# Affine Laumon spaces and iterated ${\mathcal{W}}$-algebras

Thomas Creutzig  Duiliu-Emanuel Diaconescu  Mingyang Ma

###### Abstract

A family of vertex algebras whose universal Verma modules coincide with the
cohomology of affine Laumon spaces is found. This result is based on an
explicit expression for the generating function of Poincaré polynomials of
these spaces.

There is a variant of quantum Hamiltonian reduction that realizes vertex
algebras which we call iterated ${\mathcal{W}}$-algebras and our main
conjecture is that the vertex algebras associated to the affine Laumon spaces
are subalgebras of iterated ${\mathcal{W}}$-algebras.

###### Contents

  1. 1 Introduction
  2. 2 Affine Laumon spaces
    1. 2.1 ADHM orbifolds
    2. 2.2 Torus fixed points
  3. 3 The generating function of Poincaré polynomials
    1. 3.1 The Morse index
    2. 3.2 The generating function
  4. 4 ${\mathcal{W}}$-algebras
    1. 4.1 Affine Lie algebras
      1. 4.1.1 $\mathfrak{g}=\widehat{\mathfrak{gl}}_{N}$
    2. 4.2 Quantum Hamiltonian reduction
    3. 4.3 Examples for ${\mathfrak{g}}={\mathfrak{g}}{\mathfrak{l}}_{N}$
    4. 4.4 Properties
    5. 4.5 Characters
    6. 4.6 Iterated reductions
    7. 4.7 Iterated Verma modules
    8. 4.8 ${\mathcal{W}}_{L}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))$: The iterated ${\mathcal{W}}$-algebras of ${\mathfrak{g}}{\mathfrak{l}}_{N}$
    9. 4.9 Conformal weights and refined characters
  5. 5 Comparison to partition functions of affine Laumon spaces
  6. A Some combinatorial identities
  7. B Some product identities

##  1 Introduction

Affine Laumon spaces provide a rigorous mathematical construction for the
surface defects introduced by Gukov and Witten in [GW08]. In the context of
AGT correspondence [AGT10], this predicts a vertex operator algebra action on
their equivariant Borel-Moore homology. As shown by Schiffmann and Vasserot
[SV13] and by Maulik and Okounkov [MO19], for the moduli space of rank $r$
unramified instantons, this vertex algebra is the ${\mathcal{W}}$-algebra
associated to a principal nilpotent element in ${\mathfrak{g}l}_{r}$. More
precisely, the equivariant Borel-Moore homology of the instanton moduli space
is identified to a highest weight ${\mathcal{W}}({\mathfrak{g}l}_{r})$-module.
This action is constructed using an algebraic relation between the
${\mathcal{W}}$-algebra and the Yangian of affine ${\mathfrak{g}l}_{1}$. The
latter is then shown to act on the equivariant Borel-Moore homology via Hecke
operators. The analogous construction of moduli spaces of $G$-instantons was
carried out by Braverman, Fikelberg and Nakajima [BFN16]. Earlier results on
the representation theoretic structure of the equivariant Borel-Moore homology
of affine Laumon spaces of type $(1,\ldots,1)$ include [FFNR11, Neg18, Neg11].

A similar program for ramified instantons was initiated by Finkelberg and
Tsymbaliuk [FT19] and Negut [Neg19]. Using Hecke transformations, they
constructed a shifted affine Yangian action on the equivariant Borel-Moore
homology of affine Laumon spaces, respectively a shifted quantum toroidal
action on their equivariant K-theory. The main missing piece of the puzzle is
the construction of the associated vertex operator algebras. This is
understood only for the rectangular case i.e. affine Laumon spaces of type
$(s,\ldots,s)$, which is treated in [Neg20]. Similar relations between affine
Yangians and rectangular ${\mathcal{W}}$-algebras were established in [Ued22,
Ued21].

In the physics literature, the relation between the Nekrasov partition
functions of affine Laumon spaces and ${\mathcal{W}}$-characters was studied
in [KPW10, Wyl11b, Wyl11a, KT11]. Most of these results are restricted to the
case to affine Laumon spaces of type $(1,\ldots,1)$, with the exception of
Kanno and Tachikawa [KT11] who address some aspects of the partition function
for Euler characteristics in the general case.

The present paper presents a conjectural construction of vertex operator
algebras associated to affine Laumon spaces of type ${\bf
r}=(r_{0},\ldots,r_{\ell-1})$, $\ell\geq 2$, subject to the conditions
$r_{0}\geq r_{1}\geq\ldots\geq r_{\ell-1}\geq 1$. The main result identifies
the generating function for Poincaré polynomials of affine Laumon spaces of
type ${\bf r}$ as above to the conformal weight refinement of the universal
Verma module character of the associated vertex algebra. These vertex operator
algebras are obtained as natural subalgebras of iterated
${\mathcal{W}}$-algebras.

In more detail, recall that the affine Laumon space of type
$(r_{0},\ldots,r_{\ell})$ is a moduli space of framed parabolic torsion free
sheaves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$. The parabolic structure is
specified by a flag of subsheaves of ranks $r_{0},\ldots,r_{\ell}$ along
$\mathbb{P}^{1}\times\\{0\\}$, while the framing condition rigidifies the
parabolic sheaves along
$\mathbb{P}^{1}\times\\{\infty\\}\,\cup\,\\{\infty\\}\times\mathbb{P}^{1}$.
These spaces were studied in detail in [FR14]. In particular, as shown in loc.
cit, each such space is naturally identified to a connected component of the
fixed locus for a certain ${{\boldsymbol{\mu}}}_{\ell}$-action on the moduli
space of rank $r=r_{0}+\cdots+r_{\ell}$ instantons. Moreover, using the ADHM
construction, the affine Laumon spaces are identified to moduli spaces of
cyclic framed representations of the chain-saw quiver. Section 2 provides a
summary of all these results.

For fixed type ${\bf r}=(r_{0},\ldots,r_{\ell})$, the moduli space of framed
parabolic sheaves consists of infinitely many smooth components
${\mathcal{M}}({\bf r},{\bf n})$. The index ${\bf n}\in\mathbb{Z}_{\geq
0}^{\times\ell}$ encodes the parabolic second Chern class. The generating
function

$Z_{\bf r}=\sum_{{\bf n}}P_{y}({\mathcal{M}}({\bf r},{\bf
n}))\prod_{a=0}^{\ell-1}q_{a}^{n_{a}}$

for Poincaré polynomials is computed in Section 3 using Atiyah-Bott-Morse
localization. This computation generalizes the results obtained by Nakajima
and Yoshioka [NY04] for unramified instantons, and employs similar techniques.
Similar results were obtained by Bruzzo et al in [BPT11] for framed torsion
free sheaves on Hirzebruch surfaces.

