# Category $\mathcal{O}$ for Truncated Current Lie Algebras

Matthew Chaffe and Lewis Topley<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15
2TT, UK.  Department of Mathematical Sciences, University of Bath, North Road,
Claverton Down, Bath BA2 7AY, UK.

###### Abstract.

In this paper we study an analogue of the Bernstein–Gelfand–Gelfand category
$\mathcal{O}$ for truncated current Lie algebras $\mathfrak{g}_{n}$ attached
to a complex semisimple Lie algebra. This category admits Verma modules and
simple modules, each parametrised by the dual space of the truncated currents
on a choice of Cartan subalgebra in $\mathfrak{g}$. Our main result describes
an inductive procedure for computing composition multiplicities of simples
inside Vermas for $\mathfrak{g}_{n}$, in terms of similar composition
multiplicities for $\mathfrak{l}_{n-1}$ where $\mathfrak{l}$ is a Levi
subalgebra. As a consequence, these numbers are expressed as integral linear
combinations of Kazhdan–Lusztig polynomials evaluated at 1. This generalises
recent work of the first author, where the case $n=1$ was treated.

##  1\. Introduction

Truncated current Lie algebras have appeared in numerous parts of the
literature in recent years, and a large part of their interest stems from the
fact that they interpolate between the finite dimensional simple Lie algebras
and the vacuum parabolic of the corresponding untwisted affine Lie algebra. If
$\mathfrak{g}=\operatorname{Lie}(G)$ is the Lie algebra of a complex reductive
algebraic group then the group $G[t]$ of polynomial currents in $G$ is the
(infinite dimensional) algebraic group of regular maps
$\mathbb{A}^{1}_{\mathbb{C}}\to G$. The current Lie algebra
$\mathfrak{g}[t]:=\operatorname{Lie}G[t]=\mathfrak{g}\otimes\mathbb{C}[t]$ is
isomorphic to the derived subalgebra of a maximal parabolic of the Kac–Moody
affinisation of $\mathfrak{g}$, and the $n$th truncated current Lie algebra is
the quotient $\mathfrak{g}_{n}:=\mathfrak{g}\otimes\mathbb{C}[t]/(t^{n+1})$.
Equivalently, $\mathfrak{g}_{n}$ can be described as the Lie algebra of the
$n$th jet scheme $J_{n}G$ of $G$.

The first truncated currents Lie algebra $\mathfrak{g}_{1}$ appeared in the
work of Takiff, and so they are often referred to as Takiff Lie algebras. He
showed that the symmetric invariant algebra
$S(\mathfrak{g}_{1})^{\mathfrak{g}_{1}}$ is a polynomial algebra on
$2\operatorname{rank}(\mathfrak{g})$ variables [Ta71], generalising the
classical theorem of Chevalley which describes
$S(\mathfrak{g}_{0})^{\mathfrak{g}_{0}}$. Later Raïs–Tauvel extended Takiff’s
theorem for arbitrary $n$ [RT92]. Since the latter work is a crucial
ingredient in our paper, we will briefly describe their main results in
Section 2.2. More recently, Macedo–Savage extended their work further to the
case of truncated multicurrents [MS19], and Panyushev–Yakimova showed that for
complex Lie algebras the operation $\mathfrak{g}\mapsto\mathfrak{g}_{1}$
preserves the property of having polynomial symmetric invariants, under mild
assumptions [PY20].

These works on invariant theory have important applications in the theory of
vertex algebras. Notably, the first approximation to describing the centre of
the critical level universal affine vertex algebra associated to
$\mathfrak{g}$ (a.k.a. the Feigin–Frenkel centre) is the description of the
semi-classical limit, which is equal to
$S(\mathfrak{g}[t^{-1}]t^{-1})^{\mathfrak{g}[t]}$. In turn can be described as
the direct limit of the algebras $S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$
discussed in the previous paragraph (see [Fr07, §3.4] or [Mo18, §6] for
example). This connection with the Feigin–Frenkel centre was recently used by
Arakawa–Premet to provide a positive solution to Vinberg’s problem for
centralisers, using affine $W$-algebras [AP17]. We also mention the work of
Kamgarpour [Ka16] for a discussion of the relationship with the geometric
Langlands program. Another connection with the theory of $W$-algebras is given
by the work of Brundan–Kleshchev [BK06, §12], which states that the finite
$W$-algebra for $\mathfrak{gl}_{N}$ associated to a nilpotent element with all
Jordan blocks of size $n$ is isomorphic to a truncated Yangian, which admits
$U(\mathfrak{g}_{n})$ as a filtered degeneration.

This article focuses on the representation theory of truncated current
algebras. The most famous category of modules for a complex reductive Lie
algebra $\mathfrak{g}$ is the Bernstein–Gelfand–Gelfand (BGG) category
$\mathcal{O}$, which is an abelian category containing all highest weight
modules (see [Hu08] for a survey). Wilson extended the notion of highest
weight modules to all truncated currents [Wi11], and subsequently
Mazorchuck–Sörderberg introduced a version of category $\mathcal{O}$ for
Takiff $\mathfrak{sl}_{2}$ [MSo19]. The most recent development in this field
is the work of the first author [Ch23] which made a thorough study of category
$\mathcal{O}$ for all Takiff Lie algebras, eventually showing that the
composition multiplicities of simple modules inside Verma modules can be
determined by certain formulas involving Kazhdan–Lusztig polynomials. The
results of the present paper generalise all of the main results of loc. cit.
to the case of truncated current Lie algebras $\mathfrak{g}_{n}$.

For the rest of the introduction we fix $\mathfrak{g}=\operatorname{Lie}(G)$
where $G$ is a complex reductive algebraic group and fix $n>0$. We also fix a
triangular decomposition
$\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$ and
write $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}^{+}$. This gives rise to a
triangular decomposition
$\mathfrak{g}_{n}=\mathfrak{n}^{-}_{n}\oplus\mathfrak{h}_{n}\oplus\mathfrak{n}^{+}_{n}$,
and we say that a module is highest weight if it is generated (as a
$\mathfrak{g}_{n}$-module) by a one dimensional $\mathfrak{b}_{n}$-module.
These one dimensional modules are parametrised by $\mathfrak{h}_{n}^{*}$, and
the one dimensional module afforded by $\lambda\in\mathfrak{h}_{n}^{*}$ is
denoted $\mathbb{C}_{\lambda}$. We define the universal highest weight module
or Verma module of weight $\lambda$ by

(1.1) $\displaystyle
M_{\lambda}:=U(\mathfrak{g}_{n})\otimes_{U(\mathfrak{b}_{n})}\mathbb{C}_{\lambda}.$

Since this is semisimple over $\mathfrak{h}=\mathfrak{h}\otimes
1\subseteq\mathfrak{h}_{n}$, with one dimensional
$\lambda|_{\mathfrak{h}}$-weight space, it follows that it has a unique
maximal submodule and a unique simple quotient, which we denote $L_{\lambda}$.

We study the category $\mathcal{O}(\mathfrak{g}_{n})$ of finitely generated
$\mathfrak{g}_{n}$-modules on which $\mathfrak{n}_{n}^{+}$ acts locally
nilpotently, $\mathfrak{h}$ acts semisimply and $\mathfrak{h}_{n}^{\geq
1}:=\mathfrak{h}\otimes t\mathbb{C}[t]/(t^{n+1})\subseteq\mathfrak{h}_{n}$
acts locally finitely (see Definition 3.1). This category is closed under
quotients and submodules, and the simple modules are precisely
$\\{L_{\lambda}\mid\lambda\in\mathfrak{h}_{n}^{*}\\}$.

One especially nice feature of the BGG category $\mathcal{O}$ for
$\mathfrak{g}$ it that it is artinian, however this fails for
$\mathcal{O}(\mathfrak{g}_{n})$. Therefore we define composition
multiplicities $[M:L_{\mu}]$ using formal characters: every module
$M\in\mathcal{O}(\mathfrak{g}_{n})$ has finite dimensional
$\mathfrak{h}$-weight spaces and the formal character $\operatorname{ch}M$ can
be expressed uniquely as a non-negative integral linear combinations of
characters of simple modules. We call the coefficients appearing in these sums
the composition multiplicities of $M\in\mathcal{O}(\mathfrak{g}_{n})$. We
explain this in more detail in Section 6, and give a description (see Lemma
6.2) of the $[M:L_{\lambda}]$ in the spirit of composition multiplicities for
affine Lie algebras [Kac90, Proposition 9.7].

We now state the main result of this paper, which follows directly from
Corollary 6.8

######  Theorem 1.1.

Let $n>0$, let $G$ be a connected reductive group and
$\lambda,\mu\in\mathfrak{h}_{n}^{*}$. The composition multiplicity
$[M_{\lambda}:L_{\mu}]$ can be expressed via a precise formula (6.2) in terms
of composition multiplicities of simple modules inside Verma modules for a
truncated current algebra $\mathfrak{l}_{n-1}$ where
$\mathfrak{l}=\operatorname{Lie}(L)$ for some Levi subgroup $L\subseteq G$.

Theorem 1.1 suggests an inductive procedure for calculating the composition
multiplicities $[M_{\lambda}:L_{\mu}]$ for the $n$th truncated current algebra
in terms of the analogous composition multiplicities for $\mathfrak{g}_{0}$.
By the proof of the Kazhdan–Lusztig conjecture by Beilinson–Bernstein and
Brylinski–Kashiwara (see [HTT08]) we know that for $n=0$ the composition
multiplicities $[M_{\lambda}:L_{\mu}]$ are given by the evaluation at 1 of
certain Kazhdan–Lusztig polynomials. Hence for $n>0$ the values
$[M_{\lambda}:L_{\mu}]$ can be described by formulas involving non-negative
integral linear combinations of these Kazhdan–Lusztig polynomials at 1. It is
interesting to wonder whether our formulas have a natural geometric
interpretation on the flag variety.

We remark that the methods of this article generalise those of [Ch23]. In the
rest of the introduction we highlight some of our other key results, and
describe the structure of the paper.

In Section 2 we introduce the basic notation which will be used throughout the
paper. We then recall the work of Raïs–Tauvel [RT92] on
$S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$ and use it to describe the generators
of the centre of the enveloping algebra $U(\mathfrak{g}_{n})$.

In Section 3 we describe all of the elementary properties of our category
$\mathcal{O}(\mathfrak{g}_{n})$. We also explain that the category can be
decomposed into a direct sum of abelian subcategories which are parameterised
by the generalised eigenspaces of $\mathfrak{h}\otimes t^{i}$ for $i=1,...,n$.
The generalised eigenvalues are parametrised by $(\mathfrak{h}_{n}^{\geq
1})^{*}$, and for $\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ we call the
subcategory $\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ a Jordan block of
$\mathcal{O}(\mathfrak{g}_{n})$.

In Section 4 we introduce one of our main tools for simplifying the the study
of Jordan blocks of $\mathcal{O}(\mathfrak{g}_{n})$. Let
$\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ and let
$\mu_{n}:=\mu|_{\mathfrak{h}\otimes t^{n}}$, which we identify with an element
of $\mathfrak{h}^{*}$ in the obvious fashion. The centraliser
$\mathfrak{l}:=\mathfrak{g}^{\mu_{n}}$ is a Levi subalgebra containing
$\mathfrak{h}$, and we suppose that it is the Levi factor of a standard
parabolic $\mathfrak{p}\subseteq\mathfrak{g}$ such that
$\mathfrak{r}=\operatorname{Rad}(\mathfrak{p})\subseteq\mathfrak{n}^{+}$. In
this case we can parabolically induce modules from
$\mathcal{O}^{(\mu)}(\mathfrak{l}_{n})$ to
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$. Similarly we have a functor of
$\mathfrak{r}_{n}$-invariants in the opposite direction.

A precise statement of the following result is given in Theorem 4.1.

######  Theorem 1.2.

If $\mathfrak{l}=\mathfrak{g}^{\mu_{n}}$ is a standard Levi subalgebra,
parabolic induction and $\mathfrak{r}_{n}$-invariants are quasi-inverse
equivalences between $\mathcal{O}^{(\mu)}(\mathfrak{l}_{n})$ and
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$.

In the case where $\mathfrak{g}^{\mu_{n}}$ is a standard Levi subalgebra, the
theorem allows us to reduce the study of
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ to the case where $\mu_{n}$ is
supported on the centre of $\mathfrak{g}$. Another simple reduction (Lemma
3.7) impels us to focus on the case $\mu_{n}=0$.

Theorem 1.2 was inspired by a result of Friedlander–Parshall in modular
representation theory of Lie algebras [FP88, Theorem 3.2]. Using the fact that
modules in $\mathcal{O}(\mathfrak{g}_{n})$ admit finite filtrations with
highest weight sections (Lemma 3.3) the proof quickly reduces to showing that
$(\bullet)^{\mathfrak{r}_{n}}$ is exact. This is the hardest part of the
proof, and requires a careful study of central characters of highest weight
modules (Theorem 4.2), which ultimately depends on our description of the
centre of $U(\mathfrak{g}_{n})$ given in Section 2.2. In particular, in
comparison to the Takiff case treated in [Ch23, §4] the description of the
central elements given by [RT92] is more difficult to work with.

In Section 5 we remove the requirement in Theorem 1.2 that $\mathfrak{l}$ is a
standard Levi subalgebra; this is achieved with the use of twisting functors.
These were first introduced by Arkhipov [Ar01] and were applied to category
$\mathcal{O}$ for $\mathfrak{g}$ by Andersen–Stroppel [AS03]. They were used
in the context of Takiff Lie algebras by the first author in [Ch23], whilst
proving a similar reduction to that of the present paper. The main result of
Section 5 is Theorem 5.9, which implies, together with Proposition 6.7, the
following:

######  Theorem 1.3.

Every Jordan block of $\mathcal{O}(\mathfrak{g}_{n})$ is equivalent to a
Jordan block $\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ such that
$\mathfrak{g}^{\mu_{n}}$ is a standard Levi subalgebra.

Together with Theorem 1.2 this allows us to reduce the study of Jordan blocks
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ to the case where $\mu_{n}=0$. The
proof of Theorem 1.3 is similar to [Ch23, Theorem 5.7] however several of the
proofs, especially the proofs of Lemmas 5.15 and 5.16, are significantly more
complicated. As such we provide detailed arguments for all of the proofs which
are different, and refer the reader to the appropriate part of op. cit.
otherwise.

In Section 6 we address the problem of defining composition multiplicities and
computing multiplicities of simples in Vermas. We begin by explaining how
these numbers are defined and interpreting them in terms of composition
series, which is all very similar to [Ch23, §6.1], except that the structure
of the filtrations considered in Lemma 6.2 is more complicated than the
analogus filtrations in the Takiff case. Finally we focus on the blocks
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ with $\mu_{n}=0$. The main result of
the section is Corollary 6.8 which is a precise version of Theorem 1.1.

### Acknowledgements

Both authors would like to thank Simon Goodwin for many useful conversations.
The first author is grateful to the EPSRC for studentship funding, and the
research of the second author is supported by a UKRI FLF, grant numbers
MR/S032657/1, MR/S032657/2, MR/S032657/3.

##  2\. Preliminaries

In this paper all vector spaces, algebras and algebraic varieties will be
defined over $\mathbb{C}$. Unadorned tensor products are taken over
$\mathbb{C}$.

###  2.1. Reductive Lie algebras and truncated currents

From henceforth we fix a reductive algebraic group $G$ of rank $r$, with Lie
algebra $\mathfrak{g}$, and a choice of maximal torus
$\mathfrak{h}\subseteq\mathfrak{g}$. The Weyl group
$N_{G}(\mathfrak{h})/C_{G}(\mathfrak{h})$ will be denoted $W$. Let
$\Phi\subseteq\mathfrak{h}^{*}$ be the root system of $\mathfrak{g}$ and let
$\Delta$ be a choice of simple roots for $\Phi$, which give a set of positive
roots $\Phi^{+}=\Phi\cap\sum_{\alpha\in\Delta}\mathbb{Z}_{\geq 0}\alpha$. For
each root $\alpha\in\Phi$ we have a one dimensional root space
$\mathfrak{g}_{\alpha}$ and for positive roots $\alpha$ we fix a choice of
triple $(e_{\alpha},h_{\alpha},f_{\alpha})$ such that
$e_{\alpha}\in\mathfrak{g}_{\alpha}$, $f_{\alpha}\in\mathfrak{g}_{-\alpha}$,
and $h_{\alpha}:=[e_{\alpha},f_{\alpha}]$, satisfying $\alpha(h_{\alpha})=2$.

