$ which is identical to the 
${\rm C}_3$-module 
$\nabla (\omega _1 + \omega _2)$.
4.4 $\Phi ={\rm B}_3$ and $p=2$
In [Reference Bendel, Nakano, Pillen and SobajeBNPS22] it was shown that the TMC holds for a group of type 
${\rm B}_2$ and all primes. In this section, the third method that was introduced in § 3.4 is used to show that the TMC fails for a group of type 
${\rm B}_3$ and 
$p=2$.
4.4.1 TMC
Assume that 
$T(2\rho )\mid _{G_{1}T}\cong \widehat {Q}_1(0)$. We show in the following that 
$ {\operatorname {Hom}}_{G_1}(\widehat {Q}_1(0), \nabla (2 \omega _2))^{(-1)}$ does not afford a good filtration, thereby obtaining a contradiction to [Reference Kildetoft and NakanoKN15, Theorem 9.2.3].
The dominant weights less than or equal to 
$2\omega _2$ are 
$2 \omega _2,$ 
$\omega _1+ 2 \omega _3,$ 
$\omega _1+ \omega _2,$ 
$2\omega _3,$ 
$2\omega _1,$ 
$\omega _2,$ 
$\omega _1,$ and 
$0.$ All of these are linked to 
$2\omega _2$ and also appear in 
$\nabla (2 \omega _2)$. Note that 
$\omega _3$ is minuscule. Lemma 4.3.1 provides, therefore, sufficient data to determine the characters of all simple modules with highest weights from the above list as well as all the decomposition numbers for all 
$\nabla (\sigma )$ with weights from the list. Alternatively, one could also refer to the tables in [Reference LübeckLub]. For our argument we make use of the Jantzen filtration for the Weyl module 
$\Delta (2\omega _2)$. The following table lists the multiplicities of each composition factor of 
$\Delta (2 \omega _2)$ as well as the multiplicities of each simple factor in the Jantzen sum formula. We include only weights with positive multiplicity.
\begin{align*}
\lambda \quad & \quad [ \Delta(2 \omega_2):L(\lambda)] \quad & \quad
[\sum_{i>0} \text{ch } \Delta(2 \omega_2)^i:L(\lambda)] \\
2\omega_2 \quad & \quad 1 \quad & \quad 0\\
\omega_1+2 \omega_3 \quad & \quad 1\quad & \quad 1\\
\omega_1+\omega_2 \quad & \quad 1\quad & \quad 2\\
2\omega_1\quad & \quad 2\quad & \quad 2\\
\omega_2 \quad & \quad 2\quad & \quad 4 \\
0\quad & \quad 2\quad & \quad 2\\
\end{align*}
This table indicates that any composition factor with highest weight 
$\omega _1+2\omega _3$, 
$2\omega _1,$ or 
$0$ has to appear in the second highest layer of the Jantzen filtration, that is, in 
$\Delta (2\omega _2)^1/\Delta (2\omega _2)^2$. Hence, only composition factors with highest weight 
$\omega _2$ or 
$\omega _1+\omega _2$ appear in 
$\Delta (2 \omega _2)^2$. Recall the 
$\tau$-functor as defined in § 2.1. Set 
$S=^{\tau }\!\!\!(\Delta (2\omega _2)/ \Delta (2\omega _2)^2)$. Then one obtains the short exact sequence via the 
$\tau$-functor:
\[ 0 \to S \to \nabla(2\omega_2) \to ^{\tau}\!\!\!(\Delta (2\omega_2)^2)\to 0. \]
It follows that
\[ {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0), \nabla(2 \omega_2)) \cong {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0), S). \]
In addition, there exists an embedding 
$\nabla (\omega _2)^{(1)} \hookrightarrow \nabla ( 2\omega _2)$. Since none of the composition factors of 
$\nabla (\omega _2)^{(1)}$ appear in 
$^{\tau }\!\!(\Delta (2\omega _2)^2)$ one obtains an embedding 
$\nabla (\omega _2)^{(1)} \hookrightarrow S$. Lemma 4.3.1(b) now yields embeddings
\[ L(\omega_1)^{(1)}\hookrightarrow \nabla( \omega_1)^{(1)} \hookrightarrow S/L(\omega_2)^{(1)}=^{\tau}\!\!\!(\Delta(2\omega_2)^1/ \Delta (2\omega_2)^2). \]
Note that the layers in the Jantzen filtration are 
${\tau }$-invariant (cf [Reference JantzenJan03, II.8.19(3)]). One obtains a projection 
$\pi : S\twoheadrightarrow L(\omega _1)^{(1)}$. Next we define 
$Q$ via
\[ 0 \to \nabla(\omega_2)^{(1)} \to S \to Q \to 0. \]
Since 
$\nabla (\omega _2)$ has a simple head isomorphic to the trivial module, 
$\pi$ has to factor through 
$Q$. Therefore, both 
$Q$ and 
$ {\operatorname {Hom}}_{G_1} (\widehat {Q}_1(0), Q)$ map onto 
$L(\omega _1)^{(1)}$. Since both 
$[S:L(\omega _1)^{(1)}]$ and 
$[S:k]$ are at most two, one concludes, via subtraction of the character of 
$\nabla (\omega _2)^{(1)},$ that 
$[Q: L(\omega _1)^{(1)}] =1$ and that 
$[Q: k]\leq 1$. No other composition factor of 
$Q$ can contribute to 
$ {\operatorname {Hom}}_{G_1} (\widehat {Q}_1(0), Q)$. The character of 
$ {\operatorname {Hom}}_{G_1} (\widehat {Q}_1(0), Q)^{(-1)}$ is therefore either equal to the character of 
$L(\omega _1)$ or to the character of 
$L(\omega _1)$ together with a trivial character. Since 
$ {\operatorname {Hom}}_{G_1} (\widehat {Q}_1(0), Q)^{(-1)}$ maps onto 
$L(\omega _1)$ and 
$\nabla (\omega _1) \neq L(\omega _1)$ one concludes that 
$ {\operatorname {Hom}}_{G_1} (\widehat {Q}_1(0), Q)^{(-1)}$ cannot have a good filtration.
On the other hand, by looking at
\[ 0 \to {\operatorname{Hom}}_{G_1} (\widehat{Q}_1(0), \nabla(\omega_2)^{(1)}) \to {\operatorname{Hom}}_{G_1} (\widehat{Q}_1(0), S) \to {\operatorname{Hom}}_{G_1} (\widehat{Q}_1(0), Q) \to 0 \]
which is equivalent to
\[ 0 \to \nabla(\omega_2)^{(1)} \to {\operatorname{Hom}}_{G_1} (\widehat{Q}_1(0), \nabla(2\omega_2)) \to {\operatorname{Hom}}_{G_1} (\widehat{Q}_1(0), Q)\to 0, \]
one concludes that 
$ {\operatorname {Hom}}_{G_1} (\widehat {Q}_1(0), \nabla (2\omega _2))^{(-1)}$ does not afford a good filtration.
4.4.2 JQ
We show that the induced module 
$\nabla (2 \omega _2)$ does not afford a good 
$p$-filtration. Suppose that 
$\nabla (2 \omega _2)$ has a good 
$p$-filtration. From the data given in the table of § 4.4.1, one concludes that the factors 
$\nabla (\omega _2)^{(1)}$ and 
$\nabla (\omega _1)^{(1)}$ each have to appear once in such a filtration. The first, 
$\nabla (\omega _2)^{(1)},$ appears at the very bottom of any good 
$p$-filtration. Then 
$V := \nabla (2\omega _2)/\nabla (\omega _2)^{(1)}$ also has as good 
$p$-filtration with one of the factors being 
$\nabla (\omega _1)^{(1)}$. The module 
$\nabla (\omega _1)^{(1)}$ has two composition factors: the trivial module 
$k$ and 
$L(\omega _1)^{(1)}$. Further, we would have 
$[V : L(\omega _1)^{(1)}] = 1 = [V : k]$. By the nature of 
$\nabla (\omega _1)^{(1)}$, in the radical series for 
$V$, the copy of 
$k$ must appear higher than the 
$L(\omega _1)^{(1)}$. This means that the 
$k$ must appear above the 
$L(\omega _1)^{(1)}$ in any composition series for 
$V$ (as, in general, a composition series is a refinement of the radical series). However, the argument in the previous section shows that there exists a composition series of 
$V$ in which a composition factor of the form 
$L(\omega _1)^{(1)}$ appears higher than the factor isomorphic to the trivial module; a contradiction.