In order to present the result in a concise form, for any $a\in\mathbb{Z}$,
let $q_{a}=q_{c}$ and $r_{a}=r_{c}$ where $c\in\\{0,\ldots,\ell-1\\}$ is
uniquely defined by $a\equiv c$ mod $\ell$. Then one has:

######  Theorem 1.1.

(Theorem 3.3) The following identity holds

$Z_{\bf
r}=\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{c=1}^{\ell-1}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a(\beta)-c})^{-1}({\widetilde{q}}_{a(\beta)-c-1})^{-1}\cdots({\widetilde{q}}_{a(\beta)-\ell+1})^{-1}}~{},$
(1.1)

where

${\widetilde{q}}_{a}=y^{2r_{a}}q_{a}$ (1.2)

for any $a\in\mathbb{Z}$ and

$z=\prod_{a=0}^{\ell-1}{\widetilde{q}}_{a}.$

Moreover, note that the odd degree cohomology of the spaces
${\mathcal{M}}({\bf r},{\bf n})$ is trivial since the fixed locus of the torus
action is finite for any ${\bf r}$, ${\bf n}$.

As shown in Proposition (3.4), an alternative expression for the partition
function (1.1) is

$\displaystyle Z_{\bf r}=$
$\displaystyle\bigg{(}\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}\bigg{)}$ (1.3) $\displaystyle\prod_{1\leq
a<c\leq\ell}\bigg{(}\prod_{t=1}^{r_{\ell-c}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\bigg{(}\prod_{t=1}^{r_{\ell-a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)},$

where the variables $u_{c}$, $1\leq c\leq\ell$, are given by

$u_{c}=({\widetilde{q}}_{-c})^{-1}\cdots({\widetilde{q}}_{-\ell+1})^{-1}~{},\qquad
1\leq c\leq\ell-1,\qquad u_{\ell}=1.$ (1.4)

In section 4 we find a family of vertex algebras indexed by ${\bf
r}=(r_{0},\ldots,r_{\ell-1})$, $r_{0}\geq\cdots\geq r_{\ell-1}\geq 1$, so that
the partition function (1.3) coincides with the conformal weight refinement of
the character of the associated universal Verma module. These vertex algebras
are ${\mathcal{W}}$-algebras times certain commutative vertex algebras. A
${\mathcal{W}}$-algebra is constructed from an affine vertex algebra via
quantum Hamiltonian reduction, i.e. as a BRST cohomology associated to a
nilpotent element. We are interested in a variant or better iteration of
quantum Hamiltonian reduction that realizes this family of vertex algebras.

Most of section 4 is thus devoted to the explanation of a concept that we call
iterated quantum Hamiltonian reduction. This procedure is related to the
gluing of corner vertex algebras [GR19, CG20, PR18]. Our main conjecture is
then that the families of vertex algebras that are associated to affine Laumon
spaces are subalgebras of iterated quantum Hamiltonian reductions.

In order to set the stage, sections 4.1 – 4.5 provide a review of
${\mathcal{W}}$-algebras focusing on ${\mathcal{W}}$-algebras associated to
nilpotent elements in ${\mathfrak{g}l}_{N}$. In particular Sections 4.4 and
4.5 summarize their main properties as well as the main properties of their
universal Verma modules and characters.

The construction of iterated ${\mathcal{W}}$-algebras and their universal
Verma modules, which is the main construction in this paper, is presented in
Section 4.6. The input data consists of:

  * •

a simple Lie algebra ${\mathfrak{g}}$,

  * •

an ordered sequence $({\mathfrak{a}}_{1},\ldots,{\mathfrak{a}}_{L})$ of
subalgebras of ${\mathfrak{g}}$ so that
${\mathfrak{a}}_{1}\oplus\cdots\oplus{\mathfrak{a}}_{L}$ is a subalgebra of
${\mathfrak{g}}$, and

  * •

a sequence $(f_{1},\ldots,f_{L})$ of nilpotent elements of
$({\mathfrak{a}}_{1},\ldots,{\mathfrak{a}}_{L})$

Given the above data, Section 4.6 constructs a ${\mathcal{W}}$-algebra
${\mathcal{W}}^{k}_{L}({\mathfrak{g}},({\mathfrak{a}}_{1},\ldots,{\mathfrak{a}}_{L}),(f_{1},\ldots,f_{L}))$
by iterated quantum Hamiltonian reduction. This algebra is conjectured to be
stable equivalent to the ${\mathcal{W}}$-algebra
${\mathcal{W}}^{k}({\mathfrak{g}},f)$ where $f=f_{1}+\cdots+f_{L}$. Vertex
algebras are called stable equivalent if they are isomorphic up to tensoring
with free field algebras. Section 4.7 extends the iterated construction to
Verma modules and section 4.8 presents in detail a sequence of iterated
${\mathcal{W}}$-algebras of ${\mathfrak{g}l}_{N}$ associated to an ordered
partition

$N=m_{1}s_{1}+\cdots+m_{L}s_{L}$

with $0<s_{1}<\dots<s_{L}$ and $m_{1},\ldots,m_{L}>0$. In this case the
subalgebras $({\mathfrak{a}}_{1},\ldots,{\mathfrak{a}}_{L})$ are the natural
subalgebras $({\mathfrak{g}}_{N_{1}},\ldots,{\mathfrak{g}}_{N_{L}})$,
$N_{i}=m_{i}s_{i}$, and $f_{i}$ is a principal nilpotent element in
${\mathfrak{g}}_{N_{i}}$. The main conjecture in this paper then reads

######  Conjecture 1.2.

(Conjecture 4.7) There is an isomorphism of vertex algebras

${\mathcal{W}}_{L}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))\cong{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)\otimes\bigotimes\limits_{1\leq
i<j\leq L}\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}$ (1.5)

where each $\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}$ factor denotes the vertex
algebra of a pair of symplectic bosons.