These data give us a triangular decomposition

(2.1) $\displaystyle\mathfrak{g}$ $\displaystyle=$
$\displaystyle\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+},$

where
$\mathfrak{n}^{\pm}:=\bigoplus_{\alpha\in\Phi^{+}}\mathfrak{g}_{\pm\alpha}$.
We also write $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}^{+}$ for the
corresponding Borel subalgebra of $\mathfrak{g}$.

For $n\geq 1$ we consider the truncated current algebra
$\mathfrak{g}_{n}:=\mathfrak{g}\otimes\mathbb{C}[t]/(t^{n+1})$. For any
subalgebra $\mathfrak{s}\subseteq\mathfrak{g}$, have a natural embedding
$\mathfrak{s}_{n}\subseteq\mathfrak{g}_{n}$ of truncated currents. We make the
notation $\mathfrak{s}_{n}^{i}:=\mathfrak{s}\otimes
t^{i}\subseteq\mathfrak{g}_{n}$. This gives a grading
$\mathfrak{s}_{n}=\bigoplus_{i=0}^{n}\mathfrak{s}_{n}^{i}$. We write
$\mathfrak{s}_{n}^{\geq m}$ for the sum of the graded pieces of degree
$m,m+1,...,n$.

For $x\in\mathfrak{g}$ and $\alpha\in\Phi^{+}$ then we make the notation

(2.6) $\displaystyle\begin{array}[]{rcl}x_{i}&:=&x\otimes t^{i},\\\
e_{\alpha,i}&:=&e_{\alpha}\otimes t^{i},\\\ f_{\alpha,i}&:=&f_{\alpha}\otimes
t^{i},\\\ h_{\alpha,i}&:=&h_{\alpha}\otimes t^{i}\end{array}$

We will often need to consider linear functions of $\mathfrak{g}_{n}$ and
$\mathfrak{h}_{n}$. If $\lambda\in\mathfrak{h}_{n}^{*}$ then we write

(2.9)
$\displaystyle\begin{array}[]{rcl}\lambda_{i}&:=&\lambda|_{\mathfrak{h}_{n}^{i}},\\\
\lambda_{\geq i}&:=&\lambda|_{\mathfrak{h}_{n}^{\geq i}},\end{array}$

for $i=0,...,n$, and we often view $\lambda_{i}$ as an element of
$\mathfrak{h}^{*}$ via the obvious identification
$\mathfrak{h}=\mathfrak{h}_{n}^{i}$. If $\lambda\in\mathfrak{h}_{n}^{*}$ then
we can also view it as an element of $\mathfrak{g}_{n}^{*}$ by setting
$\lambda(\mathfrak{n}_{n}^{\pm})=0$.

###  2.2. The symmetric invariants and the centre of the enveloping algebra

We write $S(\mathfrak{g}_{n})$ and $U(\mathfrak{g}_{n})$ for the symmetric
algebra and the universal enveloping algebra of $\mathfrak{g}_{n}$,
respectively. The algebra $U(\mathfrak{g}_{n})$ is equipped with the
Poincaré–Birkhoff–Witt (PBW) filtration, and the associated graded algebra is
$\operatorname{gr}U(\mathfrak{g}_{n})=S(\mathfrak{g}_{n})$ the symmetric
algebra on $\mathfrak{g}_{n}$. In the present section we describe the centre
$Z(\mathfrak{g}_{n})$ of the enveloping algebra and its semi-classical limit.

The adjoint representation of $\mathfrak{g}_{n}$ extends uniquely to an action
of $\mathfrak{g}_{n}$ on both $S(\mathfrak{g}_{n})$ and $U(\mathfrak{g}_{n})$
by derivations [Di77, Proposition 2.4.9]. The centre $Z(\mathfrak{g}_{n})$ is
equal to $U(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$ and the identification
$\operatorname{gr}U(\mathfrak{g}_{n})=S(\mathfrak{g}_{n})$ is
$\mathfrak{g}_{n}$-equivariant. The invariant algebra
$S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$ was first described by Raïs–Tauvel
[RT92], as we now recall.

We define a series of (vector space) endomorphisms
$\partial^{(0)},...,\partial^{(n)}$ of $S(\mathfrak{g}_{n})$ by putting
$\partial^{(0)}$ equal to the identity map, and then inductively defining

(2.12)
$\displaystyle\partial^{(k)}(x_{j})=\left\\{\begin{array}[]{cl}\binom{j}{k}x_{j-k}&\text{
if }j\geq k,\\\ 0&\text{ otherwise},\end{array}\right.$ (2.13)
$\displaystyle\partial^{(k)}(fg)=\sum_{i+j=k}\partial^{(i)}(f)\partial^{(j)}(g).$

for $x\in\mathfrak{g}$ and $f,g\in S(\mathfrak{g}_{n})$. These endomorphisms
will be used to construct the basic invariant generators introduced in op.
cit. We remark that the family of operators
$(\partial^{(0)},...,\partial^{(n)})$ which we define here and an instance of
higher order derivation [SH], but we shall not use this formalism.

Recall Chevalley’s restriction theorem, which states that
$S(\mathfrak{g})^{G}=S(\mathfrak{g})^{\mathfrak{g}}\cong S(\mathfrak{h})^{W}$.
Furthermore the Chevalley–Sheppard–Todd theorem implies that
$S(\mathfrak{g})^{\mathfrak{g}}$ is a graded polynomial algebra generated by
$\operatorname{rank}(\mathfrak{g})$ homogeneous elements. Write
$p_{1},...,p_{r}$ for a choice of such elements.

The adjoint representation of $\mathfrak{g}_{n}$ stabilises
$\mathfrak{g}_{n}^{n}$ and the action factors through the map
$\mathfrak{g}_{n}\twoheadrightarrow\mathfrak{g}_{n}/\mathfrak{g}_{n}^{\geq
1}=\mathfrak{g}_{0}=\mathfrak{g}$. Therefore we have a natural inclusion
$S(\mathfrak{g})^{\mathfrak{g}}\hookrightarrow
S(\mathfrak{g}_{n}^{n})^{\mathfrak{g}_{n}}$. Abusing notation we view
$p_{1},...,p_{r}$ as elements of $S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$.

######  Theorem 2.1.

[RT92, §3] The invariant algebra $S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$ is a
polynomial ring generated by $(n+1)r$ elements

(2.14) $\displaystyle\\{\partial^{(k)}p_{j}\mid j=1,...,r,\ k=0,...,n\\}.$

Let $d^{(k)}:S(\mathfrak{g}_{n}^{n})\to S(\mathfrak{g}_{n})$ be the partial
derivative $\sum_{x}x_{n-k}\frac{d}{dx_{n}}$ where the sum is taken over a
basis for $\mathfrak{g}$. Then it follows straight from the definitions
(2.12), (2.13) that there exist elements $q_{j}^{(k)}\in
S(\mathfrak{g}_{n}^{\geq n-k+1})$ such that

(2.15) $\displaystyle\partial^{(k)}p_{j}=d^{(k)}p_{j}+q_{j}^{(k)}\text{ for
}j=1,...,r,\ k=0,...,n.$

We refer the reader to [RT92, Lemma 3.2(ii)] for the proof of (2.15). The next
observation follows directly from the definitions, and we record it as a lemma
for later use.

######  Lemma 2.2.

The map $d^{(k)}$ sends $\operatorname{ad}(\mathfrak{h})$-invariants to
$\operatorname{ad}(\mathfrak{h})$-invariants. $\hfill\qed$

Finally we give a description of the centre $Z(\mathfrak{g}_{n})$ of the
enveloping algebra. There is an isomorphism of $\mathfrak{g}_{n}$-modules
$\omega:S(\mathfrak{g}_{n})\to U(\mathfrak{g}_{n})$ called the symmetrisation
map [Di77, §2.4]. It is defined by the rule

(2.16) $\displaystyle\omega:x^{1}\cdots
x^{m}\mapsto\frac{1}{m!}\sum_{\sigma\in\mathfrak{S}_{m}}x^{\sigma 1}\cdots
x^{\sigma m}$

where $x^{1},...,x^{m}\in\mathfrak{g}_{n}$ are any elements, so that
$x^{1}\cdots x^{m}\in S(\mathfrak{g}_{n})$ is a monomial of degree $m$, and
$\mathfrak{S}_{m}$ denotes the symmetric group on $m$ letters. If
$\operatorname{gr}:U(\mathfrak{g}_{n})\to S(\mathfrak{g}_{n})$ is the (non-
linear) map defined by taking the top degree component with respect to the
Poincaré–Birkoff–Witt filtration, then

(2.17) $\displaystyle\operatorname{gr}\circ\ \omega\text{ is equal to the
identity mapping on each graded piece of }S(\mathfrak{g}_{n}).$

Since the isomorphism
$\omega:S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}\overset{\sim}{\longrightarrow}U(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}=Z(\mathfrak{g}_{n})$
respects filtrations we can describe the $Z(\mathfrak{g}_{n})$.

######  Corollary 2.3.

$Z(\mathfrak{g}_{n})$ is a polynomial algebra generated by $(n+1)r$ elements

(2.18) $\displaystyle\\{\omega(\partial^{(k)}p_{j})\mid j=1,...,r,\
k=0,...,n\\}.$

###### Proof.

Let $Z\subseteq Z(\mathfrak{g}_{n})$ be the subalgebra generated by the
elements (2.18). Since the inclusion
$\operatorname{gr}Z\subseteq\operatorname{gr}Z(\mathfrak{g}_{n})$ is an
equality it follows that the inclusion $Z\subseteq Z(\mathfrak{g}_{n})$ is
also an equality. If these elements admit a non-trivial algebraic relation
then taking $\operatorname{gr}$ and using (2.17) we see that the elements
(2.14) would admit a non-trivial relation. By Theorem 2.1 we see that (2.18)
are algebraically independent. ∎

##  3\. Category $\mathcal{O}$ for Truncated Current Lie algebras

###  3.1. Definition and first properties of $\mathcal{O}$

We begin by stating the definition of category $\mathcal{O}$ for
$\mathfrak{g}_{n}$.

######  Definition 3.1.

The category $\mathcal{O}(\mathfrak{g}_{n})$ is the full subcategory of
$U(\mathfrak{g}_{n})$-Mod with objects $M$ satisfying the following:

  1. $(\mathcal{O}1)$

$M$ is finitely generated.

  2. $(\mathcal{O}2)$

$\mathfrak{h}_{n}^{0}$ acts semisimply on $M$.

  3. $(\mathcal{O}3)$

$\mathfrak{h}_{n}^{\geq 1}\oplus\mathfrak{n}_{n}^{+}$ acts locally finitely on
$M$.

Note that $\mathcal{O}(\mathfrak{g}_{0})$ is nothing but the
Bernstein–Gelfand–Gelfand (BGG) category $\mathcal{O}$ for
$\mathfrak{g}_{0}=\mathfrak{g}$.

We refer the reader to [Hu08] for a fairly comprehensive introduction to the
algebraic study of $\mathcal{O}$, and to [HTT08] for a discussion of the
relationship between $\mathcal{O}$ and $\mathcal{D}$-modules on the flag
variety.

It is not hard to see that $\mathcal{O}(\mathfrak{g}_{n})$ is closed under
submodules, quotients, and finite direct sums. Furthermore by ($\mathcal{O}1)$
every $M\in\mathcal{O}(\mathfrak{g}_{n})$ is noetherian, since
$U(\mathfrak{g}_{n})$ is noetherian.

Let $M\in\mathcal{O}(\mathfrak{g}_{n})$ and $\lambda\in\mathfrak{h}^{*}$.
Define the weight space of weight $\lambda$ by

(3.1) $\displaystyle M^{\lambda}=\\{v\in M\mid h_{0}v=\lambda(h)v\text{ for
all }h\in\mathfrak{h}\\}.$

By ($\mathcal{O}2$) we have
$M=\bigoplus_{\lambda\in\mathfrak{h}^{*}}M^{\lambda}$ and this is a module
grading of $M$, if we equip $U(\mathfrak{g}_{n})$ with its natural grading by
the root lattice. The elements of $M^{\lambda}$ are called _weight vectors of
weight $\lambda\in\mathfrak{h}^{*}$_. If $m\in M^{\lambda}$ is a weight vector
satisfying $\mathfrak{n}_{n}^{+}\cdot m=0$ then we say that $m$ is a maximal
vector of weight $\lambda$.

Now let $\lambda\in\mathfrak{h}_{n}^{*}$ and recall that
$\lambda_{i}:=\lambda|_{\mathfrak{h}_{n}^{i}}$. We say that $m\in
M^{\lambda_{0}}$ is a _highest weight vector of weight $\lambda$_ if $m$ is
maximal of weight $\lambda$ and

(3.2) $\displaystyle h\cdot m=\lambda(h)m\text{ for all
}h\in\mathfrak{h}_{n}.$

The following basic properties of weight spaces of
$M\in\mathcal{O}(\mathfrak{g}_{n})$ can be proven using the same argument as
in BGG category $\mathcal{O}$, see [Hu08, §1.1]:

(3.5) $\displaystyle\begin{array}[]{l}\dim(M^{\lambda})<\infty\text{ for all
}\lambda\in\mathfrak{h}^{*},\vspace{6pt}\\\
\\{\lambda\in\mathfrak{h}^{*}:M^{\lambda}\neq 0\\}\subseteq\bigcup_{\lambda\in
I}(\lambda-\mathbb{Z}_{\geq 0}\Phi^{+})\text{ for some finite subset
}I\subseteq\mathfrak{h}^{*}.\end{array}$

If $m\in M\in\mathcal{O}(\mathfrak{g}_{n})$ is a maximal vector then we can
find a highest weight vector in $U(\mathfrak{h}_{n})v$ thanks to
$(\mathcal{O}3)$. This proves the following result.

######  Lemma 3.2.

[Ch23, Corollary 3.3] Suppose that $M\in\mathcal{O}$ admits a nonzero maximal
vector of weight $\lambda\in\mathfrak{h}^{*}$ in $M$. Then $M$ admits a
nonzero highest weight vector of weight $\mu$ for some
$\mu\in\mathfrak{h}_{n}^{*}$ satisfying $\mu_{0}=\lambda$. $\hfill\qed$

###  3.2. Highest weight modules

We say $M$ is a highest weight module of weight
$\lambda\in\mathfrak{h}_{n}^{*}$ if $M$ is generated by a highest weight
vector of weight $\lambda$. The following result on highest weight filtrations
is analogous to the situation for BGG category $\mathcal{O}$, see [Hu08,
Corollary 1.2]. The proof is essentially the same as [Ch23, Lemma 3.4].

######  Lemma 3.3.

Let $M\in\mathcal{O}(\mathfrak{g}_{n})$. Then $M$ has a finite filtration
$0=M_{0}\subseteq M_{1}\subseteq\cdots\subseteq M_{k}=M$ such that each
$M_{i+1}/M_{i}$ is a highest weight module. We call such a filtration a
highest weight filtration. $\hfill\qed$

For $\lambda\in\mathfrak{h}_{n}^{*}$ we define the Verma module of weight
$\lambda$ via

(3.6) $\displaystyle
M_{\lambda}:=U(\mathfrak{g}_{n})\otimes_{U(\mathfrak{b}_{n})}\mathbb{C}_{\lambda},$

where $\mathbb{C}_{\lambda}$ is the one dimensional
$U(\mathfrak{b}_{n})$-module upon which $\mathfrak{h}_{n}$ acts via $\lambda$,
and $\mathfrak{n}_{n}$ acts by 0. The Verma modules are the universal highest
weight modules, in the sense that every highest weight module is a quotient of
a Verma module.

They enjoy the following nice properties, generalising the classical case
$n=0$, see [Hu08, §1]:

  1. (1)

$\dim M_{\lambda}^{\lambda_{0}}=1$, and hence $\dim M^{\lambda_{0}}=1$ for
every highest weight module of weight $\lambda\in\mathfrak{h}_{n}^{*}$.

  2. (2)

Every highest weight module $M$ admits a central character: for every
$\lambda\in\mathfrak{h}_{n}^{*}$ there is a homomorphism
$\chi_{\lambda}:Z(\mathfrak{g}_{n})\to\mathbb{C}$ such that $z\cdot
m=\chi_{\lambda}(z)m$ for all $z\in Z(\mathfrak{g}_{n})$ and all $m\in M$
where $M$ is a highest weight module, of weight $\lambda$.

  3. (3)

$M_{\lambda}$ admits a unique maximal submodule and a unique simple quotient,
which we denote $L_{\lambda}$.

  4. (4)

Every simple object in $\mathcal{O}(\mathfrak{g}_{n})$ is isomorphic to
precisely one of these simple modules (by Lemma 3.3). Thus the modules

$\\{L_{\lambda}\mid\lambda\in\mathfrak{h}_{n}^{*}\\}$

give a complete set of representatives for the isomorphism classes of simple
modules in $\mathcal{O}(\mathfrak{g}_{n})$.