4.4.3
The methods of § 3 suggest a connection between the validity of the TMC and good filtrations on 
$G_1$-extension groups. With an eye towards a more precise connection (that will be discussed further in § 7), we note that, from [Reference Dowd and SinDS96, Table II.2.5(a) on page 2632] or [Reference JantzenJan91, Proposition 6.9],
\begin{align*} \operatorname{Ext}_{G_1}^1(k, L(\omega_2))^{(-1)} \cong \nabla(\omega_1), \end{align*}
which is not tilting.
4.5 $\Phi ={\rm C}_3$ and $p=3$
We again employ the method introduced in § 3.4 to show that the TMC fails in this case.
4.5.1 TMC
By using the Jantzen filtration one obtains the following tables.
\[
\begin{align*}
\lambda\quad & \quad \sum_{i>0} \text{ch } \Delta(\lambda)^i\\
(0,0,0)\quad & \quad \emptyset\\
(0,1,0) &
\chi(0,0,0)\\
(1,0,1)\quad & \quad \chi(0,1,0) -
\chi(0,0,0)\\
(0,0,2)\quad & \quad \emptyset\\
(1,1,1)\quad & \quad \chi(0,0,2) +2\times \chi(1,0,1) - \chi(0,1,0)
+ \chi(0,0,0)\\
(0,3,0)\quad & \quad \chi(1,1,1) - \chi(0,0,2) - \chi(1,0,1)+ 2\times \chi(0,1,0) -
\chi(0,0,0) \\
(2,0,2)\quad & \quad 2 \times \chi(1,1,1) +
\chi(0,0,2) - 2 \times \chi(1,0,1) + \chi(0,1,0) -
\chi(0,0,0) \\
(2,1,2)\quad & \quad \chi(2,0,2) +2 \times \chi(0,3,0)-\chi(0,0,2)+ \chi(1,0,1)- 2 \times \chi(0,1,0)+
\chi(0,0,0) \\
\end{align*} \]
\[
\begin{align*}
\lambda \quad & \quad [\sum_{i>0} \text{ch } \Delta(\lambda)^i:L(0,0,0)] \quad & \quad \lambda \quad & \quad
[\sum_{i>0} \text{ch } \Delta(\lambda)^i:L(0,3,0)] \\
(0,0,0) \quad & \quad 0 \quad & \quad (2,0,2) \quad & \quad 0 \\
(0,1,0) \quad & \quad 1 \quad & \quad (2,1,2) \quad & \quad 2 \\
(1,0,1) \quad & \quad 0 \quad & \quad \quad & \quad \\
(0,0,2) \quad & \quad 0 \quad & \quad \quad & \quad \\
(1,1,1) \quad & \quad 0 \quad & \quad \quad & \quad \\
(0,3,0) \quad & \quad 1 \quad & \quad \quad & \quad \\
(2,0,2) \quad & \quad 0 \quad & \quad \quad & \quad \\
(2,1,2) \quad & \quad 1 \quad & \quad \quad & \quad \\
\end{align*} \]
Assume that 
$T(4 \rho ) \mid _{G_1T}\cong \widehat {Q}_1(0,0,0)$. From these tables, we conclude that
\[ \bigg[\sum_{i>0} \text{ch } \Delta(2,1,2)^i:L(0,0,0)\bigg] = [\Delta(2,1,2): L(0,0,0)] =1, \]
whereas
\[ [\Delta(2,1,2)^2:L(0,0,0)]=0. \]
Note that the only non-restricted composition factor of 
$\Delta (2,1,2)$ has highest weight 
$(0,3,0)$ and that 
$[\sum _{i>0} \text {ch } \Delta (2,1,2)^i:L(0,3,0)] = 2$. There are two possibilities:
Case 1: 
$[ \Delta (2,1,2)^2:L(0,3,0)]=0.$ This implies that 
$[\Delta (2,1,2): L(0,3,0)]=2$ and, therefore,
\begin{align*} \text{ch}{\operatorname{Hom}}_{G_1}(T(4 \rho): \nabla(2,1,2))^{(-1)} &= \text{ch}{\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0,0,0), \nabla(2,1,2))^{(-1)}\\ &= \text{ch}{\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0,0,0), \Delta(2,1,2))^{(-1)} \\ &= 2\cdot\text{ch } L(0,1,0) + \text{ch } L(0,0,0). \end{align*}
Since 
$\nabla (\omega _2)$ has composition factors 
$L(\omega _2)$ and 
$k$, 
$\text {ch} {\operatorname {Hom}}_{G_1}(T(4 \rho ): \nabla (2,1,2))^{(-1)}$ cannot be the character of a module with a good filtration. Thus, we obtain a contradiction to [Reference Kildetoft and NakanoKN15, Theorem 9.2.3].
Case 2: 
$[ \Delta (2,1,2)^2:L(0,3,0)]=1.$ This implies that 
$[\Delta (2,1,2): L(0,3,0)]=1.$ Define 
$Q$ via the exact sequence
\begin{align*} 0 \to \Delta(2,1,2)^2 \to \Delta(2,1,2)\to Q \to 0, \end{align*}
which gives rise to the short exact sequence
\[ 0 \to {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0,0,0), \Delta(2,1,2)^2)^{(-1)} \to {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0,0,0), \Delta(2,1,2))^{(-1)} \]
\[ \to {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0,0,0), Q)^{(-1)} \to 0. \]
This sequence is equivalent to
\[ 0 \to L(0,1,0) \to {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0,0,0), \Delta(2,1,2))^{(-1)} \to L(0,0,0) \to 0. \]
Its dual version is
\[ 0 \to L(0,0,0) \to {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0,0,0), \nabla(2,1,2))^{(-1)} \to L(0,1,0) \to 0. \]
Again we obtain a contradiction to [Reference Kildetoft and NakanoKN15, Theorem 9.2.3].
4.5.2 JQ
We show that the induced module 
$\nabla (2,1,2)$ does not afford a good 
$p$-filtration. Assume that 
$\nabla (2,1,2)$ has a good 
$p$-filtration. The weight 
$3\omega _2$ is maximal among the dominant weights of the form 
$3\gamma$ that appear in 
$\nabla (2,1,2)$. From the data given in the tables in § 4.5.1, one can see that 
$\nabla (\omega _2)^{(1)}$ has to appear at least once, possibly twice, in any good 
$p$-filtration of 
$\nabla (2,1,2)$. Note that 
$\nabla (\omega _2)$ has two composition factors with highest weights 
$\omega _2$ and 
$0$. Correspondingly, 
$\nabla (\omega _2)^{(1)}$ will have factors 
$L(\omega _2)^{(1)}$ (socle) and 
$k$ (head). As argued in § 4.4.2 for type 
${\rm B}_3$, for 
$\nabla (2,1,2)$ to admit a good 
$p$-filtration, every appearance of 
$L(\omega _2)^{(1)}$ must lie below an occurrence of 
$k$ in its radical series, and, hence, in any composition series. Consider cases 1 and 2 as in the argument of § 4.5.1. In the first case, 
$\nabla (2,1,2)$ would have two copies of 
$L(\omega _2)^{(1)}$, but only a single copy of 
$k$; a clear contradiction. In case 2, 
$k$ appears in 
$\Delta (2,1,2)^1$, whereas 
$L(\omega _2)^{(1)}$ appears in 
$\Delta (2,1,2)^2$ (and only once in 
$\Delta (2,1,2)$). Dualizing, there exists a composition series of 
$\nabla (2,1,2)$ for which a composition factor isomorphic to 
$L(\omega _2)^{(1)}$ appears higher than any trivial module. With this contradiction, we conclude that 
$\nabla (2,1,2)$ does not afford a good 
$p$-filtration.
4.5.3
From the table in § 4.5.1, one obtains that 
$[\nabla (1,1,1):k]=0$ and, therefore, 
$ {\operatorname {Hom}}_{G_1}(k, \nabla (1,1,1)/L(1,1,1))=0$. From [Reference JantzenJan91, Proposition 4.1 and § 4.2], one has
\[ \operatorname{Ext}_{G_1}^1(k, L(1,1,1))^{(-1)} \hookrightarrow \operatorname{Ext}_{G_1}^1(k, \nabla(1,1,1))^{(1)} \cong \nabla(\omega_2). \]
Furthermore, from [Reference JantzenJan91, Proposition 4.5],
\[ L(\omega_2) \hookrightarrow \operatorname{Ext}_{G_1}^1(k, L(1,1,1))^{(-1)}. \]
It follows that 
$\operatorname {Ext}_{G_1}^1(k, L(1,1,1))^{(-1)}$ is not tilting.