This conjecture is verified for characters in identity (4.26), which reads

$\begin{split}\text{ch}[{\mathcal{W}}_{L}^{k}&({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))]={\mathrm{ch}}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}\text{ch}[\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}],\end{split}$ (1.6)

The presentation of iterated ${\mathcal{W}}$-algebras is then concluded in
Section 4.9, which presents the conformal weight filtration [Li04] providing
the resulting refined characters. In particular, it is observed that identity
(1.6) holds for refined characters as well.

Finally, Section 5, identifies the partition function (1.3) with the refined
character of the universal Verma module of a vertex subalgebra of the tensor
product

${\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)\otimes\bigotimes\limits_{1\leq
i<j\leq L}\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}$

This subalgebra is obtained as a truncation of each $\beta\gamma$-system in
the right hand side to the subalgebra $B(m_{i},m_{j},s_{j}-s_{i})$ generated
by the fields $\beta$. A detailed discussion is provided in Example 4.8. Using
this truncation, the following identity is proven in Proposition 5.1

$Z_{\bf r}={\rm ch}_{\rm
ref}^{V}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}\ {\rm ch}_{\rm ref}[B(m_{i},m_{j},s_{j}-s_{i})]$ (1.7)

where

$(r_{0},\ldots,r_{\ell-1})=\big{(}\,\underbrace{s_{L},\ldots,s_{L}}_{m_{L}},\ldots,\underbrace{s_{1},\ldots,s_{1}}_{m_{1}}\,\big{)},\qquad
0<s_{1}<\cdots s_{L},\qquad m_{i}>0,\ 1\leq i\leq L.$

Using Conjecture 1.2, one concludes that

$Z_{\bf r}={\rm ch}_{\rm
ref}^{V}[{\mathcal{W}}_{L,\beta}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))]$
(1.8)

where
${\mathcal{W}}_{L,\beta}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))$
is the inverse image of the subalgebra

${\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)\otimes\bigotimes\limits_{1\leq
i<j\leq L}B(m_{i},m_{j},s_{j}-s_{i})$

via isomorphism (1.5). Since the odd degree cohomology is trivial, this leads
to an implicit identification of the cohomology of the affine Laumon space of
type ${\bf r}$ to the universal Verma module of the vertex algebra
${\mathcal{W}}_{L,\beta}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))$.
The main open question is whether this module structure is related to the
shifted affine Yangian action constructed in [FT19, Neg19]. This will be left
for future work.

Acknowledgements. We are very grateful to Sujay Ashok for collaboration and
sharing his insights with us at an early stage of this project. We would also
like to thank Andrei Negut, Francesco Sala, Olivier Schiffmann, Miroslav
Rapčák and Abay Zhakenov for very helpful discussions and correspondence. The
work of T.C. was partially supported by NSERC Grant Number RES0048511. The
work of D.-E.D. was partially supported by the NSF grant DMS-1802410.

##  2 Affine Laumon spaces

This section is a review of affine Laumon space following [FR14]. The
presentation will emphasize their construction in terms of representations of
the chain-saw quiver.

###  2.1 ADHM orbifolds

Given two vector spaces $V$ and $W$, the moduli space of framed
representations of the ADHM quiver is a GIT quotient ${\mathcal{A}}//GL(V)$
where

${\mathcal{A}}=\\{\,(A_{1},A_{2},I,J)\in{\rm End}(V)^{\times 2}\times{\rm
Hom}(W,V)\times{\rm Hom}(W,V)\,|\,[A_{1},A_{2}]+IJ=0\\}.$

The general linear group $GL(V)$ acts naturally on ${\mathcal{A}}$,

$g\times(A_{1},A_{2},I,J)\mapsto(gA_{1}g^{-1},gA_{2}a^{-1},gI,Jg^{-1}).$

The stable locus ${\mathcal{A}}^{s}\subset{\mathcal{A}}$ is defined by ruling
out $A_{i}$-invariant subspaces $V^{\prime}\subsetneq V$, $1\leq i\leq 2$, so
that ${\rm Im}(I)\subset V^{\prime}$. The resulting moduli space is a smooth
quasi-projective variety of dimension $2rn$, where $r={\rm dim}(W)$ and
$n={\rm dim}(V)$. In the following we will fix a basis $(w_{\alpha})$ in $W$,
$1\leq\alpha\leq r$, and write $(I_{\alpha})$, $(J_{\alpha})$ for the
associated components of $I,J$.

Any ordered partition ${\bf r}=(r_{0},\ldots,r_{\ell-1})$, where
$r_{0}+\cdots+r_{\ell-1}=r$, $r_{a}\geq 0$, $\ell\geq 2$, determines a cyclic
group action ${\boldsymbol{\mu}}_{\ell}\times{\mathcal{A}}\to{\mathcal{A}}$ as
follows. Let ${\boldsymbol{\mu}}_{\ell}\to{\rm End}(W)$, be the linear action
which maps the canonical generator $\omega=e^{2i\pi/\ell}$ to the diagonal
matrix

$\omega\mapsto M=\rm{diag}(\ \underbrace{1,\ldots
1}_{r_{0}},\underbrace{\omega,\ldots,\omega}_{r_{1}},\ldots,\underbrace{\omega^{\ell-1},\ldots,\omega^{\ell-1}}_{r_{\ell-1}}\
).$ (2.1)

Then one defines the linear action
${\boldsymbol{\mu}}_{\ell}\times{\mathcal{A}}\to{\mathcal{A}}$,

$\omega\times(A_{1},A_{2},I,J)\mapsto(A_{1},\omega A_{2},IM^{-1},\omega MJ).$

This action commutes with the $GL(V)$-action on ${\mathcal{A}}$ and preserves
the stable locus, hence it descends to a ${\boldsymbol{\mu}}_{\ell}$-action on
the moduli space.