###  3.3. Jordan decomposition for $\mathcal{O}(\mathfrak{g}_{n})$

The standard approach to studying modules in BGG category $\mathcal{O}$ is to
consider modules with a fixed central character for $U(\mathfrak{g})$. This
refinement is also useful in our more general setting (see Theorem 4.2),
however as a first approximation we decompose $\mathcal{O}(\mathfrak{g}_{n})$
in terms of generalised eigenvalues for $\mathfrak{h}_{n}^{\geq 1}$.

Fix $M\in\mathcal{O}(\mathfrak{g}_{n})$ and $\mu\in(\mathfrak{h}_{n}^{\geq
1})^{*}$. We define the generalised eigenspace of eigenvalue $\mu$ via

$M^{(\mu)}=\\{m\in M\mid(h-\mu(h))^{k}m=0\text{ for all }k\gg 0,\
h\in\mathfrak{h}_{n}^{\geq 1}\\}.$

The following result is a slight generalisation of [Ch23, Lemma 3.7], and we
supply a sketch of the proof for the reader’s convenience.

######  Lemma 3.4.

Every $M\in\mathcal{O}(\mathfrak{g}_{n})$ admits a direct sum decomposition of
$\mathfrak{g}_{n}$-modules

(3.7) $\displaystyle M=\bigoplus_{\mu\in(\mathfrak{h}^{*})^{n}}M^{(\mu)}.$

###### Proof.

Since $\mathfrak{h}_{n}$ preserves the weight spaces of $M$, which are finite
dimensional, it follows that each $M^{\lambda}$ decomposes into generalised
eigenspaces for $\mathfrak{h}_{n}^{\geq 1}$. Therefore $M$ admits a
decomposition (3.7) and it suffices to show that each summand is a
$\mathfrak{g}_{n}$-module. This follows by a direct calculation, using the
fact that $\mathfrak{g}$ admits an eigenbasis for $\mathfrak{h}$ (root space
decomposition) and $\mathfrak{h}_{n}^{\geq 1}$ acts nilpotently on
$\mathfrak{g}_{n}$. ∎

Now we define the Jordan block of $\mathcal{O}(\mathfrak{g}_{n})$ of weight
$\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ to be the full subcategory
$\mathcal{O}(\mathfrak{g}_{n})$ whose objects are the modules $M$ such that
$M=M^{(\mu)}$. We then have the following Jordan decomposition

(3.8)
$\displaystyle\mathcal{O}(\mathfrak{g}_{n})=\bigoplus_{\mu\in(\mathfrak{h}_{n}^{\geq
1})^{*}}\mathcal{O}^{(\mu)}(\mathfrak{g}_{n}).$

######  Remark 3.5.

It is not hard to see that if $\lambda\in\mathfrak{h}_{n}^{*}$ and
$\mu=\lambda|_{\mathfrak{h}_{n}^{\geq 1}}$ then both $M_{\lambda}$ and
$L_{\lambda}$ lie in $\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ (see also [Ch23,
Lemma 3.9]). Combining (3.8) with the fact that Verma modules have unique
maximal submodules, and are therefore indecomposable, it follows that
$L_{\lambda}$ cannot occur as a subquotient of $M_{\nu}$ unless $\lambda_{\geq
1}=\nu_{\geq 1}$

Let $\mathfrak{g}_{n}\to\mathfrak{g}_{n-1}$ be the natural quotient map with
kernel $\mathfrak{g}_{n}^{n}$, and consider the pull-back functor

(3.9) $\displaystyle
p:\mathcal{O}(\mathfrak{g}_{n-1})\longrightarrow\mathcal{O}(\mathfrak{g}_{n}).$

######  Lemma 3.6.

Let $\lambda\in\mathfrak{h}_{n-1}^{*}$ and define $\nu\in\mathfrak{h}_{n}^{*}$
by $\nu(h_{i})=\lambda(h_{i})$ for $i=0,...,n-1$ and $\nu(h_{n})=0$ for all
$h\in\mathfrak{h}$. Then $p(L_{\lambda})\cong L_{\nu}$ as
$\mathfrak{g}_{n}$-modules.

###### Proof.

Certainly $p(L_{\lambda})$ is a simple highest weight module of highest weight
$\nu$, and the proof follows. ∎

Now we state and prove some easy equivalences between Jordan blocks of
$\mathcal{O}(\mathfrak{g}_{n})$ which arise by tensoring with one dimensional
$\mathfrak{g}_{n}$-modules. We write
$\mathfrak{g}^{\prime}=[\mathfrak{g},\mathfrak{g}]$ for the derived
subalgebra. We note that
$(\mathfrak{g}_{n})^{\prime}=(\mathfrak{g}^{\prime})_{n}$, and so we may use
the notation $\mathfrak{g}_{n}^{\prime}$ unambiguously.

For $\lambda\in\mathfrak{h}_{n}^{*}$ we recall the notation $\lambda_{\geq
1}:=\lambda|_{\mathfrak{h}_{n}^{\geq 1}}$. Any such $\lambda$ can be extended
to an element of $\mathfrak{g}_{n}^{*}$ via
$\lambda(\mathfrak{n}_{n}^{\pm})=0$, and we may abuse notation by identifying
$\mathfrak{h}_{n}^{*}$ with a subspace of $\mathfrak{g}_{n}^{*}$. For
$\lambda\in\operatorname{Ann}_{\mathfrak{g}^{*}_{n}}(\mathfrak{g}^{\prime}_{n})$
let $\mathbb{C}_{\lambda}$ be the one dimensional $\mathfrak{g}_{n}$-module
afforded by $\lambda$.

######  Lemma 3.7.

Suppose that $\lambda,\nu\in\mathfrak{h}_{n}^{*}$ such that
$\lambda|_{\mathfrak{g}_{n}^{\prime}}=\nu|_{\mathfrak{g}_{n}^{\prime}}$. Then
$(\bullet)\otimes_{U(\mathfrak{g}_{n})}\mathbb{C}_{\lambda-\nu}$ and
$(\bullet)\otimes_{U(\mathfrak{g}_{n})}\mathbb{C}_{\nu-\lambda}$ are quasi-
inverse equivalences between $\mathcal{O}^{(\lambda_{\geq
1})}(\mathfrak{g}_{n})$ and $\mathcal{O}^{(\nu_{\geq 1})}(\mathfrak{g}_{n})$.

###### Proof.

Since $\pm(\lambda-\nu)$ vanishes on $\mathfrak{g}_{n}^{\prime}$ it defines a
one dimensional representation of $\mathfrak{g}_{n}$, and the named functors
are quasi-inverse autoequivalences of $\mathfrak{g}_{n}\operatorname{-mod}$.
To complete the proof it suffices to observe check that
$(\bullet)\otimes_{U(\mathfrak{g}_{n})}\mathbb{C}_{\lambda-\nu}$ sends
$\mathcal{O}^{(\lambda_{\geq 1})}(\mathfrak{g}_{n})$ to
$\mathcal{O}^{(\nu_{\geq 1})}(\mathfrak{g}_{n})$, which follows directly from
the definitions. ∎

##  4\. Parabolic Induction

In this section we prove Theorem 1.2, which allows us to relate the category
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ with a Jordan block of
$\mathcal{O}(\mathfrak{l}_{n})$ for a Levi subalgebra $\mathfrak{l}$. Recall
that if $\nu\in\mathfrak{h}^{*}$ then we extend $\nu$ to an element of
$\mathfrak{g}^{*}$ via $\nu(\mathfrak{n}^{\pm})=0$, and write
$\mathfrak{g}^{\nu}$ for the coadjoint centraliser.

######  Theorem 4.1.

Let $\lambda\in\mathfrak{h}_{n}^{*}$. Suppose that the centraliser
$\mathfrak{l}=\mathfrak{g}^{\lambda_{n}}$ is in standard Levi form and let
$\mathfrak{p}:=\mathfrak{l}+\mathfrak{n}^{+}=\mathfrak{l}\oplus\mathfrak{r}$
be a parabolic subalgebra with Levi factor $\mathfrak{l}$ and nilradical
$\mathfrak{r}$. Write $\mu=\lambda_{\geq 1}$ (notation (2.9)). The categories
$\mathcal{O}^{(\mu)}(\mathfrak{l}_{n})$ and
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ are equivalent. The quasi-inverse
functors inducing the equivalence are parabolic induction and
$\mathfrak{r}_{n}$-invariants:

$\displaystyle\begin{array}[]{rcccl}\operatorname{Ind}&:&\mathcal{O}^{(\mu)}(\mathfrak{l}_{n})&\longrightarrow&\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})\vspace{3pt}\\\
&&M&\longmapsto&U(\mathfrak{g}_{n})\otimes_{U(\mathfrak{p}_{n})}M\vspace{6pt}\\\
(\bullet)^{\mathfrak{r}_{n}}&:&\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})&\longrightarrow&\mathcal{O}^{(\mu)}(\mathfrak{l}_{n})\vspace{3pt}\\\
&&M&\longmapsto&M^{\mathfrak{r}_{n}}.\end{array}$

Theorem 4.1 is inspired by a category equivalence in modular representation
theory due to Friedlander and Parshall [FP88, Theorem 2.1]. The case $n=1$ is
due to the first author [Ch23] and our method here is a generalisation of loc.
cit.

We observe that $\operatorname{Ind}$ is left adjoint to
$(\bullet)^{\mathfrak{r}_{n}}$ since we have inverse isomorphisms

$\operatorname{Hom}_{\mathfrak{l}_{n}}(M,N^{\mathfrak{r}_{n}})\underset{\eta}{\overset{\theta}{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{28.45274pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{28.45274pt}{0.0pt}}\end{subarray}}}}\operatorname{Hom}_{\mathfrak{g}_{n}}(\operatorname{Ind}M,N)$

given by $\theta(f)(u\otimes m)=u\cdot f(m)$ and $\eta(g)(m)=g(1\otimes m)$
for $u\in U(\mathfrak{g}_{n})$ and $m\in M$.

Let $\operatorname{\mathbbm{1}}_{\mathcal{C}}$ denote the identity endofunctor
of a category $\mathcal{C}$. In order to show that the adjoint functors
$\operatorname{Ind}$ and $(\bullet)^{\mathfrak{r}_{n}}$ are equivalences we
consider the unit and counit of the adjunction, see [HS97, II.7] for example.
The unit is the natural transformation
$\psi:\operatorname{\mathbbm{1}}_{\mathcal{O}(\mathfrak{g}_{n})}\to(\bullet)^{\mathfrak{r}_{n}}\circ\operatorname{Ind}$
which is obtained by applying $\eta$ to the identity mapping
$N^{\mathfrak{r}_{n}}\to N^{\mathfrak{r}_{n}}$, whilst the counit is the
natural transformation
$\varphi:\operatorname{Ind}\circ(\bullet)^{\mathfrak{r}_{n}}\to\operatorname{\mathbbm{1}}_{\mathcal{O}(\mathfrak{l}_{n})}$
obtained by applying $\theta$ to the identity mapping
$\operatorname{Ind}M\to\operatorname{Ind}M$. In particular we have

(4.6)
$\displaystyle\begin{array}[]{rcl}\psi_{M}&:&M\longrightarrow(\operatorname{Ind}M)^{\mathfrak{r}_{n}}\\\
&&m\longmapsto 1\otimes m,\vspace{4pt}\\\
\varphi_{N}&:&\operatorname{Ind}(N^{\mathfrak{r}_{n}})\longrightarrow N\\\
&&u\otimes m\longmapsto u\cdot m.\end{array}$

In order to complete the proof of Theorem 4.1 it suffices to show that $\psi$
and $\varphi$ are both natural equivalences. The proof, which is given is
Section 4.2, depends heavily on the exactness of
$(\bullet)^{\mathfrak{r}_{n}}$, which will occupy the majority of Sections 4.1
and 4.2.

###  4.1. Central characters

If two $\mathfrak{g}_{n}$-modules admit different infinitesimal central
characters then there can be no extensions between them. The main step in
proving exactness of $(\bullet)^{\mathfrak{r}_{n}}$ is the following result
which leads to a vanishing criterion for extensions. For
$\nu\in\mathfrak{h}^{*}$ we make the notation

$\Phi_{\nu}:=\\{\alpha\in\Phi:\nu(h_{\alpha})=0\\}.$

For $\lambda\in\mathfrak{h}_{n}^{*}$ recall the notation $\chi_{\lambda}$ for
central characters, introduced in Section 3.2, (2).

######  Theorem 4.2.

Let $\lambda,\lambda^{\prime}\in\mathfrak{h}^{*}_{n}$ such that $\lambda_{\geq
1}=\lambda^{\prime}_{\geq 1}$ and $\mathfrak{g}^{\lambda_{n}}$ is in standard
Levi form. Then $\chi_{\lambda}=\chi_{\lambda^{\prime}}$ if and only if
$\lambda_{0}-\lambda^{\prime}_{0}\in\mathbb{C}\Phi_{\lambda_{n}}$.

######  Corollary 4.3.

If $M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ is indecomposable and
$\mathfrak{g}^{\mu_{n}}$ is in standard Levi form, then there is a unique
coset $\Xi_{M}\in\mathfrak{h}^{*}/\mathbb{C}\Phi_{\mu_{n}}$ such that if $N$
is a highest weight subquotient of $M$ of weight
$\lambda\in\mathfrak{h}_{n}^{*}$ then $\lambda_{\geq 1}=\mu$ and
$\lambda_{0}+\mathbb{C}\Phi_{\mu_{n}}=\Xi_{M}$.

Before we proceed, we explain how to deduce the Corollary from Theorem 4.2.
Using the Jordan decomposition from Lemma 3.4 along with Remark 3.5 we see
that all subquotients of an indecomposable module lie in the same Jordan block
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$, which confirms that $\lambda_{\geq
1}=\mu$. Since $M$ has finite dimensional weight spaces (3.5) and is
indecomposable it admits a generalised central character, i.e. there is a
unique maximal ideal $\mathfrak{m}$ of $Z(\mathfrak{g}_{n})$ such that
$\mathfrak{m}^{k}M=0$ for $k\gg 0$. Thus all of the highest weight
subquotients have the same central character (§3.2, (2)). Now Corollary 4.3
follows from Theorem 4.2.

######  Remark 4.4.

Later we shall see that Theorem 4.2 and Corollary 4.3 hold without the
standard Levi type hypothesis. This follows from Theorem 1.3. However since
the latter theorem relies on Theorem 4.2 we retain this hypothesis to keep
these dependences clear.

We now proceed to prove Theorem 4.2. Recall from Section 2.2 that the
symmetric invariants $S(\mathfrak{g})^{\mathfrak{g}}$ are generated by
algebraically independent homogeneous elements $p_{1},...,p_{r}$. Furthermore
there is an embedding $S(\mathfrak{g})^{\mathfrak{g}}\hookrightarrow
S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$ such that
$S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}$ is generated by $r(n+1)$ elements
$\partial^{(k)}p_{j},\ j=1,...,r,\ k=0,...,n$. There is a linear map
$\omega:S(\mathfrak{g}_{n})^{\mathfrak{g}_{n}}\to Z(\mathfrak{g}_{n})$ and we
will be especially interested in the central elements

$z_{i}^{(j)}:=\omega(\partial^{(j)}p_{i})\ \ \text{ for }i=1,...,r.$

Now let $U(\mathfrak{g}_{n})^{\mathfrak{h}}$ be the invariant subalgebra under
the adjoint action of $\mathfrak{h}$. Also let
$U(\mathfrak{g}_{n})\mathfrak{n}_{n}^{+}$ be the left ideal generated by
$\mathfrak{n}_{n}^{+}$. The intersection
$U(\mathfrak{g}_{n})\mathfrak{n}_{n}^{+}\cap
U(\mathfrak{g}_{n})^{\mathfrak{h}}$ is actually an ideal of
$U(\mathfrak{g}_{n})^{\mathfrak{h}}$, and it is not hard to see that the
quotient by this ideal is isomorphic to $U(\mathfrak{h}_{n})$. Following the
observations of [Hu08, §1.7] verbatim we see that $\chi_{\lambda}$ coincides
with the composition

(4.7) $\displaystyle U(\mathfrak{g}_{n})^{\mathfrak{h}}\to
U(\mathfrak{h}_{n})=\mathbb{C}[\mathfrak{h}_{n}^{*}]\overset{\operatorname{ev}_{\lambda}}{\longrightarrow}\mathbb{C}$

where $\operatorname{ev}_{\lambda}$ denotes evaluation at $\lambda$. This
allows us to consider $\chi_{\lambda}(p)$ for any $p\in
U(\mathfrak{g}_{n})^{\mathfrak{h}}$, not just $p\in Z(\mathfrak{g}_{n})$.