Remark 4.5.1 More generally, consider the case of type 
${\rm C}_p$ for 
$p \geq 3$. The module 
$\nabla (\omega _2)$ is not simple. It has two composition factors: 
$L(\omega _2)$ and the trivial module 
$k$. It follows from [Reference JantzenJan91, Proposition 4.1, § 4.2, and Proposition 4.5] that 
$\operatorname {Ext}_{G_1}^1(k,\nabla (p\omega _2 - \alpha _2))^{(-1)} \cong \nabla (\omega _2)$ and one has an exact sequence
\[ 0 \to (\nabla(p\omega_2 - \alpha_2)/L(p\omega_2 - \alpha_2))^{G_1} \to \operatorname{Ext}_{G_1}^1(k,L(p\omega_2 - \alpha_2)) \to \operatorname{Ext}_{G_1}^1(k,\nabla(p\omega_2 - \alpha_2)), \]
where the image of the last map contains 
$L(\omega _2)^{(1)}$. By direct computation, the only dominant weight 
$\gamma$ for which 
$p\gamma$ is a weight of 
$\nabla (p\omega _2 - \alpha _2)$ is 
$\gamma = 0$. As such, 
$(\nabla (p\omega _2 - \alpha _2)/L(p\omega _2 - \alpha _2))^{G_1}$ is a (possibly empty) sum of trivial modules. If this term vanishes, we would see that 
$\operatorname {Ext}_{G_1}^1(k,L(p\omega _2 - \alpha _2))^{(-1)}$ is not a tilting module, but we have been unable to confirm this. We speculate that this is the case and that the TMC fails in type 
${\rm C}_p$ for 
$p \geq 3$.
4.6 $\Phi ={\rm D}_4$ and $p=2$
In this section the first method introduced in § 3.1 is used to show that the TMC fails for 
${\rm D}_4$ and 
$p=2$.
4.6.1 TMC
We demonstrate that the tilting module 
$T(2\rho )$ is not isomorphic to 
$\widehat {Q}_1(0)$ as a 
$G_1T$-module. According to [Reference Dowd and SinDS96, 3.4(b), p. 2659] (cf. also [Reference SinSin94]),
\begin{equation} \operatorname{Ext}_{G_1}^1(k, L(\omega_1 +\omega_3+ \omega_4)) \cong k \oplus L(\omega_2)^{(1)}. \end{equation}Assume that the TMC holds. Then, according to Proposition 3.1.1,
\begin{equation} L(\omega_2)^{(1)}\hookrightarrow \operatorname{Ext}_{G_1}^1(L(0), L(\omega_1 +\omega_3+ \omega_4)) \hookrightarrow {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(0), \widehat{Q}_1(\omega_1 + \omega_3+ \omega_4)). \end{equation}
Moreover, according to Theorem 3.1.2(b), 
$L(\omega _2)$ must appear in the socle of a Weyl module 
$\Delta (\gamma )$ with highest weight less than or equal to 
$2\rho - (\omega _1+\omega _3+\omega _4) = \rho + \omega _2$. Any weight 
$\gamma$ that is linked to 
$\omega _2$ and satisfies 
$2 \gamma \leq \rho + \omega _2$ is contained in the following list:
\[ \omega_1 + \omega_3+ \omega_4, 2\omega_1, 2\omega_3, 2\omega_4, \omega_2, 0. \]
The module 
$\Delta (\omega _2)$ is the dual of the adjoint representation. Its radical consists of two copies of the trivial module. The modules 
$\Delta (2\omega _i)$ with 
$i \in \{1, 3, 4\}$ are uniserial with the trivial module as their socle and a middle consisting of 
$L(\omega _2)$ (cf. [Reference SinSin94, Lemma 4.3]).
Next we observe that 
$L(\omega _2)$ does not appear in the socle of 
$\Delta (\omega _1 + \omega _3 + \omega _4)$. We embed 
$\Delta (\omega _1+\omega _3+\omega _4)$ in 
$\Delta (\omega _1) \otimes \Delta (\omega _3+\omega _4)$ and show that
\[ {\operatorname{Hom}}_G(L(\omega_2), \Delta(\omega_1) \otimes \Delta(\omega_3+\omega_4))\cong {\operatorname{Hom}}_G(\nabla(\omega_3+\omega_4), L(\omega_2) \otimes \Delta(\omega_1))=0. \]
Note that 
$\Delta (\omega _1) \cong L(\omega _1)$ while 
$\nabla (\omega _3+\omega _4)$ has length two with simple head 
$L(\omega _1)$. Moreover, a straightforward character calculation shows that 
$L(\omega _2) \otimes L(\omega _1)$ has composition factors 
$L(\omega _1 + \omega _2)$ and 
$L(\omega _3 + \omega _4).$ Since 
$L(\omega _1)$ does not appear as a factor, the socle of 
$\Delta (\omega _1 + \omega _3 + \omega _4)$ does not contain 
$L(\omega _{2})$, giving a contradiction to Theorem 3.1.2.
Thus, at least one of 
$\widehat {Q}_1(0)$ or 
$\widehat {Q}_1(\omega _1 + \omega _3+ \omega _4))$ is not a tilting module. From character data (particularly, that the dominant weights of 
$\nabla (\omega _2)$ are only 
$\omega _2$ and 0), 
$T(\rho +\omega _2) \mid _{G_1T} \cong \widehat {Q}_1(\omega _1 + \omega _3 + \omega _4)$. Therefore, 
$T(2 \rho )$ cannot be isomorphic to 
$\widehat {Q}_1(0)$, as a 
$G_1T$-module.
4.6.2 JQ
It was shown in [Reference Bendel, Nakano, Pillen and SobajeBNPS20a, 5.6] that Conjecture 2.3.3 holds in this case. Moreover, from [Reference SobajeSob18] we know that an affirmative answer to JQ (i.e. Question 2.3.1) together with Conjecture 2.3.3 implies that the TMC holds. Since the TMC fails here, one concludes that there exists a dominant weight for which JQ does not have a positive answer.
5. Proof of the main theorem
5.1 Proof
The following is a list of groups and their root systems for which the small rank groups of Theorem 4.1.1 appear naturally as Levi subgroups:
–
${\rm B}_3$, appears in ${\rm B}_n,$ $n \geq 3,$ and in ${\rm F}_4;$–
${\rm C}_3$, appears in ${\rm C}_n,$ $n \geq 3,$ and in ${\rm F}_4;$–
${\rm D}_4$, appears in ${\rm D}_n,$ $n \geq 4,$ and in ${\rm E}_n$, $n=6, 7, 8.$
This allows one to apply the contrapositive of Theorem 2.6.1 and extend the statement concerning the TMC of Theorems 4.1.1 to the root systems listed in § 1.2, for their respective primes. Similarly, the contrapositive of Theorem 2.4.1 together with §§ 4.4.2, 4.5.2, and 4.6.2 immediately verify the part of the main theorem concerning JQ.
The following section gives a more precise statement concerning the failure of the TMC.
5.2 Statement with weights
Keeping track of the weights in Theorem 4.1.1 as well as including the counterexamples of Theorems 6.1.2 and 6.5.1, provides for a more extensive list of cases in which the TMC fails. We also include the original counterexample in [Reference Bendel, Nakano, Pillen and SobajeBNPS20b].