The components of the ${\boldsymbol{\mu}}_{\ell}$-fixed locus in the moduli
space are indexed by ordered partitions ${\bf n}=(n_{0},\cdots,n_{\ell-1})$,
$n_{0}+\cdots+n_{\ell-1}=n$, $n_{a}\geq 0$, $0\leq a\leq\ell-1$. The fixed
component ${\mathcal{M}}({\bf r},{\bf n})$ corresponding to an ordered
partition ${\bf n}$ consists of isomorphism classes of ADHM data
$(A_{1},A_{2},I,J)$ satisfying

$A_{1}=U_{\bf n}A_{1}U_{\bf n}^{-1},\qquad\omega A_{2}=U_{\bf n}A_{2}U_{\bf
n}^{-1},\qquad IM^{-1}=U_{\bf n}^{-1}I,\qquad\omega MJ=JU_{\bf n}$ (2.2)

where

$U_{\bf n}=\rm{diag}(\ \underbrace{1,\ldots
1}_{n_{0}},\underbrace{\omega,\ldots,\omega}_{n_{1}},\ldots,\underbrace{\omega^{\ell-1},\ldots,\omega^{\ell-1}}_{n_{\ell-1}}\
).$

As explained in [FR14, Section. 2.3], each component ${\mathcal{M}}({\bf
r},{\bf n})$ is identified to a moduli space of stable representations of the
chain-saw quiver. Moreover, it is a smooth quasi-projective variety.

###  2.2 Torus fixed points

The torus ${\bf
T}=\mathbb{C}^{\times}\times\mathbb{C}^{\times}\times{\mathbb{C}}^{\times r}$
acts on the moduli space of ADHM data by

$(t_{1},t_{2},\zeta_{1},\ldots,\zeta_{r})\times(A_{1},A_{2},I,J)\mapsto(t_{1}A_{1},t_{2}A_{2},\zeta_{\alpha}^{-1}I_{\alpha},t_{1}t_{2}\zeta_{\alpha}J),\qquad
1\leq\alpha\leq r.$

The ${\bf T}$-fixed locus consists of a finite set of points in one-to-one
correspondence to $r$-uples ${\underline{\mu}}=(\mu_{1},\ldots,\mu_{r})$ of
Young diagrams so that $\sum_{\alpha=1}^{r}|\mu_{\alpha}|=n$. The ${\bf
T}$-fixed locus is moreover contained in the ${\boldsymbol{\mu}}_{\ell}$-fixed
locus.

In order to fix conventions, note that a partition $n=n_{1}+\cdots+n_{k}$,
$n_{1}\geq\cdots\geq n_{k}$ will be identified with the set

$\bigcup_{j=1}^{k}\\{(i,j)\in\mathbb{R}^{2}\,|\,i\in\mathbb{Z},\ 1\leq i\leq
n_{j}\\}\subset\mathbb{R}^{2}$

This set consists of $k$ rows of $n_{j}$ boxes each. Note that our conventions
for partitions are different from those used in [Nak99, NY05a, NY05b, NY04].
In loc. cit. the partition associated to a set as above is determined by
columns rather than rows.

As explained next, the set of ${\bf T}$-fixed points contained in
${\mathcal{M}}({\bf r},{\bf n})$ is naturally identified with a set of
$r$-uples ${\underline{\mu}}$ of colored Young diagrams. A Young diagram $\mu$
will be said to be $a$-colored, $0\leq a\leq\ell-1$, if and only if each box
$(i,j)\in\mu$ is marked by $c_{a}(i,j)\in\\{0,\ldots,\ell-1\\}$ which is
uniquely determined by

$c_{a}(i,j)\equiv a-j+1\ {\rm mod}\ \ell.$ (2.3)

For a given ${\bf T}$-fixed point ${\underline{\mu}}$, each partition
$\mu_{\alpha}$, $1\leq\alpha\leq r$, will be $a(\alpha)$-colored, where $0\leq
a(\alpha)\leq\ell-1$ is is determined from the fixed point conditions (2.2) as
explained below.

Note that the ${{\boldsymbol{\mu}}}_{\ell}$-module structure on $W$ determined
by the action (2.1) is

$W=\underbrace{{\bf 1}\oplus\cdots{\bf
1}}_{r_{0}}\oplus\underbrace{\Omega\oplus\cdots\Omega}_{r_{1}}\oplus\cdots\oplus\underbrace{\Omega^{\ell-1}\oplus\cdots\Omega^{\ell-1}}_{r_{\ell-1}}~{},$
(2.4)

where $\Omega$ is the canonical one dimensional representation of
${\boldsymbol{\mu}}_{\ell}$. Hence

$W=\bigoplus_{\alpha=1}^{r}\Omega^{a(\alpha)}$

where $0\leq a(\alpha)\leq\ell-1$ is uniquely determined by the conditions

$r_{0}+\cdots+r_{a(\alpha)-1}+1\leq\alpha\leq r_{0}+\cdots+r_{a(\alpha)}$

for each $1\leq\alpha\leq r$. By convention, $r_{-1}=0$.

Then a fixed point ${\underline{\mu}}$ belongs to the component
${\mathcal{M}}({\bf r},{\bf n})$ if and only if

$\sharp\\{(i,j)\in\mu_{\alpha},\ 1\leq\alpha\leq
r\,|\,c_{a(\alpha)}(i,j)=b\,\\}=n_{b}$ (2.5)

for each $0\leq b\leq\ell-1$.

Let $(T_{1},T_{2},Z_{\alpha})$, $1\leq\alpha\leq r$, be the canonical
generators of the representation ring $R_{\bf T}$. For any $r$-uple
${\underline{\mu}}$ let $T_{\underline{\mu}}$ denote the ${\bf T}$-equivariant
tangent space to the moduli space at the fixed point indexed by
${\underline{\mu}}$. As shown in [NY05a, Theorem 2.11], the following identity
holds in $R_{\bf T}$

$T_{{\underline{\mu}}}=\sum_{1\leq\alpha,\beta\leq
r}\,Z_{\beta}Z_{\alpha}^{-1}\bigg{(}\sum_{(i,j)\in\mu_{\alpha}}T_{1}^{-\mu_{\beta,j}+i}T_{2}^{\mu^{\prime}_{\alpha,i}-j+1}+\sum_{(i,j)\in\mu_{\beta}}T_{1}^{\mu_{\alpha,j}-i+1}T_{2}^{-\mu^{\prime}_{\beta,i}+j}\bigg{)},$
(2.6)

where $\mu_{j}$ denotes the length of the $j$-th row of the partition $\mu$
while $\mu^{\prime}_{i}$ denotes the height of the $i$-th column of $\mu$.