Now for $\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ we define two maps
$\mathfrak{h}^{*}\to\mathbb{C}^{r}$. For $\nu\in\mathfrak{h}^{*}$ we use the
notation $(\nu,\mu)$ to denote the element of $\mathfrak{h}_{n}^{*}$ which
restricts as $\nu$ on $\mathfrak{h}_{n}^{0}$ and to $\mu$ on
$\mathfrak{h}_{n}^{\geq 1}$. The first map

$\displaystyle\xi_{\mu}(\nu):=(\chi_{(\nu,\mu)}\circ\omega\circ
d^{(n)}p_{1},...,\chi_{(\nu,\mu)}\circ\omega\circ
d^{(n)}p_{r})\in\mathbb{C}^{r}$

whilst the second is

$\displaystyle\eta_{\mu}(\nu):=(\theta_{(\nu,\mu)}(p_{1}),...,\theta_{(\nu,\mu)}(p_{r}))\in\mathbb{C}^{r}$

where $\theta_{(\nu,\mu)}$ denotes the composition

$S(\mathfrak{g})=\mathbb{C}[\mathfrak{g}^{*}]\to\mathbb{C}[\mathfrak{h}^{*}]\to\mathbb{C}[\mathfrak{h}_{n}^{*}]\to\mathbb{C}$

and $\mathbb{C}[\mathfrak{g}^{*}]\to\mathbb{C}[\mathfrak{h}^{*}]$ is
restriction across the triangular decomposition of $\mathfrak{g}^{*}$, whilst
the second map is the restriction of $d^{(n)}$ to
$\mathbb{C}[\mathfrak{h}^{*}]$ and the third map is evaluation
$\operatorname{ev}_{(\nu,\mu)}$.

######  Lemma 4.5.

$\xi_{\mu}(\nu)-\eta_{\mu}(\nu)$ depends only on $\mu$.

###### Proof.

Note that we a Lie algebra embedding
$\mathfrak{g}_{1}\hookrightarrow\mathfrak{g}_{n}$ given by $x_{0}\mapsto
x_{0}$ and $x_{1}\mapsto x_{n}$ for all $x\in\mathfrak{g}$. We have
$\xi_{\mu}(\nu)-\eta_{\mu}(\nu)\in S(\mathfrak{g}_{1})\subseteq
S(\mathfrak{g}_{n})$ and this reduces the claim to the case $n=1$. This was
proven in [Ch23, Lemma 4.6(b)]. ∎

######  Lemma 4.6.

For $\lambda,\lambda^{\prime}\in\mathfrak{h}_{n}^{*}$ satisfying
$\lambda_{\geq 1}=\lambda^{\prime}_{\geq 1}$, we have
$\chi_{\lambda}=\chi_{\lambda^{\prime}}$ if and only if
$\chi_{\lambda}(z_{i}^{(n)})=\chi_{\lambda^{\prime}}(z_{i}^{(n)})$ for
$i=1,...,r$.

###### Proof.

Since the centre $Z(\mathfrak{g}_{n})$ is generated by the elements
$\omega(\partial^{(k)}p_{j})$ it follows that
$\chi_{\lambda}=\chi_{\lambda^{\prime}}$ if and only if the characters
coincide on these elements. Thus if we can show that
$\chi_{\lambda}(z_{i}^{(j)})=\chi_{\lambda^{\prime}}(z_{i}^{(j)})$ for all
$j=0,...,n-1$ then the proof will be complete.

We have $p_{i}\in S(\mathfrak{g}_{n}^{n})$ and it follows from (2.12) and
(2.13) that $\partial^{(j)}p_{i}\in S(\mathfrak{g}_{n}^{\geq n-j})$. By the
definition of $\omega$ (2.16) we have $z_{i}^{(j)}\in U(\mathfrak{g}_{n}^{\geq
n-j})$. Therefore $\chi_{\lambda}(z_{i}^{(j)})$ only depends on $\lambda_{\geq
n-j}$. Since $\chi_{\lambda}$ is precisely the composition (4.7), we have
reached the desired conclusion. This completes the proof. ∎

######  Proof of Theorem 4.2.

Let $\lambda,\lambda^{\prime}$ satisfy the assumptions of the Theorem. Thanks
to Lemma 4.6 we must show that
$\chi_{\lambda}(z_{i}^{(n)})=\chi_{\lambda^{\prime}}(z_{i}^{(n)})$ for all $i$
is equivalent to the condition on $\lambda_{0},\lambda_{0}^{\prime}$.

Thanks to (2.15) we have $z_{i}^{(n)}=\omega(d^{(n)}p_{j})+\omega(q_{i}^{n})$.
Since $q_{i}^{n}\in S(\mathfrak{g}_{n}^{\geq 1})$ it follows from (2.16) that
$\chi_{\lambda}(z_{i}^{(n)})=\chi_{\lambda^{\prime}}(z_{i}^{(n)})$ if and only
if $\chi_{\lambda}\circ\omega\circ
d^{(n)}(p_{j})=\chi_{\lambda^{\prime}}\circ\omega\circ d^{(n)}(p_{j})$ for all
$i=1,...,r$. We note that this second equality is well-defined because
$d^{(n)}$ sends $\mathfrak{h}$-invariants to $\mathfrak{h}$-invariants (Lemma
2.2), and $\omega$ is $\mathfrak{h}_{n}$-equivariant, and $\chi_{\lambda}$
coincides with the composition (4.7).

Since $\mu:=\lambda_{\geq 1}=\lambda^{\prime}_{\geq 1}$ we can apply Lemma 4.5
to see that the central characters coincide if and only if
$\eta_{\mu}(\lambda_{0})=\eta_{\mu}(\lambda_{0}^{\prime})$. Let
$\pi:\mathfrak{h}^{*}=\mathfrak{h}^{*}/W\cong\mathbb{C}^{r}$ be the quotient
map. It follows from the Chevalley restriction theorem [CGi10, 3.1.37] that we
can write this in coordinates as
$\pi(\nu)=(p_{1}|_{\mathfrak{h}^{*}}(\nu),...,p_{r}|_{\mathfrak{h}^{*}}(\nu))$.
Now it is easy to see by a direct comparison of the two definitions that
$\eta_{\mu}(\lambda_{0})$ coincides with the differential
$d_{\lambda_{n}}\pi(\lambda_{0})$ of the quotient map $\pi$.

By [Ch23, Lemma 4.6(e)] we see that $\ker\eta_{\mu}=\ker
d_{\lambda_{n}}\pi=\mathbb{C}\Phi_{\lambda_{n}}$. This implies that
$\eta_{\mu}(\lambda_{0})=\eta_{\mu}(\lambda_{0}^{\prime})$ if and only if
$\lambda_{0}-\lambda_{0}^{\prime}\in\mathbb{C}\Phi_{\mu_{n}}$ (this is where
we use the fact that the centraliser in a standard Levi subalgebra). This
completes the proof. $\hfill\qed$

###  4.2. Exactness of the invariants functor

In this section we fix a standard parabolic subalgebra $\mathfrak{p}$ of
$\mathfrak{g}$, so that our fixed choice of positive root spaces are contained
in $\mathfrak{p}$. Pick a Levi decomposition
$\mathfrak{p}=\mathfrak{l}\oplus\mathfrak{r}$. The main result of this section
is that $(\bullet)^{\mathfrak{r}_{n}}$ is exact on
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ provided
$\mathfrak{g}^{\mu_{n}}=\mathfrak{l}$. First we need the following Lemma.

######  Proposition 4.7.

Let $M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ be indecomposable module and
let $\Xi_{M}\in\mathfrak{h}^{*}/\mathbb{C}\Phi_{\mu_{n}}$ be the coset
determined by $M$ in Corollary 4.3. If $\mathfrak{l}=\mathfrak{g}^{\mu_{n}}$
is a standard Levi subalgebra then

$M^{\mathfrak{r}_{n}}=\bigoplus_{\nu\in\Xi_{M}}M^{\nu}.$

###### Proof.

Since $M$ is indecomposable and admits a highest weight subquotient of weight
$\lambda$ we have $\lambda_{n}=\mu_{n}$. Since $\mathfrak{g}^{\mu_{n}}$ is the
Levi factor of a standard parabolic, we have
$\mathfrak{r}=\operatorname{span}\\{e_{\alpha}:\alpha\in\Phi^{+}\backslash\Phi_{\lambda_{n}}\\}$.
By Lemma 3.3 we have a finite filtration $0\subseteq
M_{1}\subseteq\dots\subseteq M_{k}=M$ such that $M_{i}/M_{i-1}$ has highest
weight $\lambda^{(i)}\in\mathfrak{h}_{n}^{*}$. By (3.5) the weights of $M$ lie
in the set
$\bigcup_{i=1}^{k}\\{\lambda^{(i)}_{0}-\sum_{\beta\in\Phi^{+}}k_{\beta}\beta\mid
k_{\beta}\in\mathbb{Z}_{\geq 0}\\}$. Furthermore by Corollary 4.3 there is an
element $\Xi_{M}\in\mathfrak{h}^{*}/\mathbb{C}\Phi_{\mu_{n}}$ such that
$\lambda_{0}^{(i)}+\mathbb{C}\Phi_{\mu_{n}}=\Xi_{M}$ for all $i$. It follows
that the weights of $M$ actually lie in the set
$\lambda^{(i)}+\mathbb{C}\Phi_{\mu_{n}}-\sum_{\beta\in\Phi^{+}}\mathbb{Z}_{\geq
0}\beta$, for any choice of $i$.

In particular if $\nu\in\mathfrak{h}^{*}$ satisfies
$\nu\in\lambda_{0}+\Phi_{\lambda_{n}}$ then $\nu+\alpha$ does not lie in
$\lambda^{(i)}_{0}-\sum_{\beta\in\Phi^{+}}\mathbb{Z}_{\geq 0}\beta$ for any
$\alpha\in\Phi^{+}\setminus\Phi_{\lambda_{n}}$ and for any $i$. Therefore
$\mathfrak{r}_{n}\cdot M^{\nu}=0$.

Conversely, suppose $v\in M^{\mathfrak{r}_{n}}$ is of weight
$\nu\in\mathfrak{h}^{*}$. Since $\mathfrak{h}_{n}$ acts locally finitely and
preserves weight spaces we can find a common eigenvector for
$\mathfrak{h}_{n}$ of weight $\nu$ in $U(\mathfrak{h}_{n})\cdot v$. Suppose
the eigenvalue is $\lambda^{\prime}\in\mathfrak{h}_{n}^{*}$ (by assumption
$\lambda^{\prime}_{0}=\nu$). Then a quotient of $M_{\lambda^{\prime}}$ occurs
as a submodule of $M$. All highest weight modules are indecomposable (they
admit unique maximal submodules) and this forces the generalised central
character of $M$ to be $\chi_{\lambda^{\prime}}$. Now Theorem 4.2 implies that
$\nu\in\lambda_{0}^{\prime}+\mathbb{C}\Phi_{\mu_{n}}$. By Corollary 4.3 we see
that $\lambda_{0}^{\prime}$ and $\lambda_{0}$ lie in the same coset of
$\mathfrak{h}^{*}$ modulo $\mathbb{C}\Phi_{\mu_{n}}$, and so
$\nu\in\lambda_{0}+\mathbb{C}\Phi_{\mu_{n}}$. ∎

######  Corollary 4.8.

Suppose that $\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ such that
$\mathfrak{g}^{\mu_{n}}=\mathfrak{l}$ is a standard Levi subalgebra. Then the
functor
$(\bullet)^{\mathfrak{r}_{n}}:\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})\to\mathcal{O}^{(\mu)}(\mathfrak{g}_{n}^{\mu_{n}})$
is exact.

###### Proof.

It suffices to take a surjective morphism $M\to N$ in
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$, and show that the restriction
$M^{\mathfrak{r}_{n}}\rightarrow N^{\mathfrak{r}_{n}}$ is surjective. Without
loss of generality we can assume that $M,N$ are indecomposable. The existence
of a nonzero map $M\to N$ forces $\Xi_{M}=\Xi_{N}$. Since objects of
$\mathcal{O}(\mathfrak{g}_{n})$ are $\mathfrak{h}$-semisimple, it follows that
$M\to N$ is surjective on $\mathfrak{h}$-weight spaces, and now the result
follows immediately from Proposition 4.7. ∎

###### Proof of Theorem 4.1.

We let $\varphi_{M}$ and $\psi_{N}$ be the adjunction morphisms from (4.6).

Note that $\operatorname{Ind}$ is an exact functor because
$U(\mathfrak{g}_{n})$ is free over $U(\mathfrak{p}_{n})$, thanks to the PBW
theorem. Furthermore $(\bullet)^{\mathfrak{r}_{n}}$ is exact by Corollary 4.8.
If we can check that $\varphi_{M}$ and $\psi_{N}$ are isomorphisms on highest
weight modules, then a standard argument using the length of a highest weight
filtration (Lemma 3.3) can be used to conclude that $\varphi_{M}$ and
$\psi_{N}$ are isomorphisms for all
$M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ and all
$N\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n}^{\mu_{n}})$

The map $\psi_{M}$ is an isomorphism for highest weight modules $M$, thanks to
Proposition 4.7.

Now suppose that $N\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ is a highest
weight module with highest weight generator $v$. Since $\mathfrak{r}_{n}\cdot
v=0$ it follows that $1\otimes v$ lies in the image of $\varphi_{N}$ and so
$\varphi_{N}$ is surjective. To prove injectivity let $K=\ker(\varphi_{N})$
and consider the short exact sequence
$K\to\operatorname{Ind}(N^{\mathfrak{r}_{n}})\to N$. By Corollary 4.8 we have
another short exact sequence
$K^{\mathfrak{r}_{n}}\to\operatorname{Ind}(N^{\mathfrak{r}_{n}})^{\mathfrak{r}_{n}}\to
N^{\mathfrak{r}_{n}}$.

Now set $M=N^{\mathfrak{r}_{n}}$, which is a highest weight
$\mathfrak{g}^{\mu_{n}}_{n}$-module generated by $v$. The map
$\operatorname{Ind}(M)^{\mathfrak{r}_{n}}\to M$ is
$U(\mathfrak{g}_{n})$-equivariant map uniquely determined by $1\otimes
m\mapsto m$. Therefore it is the left inverse of $\psi_{M}$, which we have
already shown to be bijective. It follows that $K^{\mathfrak{r}_{n}}=0$, but
since every nonzero object in $\mathcal{O}(\mathfrak{g}_{n})$ admits a nonzero
highest weight vector, it follows that $K=0$ and so $\varphi_{N}$ is an
isomorphism for highest weight $N$.

This concludes the proof. ∎

##  5\. Twisting Functors

###  5.1. Definition of twisting functors

Now we proceed to proving Theorem 1.3, extending the results of the first
author [Ch23, §5]. Throughout this section we fix a simple root
$\alpha\in\Delta$ and make the notation $U:=U(\mathfrak{g}_{n})$ for the sake
of brevity.

Recall that a right Ore set $S$ in a non-commutative ring $R$ is a
multiplicatively closed set of elements such that for all $r\in R$ and $s\in
S$ there exists $r^{\prime}\in R$ and $s^{\prime}\in S$ such that
$rs^{\prime}=sr^{\prime}$. A left Ore set is defined dually. In order for $R$
to admit a right ring of quotients with respect to $S$ it is necessary that
$S$ is a right Ore set. When $R$ has no zero divisors the condition is also
sufficient, and when $S$ is both a right and left Ore set the left and right
fraction fields are isomorphic; see [MR01, §2.1] for a survey of these facts.

Also recall the notation $e_{\alpha,i},h_{\alpha,i},h_{\alpha,i}$ from (2.6).
Let $F_{\alpha}$ be the multiplicative set generated by
$\\{f_{\alpha,0},...,f_{\alpha,n}\\}$ and note that these elements commute
amongst themselves. The proof of the following result is almost identical to
[Ch23, Lemma 5.1].

######  Lemma 5.1.

$F_{\alpha}$ is both a left and right Ore set in $U$. $\hfill\qed$

We now wish to explicitly construct the localisation $U$ with respect to
$F_{\alpha}$. It certainly exists, by our previous remarks. Consider the
$U$-algebra $U_{\alpha}:=U[F_{\alpha}^{-1}]$ which is freely generated by
symbols $\\{f_{\alpha,i}^{-1}\mid i=0,1,...,n\\}$, subject to the relations
$f_{\alpha,i}f_{\alpha,i}^{-1}=1$.