Theorem 5.2.1 Let 
$G$ be a simple algebraic group over an algebraically closed field of characteristic 
$p>0$ with underlying root system 
$\Phi$ and 
$\lambda$ a 
$p$-restricted weight. Then
\[ T((p-1)\rho+\lambda)|_{G_{1}T} \neq \widehat{Q}_{1}((p-1)\rho+w_{0}\lambda), \]
provided the triple 
$(\Phi, p, \lambda )$ appears in the following list:
–
$\Phi ={\rm B}_{n}$, $n\geq 3$, $p=2,$ and
(i)
$\langle \lambda, \alpha _i^{\vee } \rangle = p-1,$ for $n-2 \leq i \leq n,$ or(ii) for some
$k$ with $1 \leq k \leq n-2$, $\langle \lambda, \alpha _i^{\vee } \rangle = p-1,$ for $i \in \{k, k+1\},$ and $\langle \lambda, \alpha _i^{\vee } \rangle = 0,$ for $k+2 \leq i;$
–
$\Phi ={\rm C}_{n}$, $n\geq 3$, $p=3,$ and$\langle \lambda, \alpha _i^{\vee } \rangle = p-1,$ for $n-1 \leq i \leq n,$ and $\langle \lambda, \alpha _{n-2}^{\vee } \rangle \in \{p-2,p-1\};$–
$\Phi ={\rm D}_{n}$, $n\geq 4$, $p=2,$ and $\langle \lambda, \alpha _i^{\vee } \rangle = p-1,$ for $n-3 \leq i \leq n$;–
$\Phi ={\rm E}_{n}$, $n=6,7,8$, $p=2,$ and $\langle \lambda, \alpha _i^{\vee } \rangle = p-1,$ for $2 \leq i \leq 5$;–
$\Phi ={\rm F}_{4}$,(i)
$p=2$ and $\langle \lambda, \alpha _i^{\vee } \rangle = p-1,$ for $1 \leq i \leq 2,$ or(ii)
$p=3,$ $\langle \lambda, \alpha _i^{\vee } \rangle = p-1,$ for $2 \leq i \leq 3,$ and $\langle \lambda, \alpha _4^{\vee } \rangle \in \{p-2,p-1\};$
–
$\Phi ={\rm G}_{2}$, $p=2,$ and $\lambda =(p-1)\rho.$
Proof. For type 
${\rm B}_n$ and 
$p = 2$, when 
$n = 3$, case (i) follows from § 4.4.1, where 
$\lambda = \rho = (1,1,1)$. For 
$n > 3$, one may consider the Levi subgroup 
$L_J$ associated to 
$J = \{\alpha _{n-2},\alpha _{n-1},\alpha _{n}\}$. Let 
$\lambda = (*,\ldots,*,1,1,1)$ be any 
$p$-restricted weight. Then the TMC fails for 
$T_J((p-1)\rho _J + \lambda )$ over 
$L_J$. Applying the contrapositive of Theorem 2.6.1, the TMC must fail for 
$T((p-1)\rho + \lambda )$ over 
$G$. In case (ii), for any 
$n \geq 3$, the base case of 
$\lambda = (1,1,0,\ldots,0)$ follows from Theorem 6.1.2. For 
$n > 4$ and 
$\lambda = (*,\ldots,*,1,1,0,\ldots,0)$, where the 1s lie in the 
$k$th and 
$(k+1)$th spots (with 
$k \leq n-2$), one uses a Levi subgroup 
$L_J$ associated to 
$J = \{\alpha _k,\alpha _{k+1},\ldots,\alpha _n\}$.
For type 
${\rm C}_n$ and 
$p = 3$, when 
$n = 3$, one base case is 
$\lambda = (2,2,2)$, given in § 4.5.1. For 
${n > 3}$, the case of 
$\lambda = (*,\ldots,*,2,2,2)$ follows by using the Levi subgroup 
$L_J$ associated to 
$J = \{\alpha _{n-2},\alpha _{n-1},\alpha _n\}$. The second type 
${\rm C}_3$ base case of 
$\lambda = (1,2,2)$ is given in Theorem 6.5.1, from which the general case of 
$\lambda = (*,\ldots,*,1,2,2)$ similarly follows.
For type 
${\rm D}_n$ and 
$p = 2$, when 
$n = 4$, the base case of 
$\lambda = \rho = (1,1,1,1)$ is given in § 4.6.1. For 
$n > 4$, the case of 
$\lambda = (*,\ldots,*,1,1,1,1)$ follows using a Levi subgroup 
$L_J$ associated to 
$J = \{\alpha _{n-3},\alpha _{n-2},\alpha _{n-1},\alpha _n\}$. This type 
${\rm D}_4$ case also gives the type 
${\rm E}_n$ cases by using a Levi subgroup 
$L_J$ associated to 
$J = \{\alpha _2,\alpha _3,\alpha _4,\alpha _5\}$.
For type 
${\rm F}_4$ and 
$p = 2$, we start with the base case of 
$\lambda = (1,1,0)$ for a type 
${\rm B}_3$ group, as above. Using a Levi subgroup 
$L_J$ associated to 
$J = \{\alpha _1,\alpha _2,\alpha _3\}$ gives the 
${\rm F}_4$ case of 
$\lambda = (1,1,*,*)$. Note that starting with the 
$\lambda = (1,1,1)$ case for type 
${\rm B}_3$ does not generate any additional examples in type 
${\rm F}_4$. For 
$p = 3$, we use the two type 
${\rm C}_3$ cases of 
$\lambda = (2,2,2)$ or 
$\lambda = (1,2,2)$ and a Levi subgroup associated to 
$J = \{\alpha _2,\alpha _3,\alpha _4\}$ to give the type 
${\rm F}_4$ cases of 
$\lambda = (*,2,2,2)$ or 
$(*,2,2,1)$, respectively. Note the ordering swap when changing from type 
${\rm C}_3$ to 
${\rm F}_4$, as the root 
$\alpha _2$ is the long root, whereas 
$\alpha _3$, 
$\alpha _4$ are the short roots in the type 
${\rm C}_3$ subroot system.
Lastly, the type 
${\rm G}_2$ case follows from [Reference Bendel, Nakano, Pillen and SobajeBNPS20b].
Remark 5.2.2 For all pairs 
$(\Phi, p)$ that appear above one always has
\[ T(2(p-1)\rho)|_{G_{1}T} \ncong \widehat{Q}_{1}(0). \]
5.3 Remarks about the proof
Our goal was to keep the calculations for the low rank groups in § 4 fairly self-contained. We make use of Ext-data that appears in the literature, mainly due to Jantzen [Reference JantzenJan91], and to Dowd and Sin [Reference Dowd and SinDS96]. In addition, we apply Weyl's character formula, via the computer algebra package [Reference van Leeuwen, Cohen and LisserLCL92], and perform explicit calculations of the multiplicities appearing in the Jantzen filtration. Some of these calculations are obtained via a short computer program written for LiE. However, we owe a debt of gratitude to Frank Lübeck for his tables of weight multiplicites [Reference LübeckLub] and to Stephen Doty for his [GAP21] package WeylModules [Reference DotyDot09]. These enabled us to look at various examples and observe many of the phenomena that are described in the paper.
6. More counterexamples
6.1 Type ${\rm B}_n$ revisited
With the use of Theorem 2.6.1 and Levi subgroups, we have already observed in Theorem 5.2.1 that the TMC fails for type 
${\rm B}_n$, 
$n \geq 3$, and 
$p = 2$, based on a counterexample in type 
${\rm B}_3$. We present here a direct proof of the failure of the TMC for all 
$n \geq 3$, noting that these counterexamples do not arise from Levi subgroups, thus demonstrating further subtleties in the question of when the TMC holds. Here we make use of the construction outlined in § 3.3. As observed in Theorem 5.2.1, Levi subgroups may also be used to generate further examples from these examples.
A key result that is used involves having some information about the socles of Weyl modules. The proof of the next proposition is technical and is provided in § 6.4.
Proposition 6.1.1 Let 
$G$ be of type 
${\rm B}_n,$ 
$n \geq 3$, and 
$p=2.$ Let 
$\sigma \in X^+$. If 
$\sigma < \rho -\omega _1$ and 
$L(\sigma )$ appears as a composition factor of 
$\nabla (\rho - \omega _1)$, then 
$\sigma < \rho - \omega _1 -\omega _2$.
With Proposition 6.1.1, one can produce new counterexamples to the TMC for type 
${\rm B}_n$, 
$n\geq 3$, and 
$p=2$.
Theorem 6.1.2 Let 
$G$ be a simple algebraic group of type 
${\rm B}_n$, 
$n \geq 3,$ and 
$p=2$. The tilting module 
$T(\rho + \omega _1 + \omega _2)$ is not isomorphic to 
$\widehat {Q}_1(\rho -\omega _1-\omega _2)$ as a 
$G_1T$-module.