Suppose the fixed point ${\underline{\mu}}$ belongs to ${\mathcal{M}}({\bf
r},{\bf n})$ for some ordered partition ${\bf n}$. Then $T_{\underline{\mu}}$
has a natural ${\bf T}\times{\boldsymbol{\mu}}_{\ell}$-module structure and
the ${\bf T}$-equivariant tangent space $T_{\underline{\mu}}{\mathcal{M}}_{\bf
n}$ is given by the ${\boldsymbol{\mu}}_{\ell}$-fixed part
$T_{\underline{\mu}}^{{\boldsymbol{\mu}}_{\ell}}$. In complete analogy to
[NY05a, Theorem 2.11], the ${\bf T}\times{\boldsymbol{\mu}}_{\ell}$-module
structure of $T_{\underline{\mu}}$ is given by:

######  Lemma 2.1.

For any $1\leq\alpha,\beta\leq r$ let $T_{\alpha,\beta}\in R_{{\bf
T}\times{\boldsymbol{\mu}}_{\ell}}$ be defined by

$T_{\alpha,\beta}=\Omega^{a(\beta)-a(\alpha)}Z_{\beta}Z_{\alpha}^{-1}\bigg{(}\sum_{(i,j)\in\mu_{\alpha}}T_{1}^{-\mu_{\beta,j}+i}(\Omega
T_{2})^{\mu^{\prime}_{\alpha,i}-j+1}+\sum_{(i,j)\in\mu_{\beta}}T_{1}^{\mu_{\alpha,j}-i+1}(\Omega
T_{2})^{-\mu^{\prime}_{\beta,i}+j}\bigg{)}~{},$ (2.7)

where $\Omega$ is the canonical one dimensional representation of
${\boldsymbol{\mu}}_{\ell}$. Then

$T_{\underline{\mu}}=\sum_{1\leq\alpha,\beta\leq r}T_{\alpha,\beta}.$ (2.8)

in the representation ring $R_{{\bf T}\times{\boldsymbol{\mu}}_{\ell}}$.

##  3 The generating function of Poincaré polynomials

The goal of this section is to find a closed expression for the generating
function of the Poincaré polynomials of the affine Laumon spaces
${\mathcal{M}}({\bf r},{\bf n})$ for fixed invariants ${\bf
r}=(r_{0},\ldots,r_{\ell-1})$. The computation will employ Atiyah-Bott-Morse
localization by analogy to [NY04, Theorem 3.8].

Let $\lambda:\mathbb{C}^{\times}\to{\bf T}$ be a one parameter group of the
form

$t\mapsto(t^{M_{1}},t^{M_{2}},t_{\alpha}^{N_{\alpha}})$

where

$M_{2}>>N_{1}>\cdots>N_{r}>>M_{1}>0.$

As shown in the proof of [NY04, Theorem 3.7], the resulting
$\mathbb{C}^{\times}$-action on the ADHM moduli space is compact. The same
holds for the induced action on ${\mathcal{M}}({\bf r},{\bf n})$ since the
latter is a closed subscheme of the ADHM moduli space. Therefore the first
step is the computation of the Morse index for an arbitrary fixed point
${\underline{\mu}}\in{\mathcal{M}}({\bf r},{\bf n})$.

###  3.1 The Morse index

Let ${\underline{\mu}}=(\mu_{1},\ldots,\mu_{r})$ be a ${\bf T}$-fixed point in
${\mathcal{M}}({\bf r},{\bf n})$. Recall that any integer $1\leq\beta\leq r$
determines a unique $0\leq a(\beta)\leq\ell-1$ so that

$r_{0}+\cdots+r_{a(\beta)-1}+1\leq\beta\leq
r_{1}+\cdots+r_{a(\beta)-1}+r_{a(\beta)}.$

As explained in Section 2.2, this determines the coloring function for the
partition $\mu_{\beta}$ to be

$c_{a(\beta)}(i,j)\equiv a(\beta)-j+1\ (\ell).$

For any $0\leq c\leq\ell-1$ and any $1\leq\beta\leq r$ let
$n_{c}(\mu_{\beta})$ be the total number of $c$-marked boxes in $\mu_{\beta}$.
Let also ${\rm col}(\mu_{\beta})$ denote the total number of columns of
$\beta$. Then the main result of this section is the following.

######  Lemma 3.1.

The Morse index associated to the fixed point
${\underline{\mu}}\in{\mathcal{M}}({\bf r},{\bf n})^{\bf T}$ is given by

$w({\underline{\mu}})=\sum_{\beta=1}^{r}w(\mu_{\beta})$ (3.1)

where

$w(\mu_{\beta})=\sum_{c=0}^{\ell-1}n_{c}(\mu_{\beta})r_{c}-{\rm
col}(\mu_{\beta})(r_{0}+\cdots+r_{a(\beta)}-\beta+1).$ (3.2)

Proof. The tangent space to ${\mathcal{M}}({\bf r},{\bf n})$ at the fixed
point ${\underline{\mu}}$ is the ${\boldsymbol{\mu}}_{\ell}$-fixed part of the
right hand side of equation (2.8),

$T_{{\underline{\mu}}}^{{\boldsymbol{\mu}}_{\ell}}=\sum_{1\leq\alpha,\beta\leq
r}T^{{\boldsymbol{\mu}}_{\ell}}_{\alpha,\beta}~{}.$

Using equation (2.7), one obtains

$\displaystyle T^{{\boldsymbol{\mu}}_{\ell}}_{\alpha,\beta}=$ $\displaystyle\
\sum_{\begin{subarray}{c}(i,j)\in\mu_{\alpha}\\\
\mu^{\prime}_{\alpha,i}-j+1+a(\beta)-a(\alpha)\equiv 0\,(\ell)\end{subarray}}\
T_{1}^{-\mu_{\beta,j}+i}T_{2}^{\mu^{\prime}_{\alpha,i}-j+1}$ (3.3)
$\displaystyle+\sum_{\begin{subarray}{c}(i,j)\in\mu_{\beta}\\\
-\mu^{\prime}_{\beta,i}+j+a(\beta)-a(\alpha)\equiv 0\,(\ell)\end{subarray}}\
T_{1}^{\mu_{\alpha,j}-i+1}T_{2}^{-\mu^{\prime}_{\beta,i}+j}$

As in the proof of [NY04, Theorem 3.8], the Morse index of the fixed point
${\underline{\mu}}$ is obtained by counting the number of monomials in the
expression of $T_{{\underline{\mu}}}^{{\boldsymbol{\mu}}_{\ell}}$ satisfying
the following conditions:

  * $(i)$

If $\alpha\geq\beta$ then the weight of $T_{2}$ is negative.