We introduce some notation to describe a basis for $U_{\alpha}$. If

(5.4)
$\displaystyle\begin{array}[]{l}k:\Phi^{+}\times\\{0,1,...,n\\}\to\mathbb{Z}_{\geq
0},\\\ l:\Delta\times\\{0,1,...,n\\}\to\mathbb{Z}_{\geq 0},\\\
m:\Phi^{+}\setminus\\{\alpha\\}\times\\{0,1,...,n\\}\to\mathbb{Z}_{\geq
0}\end{array}$

are arbitrary maps of sets then we let

(5.5) $\displaystyle
v(k,l,m)=\Big{(}\prod_{i=0}^{n}\prod_{\beta\in\Phi^{+}}e_{\beta,i}^{k_{\beta,i}})\Big{(}\prod_{i=0}^{n}\prod_{\beta\in\Delta}h_{\beta,i}^{l_{\beta,i}})\Big{(}\prod_{i=0}^{n}\prod_{\beta\in\Phi^{+}\setminus\\{\alpha\\}}f_{\beta,i}^{m_{\beta,i}}\Big{)}\in
U$

where the product is taken with respect to some fixed choice of ordering on
the basis of $\mathfrak{g}$. More succinctly these elements are precisely the
PBW monomials in $U$ which have no factor in $F_{\alpha}$.

######  Lemma 5.2.

  1. (1)

$U_{\alpha}$ is the (left and right) localisation of $U$ at the Ore set
$F_{\alpha}$.

  2. (2)

A basis for $U_{\alpha}$ is given by the elements $f_{\alpha,0}^{i_{0}}\cdots
f_{\alpha,n}^{i_{n}}v(k,l,m)$ where $i_{j}\in\mathbb{Z}$ and $k,l,m$ are as in
(5.4).

  3. (3)

A basis for $U_{\alpha}$ is given by the elements
$v(k,l,m)f_{\alpha,0}^{i_{0}}\cdots f_{\alpha,n}^{i_{n}}$ where
$i_{j}\in\mathbb{Z}$ and $k,l,m$ are as in (5.4).

###### Proof.

$U_{\alpha}$ satisfies the universal property of the localisation by
construction, and so it is isomorphic to both the left and right localisation,
thanks to [MR01, Corollary 2.1.4]. This proves (1), and also implies that $U$
embeds inside $U_{\alpha}$, and $U_{\alpha}$ is an integral domain.

Note that $U_{\alpha}$ is spanned by unordered monomials in $\mathfrak{g}_{n}$
and $F_{\alpha}^{-1}$. Using the left Ore condition we can rewrite any such
monomial as a span of monomials of the form described in (2). Furthermore, if
there is a linear dependence between the latter monomials, we can left
multiply by appropriate elements of $F_{\alpha}$ to obtain a linear dependence
between PBW monomials in $U$, which must be zero. This proves (2), and (3)
follows by a symmetrical argument. ∎

We will need more precise relations between generators of $U_{\alpha}$.

######  Lemma 5.3.

For any $i,j\in\mathbb{Z}$, the following relations hold in $U_{\alpha}$, for
any $h\in\mathfrak{h}$ and any $\beta\in\Phi^{+}\setminus\\{\alpha\\}$:

(5.6) $\displaystyle[e_{\alpha,i},f_{\alpha,j}^{-1}]$
$\displaystyle=-f_{\alpha,j}^{-2}h_{\alpha,i+j}-2f_{\alpha,j}^{-3}f_{\alpha,i+2j}.$
(5.7) $\displaystyle[h_{i},f_{\alpha,j}^{-1}]$
$\displaystyle=\alpha(h)f_{\alpha,j}^{-2}f_{\alpha,i+j}.$ (5.8)
$\displaystyle[e_{\beta,i},f_{\alpha,j}^{-1}]$
$\displaystyle=af_{\alpha,j}^{-2}e_{\beta-\alpha,i+j}+bf_{\alpha,j}^{-3}e_{\beta-2\alpha,i+2j}+cf_{\alpha,j}^{-4}e_{\beta-3\alpha,i+3j}$

for some $a,b,c\in\mathbb{C}$. We adopt the convention $e_{\gamma,i}=0$ if
$\gamma\notin\Phi$ or if $i>n$.

###### Proof.

These can be verified by multiplying by powers of $f_{\alpha,j}$ to obtain an
expression which holds in $U$. We show the calculation for (5.6); the other
relations are proved similarly. Using the relations in $U$ we have:

$\displaystyle
f_{\alpha,j}^{3}e_{\alpha,i}=f_{\alpha,j}^{2}e_{\alpha,i}f_{\alpha,j}-f_{\alpha,j}h_{\alpha,i+j}f_{\alpha,j}-2f_{\alpha,i+2j}f_{\alpha,j}.$

We then multiply on the left by $f_{\alpha,j}^{-3}$ and on the right by
$f_{\alpha,j}^{-1}$ to obtain (5.6). ∎

Let $V_{\alpha}\subseteq U_{\alpha}$ be the span of the monomials appearing in
Lemma 5.2(2) such that $i_{j}\geq 0$ for some $j=0,...,n$. Using an argument
identical to the one used in the second half of the proof of [Ch23, Lemma 5.3]
we see that $V_{\alpha}$ can be defined symmetrically as the the span of the
monomials appearing in Lemma 5.2(3), subject to the condition $i_{j}\geq 0$
for some $j$. Using an argument identical to loc. cit. once again we obtain
the following.

######  Lemma 5.4.

$V_{\alpha}$ is a $U$-$U$-sub-bimodule of $U_{\alpha}$. $\hfill\qed$

We now consider the $U$-$U$-bimodule

(5.9) $\displaystyle S_{\alpha}:=U_{\alpha}/V_{\alpha}.$

By Lemma 5.2, we see that $S_{\alpha}$ has a basis given by

(5.10) $\displaystyle\\{f_{\alpha,0}^{i_{0}}\dots
f_{\alpha,n}^{i_{n}}v(k,l,m)\mid\text{ for }v,k,l,m\text{ as per
\eqref{e:introduceklm}, \eqref{e:introducev} and }i_{j}<0\text{ for all }j\\}$

and another basis given by:

(5.11) $\displaystyle\\{v(k,l,m)f_{\alpha,0}^{i_{0}}\dots
f_{\alpha,n}^{i_{n}}\mid\text{ for }v,k,l,m\text{ as per
\eqref{e:introduceklm}, \eqref{e:introducev} and }i_{j}<0\text{ for all }j\\}$

As a slight abuse of notation we denote an element of $U_{\alpha}$ and its
coset in $S_{\alpha}$ by the same symbol.

Now we pick a special automorphism of $\mathfrak{g}$. The simple root $\alpha$
which we have fixed throughout this section gives rise to a reflection
$s_{\alpha}\in W=N_{G}(\mathfrak{h})/\mathfrak{h}$. We lift $s_{\alpha}$
arbitrarily to an element of $N_{G}(\mathfrak{h})$, which defines an
automorphism of $\mathfrak{g}$ via the adjoint representation. We denote this
automorphism by $\phi_{\alpha}$.

Note that $\phi_{\alpha}$ acts on the root spaces as $s_{\alpha}$, i.e. it
sends $\mathfrak{g}_{\beta}$ to $\mathfrak{g}_{s_{\alpha}(\beta)}$ and
preserves $\mathfrak{h}$. Furthermore, after rescaling $e_{\alpha}$ and
$f_{\alpha}$ if necessary, we may assume that:

$\displaystyle\phi_{\alpha}(e_{\alpha})$ $\displaystyle=f_{\alpha}$
$\displaystyle\phi_{\alpha}(f_{\alpha})$ $\displaystyle=e_{\alpha}.$

We extend $\phi_{\alpha}$ to an automorphism of $\mathfrak{g}_{n}$ by the rule
$\phi_{\alpha}(x_{i})=\phi_{\alpha}(x)_{i}$ for all $x\in\mathfrak{g}$.

If $M$ is a left $U$-module, we denote by $\phi_{\alpha}(M)$ the module
obtained by twisting the left action on $M$ by $\phi_{\alpha}$ and write
$\cdot_{\alpha}$ for this action. More precisely, if $m\in M$ and $u\in U$
then $u\cdot_{\alpha}m:=\phi_{\alpha}(u)\cdot m$. Similarly, we can similarly
twist the action by $\phi_{\alpha}^{-1}$, and use notation
$\cdot_{\alpha^{-1}}$ in this case.

Let $\mathcal{C}$ be the full subcategory of $U$-mod whose objects are the
$\mathfrak{h}$-semisimple modules. The proof of the next result is the same as
[Ch23, Lemma 5.9].

######  Lemma 5.5.

$\mathcal{O}(\mathfrak{g}_{n})$ is a Serre subcategory of the category of
$\mathcal{C}$. $\hfill\qed$

We define a functor $\mathcal{H}$ from $U$-mod to $\mathcal{C}$ by letting
$\mathcal{H}(M)$ be the sum of the $\mathfrak{h}$-weight spaces in $M$.

Now we define two endofunctors of $\mathcal{C}$ by setting

(5.12) $\displaystyle T_{\alpha}M$
$\displaystyle:=\phi_{\alpha}(S_{\alpha}\otimes_{U}M)$ (5.13) $\displaystyle
G_{\alpha}M$
$\displaystyle:=\mathcal{H}(\operatorname{Hom}_{U}(S_{\alpha},\phi_{\alpha}^{-1}(M))).$

Note that the action of $U$ on $G_{\alpha}M$ is given by

$(u\cdot f)(s)=f(s\cdot u)\text{ for }u\in U,\ f\in G_{\alpha}M,\ s\in
S_{\alpha}.$

Also note that if $M,N\in\mathcal{C}$ and
$\chi\in\operatorname{Hom}_{\mathcal{C}}(M,N)$, then
$T_{\alpha}(\chi):T_{\alpha}M\rightarrow T_{\alpha}N$ and
$G_{\alpha}(\chi):G_{\alpha}M\rightarrow G_{\alpha}N$ are given by

$\displaystyle T_{\alpha}(\chi)(s\otimes m)=s\otimes\chi(m)\text{ for }s\in
S_{\alpha},\ m\in M,$ $\displaystyle
G_{\alpha}(\chi)(\rho)=\chi\circ\rho\text{ for
}\rho\in\operatorname{Hom}_{U}(S_{\alpha},\phi_{\alpha}^{-1}(M))).$

In order to see these functors are well defined, the only non-trivial check is
that $T_{\alpha}M\in\mathcal{C}$ for any $M\in\mathcal{C}$. For
$M\in\mathcal{C}$, we see that $T_{\alpha}M$ is spanned by
$\\{f_{\alpha,0}^{-i_{0}}\dots f_{\alpha,n}^{-i_{n}}\otimes
w:i_{j}>0,w\in\mathcal{H}(M)\\}$. For $w\in M^{\lambda}$ and
$h\in\mathfrak{h}$ we have

$\displaystyle h\cdot_{\alpha}(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes w)$
$\displaystyle=(s_{\alpha}(h)f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}})\otimes w$ $\displaystyle=f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}(s_{\alpha}(h)-(i_{0}+\dots+i_{n})\alpha(h))\otimes w$
$\displaystyle=f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes(s_{\alpha}(h)-(i_{0}+\dots+i_{n})\alpha(h))w$
$\displaystyle=f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes(s_{\alpha}(\lambda)-(i_{0}+\dots+i_{n})\alpha)(h)w$
$\displaystyle=(s_{\alpha}(\lambda)-(i_{0}+\dots+i_{n})\alpha)(h)(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes w).$

To summarise, for any $w\in M$ of weight $\lambda$ we have

(5.14) $\displaystyle f_{\alpha,0}^{-i_{0}}\dots f_{\alpha,n}^{-i_{n}}\otimes
w\in(T_{\alpha}M)^{s_{\alpha}(\lambda)-(i_{0}+\dots+i_{n})\alpha}.$

In particular, $T_{\alpha}M$ is spanned by weight vectors, and so is
$\mathfrak{h}$-semisimple. The following result can be proven by a calculation
almost identical to [Ch23, Lemma 5.4].

######  Lemma 5.6.

Let $g\in\operatorname{Hom}_{U}(S_{\alpha},\phi_{\alpha}^{-1}(M))$. Then $g$
has weight $\lambda\in\mathfrak{h}^{*}$ if and only if
$g(f_{\alpha,0}^{-i_{0}}\dots f_{\alpha,n}^{-i_{n}})$ has weight
$\lambda+(i_{0}+\dots+i_{n})\alpha$ in $\phi^{-1}_{\alpha}(M)$ for all
$i_{0},...,i_{n}\in\mathbb{Z}_{\geq 0}$, which is if and only if this vector
has weight $s_{\alpha}(\lambda)-(i_{0}+\dots+i_{n})\alpha$ in $M$.

We immediately obtain the following consequence.

######  Corollary 5.7.

Let $g\in\operatorname{Hom}_{U}(S_{\alpha},\phi_{\alpha}^{-1}(M))$, Then $g$
is a weight vector if and only if $g(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}})$ is a weight vector for all $i,j\geq 0$. In particular,
$G_{\alpha}(M)$ is the direct sum of such vectors $g$.

######  Lemma 5.8.

$T_{\alpha}$ is right exact and $G_{\alpha}$ is left exact.

###### Proof.

For any module $M\in U\operatorname{-mod}$

$\phi_{\alpha}(S_{\alpha}\otimes_{U}M)\cong\phi_{\alpha}(S_{\alpha})\otimes_{U}M$

and $(\bullet)\otimes_{U}M$ is right exact, hence $T_{\alpha}$ is right exact.

Similarly $G_{\alpha}$ is a composition of two left exact functors,
$\operatorname{Hom}_{U}(S_{\alpha},\phi^{-1}_{\alpha}(\bullet))$ and
$\mathcal{H}$, hence it is also left exact. ∎

###  5.2. Twisting functors between blocks of category $\mathcal{O}$

Note that the Weyl group action on $\mathfrak{h}^{*}$ extends naturally to an
action on $(\mathfrak{h}_{n}^{\geq 1})^{*}$ acting diagonally through the
identification $(\mathfrak{h}_{n}^{\geq 1})^{*}=(\mathfrak{h}^{*})^{\oplus
n}$, and this vector space parameterises the Jordan blocks of
$\mathcal{O}(\mathfrak{g}_{n})$. Retaining the notation of the previous
section, we can now precisely state Theorem 1.3.

######  Theorem 5.9.

Let $\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ be such that
$\mu(h_{\alpha,n})\neq 0$. The functors $T_{\alpha}$ and $G_{\alpha}$ from
(5.12) restrict to functors

$\displaystyle T_{\alpha}$ $\displaystyle:$
$\displaystyle\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})\longrightarrow\mathcal{O}^{(s_{\alpha}(\mu))}(\mathfrak{g}_{n})$
$\displaystyle G_{\alpha}$ $\displaystyle:$
$\displaystyle\mathcal{O}^{(s_{\alpha}(\mu))}(\mathfrak{g}_{n})\longrightarrow\mathcal{O}^{(\mu)}(\mathfrak{g}_{n}).$

These form a quasi-inverse pair of equivalences.

The proof of Theorem 5.9 will be broken down into a series of lemmas, which we
record and prove over the course of this section. To be more precise, the
theorem will follow directly from Lemmas 5.10, 5.13, 5.16, 5.17 and 5.18.

For the rest of the section we keep $\alpha$ fixed and let
$\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ be such that $\mu(h_{\alpha,n})\neq
0$.

The next result is the first step in the proof of Theorem 5.9, and is a
generalisation of [Ch23, Lemma 5.3]. The proof given here is an alternative,
shorter argument.

######  Lemma 5.10.

$T_{\alpha}$ restricts to a functor
$\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})\rightarrow\mathcal{O}^{(s_{\alpha}(\mu))}(\mathfrak{g}_{n})$.

###### Proof.

For $M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ we let $l(M)$ denote the
minimal length of a filtration $0=M_{0}\subseteq M_{1}\subseteq\dots\subseteq
M_{k-1}\subseteq M_{k}=M$ such that the sections are highest weight modules
(Cf. Lemma 3.3). We have an exact sequence:

$0\rightarrow M_{1}\rightarrow M\rightarrow M/M_{1}\rightarrow 0$

and since $T_{\alpha}$ is right exact, we have an exact sequence

$T_{\alpha}M_{1}\rightarrow T_{\alpha}M\rightarrow
T_{\alpha}(M/M_{1})\rightarrow 0$

Using Lemma 5.5 and the fact that $T_{\alpha}M_{1}$ is a highest weight
module, we can reduce the claim that
$T_{\alpha}(M)\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ to the case where
$l(M)=1$, i.e. $M$ is a highest weight module.

For the rest of the proof we fix $M$ highest weight of weight
$\lambda\in\mathfrak{h}_{n}^{*}$, and let $v\in M$ be a highest weight
generator of $M$. We will show that $f_{\alpha,0}^{-1}\dots
f_{\alpha,n}^{-1}\otimes v$ is highest weight and generates $T_{\alpha}M$,
which will complete the proof of the lemma.