Proof. Assume that the TMC holds for the tilting module 
$T(\rho + \omega _1 + \omega _2)$. Set 
$\lambda = \rho -\omega _1-\omega _2$ and 
$\mu = \rho - \omega _1.$ Note that 
$\lambda + 2 \omega _1= \mu + \alpha _1$ and that 
$\langle \lambda, \alpha _1^{\vee } \rangle =0.$ It follows from Proposition 3.3.3 that
\[ L(\omega_1) \hookrightarrow \nabla (\omega_1) \hookrightarrow \operatorname{Ext}_{G_1}^1(L(\lambda), \nabla(\mu))^{(-1)}. \]
In addition, Proposition 6.1.1 implies that 
$ {\operatorname {Hom}}_{G_1}(\widehat {Q}_1(\lambda ), \nabla (\mu ))=0$. From Proposition 3.3.1 one concludes that
\[ L(\omega_1) \hookrightarrow {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(\lambda), \widehat{Q}_1(\mu))^{(-1)}. \]
Using Theorem 3.3.2, one concludes that 
$L(\omega _1)$ is a submodule of some 
$\Delta (\gamma )$ with 
$2\gamma \leq 2 \rho -(\rho - \omega _1)-(\rho -\omega _1 - \omega _2)= 2 \omega _1 + \omega _2$. The only possibilities for 
$\gamma$ with 
$2\omega _1 \leq 2\gamma \leq 2\omega _1 + \omega _2$ are 
$\omega _1$ and 
$\omega _2$. Now Lemma 4.3.1 implies that the corresponding Weyl modules 
$\Delta (\omega _1)$, and 
$\Delta (\omega _2)$ both have the trivial module as their simple socle, a contradiction. It follows from character data (the only dominant weights of 
$\nabla (\omega _1)$ are 
$\omega _1$ and 
$0$) that 
$T(\rho + \omega _1)$ is isomorphic to 
$\widehat {Q}_1(\rho -\omega _1)$ as a 
$G_{1}T$-module. From this argument, we can conclude that 
$T(\rho + \omega _1 + \omega _2)$ is not isomorphic 
$\widehat {Q}_1(\rho -\omega _1-\omega _2)$ as a 
$G_1T$-module.
6.2 Euler characteristic
To prove Proposition 6.1.1, we consider the Jantzen filtration on 
$\Delta (\rho - \omega _1)$. Recall from § 4.2 that for any 
$\lambda \in X^+$
\[ \sum_{i > 0} \operatorname{ch} \Delta(\lambda)^i = \sum_{\alpha \in \Phi^+} \sum_{0 < mp < \langle\lambda + \rho,\alpha^{\vee}\rangle} \nu_p(mp)\chi(s_{\alpha,mp}\cdot \lambda). \]
The goal is to show that (for 
$\lambda = \rho - \omega _1$) most of the 
$\chi (s_{\alpha,mp}\cdot \lambda )$ appearing in the sum, in fact, vanish.
We first recall some general facts about the Euler characteristic 
$\chi (\mu )$ for a weight 
$\mu$, which is defined as
\[ \chi(\mu) = \sum_{i \ge 0} (-1)^i \text{ch} (R^i \operatorname{ind}_B^G \mu). \]
The statements and the proofs can be found in [Reference JantzenJan03, II 5.4, 5.9, 8.19].
Lemma 6.2.1 Let 
$\mu \in X$.
(a) If there exists
$\alpha \in \Delta$ with $\langle \mu,\alpha ^{\vee }\rangle = -1$, then $\chi (\mu ) = 0$.(b) For
$w \in W$, $\chi (w\cdot \mu ) = (-1)^{\ell (w)}\chi (\mu )$.(c) In particular, for
$\alpha \in \Delta$, $\chi (s_{\alpha }\cdot \mu ) = -\chi (\mu )$.(d) For
$\alpha \in \Phi ^+$, $\chi (s_{\alpha,mp}\cdot \lambda ) = -\chi (\lambda - mp\alpha )$.
6.3 Vanishing of $\chi$
We next identify a condition for vanishing of 
$\chi (\mu )$ in type 
${\rm B}_n$ that is used repeatedly in § 6.4. The condition is stated in terms of the epsilon-basis for weights. To this point in the paper, per standard convention, weights have been given in the omega-basis, that is, expressing a weight as a linear combination of the fundamental dominant weights 
$\omega _i$. By the epsilon-basis, we mean expressing a weight in terms of the standard basis vectors 
$\epsilon _i$ of the underlying Euclidean space 
$\mathbb {E}$. For clarity in this section, we are explicit about which basis is being used. When weights are written in component notation, let 
$\mu _{\epsilon }$ denote the weight 
$\mu$ in the epsilon-basis and 
$\mu _{\omega }$ denote the weight the omega-basis. More precisely, for 
$\mu \in X$, the notation 
$\mu _{\epsilon } = (c_1, c_2, \ldots, c_n)$ for integers 
$\{c_s\}$ means 
$\mu = \sum _{s = 1}^nc_s\epsilon _s$, whereas 
$\mu _{\omega } = (c_1, c_2, \ldots, c_n)$ for integers 
$\{c_s\}$ means 
$\mu = \sum _{s = 1}^nc_s\omega _s$. For example, in type 
$B_3$, 
$\rho = \frac 52\epsilon _1 + \frac 32\epsilon _2 + \frac 12\epsilon _3 = \omega _1 + \omega _2 + \omega _3$, and so we write 
$\rho _{\epsilon } = (\frac 52,\frac 32,\frac 12)$ and 
$\rho _{\omega } = (1,1,1)$. In the epsilon-basis, the positive roots for 
$\Phi ={\rm B}_{n}$ consist of 
$\{\epsilon _i + \epsilon _j, 1 \leq i < j \leq n\}$, 
$\{\epsilon _i - \epsilon _j, 1 \leq i < j \leq n\}$, and 
$\{\epsilon _i, 1 \leq i \leq n\}$.
Lemma 6.3.1 Assume that the root system 
$\Phi$ is of type 
${\rm B}_n$ or 
${\rm C}_n$. Let 
$\mu \in X$ with 
$\mu + \rho = \sum _{s = 1}^nm_s\epsilon _s$ for integers 
$\{m_s\}$. If there exist 
$1 \leq i < j \leq n$ with 
$|m_i| = |m_j|$, then 
$\chi (\mu ) = 0$.
Proof. Let 
$\Phi$ be a root system of type 
${\rm B}$ or 
${\rm C}$. Note that the condition on 
$\mu + \rho$ implies that there is a root 
$\beta \in \Phi$ such that 
$\langle \mu + \rho, \beta ^{\vee } \rangle =0$ (e.g. 
$\beta = \epsilon _i \pm \epsilon _j$ as appropriate). Using the 
$W$-invariance of the inner product we can find 
$w \in W$ and 
$\alpha \in \Delta$ such that 
$\langle w \cdot \mu, \alpha ^{\vee } \rangle =-1$. It follows from Lemma 6.3.1(a) and (b) that 
$\chi (\mu ) = 0$.
6.4 Proof of Proposition 6.1.1
We are now ready to compute the Jantzen sum formula for 
$\Delta (\rho - \omega _1)$ in type 
${\rm B}_n$ and 
$p = 2$. Proposition 6.1.1 will follow from the result below, since 
$\rho -\omega _m - \omega _{m + 1}< \rho -\omega _{1}-\omega _{2}$ for 
$m\geq 2$.
Proposition 6.4.1 Let 
$\Phi$ be of type 
${\rm B}_n$ with 
$n \geq 3$ and 
$p = 2$. In the Jantzen sum formula,
\[ \sum_{i >0}\operatorname{ch}\Delta(\rho - \omega_1)^i = \sum m_{\mu}\chi(\mu), \]
for integers 
$m_{\mu }$, where 
$\mu = \rho - \omega _m - \omega _{m + 1}$ for 
$m$ being even and at least 
$2$.
Proof. Set 
$\lambda = \rho - \omega _1$. Then 
$\lambda + \rho = 2\rho - \omega _1 = (2n-2)\epsilon _1 + \sum _{s = 2}^{n}(2(n-s) + 1)\epsilon _s$ or
\begin{equation} (\lambda + \rho)_{\epsilon} = (2n-2, 2n-3, 2n-5, \ldots, 3, 1). \end{equation}
For each positive root 
$\alpha$, which will be considered in the epsilon-basis, we consider all 
$\chi (s_{\alpha,2m}\cdot \lambda )$ that may occur. More precisely, using Lemma 6.2.1(d), we consider 
$\chi (\lambda - 2m\alpha )$. For many 
$\alpha$, by the nature of 
$\lambda - 2m\alpha + \rho$, we may apply Lemma 6.3.1 to conclude that 
$\chi (\lambda - 2m\alpha )$ (and, hence, 
$\chi (s_{\alpha,2m}\cdot \lambda )$) is zero. In some other cases, we show that while terms initially survive, they appear more than once and will cancel, in the end leaving only the stated weights.