  * $(ii)$

If $\alpha<\beta$ then the weight of $T_{2}$ is nonpositive.

Note that all such monomials are contained in the second term in the right
hand side of equation (3.3). All monomials in the first term have positive
$T_{2}$-degree. Equation (3.3) shows that $w({\underline{\mu}})$ has a
decomposition

$w({\underline{\mu}})=\sum_{\beta=1}^{r}\sum_{\alpha=1}^{r}w_{\beta,\alpha}({\underline{\mu}})$

where

$w_{\beta,\alpha}({\underline{\mu}})=\left\\{\begin{array}[]{ll}\sharp\\{(i,j)\in\beta\,|\,\mu^{\prime}_{i}-j\equiv
a(\beta)-a(\alpha)\ (\ell)\\},&{\rm for}\ \alpha<\beta,\\\ &\\\
\sharp\\{(i,j)\in\beta\,|\,j\leq\mu_{i}^{\prime}-1,\ \mu^{\prime}_{i}-j\equiv
a(\beta)-a(\alpha)\ (\ell)\\},&{\rm for}\ \alpha\geq\beta.\end{array}\right.$

As shown in Lemma A.1, equation (A.5),

$\sharp\\{(i,j)\in\beta\,|\,\mu^{\prime}_{i}-j\equiv a(\beta)-a(\alpha)\
(\ell)\\}=n_{a(\alpha)}(\mu_{\beta}).$ (3.4)

Furthermore, Lemma A.2, equation (A.3), shows that

$\sharp\\{(i,j)\in\beta\,|\,j\leq\mu_{i}^{\prime}-1,\ \mu^{\prime}_{i}-j\equiv
a(\beta)-a(\alpha)\
(\ell)\\}=\left\\{\begin{array}[]{ll}n_{a(\alpha)}(\mu_{\beta}),&{\rm if}\
a(\alpha)\neq a(\beta)\\\ &\\\ n_{a(\alpha)}(\mu_{\beta})-{\rm
col}(\mu_{\beta}),&{\rm if}\ a(\alpha)=a(\beta).\\\ \end{array}\right.$ (3.5)

Then

$w({\underline{\mu}})=\sum_{\beta=1}^{r}\left(w_{<}(\mu_{\beta})+w_{0}(\mu_{\beta})+w_{>}(\mu_{\beta})\right),$

where

$\displaystyle w_{<}(\mu_{\beta})$ $\displaystyle\
=\sum_{\alpha=1}^{r_{0}+\cdots+r_{a(\beta)-1}}n_{a(\alpha)}(\mu_{\beta})$
$\displaystyle\ =\sum_{c=0}^{a(\beta)-1}n_{c}(\mu_{\beta})r_{c}~{},$
$\displaystyle w_{0}(\mu_{\beta})$ $\displaystyle\
=\sum_{\alpha=r_{0}+\cdots+r_{a(\beta)-1}+1}^{\beta-1}n_{a(\alpha)}(\mu_{\beta})+\sum_{\alpha=\beta}^{r_{0}+\cdots+r_{a(\beta)}}\left(n_{a(\alpha)}(\mu_{\beta})-{\rm
col}(\mu_{\beta})\right)$ $\displaystyle\
=n_{a(\beta)}(\mu_{\beta})r_{a(\beta)}-{\rm
col}(\mu_{\beta})(r_{0}+\cdots+r_{a(\beta)}-\beta+1)~{},$ $\displaystyle
w_{<}(\mu_{\beta})$ $\displaystyle\
=\sum_{\alpha=r_{0}+\cdots+r_{a(\beta)}+1}^{r}n_{a(\alpha)}(\mu_{\beta})$
$\displaystyle\ =\sum_{c=a(\beta)+1}^{\ell-1}n_{c}(\mu_{\beta})r_{c}~{}.$

Then

$\displaystyle w(\mu_{\beta})$ $\displaystyle\
=w_{<}(\mu_{\beta})+w_{0}(\mu_{\beta})+w_{>}(\mu_{\beta})$ $\displaystyle\
=\sum_{c=0}^{\ell-1}n_{c}(\mu_{\beta})r_{c}-{\rm
col}(\mu_{\beta})(r_{0}+\cdots+r_{a(\beta)}-\beta+1).$

$\Box$

###  3.2 The generating function

In order to compute the generating function, first note the following
partition identity derived in [KT11, Section 3.2].

######  Lemma 3.2.

Let $v$ and $\\{X_{a}\\}$, $a\in\mathbb{Z}$, be formal variables so that
$X_{a}=X_{a+\ell}$ for any $a\in\mathbb{Z}$. Then the following identity holds
for any partition $\mu$ and any $a\in\mathbb{Z}$.

$\sum_{\mu}v^{{\rm
col}(\mu)}\prod_{(i,j)\in\mu}X_{a-j+1}=\prod_{k=1}^{\infty}{1\over
1-vz^{k}}\prod_{c=1}^{\ell-1}{1\over
1-vz^{k}\prod_{b=c}^{\ell-1}X^{-1}_{a-b}}~{},$ (3.6)

where

$z=\prod_{c=0}^{\ell-1}X_{c}.$

Proof. The proof is straightforward once one notices that

$v^{{\rm col}(\mu)}\prod_{(i,j)\in\mu}X_{a-j+1}=\prod_{i=1}^{{\rm
col}(\mu)}\bigg{(}v\prod_{j=1}^{\mu^{\prime}_{i}}X_{a-j+1}\bigg{)},$

where $\mu^{\prime}_{i}$ denotes the height of the $i$-th column. This yields
the identity

$\sum_{\mu}v^{{\rm
col}(\mu)}\prod_{(i,j)\in\mu}X_{a-j+1}=\prod_{k=1}^{\infty}{1\over
1-v\prod_{j=1}^{k}X_{a-j+1}}~{}.$ (3.7)

Identity (3.6) is obtained by rewriting the right hand side of (3.7) using the
relations $X_{a}=X_{a+\ell}$.

$\Box$

Now let

$Z_{\bf r}=\sum_{{\bf n}}P_{y}({\mathcal{M}}_{\bf
n})\prod_{a=0}^{\ell-1}q_{a}^{n_{a}}$

be the generating function of Poincaré polynomials of affine Laumon spaces
with fixed invariants $(r_{0},\ldots,r_{\ell-1})$. For any $a\in\mathbb{Z}$
let $q_{a}=q_{c}$ and $r_{a}=r_{c}$, where $c\in\\{0,\ldots,\ell-1\\}$ is
uniquely determined by $a\equiv c$ mod $\ell$.

