# Quasi-simple modules and Loewy lengths in modular representations of
reductive Lie algebras

Yi-Yang Li, Bin Shu and Yu-Feng Yao  School of Mathematics, Physics and
Statistics, Shanghai University of Engineering Science, Shanghai 201620,
China<EMAIL_ADDRESS>Department of Mathematics, East China Normal
University, Shanghai 200241, China<EMAIL_ADDRESS>Department of
Mathematics, Shanghai Maritime University, Shanghai 201306, China.
<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathfrak{g}}$ be a reductive Lie algebra over an algebraically closed
field of characteristic $p>0$. In this paper, we study the representations of
${\mathfrak{g}}$ with a $p$-character $\chi$ of standard Levi form associated
with a given subset $I$ of the simple root system $\Pi$ of ${\mathfrak{g}}$.
Let $U_{\chi}({\mathfrak{g}})$ be the reduced enveloping algebra of
${\mathfrak{g}}$. A notion “quasi-simple module” (denoted by
$\mathcal{L}_{\chi}(\lambda)$) is introduced. The properties of such a module
turn out to be better than those of the corresponding simple module
$\widehat{L}_{\chi}(\lambda)$. It enables us to investigate the
$U_{\chi}({\mathfrak{g}})$-modules from a new point of view, and
correspondingly gives rise new consequences. First, we show that the first
self extension of $\mathcal{L}_{\chi}(\lambda)$ is zero, and the projective
dimension of $\mathcal{L}_{\chi}(\lambda)$ is finite when $\lambda$ is
$p$-regular. These properties make it significant to rewrite the formula of
Lusztig’s Hope (Lusztig’s conjecture on the irreducible characters in the
category of $U_{\chi}({\mathfrak{g}})$-modules) by replacing
$\widehat{L}_{\chi}(\lambda)$ by $\mathcal{L}_{\chi}(\lambda)$. Second, with
the aid of quasi-simple modules, we get a formula on the Loewy lengths of
standard modules and proper standard modules over $U_{\chi}({\mathfrak{g}})$.
And by studying some examples, we formulate some conjectures on the Loewy
lengths of indecomposable projective ${\mathfrak{g}}$-modules, standard
modules and proper standard modules.

###### Key words and phrases:

reductive Lie algebras, standard Levi form, quasi-simple modules, Loewy length

###### 2010 Mathematics Subject Classification:

17B10; 17B20; 17B35; 17B50

This work is supported by the NSF of China (Grant Nos: 11771279 and 12071136)

## Introduction

### 0.1.

Let $G$ be a connected reductive algebraic group over an algebraically closed
field $k$ of prime characteristic $p$, and ${\mathfrak{g}}=\mbox{\rm Lie}(G)$.
Then ${\mathfrak{g}}$ admits a natural $p$-mapping $[p]$. For any simple
${\mathfrak{g}}$-module $L$, there exists a linear form
$\chi_{L}\in{\mathfrak{g}}^{*}$ such that the action of $x^{p}-x^{[p]}$ on $L$
is just multiplication by the scalar $\chi_{L}(x)^{p}$. Associated with any
$\chi\in{\mathfrak{g}}^{*}$, $U_{\chi}({\mathfrak{g}})$ is defined as the
quotient of the universal enveloping algebra $U({\mathfrak{g}})$ by the ideal
generated by $x^{p}-x^{[p]}-\chi(x)^{p},\forall x\in{\mathfrak{g}}$. Each
simple ${\mathfrak{g}}$-module is attached with a unique $p$-character $\chi$.
The Morita equivalence due to Kac-Weisfeiler and Friedlander-Parshall (cf.
[22], [7, Thms. 3.2; 8.5]) enables us to consider the representations of
$U_{\chi}({\mathfrak{g}})$ just with nilpotent $\chi$.

In this paper, we study the representations of $U_{\chi}({\mathfrak{g}})$ for
$\chi$ having standard Levi form associated with an subset $I$ of $\Pi$, where
$\Pi$ is the set of all simple roots ($\chi$ is regular nilpotent on the Levi
factor ${\mathfrak{g}}_{I}$).

Let ${\mathfrak{g}}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$,
$\mathfrak{h}$ and $\mathfrak{b}^{+}=\mathfrak{h}\oplus\mathfrak{n}^{+}$ be
the triangular decomposition of ${\mathfrak{g}}$, the Cartan subalgebra of
${\mathfrak{g}}$ and the standard Borel subalgebra of ${\mathfrak{g}}$,
respectively. Let $R$, $R^{+}$, $R_{I}$ and $R_{I}^{+}$ be the root system of
${\mathfrak{g}}$, the set of positive roots of ${\mathfrak{g}}$, the root
system of ${\mathfrak{g}}_{I}$ and the set of positive roots of
${\mathfrak{g}}_{I}$, respectively.

Denote
${\mathfrak{g}}_{I}=\mathfrak{n}_{I}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{I}^{+}$
and $\mathfrak{p}_{I}={\mathfrak{g}}_{I}\oplus\mathfrak{u}^{+}$ (resp.
$\mathfrak{p}_{I}^{\prime}={\mathfrak{g}}_{I}\oplus{\mathfrak{u}}^{-}$) by the
triangular decomposition of ${\mathfrak{g}}_{I}$ and the parabolic subalgebra
of ${\mathfrak{g}}$ containing ${\mathfrak{g}}_{I}$ where
${\mathfrak{u}}^{+}=\sum_{\alpha\in R^{+}\backslash
R^{+}_{I}}{\mathfrak{g}}_{\alpha}$ (resp. ${\mathfrak{u}}^{-}=\sum_{\alpha\in
R^{+}\backslash R^{+}_{I}}{\mathfrak{g}}_{-\alpha}$) is the nilpotent radicals
of ${\mathfrak{p}}_{I}$ (resp. ${\mathfrak{p}}_{I}^{\prime}$). Denote $w_{I}$
by the longest element in the Weyl group $W_{I}$ of ${\mathfrak{g}}_{I}$ and
$w_{0}$ the longest element in the Weyl group $W$ of ${\mathfrak{g}}$. Set
$w^{I}=w_{I}w_{0}$.

In [10, §11], a $X(T)/\mathbb{Z}I$-graded $U_{\chi}({\mathfrak{g}})$-modules
category $\mathcal{C}$ was introduced by Jantzen. We write $\widehat{M}$ for a
$U_{\chi}({\mathfrak{g}})$-module $M$ if it is endowed with
$X(T)/\mathbb{Z}I$-graded structure, this is to say, $\widehat{M}$ becomes a
canonical object of $\mathcal{C}$. Denote by $[\widehat{M}]$ the generator
corresponding to a module $\widehat{M}\in\mathcal{C}$ in the Grothendieck
group.

Our purposes in the paper are double. The first one is to introduce quasi-
simple modules and study their properties. The second one is to study the
Loewy lengths for typical modules.

### 0.2.

The first theme of this paper is to introduce the quasi-simple module
$\mathcal{L}_{\chi}(\lambda),\lambda\in X(T)$ (see Definition 2.5) in
$\mathcal{C}$.

Set $\widehat{Q}^{I}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I})}{{\widehat{Q}}}_{\chi,I}(\lambda)\,\,\text{and}\,\,\widehat{Q}^{w^{I}}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I}^{\prime})}{{\widehat{Q}}}_{\chi,I}(\lambda)$
where ${{\widehat{Q}}}_{\chi,I}(\lambda)$ is the indecomposable projective
$U_{\chi}({\mathfrak{g}}_{I})$-module. We call
$\widehat{Q}^{I}_{\chi}(\lambda)$ (resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda)$) the standard (resp. costandard) module.

Let $\widehat{L}_{\chi}(\lambda)$ be a irreducible object in $\mathcal{C}$ and
$\widehat{Q}_{\chi}(\lambda)$ be the projective cover of
$\widehat{L}_{\chi}(\lambda)$. By [11, Prop. 2.1],
$\widehat{Q}_{\chi}(\lambda)$ admit both a filtration with sub-quotients
$\widehat{Q}^{I}_{\chi}(\lambda_{i}),1\leq i\leq r$ and a filtration with sub-
quotients $\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}_{i}),1\leq i\leq r$
where $\lambda_{i}^{w^{I}}=\lambda_{i}-(p-1)(\rho-w^{I}\rho)$ (see §2.2).
$\widehat{L}_{\chi}(\lambda)$ is not only the socle of
$Q^{w^{I}}_{\chi}(\lambda^{w^{I}})$ but also the socle of
$Q^{I}_{\chi}(\lambda_{r}^{I})$ (see §2.3). We prove
$\mathcal{L}_{\chi}(\lambda)\cong Q^{I}_{\chi}(\lambda_{r}^{I})\cap
Q^{w^{I}}_{\chi}(\lambda^{w^{I}})$ (see §2.6 and Proposition 2.9).

Each composition factor of $\mathcal{L}_{\chi}(\lambda)$ is isomorphic to
$\widehat{L}_{\chi}(\lambda)$ (see Lemma 2.3), and
$\mathcal{L}_{\chi}(\lambda)$ is not only a quotient module but also a
submodule of $\widehat{Q}_{\chi}(\lambda)$ (see Theorem 2.11).

In comparison with the category of $G_{1}T$-modules (here, $G_{1}$ stands for
the first Frobenius kernel of $G$, and $T$ a maximal torus of $G$. This
category is actually a category of ${\mathbb{Z}}$-graded
$U_{0}({\mathfrak{g}})$-modules compatible with $T$-action), the counterpart
study in the category $\mathcal{C}$ encounters some difficulties on dealing
with the representations of the Levi factor ${\mathfrak{g}}_{I}$. And some
properties of simple objects in $\mathcal{C}$ are not as good as in the
category of $G_{1}T$-modules. Such as
$\mbox{Ext}^{1}_{\mathcal{C}}(\widehat{L}_{\chi}(\lambda),\widehat{L}_{\chi}(\lambda))\neq
0$ in general (cf. [16, Prop.2.1]) but
$\mbox{Ext}^{1}_{G_{1}T}(\widehat{L}_{0}(\lambda),\widehat{L}_{0}(\lambda))=0$
(cf. [12, II 9.19(5)]) where $\widehat{L}_{\chi}(\lambda)$ (resp.
$\widehat{L}_{0}(\lambda)$) is the simple module in $\mathcal{C}$ (resp. the
category of $G_{1}T$-modules). And the projective dimension of
$\widehat{L}_{\chi}(\lambda)$ is infinite in general but the projective
dimension of $\widehat{L}_{0}(\lambda)$ is finite.

The cohomological properties of the quasi-simple module
$\mathcal{L}_{\chi}(\lambda)$ are shown to be better than
$\widehat{L}_{\chi}(\lambda)$’s, that is,
$\hbox{Ext}_{\mathcal{C}}^{1}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0$
(see Theorem 3.2), and the projective dimension of the quasi-simple module
$\mathcal{L}_{\chi}(\lambda)$ is finite when $\lambda$ is $p$-regular (see
Theorem 5.1).

Moreover, when $\lambda$ is $p$-regular, we prove that the standard module
$\widehat{Q}^{I}_{\chi}(\lambda)$ has a filtration with sub-quotients
isomorphic to quasi-simple modules(see Theorem 4.2). Based on this result,
Lusztig’s Hope (Lusztig’s conjecture on the irreducible characters in
$\mathcal{C}$) can be rewritten as (see (5.2))

$(\widehat{Q}^{I}_{\chi}(v.\lambda_{0}):\mathcal{L}_{\chi}(w.\lambda_{0}))=[\widehat{Z}_{\chi}(v.\lambda_{0}):\widehat{L}_{\chi}(w.\lambda_{0})]=P_{\kappa_{I}(v),\kappa_{I}(w)}(1).$

Since the cohomological properties of quasi-simple module
$\mathcal{L}_{\chi}(\lambda)$ are better than $\widehat{L}_{\chi}(\lambda)$’s,
the new expression might make the proof of Lusztig’s Hope easier.

### 0.3.

In [9], the Loewy series of Verma modules and projective modules in the
category $\mathcal{O}$ are determined. For the category of $G_{1}T$-modules,
under the assumption that Lusztig’s conjecture on the irreducible characters
of $G$ holds, Andersen and Kaneda proved that the baby Verma modules
$\widehat{Z}_{0}(\lambda)$ for $p$-regular weights $\lambda$ and all
projective modules are rigid (cf [3]). Their Loewy lengths are $N+1$ and
$2N+1$, respectively ($N$ is the dimension of $G/B$ where $B$ is the Borel
subgroup of $G$).

Our second theme is the study of the Loewy lengths of standard module
$\widehat{Q}^{I}_{\chi}(\lambda)$, proper standard module
$\widehat{Z}_{\chi}(\lambda)$ (baby Verma module) and indecomposable
projective module $\widehat{Q}_{\chi}(\lambda)$ in $\mathcal{C}$ when
$\lambda$ is $p$-regular (see the first 6 subsections of §1 for these
notations). We denote $ll(\widehat{M})$ by the Loewy length of
$\widehat{M}\in\mathcal{C}$.

With the aid of quasi-simple module, we get a formula on the Loewy lengths of
standard module $\widehat{Q}^{I}_{\chi}(\lambda)$ and proper standard module
when $\lambda$ is $p$-regular (see Theorem 6.2)

$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\mathcal{L}_{\chi}(\lambda))+ll(\widehat{Z}_{\chi}(\lambda))-1.$

In the category of graded $U_{\chi}({\mathfrak{g}}_{I})$-modules, any proper
standard module $\widehat{Z}_{\chi,I}(\lambda)$ is simple and its Loewy length
is $1$ (cf. [10, §10]). Let $\widehat{Q}_{\chi,I}(\lambda)$ be the projective
cover of $\widehat{Z}_{\chi,I}(\lambda)$. In Theorem 6.1, we prove
$ll(\mathcal{L}_{\chi}(\lambda))=ll(\widehat{Q}_{\chi,I}(\lambda))$ when
$\lambda$ is $p$-regular.

In [20], the Loewy length of $\widehat{Q}_{\chi,I}(\lambda)$ was determined
for ${\mathfrak{g}}_{I}=\mathfrak{sl}(3,k)$. According to this result, when
$\lambda$ is $p$-regular, we conjecture that the Loewy length of
$\widehat{Q}_{\chi,I}(\lambda)$ or $\mathcal{L}_{\chi}(\lambda)$ equals to
$l(w_{I})+1$ where $l(w_{I})$ is the length of $w_{I}$(see Conjecture 7.1).
And show that it is true for ${\mathfrak{g}}=\mathfrak{sl}(2,k)$ and
${\mathfrak{g}}=\mathfrak{sl}(3,k)$ when $\chi$ is regular nilpotent.

When $\chi$ has standard Levi form and $\chi$ is subregular nilpotent, the
Loewy length of proper standard module $\widehat{Z}_{\chi}(\lambda)$ is
$l(w_{0})-l(w_{I})+1$ by the results of Jantzen in [13], where $l(w_{0})$ is
the length of $w_{0}$. For ${\mathfrak{g}}=\mathfrak{sl}(3,k)$ with $I$
containing only a simple root and ${\mathfrak{g}}=\mathfrak{s0}(5,k)$ with $I$
containing only the short simple root, according to the results of Jantzen(cf.
[13]), when $\lambda$ is $p$-regular, we get the Loewy lengths of standard
modules $\widehat{Q}^{I}_{\chi}(\lambda)$ and indecomposable projective
modules $\widehat{Q}_{\chi}(\lambda)$ (see Theorem 7.4).

In general, similar to [3, §2; §3], we give the lower bounds for the Loewy
length of proper standard module $\widehat{Z}_{\chi}(\lambda)$, standard
module $\widehat{Q}^{I}_{\chi}(\lambda)$ and indecomposable projective module
$\widehat{Q}_{\chi}(\lambda)$ when $\lambda$ is $p$-regular (see Propositions
8.1, 8.2 and 8.3).

Based on the above results, we conjecture the Loewy length of proper standard
module $\widehat{Z}_{\chi}(\lambda)$ is $l(w_{0})-l(w_{I})+1$, the Loewy
lengths of the standard module $\widehat{Q}^{I}_{\chi}(\lambda)$ is
$l(w_{0})+1$ and the Loewy lengths of the projective module
$\widehat{Q}_{\chi}(\lambda)$ is $2l(w_{0})-l(w_{I})+1$ (see Conjecture 8.4).

In [19], with aid of quasi-simple modules and under the assumption that
Conjecture 7.1 and Vogan’s Conjecture hold (cf. [19, Conj. 4.6]), we prove
Conjecture 8.4.

##  1\. Preliminaries

###  1.1. Some preliminary notations and convention

All the following notations are standard.

(1) $G$, ${\mathfrak{g}}$: a connected reductive group over an algebraically
closed field $k$ of prime characteristic $p$, the Lie algebra of $G$.

(2) $T$, $X(T)$: a maximal torus of $G$, the character group of $T$.

(3) $\chi$: a $p$-character of ${\mathfrak{g}}$.

(4) $U({\mathfrak{g}})$, $U_{\chi}({\mathfrak{g}})$, $U_{0}({\mathfrak{g}})$:
the universal enveloping algebra of ${\mathfrak{g}}$, the reduced enveloping
algebra $U_{\chi}({\mathfrak{g}})$ of ${\mathfrak{g}}$, i.e.,
$U_{\chi}({\mathfrak{g}})=U({\mathfrak{g}})/(x^{p}-x^{[p]}-\chi{(x)^{p}}|x\in{\mathfrak{g}})$,
the restricted enveloping algebra $U_{0}({\mathfrak{g}})$ of ${\mathfrak{g}}$,
i.e.,
$U_{0}({\mathfrak{g}})=U({\mathfrak{g}})/(x^{p}-x^{[p]}|x\in{\mathfrak{g}})$.

(5) ${\mathfrak{g}}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$,
$\mathfrak{h}$, $\mathfrak{b}^{+}$: the triangular decomposition of
${\mathfrak{g}}$, the Cartan subalgebra of ${\mathfrak{g}}$, the standard
Borel subalgebra of ${\mathfrak{g}}$, i.e, the span of positive root vectors
and $\mathfrak{h}$.