To see that the vector is maximal, use (5.14) to see that
$e_{\alpha,i}f_{\alpha,0}^{-1}\dots f_{\alpha,n}^{-1}\otimes v\in T_{\alpha}M$
lies in an $\mathfrak{h}$-eigenspace which is not a weight of $T_{\alpha}M$
(for any $i\geq 0$ and $\alpha\in\Phi^{+}$). To see that it is a genuine
highest weight vector, one can use (5.7) to show that $h_{i}$ acts via
$s_{\alpha_{i}}(h_{i})$.

To see that $f_{\alpha,0}^{-1}\dots f_{\alpha,n}^{-1}\otimes v$ generates
$T_{\alpha}M$ we use the fact that $Uv=M$ and that $S_{\alpha}$ has two bases,
(5.10) and (5.11), to check that every element of $T_{\alpha}M$ lies in the
submodule generated by the set $\\{f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes v\mid i_{k}>0\\}$. Let $L$ denote the span of
this set. Note that it is an $(\mathfrak{sl}_{2})_{n}$-module, where
$(\mathfrak{sl}_{2})_{n}$ is the truncated current algebra on
$\mathfrak{sl}_{2}=\langle e_{\alpha},h_{\alpha},f_{\alpha}\rangle$. To
complete the current proof we show that $L$ is a simple
$(\mathfrak{sl}_{2})_{n}$-module.

Let $\mathfrak{t}_{n}\subseteq(\mathfrak{sl}_{2})_{n}$ denote the span on
$h_{\alpha,0},...,h_{\alpha,n}$ and let
$\gamma=\lambda|_{\mathfrak{t}_{n}}\in\mathfrak{t}_{n}^{*}$ give the action on
$f_{\alpha,0}^{-1}\dots f_{\alpha,n}^{-1}\otimes v$. If $M_{\gamma}$ denotes
the Verma module of highest weight $\gamma$ then there is a nonzero
homomorphism $M_{\gamma}\to L$ and the dimensions of the weight spaces are the
same. Therefore it remains to show that the Verma module $M_{\gamma}$ is
simple. By (5.7) we see that
$\gamma(h_{\alpha,n})=(s_{\alpha}\lambda)(h_{\alpha,n})\neq 0$ and so we can
apply Theorem 4.1 to see that $\mathcal{O}^{(\gamma_{\geq
1})}((\mathfrak{sl}_{2})_{n})$ is equivalent to $\mathcal{O}^{(\gamma_{\geq
1})}(\mathfrak{t}_{n})$. In the latter category, all highest weight modules
are simple, and it follows that $M_{\gamma}$ is simple, as required. This
completes the proof. ∎

Consider the abelian Lie algebra $\mathfrak{a}=\langle
f_{\alpha,0},\dots,f_{\alpha,n}\rangle$ contained in $\mathfrak{g}_{n}$, and
let $A:=\operatorname{span}\\{f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha}^{-i_{n}}:i_{0},\dots,i_{n}>0\\}\subseteq S_{\alpha}$, which is a
$U(\mathfrak{a})$-$U(\mathfrak{a})$-subbimodule of $S_{\alpha}$. Thanks to the
description of the two bases (5.10) and (5.11) of $S_{\alpha}$, we have an
isomorphism of $U$-$U(\mathfrak{a})$-bimodules

$S_{\alpha}\cong U\otimes_{U(\mathfrak{a})}A,$

and and an isomorphism of $U(\mathfrak{a})$-$U$-bimodules

$S_{\alpha}\cong A\otimes_{U(\mathfrak{a})}U.$

In particular, for any left $U$-module $M$, we have an isomorphism of left
$U(\mathfrak{a})$-modules

$\displaystyle
S_{\alpha}\otimes_{U}M\cong(A\otimes_{U(\mathfrak{a})}U)\otimes_{U}M\cong
A\otimes_{U(\mathfrak{a})}M.$

Recall that $\mu\in\mathfrak{h}_{n}^{*}$ satisfies $\mu(h_{\alpha,n})\neq 0$.

######  Lemma 5.11.

[Ch23, Lemma 5.13] Let $M\in\mathcal{O}^{(\mu)}$. Let $m\in
M^{\lambda}\backslash\\{0\\}$ for some $\lambda\in\mathfrak{h}^{*}$. Then the
following are equivalent:

  1. (a)

In $T_{\alpha}M$, we have $f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes m=0$.

  2. (b)

For any vector space $V$ and $U(\mathfrak{a})$-balanced map $\varphi:A\times
M\rightarrow V$ we have $\varphi(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}},m)=0$.

  3. (c)

There exist $m_{0},\dots m_{n}\in M$ such that $m=f_{\alpha,0}^{i_{0}}\cdot
m_{0}+\dots+f_{\alpha,n}^{i_{n}}\cdot m_{n}$.

The following fact only depends on the existence of inverses for
$f_{\alpha,i}$ in $S_{\alpha}$. We omit the proof.

######  Lemma 5.12.

[Ch23, Lemma 5.15] Let $M\in U$-mod. Then any element of $S_{\alpha}\otimes M$
can be written in the form $f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes m$, for some $i_{0},\dots,i_{n}>0$ and $m\in M$.

######  Lemma 5.13.

[Ch23, Lemma 5.7(b)] For any $M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$, the
map $\psi_{M}:M\rightarrow G_{\alpha}T_{\alpha}M$ given by
$\psi_{M}(m)(s)=s\otimes m$ is an isomorphism.

######  Lemma 5.14.

[Ch23, Lemma 5.16] Let $M$ be a $U$-module, and let $A\subseteq S_{\alpha}$
and $\mathfrak{a}\subseteq\mathfrak{g}$ be as in Lemma 5.11. Let
$\varphi:A\rightarrow M$ be a $U(\mathfrak{a})$-homomorphism. Then $\varphi$
extends uniquely to a $U$-homomorphism $\varphi:S_{\alpha}\rightarrow M$.

This next result is a generalisation of [Ch23, Lemma 5.17], but we provide the
proof which is more complicated in the present setting.

######  Lemma 5.15.

Let $M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ and let $\mathcal{I}$ be a
subset of $\mathbb{Z}^{n+1}$ satisfying:

  1. (1)

$(i_{0},\dots,i_{n})\in\mathcal{I}$ whenever any $i_{k}\leq 0$.

  2. (2)

If $(i_{0},\dots,i_{n})\in\mathcal{I}$, then
$(i_{0},\dots,i_{k}-1,\dots,i_{n})\in\mathcal{I}$ for any $0\leq k\leq n$.

and let $\\{m_{i_{0},\dots,i_{n}}\in M:(i_{0},\dots,i_{n})\in\mathcal{I}\\}$
be a collection of elements of $M$ satisfying:

  1. (i)

$m_{i_{0},\dots,i_{n}}=0$ whenever any $i_{k}\leq 0$.

  2. (ii)

$e_{\alpha,k}\cdot m_{i_{0},\dots,i_{n}}=m_{i_{0},\dots,i_{k}-1,\dots,i_{n}}$
whenever $(i_{0},\dots,i_{n})\in\mathcal{I}$

Then there exists a $U(\mathfrak{a})$-homomorphism
$\varphi:A\rightarrow\phi_{\alpha}^{-1}(M)$ such that
$\varphi(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}})=m_{i_{0},\dots,i_{n}}$, which by Lemma 5.14 extends to
a $U$-homomorphism $\varphi:S_{\alpha}\rightarrow\phi_{\alpha}^{-1}(M)$.
Moreover, if there exists $\lambda\in\mathfrak{h}^{*}$ such that the weight of
$m_{i_{0},\dots,i_{n}}$ is $\lambda-(i_{0}+\dots+i_{n})\alpha$ for all
$(i_{0},\dots,i_{n})\in\mathcal{I}$, then we can choose $\varphi$ to also be
weight and hence in $G_{\alpha}M$.

###### Proof.

We construct elements $m_{i_{0},\dots,i_{n}}$ for
$(i_{0},\dots,i_{n})\in\mathbb{Z}^{n+1}\backslash\mathcal{I}$ inductively such
that the $m_{i_{0},\dots,i_{n}}$ satisfy conditions (i) and (ii) above for any
$(i_{0},\dots,i_{n})\in\mathbb{Z}^{n+1}$. Then observe that (i) and (ii)
ensure that setting $\varphi(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}})=m_{i_{0},\dots,i_{n}}$ defines a
$U(\mathfrak{a})$-homomorphism $\varphi:A\rightarrow\phi_{\alpha}^{-1}(M)$,
since it is enough to check that $m_{i_{0},\dots,i_{k}-1,\dots
i_{n}}=f_{\alpha,k}\cdot_{\alpha^{-1}}\varphi(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}})=f_{\alpha,k}\cdot_{\alpha^{-1}}m_{i_{0},\dots,i_{n}}=e_{\alpha,k}\cdot
m_{i_{0},\dots,i_{n}}$ for any $0\leq k\leq n$ and
$(i_{0},\dots,i_{n})\in\mathbb{Z}^{n+1}$.

To construct such $m_{i_{0},\dots,i_{n}}$, let
$(i_{0},\dots,i_{n})\in\mathbb{Z}^{n+1}\backslash\mathcal{I}$ be such that
$i_{0}+\dots+i_{n}$ is minimal among elements of
$\mathbb{Z}^{n+1}\backslash\mathcal{I}$. Then in particular,
$(i_{0},\dots,i_{k}-1,\dots,i_{n})\in\mathcal{I}$ for each $0\leq k\leq n$.
Let $(\mathfrak{sl}_{2})_{n}$ be the truncated current algebra on the copy of
$\mathfrak{sl}_{2}$ spanned by $e_{\alpha},h_{\alpha},f_{\alpha}$ and let $N$
be the $(\mathfrak{sl}_{2})_{n}$-module generated by
$\\{m_{i_{0},\dots,i_{k}-1,\dots,i_{n}}:0\leq k\leq n\\}$, so
$N\in\mathcal{O}^{(\mu)}((\mathfrak{sl}_{2})_{n})$. Here we make a slight
abuse of notation, identifying $\mu$ with the restriction to
$\mathfrak{h}_{n}\cap(\mathfrak{sl}_{2})_{n}$. Then we use the following claim
to construct $m_{i_{0},\dots,i_{n}}\in N\subseteq M$ such that
$e_{\alpha,k}\cdot m_{i_{0},\dots,i_{n}}=m_{i_{0},\dots,i_{k}-1,\dots,i_{n}}$.

######  Claim $(\ast)$.

Consider the maps:

$N\overset{\theta_{1}}{\longrightarrow}N^{n+1}\overset{\theta_{2}}{\longrightarrow}N^{\frac{1}{2}n(n+1)}$

given by $\theta_{1}(x)=(e_{\alpha,0}\cdot x,\dots,e_{\alpha,n}\cdot x)$ and
$\theta_{2}(y_{0},\dots,y_{n})=(e_{\alpha,k}\cdot y_{l}-e_{\alpha,l}\cdot
y_{k})_{0\leq k<l\leq n}$. Then
$\ker(\theta_{2})=\operatorname{im}(\theta_{1})$. Furthermore, if
$(y_{0},\dots,y_{n})\in\ker(\theta_{2})$ and $y_{k}\in N\cap M^{\lambda}$ for
all $0\leq k\leq n$, then there exists $x\in M^{\lambda-\alpha}$ such that
$\theta_{1}(x)=(y_{0},\dots,y_{n})$.

We now prove the claim, which will take several steps.

First observe that $\theta_{2}\circ\theta_{1}=0$, so certainly
$\ker(\theta_{2})\supseteq\operatorname{im}(\theta_{1})$. Hence we only need
to show that $\operatorname{im}(\theta_{1})\supseteq\ker(\theta_{2})$.
Throughout the proof of the claim, we write $e_{k}$ for $e_{\alpha,k}$.

We first deal with the case where $n=1$ and $N=M_{\gamma}$ is a Verma module.
In this case, we consider the restriction of these maps to certain weight
spaces in the following way (for any $\lambda\in\mathfrak{h}^{*}$):

$N^{\lambda}\overset{\theta_{1}}{\longrightarrow}(N^{\lambda+\alpha})^{2}\overset{\theta_{2}}{\longrightarrow}N^{\lambda+2\alpha}$

Now, either $\dim(N^{\lambda+\alpha})=0$, in which case
$\operatorname{im}(\theta_{1})=\ker(\theta_{2})$ automatically, or the
dimensions of these weight spaces satisfy $\dim(N^{\lambda})=n+1$,
$\dim(N^{\lambda+\alpha})=n$, and $\dim(N^{\lambda+2\alpha})=n-1$. Hence by
considering dimensions and the fact that
$\ker(\theta_{2})\supseteq\operatorname{im}(\theta_{1})$, it is enough to show
that $\theta_{1}$ is injective and $\theta_{2}$ is surjective. We can compute
that $e_{1}$ acts on the basis vectors $f_{1}^{i}f_{0}^{j}\otimes 1$ by:

(5.15) $\displaystyle e_{1}\cdot(f_{1}^{i}f_{0}^{j}\otimes 1)=\mu
jf_{1}^{i}f_{0}^{j-1}\otimes 1-j(j-1)f_{1}^{i+1}f_{0}^{j-2}\otimes 1$

so by considering $\theta_{2}(x,0)$, we see $\theta_{2}$ is surjective using
(5.15) and an inductive argument. We also see that $e_{1}\cdot v=0$ if and
only if $v=f_{1}^{i}\otimes 1$. Since $\mu\neq 0$, we can show that
$e_{0}\cdot f_{1}^{i}\otimes 1\neq 0$, so $\theta_{1}$ is injective as
required.

We now deal with the case where $n\geq 1$ and $N=M_{\lambda}$ is a Verma in
the $\mu$-Jordan block, so $\lambda|_{\geq 1}=\mu$. We use the following
facts, which can be verified by computing the action of $e_{n}$ on the basis
elements $f_{0}^{i_{0}}\dots f_{n}^{i_{n}}\otimes 1$ of $M_{\lambda}$ and
recalling that $\mu(h_{\alpha})\neq 0$:

  1. (a)

$e_{n}\cdot N=N$, which uses an inductive argument with (5.15).

  2. (b)

$e_{n}\cdot m=0$ if and only if $m\in\operatorname{span}\\{f_{1}^{i_{1}}\dots
f_{n}^{i_{n}}\otimes 1:i_{1},\dots,i_{n}\geq 0\\}$. The latter is isomorphic
as a $U(\langle e_{0},\dots,e_{n-1}\rangle)$-module to the Verma module
$M_{\gamma}$ over
$\operatorname{span}\\{e_{\alpha,i},h_{\alpha,i},f_{\alpha,i}\mid
i=0,...,n-1\\}\cong((\mathfrak{sl}_{2})_{\alpha})_{n-1}$ where
$\gamma(h_{\alpha,i}):=\mu(h_{\alpha,i+1})$.

  3. (c)

Applying (b) repeatedly, we see that for any $k\geq 1$, we have that
$e_{l}\cdot m=0$ for all $k\leq l\leq n$ if and only if
$m\in\operatorname{span}\\{f_{n-k+1}^{i_{n-k+1}}\dots f_{n}^{i_{n}}\otimes
1:i_{n-k+1},\dots,i_{n}\geq 0\\}$.

To ease notation slightly we will write $\mathfrak{a}_{(i)}=\langle
e_{0},...,e_{i}\rangle$, write $(\mathfrak{sl}_{2})_{(i)}$ for the Lie algebra
$((\mathfrak{sl}_{2})_{\alpha})_{i}$ and $\gamma_{(i)}$ for the character of
$\operatorname{span}\\{h_{\alpha,0},...,h_{\alpha,i}\\}$ given by
$\gamma_{(i)}(h_{\alpha,j}):=\mu(h_{\alpha,n-i})$.

Now let $(y_{0},\dots,y_{n})\in\ker(\theta_{2})$, i.e. $e_{k}\cdot
y_{l}-e_{l}\cdot y_{k}=0$ for all $0\leq k,l\leq n$. By fact (a), there
certainly exists an $x_{n}$ such that $e_{n}\cdot x_{n}=y_{n}$. We then
inductively construct $x_{k}$ for $2\leq k\leq n$ such that $e_{l}\cdot
x_{k}=y_{l}$ for all $k\leq l\leq n$. Note that the cases $k=0,1$ will be
dealt with by an additional argument immediately afterwards, and we will then
obtain $x_{0},...,x_{n}$ such that
$\theta(x_{0},...,x_{n})=(y_{0},...,y_{n})$.