######  Theorem 3.3.

The following identity holds

$Z_{\bf
r}=\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{c=1}^{\ell-1}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}~{},$
(3.8)

where

${\widetilde{q}}_{a}=y^{2r_{a}}q_{a}$ (3.9)

for any $a\in\mathbb{Z}$, and

$z=\prod_{a=0}^{\ell-1}{\widetilde{q}}_{a}.$

Proof. By Atiyah-Bott-Morse localization, the generating function is given by

$\displaystyle\sum_{(\mu_{1},\ldots,\mu_{r})}\prod_{\beta=1}^{r}y^{2w(\mu_{\beta})}\prod_{c=0}^{\ell-1}q_{c}^{n_{c}(\mu_{\beta})}=$
$\displaystyle\
\prod_{\beta=1}^{r}\sum_{\mu_{\beta}}y^{2w(\mu_{\beta})}\prod_{c=0}^{\ell-1}q_{c}^{n_{c}(\mu_{\beta})}.$
(3.10)

where $w(\mu_{\beta})$ is given by (3.2). For each $1\leq\beta\leq r$, the sum
in the right hand side of (3.10) is over all $a(\beta)$-colored partitions
$\mu_{\beta}$. Note that

$\displaystyle\sum_{\mu_{\beta}}y^{2w(\mu_{\beta})}\prod_{c=0}^{\ell-1}q_{c}^{n_{c}(\mu_{\beta})}=$
$\displaystyle\ \sum_{\mu_{\beta}}y^{-2{\rm
col}(\mu_{\beta})(r_{0}+\cdots+r_{a(\beta)}-\beta+1)}\prod_{c=0}^{\ell-1}\big{(}y^{2r_{c}}q_{c}\big{)}^{n_{c}(\mu_{\beta})}$
$\displaystyle=$ $\displaystyle\ \sum_{\mu_{\beta}}y^{-2{\rm
col}(\mu_{\beta})(r_{0}+\cdots+r_{a(\beta)}-\beta+1)}\prod_{c=0}^{\ell-1}({\widetilde{q}}_{c})^{n_{c}(\mu_{\beta})}$
$\displaystyle\ \sum_{\mu_{\beta}}y^{-2{\rm
col}(\mu_{\beta})(r_{0}+\cdots+r_{a(\beta)}-\beta+1)}\prod_{(i,j)\in\mu_{\beta}}{\widetilde{q}}_{a(\beta)-j+1}$

For any $1\leq\beta\leq r$ let

$v_{\beta}=y^{2(r_{0}+\cdots+r_{a(\beta)}-\beta+1)}.$

Then identity (3.6) yields

$\displaystyle\sum_{\mu_{\beta}}y^{2w(\mu_{\beta})}\prod_{c=0}^{\ell-1}q_{c}^{n_{c}(\mu_{\beta})}=$
(3.11) $\displaystyle\prod_{k=1}^{\infty}{1\over
1-v_{\beta}^{-1}z^{k}}\prod_{c=1}^{\ell-1}{1\over
1-v_{\beta}^{-1}z^{k}({\widetilde{q}}_{a(\beta)-c})^{-1}({\widetilde{q}}_{a(\beta)-c-1})^{-1}\cdots({\widetilde{q}}_{a(\beta)-\ell+1})^{-1}}~{}.$

In conclusion, one obtains

$Z_{\bf r}=\prod_{\beta=1}^{r}\prod_{k=1}^{\infty}{1\over
1-v_{\beta}^{-1}z^{k}}\prod_{c=1}^{\ell-1}{1\over
1-v_{\beta}^{-1}z^{k}({\widetilde{q}}_{a(\beta)-c})^{-1}({\widetilde{q}}_{a(\beta)-c-1})^{-1}\cdots({\widetilde{q}}_{a(\beta)-\ell+1})^{-1}}$
(3.12)

where

$z=\prod_{a=0}^{\ell-1}{\widetilde{q}}_{a}.$

Next note the set isomorphism

$\\{r_{0}+\cdots+r_{a(\beta)-1}+1\leq\beta^{\prime}\leq
r_{0}+\cdots+r_{a(\beta)}\\}\xrightarrow{\sim}\\{1,\ldots,r_{a(\beta)}\\}$

sending

$\beta^{\prime}\mapsto r_{0}+\cdots+r_{a(\beta)}-\beta^{\prime}+1.$

Since $v_{\beta}=y^{2(r_{0}+\cdots+r_{a(\beta)}-\beta+1)}$, this yields the
following equivalent expression for identity (3.12)

$Z_{\bf
r}=\prod_{a=0}^{\ell-1}\prod_{c=1}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}~{}.$

$\Box$

The next goal is to express the generating function (3.12) in terms of the new
variables

$u_{c}=({\widetilde{q}}_{-c})^{-1}\cdots({\widetilde{q}}_{-\ell+1})^{-1}~{},\qquad
1\leq c\leq\ell-1.$ (3.13)

######  Proposition 3.4.

The following identity holds

$\displaystyle Z_{\bf r}=$
$\displaystyle\bigg{(}\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}\bigg{)}$ (3.14) $\displaystyle\prod_{1\leq a<c\leq\ell}\
\bigg{(}\prod_{t=1}^{r_{\ell-c}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)},$

where the variables $u_{c}$, $1\leq c\leq\ell-1$, are defined in (3.13) and
$u_{\ell}=1$.