(6)$\Pi$, $I$: the set of simple roots, a subset of $\Pi$.

(7) $R$, $R^{+}$, $R_{I}$, $R_{I}^{+}$: the root system of ${\mathfrak{g}}$,
the set of positive roots, the root system of ${\mathfrak{g}}_{I}$, the set of
positive roots in ${\mathfrak{g}}_{I}$.

(8)
${\mathfrak{g}}_{I}=\mathfrak{n}_{I}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{I}^{+}$,
$\mathfrak{p}_{I}={\mathfrak{g}}_{I}\oplus\mathfrak{u}^{+}$,
$\mathfrak{p}_{I}^{\prime}={\mathfrak{g}}_{I}\oplus{\mathfrak{u}}^{-}$: the
Levi subalgebra of ${\mathfrak{g}}$ corresponding to $I$, the parabolic
subalgebras of ${\mathfrak{g}}$ containing ${\mathfrak{g}}_{I}$ where
${\mathfrak{u}}^{+}=\sum_{\alpha\in R^{+}\backslash
R^{+}_{I}}{\mathfrak{g}}_{\alpha}$ and ${\mathfrak{u}}^{-}=\sum_{\alpha\in
R^{+}\backslash R^{+}_{I}}{\mathfrak{g}}_{-\alpha}$ are the nilpotent radicals
of ${\mathfrak{p}}_{I}$ and ${\mathfrak{p}}_{I}^{\prime}$, respectively.

(9) $E$: the Euclidean space spanned by $R$ with inner product
$\langle,\rangle$ normalized so that $\langle\alpha,\alpha\rangle=2$ for
$\alpha\in\Pi$.

(10) $\rho,\alpha^{\vee}$: half the sum of positive roots, the coroot of
$\alpha$.

(11) $s_{\alpha}$, $s_{\alpha,rp}$: the reflection with
$s_{\alpha}(\mu)=\mu-(\langle\mu,\alpha^{\vee}\rangle)\alpha,\forall\mu\in{\mathfrak{h}}^{*}$,
the affine reflection with
$s_{\alpha,rp}(\mu)=\mu-(\langle\mu,\alpha^{\vee}\rangle-
rp)\alpha,\forall\mu\in{\mathfrak{h}}^{*}$.

(12) $W$, $W_{I}$; $W_{p}$, $W_{I,p}$: the Weyl groups generated by
$\\{s_{\alpha}\mid\alpha\in R\\}$, $\\{s_{\alpha}\mid\alpha\in I\\}$,
respectively; the affine Weyl groups generated by
$\\{s_{\alpha,rp}\mid\alpha\in R,r\in\mathbb{Z}\\}$,
$\\{s_{\alpha,rp}\mid\alpha\in I,r\in\mathbb{Z}\\}$, respectively.

(13) $w.\lambda=w(\lambda+\rho)-\rho,w\in W;\lambda\in{\mathfrak{h}}^{*}$: the
dot action of $w$ on $\lambda$.

(14)$C_{0},C_{I}$: the first dominant alcove of $X(T)$ for the action of
$W_{p}$ given by $\\{\lambda\in X(T)\otimes_{\mathbb{Z}}\mathbb{R}\mid
0\leq\langle\lambda+\rho,\alpha^{\vee}\rangle\leq p\mbox{ for all }\alpha\in
R^{+}\\}$, the dominant alcove of $X(T)\otimes_{\mathbb{Z}}\mathbb{R}$ for the
action of $W_{I,p}$ given by $\\{\lambda\in
X(T)\otimes_{\mathbb{Z}}\mathbb{R}\mid
0\leq\langle\lambda+\rho,\alpha^{\vee}\rangle\leq p\mbox{ for all }\alpha\in
R^{+}\cap\mathbb{Z}I\\}$.

(15) $|S|$: the number of elements in a finite set $S$.

###  1.2. Assumptions

Let $G$ be a connected reductive algebraic group over $k$ and
${\mathfrak{g}}=\mbox{\rm Lie}(G)$, satisfying the following three hypotheses
as in [10, §6.3]:

  1. (H1)

The derived group $\mathcal{D}G$ of $G$ is simply connected;

  2. (H2)

The prime $p$ is good for ${\mathfrak{g}}$;

  3. (H3)

There exists a $G$-invariant non-degenerate bilinear form on ${\mathfrak{g}}$,

Throughout this paper, we always assume that $\chi({\mathfrak{b}}^{+})=0$,
i.e. $\chi$ is nilpotent.

###  1.3. Baby Verma modules

For $\chi\in{\mathfrak{g}}^{*}$, set
$\Lambda_{\chi}=\\{\lambda\in\mathfrak{h}^{*}\mid\lambda(h)^{p}-\lambda(h^{[p]})=\chi(h)^{p},~{}\forall~{}h\in\mathfrak{h}\\}$.
When $\chi$ is nilpotent,
$\Lambda_{\chi}=\Lambda_{0}:=\\{\lambda\in{\mathfrak{h}}^{*}\mid\lambda(h)^{p}=\lambda(h^{[p]}),\forall
h\in{\mathfrak{h}}\\}$.

Since $\chi({\mathfrak{b}}^{+})=0$, any simple $U_{0}({\mathfrak{h}})$-module
is one-dimensional $k_{\lambda}=k$ for $\lambda\in\Lambda_{0}$, where $h\cdot
k_{\lambda}=\lambda(h)k_{\lambda}$ for any $h\in{\mathfrak{h}}$. Each
$k_{\lambda}$ can be extended to be a $U_{0}({\mathfrak{b}}^{+})$-module by
trivial ${\mathfrak{n}}^{+}$-action. Hence, one can get an induced module
${Z}_{\chi}(\lambda)=U_{\chi}({\mathfrak{g}})\otimes_{U_{0}(\mathfrak{b}^{+})}k_{\lambda}$
which is called a baby Verma module. Each simple
$U_{\chi}({\mathfrak{g}})$-module is the homomorphic image of some baby Verma
module ${Z}_{\chi}(\lambda),\lambda\in\Lambda_{0}$ (cf. [10]).

###  1.4. Standard Levi form

We say a $p$-character $\chi$ has standard Levi form if
$\chi({\mathfrak{b}}^{+})=0$ and there exists a subset $I$ of the set $\Pi$ of
all simple roots such that $\chi({\mathfrak{g}}_{-\alpha})\neq
0,\forall\alpha\in{I}$ and
$\chi({\mathfrak{g}}_{-\alpha})=0,\forall\alpha\in{R^{+}\backslash I}$(cf. [8]
or [10, §10]).

When $I=\Pi$, $\chi$ is called regular nilpotent (cf. [10, §10]). When
$I=\emptyset$ which implies $\chi=0$,
$U_{\chi}({\mathfrak{g}})=U_{0}({\mathfrak{g}})$ is the restricted enveloping
algebra of ${\mathfrak{g}}$. Let $W_{I}$ be the Weyl group generated by all
the $s_{\alpha}$ with $\alpha\in I$. Denote by $w_{I}$ the longest element in
$W_{I}$ and $w_{0}$ the longest element in $W$. Set $w^{I}=w_{I}w_{0}$.

From now on, we will always assume that $\chi$ has standard Levi form.

###  1.5. Twisted baby Verma modules

The induced module

${Z}^{w^{I}}_{\chi}(\lambda):=U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}(w^{I}\mathfrak{b}^{+})}k_{\lambda},\lambda\in\Lambda_{0}$

is called a twist baby Verma module (cf. [10, §11.2]).

Let
${{Z}}_{\chi,I}(\lambda)=U_{\chi}({\mathfrak{g}}_{I})\otimes_{U_{\chi}({\mathfrak{g}}_{I}\bigcap\mathfrak{b}^{+})}k_{\lambda},\lambda\in\Lambda_{0}$.
Then $\chi|_{{\mathfrak{g}}_{I}}$ is regular nilpotent. It follows from [7,
§4.2/3] that ${{Z}}_{\chi,I}(\lambda)$ is a simple
$U_{\chi}({\mathfrak{g}}_{I})$-module. Then ${{Z}}_{\chi,I}(\lambda)$ can be
regarded as a $U_{\chi}({\mathfrak{p}}_{I})$(resp.
$U_{\chi}({\mathfrak{p}}_{I}^{\prime})$)-module with trivial
${\mathfrak{u}}^{+}$(resp. ${\mathfrak{u}}^{-}$)-action. We have

$Z_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I})}{{Z}}_{\chi,I}(\lambda)\,\mbox{and}\,Z^{w^{I}}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I}^{\prime})}{{Z}}_{\chi,I}(\lambda).$

###  1.6. Standard modules

Let $Q_{\chi,I}(\lambda),\lambda\in\Lambda_{0}$ be the projective cover of the
$U_{\chi}({\mathfrak{g}}_{I})$-module ${{Z}}_{\chi,I}(\lambda)$. Then
$[Q_{\chi,I}(\lambda):{{Z}}_{\chi,I}(\lambda)]=|W_{I}.\lambda|$ and
$Q_{\chi,I}(\lambda)$ has a filtration with quotients isomorphic to
$Z_{\chi,I}(\lambda)$ (cf. [10, §10.10]). Set

$Q^{I}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I})}{{Q}}_{\chi,I}(\lambda)\,\,\text{and}\,\,Q^{w^{I}}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I}^{\prime})}{{Q}}_{\chi,I}(\lambda).$

Then $Q^{I}_{\chi}(\lambda)$ (resp. $Q^{w^{I}}_{\chi}(\lambda)$) has a
filtration with sub-quotients isomorphic to $Z_{\chi}(\lambda)$ (resp.
$Z_{\chi}^{w^{I}}(\lambda)$) (cf. [10, §10]).

We call $Q^{I}_{\chi}(\lambda)$ (resp. $Q^{w^{I}}_{\chi}(\lambda)$) standard
(resp. costandard) modules, and $Z_{\chi}(\lambda)$ (resp.
${Z}^{w^{I}}_{\chi}(\lambda)$) proper standard (resp. proper costandard)
modules.

Let $Q_{\chi}(\lambda)$ ($\lambda\in\Lambda_{0}$) be the projective cover of
$L_{\chi}(\lambda)$. Then $Q_{\chi}(\lambda)$ has a filtration with sub-
quotient factors isomorphic to $Q^{I}_{\chi}(\lambda)$, and has a filtration
with sub-quotient factors isomorphic to $Q^{w^{I}}_{\chi}(\lambda)$ (cf. [11,
Prop. 1.21]).

###  1.7. Loewy series

Suppose that $M$ is a module of finite length (over some algebra), the radical
series of $M$ is a filtration

$0=\mbox{rad}^{t}M\subseteq\mbox{rad}^{t-1}M\subseteq...\subseteq\mbox{rad}^{0}M=M,$

such that $~{}\mbox{rad}^{j}M=\mbox{rad}(\mbox{rad}^{j-1}M)$ ,
$~{}\\{\mbox{rad}^{r}M,0\leq r<+\infty\\}$ is called the radical series of
$M$. Then $\mbox{rad}^{j}M/\mbox{rad}^{j+1}M,\forall j$ is semisimple. We have
$\mbox{rad}^{1}M=\mbox{rad}M$.

The socle series of $M$ is a filtration

$0=\mbox{soc}^{0}M\subseteq\mbox{soc}^{1}M\subseteq...\subseteq\mbox{soc}^{t}M=M,$

such that
$\texttt{soc}(M/\mbox{soc}^{j-1}M)=\mbox{soc}^{j}M/\mbox{soc}^{j-1}M$ ,
$\\{\mbox{soc}^{r}M,0\leq r<+\infty\\}$ is called the socle series of $M$.
Then $\mbox{soc}^{j}M/\mbox{soc}^{j-1}M$ is semisimple for any $j$. We have
$\mbox{soc}^{1}M=\mbox{soc}M$.

If $0=M^{0}\subseteq M^{1}\subseteq...\subseteq M^{n}=M$ is a filtration with
$M^{j}/M^{j-1}$ is semisimple for any $j$, then $\mbox{rad}^{n-j}M\subset
M^{j}\subset\mbox{soc}^{j}M$ for any $j$.

The length of the socle series equals the length of the radical series. Their
common length is called the Loewy length of $M$, and it is denoted by $ll(M)$.
Define $\mbox{head}^{j}M=M/\mbox{rad}^{j}M$. The head of $M$ is also called
the cap of $M$. Then $\mbox{head}M=\mbox{head}^{1}M$.

###  1.8. Graded $U_{\chi}({\mathfrak{g}})$-modules

In [10, §11], an $X(T)/\mathbb{Z}I$-graded $U_{\chi}({\mathfrak{g}})$-module
category $\mathcal{C}$ was introduced by Jantzen. For $\lambda\in X(T)$,
denote by $\widehat{{L}}_{\chi}(\lambda)$, $\widehat{{Z}}_{\chi}(\lambda)$ and
$\widehat{Q}^{I}_{\chi}(\lambda)$ the simple module, baby Verma module and
standard module in $\mathcal{C}$, respectively.

######  Lemma 1.1.

([10, Prop. 11.9]) Let $\lambda,\mu\in X(T)$. Then
$\widehat{{L}}_{\chi}(\lambda)\simeq\widehat{{L}}_{\chi}(\mu)$ if and only if
$\widehat{{Z}}_{\chi}(\lambda)\simeq\widehat{{Z}}_{\chi}(\mu)$, if and only if
$\mu\in W_{I,p}\cdot\lambda$.

For $\widehat{M}\in\mathcal{C}$, it has a direct sum decomposition
$M=\bigoplus_{\nu\in X(T)/\mathbb{Z}I}M^{\nu+\mathbb{Z}I}$ such that
${\mathfrak{h}}.M^{\nu+\mathbb{Z}I}\subset M^{\nu+\mathbb{Z}I}$ and
$x_{\alpha}.M^{\nu++\mathbb{Z}I}\subset M^{\nu+\alpha+\mathbb{Z}I}$ for all
$\alpha\in R$.

###  1.9. Duality

By [10, § 11.15], there is an automorphism $\tau$ of $G$ satisfying that
$\tau(T)=T$, $\chi\circ\tau^{-1}=-\chi$,
$\tau(x_{\alpha})=x_{-w_{I}\alpha}\mbox{, }\tau(h_{\alpha})=h_{-w_{I}\alpha}$
and $\lambda\circ\tau^{-1}=-w_{I}(\lambda)$.

Let $M\in\mathcal{C}$. Denote by ${}^{\tau}M:=\hbox{Hom}_{\mathcal{C}}(M,k)$
the $\tau$-$dual$ of $M$ (cf. [15, §11]), in which the module structure is
defined via $(u\cdot f)(m):=-f(\tau^{-1}(u)\cdot m)$ for
$u\in{\mathfrak{g}},f\in\,^{\tau}M,m\in M.$ This duality is the same as that
in [10, §11] (which is denoted by ${}^{\tau}(-^{*})$ therein).

Set $\lambda^{w^{I}}=\lambda-(p-1)(\rho-w^{I}\rho)$. According to [10, §11.16]
and [17, Lem. 2.6], we have
$\widehat{{Z}}_{\chi}(\lambda)\cong{{}^{\tau}\widehat{{Z}}^{w^{I}}_{\chi}(\lambda^{w^{I}})}$
and
${{}^{\tau}\widehat{{Z}}_{\chi}(\lambda)}\cong\widehat{{Z}}^{w^{I}}_{\chi}(\lambda^{w^{I}})$.
This implies

(1.1)
$[\widehat{{Z}}_{\chi}(\lambda)]=[\widehat{{Z}}^{w^{I}}_{\chi}(\lambda^{w^{I}})]$

in the Grothendieck group.

Since
${}^{\tau}\widehat{Z}_{\chi,I}(\lambda)\cong\widehat{Z}_{\chi,I}(\lambda)$
(cf. [10, §11.16]) and $\widehat{Q}_{\chi,I}(\lambda)$ is the projective cover
of $\widehat{Z}_{\chi,I}(\lambda)$, we have
${}^{\tau}\widehat{Q}_{\chi,I}(\lambda)\cong\widehat{Q}_{\chi,I}(\lambda)$.
Because $\widehat{Q}^{I}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I})}\widehat{{Q}}_{\chi,I}(\lambda)$,
using the method in [10, §11.17], we can get

(1.2) $^{\tau}\widehat{Q}^{I}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}^{\prime}_{I})}\widehat{{Q}}_{\chi,I}(\lambda^{w^{I}})\cong\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}).$

Hence, we have
$ll(\widehat{Z}_{\chi}(\lambda))=ll(\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}}))$
and
$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))$.

Because $\widehat{Q}_{\chi}(\lambda)$ is the projective cover and injective
hull of $\widehat{L}_{\chi}(\lambda)$ (cf. [10, §10.9]), we have that
${}^{\tau}\widehat{Q}_{\chi}(\lambda)$ is the projective cover and injective
hull of ${}^{\tau}\widehat{L}_{\chi}(\lambda)\cong\widehat{L}_{\chi}(\lambda)$
(cf. [10, §11.16]). This means

(1.3) $^{\tau}\widehat{Q}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}(\lambda).$

###  1.10. Homomorphisms

Let $\lambda\in X(T)$. By [11, Lem. 3.11] and [17, Lem. 3.2], there exists a
unique homomorphism (up to scalars) $\varphi$ (resp. $\varphi^{\prime}$)
between $\widehat{{Z}}_{\chi}(\lambda)$ and
$\widehat{{Z}}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ (resp.
$\widehat{{Z}}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ and
$\widehat{{Z}}_{\chi}(\lambda)$) with the image of $\varphi$ (resp.
$\varphi^{\prime}$) is the socle of $\widehat{{Z}}_{\chi}(\lambda)$ (resp.
$\widehat{{Z}}^{w^{I}}_{\chi}(\lambda^{w^{I}})$.