Case 1: 
$\alpha = \epsilon _i + \epsilon _j$ for 
$1 < i < j \leq n$. Here 
$\langle \lambda + \rho,\alpha ^{\vee }\rangle = 2(n-i) + 2(n-j) + 2$. We show that 
$\chi (\lambda - 2m\alpha ) = 0$ for 
$2 \leq 2m \leq 2(n - i) + 2(n - j)$ by considering the components of 
$(\lambda + \rho - 2m\alpha )_{\epsilon }$ and applying Lemma 6.3.1. Write 
$(\lambda + \rho - 2m\alpha )_{\epsilon } = (c_1, c_2, \ldots, c_n)$. Note that the 
$c_s$ match those in (6.4.1) with two exceptions: 
$c_i = 2(n-i) + 1 - 2m$ and 
$c_j = 2(n-j) + 1 - 2m$. Consider 
$c_i$. Based on the possible values for 
$m$, we have
\begin{align*} -2(n-(j+1)) - 1 &= 2(n-i) + 1 - [2(n-i) + 2(n-j)] \leq c_i \leq 2(n-i) + 1 - 2 \\ &= 2(n - (i+1)) + 1. \end{align*}
When 
$c_i < 0$, we have 
$-2(n - (j+1)) - 1 \leq c_i \leq -1$. Therefore, 
$|c_i| = c_t$ for some 
$j + 1 \leq t \leq n$, and the vanishing follows as claimed. When 
$c_i \geq 0$, we have 
$1 \leq c_i \leq 2(n-(i+1)) + 1$. Here 
$c_i = c_t$ for some 
$i + 1 \leq t \leq n$, with one exception: when 
$t = j$ or 
$c_i = 2(n-j) + 1$. That occurs when 
$2(n-i) + 1 - 2m = 2(n-j) + 1$ or when 
$2m = 2j - 2i$. In that situation,
\[ c_j = 2(n-j) + 1 - 2m = 2(n-j) + 1 - (2j - 2i) = 2(n - 2j + i) + 1. \]
Since 
$j > i$, 
$c_j \neq 2(n-j) + 1$. However, it is possible that 
$c_j = -2(n-j) - 1$. In that case, 
$|c_j| = c_i$, and we are done. In general, since 
$i + 1 \leq j \leq n$, we have
\[ -2(n - (i + 1)) - 1 = 2(n - 2n + i) + 1 \leq c_j \leq 2(n - 2(i+1) + i) + 1 = 2(n - (i + 2)) + 1. \]
If 
$|c_j| \neq 2(n-j) + 1$, then we see that 
$|c_j| = c_t$ for some 
$i + 1 \leq t \leq n$, 
$t \neq j$, and the claim follows.
Case 2: 
$\alpha = \epsilon _i - \epsilon _j$ for 
$1 \leq i < j \leq n$. Here
\[ \langle \lambda + \rho,\alpha^{\vee}\rangle = \begin{cases} 2n - 2 - 2(n - j) - 1 = 2(j - 1) - 1 & \text{ if } i = 1,\\ 2(n-i) + 1 - 2(n-j) - 1 = 2(j - i) & \text{ if } i > 1. \end{cases} \]
We need to show that 
$\chi (\lambda - 2m\alpha ) = 0$ for 
$2 \leq 2m \leq 2(j - i) - 2$ (which is vacuous if 
$j = i + 1$). We proceed as in case 1 to apply Lemma 6.3.1. Write 
$(\lambda + \rho - 2m\alpha )_{\epsilon } = (c_1, c_2, \ldots, c_n)$. Note that the 
$c_s$ match those in (6.4.1) with two exceptions: 
$c_i$ and 
$c_j$. Here we need to consider only 
$c_j = 2(n - j) + 1 + 2m$. We have
\[ 2(n - (j - 1)) + 1 = 2(n-j) + 1 + 2 \leq c_j \leq 2(n-j) + 1 + 2(j-i) - 2 = 2(n - (i + 1)) + 1. \]
Therefore, 
$c_j = c_t$ for some 
$i + 1 \leq t \leq j - 1$, and the claim follows.
Case 3: 
$\alpha = \epsilon _i$ for 
$i > 1$. Here 
$\langle \lambda + \rho,\alpha ^{\vee }\rangle = 2(2(n-i) + 1) = 4(n-i) + 2$. We need to show that 
$\chi (\lambda - 2m\alpha ) = 0$ for 
$2 \leq 2m \leq 4(n-i)$ (which is vacuous if 
$i = n$). We proceed as above and apply Lemma 6.3.1. Write 
$(\lambda + \rho - 2m\alpha )_{\epsilon } = (c_1, c_2, \ldots, c_n)$. Note that the 
$c_s$ match those in (6.4.1) with one exception: 
$c_i = 2(n-i) + 1 - 2m$. Based on the possible values for 
$m$, we have
\[ -2(n -(i + 1)) - 1 = 2(n-i) + 1 - 4(n-i) \leq c_i \leq 2(n-i) + 1 - 2 = 2(n - (i + 1)) + 1. \]
Therefore, 
$|c_i| = c_t$ for some 
$i + 1 \leq t \leq n$.
Case 4: 
$\alpha = \epsilon _1$. Here 
$\langle \lambda + \rho,\alpha ^{\vee }\rangle = 2(2n - 2) = 4(n-1)$. We need to consider 
$\chi (s_{\alpha,2m}\cdot \lambda ) = - \chi (\lambda - 2m\alpha )$ for 
$2 \leq 2m \leq 4(n-1) - 2$. Unlike the previous cases, these do not all vanish, although some will be seen to cancel. Consider first the case 
$2m = 2(n-1)$ (or 
$m = n - 1$). By definition,
\[ \chi(s_{\alpha,2m}\cdot\lambda) = \chi(\lambda - (4(n-1) - 2(n-1))\alpha) = \chi(\lambda - 2(n-1)\alpha) = \chi(\lambda - 2m\alpha). \]
Since this also equals (as noted previously) 
$-\chi (\lambda - 2m\alpha )$, we must have 
$\chi (\lambda - 2m\alpha ) = 0$.
The remaining cases are considered in pairs: 
$2m$ and 
$4(n-1) - 2m$ for 
$2 \leq 2m \leq 2(n - 2)$ or 
$1 \leq m \leq n - 2$. Write 
$(\lambda + \rho - 2m\alpha )_{\epsilon } = (c_1,c_2,\ldots,c_n)$ as before, again, this agrees with (6.4.1) except in the first component, where 
$c_1 = 2n - 2 - 2m$. On the other hand, the first component of 
$\lambda + \rho - [4(n - 1) - 2m]\alpha$ is
\[ 2n - 2 - [4(n-1) - 2m] = -2n + 2 + 2m = -(2n -2 - 2m). \]
Let 
$w$ be the reflection in the 
$\epsilon _1$-hyperplane, then 
$w(\lambda + \rho - 2m\alpha ) = \lambda + \rho - (4(n-1) - 2m))\alpha$. Since the length of 
$w$ is odd, by Lemma 6.2.1(b), 
$\chi (\lambda - 2m\alpha ) = -\chi (\lambda - (4(n-1)-2m)\alpha )$. If 
$m$ is odd, 
$\nu _2(2m) = 1 = \nu _2(2(2(n-1) - m)) = \nu _2(4(n-1) - 2m)$, and so the two characters cancel. If 
$m$ is even, the characters do not necessarily cancel.