Suppose we have constructed $x_{k+1}$ such that $e_{l}\cdot x_{k+1}=y_{l}$ for
all $k+1\leq l\leq n$. For any $k+1\leq l\leq n$, consider
$e_{l}\cdot(e_{k}\cdot x_{k+1}-y_{k})=e_{k}\cdot(e_{l}\cdot
x_{k+1})-e_{l}\cdot y_{k}=e_{k}\cdot y_{l}-e_{l}\cdot y_{k}=0$. Hence by fact
(c), we have $e_{k}\cdot
x_{k+1}-y_{k}\in\operatorname{span}\\{f_{n-k}^{i_{n-k}}\dots
f_{n}^{i_{n}}\otimes 1:i_{n-k},\dots,i_{n}\geq 0\\}$. But as a
$U(\mathfrak{a}_{(k)})$-module, this is isomorphic to $M_{\gamma_{(k)}}$, so
by fact (a) there exists
$x_{k}^{\prime}\in\operatorname{span}\\{f_{n-k}^{i_{n-k}}\dots
f_{n}^{i_{n}}\otimes 1:i_{n-k},\dots,i_{n}\geq 0\\}$ such that $e_{k}\cdot
x_{k}^{\prime}=e_{k}\cdot x_{k+1}-y_{k}$, and by fact (c) $e_{l}\cdot
x_{k}^{\prime}=0$ for all $k+1\leq l\leq n$. Setting
$x_{k}=x_{k+1}-x_{k}^{\prime}$, we see that $e_{k}\cdot x_{k}=e_{k}\cdot
x_{k+1}-e_{k}\cdot x_{k}^{\prime}=e_{k}\cdot x_{k+1}-(e_{k}\cdot
x_{k+1}-y_{k})=y_{k}$, and for $k+1\leq l\leq n$ we have $e_{l}\cdot
x_{k}=e_{l}\cdot x_{k+1}-e_{l}\cdot x_{k}^{\prime}=y_{l}-0$. Hence we have
constructed $x_{k}$ with the desired properties.

Now consider $y_{1}^{\prime}=e_{1}\cdot x_{2}-y_{1}$ and
$y_{0}^{\prime}=e_{0}\cdot x_{2}-y_{0}$. For $2\leq l\leq n$, we have
$e_{l}\cdot y_{1}^{\prime}=e_{l}\cdot(e_{1}\cdot x_{2}-y_{1})=e_{1}\cdot
y_{1}-e_{l}\cdot y_{1}=0$, and similarly $e_{l}\cdot y_{0}^{\prime}=0$, so
$y_{0}^{\prime},y_{1}^{\prime}\in\operatorname{span}\\{f_{n-1}^{i_{n-1}}f_{n}^{i_{n}}\otimes
1:i_{n-1},i_{n}\geq 0\\}$ which is isomorphic to $M_{\gamma_{(1)}}$ (Verma
module over $((\mathfrak{sl}_{2})_{(1)}$) as a $U(\langle
e_{0},e_{1}\rangle)$-module. In addition, since
$(y_{0},...,y_{n})\in\ker\theta$ we have $e_{0}\cdot y_{1}^{\prime}-e_{1}\cdot
y_{0}^{\prime}=0$, so we can apply the case where $N$ is a Verma module and
$n=1$, proved earlier, to find $x^{\prime}$ such that $e_{0}\cdot
x^{\prime}=y_{0}^{\prime}$, $e_{1}\cdot x^{\prime}=y_{1}^{\prime}$, and
$e_{l}\cdot x^{\prime}=0$ for $2\leq l\leq n$. Then setting
$x=x_{2}-x^{\prime}$, we have that $\theta_{1}(x)=(y_{0},\dots,y_{n})$.

If $N$ is not a Verma module, let $0=N_{0}\subseteq
N_{1}\subseteq\dots\subseteq N_{k-1}\subseteq N_{k}=N$ be a filtration of $N$
such that each quotient is a highest weight module. Since $\mu(h_{\alpha})\neq
0$, all these highest weight modules must in fact be Verma modules by [Wi11,
7.1]. Then, given $(y_{0},\dots,y_{n})\in\ker(\theta_{2})$, in the quotient
$N/N_{k-1}$ we have that there exists $x\in N$ such that $e_{l}\cdot
x+N_{k-1}=y_{l}+N_{k-1}$ for all $0\leq l\leq n$. Hence $e_{l}\cdot x-y_{l}\in
N_{k-1}$ for all $0\leq l\leq n$. But $e_{l^{\prime}}\cdot(e_{l}\cdot
x-y_{l})=e_{l}\cdot(e_{l^{\prime}}\cdot x-y_{l^{\prime}})$ for all $0\leq
l,l^{\prime},\leq n$, so by induction on $k$, there exists $x^{\prime}\in
N_{k-1}$ such that $e_{l}\cdot x^{\prime}=e_{l}\cdot x-y_{l}$ for all $0\leq
l\leq n$. Note that in this final argument we have used the fact that
$y_{l}^{\prime}:=e_{l}\cdot x-y_{l}$ gives a collection of elements lying in
the kernel of $\theta$, which allows us to apply the inductive hypothesis.
Hence $e_{l}\cdot(x-x^{\prime})=y_{l}$ for all $0\leq l\leq n$, so
$\operatorname{im}(\theta_{1})\supseteq\ker(\theta_{2})$ as required.

Finally, if $y_{0},\dots,y_{n}\in M^{\lambda}\cap N$, then given $x\in
N\subseteq M$ such that $\theta_{1}(x)=(y_{0},\dots,y_{n})$, by considering
weight spaces we may replace $x$ with its component $x^{\lambda-\alpha}$ in
the $\lambda-\alpha$ weight space and
$\theta_{1}(x)=\theta_{1}(x^{\lambda-\alpha})$, proving the final part of
Claim $(\ast)$.

Since $e_{\alpha,k}\cdot
m_{i_{0},\dots,i_{l}-1,\dots,i_{n}}=m_{i_{0},\dots,i_{k}-1,\dots,i_{l}-1,\dots
i_{n}}=e_{\alpha,l}\cdot m_{i_{0},\dots,i_{k}-1,\dots i_{n}}$ for all $0\leq
k<l\leq n$, this claim then allows us to pick $m_{i_{0},\dots,i_{n}}$ such
that $e_{\alpha,k}\cdot m_{i_{0},\dots,i_{n}}=m_{i_{0},\dots,i_{k}-1,\dots
i_{n}}$ for all $0\leq k\leq n$, and if $m_{i_{0},\dots,i_{k}-1,\dots,i_{n}}$
has weight $\lambda$ for all $0\leq k\leq n$, then we can choose
$m_{i_{0},\dots,i_{n}}$ to have weight $\lambda-\alpha$. Hence applying this
inductively, we can construct for all $(i_{0},\dots,i_{n})\in\mathbb{Z}^{n+1}$
elements $m_{i_{0},\dots,i_{n}}\in M$ satisfying conditions (i) and (ii).

Suppose there exists $\lambda\in\mathfrak{h}^{*}$ such that
$\operatorname{wt}(m_{i_{0},\dots,i_{n}})=\lambda-(i_{0}+\dots+i_{n})\alpha$
for all $(i_{0},\dots,i_{n})\in\mathcal{I}$. Then by construction
$m_{i_{0},\dots,i_{n}}$ is weight for all
$(i_{0},\dots,i_{n})\in\mathbb{Z}^{n+1}$, so by Corollary 5.7 we have
constructed is a weight vector. ∎

We are now ready to establish one of the main ingredients towards the proof of
Theorem 5.9.

######  Lemma 5.16.

For any $N\in\mathcal{O}^{(s_{\alpha}(\mu))}(\mathfrak{g}_{n})$, the map
$\epsilon_{N}:T_{\alpha}G_{\alpha}N\rightarrow N$ given by
$\epsilon_{N}(s\otimes g)=g(s)$ is an isomorphism.

###### Proof.

First we show $\epsilon_{N}$ is a homomorphism. Let $u\in U$, let $s\in
S_{\alpha}$, and let $g\in G_{\alpha}N$. Then

$u\cdot\epsilon_{N}(s\otimes g)=u\cdot
g(s)=\phi_{\alpha}^{-1}\phi_{\alpha}(u)\cdot g(s)=g(\phi_{\alpha}(u)\cdot
s)=\epsilon_{N}((\phi_{\alpha}(u)\cdot s)\otimes
g)=\epsilon_{N}(u\cdot(s\otimes g))$

so $\epsilon_{N}$ is certainly a $U$-homomorphism.

Now let $m\in N$ be a weight vector, and choose $a_{0},\dots,a_{n}$ such that
$e_{\alpha,k}^{a_{k}}\cdot m=0$ for each $k$. Let

$\displaystyle\mathcal{I}$ $\displaystyle=\\{(i_{0},\dots,i_{n}):i_{k}\leq
0\mbox{ for some }k\mbox{ or }i_{k}\leq a_{k}\mbox{ for all
}k\\}\subseteq\mathbb{Z}^{n+1}$ $\displaystyle m_{i_{0},\dots,i_{n}}$
$\displaystyle=e_{\alpha,0}^{a_{0}-i_{0}}\dots e_{\alpha,n}^{a_{n}-i_{n}}\cdot
m\mbox{ if }i_{k}\leq a_{k}\mbox{ for all }k$ $\displaystyle
m_{i_{0},\dots,i_{n}}$ $\displaystyle=0\mbox{ otherwise.}$

Then we can use Lemma 5.15 to construct
$\varphi\in\operatorname{Hom}_{U}(S_{\alpha},\phi_{\alpha}^{-1}(M))$ which is
a weight vector and therefore in $G_{\alpha}N$ such that
$\varphi(f_{\alpha,0}^{-a_{0}}\dots f_{\alpha,n}^{-a_{n}})=m$. Hence
$\epsilon_{N}(f_{\alpha,0}^{-a_{0}}\dots
f_{\alpha,n}^{-a_{n}}\otimes\varphi)=m$, so $\epsilon_{N}$ is surjective.

We now show $\epsilon_{N}$ is injective. By Lemma 5.12 any element of
$T_{\alpha}G_{\alpha}N$ may be written as $f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}}\otimes g$ for some $g\in G_{\alpha}N$. Suppose
$\epsilon_{N}(f_{\alpha,0}^{-i_{0}}\dots f_{\alpha,n}^{-i_{n}}\otimes
g)=g(f_{\alpha,0}^{-i_{0}}\dots f_{\alpha,n}^{-i_{n}})=0$. We may assume $g$
is weight (i.e. an $\mathfrak{h}$-eigenvector). If not, we may write $g$ as a
sum of $g_{\lambda}\in(G_{\alpha}N)^{\lambda}$, which by considering weight
spaces must all satisfy $g_{\lambda}(f_{\alpha,0}^{-i_{0}}\dots
f_{\alpha,n}^{-i_{n}})=0$, and then apply the following argument to each
$g_{\lambda}$.

Observe that

(5.16) $\displaystyle g(f_{\alpha,0}^{-j_{0}}\dots
f_{\alpha,n}^{-j_{n}})=0\text{ if either }j_{k}\leq 0\text{ for some }0\leq
k\leq n\text{ or all }j_{k}\leq i_{k}.$

Now choose

$\displaystyle\mathcal{I}$
$\displaystyle=\Big{\\{}(j_{0},\dots,j_{n})\in\mathbb{Z}^{n+1}:\begin{array}[]{c}j_{k}\leq
i_{k}\mbox{ for all }0\leq k\leq n-1\mbox{ or }j_{n}\leq i_{n}\\\ \mbox{ or
}j_{k}\leq 0\mbox{ for some }0\leq k\leq
n\end{array}\Big{\\}}\subseteq\mathbb{Z}^{n+1}$ $\displaystyle
m_{j_{0},\dots,j_{n}}$ $\displaystyle=g(f_{\alpha,0}^{-j_{0}}\dots
f_{\alpha,n}^{-j_{n}})\mbox{ if }j_{k}\leq i_{k}\mbox{ for all }0\leq k\leq
n-1$ $\displaystyle m_{j_{0},\dots,j_{n}}$ $\displaystyle=0\mbox{ if
}j_{n}\leq i_{n}\mbox{ or }j_{k}\leq 0\mbox{ for some }0\leq k\leq n$

Then applying Lemma 5.15 to this, we construct
$g_{n}\in\operatorname{Hom}_{U}(S_{\alpha},\phi_{\alpha}^{-1}(M))$ which is a
weight vector (and hence in $G_{\alpha}N$) such that:

  1. (a)

$g_{n}(f_{\alpha,0}^{-j_{0}}\dots f_{\alpha,n}^{-j_{n}})=0$ if $j_{n}\leq
i_{n}$

  2. (b)

$g_{n}(f_{\alpha,0}^{-j_{0}}\dots
f_{\alpha,n}^{-j_{n}})=g(f_{\alpha,0}^{-j_{0}}\dots f_{\alpha,n}^{-j_{n}})$ if
$j_{k}\leq i_{k}$ for all $0\leq k\leq n-1$

Note that these two conditions are not mutually exclusive but they are
consistent by (5.16).

By (a), we can define $g_{n}^{\prime}\in G_{\alpha}N$ by setting
$g_{n}^{\prime}(f_{\alpha,0}^{-j_{0}}\dots
f_{\alpha,n-1}^{-j_{n-1}}f_{\alpha,n}^{-j_{n}+i_{n}})=g_{n}(f_{\alpha,0}^{-j_{0}}\dots
f_{\alpha,n}^{-j_{n}})$ and so $g_{n}=f_{\alpha,n}^{i_{n}}\cdot
g_{n}^{\prime}$. By (b) we have $(g-g_{n})(f_{\alpha,0}^{-j_{0}}\dots
f_{\alpha,n}^{-j_{n}})=0$ if either some $j_{k}\leq 0$ or if $j_{k}\leq i_{k}$
for all $0\leq k\leq n-1$.

We now construct $g_{s}\in G_{\alpha}N$ inductively by setting

$\displaystyle\mathcal{I}$
$\displaystyle=\\{(j_{0},\dots,j_{n})\in\mathbb{Z}^{n+1}:j_{k}\leq i_{k}\mbox{
for all }0\leq k\leq s-1\mbox{ or }j_{s}\leq i_{s}\mbox{ or }j_{k}\leq 0\mbox{
for some }0\leq k\leq n\\}\subseteq\mathbb{Z}^{n+1}$ $\displaystyle
m_{j_{0},\dots,j_{n}}$ $\displaystyle=(g-g_{n}-\dots-
g_{s+1})(f_{\alpha,0}^{-j_{0}}\dots f_{\alpha,n}^{-j_{n}})\mbox{ if }j_{k}\leq
i_{k}\mbox{ for all }0\leq k\leq s-1$ $\displaystyle m_{j_{0},\dots,j_{n}}$
$\displaystyle=0\mbox{ if }j_{s}\leq i_{s}\mbox{ or }j_{k}\leq 0\mbox{ for
some }0\leq k\leq n$

The element $g-g_{n}-\dots-g_{s+1}$ satisfies a condition analogous to (5.16),
which means that the definition of $m_{j_{0},...,j_{n}}$ is consistent.

This $g_{s}$ satisfies:

  1. (a)

$g_{s}(f_{\alpha,0}^{-j_{0}}\dots f_{\alpha,n}^{-j_{n}})=0$ if $j_{s}\leq
i_{s}$

  2. (b)

$g_{s}(f_{\alpha,0}^{-j_{0}}\dots f_{\alpha,n}^{-j_{n}})=(g-g_{n}-\dots-
g_{s+1})(f_{\alpha,0}^{-j_{0}}\dots f_{\alpha,n}^{-j_{n}})$ if $j_{k}\leq
i_{k}$ for all $0\leq k\leq s-1$

By (a), we can write this $g_{s}$ in the form $g_{s}=f_{\alpha,s}^{i_{s}}\cdot
g_{s}^{\prime}$ for some $g_{s}^{\prime}\in G_{\alpha}N$. By (b) we have that
$(g-g_{n}-\dots-g_{s})(f_{\alpha,0}^{-j_{0}}\dots f_{\alpha,n}^{-j_{n}})=0$ if
$j_{k}\leq i_{k}$ for all $0\leq k\leq s-1$. Hence we see that
$g=f_{\alpha,0}^{-i_{0}}\cdot g_{0}^{\prime}+\dots+f_{\alpha,n}^{-i_{n}}\cdot
g_{n}^{\prime}$ for some $g_{0}^{\prime},\dots,g_{n}^{\prime}\in G_{\alpha}N$,
so $f_{\alpha,0}^{-i_{0}}\dots f_{\alpha,n}^{-i_{n}}\otimes
g=\sum(f_{\alpha,0}^{-i_{0}}\dots f_{\alpha,n}^{-i_{n}})\otimes
f_{\alpha,k}^{i_{k}}g_{k}^{\prime}=0$. Hence $\epsilon_{N}$ is injective. ∎

The proof of this next lemma is almost identical to [Ch23, Lemma 5.8], and so
we skip it for the sake of brevity.