###  1.11. Order

According to [10, §11.6], there exists an order “$\leq$” on $X(T)/\mathbb{Z}I$
such that $\mu\leq\nu$ if and only if there are integers $m_{\alpha}\geq 0$
with $\nu-\mu=\sum_{\alpha}m_{\alpha}\alpha+\mathbb{Z}I$ where $\alpha$ runs
over all the simple roots not in $I$. If there exist at least one
$m_{\alpha}>0$ with $\alpha\notin I$ in $\sum_{\alpha}m_{\alpha}\alpha$, we
write $\mu<\nu$.

##  2\. Quasi-simple modules

### 2.1.

We first present the following definition of filtrations of (twist) baby Verma
modules and (co)standard modules in the category $\mathcal{C}$.

######  Definition 2.1.

Let $\widehat{M}\in{\mathcal{C}}$. Then

  * (1)

$\widehat{M}$ is said to have a $\widehat{Z}_{\chi}$-filtration (resp.
$\widehat{Z}^{w^{I}}_{\chi}$-filtration) if there is a filtration
$\widehat{M}=\widehat{M}^{0}\supset\widehat{M}^{1}\supset\cdot\cdot\cdot\supset\widehat{M}^{r}=0$
of $\widehat{M}$ with
$\widehat{M}^{i-1}/\widehat{M}^{i}\cong\widehat{Z}_{\chi}(\lambda_{i})$ (resp.
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i}^{w^{I}})$) for some $\lambda_{i}\in
X(T),i=1,...,r$.

  * (2)

$\widehat{M}$ is said to have a $\widehat{Q}_{\chi}^{I}$-filtration (resp.
$\widehat{Q}_{\chi}^{w^{I}}$-filtration) if there is a filtration
$\widehat{M}=\widehat{M}^{0}\supset\widehat{M}^{1}\supset\cdot\cdot\cdot\supset\widehat{M}^{s}=0$
of $\widehat{M}$ with
$\widehat{M}^{i-1}/\widehat{M}^{i}\cong\widehat{Q}_{\chi}^{I}(\lambda_{i})$
(resp. $\widehat{Q}_{\chi}^{w^{I}}(\lambda_{i}^{w^{I}})$) for some
$\lambda_{i}\in X(T),i=1,...,s$.

Similar to [2, Lem. 2.14], we have

######  Lemma 2.2.

If
${Ext}^{1}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\widehat{Q}^{I}_{\chi}(\mu))\cong{Ext}^{1}_{\mathcal{C}}(\widehat{Q}_{\chi}^{w^{I}}(\mu^{w^{I}}),\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}}))\neq
0,$ then $\mu>\lambda$.

###### Proof.

Consider the exact sequence
$0\rightarrow\widehat{Q}^{I}_{\chi}(\mu)\rightarrow\widehat{M}\rightarrow\widehat{Q}^{I}_{\chi}(\lambda)\rightarrow
0$ in $\mathcal{C}$, recall $\widehat{Q}^{I}_{\chi}(\lambda)\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}_{I})}\widehat{Q}_{\chi,I}(\lambda)$
and the action of ${\mathfrak{u}}^{+}$ on $\widehat{Q}_{\chi,I}(\lambda)$ is
trivial where ${\mathfrak{u}}^{+}=\sum_{\alpha\in R^{+}\backslash
R^{+}_{I}}{\mathfrak{g}}_{\alpha}$, similar to [10, §11.7(1)], we have

(2.1)
$\widehat{Q}^{I}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Q}_{\chi,I}(\lambda).$

Since $\widehat{Q}_{\chi,I}(\lambda)$ is projective, as
$U_{\chi}({\mathfrak{g}}_{I})$-module, we can get the homomorphism
$\bar{\varphi}:\widehat{Q}_{\chi,I}(\lambda)\rightarrow\widehat{M}|_{{\mathfrak{g}}_{I}}$.
For $v\in\widehat{Q}_{\chi,I}(\lambda)$, then $1\otimes
v\in\widehat{Q}^{I}_{\chi}(\lambda)$ and $1\otimes
v\in\widehat{Q}^{I}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}$ .

Denote $\psi$ by the epimorphism
$\widehat{M}\rightarrow\widehat{Q}^{I}_{\chi}(\lambda)$. Let
$v^{\prime}\in\widehat{M}^{\lambda+\mathbb{Z}I}$ be an inverse image of
$1\otimes v$, then $\psi(v^{\prime})=1\otimes v$. Let
$v^{\prime}|_{{\mathfrak{g}}_{I}}=\bar{\varphi}(v)$. Note that
$x_{\alpha}.v=0,\forall x_{\alpha}\in{\mathfrak{u}}^{+}$, $\bar{\varphi}$ can
be extended to be a $U_{\chi}({\mathfrak{p}}_{I})$-homomorphism. This implies
$v^{\prime}|_{{\mathfrak{g}}_{I}}\in\widehat{M}^{{\mathfrak{u}}^{+}}$ where
$\widehat{M}^{{\mathfrak{u}}^{+}}$ is the maximal submodule of $\widehat{M}$
with with ${\mathfrak{u}}^{+}$ acting trivially (cf. [11, §1]). It is possible
that $x_{\alpha}.v^{\prime}\neq 0$ for some $x_{\alpha}\in{\mathfrak{u}}^{+}$.

Similarly to [10, §11.19(3)], we can write

$\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)}\texttt{pr}_{\nu}(\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})$

where for each $\nu$ all composition factors of
$\texttt{pr}_{\nu}(\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})$ are
the form $\widehat{Z}_{\chi,I}(\mu)$ with $\mu\in W_{I,p}.\nu$. Then there
exists a weight $\lambda^{\prime}\in C_{I}\cap X(T)$ such that $\lambda\in
W_{I,p}.\lambda^{\prime}$, and
$\texttt{pr}_{\lambda^{\prime}}(\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})\cong\widehat{Q}_{\chi,I}(\lambda)$(cf.
(2.1)).

So we have

(2.2)
$\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)\setminus\lambda^{\prime}}\texttt{pr}_{\nu}(\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})\oplus\widehat{Q}_{\chi,I}(\lambda).$

If $x_{\alpha}.v^{\prime}=0,\forall x_{\alpha}\in{\mathfrak{u}}^{+}$, there is
a homomorphism $\varphi:\widehat{Q}^{I}_{\chi}(\lambda)\rightarrow\widehat{M}$
with $\varphi(u\otimes v)=uv^{\prime}$ and
$\varphi|_{{\mathfrak{g}}_{I},\widehat{Q}_{\chi,I}(\lambda)}=\bar{\varphi}$
where $\varphi|_{{\mathfrak{g}}_{I},\widehat{Q}_{\chi,I}}$ is denoted by
$\varphi$ being considered as ${\mathfrak{g}}_{I}$-module homomorphism and
restricting to ${\mathfrak{g}}_{I}$-submodule $\widehat{Q}_{\chi,I}(\lambda)$.
Then $\psi\circ\varphi(u\otimes
v)=\psi(uv^{\prime})=u\psi(v^{\prime})=u\otimes v$. Then $\varphi$ split the
above exact sequence. So, if the above exact sequence does not split, there
must exists $x_{\alpha}\in{\mathfrak{u}}^{+}$ such that
$x_{\alpha}.v^{\prime}\neq 0$. This implies that $\lambda+\alpha$ is a weight
of $M$, thereby is a weight of $\widehat{Q}^{I}_{\chi}(\mu)$. Hence
$\mu>\lambda$. ∎

###  2.2. Filtrations in $\mathcal{C}$

According to [11, Prop. 1.21], $\widehat{Q}_{\chi}(\lambda)$ has both a
$\widehat{Q}^{I}_{\chi}$\- and a $\widehat{Q}_{\chi}^{w^{I}}$-filtration.

According to Lemma 2.2 and [2, Lem. 2.14], we have the following
$\widehat{Q}^{I}_{\chi}$ (resp. $\widehat{Q}_{\chi}^{w^{I}}$)-filtration of
$\widehat{Q}_{\chi}(\lambda)$

$\widehat{Q}_{\chi}(\lambda)=\widetilde{Q}^{0}_{\chi}(\lambda)\supset\widetilde{Q}^{1}_{\chi}(\lambda)\supset\cdot\cdot\cdot\supset\widetilde{Q}^{r}_{\chi}(\lambda)=0$

with

(2.3)
$\widetilde{Q}^{i-1}_{\chi}(\lambda)/\widetilde{Q}^{i}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}^{I}(\lambda_{i})^{m_{i}}\,(\text{resp.}\,\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r-(i-1)}^{w^{I}})^{m_{i}}),m_{i}=(\widehat{Q}_{\chi}(\lambda):\widehat{Q}_{\chi}^{I}(\lambda_{i}))$

and

(2.4)
$j>i,\,\,\text{if}\,\,\lambda_{j}>\lambda_{i}\,\,(\text{resp.}\,\lambda_{j}<\lambda_{i}).$

### 2.3.

By [10, §11.19],
$(\widehat{Q}_{\chi}(\lambda):\widehat{Q}_{\chi}^{I}(\mu))=[\widehat{Z}_{\chi}(\mu):\widehat{L}_{\chi}(\lambda)]$.
Similarly, one can get
$(\widehat{Q}_{\chi}(\lambda):\widehat{Q}_{\chi}^{w^{I}}(\mu))=[\widehat{Z}_{\chi}^{w^{I}}(\mu):\widehat{L}_{\chi}(\lambda)]$.
As $\widehat{L}_{\chi}(\lambda)$ is the simple head of
$\widehat{Z}_{\chi}(\lambda)$ and the simple socle of
$\widehat{Z}_{\chi}^{w^{I}}(\lambda^{w^{I}})$ (cf. [10, §11.13]), we have
$(\widehat{Q}_{\chi}(\lambda):\widehat{Q}_{\chi}^{I}(\lambda))=(\widehat{Q}_{\chi}(\lambda):\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}}))=1$.

In the $\widehat{Q}^{I}_{\chi}$-filtration of $\widehat{Q}_{\chi}(\lambda)$,
according to (2.3), we have
$\widetilde{Q}^{0}_{\chi}(\lambda)/\widetilde{Q}^{1}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}^{I}(\lambda_{1})\cong\widehat{Q}_{\chi}^{I}(\lambda)$.
Furthermore, as $\widehat{L}_{\chi}(\lambda)$ is the socle of
$\widehat{Q}_{\chi}(\lambda)$, we have $\widehat{L}_{\chi}(\lambda)$ is the
socle of $\widehat{Q}_{\chi}^{I}(\lambda_{r})$ which implies
$\widehat{L}_{\chi}(\lambda)$ is the socle of
$\widehat{Z}_{\chi}(\lambda_{r})$. Then
$(\widehat{Q}_{\chi}(\lambda):\widehat{Q}_{\chi}^{I}(\lambda_{r}))=[\widehat{Z}_{\chi}(\lambda_{r}):\widehat{L}_{\chi}(\lambda)]=1$
and
$\widetilde{Q}^{r-1}_{\chi}(\lambda)/\widetilde{Q}^{r}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}^{I}(\lambda_{r})$.

Similarly, in the $\widehat{Q}^{w^{I}}_{\chi}$-filtration of
$\widehat{Q}_{\chi}(\lambda)$, we get
$\widetilde{Q}^{r-1}_{\chi}(\lambda)/\widetilde{Q}^{r}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}}_{1})\cong\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}})$
and
$\widetilde{Q}^{0}_{\chi}(\lambda)/\widetilde{Q}^{1}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}}_{r})$.
Hence $\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}}_{r})$.

### 2.4.

As $\widehat{Q}_{\chi}^{I}(\lambda)$ is the quotient module of
$\widehat{Q}_{\chi}(\lambda)$, there exists a submodule $M^{I}$ with
$\widehat{Q}_{\chi}^{I}(\lambda)\cong\widehat{Q}_{\chi}(\lambda)/M^{I}$.

In the $\widehat{Q}_{\chi}^{w^{I}}$-filtration of
$\widehat{Q}_{\chi}(\lambda)$,
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ is the quotient module of
$\widehat{Q}_{\chi}(\lambda)$ (cf. §2.3). And $\widehat{L}_{\chi}(\lambda)$ is
the head of $\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ which implies
$\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$. Then there exists a
submodule $M^{w^{I}}$ with
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})\cong\widehat{Q}_{\chi}(\lambda)/M^{w^{I}}$.

Then $M^{I}+M^{w^{I}}$ is a submodule of $\widehat{Q}_{\chi}(\lambda)$. Denote
$\mathcal{L}$ by $\widehat{Q}_{\chi}(\lambda)/(M^{I}+M^{w^{I}})$. We will show
that such a $\mathcal{L}$ satisfies a certain uniqueness in the following.

### 2.5.

Based on $\widehat{L}_{\chi}(\lambda_{r})$ is a composition factor of
$\widehat{Q}_{\chi}(\lambda)$, we have $\lambda_{r}$ lies in $W_{p}.\lambda$.
When $\lambda$ is $p$-regular (the stabilizer of $\lambda$ in $W_{p}$ is
trivial), $\lambda_{r}$ is $p$-regular and
$|W_{I}.d\lambda|=|W_{I}.d\lambda_{r}|$. When $\lambda$ is not $p$-regular, it
is possible that $|W_{I}.d\lambda|\neq|W_{I}.d\lambda_{r}|$. Let
$\emph{m}=\hbox{min}\\{|W_{I}.d\lambda|,|W_{I}.d\lambda_{r}|\\}$. Then

######  Lemma 2.3.

Each composition factor of $\mathcal{L}$ is isomorphic to
$\widehat{L}_{\chi}(\lambda)$ and

$[\mathcal{L}]=\emph{m}[\widehat{L}_{\chi}(\lambda)].$

In particular, when $\lambda$ is $p$-regular,
$[\mathcal{L}]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$.

###### Proof.

We know $\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{Z}_{\chi}(\lambda)$ and
$[\widehat{Z}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]=1$. Then for any
composition factor $\widehat{L}_{\chi}(\mu)$ of
$\widehat{Q}_{\chi}^{I}(\lambda)$ except $\widehat{L}_{\chi}(\lambda)$, we
have $\mu<\lambda$ (cf. §1.6).

Note that $\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ and
$[\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}):\widehat{L}_{\chi}(\lambda)]=1$,
by (1.1), for any composition factor $\widehat{L}_{\chi}(\xi)$ of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ except
$\widehat{L}_{\chi}(\lambda)$, we have $\xi>\lambda$.

Furthermore, as $\widehat{Q}^{w^{I}}_{\chi}(\lambda)$ has a filtration with
sub-quotients isomorphic to $\widehat{Z}_{\chi}^{w^{I}}(\lambda)$, for any
composition factor $\widehat{L}_{\chi}(\xi)$ of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ except
$\widehat{L}_{\chi}(\lambda)$, we have $\xi>\lambda$. Thus all the composition
factors of $\widehat{Q}_{\chi}^{I}(\lambda)$ except for
$\widehat{L}_{\chi}(\lambda)$ are not the composition factors of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$. By
$(\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}):\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}))=|W_{I}.d\lambda_{r}|$,
we have $M^{w^{I}}$ contains all the composition factors of
$\widehat{Q}_{\chi}^{I}(\lambda)$ except $\widehat{L}_{\chi}(\lambda)$ which
appear $|W_{I}.d\lambda|$ times. Similarly, we have $M^{I}$ contains all the
composition factors of $\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$
except $\widehat{L}_{\chi}(\lambda)$ which appear $|W_{I}.d\lambda|$ times
because
$(\widehat{Q}_{\chi}^{I}(\lambda):\widehat{Z}_{\chi}(\lambda))=|W_{I}.d\lambda|$.

Furthermore, we have the common composition factors of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ and
$\widehat{Q}_{\chi}^{I}(\lambda)$ are
min$\\{|W_{I}.d\lambda|,|W_{I}.d\lambda_{r}|\\}=\emph{m}$ number
$\widehat{L}_{\chi}(\lambda)$s.

Recall
$\widehat{Q}_{\chi}^{I}(\lambda)\cong\widehat{Q}_{\chi}(\lambda)/M^{I}$,
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})\cong\widehat{Q}_{\chi}(\lambda)/M^{w^{I}}$,
we get $M^{I}+M^{w^{I}}$ contains all the composition factors of
$\widehat{Q}_{\chi}(\lambda)$ except m times $\widehat{L}_{\chi}(\lambda)$s.
Then each composition factor of $\mathcal{L}$ is isomorphic to
$\widehat{L}_{\chi}(\lambda)$, and
$[\mathcal{L}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]=\emph{m}$. Hence,
$[\mathcal{L}]=\emph{m}[\widehat{L}_{\chi}(\lambda)]$.

In particular, when $\lambda$ is $p$-regular which implies
$|W_{I}.d\lambda|=|W_{I}.d\lambda_{r}|$, we have
$[\mathcal{L}]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$. ∎

By the above proof, we can get

######  Corollary 2.4.

$\mathcal{L}$ is a quotient module of $\widehat{Q}_{\chi}^{I}(\lambda)$ and
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$.

###  2.6. Quasi-simple modules

Note that
${}^{\tau}\widehat{Q}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}(\lambda)$,
${}^{\tau}\widehat{Q}_{\chi}^{I}(\lambda)\cong\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$
(cf. (1.2); (1.3)), $\mathcal{L}$ is the quotient module of
$\widehat{Q}_{\chi}^{I}(\lambda)$ (resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$) yields
${}^{\tau}\mathcal{L}$ is a submodule of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ (resp.
$\widehat{Q}^{I}_{\chi}(\lambda_{r})$) with
$[^{\tau}\mathcal{L}]=[\mathcal{L}]=\emph{m}[\widehat{L}_{\chi}(\lambda)]$(cf.
Lemma 2.3 and §1.9). Hence, ${}^{\tau}\mathcal{L}$ is a submodule
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})\cap\widehat{Q}^{I}_{\chi}(\lambda_{r})$.