Suppose that 
$m$ is even and 
$2 \leq m \leq n - 2$. Let 
$w$ be the Weyl group element such that 
$w\cdot (\lambda - 2m\alpha ) = w(\lambda +\rho - 2m\alpha ) - \rho$ is dominant. Then 
$\chi (\lambda - 2m\alpha ) = (-1)^{\ell (w)}\chi (w\cdot (\lambda - 2m\alpha ))$. We have
\begin{align*} (\lambda + \rho - 2m\alpha)_{\epsilon} = (2n - 2 - 2m, 2n - 3, 2n - 5, \ldots, 3, 1). \end{align*}
Note that all components are necessarily positive. As discussed in the proof of Lemma 6.3.1, we obtain the components of 
$(w(\lambda + \rho - 2m\alpha ))_{\epsilon }$ by placing these in decreasing order (left to right). Thus, we obtain
\[ (w(\lambda + \rho - 2m\alpha))_{\epsilon} = (2n - 3, 2n - 5, \ldots, 2n - 1 - 2m , 2n - 2 - 2m, 2n - 3 - 2m, \ldots, 3, 1). \]
Hence,
\[ (w(\lambda + \rho - 2m\alpha))_{\omega} = (2, 2, \ldots, 2,1,1,2,\ldots,2,2), \]
where the ones are in the 
$m$th and 
$(m + 1)$th components. Subtracting 
$\rho$ gives
\[ (w\cdot(\lambda - 2m\alpha))_{\omega} = [w(\lambda + \rho - 2m\alpha) - \rho]_{\omega} = (1,\ldots,1,0,0,1,\ldots, 1), \]
with the zeros in the same locations as above. Hence, 
$w\cdot (\lambda - 2m\alpha ) = \rho - \omega _m - \omega _{m+1}$ and multiples of 
$\chi (\rho -\omega _m - \omega _{m+1})$ may appear in the sum. There are no other contributions in this case.
Case 5: 
$\alpha = \epsilon _1 + \epsilon _j$ for 
$2 \leq j \leq n$. Here 
$\langle \lambda + \rho,\alpha ^{\vee }\rangle = 2(n - 1) + 2(n - j) + 1$. We need to consider 
$\chi (s_{\alpha,2m}\cdot \lambda ) = - \chi (\lambda - 2m\alpha )$ for 
$2 \leq 2m \leq 2(n - 1) + 2(n - j)$. We use Lemma 6.3.1 to show that most of these vanish, with some remaining cases seen to cancel, so that these terms give no further contribution to the sum formula. Write 
$(\lambda + \rho - 2m\alpha )_{\epsilon } = (c_1, c_2, \ldots, c_n)$. Note that the 
$c_s$ match those in (6.4.1) with two exceptions: 
$c_1$ and 
$c_j$. Consider 
$c_j = 2(n-j) + 1 - 2m$. Based on the values of 
$m$, we have
\[ -2(n-2) \,{-}\, 1 = 2(n-j) + 1 \,{-}\, 2(n-1) \,{-}\, 2(n-j) \leq c_j \leq 2(n-j) + 1 \,{-}\, 2 = 2(n \,{-}\, (j+1)) + 1. \]
If 
$c_j \geq 0$, then we have 
$1 \leq c_j \leq 2(n - (j + 1)) + 1$. Thus, 
$c_j = c_t$ for some 
$j + 1 \leq t \leq n$ and the terms vanish by Lemma 6.3.1. If 
$c_j < 0$, we have 
$-2(n-2) - 1 < c_j < -1$ and 
$|c_j| = c_t$ for some 
$2 \leq t \leq n$, unless 
$t = j$. That is, when 
$2(n-j) + 1 - 2m = -2(n-j) - 1$ or 
$2m = 4(n-j) + 2$.
Suppose 
$2m = 4(n-j) + 2$ and consider the first component: 
$c_1 = 2n - 2 - 4(n-j) - 2 = -2n + 4j - 4 = -2(n - 2j + 2)$. If 
$n$ is even, this is zero when 
$n = 2j - 2$ or 
$j = ({n+2})/{2}$. Let 
$w \in W$ be such that 
$w(\lambda + \rho - 2m\alpha )$ is dominant. Since 
$(\lambda + \rho - 2m\alpha )_{\epsilon }$ has a zero in the first component, 
$(w(\lambda + \rho - 2m\alpha ))_{\epsilon }$ has a zero in the last component. Hence, the last component of 
$(w(\lambda + \rho - 2m\alpha ))_{\omega }$ is also zero, and so the last component of 
$(w\cdot (\lambda - 2m\alpha ))_{\omega } = (w(\lambda + \rho - 2m\alpha ) - \rho )_{\omega }$ is 
$-1$. From Lemma 6.2.1(a) and (b), it follows that 
$\chi (\lambda - 2m\alpha ) = 0$. Thus, we are done for 
$j = ({n + 2})/{2}$.
Recapping: for any 
$2 \leq j \leq n$ with 
$j \neq ({n + 2})/{2}$, we know that 
$\chi (\lambda - 2m\alpha ) = 0$ except when 
$2m = 4(n-j) + 2$ (or 
$m = 2(n-j) + 1$). Split 
$\{j : 2 \leq j \leq n, j \neq ({n+2})/{2}\}$ into pairs 
$j$ and 
$n + 2 - j$ for 
$2 \leq j \leq \lfloor ({n+1})/{2} \rfloor$. For each such 
$j$, we show that 
$\nu _{2}(2m)\chi (\lambda - 2m\alpha ) = -\nu _{2}(2\tilde {m})\chi (\lambda - 2\tilde {m}\tilde {\alpha })$, where 
$\tilde {m} = 2(j -2) + 1$ is the relevant ‘
$m$’ for the index 
$n + 2 - j$ and 
$\tilde {\alpha } = \epsilon _1 + \epsilon _{n+2 - j}$. This shows that these remaining characters cancel, so there is no contribution from the 
$\epsilon _1 + \epsilon _j$.
First, observe that both 
$m$ and 
$\tilde {m}$ are odd, so 
$\nu _2(2m) = 1 = \nu _2(2\tilde {m})$. As previously, write 
$(\lambda + \rho - 2m\alpha )_{\epsilon } = (c_1,c_2, \ldots,c_n)$. Then the 
$c_s$ match those in (6.4.1) with two exceptions:
\[ c_1 = -2(n - 2j + 2) \quad \text{and}\quad c_j = 2(n-j) + 1 - 4(n-j) - 2 = -2(n - j) - 1. \]
Similarly, write 
$(\lambda + \rho -2\tilde {m}\tilde {\alpha })_{\epsilon } = (d_1, d_2, \ldots,d_n)$. Again, the 
$d_s$ agree with those in (6.4.1) with two exceptions:
\begin{align*}
d_1 = 2n - 2 - 4(j-2) - 2 = 2(n-2j + 2)
\end{align*}
and
\begin{align*}
d_{n + 2 - j} = 2(j - 2) + 1 - 4(j-2) - 2 = -2(j-2) - 1.
\end{align*}
We see that 
$c_s = d_s$ for 
$s \neq 1, j, n + 2 - j$, whereas 
$c_1 = -d_1$, 
$c_j = -d_j$, and 
$c_{n+2-j} = -d_{n+2-j}$. Let 
$w_i$ denote the reflection in the 
$\epsilon _i$-hyperplane and set 
$w = w_1w_jw_{n+2-j}$ (the ordering being irrelevant). Then 
$w(\lambda + \rho - 2m\alpha ) = \lambda + \rho - 2\tilde {m}\tilde {\alpha }$, and so 
$\chi (\lambda - 2m\alpha ) = (-1)^{\ell (w)}\chi (w\cdot (\lambda - 2m\alpha ) = (-1)^{\ell (w)}\chi (\lambda - 2\tilde {m}\tilde {\alpha })$. Since each 
$w_i$ has odd length, so does 
$w$, and the claim that 
$\nu _{2}(2m)\chi (\lambda - 2m\alpha ) = -\nu _{2}(2\tilde {m})\chi (\lambda - 2\tilde {m}\tilde {\alpha })$ follows.
6.5 A second failure of the TMC for type ${\rm C}_3$
Let 
$G$ be of type 
${\rm C}_3$ with 
$p = 3$. In § 4.5.1, we saw that 
$T(4\rho )$ fails to be indecomposable upon restriction to 
$G_1T$. In this section, we show that the tilting module 
$T(4\rho -\omega _1)$ also fails to remain indecomposable when restricted to 
$G_1T.$ Note that, in this subsection, we return to expressing weights solely in the omega-basis.
Theorem 6.5.1 Let 
$G$ be a simple algebraic group of type 
${\rm C}_3$ and 
$p=3$. The tilting module 
$T(4 \rho -\omega _1)$ is not isomorphic to 
$\widehat {Q}_1(\omega _1)$ as a 
$G_1T$-module.