######  Lemma 5.17.

$G_{\alpha}$ restricts to a functor
$\mathcal{O}^{(s_{\alpha}(\mu))}(\mathfrak{g}_{n})\rightarrow\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$.

This next result can be checked by following the argument for [Ch23, Lemma
5.8(e)] verbatim.

######  Lemma 5.18.

The transformations

$\psi:\operatorname{id}_{\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})}\rightarrow
G_{\alpha}T_{\alpha}$

and

$\epsilon:T_{\alpha}G_{\alpha}\rightarrow\operatorname{id}_{\mathcal{O}^{(s_{\alpha}(\mu))}(\mathfrak{g}_{n})}$

are natural.

Finally Theorem 5.9 follows by combining Lemmas 5.10, 5.13, 5.16, 5.17 and
5.18.

##  6\. Composition Multiplicities

###  6.1. Definition of composition multiplicity

Let $M\in\mathcal{O}(\mathfrak{g}_{n})$. We define the _character_ of $M$ to
be the function
$\operatorname{ch}(M):\mathfrak{h}^{*}\rightarrow\mathbb{Z}_{\geq 0}$ which
sends $\lambda$ to $\operatorname{dim}(M^{\lambda})$, and the _support_ of
$M$, denoted $\operatorname{supp}(M)$, to be the set
$\operatorname{supp}(M)=\\{\lambda\in\mathfrak{h}^{*}:M^{\lambda}\neq 0\\}$.

We define the composition multiplicities $k_{\lambda}(M)$ of $M$ for
$\lambda\in\mathfrak{h}^{*}$ using the following result, whose statement and
proof is very similar to [Ch23, Lemma 6.1] or [MSo19, Proposition 8].

######  Lemma 6.1.

Let $\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$. For any
$M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$, there are unique
$\\{k_{\lambda}(M)\in\mathbb{Z}_{\geq
0}:\lambda\in\mathfrak{h}_{n}^{*},\lambda_{\geq 1}=\mu\\}$, which we refer to
as composition multiplicities, such that

$\operatorname{ch}(M)=\sum k_{\lambda}(M)\operatorname{ch}(L_{\lambda})$

We give another interpretation of these which is closer to the notion of
composition multiplicities in artinian categories. The following result is a
natural generalisation of [Ch23, Lemma 6.2].

######  Lemma 6.2.

Let $\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ and
$M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$. Then there exists
$\mathcal{I}\subseteq(\mathbb{Z}_{\geq 0})^{n}$ and a descending filtration of
$M$ indexed by by $\mathcal{I}$ with the lexicographic ordering, such that:

  1. (a)

If $(i_{1},\dots,i_{k-1},i_{k},\dots,i_{n})\in\mathcal{I}$ with $i_{k}>0$ then
$(i_{1},\dots,i_{k-1},i_{k}-1,0,\dots,0)\in\mathcal{I}$.

  2. (b)

Each quotient $M_{(i_{1},\dots,i_{n}-1)}/M_{(i_{1},\dots,i_{n})}$ is simple.

  3. (c)

The intersection of all the $M_{(i_{1},\dots,i_{n})}$ is 0.

Furthermore, for any such filtration $L_{\lambda}$ appears as a quotient
$M_{(i_{1},\dots,i_{n}-1)}/M_{(i_{1},\dots,i_{n})}$ precisely $k_{\lambda}(M)$
times.

###### Proof.

First observe that by Theorems 4.1 and 5.9 we may reduce to the case
$\mu(\mathfrak{h}_{n}^{n})=0$.

Recall the notation $\mathfrak{g}_{n}^{n}=\mathfrak{g}\otimes
t^{n}\subseteq\mathfrak{g}_{n}$.

First suppose $n=1$. In this case $M$ has a filtration
$M\supseteq\mathfrak{g}_{1}^{1}M\supseteq(\mathfrak{g}_{1}^{1})^{2}M\supseteq\cdots$.
Each quotient $(\mathfrak{g}_{1}^{1})^{i}M/(\mathfrak{g}_{1}^{1})^{i+1}M$ lies
in BGG category $\mathcal{O}$ for $\mathfrak{g}$ and hence has finite length,
so this may be refined to a filtration satisfying (a) and (b).

For $n>1$ consider the filtration
$M\supseteq\mathfrak{g}_{n}^{n}M\supseteq(\mathfrak{g}_{n}^{n})^{2}M\supseteq\cdots$.
Each quotient $(\mathfrak{g}_{n}^{n})^{i}M/(\mathfrak{g}_{n}^{n})^{i+1}M$ lies
in the category $\mathcal{O}^{(\mu)}(\mathfrak{g}_{n-1})$, where we identify
$\mu$ with an element of $(\mathfrak{h}_{n-1}^{\geq 1})^{*}$ in the obvious
manner, so we may set $M_{(i,0,\dots,0)}=(\mathfrak{g}_{n}^{n})^{i}M$ and
argue by induction that this may be refined to a filtration satisfying (a) and
(b).

To show this filtration satisfies (c), it is enough to show that whenever
$\mu(\mathfrak{h}_{n}^{n})=0$ every
$M\in\mathcal{O}^{(\mu)}(\mathfrak{g}_{n})$ has the property that
$\bigcap_{i\geq 0}(\mathfrak{g}_{n}^{n})^{i}M=0$. By considering weight
spaces, this property is preserved by taking quotients and extensions, so it
is enough to verify this in the case where $M=M_{\lambda}$ is a Verma module.
Since $\lambda_{n}=\mu_{n}=0$ this Verma module is graded: we place a grading
on $U(\mathfrak{g}_{n})$ by setting, for $x\in\mathfrak{g}$,

$\displaystyle\deg x_{i}=\left\\{\begin{array}[]{rl}0&\text{ if
}i=0,...,n-1,\\\ 1&\text{ if }i=n.\end{array}\right.$

We transfer the induced grading on $U(\mathfrak{n}^{-}_{n})$ to $M_{\lambda}$
via the isomorphism of $\mathfrak{n}^{-}_{n}$-modules
$U(\mathfrak{n}^{-}_{n})\cong M_{\lambda}$. Now $M_{\lambda}$ is a positively
graded $\mathfrak{g}_{n}$-module and $\bigcap_{i\geq
0}(\mathfrak{g}_{n}^{n})^{i}M_{\lambda}$ is contained in the intersection of
all graded components, which is zero.

Finally, we observe that
$\operatorname{ch}(M)=\sum_{(i_{0},\dots,i_{n})\in\mathcal{I},i_{n}>0}\operatorname{ch}(M_{i_{0},\dots,i_{n}-1}/M_{i_{0},\dots,i_{n}})$,
so by uniqueness in Lemma 6.1 the final part of the Lemma holds. ∎

This result justifies the use of the terminology composition multiplicities,
for $k_{\lambda}(M)$. From now on we use the notation
$[M:L_{\lambda}]:=k_{\lambda}(M)$.

######  Corollary 6.3.

The parabolic induction and restriction functors and the twisting functors
$T_{\alpha},G_{\alpha}$ preserve composition multiplicities. $\hfill\qed$

###  6.2. Computation of composition multiplicities for Verma modules

We now wish to compute the composition multiplicities $[M_{\lambda}:L_{\nu}]$.
Thanks to Lemma 3.4 and Lemma 6.2 we know that $[M_{\lambda}:L_{\nu}]=0$
unless $\lambda_{n}=\nu_{n}$ and so, using a combination of twisting functors
and parabolic induction functors, we can reduce to the case
$\lambda_{n}=\nu_{n}=0$, and this is our strategy.

Let $p:\mathbb{Z}_{\geq 0}\Phi^{+}\to\mathbb{Z}_{\geq 0}$ be Kostant’s
partition function. For $\alpha\in\mathbb{Z}\Phi$ we write
$\alpha_{0}\in\mathfrak{h}_{n-1}^{*}$ to be the element satisfying
$\alpha_{0}(h_{i})=0$ for $i>0$ and $\alpha_{0}(h_{0})=\alpha(h)$, where
$h\in\mathfrak{h}$. Also, for any $\lambda\in\mathfrak{h}_{n}^{*}$ we identify
$\lambda_{\leq n-1}$ with an element of $\mathfrak{h}_{n-1}^{*}$ in the
obvious manner.

######  Lemma 6.4.

If $M_{\lambda_{\leq n-1}-\alpha_{0}}(\mathfrak{g}_{n-1})$ is the Verma module
over $\mathfrak{g}_{n-1}$ then

$\operatorname{ch}M_{\lambda}=\sum_{\alpha\in\mathbb{Z}_{\geq
0}\Phi^{+}}p(\alpha)\operatorname{ch}(M_{\lambda_{\leq
n-1}-\alpha_{0}}(\mathfrak{g}_{n-1})).$

###### Proof.

Note that characters can be defined for any semisimple $\mathfrak{h}$-module
with finite dimensional weight spaces. Let $\mathbb{C}_{\lambda_{0}}$ be the
$\mathfrak{h}$-module of weight $\lambda_{0}$.

Since $M_{\lambda}\cong
U(\mathfrak{n}^{-}_{n})\otimes\mathbb{C}_{\lambda_{0}}$ as
$\mathfrak{h}$-modules the lemma follows from the facts:

  1. (i)

The character of $U(\mathfrak{n}^{-}_{n})$ is equal to the character of
$S(\mathfrak{n}^{-}_{n})$.

  2. (ii)

$S(\mathfrak{n}^{-}_{n})$ is a free module over $S(\mathfrak{n}^{-}_{n-1})$
and, for $\alpha\in\mathbb{Z}_{\geq 0}\Phi^{+}$, there are $p(\alpha)$ basis
vectors of weight $-\alpha$.

∎

######  Corollary 6.5.

If $\lambda,\nu\in\mathfrak{h}_{n}^{*}$ satisfy $\lambda_{n}=\nu_{n}=0$, and
$\lambda_{\geq 1}=\nu_{\geq 1}$, and $L_{\nu_{\leq n-1}}(\mathfrak{g}_{n-1})$
denotes the simple $\mathfrak{g}_{n-1}$-module of highest weight $\nu_{\leq
n-1}\in\mathfrak{h}_{n}^{*}$ then

$[M_{\lambda}:L_{\nu}]=\sum_{\alpha\in\mathbb{Z}_{\geq
0}\Phi^{+}}p(\alpha)[M_{\lambda_{\leq
n-1}-\alpha_{0}}(\mathfrak{g}_{n-1}):L_{\nu_{\leq n-1}}(\mathfrak{g}_{n-1})].$

###### Proof.

By Lemma 3.6 we have $\operatorname{ch}L_{\nu}=\operatorname{ch}L_{\nu_{\leq
n-1}}$ for all $\nu\in\mathfrak{h}_{n}^{*}$ satisfying $\nu_{n}=0$, and so the
claim follows from Lemma 6.4. ∎

For $\mu_{n}\in\mathfrak{h}^{*}$ we write
$\mathfrak{g}^{\mu_{n}}_{n-1}:=(\mathfrak{g}^{\mu_{n}})_{n-1}=(\mathfrak{g}_{n-1})^{\mu_{n}}$.

######  Corollary 6.6.

Let $\mu\in(\mathfrak{h}_{n}^{\geq 1})^{*}$ such that $\mathfrak{g}^{\mu_{n}}$
is the Levi factor of a standard parabolic. Then for any
$\lambda,\nu\in\mathfrak{h}_{n}^{*}$ such that $\lambda_{\geq 1}=\nu_{\geq
1}=\mu$ we have

$[M_{\lambda}:L_{\nu}]=\sum_{\alpha\in\mathbb{Z}_{\geq
0}\Phi^{+}}p(\alpha)[M_{\lambda_{\leq
n-1}-\alpha_{0}}(\mathfrak{g}^{\mu_{n}}_{n-1}):L_{\nu_{\leq
n-1}}(\mathfrak{g}^{\mu_{n}}_{n-1})].$

###### Proof.

This follows from Lemma 3.7 and Theorem 4.1, Corollary 6.2, Corollary 6.5. In
order to apply these results one should check that the functors appearing in
the first two cited results send highest weight modules to highest weight
modules of the corresponding weight, and we leave this verification to the
reader. ∎

To complete the computation of composition multiplicities for all Verma
modules we must show that to any Jordan block we can apply twisting functors
to obtain a Jordan block $\mathcal{O}^{(\mu)}$ such that $\mathfrak{g}^{\mu}$
is the Levi factor of a standard parabolic. This is possible thanks to the
following result (see [Ch23, Lemma 2.1] for a proof).

######  Proposition 6.7.

Let $\mu\in\mathfrak{h}^{*}$. Then there exists $w\in W$ and a Levi factor
$\mathfrak{l}$ of a standard parabolic $\mathfrak{p}$ such that
$\mathfrak{g}^{w(\mu)}=\mathfrak{l}$. Furthermore, if $w$ is chosen to be of
minimal length subject to this condition and
$w=s_{\alpha_{n}}s_{\alpha_{n-1}}\dots s_{\alpha_{1}}$ is a reduced expression
for $w$, then for each $1\leq i\leq n$, we have $((s_{\alpha_{i-1}}\dots
s_{\alpha_{1}})\mu)(h_{\alpha_{i}})\neq 0$. $\hfill\qed$

We now define an action $\bullet_{n}$ of $W$ on $\mathfrak{h}^{*}$ by
$w\bullet_{n}\lambda:=w(\lambda+n\rho)-n\rho$ where $\rho$ is half the sum of
the positive roots. Note this generalises the dot action of $W$ on
$\mathfrak{h}^{*}$, which corresponds to the case $n=1$, and which controls
the central characters of $\mathfrak{g}_{0}$ [Hu08, §1.9]. We remark that this
$n$-dot action appeared in the work of Geoffriau on the centre of the
enveloping algebra of $\mathfrak{g}_{n}$ [Ge95].

If $\alpha$ is any simple root, then since $s_{\alpha}(\alpha)=-\alpha$ and
$s_{\alpha}$ permutes the other positive roots we have
$s_{\alpha}(\rho)=\rho-\alpha$. We then have

$\displaystyle s_{\alpha}\bullet_{n}\lambda$
$\displaystyle=s_{\alpha}(\lambda+n\rho)-n\rho$
$\displaystyle=s_{\alpha}(\lambda)+n\rho-n\alpha-n\rho$
$\displaystyle=s_{\alpha}(\lambda)-n\alpha.$

We extend this to a $W$-action on $\mathfrak{h}_{n}^{*}$ by setting

$(s_{\alpha}\bullet_{n}\lambda)(h_{i})=\left\\{\begin{array}[]{cc}(s_{\alpha}\bullet_{n}\lambda_{i})(h_{i})&\text{
for }i=0\\\ (s_{\alpha}\lambda_{i})(h_{i})&\text{ for }i\neq
0.\end{array}\right.$

By our calculations in the proof of Lemma 5.10, the twisting functors
$T_{\alpha}$ take highest weight modules of weight
$\lambda\in\mathfrak{h}_{n}^{*}$ to highest weight modules of weight
$s_{\alpha}\bullet_{n}\lambda$, and hence take $M_{\lambda}$ to
$M_{s_{\alpha}\bullet_{n}\lambda}$ and similar for $L_{\lambda}$. Applying
Corollary 6.3 again we obtain the following.

######  Corollary 6.8.

Let $\lambda,\nu\in\mathfrak{h}_{n}^{*}$ such that $\lambda_{\geq 1}=\nu_{\geq
1}$. Let $w\in W$ have a reduced expression
$w=s_{\alpha_{k}}s_{\alpha_{k-1}}\cdots s_{\alpha_{1}}$ for simple reflections
$s_{\alpha_{i}}$ such that:

  1. (a)

$\mathfrak{g}^{w(\mu_{n})}$ is the Levi factor of a standard parabolic.

  2. (b)

For each $1\leq i\leq n$, we have $((s_{\alpha_{i-1}}\cdots
s_{\alpha_{1}})\mu)(h_{\alpha_{i}})\neq 0$.

Then we have

(6.2)
$\displaystyle[M_{\lambda}:L_{\nu}]=[M_{w\bullet_{n}\lambda}:L_{w\bullet_{n}\nu}]=\sum_{\alpha\in\mathbb{Z}_{\geq
0}\Phi^{+}}p(\alpha)[M_{(w\bullet_{n}\lambda)_{\leq
n-1}-\alpha_{0}}(\mathfrak{g}^{w(\lambda_{n})}_{n-1}):L_{(w\bullet_{n}\nu)_{\leq
n-1}}(\mathfrak{g}^{w(\mu_{n})}_{n-1})].$

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