Furthermore, as $\widehat{Q}^{I}_{\chi}(\lambda_{r})$(resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$) has a filtration with sub-
quotients isomorphic to $\widehat{Z}_{\chi}(\lambda_{r})$ (resp.
$\widehat{Z}_{\chi}^{w^{I}}(\lambda)$), for any composition factor
$\widehat{L}_{\chi}(\xi)$ (resp. $\widehat{L}_{\chi}(\zeta)$) of
$\widehat{Q}^{I}_{\chi}(\lambda_{r})$ (resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$) except
$\widehat{L}_{\chi}(\lambda)$, we have $\xi>\lambda$ (resp. $\zeta<\lambda$).
Hence,
$[\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})\cap\widehat{Q}^{I}_{\chi}(\lambda_{r}):\widehat{L}_{\chi}(\lambda)]\leq\emph{m}[\widehat{L}_{\chi}(\lambda)]$.

Based on the above arguments, we can get
${}^{\tau}\mathcal{L}=\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})\cap\widehat{Q}^{I}_{\chi}(\lambda_{r})$.
This yields ${}^{\tau}\mathcal{L}$ is unique under isomorphism. Hence
$\mathcal{L}$ is unique under isomorphism. And we can give the following
definition

######  Definition 2.5.

The module
$\mathcal{L}_{\chi}(\lambda):=\widehat{Q}_{\chi}(\lambda)/(M^{I}+M^{w^{I}})$
is called a quasi-simple module.

According to the above arguments, we can get

######  Remark 2.6.

If a quotient module $\mathcal{M}$ of $\widehat{Q}_{\chi}^{I}(\lambda)$ (resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$) satisfies
$[\mathcal{M}]=\emph{m}[\widehat{L}_{\chi}(\lambda)]$,
$\mathcal{L}_{\chi}(\lambda)$ is unique under isomorphism and is a quotient
module of $\widehat{Q}_{\chi}^{I}(\lambda)$ (resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$) with
$[\mathcal{L}_{\chi}(\lambda)]=\emph{m}[\widehat{L}_{\chi}(\lambda)]$ yields
$\mathcal{M}\cong\mathcal{L}_{\chi}(\lambda)$.

######  Lemma 2.7.

(i). The quasi-simple module $\mathcal{L}_{\chi}(\lambda)$ is unique in the
sense of Remark 2.6.

(ii). $\mathcal{L}_{\chi}(\lambda)$ is a quotient module of
$\widehat{Q}_{\chi}^{I}(\lambda)$,
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ and
$\widehat{Q}_{\chi}(\lambda)$.

(iii). $\widehat{L}_{\chi}(\lambda)$ is the head of
$\mathcal{L}_{\chi}(\lambda)$.

### 2.7.

For a module $\widehat{M}$, denote by $\widehat{M}^{\lambda+\mathbb{Z}I}$ the
homogeneous part of $\widehat{M}$ with degree $\lambda+\mathbb{Z}I$ (cf. [10,
§11.4] or §1.8).

######  Lemma 2.8.

(i).When $|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$, we have

(2.5)
$\mathcal{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}(^{\tau}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Q}_{\chi,I}(\lambda).$

(ii). When $|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|$, we have

(2.6)
$\mathcal{L}_{\chi}(\lambda)^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}(^{\tau}\mathcal{L}_{\chi}(\lambda))^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}}).$

###### Proof.

When $|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$,
$[\widehat{Q}^{I}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]=[\mathcal{L}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]$
(cf. Lemma 2.7). As
$\widehat{Q}^{I}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Q}_{\chi,I}(\lambda)$
(cf. (2.1)),
$\widehat{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Z}_{\chi,I}(\lambda)$
and $\mathcal{L}_{\chi}(\lambda)$ is the quotient module of
$\widehat{Q}^{I}_{\chi}(\lambda)$, we have
$\widehat{Q}^{I}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}=\mathcal{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}$.
Furthermore, the surjective
$\widehat{Q}^{I}_{\chi}(\lambda)\twoheadrightarrow\mathcal{L}_{\chi}(\lambda)$
is an isomorphism as ${\mathfrak{g}}_{I}$-module when restricted to the
homogeneous part of degree $\lambda+\mathbb{Z}I$.

Then
$\mathcal{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}(^{\tau}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Q}_{\chi,I}(\lambda)$,
as desired.

When $|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|$, by Lemma 2.7,
$[\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}):\widehat{L}_{\chi}(\lambda)]=[\mathcal{L}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]$.
Recall $\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})\cong
U_{\chi}({\mathfrak{g}})\otimes_{U_{\chi}({\mathfrak{p}}^{\prime}_{I})}\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}})$,
then

$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}}).$

And the lowest weight of $\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ is
$\lambda_{r}^{w^{I}}$.

As $\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$, by [10, §11.7] we can get

(2.7)
$\widehat{L}_{\chi}(\lambda)^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{{Z}}_{\chi,I}(\lambda_{r}^{w^{I}}).$

This implies $\lambda_{r}^{w^{I}}$ is the lowest weight of
$\widehat{L}_{\chi}(\lambda)$ but not the weight of the other composition
factors of $\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$.

Note that $\mathcal{L}_{\chi}(\lambda)$ is the quotient module of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ and
$[\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}):\widehat{L}_{\chi}(\lambda)]=[\mathcal{L}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]$
(cf. Lemma 2.7), we can get
$\mathcal{L}_{\chi}(\lambda)^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}=\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}$.
The surjective
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})\twoheadrightarrow\mathcal{L}_{\chi}(\lambda)$
is an isomorphism as ${\mathfrak{g}}_{I}$-module when restricted to the
homogeneous part of degree $\lambda_{r}^{w^{I}}+\mathbb{Z}I$. And
$\mathcal{L}_{\chi}(\lambda)^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}}).$

Consequently, the desired ${\mathfrak{g}}_{I}$-isomorphisms in (2.6) hold. ∎

Thanks to Lemma 2.8, we have the following decompositions similar to (2.2),

(2.8) $\mbox{when
}|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|,\,\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)\setminus\lambda^{\prime}}\texttt{pr}_{\nu}(\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})\oplus\widehat{Q}_{\chi,I}(\lambda)$

where $\lambda^{\prime}\in C_{I}\cap X(T)$ and $\lambda\in
W_{I,p}.\lambda^{\prime}$. And

(2.9) $\mbox{when
}|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|,\,\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)\setminus\lambda^{\prime}}\texttt{pr}_{\nu}(\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})\oplus\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})$

where $\lambda^{\prime}\in C_{I}\cap X(T)$ and $\lambda_{r}^{w^{I}}\in
W_{I,p}.\lambda^{\prime}$.

### 2.8.

######  Proposition 2.9.

${}^{\tau}\mathcal{L}_{\chi}(\lambda)\cong\mathcal{L}_{\chi}(\lambda).$

###### Proof.

(1). When $|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$, by Lemmas 2.3 and 2.7,
${}^{\tau}\mathcal{L}_{\chi}(\lambda)$ is the submodule of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ with
$[^{\tau}\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$.

By [18, Lem.4.2], we have functorial isomorphism as follows.

(2.10)
$\Phi:\hbox{Hom}_{\mathcal{C}_{I}}(\widehat{{Q}}_{\chi,I}(\lambda),\widehat{{Q}}_{\chi,I}(\lambda))\cong\hbox{Hom}_{\mathcal{C}}(\widehat{{Q}}^{I}_{\chi}(\lambda),\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))$

where $\mathcal{C}_{I}$ is the category of $X(T)/\mathbb{Z}I$-graded
$U_{\chi}({\mathfrak{g}}_{I})$-modules.

Let
$\bar{\varphi}=\hbox{id}_{\widehat{{Q}}_{\chi,I}(\lambda)}\in\hbox{Hom}_{\mathcal{C}_{I}}(\widehat{{Q}}_{\chi,I}(\lambda),\widehat{{Q}}_{\chi,I}(\lambda))$.
Denote $\varphi=\Phi(\bar{\varphi})$. According to (2.2), we have

$\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)\setminus\lambda^{\prime}}\texttt{pr}_{\nu}(\widehat{Q}^{I}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})\oplus\widehat{Q}_{\chi,I}(\lambda).$

where $\lambda^{\prime}\in C_{I}\cap X(T)$ and $\lambda\in
W_{I,p}.\lambda^{\prime}$. Then
$\Phi^{-1}(\varphi)=\varphi|_{\widehat{Q}_{\chi,I}(\lambda)}=\bar{\varphi}$
where $\varphi|_{\widehat{Q}_{\chi,I}(\lambda)}$ is denoted by $\varphi$ being
considered as ${\mathfrak{g}}_{I}$-module homomorphism and restricting to
${\mathfrak{g}}_{I}$-submodule $\widehat{Q}_{\chi,I}(\lambda)$. This implies
$\hbox{Im}\bar{\varphi}=\hbox{Im}\varphi|_{\widehat{Q}_{\chi,I}(\lambda)}=\widehat{Q}_{\chi,I}(\lambda)$.

As
$[\widehat{Q}_{\chi,I}(\lambda):\widehat{Z}_{\chi,I}(\lambda)]=|W_{I}.d\lambda|$
and
$\widehat{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Z}_{\chi,I}(\lambda)$
(cf. [10, §11.7]), we have $\lambda$ is the weight of
$\widehat{L}_{\chi}(\lambda)$ but not the weight of the other composition
factors of $\widehat{Q}_{\chi,I}(\lambda)$. By
$[\widehat{Q}^{I}_{\chi}(\lambda)]=[\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})]$,
$\lambda$ is the weight of $\widehat{L}_{\chi}(\lambda)$ but not the weight of
the other composition factors of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$. Furthermore, because
$\hbox{Im}\varphi$ is the submodule of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$,
$\hbox{Im}\varphi|_{\widehat{Q}_{\chi,I}(\lambda)}=\widehat{Q}_{\chi,I}(\lambda)$
yields $[\hbox{Im}\varphi:\widehat{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|$.

Recall ${}^{\tau}\mathcal{L}_{\chi}(\lambda)$ is the submodule of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ with
$[^{\tau}\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$,
it follows that ${}^{\tau}\mathcal{L}_{\chi}(\lambda)$ is the submodule of
$\hbox{Im}\varphi$.

Note that
$\widehat{{Q}}^{I}_{\chi}(\lambda)/\hbox{ker}(\varphi)\cong\hbox{Im}(\varphi)$
and $\mathcal{L}_{\chi}(\lambda)$ is a quotient module of
$\widehat{{Q}}^{I}_{\chi}(\lambda)$, we have $\mathcal{L}_{\chi}(\lambda)$ is
a quotient module of $\hbox{Im}(\varphi)$. As
$[\hbox{Im}\varphi:\widehat{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|$, one can get

$\mathcal{L}_{\chi}(\lambda)\cong\hbox{Im}(\varphi)\cong^{\tau}\mathcal{L}_{\chi}(\lambda).$

(2). When $|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|$, by Lemmas 2.3 and 2.7,
$\mathcal{L}_{\chi}(\lambda)$ is the quotient module of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ with
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda_{r}|[\widehat{L}_{\chi}(\lambda)]$.
And ${}^{\tau}\mathcal{L}_{\chi}(\lambda)$ is the submodule of
$\widehat{Q}^{I}_{\chi}(\lambda_{r})$.

Using the same method as [18, Lem.4.2], one can prove

$\Phi^{\prime}:\hbox{Hom}_{\mathcal{C}_{I}}(\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}}),\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}}))\cong\hbox{Hom}_{\mathcal{C}}(\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}),\widehat{{Q}}^{I}_{\chi}(\lambda_{r})).$

Similar to (2.2), we have

$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)\setminus\lambda^{\prime}}\texttt{pr}_{\nu}(\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})|_{{\mathfrak{g}}_{I}})\oplus\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}}).$

where $\lambda^{\prime}\in C_{I}\cap X(T)$ and $\lambda_{r}^{w^{I}}\in
W_{I,p}.\lambda^{\prime}$.

Let
$\bar{\varphi}^{\prime}=\hbox{id}_{\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}})}\in\hbox{Hom}_{\mathcal{C}_{I}}(\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}}),\widehat{{Q}}_{\chi,I}(\lambda_{r}^{w^{I}}))$
and $\varphi^{\prime}=\Phi^{\prime}(\bar{\varphi}^{\prime})$. Then
$\Phi^{\prime-1}(\varphi^{\prime})=\varphi^{\prime}|_{\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})}=\bar{\varphi}^{\prime}$
where $\varphi^{\prime}|_{\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})}$ is
denoted by $\varphi^{\prime}$ being considered as ${\mathfrak{g}}_{I}$-module
homomorphism and restricting to ${\mathfrak{g}}_{I}$-submodule
$\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})$. This implies
$\hbox{Im}\bar{\varphi}^{\prime}=\hbox{Im}\varphi^{\prime}|_{\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})}=\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})$.

By calculating,
$\langle\lambda_{r}^{w^{I}},\alpha_{r}^{\vee}\rangle=\langle\lambda_{r},\alpha^{\vee}\rangle,\forall\alpha\in
R_{I}$. Hence,
$\widehat{{Q}}_{\chi,I}(\lambda_{r})\cong_{{\mathfrak{g}}_{I}}\widehat{{Q}}_{\chi,I}(\lambda^{w^{I}}_{r})$
and
$[\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}}):\widehat{Z}_{\chi,I}(\lambda_{r}^{w^{I}})]=|W_{I}.d\lambda_{r}^{w^{I}}|=|W_{I}.d\lambda_{r}|$.

As
$\widehat{L}_{\chi}(\lambda)^{\lambda_{r}^{w^{I}}+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Z}_{\chi,I}(\lambda_{r}^{w^{I}})$
(cf. (2.7)) and $\lambda_{r}^{w^{I}}$ is the weight of
$\widehat{L}_{\chi}(\lambda)$ but not the weight of the other composition
factors of $\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ (cf. the proof of
Lemma 2.8), we get $\lambda_{r}^{w^{I}}$ is the weight of
$\widehat{L}_{\chi}(\lambda)$ but not the weight of the other composition
factors of $\widehat{Z}_{\chi}(\lambda_{r})$ because of
$[\widehat{Z}_{\chi}(\lambda_{r})]=[\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})]$.
Furthermore,
$[\hbox{Im}\varphi^{\prime}:\widehat{L}_{\chi}(\lambda)]=|W_{I}.d\lambda_{r}|$
because $\hbox{Im}\varphi^{\prime}$ is the submodule of
$\widehat{Q}^{I}_{\chi}(\lambda_{r})$ and
$\hbox{Im}\varphi^{\prime}|_{{\mathfrak{g}}_{I},\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})}=\widehat{Q}_{\chi,I}(\lambda_{r}^{w^{I}})$.

According to the above arguments, similar to the case
$|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$, we can prove
$\mathcal{L}_{\chi}(\lambda)\cong\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})/\hbox{ker}(\varphi^{\prime})\cong\hbox{Im}(\varphi^{\prime})\cong{{}^{\tau}\mathcal{L}_{\chi}(\lambda)}$.

Based on (1) and (2), we complete the proof. ∎

Then as a consequence, we have

######  Corollary 2.10.

There exists a submodule in $\widehat{Q}_{\chi}(\lambda)$ isomorphic to
$\mathcal{L}_{\chi}(\lambda)$.

###### Proof.

For the quotient module $\mathcal{L}_{\chi}(\lambda)$ of
$\widehat{Q}_{\chi}(\lambda)$, ${}^{\tau}\mathcal{L}_{\chi}(\lambda)$ is a
submodule of ${}^{\tau}\widehat{Q}_{\chi}(\lambda)$. As
${}^{\tau}\widehat{Q}_{\chi}(\lambda)\cong\widehat{Q}_{\chi}(\lambda)$ (cf.
(1.3)), by Proposition 2.9, $\mathcal{L}_{\chi}(\lambda)$ is a submodule of
$\widehat{Q}_{\chi}(\lambda)$. ∎

Based on Lemma 2.7 and Corollary 2.10, we have

######  Theorem 2.11.

The quasi-simple module $\mathcal{L}_{\chi}(\lambda)$ is not only a quotient
module but also isomorphism to a submodule of $\widehat{Q}_{\chi}(\lambda)$.

Theorem 2.11 implies $\mathcal{L}_{\chi}(\lambda)$ has a simple head and a
simple socle.

##  3\. Cohomological properties of quasi-simple modules

### 3.1.

Keeping the notations as above. We can get

######  Lemma 3.1.

(i) If $|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$, $\hbox{Ext
}^{1}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0.$

(ii) If $|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|$, $\hbox{Ext
}^{1}_{\mathcal{C}}(\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}),\mathcal{L}_{\chi}(\lambda))=0.$

###### Proof.

By Lemma 2.4 and Proposition 2.9, $\mathcal{L}_{\chi}(\lambda)$ is a submodule
of $\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}})$.

Set $\mathcal{L}^{w^{I}}$ is the quotient module of
$\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}})$ with
$\mathcal{L}^{w^{I}}\cong\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}})/\mathcal{L}_{\chi}(\lambda)$.
When $|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$, by Lemma 2.3, we have
$\widehat{L}_{\chi}(\lambda)$ is not the composition factor of
$\mathcal{L}^{w^{I}}$ and the highest weight of $\mathcal{L}^{w^{I}}$ is
smaller than $\lambda$. Thus
$\hbox{Hom}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\mathcal{L}^{w^{I}})=0$.

The short exact sequence
$0\rightarrow\mathcal{L}_{\chi}(\lambda)\rightarrow\widehat{Q}_{\chi}^{w^{I}}(\lambda^{w^{I}})\rightarrow\mathcal{L}^{w^{I}}\rightarrow
0$ induces a long exact sequence

$0\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}_{\chi}^{I}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\mathcal{L}^{w^{I}})\rightarrow$
$\hbox{Ext}_{\mathcal{C}}^{1}(\widehat{Q}_{\chi}^{I}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Ext}_{\mathcal{C}}^{1}(\widehat{Q}_{\chi}^{I}(\lambda),\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))\rightarrow\cdot\cdot\cdot.$

Since
$\hbox{Ext}_{\mathcal{C}}^{1}(\widehat{Q}_{\chi}^{I}(\lambda),\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))=0$
(cf. [21, Coro. 2.1]) and
$\hbox{Hom}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\mathcal{L}^{w^{I}})=0$,
it follows that
$\hbox{Ext}^{1}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0$.