Data obtained via the Jantzen filtration yield the following table.
\[
\begin{align*}
\lambda \quad & \quad \sum_{i>0} \text{ch } \Delta(\lambda)^i\\
(1,0,0) \quad & \quad \emptyset \\
(0,1,1) \quad & \quad \emptyset \\
(2,1,1) \quad & \quad \chi(0,1,1) \\
(1,3,0) \quad & \quad \chi(2,1,1))- \chi(0,1,1)+ \chi(1,0,0) \\
(3,2,0) \quad & \quad \chi(1,3,0) + \chi(2,1,1))+ \chi(1,0,0) \\
(2,2,1)\quad & \quad \chi(3,2,0) + 2 \cdot \chi(1,3,0) + \chi(2,1,1) - 2\cdot \chi(1,0,0) \\
\end{align*} \]
It follows that 
$\Delta (1,0,0)$ and 
$\Delta (0,1,1)$ are simple and that 
$\Delta (2,1,1)$ has length two, the second factor having highest weight 
$(0,1,1)$. Character considerations now show that 
$\Delta (1,3,0)$ is multiplicity free with three composition factors, including factors with highest weights 
$(2,1,1)$ and 
$(1,0,0)$. Similarly one concludes from character data that 
$\Delta (3,2,0)$ has six composition factors, namely the simple modules with highest weights 
$(3,2,0)$, 
$(1,3,0)$, 
$(2,1,1)$, 
$(0,1,1)$, and 
$(1,0,0)$ twice. One concludes that
\begin{align*} \sum_{i>0} \text{ch } \Delta(2,2,1)^i &= \text{ch }L(3,2,0) + 2 \cdot \text{ch }L(0,1,1) + 2 \cdot \text{ch }L(1,0,0) \\ &\quad + 3\cdot \text{ch }L(1,3,0) + 4 \cdot \text{ch }L(2,1,1). \end{align*}
Suppose that 
$T(4\rho - \omega _1) \cong \widehat {Q}_1(\omega _1)$ as 
$G_{1}T$-module. Then
\[ V := {\operatorname{Hom}}_{G_1}(\widehat{Q}_1(1,0,0),\nabla(2,2,1))^{(-1)} \]
admits a good filtration. We argue in a similar manner as in § 4.5.1. From above, we see that 
$V$ has composition factors of 
$L(\omega _2)$ and 
$k$, with the number (and arrangement) of such factors determined by the number (and arrangement) of copies of 
$L(1,3,0)$ and 
$L(1,0,0)$, respectively, in 
$\nabla (2,2,1)$. From the above, we have that 
$\Delta (2,2,1)$ (and, hence, 
$\nabla (2,2,1)$) contains between one and three copies of 
$L(1,3,0)$ and one or two copies of 
$L(1,0,0)$. Analogous to the argument in case 1 of § 4.5.1, for 
$V$ to admit a good filtration, the number of copies of 
$L(1,0,0)$ appearing must be at least the number of copies of 
$L(1,3,0)$ that appear. In particular, there can, in fact, be at most 2 copies of 
$L(1,3,0)$. Suppose that 
$[\Delta (2,2,1) : L(1,0,0)] = 2$. Then 
$[\Delta (2,2,1)^2 : L(1,0,0)] = 0$. Furthermore, 
$[\Delta (2,2,1) : L(1,3,0)] = 1 \text { or } 2$, and, in either case, 
$[\Delta (2,2,1)^2 : L(1,3,0)] = 1$. One may now argue as in case 2 of § 4.5.1, starting from an exact sequence
\[ 0 \to \Delta(2,2,1)^2 \to \Delta(2,2,1) \to Q \to 0, \]
to obtain a similar contradiction. Lastly, suppose 
$[\Delta (2,2,1) : L(1,0,0)] = 1$, then
\begin{align*} [\Delta(2,2,1)^2 :
L(1,0,0)] &= 1,\\ [\Delta(2,2,1) : L(1,3,0)] &= 1,\\
[\Delta(2,2,1)^2 : L(1,3,0)] &= 0,\\
[\Delta(2,2,1)^3 : L(1,3,0)] &= 1.
\end{align*}
In this case, one starts from an exact sequence
\begin{align*} 0 \to \Delta(2,2,1)^3 \to \Delta(2,2,1) \to Q \to 0 \end{align*}
to obtain a contradiction.
7. Further questions
7.1 $G_r$-extensions being tilting
We define the condition (ET) as follows.
(ET) 
$\operatorname {Ext}_{G_r}^1(L(\lambda ),L(\mu ))^{(-r)}$ is tilting as a 
$G$-module for all 
$\lambda,\mu \in X_r$.
The evidence thus far suggests that there is some connection between this condition and the TMC. Indeed, (ET) fails in each of the (known) low-rank counterexamples to the TMC, and in the present paper, the former served to help detect the latter. At the same time, in every case in which we know that the TMC holds, we also know that (ET) holds.
In the course of this discussion, one would like to know the answer to the following question, which is of interest in its own right.
Question 7.1.1 Under what conditions does (ET) hold?
In [Reference Bendel, Nakano, Pillen and SobajeBNPS23, Theorem 4.3.1], the authors proved that if 
$p \ge 2h-4$, then (ET) holds for 
$r = 1$. Moreover, the structure is semi-simple (so that the tilting factors are all simple 
$G$-modules). This improved an earlier confirmation of (ET) for 
$r = 1$ by Andersen [Reference AndersenAnd84] for 
$p \ge 3h-3$ and by Bendel, Nakano, and Pillen [Reference Bendel, Nakano and PillenBNP04, § 5.5] for 
$p \geq 2h - 2$. In the recent work [Reference Bendel, Nakano, Pillen and SobajeBNPS23] of the present authors, it was also shown that the TMC holds when 
$p\geq 2h-4$. Further sharpening of this bound in (ET) could be a key step in lowering the bound of validity for the TMC.
7.2 $G_r$-extensions and the TMC
Another mystery to unravel is precisely how (ET) and the TMC relate to each other.
Question 7.2.1 Is the TMC equivalent to (ET)? If not, does one imply the other?
A revised problem is to consider this for each restricted 
$\lambda$.
Question 7.2.2 Is 
$T(2(p^r-1)\rho +w_0\lambda )$ indecomposable over 
$G_r$ if and only if
\begin{align*}
\operatorname {Ext}_{G_r}^1(L(\lambda ), L(\mu ))^{(-r)}
\end{align*}
is tilting for all 
$\mu \in X_r$ where 
$\mu \ge _{\mathbb {Q}} \lambda$?
To motivate this question, as observed in § 3.1, if 
$Q_r(\lambda )$ and 
$Q_r(\mu )$ admit a 
$G$-structure, then we have an embedding of 
$G$-modules
\[ \operatorname{Ext}_{G_r}^1(L(\lambda),L(\mu))^{(-r)} \subseteq {\operatorname{Hom}}_{G_r}(Q_r(\lambda),Q_r(\mu))^{(-r)}. \]
The idea then is that if the submodule is not tilting over 
$G$, potentially the larger Hom-set cannot carry such a structure either, in which case one or both of 
$Q_r(\lambda )$ and 
$Q_r(\mu )$ cannot lift to a tilting module for 
$G$.
The final stipulation, that 
$\mu \ge _{\mathbb {Q}} \lambda$, is to isolate the larger of the two projective covers, 
$Q_r(\lambda )$, as the one that must fail to be tilting if only one of two does.
7.3 Twice the Steinberg weight
We also have observed that the 
$G_1T$-projective cover of the trivial module fails to be isomorphic to 
$T(2(p-1)\rho )$ in all cases in which the TMC fails. This raises another question as to whether the TMC is equivalent to 
$T(2(p-1)\rho )\mid _{G_{1}T}\cong \widehat {Q}(0)$.
Question 7.3.1 Does the TMC hold if and only if 
$T(2(p-1)\rho )$ is indecomposable over 
$G_1$?
We note that if there is an affirmative answer to this question and to the questions raised in the previous subsections, then in such a case the problem of checking the validity of the TMC would reduce to checking if 
$\text {H}^1(G_1,L(\lambda ))^{(-1)}$ is tilting for all 
$\lambda \in X_1$.
Acknowledgements
We thank Henning Andersen for comments and suggestions on an earlier version of our manuscript. The authors also thank the anonymous referees for their suggestions that were incorporated into the current version of the paper.
Conflicts of Interest
None.