When $|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|$, by Lemma 2.7 and Proposition
2.9, we have $\mathcal{L}_{\chi}(\lambda)$ is the submodule of
$\widehat{Q}_{\chi}^{I}(\lambda_{r})$ and
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda_{r}|[\widehat{L}_{\chi}(\lambda)]$.
Set $\mathcal{L}_{1}^{I}$ is the quotient module of
$\widehat{Q}_{\chi}^{I}(\lambda_{r})$ with
$\mathcal{L}_{1}^{I}\cong\widehat{Q}_{\chi}^{I}(\lambda_{r})/\mathcal{L}_{\chi}(\lambda)$.
Then $\widehat{L}_{\chi}(\lambda)$ is not the composition factor of
$\mathcal{L}_{1}^{I}$. As $\widehat{L}_{\chi}(\lambda)$ is the socle (resp.
head) of $\widehat{Q}_{\chi}^{I}(\lambda_{r})$ (resp.
$\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}})$), we have the highest weight
of each composition factor of $\mathcal{L}_{1}^{I}$ is higher than $\lambda$.
And
$\hbox{Hom}_{\mathcal{C}}(\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\mathcal{L}_{1}^{I})=0$.

The short exact sequence
$0\rightarrow\mathcal{L}_{\chi}(\lambda)\rightarrow\widehat{Q}_{\chi}^{I}(\lambda_{r})\rightarrow\mathcal{L}_{1}^{I}\rightarrow
0$ induces a long exact sequence

$0\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),Q_{\chi}^{I}(\lambda_{r}))\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\mathcal{L}_{1}^{I})$
$\rightarrow\hbox{Ext}_{\mathcal{C}}^{1}(\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Ext}_{\mathcal{C}}^{1}(Q_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\widehat{Q}^{I}_{\chi}(\lambda_{r}))\rightarrow\cdot\cdot\cdot.$

Since
$\hbox{Ext}_{\mathcal{C}}^{1}(Q_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\widehat{Q}^{I}_{\chi}(\lambda_{r}))=0$
(cf. [21, Coro. 2.1]) and
$\hbox{Hom}_{\mathcal{C}}(\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\mathcal{L}_{1}^{I})=0$,
it follows that
$\hbox{Ext}^{1}_{\mathcal{C}}(\widehat{Q}_{\chi}^{w^{I}}(\lambda_{r}^{w^{I}}),\mathcal{L}_{\chi}(\lambda))=0$.

∎

Applying the duality $"\tau"$ on (i) and (ii) of Lemma 3.1, we can get

$\hbox{Ext
}^{1}_{\mathcal{C}}(\mathcal{L}_{\chi}(\lambda),\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))=0,\mbox{if
}|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|;$ $\hbox{Ext
}^{1}_{\mathcal{C}}(\mathcal{L}_{\chi}(\lambda),\widehat{Q}^{I}_{\chi}(\lambda_{1}))=0,\mbox{if
}|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|.$

Based on Lemma 3.1, we can get

######  Theorem 3.2.

$\hbox{Ext
}_{\mathcal{C}}^{1}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0.$

###### Proof.

(1). If $|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$,
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$(cf.
Lemma 2.7). Set $\mathcal{L}^{I}$ is the submodule of
$\widehat{Q}^{I}_{\chi}(\lambda)$ with
$\mathcal{L}_{\chi}(\lambda)\cong\widehat{Q}^{I}_{\chi}(\lambda)/\mathcal{L}^{I}$.
Then $\mathcal{L}^{I}$ does not contain $\widehat{L}_{\chi}(\lambda)$ as a
composition factor. This implies
$\hbox{Hom}_{\mathcal{C}}(\mathcal{L}^{I},\mathcal{L}_{\chi}(\lambda))=0$.
There exists a long exact sequence

$0\rightarrow\hbox{Hom}_{\mathcal{C}}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Hom}_{\mathcal{C}}(\mathcal{L}^{I},\mathcal{L}_{\chi}(\lambda))\rightarrow$
$\hbox{Ext}_{\mathcal{C}}^{1}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Ext}_{\mathcal{C}}^{1}(\widehat{Q}_{\chi}^{I}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\cdot\cdot\cdot.$

As $\hbox{Hom}_{\mathcal{C}}(\mathcal{L}^{I},\mathcal{L}_{\chi}(\lambda))=0$
and
$\hbox{Ext}^{1}_{\mathcal{C}}(\widehat{Q}^{I}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0$
(cf. Lemma 3.1), we have
$\hbox{Ext}_{\mathcal{C}}^{1}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0.$

(2). If $|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|$, we have
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda_{r}|[\widehat{L}_{\chi}(\lambda)]$
(cf. Lemma 2.7). As $\mathcal{L}_{\chi}(\lambda)$ is the quotient module of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$, we can denote
$\mathcal{L}_{1}^{w^{I}}$ by the submodule of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ with
$\mathcal{L}_{\chi}(\lambda)\cong\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})/\mathcal{L}_{1}^{w^{I}}$.
Then $\mathcal{L}_{1}^{w^{I}}$ does not contain $\widehat{L}_{\chi}(\lambda)$
as composition factor and
$\hbox{Hom}_{\mathcal{C}}(\mathcal{L}_{1}^{w^{I}},\mathcal{L}_{\chi}(\lambda))=0$.
There exists a long exact sequence

$0\rightarrow\hbox{Hom}_{\mathcal{C}}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Hom}_{\mathcal{C}}(\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Hom}_{\mathcal{C}}(\mathcal{L}_{1}^{w^{I}},\mathcal{L}_{\chi}(\lambda))\rightarrow$
$\hbox{Ext}_{\mathcal{C}}^{1}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))\rightarrow\hbox{Ext}_{\mathcal{C}}^{1}(\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}),\mathcal{L}_{\chi}(\lambda))\rightarrow\cdot\cdot\cdot.$

It follows from
$\hbox{Hom}_{\mathcal{C}}(\mathcal{L}_{1}^{w^{I}},\mathcal{L}_{\chi}(\lambda))=0$
and
$\hbox{Ext}^{1}_{\mathcal{C}}(\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}}),\mathcal{L}_{\chi}(\lambda))=0$
(cf. Lemma 3.1) that
$\hbox{Ext}_{\mathcal{C}}^{1}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0.$

Then (1) and (2) yield the theorem. ∎

##  4\. The $\mathcal{L}_{\chi}$-filtration

### 4.1.

######  Definition 4.1.

$\widehat{M}$ is said to have an $\mathcal{L}_{\chi}$-filtration if there is a
filtration
$\widehat{M}=\widehat{M}^{0}\supset\widehat{M}^{1}\supset\cdot\cdot\cdot\supset\widehat{M}^{r}=0$
of $\widehat{M}$ with
$\widehat{M}^{i-1}/\widehat{M}^{i}\cong\mathcal{L}_{\chi}(\lambda_{i})$ for
some $\lambda_{i}\in X(T),i=1,...,r$.

### 4.2.

Assume that $\widehat{M}$ has simple head (resp. socle)
$\widehat{L}_{\chi}(\lambda)$. If $\mathcal{L}_{\chi}(\lambda)$ is the
quotient module (resp. submodule) of $\widehat{M}$, we call
$\mathcal{L}_{\chi}(\lambda)$ is the quasi-head (resp. quasi-socle) of
$\widehat{M}$.

######  Theorem 4.2.

When $\lambda\in X(T)$ is $p$-regular, $\widehat{Q}^{I}_{\chi}(\lambda)$
(resp. $\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$) has an
$\mathcal{L}_{\chi}$-filtration.

###### Proof.

By (2.3) and (2.4), $\widehat{Q}_{\chi}(\lambda)$ has a
$\widehat{Q}^{I}_{\chi}$-filtration (resp.
$\widehat{Q}^{w^{I}}_{\chi}$-filtration) with the sub-quotients
$\widehat{Q}^{I}_{\chi}(\lambda_{i}),1\leq i\leq r$ (resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r-(i-1)}^{w^{I}}),1\leq i\leq r$).

Then $\widehat{Q}_{\chi}(\lambda)$ also has a $\widehat{Z}_{\chi}$-filtration
(resp. $\widehat{Z}^{w^{I}}_{\chi}$-filtration) with the sub-quotients
$\widehat{Z}_{\chi}(\lambda_{i}),1\leq i\leq r$ (resp.
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{r-(i-1)}^{w^{I}}),1\leq i\leq r$).

By §1.6, there exists a $\widehat{Z}_{\chi}$-filtration of
$\widehat{Q}_{\chi}^{I}(\lambda)$ as follows

$0=\widehat{Q}^{I}_{\chi}(\lambda)^{(|W_{I}.d\lambda|+1)}\subset\widehat{Q}^{I}_{\chi}(\lambda)^{(|W_{I}.d\lambda|)}\subset\widehat{Q}^{I}_{\chi}(\lambda)^{(|W_{I}.d\lambda|-1)}\subset\dots\subset\widehat{Q}^{I}_{\chi}(\lambda)^{(1)}=\widehat{Q}^{I}_{\chi}(\lambda)$

with
$\widehat{Q}^{I}_{\chi}(\lambda)^{(i)}/\widehat{Q}^{I}_{\chi}(\lambda)^{(i+1)}=\widehat{Z}_{i}(\lambda)\cong\widehat{Z}_{\chi}(\lambda),1\leq
i\leq|W_{I}.d\lambda|$. Then
$\widehat{Z}_{1}(\lambda)\cong\widehat{Q}^{I}_{\chi}(\lambda)^{(1)}/\widehat{Q}^{I}_{\chi}(\lambda)^{(2)}$
is the quotient module of $\widehat{Q}^{I}_{\chi}(\lambda)$.

Hence, $\widehat{Z}_{1}(\lambda)$ is a quotient module of
$\widehat{Q}_{\chi}(\lambda)$ with
$\widehat{Z}_{1}(\lambda)\cong\widehat{Q}_{\chi}(\lambda)/\widehat{Z}$ where
$\widehat{Z}$ is a submodule of $\widehat{Q}_{\chi}(\lambda)$. In
$\widehat{Q}_{\chi}(\lambda)$, as $\widehat{Q}_{\chi}(\lambda)$ has a
$\widehat{Z}^{w^{I}}_{\chi}$-filtration, each composition factor of
$\widehat{Z}_{1}(\lambda)$ is also the composition factor of some
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i}^{w^{I}})$ where
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i}^{w^{I}})$ is a sub-quotient factor in
the $\widehat{Z}^{w^{I}}_{\chi}$-filtration of $\widehat{Q}_{\chi}(\lambda)$.

Let $ll(\widehat{Z}_{1}(\lambda))=N$.

(1). Denote
$\widehat{L}_{\chi}(\lambda_{(1,1)})=\widehat{L}_{\chi}(\lambda)\cong\mbox{soc}^{N}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-1}\widehat{Z}_{1}(\lambda)$
by the head of $\widehat{Z}_{1}(\lambda)$. By Lemma 2.7, the quasi-simple
module $\mathcal{L}_{\chi}(\lambda_{(1,1)})\cong\mathcal{L}_{\chi}(\lambda)$
is a quotient module of $\widehat{Q}_{\chi}^{I}(\lambda)$. When $\lambda\in
X(T)$ is $p$-regular, by Lemma 2.3,
$[\mathcal{L}_{\chi}(\lambda)]=|w_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]=[\widehat{Q}_{\chi}^{I}(\lambda):\widehat{Z}_{\chi}(\lambda)]$.
Hence, $\mathcal{L}_{\chi}(\lambda)$ consist of the heads of all
$\widehat{Z}_{i}(\lambda),1\leq i\leq|w_{I}.d\lambda|.$

(2). By §1.7,
$\mbox{soc}^{N-1}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-2}\widehat{Z}_{1}(\lambda)$
is the second layer in the structure diagram (about composition factors) of
$\widehat{Z}_{1}(\lambda)$. Hence, in the structure diagram of
$\widehat{Z}_{1}(\lambda)$ there does not exist composition factor except for
$\widehat{L}_{\chi}(\lambda)$ whose position is higher than the composition
factors of
$\mbox{soc}^{N-1}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-2}\widehat{Z}_{1}(\lambda)$.

Let
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})\in\mbox{soc}^{N-1}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-2}\widehat{Z}_{1}(\lambda)$
and $\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ be the proper
costandard module which contains $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$. We
know $\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ is a sub-quotient
factor in the $\widehat{Z}^{w^{I}}_{\chi}$-filtration of
$\widehat{Q}_{\chi}(\lambda)$.

(2.1). We claim $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ is the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i}^{w^{I}})$.

If not, suppose $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ is not the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$. Denote
$\widehat{L}_{\chi}(\mu_{i_{1}})$ by the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$. Then, in
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$,
$\mu_{i_{1}}<\lambda_{(1,2_{1})}$. Recall
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ is a sub-quotient factor
in the $\widehat{Z}^{w^{I}}_{\chi}$-filtration of
$\widehat{Q}_{\chi}(\lambda)$, the structure diagram (about composition
factors) of $\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ is a part of
$\widehat{Q}_{\chi}(\lambda)$’s. As $\widehat{L}_{\chi}(\mu_{i_{1}})$ is the
head of $\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$, the position of
$\widehat{L}_{\chi}(\mu_{i_{1}})$ is higher than
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ in the structure diagram of
$\widehat{Q}_{\chi}(\lambda)$. And $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$
is a composition factor of the submodule of $\widehat{Q}_{\chi}(\lambda)$
which contains $\widehat{L}_{\chi}(\mu_{i_{1}})$.

Suppose that $\widehat{L}_{\chi}(\mu_{i_{1}})$ is a composition factor of
$\widehat{Z}$, because $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ lies in any
submodule of $\widehat{Q}_{\chi}(\lambda)$ which contains
$\widehat{L}_{\chi}(\mu_{i_{1}})$, we have
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ is a composition factor of
$\widehat{Z}$. It contradict to
$\widehat{Z}_{1}(\lambda)\cong\widehat{Q}_{\chi}(\lambda)/\widehat{Z}$. Hence
$\widehat{L}_{\chi}(\mu_{i_{1}})$ is a composition factor of
$\widehat{Z}_{1}(\lambda)$.

In $\widehat{Z}_{1}(\lambda)$, as $\mu_{i_{1}}<\lambda_{(1,2_{1})}<\lambda$
and $\widehat{L}_{\chi}(\lambda)$ is the unique composition factor whose
position is higher than $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$, the
position of $\widehat{L}_{\chi}(\mu_{i_{1}})$ does not higher than
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ in the structure diagram of
$\widehat{Z}_{1}(\lambda)$. We know the structure diagram of
$\widehat{Z}_{1}(\lambda)$ is a part of $\widehat{Q}_{\chi}(\lambda)$’s
($\widehat{Z}_{1}(\lambda)$ is the quotient module of
$\widehat{Q}_{\chi}(\lambda)$), it contradict to that the position of
$\widehat{L}_{\chi}(\mu_{i_{1}})$ is higher than
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ in the structure diagram of
$\widehat{Q}_{\chi}(\lambda)$.

Hence, $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ is the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$. Our claim is correct.

(2.2). As $\widehat{Z}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ is a sub-
quotient factor in the $\widehat{Z}^{w^{I}}_{\chi}$-filtration of
$\widehat{Q}_{\chi}(\lambda)$, then
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ is a sub-quotient factor
in the $\widehat{Q}^{w^{I}}_{\chi}$-filtration of
$\widehat{Q}_{\chi}(\lambda)$.

Recall $M^{w^{I}}$ is the submodule of $\widehat{Q}_{\chi}(\lambda)$ with
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})\cong\widehat{Q}_{\chi}(\lambda)/M^{w^{I}}$
and $\mathcal{L}^{I}$ is the submodule of $\widehat{Q}^{I}_{\chi}(\lambda)$
with
$\mathcal{L}_{\chi}(\lambda)\cong\widehat{Q}^{I}_{\chi}(\lambda)/\mathcal{L}^{I}$.
According to $\mathcal{L}_{\chi}(\lambda)$ is the common quotient module of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ and
$\widehat{Q}^{I}_{\chi}(\lambda)$, there exists a submodule $M^{I}_{1}$ of
$M^{w^{I}}$ such that $\mathcal{L}^{I}\cong M^{w^{I}}/M^{I}_{1}$.

Note that
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})\in\mbox{soc}^{N-1}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-2}\widehat{Z}_{1}(\lambda)$
lies in the second layer in the structure diagram of
$\widehat{Z}_{1}(\lambda)$, then it also lies in the second layer in the
structure diagram of $\widehat{Q}_{\chi}(\lambda)$ because
$\widehat{Z}_{1}(\lambda)$ is the quotient module of
$\widehat{Q}_{\chi}(\lambda)$. We know
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ is not the composition factor of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ because
$\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{r}^{w^{I}})$ and
$\lambda_{(1,2_{1})}<\lambda$, Then $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$
is the quotient module of $M^{w^{I}}$.

As $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ is the head of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ and $M^{w^{I}}$ has the
$\widehat{Q}_{\chi}^{w^{I}}$-filtration, we get
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ is the quotient module
of $M^{w^{I}}$. And there exists a submodule $M_{1}^{w^{I}}$ of $M^{w^{I}}$
such that $\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})\cong
M^{w^{I}}/M_{1}^{w^{I}}$. It is easy to see
$M^{w^{I}}/(M^{I}_{1}+M_{1}^{w^{I}})$ is the common quotient module of
$\mathcal{L}^{I}$ and $\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$.

(2.3). By Lemma 2.7, when $\lambda$ is $p$-regular, quasi-simple module
$\mathcal{L}_{\chi}(\lambda)$ is a quotient module of
$\widehat{Q}^{I}_{\chi}(\lambda)$ with
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$.

As $\lambda$ is the unique weight which is higher than $\lambda_{(1,2_{1})}$
among the highest weights of all composition factors of
$\widehat{Q}^{I}_{\chi}(\lambda)$, we can get the highest weight of any
composition factor of $\mathcal{L}^{I}$ is not higher than
$\lambda_{(1,2_{1})}$. Because $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ is
the head of $\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$, we have the
highest weight of any composition factor except
$\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$ of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$ is higher than
$\lambda_{(1,2_{1})}$. It follows from $\lambda$ is $p$-regular that
$\lambda_{(1,2_{1})}$ is $p$-regular. Hence,
$M^{w^{I}}/(M^{I}_{1}+M_{1}^{w^{I}})=|W_{I}.d\lambda_{(1,2_{1})}|[\widehat{L}_{\chi}(\lambda_{(1,2_{1})})]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda_{(1,2_{1})})]$.

Note that $M^{w^{I}}/(M^{I}_{1}+M_{1}^{w^{I}})$ is the quotient module of
$\widehat{Q}^{w^{I}}_{\chi}(\lambda_{i_{1}}^{w^{I}})$, according to the quasi-
head $\mathcal{L}_{\chi}(\lambda_{(1,2_{1})})$ is unique under isomorphism
(cf. Lemma 2.7), we get $\mathcal{L}_{\chi}(\lambda_{(1,2_{1})})\cong
M^{w^{I}}/(M^{I}_{1}+M_{1}^{w^{I}})$.

Hence, $\mathcal{L}_{\chi}(\lambda_{(1,2_{1})})$ is the quotient module of
$\mathcal{L}^{I}$. Furthermore, $\mathcal{L}_{\chi}(\lambda_{(1,2_{1})})$ is a
sub-quotient module of $\widehat{Q}^{I}_{\chi}(\lambda)$.

(2.4). The set of all the composition factors of
$\mbox{soc}^{N-1}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-2}\widehat{Z}_{1}(\lambda)$
is denoted by $\\{\widehat{L}_{\chi}(\lambda_{(1,2_{i})})|1\leq i\leq
t_{2}\\}$. Similarly, one can get the corresponding quasi-simple modules
$\\{\mathcal{L}_{\chi}(\lambda_{(1,2_{i})})|1\leq i\leq t_{2}\\}$ are the sub-
quotient modules of $\widehat{Q}^{I}_{\chi}(\lambda)$.

Then $\widehat{Q}^{I}_{\chi}(\lambda)$ has a quotient module
$\widehat{Q}^{I}_{\chi,2}(\lambda)$ consist of $\mathcal{L}_{\chi}(\lambda)$
and $\mathcal{L}_{\chi}(\lambda_{(1,2_{i})}),1\leq i\leq t_{2}$ with
$\mathcal{L}_{\chi}(\lambda)$ the quasi-head.

(3). Next, we consider the composition factors of
$\mbox{soc}^{N-2}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-3}\widehat{Z}_{1}(\lambda)$.

Denote by $\\{\widehat{L}_{\chi}(\lambda_{(1,3_{i})})|1\leq i\leq t_{3}\\}$
the set of all the composition factors of
$\mbox{soc}^{N-2}\widehat{Z}_{1}(\lambda)/\mbox{soc}^{N-3}\widehat{Z}_{1}(\lambda)$.
Similar to the case $\widehat{L}_{\chi}(\lambda_{(1,2_{1})})$, we can prove
the corresponding quasi-simple modules
$\\{\mathcal{L}_{\chi}(\lambda_{(1,3_{i})})|1\leq i\leq t_{3}\\}$ are the sub-
quotient modules of $\widehat{Q}^{I}_{\chi}(\lambda)$.

Then $\widehat{Q}^{I}_{\chi}(\lambda)$ has a quotient module
$\widehat{Q}^{I}_{\chi,3}(\lambda)$ consist of $\mathcal{L}_{\chi}(\lambda)$,
$\mathcal{L}_{\chi}(\lambda_{(1,2_{i})}),1\leq i\leq t_{1}$ and
$\mathcal{L}_{\chi}(\lambda_{(1,3_{i})}),1\leq i\leq t_{3}$ with
$\mathcal{L}_{\chi}(\lambda)$ the quasi-head.

(4). Repeating the above procedure, we can prove
$\widehat{Q}^{I}_{\chi}(\lambda)$ has an $\mathcal{L}_{\chi}$-filtration.

Similar arguments yield that $\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ has
an $\mathcal{L}_{\chi}$-filtration.

∎

### 4.3.

As $\widehat{Q}_{\chi}(\lambda)$ has an $\widehat{Q}^{I}_{\chi}$-filtration,
by Theorem 4.2, we have

######  Corollary 4.3.

When $\lambda\in X(T)$ is $p$-regular, $\widehat{Q}_{\chi}(\lambda)$ has an
$\mathcal{L}_{\chi}$-filtration.

For any $\lambda\in X(T)$, we give the following conjecture.

######  Conjecture 4.4.

$\widehat{Q}^{I}_{\chi}(\lambda),\lambda\in X(T)$ (resp.
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}),\lambda\in X(T)$) has an
$\mathcal{L}_{\chi}$-filtration.

### 4.4.

######  Example 4.5.

Assume that ${\mathfrak{g}}=\mathfrak{sl}_{3}(k)$. Let
$\Pi=\\{\alpha_{1},\alpha_{2}\\}$ be the set of simple roots. Let $\chi$ be of
standard Levi form with $I=\\{\alpha_{1}\\}$, and $\chi$ lies in the
subregular nilpotent orbit, $w_{I}=s_{\alpha_{1}}$ with $l(w_{I})=1$ and
$w^{I}=s_{\alpha_{2}}s_{\alpha_{1}}$ with $l(w^{I})=2$. Denote
$\alpha=\alpha_{1}+\alpha_{2}$.

Fix a wight $\xi_{0}\in C_{0}$ with
$\xi_{0}+\rho=r_{1}\varpi_{1}+r_{2}\varpi_{2}=(r_{1},r_{2})$ where
$\varpi_{1}$ and $\varpi_{2}$ are fundamental weights. Set
$r_{0}=p-r_{1}-r_{2}$. Set $\sigma=s_{\alpha_{1}}s_{\alpha_{2}}$, and
$\xi_{i}=\sigma^{i}.\xi_{0},i=1,2$. Then $\xi_{1}+\rho=(-(r_{1}+r_{2}),r_{1})$
and $\xi_{2}+\rho=(r_{2},-(r_{1}+r_{2}))$. By [13, §2.3], we have
$\\{\widehat{L}_{\chi}(\xi_{i})\mid 0\leq i\leq 2\\}$ is the set of
isomorphism classes of simple modules in the block containing
$\widehat{L}_{\chi}(\xi_{0})$.

#### 4.4.1.

When $\xi_{0}$ is $p$-regular, according to the discussion in [13], the
structure diagram about the composition factor of
$\widehat{Z}_{\chi}(\xi_{0})$ and $\mathcal{L}_{\chi}(\xi_{0})$ are

$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$and$\textstyle{\widehat{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

respectively. (The solid line indicate the connection relationships among the
composition factors of proper standard module and quasi-simple module.)

The structure diagrams of $\widehat{Q}^{I}_{\chi}(\xi_{0})$ and
$\widehat{Q}^{w^{I}}_{\chi}(\xi^{w^{I}}_{0})$ about
$\mathcal{L}_{\chi}$-filtration are

$\textstyle{\mathcal{L}_{\chi}(\xi_{0})}$$\textstyle{\mathcal{L}_{\chi}(\xi_{2})}$$\textstyle{\mathcal{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
and
$\textstyle{\mathcal{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

respectively. (In the diagrams above and below, the solid line indicate the
connection relationships among the $\mathcal{L}_{\chi}$-filtration factors of
standard module and dotted lines indicate the connection relationships among
the $\mathcal{L}_{\chi}$-filtration factors of costandard module.)

The structure diagram of $\widehat{Q}_{\chi}(\xi_{0})$ about
$\mathcal{L}_{\chi}$-filtration is

---
$\textstyle{\mathcal{L}_{\chi}(\xi_{0})}$$\textstyle{\mathcal{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{2}+p\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{1}+p\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{2}+p\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

#### 4.4.2.

When $\xi_{0}$ is not $p$-regular, we choose $\xi_{0}+\rho=(p-1,1)$. Then
$2=|W_{I}.d\xi_{0}|>|W_{I}.d\xi_{1}|=1$ and
$\widehat{Q}^{I}_{\chi}(\xi_{1}+p\alpha)\cong\widehat{Z}_{\chi}(\xi_{1}+p\alpha)$.
By Lemma 2.3, we have
$\mathcal{L}_{\chi}(\xi_{0})\cong\widehat{L}_{\chi}(\xi_{0})$ and
$\mathcal{L}_{\chi}(\xi_{1})\cong\widehat{L}_{\chi}(\xi_{1})$.

The structure diagrams about composition factors of
$\widehat{Z}_{\chi}(\xi_{0})$, $\mathcal{L}_{\chi}(\xi_{0})$,
$\widehat{Z}_{\chi}(\xi_{1}+p\alpha)$, $\mathcal{L}_{\chi}(\xi_{1}+p\alpha)$,
$\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{0})$ and
$\widehat{Z}^{w^{I}}_{\chi}((\xi_{1}+p\alpha)^{w^{I}})$ are listed as follows.

$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}+p\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The structure diagrams about composition factors of
$\widehat{Q}^{I}_{\chi}(\xi_{0})$ is

---
$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The structure diagram about composition factors of
$\widehat{Q}_{\chi}(\xi_{0})$ is

---
$\textstyle{\widehat{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

(In the three diagrams above, the solid line indicate the connection
relationships among the composition factors of proper standard module and
dotted lines indicate the connection relationships among the composition
factors of proper costandard module.)

##  5\. Projective dimensions

### 5.1.

Let $\Lambda$ be a set of weights in $X(T)$ and $h(\Lambda)$ be the maximal
number $m$ such that there exists a chain
$\lambda_{0}<\lambda_{1}<\cdot\cdot\cdot<\lambda_{m}$ in $\Lambda$, with all
$\lambda_{i}\in\Lambda$.

Denote $[\lambda,\mu]$ by the subset $\\{\nu|\lambda\leq\nu\leq\mu\\}$. Denote
$[\lambda,-]$ and $[-,\mu]$ by the subset $\\{\nu|\lambda\leq\nu\\}$ and
$\\{\nu|\nu\leq\mu\\}$, respectively.

Denote $\mbox{{Proj.dim}}~{}\widehat{M}$ by the projective dimension of
$\widehat{M}$. If $\widehat{Q}_{\chi}(\lambda)$ has
$\mathcal{L}_{\chi}$-filtration, there is a filtration
$\widehat{Q}_{\chi}(\lambda)=\widehat{Q}_{\chi}(\lambda)^{0}\supset\widehat{Q}_{\chi}(\lambda)^{1}\supset\cdot\cdot\cdot\supset\widehat{Q}_{\chi}(\lambda)^{r}=0$
of $\widehat{Q}_{\chi}(\lambda)$ with
$\widehat{Q}_{\chi}(\lambda)^{i-1}/\widehat{Q}_{\chi}(\lambda)^{i}\cong\mathcal{L}_{\chi}(\lambda_{i})$
for some $\lambda_{i}\in X(T),i=1,...,r$. Denote $\Lambda^{\prime}$ by the set
of $\lambda_{i}\in X(T),i=1,...,r$ appear in all the composition factors
$\widehat{L}_{\chi}(\lambda_{i})$ of $\widehat{Q}_{\chi}(\lambda)$.

######  Theorem 5.1.

Let $\lambda\in X(T)$. we have

(i) $\mbox{{Proj.dim}}~{}\widehat{Q}^{I}_{\chi}(\lambda)\leq h([\lambda,-]).$

(ii) When $\lambda\in X(T)$ is $p$-regular,
$\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\lambda)\leq
h(\Lambda^{\prime})+h([-,\lambda]).$

(iii) Assume that Conjecture 4.4 holds,
$\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\lambda)\leq
h(\Lambda^{\prime})+h([-,\lambda]).$

###### Proof.

The proof is similar to [6, Lem. 2.2].

For (i), if $\widehat{Q}^{I}_{\chi}(\lambda)$ is not projective, there is a
submodule $\widetilde{Q}_{\chi}^{1}(\lambda)$ of $\widehat{Q}_{\chi}(\lambda)$
such that
$\widehat{Q}^{I}_{\chi}(\lambda)=\widehat{Q}_{\chi}(\lambda)/\widetilde{Q}_{\chi}^{1}(\lambda)$
and $\widetilde{Q}_{\chi}^{1}(\lambda)$ has a
$\widehat{Q}^{I}_{\chi}$-filtration with sub-quotients
$\widehat{Q}^{I}_{\chi}(\mu)$ satisfying $\mu>\lambda$ (cf. (2.4)). For
$\mu>\lambda$ we have $h([\mu,-])<h([\lambda,-])$, by induction one can get

$\mbox{{Proj.dim}}\widetilde{Q}_{\chi}^{1}(\lambda)\leq\textbf{max}\\{\mbox{{Proj.dim}}\widehat{Q}^{I}_{\chi}(\mu)|\mu>\lambda\\}\leq\mbox{{max}}\\{h([\mu,-])|\mu>\lambda\\}<h([\lambda,-]).$

As
$0\rightarrow\widetilde{Q}_{\chi}^{1}(\lambda)\rightarrow\widehat{Q}_{\chi}(\lambda)\rightarrow\widehat{Q}^{I}_{\chi}(\lambda)\rightarrow
0$ is a short exact series, it follows that

$\mbox{{Proj.dim}}~{}\widehat{Q}^{I}_{\chi}(\lambda)\leq
1+\mbox{{Proj.dim}}~{}\widetilde{Q}_{\chi}^{1}(\lambda)\leq h([\lambda,-]).$

Then (i) is proved.

For (ii), Assume that $\mathcal{L}_{\chi}(\lambda)$ is not projective. As
$\lambda$ is $p$-regular, we have $|W_{I}.d\lambda|=|W_{I}.d\lambda_{r}|$. By
Lemma 2.7, we have
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$.
Following the notations in the proof of Theorem 3.2, $\mathcal{L}^{I}$ is the
submodule of $\widehat{Q}^{I}_{\chi}(\lambda)$ with
$\mathcal{L}_{\chi}(\lambda)\cong\widehat{Q}^{I}_{\chi}(\lambda)/\mathcal{L}^{I}$
and $\mathcal{L}^{I}$ does not contain $\widehat{L}_{\chi}(\lambda)$ as a
composition factor. By Theorem 4.2, $\widehat{Q}^{I}_{\chi}(\lambda)$ has an
$\mathcal{L}_{\chi}$-filtration and the sub-quotients in the
$\mathcal{L}_{\chi}$-filtrations of $\mathcal{L}^{I}$ are the form
$\mathcal{L}_{\chi}(\nu)$ with $\nu<\lambda$, we have
$h([-,\nu])<h([-,\lambda])$. Thus by induction
$\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\nu)\leq
h(\Lambda^{\prime})+h([-,\nu])<h(\Lambda^{\prime})+h([-,\lambda])$, and
$\mbox{{Proj.dim}}~{}\widehat{Q}^{I}_{\chi}(\lambda)\leq h([\lambda,-])\leq
h(\Lambda^{\prime})$. As
$0\rightarrow\mathcal{L}^{I}\rightarrow\widehat{Q}^{I}_{\chi}(\lambda)\rightarrow\mathcal{L}_{\chi}(\lambda)\rightarrow
0$ is a short exact series, we can get
$\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\lambda)\leq
h(\Lambda^{\prime})+h([-,\lambda])$.

For (iii), when $\lambda$ is not $p$-regular, it is possible that
$|W_{I}.d\lambda|\neq|W_{I}.d\lambda_{r}|$.

If $|W_{I}.d\lambda|\leq|W_{I}.d\lambda_{r}|$, by Lemma 2.7, we have
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda|[\widehat{L}_{\chi}(\lambda)]$.
Similar to the case $\lambda$ is $p$-regular, under the assumption that
Conjecture 4.4 holds, we can prove
$\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\lambda)\leq
h(\Lambda^{\prime})+h([-,\lambda])$.

If $|W_{I}.d\lambda|>|W_{I}.d\lambda_{r}|$, by Lemma 2.7, we have
$[\mathcal{L}_{\chi}(\lambda)]=|W_{I}.d\lambda_{r}|[\widehat{L}_{\chi}(\lambda)]$.
Following the notations in the proof of Lemma 3.1, $\mathcal{L}_{1}^{I}$ is
the quotient module of $\widehat{Q}_{\chi}^{I}(\lambda_{r})$ with
$\mathcal{L}_{1}^{I}\cong\widehat{Q}_{\chi}^{I}(\lambda_{r})/\mathcal{L}_{\chi}(\lambda)$.
Then $\widehat{L}_{\chi}(\lambda)$ is not the composition factor of
$\mathcal{L}_{1}^{I}$ and the highest weight of each composition factor of
$\mathcal{L}_{1}^{I}$ is higher than $\lambda$. If Conjecture 4.4 holds, we
have the sub-quotients in the $\mathcal{L}_{\chi}$-filtration of
$\mathcal{L}_{1}^{I}$ are the form $\mathcal{L}_{\chi}(\nu)$ with
$\nu>\lambda$, we have $h([\nu,-])<h([\lambda,-])$.

By induction $\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\nu)\leq
h(\Lambda^{\prime})+h([\nu,-])<h(\Lambda^{\prime})+h([\lambda,-])$, and
$\mbox{{Proj.dim}}~{}\widehat{Q}^{I}_{\chi}(\lambda_{r})\leq
h([\lambda_{r},-])\leq h(\Lambda^{\prime})$. Since
$0\rightarrow\mathcal{L}_{\chi}(\lambda)\rightarrow\widehat{Q}^{I}_{\chi}(\lambda_{r})\rightarrow\mathcal{L}_{1}^{I}\rightarrow
0$ is a short exact series, we can get
$\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\lambda)\leq
h(\Lambda^{\prime})+h([\lambda,-])$. ∎

######  Remark 5.2.

(1) In general,
$\mbox{Ext}^{1}_{\mathcal{C}}(\widehat{L}_{\chi}(\lambda),\widehat{L}_{\chi}(\lambda))\neq
0$ (cf. [16, Prop.2.1]), but in $G_{1}T$-modules category,
$\mbox{Ext}^{1}_{G_{1}T}(\widehat{L}_{0}(\lambda),\widehat{L}_{0}(\lambda))=0$
(cf. [12, II 9.19(5)]) where $\widehat{L}_{\chi}(\lambda)$ (resp.
$\widehat{L}_{0}(\lambda)$) is the simple module in $\mathcal{C}$ (resp. the
category of $G_{1}T$-modules). Theorem 3.2 tells us
$\hbox{Ext}_{\mathcal{C}}^{1}(\mathcal{L}_{\chi}(\lambda),\mathcal{L}_{\chi}(\lambda))=0$.

(2) We know the category of $G_{1}T$-modules is a highest weight category
corresponding to the category of a quasi-hereditary algebra $(A,\leq)$ (cf.
[4]). By [1, Thm. 2.4], the projective dimension of any simple module
$\widehat{L}_{0}(\lambda),\lambda\in X(T)$ is finite.

If $I\neq\emptyset$, by §2.6, the category $\mathcal{C}$ corresponds to the
category of a standard stratified algebra $(B,\leq)$ (cf. [5]) but is not a
highest weight category (cf. [1]). By [1, Thm. 2.4], the projective dimension
of simple module $\widehat{L}_{\chi}(\lambda)$ is infinite in general.
According to Theorem 5.1, $\mbox{{Proj.dim}}~{}\mathcal{L}_{\chi}(\lambda)$ is
finite when $\lambda$ is $p$-regular or Conjecture 4.4 holds.

Based on (1) and (2), we get the cohomological properties of quasi-simple
module $\mathcal{L}_{\chi}(\lambda)$ are better than
$\widehat{L}_{\chi}(\lambda)$’s.

### 5.2.

According to Theorem 4.2 and [10, Prop. 11.18], when $\lambda\in X(T)$ is
$p$-regular, we can get

(5.1)
$(\widehat{Q}_{\chi}(\lambda):\widehat{Q}^{I}_{\chi}(\mu))=(\widehat{Q}^{I}_{\chi}(\mu):\mathcal{L}_{\chi}(\lambda))=[\widehat{Z}_{\chi}(\mu):\widehat{L}_{\chi}(\lambda)].$

### 5.3.

Set $W^{I,p}=\\{w\in W_{p}|w.C_{0}\subset C_{I}\\}.$ For $v,w\in W^{I,p}$, the
polynomial $P_{v,w}\in\mathbb{Z}[t^{-1}]$ was constructed in [21, §11] which
generalise the classical Kazhdan-Lusztig polynomials. In [21, §13.17], Lusztig
gave a hope which was formularized by Jantzen in [10, 11.24]:

$[\widehat{Z}_{\chi}(v.\lambda_{0}):\widehat{L}_{\chi}(w.\lambda_{0})]=P_{\kappa_{I}(v),\kappa_{I}(w)}(1)$

where $\lambda_{0}$ is a weight in the interior of $C_{0}$ ($\lambda_{0}$ is
$p$-regular) and $\kappa_{I}$ is an involution on $W^{I,p}$. Jantzen proved
Lusztig’s Hope is correct in some case (cf. [11, §5.1]).

Based on (5.1), Lusztig’s Hope can be rewritten as

(5.2)
$(\widehat{Q}^{I}_{\chi}(v.\lambda_{0}):\mathcal{L}_{\chi}(w.\lambda_{0}))=[\widehat{Z}_{\chi}(v.\lambda_{0}):\widehat{L}_{\chi}(w.\lambda_{0})]=P_{\kappa_{I}(v),\kappa_{I}(w)}(1).$

Since the cohomological properties of quasi-simple module
$\mathcal{L}_{\chi}(\lambda)$ are better than $\widehat{L}_{\chi}(\lambda)$’s
(cf. Remark 5.2), the new expression (5.2) might make the proof of Lusztig’s
Hope easier.

##  6\.  The Loewy lengths of quasi-simple modules

### 6.1.

Furthermore, we have the following result on the Loewy length of a quasi-
simple module.

######  Theorem 6.1.

When $\lambda\in X(T)$ is $p$-regular, we have
$ll(\mathcal{L}_{\chi}(\lambda))=ll(\widehat{Q}_{\chi,I}(\lambda))$.

###### Proof.

Let $r=ll(\widehat{Q}_{\chi,I}(\lambda))$ and

(6.1)
$0=\mbox{soc}^{0}\widehat{{Q}}_{\chi,I}(\lambda)\subseteq\mbox{soc}^{1}\widehat{{Q}}_{\chi,I}(\lambda)\subseteq...\subseteq\mbox{soc}^{r}\widehat{{Q}}_{\chi,I}(\lambda)=\widehat{{Q}}_{\chi,I}(\lambda)$

such that
$\mbox{soc}^{i}\widehat{Q}_{\chi,I}(\lambda)/\mbox{soc}^{i-1}\widehat{Q}_{\chi,I}(\lambda)\cong\bigoplus\limits_{j=1}^{n_{i}}\widehat{Z}_{i_{j},I}$
with $\widehat{Z}_{i_{j},I}\cong\widehat{Z}_{\chi,I}(\lambda),\forall j$ be
the socle series of $\widehat{Q}_{\chi,I}(\lambda)$. Then
$\mbox{soc}^{1}\widehat{Q}_{\chi,I}(\lambda)\cong\widehat{Z}_{\chi,I}(\lambda)$.

Let $r^{\prime}=ll(\mathcal{L}_{\chi}(\lambda))$ and

(6.2)
$0=\mbox{soc}^{0}\mathcal{L}_{\chi}(\lambda)\subseteq\mbox{soc}^{1}\mathcal{L}_{\chi}(\lambda)\subseteq...\subseteq\mbox{soc}^{r^{\prime}}\mathcal{L}_{\chi}(\lambda)=\mathcal{L}_{\chi}(\lambda)$

such that
$\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda)/\mbox{soc}^{i-1}\mathcal{L}_{\chi}(\lambda)\cong\bigoplus\limits_{j=1}^{n^{\prime}_{i}}\widehat{L}_{i_{j}}$
with $\widehat{L}_{i_{j}}\cong\widehat{L}_{\chi}(\lambda),\forall j$ be the
socle series of $\mathcal{L}_{\chi}(\lambda)$. Then
$\mbox{soc}^{1}\mathcal{L}_{\chi}(\lambda)\cong\widehat{L}_{\chi}(\lambda)$.

$\lambda$ is $p$-regular implies $|W_{I}.d\lambda|=|W_{I}.d\lambda_{r}|$. By
(2.8) we have

(6.3)
$\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)\setminus\lambda^{\prime}}\texttt{pr}_{\nu}(\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})\oplus\widehat{Q}_{\chi,I}(\lambda).$

where $\lambda^{\prime}\in C_{I}\cap X(T)$ such that $\lambda\in
W_{I,p}.\lambda^{\prime}$.

Similar to (2.8), one can get

$\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}=\bigoplus\limits_{\nu\in
C_{I}\cap
X(T)\setminus\lambda^{\prime}}\texttt{pr}_{\nu}(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})\oplus\texttt{pr}_{\lambda^{\prime}}(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})$

where $\lambda^{\prime}\in C_{I}\cap X(T)$ such that $\lambda\in
W_{I,p}.\lambda^{\prime}$.

Then
$(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\cong\texttt{pr}_{\lambda^{\prime}}(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}})$.
Because $\texttt{pr}_{\lambda^{\prime}}$ is an exact functor and
$\mbox{soc}^{i-1}\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}$ is the
submodule of
$\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda)|_{{\mathfrak{g}}_{I}}$, we have
$(\mbox{soc}^{i-1}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}$ is a
submodule of
$(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}$. As
$\mathcal{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{Q}_{\chi,I}(\lambda)$
(cf. (2.8)), Then
$(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I},\forall i$
is a submodule of $\widehat{Q}_{\chi,I}(\lambda)$. Hence,

$(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda)/\mbox{soc}^{i-1}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\cong(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}/(\mbox{soc}^{i-1}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}.$

By (6.2) and
$\widehat{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\cong\widehat{Z}_{\chi,I}(\lambda)$,
we have

$(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}/(\mbox{soc}^{i-1}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\cong(\bigoplus\limits_{j=1}^{n^{\prime}_{i}}\widehat{L}_{i_{j}})^{\lambda+\mathbb{Z}I}\cong\bigoplus\limits_{j=1}^{n^{\prime}_{i}}\widehat{Z}_{i_{j},I}$

where $\widehat{Z}_{i_{j},I}\cong\widehat{Z}_{\chi,I}(\lambda),\forall j$.

Based on the above arguments, we can get

$0=(\mbox{soc}^{0}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\subseteq(\mbox{soc}^{1}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\subseteq...\subseteq(\mbox{soc}^{r^{\prime}}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}=\mathcal{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong\widehat{Q}_{\chi,I}(\lambda)$

such that
$(\mbox{soc}^{i}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}/(\mbox{soc}^{i-1}\mathcal{L}_{\chi}(\lambda))^{\lambda+\mathbb{Z}I}\cong\bigoplus\limits_{j=1}^{n^{\prime}_{i}}\widehat{Z}_{i_{j},I}$
with $\widehat{Z}_{i_{j},I}\cong\widehat{Z}_{\chi,I}(\lambda),\forall j$ is
the socle series of
$\mathcal{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong\widehat{Q}_{\chi,I}(\lambda)$.

Since the socle series of $\widehat{Q}_{\chi,I}(\lambda)$ is unique, we have
$r=r^{\prime}$. This implies

$ll(\mathcal{L}_{\chi}(\lambda))=ll(\widehat{Q}_{\chi,I}(\lambda))=r.$

∎

### 6.2.

Furthermore,

######  Theorem 6.2.

When $\lambda$ is $p$-regular, we have

$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))=ll(\mathcal{L}_{\chi}(\lambda))+ll(\widehat{Z}_{\chi}(\lambda))-1.$

###### Proof.

Let $\widehat{{Z}}_{r}\cong\widehat{{Z}}_{\chi}(\lambda)$ be the submodule of
$\widehat{Q}^{I}_{\chi}(\lambda)$ and
$\widehat{L}_{r}\cong\widehat{L}_{\chi}(\lambda)$ be the head of
$\widehat{{Z}}_{r}$. Let $t=ll(\widehat{Q}^{I}_{\chi}(\lambda))$,
$ll(\mathcal{L}_{\chi}(\lambda))=r$ and $m=ll(\widehat{Z}_{\chi}(\lambda))$.
Then

$\widehat{L}_{r}\cong\mbox{soc}^{m}\widehat{Z}_{r}/\mbox{soc}^{m-1}\widehat{Z}_{r}\mbox{
and
}\widehat{L}_{r}\in\mbox{soc}^{m}\widehat{Q}^{I}_{\chi}(\lambda)/\mbox{soc}^{m-1}\widehat{Q}^{I}_{\chi}(\lambda).$

As
$[\mathcal{L}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]=[\widehat{Q}^{I}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]$
and $\mathcal{L}_{\chi}(\lambda)$ is the quotient module of
$\widehat{Q}^{I}_{\chi}(\lambda)$, $\widehat{L}_{r}$ is the socle of
$\mathcal{L}_{\chi}(\lambda)$ and
$\widehat{L}_{r}\in\mbox{soc}^{t-r+1}\widehat{Q}^{I}_{\chi}(\lambda)/\mbox{soc}^{t-r}\widehat{Q}^{I}_{\chi}(\lambda)$.
Hence, $m=t-r+1$ and
$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\mathcal{L}_{\chi}(\lambda))+ll(\widehat{Z}_{\chi}(\lambda))-1$.

Furthermore, by (1.2), we get

$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))=ll(\mathcal{L}_{\chi}(\lambda))+ll(\widehat{Z}_{\chi}(\lambda))-1.$

∎

##  7\. The cases of regular nilpotent and sub-regular nilpotent

###  7.1. A conjecture on the Loewy length of projective
$U_{\chi}({\mathfrak{g}}_{I})$-modules

In the category of $U_{\chi}({\mathfrak{g}}_{I})$-modules, the baby Verma
module $\widehat{Z}_{\chi,I}(\lambda),\lambda\in X(T)$ is irreducible (cf. [7,
§4.2/3]). Then $ll(\widehat{Z}_{\chi,I}(\lambda))=1$. Let
$\widehat{Q}_{\chi,I}(\lambda)$ be the projective cover of
$\widehat{Z}_{\chi,I}(\lambda)$. We propose the following conjecture on the
Loewy length of $\widehat{Q}_{\chi,I}(\lambda)$.

######  Conjecture 7.1.

Assume that $\chi$ has standard Levi form with $I=\Pi$ and $\lambda$ is
$p$-regular, we conjecture that

$ll(\widehat{Q}_{\chi,I}(\lambda))=ll(\mathcal{L}_{\chi}(\lambda))=l(w_{I})+1=|R_{I}^{+}|+1.$

###  7.2. The cases of type $A_{n}$ for $n=1,2$

######  Example 7.2.

Suppose that the root system of ${\mathfrak{g}}$ is of type $A_{1}$, and
$\chi$ has standard Levi form with $I=\Pi$ ($\chi$ lie in the regular
nilpotent orbit). We have
$[\widehat{Q}_{\chi,I}(\lambda):\widehat{Z}_{\chi,I}(\lambda)]=2$ and
$l(w_{I})=1$. Since $\widehat{Q}_{\chi,I}(\lambda)$ is indecomposable, we get
$ll(\widehat{Q}_{\chi,I}(\lambda))=2=l(w_{I})+1.$ This implies that Conjecture
7.1 holds in this case.

######  Example 7.3.

Suppose that the root system of ${\mathfrak{g}}$ is of type $A_{2}$, and
$\chi$ has standard Levi form with $I=\Pi$. By [20], the socle series of
$\widehat{Q}_{\chi,I}(\lambda)$ is

$\displaystyle\widehat{Z}_{\chi,I}(\lambda)$
$\displaystyle\widehat{Z}_{\chi,I}(\lambda)$
$\displaystyle\widehat{Z}_{\chi,I}(\lambda)$
$\displaystyle\widehat{Z}_{\chi,I}(\lambda)$
$\displaystyle\widehat{Z}_{\chi,I}(\lambda)$
$\displaystyle\widehat{Z}_{\chi,I}(\lambda)$

Then $ll(\widehat{Q}_{\chi,I}(\lambda))=4=l(w_{I})+1$ which implies that
Conjecture 7.1 holds.

### 7.3.

In this subsection, we show the following result on Loewy lengths of standard
modules and projective modules for some special cases.

######  Theorem 7.4.

Assume that $\chi$ is subregular nilpotent and has standard Levi form with
$|I|=1$. If $\lambda\in X(T)$ is $p$-regular, we have

  * (1)

$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))=l(w^{I})+l(w_{I})+1;$

  * (2)

$ll(\widehat{Q}_{\chi}(\lambda))=2l(w^{I})+l(w_{I})+1.$

###### Proof.

The cases which satisfy the conditions of Theorem 7.4 are
${\mathfrak{g}}=\mathfrak{sl}(3,k)$ with $I$ being any simple root and
${\mathfrak{g}}=\mathfrak{s0}(5,k)$ with $I$ being the short simple root. We
will only need to prove this theorem case by case in the following
subsections.

Case 1: ${\mathfrak{g}}=\mathfrak{sl}_{3}(k)$ Keeping the notations as in
Exalple 4.5. According to [13], the socle series of
$\widehat{Z}_{\chi}(\xi_{0})$, $\widehat{Z}_{\chi}(\xi_{1}+p\alpha)$ and
$\widehat{Z}_{\chi}(\xi_{2}+p\alpha)$ are listed as follows.

$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{2}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$

Hence, $ll(\widehat{Z}_{\chi}(\xi_{0}))=3=l(w^{I})+1$.

Since $\widehat{Q}_{\chi}^{I}(\xi_{0})$ has a simple head and a simple socle,
according to the socle series of $\widehat{Z}_{\chi}(\xi_{0})$ listed above,
we get the following socle series of $\widehat{Q}^{I}_{\chi}(\xi_{0})$:

---
$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$

Then $ll(\widehat{Q}_{\chi}^{I}(\xi_{0}))=4=l(w^{I})+l(w_{I})+1$.

Similarly, the socle series of $\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{0})$,
$\widehat{Z}^{w^{I}}_{\chi}((\xi_{1}+p\alpha)^{w^{I}})$ and
$\widehat{Z}_{\chi}((\xi_{2}+p\alpha))^{w^{I}})$ are listed as follows.

$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}+p\alpha)}$$\textstyle{\widehat{L}_{\chi}(\xi_{2}+p\alpha)}$

Then $ll(\widehat{Z}_{\chi}^{w^{I}}(\xi^{w^{I}}_{0}))=3=l(w^{I})+1$.

Since $\widehat{Q}_{\chi}^{w^{I}}(\xi^{w^{I}}_{0})$ has a simple head and a
simple socle, according to the composition series of
$\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{0})$ listed above, we get the
following socle series of $\widehat{Q}^{w^{I}}_{\chi}(\xi^{w^{I}}_{0})$

$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$

Then $ll(\widehat{Q}_{\chi}^{w^{I}}(\xi^{w^{I}}_{0}))=4=l(w^{I})+l(w_{I})+1$.

Based on the above results, since $\widehat{Q}_{\chi}(\xi_{0})$ has both a
$\widehat{Q}^{I}_{\chi}$\- and a $\widehat{Q}_{\chi}^{w^{I}}$-filtration, we
then get the socle series of $\widehat{Q}_{\chi}(\xi_{0})$

---
$\textstyle{\xi_{0}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{2}+p\alpha}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{2}+p\alpha}$$\textstyle{\xi_{1}+p\alpha}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{2}+p\alpha}$$\textstyle{\xi_{1}+p\alpha}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{2}+p\alpha}$$\textstyle{\xi_{0}}$

where the simple module $\widehat{L}_{\chi}(\xi_{i})$ is denoted by $\xi_{i}$
for brevity. Then $ll(\widehat{Q}_{\chi}(\xi_{0}))=6=2l(w^{I})+l(w_{I})+1$.

Similar arguments yield the Loewy series of the other standard modules,
costandard modules and indecomposable projective modules. The proof of Theorem
7.4 for ${\mathfrak{g}}=\mathfrak{sl}_{3}(k)$ is done.

Case 2: ${\mathfrak{g}}=\mathfrak{s}o(5,k)$ In this subsection, we always
assume that ${\mathfrak{g}}=\mathfrak{s}o(5,k)$ and $\\{\alpha,\beta\\}$ is
the set of simple roots with $\beta$ the short root. Let $\chi$ be of standard
Levi form with $I=\\{\beta\\}$. Then $\chi$ lies in the subregular nilpotent
orbit, $w_{I}=s_{\beta}$ with $l(w_{I})=1$, and
$w^{I}=s_{\alpha}s_{\beta}s_{\alpha}$ with $l(w^{I})=3$.

Denote by $\varpi_{1}$ and $\varpi_{2}$ the fundamental weights. Then
$\alpha=2\varpi_{1}-2\varpi_{2}=(2,-2)$,
$\beta=-\varpi_{1}+2\varpi_{2}=(-1,2)$. Let $\xi_{1}=(r_{1},r_{2})\in C_{0}$,
$\xi_{2}=(-r_{1},2r_{1}+r_{2})$, $\xi_{3}=(-(r_{1}+r_{2}),2r_{1}+r_{2})$,
$\xi_{4}=(-(r_{1}+r_{2},r_{2})$. Then $\\{\widehat{L}_{\chi}(\xi_{i})|1\leq
i\leq 4\\}$ is the set of isomorphism classes of simple modules in the block
containing $\widehat{L}_{\chi}(\xi_{1})$ (cf. [13, § 3.8]). Set
$\xi_{0}=\xi_{4}+p\alpha$, $\xi_{-1}=\xi_{3}+p\alpha$,
$\xi_{-2}=\xi_{2}+p\alpha$, $~{}\xi_{-3}=\xi_{1}+p\alpha$.

By [13, §3.10], we have the following socle series of
$\widehat{Z}_{\chi}(\xi_{1})$, $\widehat{Z}_{\chi}(\xi_{0})$,
$\widehat{Z}_{\chi}(\xi_{-1})$ and $\widehat{Z}_{\chi}(\xi_{-2})$

$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{-1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{-2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}))}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{-1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{4})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1}).}$

Then $ll(\widehat{Z}_{\chi}(\xi_{1}))=4=l(w^{I})+1$.

Similarly, the socle series of $\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{1})$,
$\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{0})$,
$\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{-1})$ and
$\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{-2})$ are listed as follows

$\textstyle{\widehat{L}_{\chi}(\xi_{4})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{-1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{0})}$$\textstyle{\widehat{L}_{\chi}(\xi_{-1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{-2}).}$

Then $ll(\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}_{1}))=4=l(w^{I})+1$.

Based on the above results, we can get the following socle series of
$\widehat{Q}^{I}_{\chi}(\xi_{1})$

---
$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{4})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{4})}$

Then $ll(\widehat{Q}_{\chi}^{I}(\xi_{1}))=5=l(w^{I})+l(w_{I})+1$.

Similarly, we have the following socle series of
$\widehat{Q}^{w^{I}}_{\chi}(\xi^{w^{I}}_{1})$

$\textstyle{\widehat{L}_{\chi}(\xi_{4})}$$\textstyle{\widehat{L}_{\chi}(\xi_{4})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{3})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{2})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$$\textstyle{\widehat{L}_{\chi}(\xi_{1})}$

Then $ll(\widehat{Q}_{\chi}^{w^{I}}(\xi^{w^{I}}_{1}))=5=l(w^{I})+l(w_{I})+1$.

Finally, we have the socle series of $\widehat{Q}_{\chi}(\xi_{1})$

|
---|---
$\textstyle{\xi_{1}}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{3}}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{-1}}$$\textstyle{\xi_{4}}$$\textstyle{\xi_{3}}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{-1}}$$\textstyle{\xi_{-2}}$$\textstyle{\xi_{4}}$$\textstyle{\xi_{3}}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{-1}}$$\textstyle{\xi_{-2}}$$\textstyle{\xi_{3}}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{-1}}$$\textstyle{\xi_{2}}$$\textstyle{\xi_{1}}$$\textstyle{\xi_{0}}$$\textstyle{\xi_{1}}$

where the simple module $\widehat{L}_{\chi}(\xi_{i})$ is denoted by $\xi_{i}$
for brevity. Then $ll(\widehat{Q}_{\chi}(\xi_{1}))=8=2l(w^{I})+l(w_{I})+1$.

Similar arguments yield the Loewy series of the other standard modules,
costandard modules and indecomposable projective modules. The proof of Theorem
7.4 for ${\mathfrak{g}}=\mathfrak{s}o(5,k)$ is done. ∎

##  8\. The general case

###  8.1. Some inequalities on the Loewy lengths

Keep notations as before, in particular, ${\mathfrak{g}}$ is a reductive Lie
algebra, $\chi$ is of standard Levi form and $\lambda$ is $p$-regular. Similar
to [3, Proposition 2.4], we have

######  Proposition 8.1.

$ll(\widehat{Z}_{\chi}(\lambda))=ll(\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}}))\geq
l(w^{I})+1$.

###### Proof.

The proof is similar to that in [3, Propo. 2.3]. Choose a reduced
decomposition $w^{I}=r_{1}r_{2}...r_{t}$ with $r_{i}$ being the simple
reflection for $1\leq i\leq t=l(w^{I})$. Set $s_{i}=r_{1}r_{2}...r_{i}$ and
$\lambda_{i}=\lambda-(p-1)(\rho-s_{i}\rho)$ for $1\leq i\leq t$. By [10,
§11.13] or [11, §3.10], there exists a nonzero homomorphism
$\phi_{i}:\widehat{Z}_{\chi}^{s_{i}}(\lambda_{i})\rightarrow\widehat{Z}_{\chi}^{s_{i+1}}(\lambda_{i+1}).$
The composition $\phi=\phi_{t}\circ\cdot\cdot\cdot\circ\phi_{2}\circ\phi_{1}:$

(8.1)
$\begin{CD}\widehat{Z}_{\chi}(\lambda)@>{\phi_{1}}>{}>\widehat{Z}^{s_{1}}_{\chi}(\lambda_{r})@>{\phi_{2}}>{}>\cdots
@>{\phi_{t}}>{}>\widehat{Z}_{\chi}^{w^{I}}(\lambda^{w^{I}}).\end{CD}$

is a homomorphism which maps the head $\widehat{L}_{\chi}(\lambda)$ of
$\widehat{Z}_{\chi}(\lambda)$ to the socle of
$\widehat{Z}_{\chi}^{w^{I}}(\lambda^{w^{I}})$ (cf. [11, Lem. 3.11]). Since
each module involved in (8.1) has a simple socle, each $\phi_{i}$ either kills
the socle or is an isomorphism (because the dimension of all baby Verma
modules are equal). Since $\lambda$ is $p$-regular, by [11, Lem. 3.5], all
$\phi_{i}$’s are not isomorphism. Then
$ll(\texttt{Im}\,\phi_{i}\circ\cdots\circ\phi_{2}\circ\phi_{1})\leq
ll(\texttt{Im}\,\phi_{i-1}\circ\cdots\circ\phi_{2}\circ\phi_{1})-1$. Then
$ll(\widehat{L}_{\chi}(\lambda))=ll(\texttt{Im}\,\phi_{t}\circ\cdots\circ\phi_{2}\circ\phi_{1})\leq
ll(\widehat{Z}_{\chi}(\lambda))-t$. This implies that
$ll(\widehat{Z}_{\chi}(\lambda))\geq l(w^{I})+1$.

Similarly, there exists a nonzero homomorphism
$\varphi_{t-i}:\widehat{Z}_{\chi}^{s_{i+1}}(\lambda_{i+1})\rightarrow\widehat{Z}_{\chi}^{s_{i}}(\lambda_{i}).$
The composition
$\varphi=\varphi_{t}\circ\cdot\cdot\cdot\varphi_{2}\circ\varphi_{1}:$

(8.2)
$\begin{CD}\widehat{Z}_{\chi}^{w^{I}}(\lambda^{w^{I}})@>{\varphi_{1}}>{}>\widehat{Z}^{s_{t-1}}_{\chi}(\lambda_{t-1})@>{\varphi_{2}}>{}>\cdots
@>{\varphi_{t}}>{}>\widehat{Z}_{\chi}(\lambda).\end{CD}$

is a homomorphism which maps the head $\widehat{L}_{\chi}(\lambda)$ of
$\widehat{Z}_{\chi}^{w^{I}}(\lambda^{w^{I}})$ to the socle of
$\widehat{Z}_{\chi}(\lambda)$. So we have
$ll(\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}}))\geq l(w^{I})+1$. ∎

######  Proposition 8.2.

$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))\geq
l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))+1.$

###### Proof.

By (1.2), we have
$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}}))$.
By Theorem 6.2 and Proposition 8.1, we have
$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Z}_{\chi}(\lambda))+ll(\mathcal{L}_{\chi}(\lambda))-1$.
∎

If Conjecture 7.1 holds, we have $ll(\widehat{Q}^{I}_{\chi}(\lambda))\geq
l(w^{I})+l(w_{I})+1.$

######  Proposition 8.3.

$ll(\widehat{Q}_{\chi}(\lambda))\geq
2l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))+1$.

###### Proof.

According to [11, Prop. 1.21], $\widehat{Q}_{\chi}(\lambda)$ has both a
$\widehat{Q}^{I}_{\chi}$\- and a $\widehat{Q}_{\chi}^{w^{I}}$-filtration.
Furthermore, it has both a $\widehat{Z}_{\chi}$\- and a
$\widehat{Z}^{w^{I}}_{\chi}$-filtration. Since the head and socle of
$\widehat{Q}_{\chi}(\lambda)$ are isomorphism $\widehat{L}_{\chi}(\lambda)$,
it follows that $\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{Z}_{\chi}(\lambda)$ and $\widehat{Q}^{I}_{\chi}(\lambda)$. Moreover,
it is also the socle of $\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ and
$\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})$. Then we have two injections
$i_{1}:\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}})\hookrightarrow\widehat{Q}_{\chi}(\lambda)$
and
$i_{2}:\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})\hookrightarrow\widehat{Q}_{\chi}(\lambda)$,
and two surjections
$\pi_{1}:\widehat{Q}_{\chi}(\lambda)\twoheadrightarrow\widehat{Z}_{\chi}(\lambda)$
and
$\pi_{2}:\widehat{Q}_{\chi}(\lambda)\twoheadrightarrow\widehat{Q}^{I}_{\chi}(\lambda)$.

By §1.10, the composite $\pi_{1}\circ
i_{1}:\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}})\hookrightarrow\widehat{Q}_{\chi}(\lambda)\twoheadrightarrow\widehat{Z}_{\chi}(\lambda)$
is nonzero which maps the head of
$\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ to the socle of
$\widehat{Z}_{\chi}(\lambda)$. Furthermore, the composite $\pi_{2}\circ
i_{2}:\widehat{Q}^{w^{I}}_{\chi}(\lambda^{w^{I}})\hookrightarrow\widehat{Q}_{\chi}(\lambda)\twoheadrightarrow\widehat{Q}^{I}_{\chi}(\lambda)$
is nonzero and the image of $\pi_{2}\circ i_{2}$ contains
$\widehat{L}_{\chi}(\lambda)$. Suppose that
$[\mbox{rad}^{l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))}\widehat{Q}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]=0$.
We have $\pi_{2}$ vanishes on
$\mbox{rad}^{l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))}\widehat{Q}_{\chi}(\lambda)$,
which means $\pi_{2}$ factors through
$\mbox{head}^{l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))}\widehat{Q}_{\chi}(\lambda)$,
and
$l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))=ll(\mbox{head}^{l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))}\widehat{Q}_{\chi}(\lambda))\geq
ll(\widehat{Q}^{I}_{\chi}(\lambda))$. This contradicts with Proposition 8.2.
Hence
$[\mbox{rad}^{l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))}\widehat{Q}_{\chi}(\lambda):\widehat{L}_{\chi}(\lambda)]\neq
0$ and $i_{1}$ maps $\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}})$ into
$\mbox{rad}^{l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))}\widehat{Q}_{\chi}(\lambda)$.
So we have

$ll(\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}}))\leq
ll(\mbox{rad}^{l(w^{I})+ll(\mathcal{L}_{\chi}(\lambda))}\widehat{Q}_{\chi}(\lambda))=ll(\widehat{Q}_{\chi}(\lambda))-l(w^{I})-ll(\mathcal{L}_{\chi}(\lambda)).$

This together with Proposition 8.1 yields the desired assertion. ∎

If Conjecture 7.1 holds, then $ll(\widehat{Q}_{\chi}(\lambda))\geq
2l(w^{I})+l(w_{I})+1.$

###  8.2. A conjecture on Loewy lengths

Based on the above results, we present the following conjecture on Loewy
lengths of proper standard modules, standard modules and indecomposable
projective modules

######  Conjecture 8.4.

Assume that $\chi$ has standard Levi form. When $\lambda\in X(T)$ is
$p$-regular, we conjecture that

$ll(\widehat{Z}_{\chi}(\lambda))=ll(\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}}))=l(w^{I})+1;$

$ll(\widehat{Q}^{I}_{\chi}(\lambda))=ll(\widehat{Q}^{w^{I}}_{\chi}((\lambda)^{w^{I}}))=l(w^{I})+l(w_{I})+1;$

$ll(\widehat{Q}_{\chi}(\lambda))=2l(w^{I})+l(w_{I})+1.$

######  Remark 8.5.

(1) When $\chi$ has standard Levi form and is regular nilpotent,
$\widehat{Z}_{\chi}(\lambda)\cong\widehat{Z}^{w^{I}}_{\chi}(\lambda^{w^{I}})$
is simple and $l(w^{I})=0$. Conjecture 8.4(1) holds.

(2) When $\chi$ has standard Levi form and is subregular nilpotent, by the
results of Jantzen(cf. [13, §2.6]), Conjecture 8.4(1) is true.

(3) When $\chi=0$, $l(w^{I})=l(w_{0})$. By [3, §5], when Lusztig’s conjecture
on the irreducible characters for $G$ holds, Conjecture 8.4(1) and (3) are
correct.

(4) Based on Conjecture 7.1 and Theorem6.2, Conjecture 8.4(1) implies
Conjecture 8.4(2).

###  8.3. An example

######  Example 8.6.

Let ${\mathfrak{g}}=\mathfrak{sl}(4,k)$ and
$\Pi=\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}$ be the the set of simple roots.
Assume $\chi$ has standard Levi form with $I=\\{\alpha_{1},\alpha_{2}\\}$. So
$\chi$ is subregular nilpotent. Then $w_{I}=s_{\alpha_{1}}s_{\alpha_{1}}$ with
$l(w_{I})=2$ and $w^{I}=s_{\alpha_{3}}s_{\alpha_{2}}s_{\alpha_{1}}$ with
$l(w^{I})=3$.

Fix a wight $\xi_{0}\in C_{0}$ with
$\xi_{0}+\rho=r_{1}\varpi_{1}+r_{2}\varpi_{2}+r_{3}\varpi_{3}=(r_{1},r_{2},r_{3})$
where $\varpi_{1}$, $\varpi_{2}$ and $\varpi_{3}$ are fundamental weights. Set
$r_{0}=p-r_{1}-r_{2}-r_{3}$. Let
$\sigma=s_{\alpha_{1}}s_{\alpha_{2}}s_{\alpha_{3}}$, and
$\xi_{i}=\sigma^{i}.\xi_{0},i=1,2,3$. Then
$\xi_{1}+\rho=(-(r_{1}+r_{2}+r_{3}),r_{1},r_{2})$,
$\xi_{2}+\rho=(r_{3},-(r_{1}+r_{2}+r_{3}),r_{1})$ and
$\xi_{3}+\rho=(r_{2},r_{3},-(r_{1}+r_{2}+r_{3}))$ . By [13, §2.3],
$\\{\widehat{L}_{\chi}(\xi_{i})\mid 0\leq i\leq 3\\}$ is the set of
isomorphism classes of simple modules in the block containing
$\widehat{L}_{\chi}(\xi_{0})$. For $0\leq i\leq 3$, according to [13, §2.6],
the baby Verma module $\widehat{Z}_{\chi}(\xi_{i})$ is uniserial and its
composition factors (from top to bottom) are

(8.3)
$\widehat{L}_{\chi}(\xi_{i}),...,\widehat{L}_{\chi}(\xi_{3}),\widehat{L}_{\chi}(\xi_{0})\otimes(-3p),...,\widehat{L}_{\chi}(\xi_{i-1})\otimes(-3p),i=1,2,3.$

When $\xi\in X(T)$ is $p$-regular,
$ll(\widehat{Z}_{\chi}(\xi))=ll(\widehat{Z}^{w^{I}}_{\chi}(\xi^{w^{I}}))=4=l(w^{I})+1$(Conjecture
8.4(1) holds). Because the root system of ${\mathfrak{g}}_{I}$ is of type
$A_{2}$, by Example 7.3, we know Conjecture 7.1 holds. Hence, Theorem 6.2
implies that

$ll(\widehat{Q}^{I}_{\chi}(\xi))=ll(\widehat{Q}^{w^{I}}_{\chi}((\xi)^{w^{I}}))=l(w^{I})+l(w_{I})+1.$

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