Proof. The right hand side of (3.8) is rewritten as follows

$\displaystyle\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{c=1}^{\ell-1}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}=$
(3.15)
$\displaystyle\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}$
$\displaystyle\prod_{t=1}^{r_{0}}\prod_{c=1}^{\ell-1}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{-c})^{-1}({\widetilde{q}}_{-c-1})^{-1}\cdots({\widetilde{q}}_{-\ell+1})^{-1}}$
$\displaystyle\prod_{c=1}^{\ell-1}\prod_{t=1}^{r_{c}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{0})^{-1}({\widetilde{q}}_{-1})^{-1}\cdots({\widetilde{q}}_{c-\ell+1})^{-1}}$
$\displaystyle\prod_{a=1}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{\begin{subarray}{c}c=1\\\
c\neq a\end{subarray}}^{\ell-1}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}$

Using the variables (3.13), the first two lines in the right hand of (3.15)
become

$\displaystyle\bigg{(}\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}\bigg{)}\ \bigg{(}\prod_{c=1}^{\ell-1}\
\prod_{t=1}^{r_{0}}\prod_{k=1}^{\infty}{1\over 1-y^{-2t}z^{k}u_{c}}\bigg{)}$
(3.16)

In order to express the third line in terms of the variables (3.13) note the
identity

$\displaystyle({\widetilde{q}}_{0}{\widetilde{q}}_{-1}\cdots{\widetilde{q}}_{c-\ell+1})^{-1}\
\prod_{b=0}^{\ell-1}{\widetilde{q}}_{b}$
$\displaystyle={\widetilde{q}}_{0}^{-1}({\widetilde{q}}_{\ell-1}\cdots{\widetilde{q}}_{c+1})^{-1}\prod_{b=0}^{\ell-1}{\widetilde{q}}_{b}$
$\displaystyle={\widetilde{q}}_{1}\cdots{\widetilde{q}}_{c}$
$\displaystyle={\widetilde{q}}_{c-\ell}\cdots{\widetilde{q}}_{-\ell+1}$
$\displaystyle=u_{\ell-c}^{-1}$

Then the third line in the right hand side of (3.15) becomes

$\displaystyle\bigg{(}\prod_{c=1}^{\ell-1}\prod_{t=1}^{r_{\ell-c}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k-1}u_{c}^{-1}}\bigg{)}$ (3.17)

The remaining factors are

$\displaystyle\prod_{a=1}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{\begin{subarray}{c}c=1\\\
c\neq
a\end{subarray}}^{\ell-1}\prod_{k=1}^{\infty}{1\over(1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1})}~{}.$
(3.18)

As shown in Appendix B the following identity holds

$\displaystyle\prod_{a=1}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{\begin{subarray}{c}c=1\\\
c\neq
a\end{subarray}}^{\ell-1}{1\over(1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1})}=$
(3.19) $\displaystyle\prod_{1\leq a<c\leq\ell-1}\
\bigg{(}\prod_{t=1}^{r_{\ell-c}}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a}}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)}.$

Then, using equations (3.16) and (3.17), the right hand side of (3.15) is
finally equal to

$\displaystyle\bigg{(}\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}\bigg{)}\ \bigg{(}\prod_{a=1}^{\ell-1}\
\prod_{t=1}^{r_{0}}\prod_{k=1}^{\infty}{1\over 1-y^{-2t}z^{k}u_{a}}\bigg{)}$
(3.20)
$\displaystyle\bigg{(}\prod_{a=1}^{\ell-1}\prod_{t=1}^{r_{\ell-a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}}\bigg{)}$ $\displaystyle\prod_{1\leq
a<c\leq\ell-1}\ \bigg{(}\prod_{t=1}^{r_{\ell-c}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)}.$

Setting $u_{\ell}=1$, this can be further rewritten as

$\displaystyle\bigg{(}\prod_{a=0}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}\bigg{)}\prod_{1\leq a<c\leq\ell}\
\bigg{(}\prod_{t=1}^{r_{\ell-c}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)}.$

as claimed in equation (3.14).

$\Box$

##  4 ${\mathcal{W}}$-algebras

${\mathcal{W}}$-algebras are obtained by affine vertex algebras via quantum
Hamiltonian reduction. In this section we will study an iterated version.

###  4.1 Affine Lie algebras

This section is a brief reminder of universal Verma modules for affine Lie
algebras. Let $\mathfrak{g}$ a Lie algebra, $\mathfrak{h}$ a Cartan subalgebra
of rank $r+1$ and $\alpha_{0},\alpha_{1},\dots,\alpha_{r}$ a set of positive
simple roots. Let

$\mathfrak{g}=\mathfrak{g}_{+}\oplus\mathfrak{h}\oplus\mathfrak{g}_{-}$

the usual triangular decomposition where $\mathfrak{g}_{+}$ is the sum of all
positive root spaces and similarly $\mathfrak{g}_{-}$ is the sum of all
negative ones. Let $R=\mathbb{C}[x_{1},\dots,x_{r},x_{0}]$ and
$K=\mathbb{C}(x_{1},\dots,x_{r},x_{0})$. The universal Verma module of
$\mathfrak{g}$ is a usual Verma module over $R$ and so after base change over
$K$. This means, let $h_{0},\dots,h_{n}$ be a basis of $\mathfrak{h}$, e.g.
the one corresponding to the simple roots $\alpha_{0},\dots,\alpha_{r}$. Then
one considers the one dimensional module $\mathbb{C}_{x}$ over $\mathfrak{h}$
on which $h_{i}$ acts by multiplication by $x_{i}$. This extends to a
$\mathfrak{g}_{\geq 0}=\mathfrak{h}\oplus\mathfrak{g}_{+}$-module by letting
$\mathfrak{g}_{+}$ act trivially. The universal Verma module is then

$V:=U(\mathfrak{g})\otimes_{U(\mathfrak{g}_{\geq 0})}\mathbb{C}_{x}.$

In particular as a $\mathfrak{h}$-graded vector space $M$ is isomorphic to
$\mathbb{C}[B_{-}]$ for $B_{-}$ a homogeneous basis (every basis vector is an
$\mathfrak{h}$-eigenvector) of $\mathfrak{g}_{-}$.

Let us now specialize to $\mathfrak{g}$ being the affinization of a finite
dimensional reductive Lie algebra $\mathfrak{g}^{0}$. Let
$\alpha_{1},\dots,\alpha_{r}$ be a set of positive simple roots for the finite
dimensional subalgebra, let $\theta$ be the longest root and let $\delta$ be
the imaginary root of $\mathfrak{g}$. Then $\alpha_{0}=\delta-\theta$. Let
$\Delta_{\pm}^{0},\Delta^{0}$ denote the sets of positive, negative roots and
roots of $\mathfrak{g}^{0}$ . Then the positive root space is

$\Delta_{+}=\Delta_{+}^{0}\cup\\{\alpha+n\delta,n\delta|n\in\mathbb{Z}_{>0},\alpha\in\Delta^{0}\\}$

and similarly

$\Delta_{+}=\Delta_{-}^{0}\cup\\{\alpha+n\delta,n\delta|n\in\mathbb{Z}_{<0},\alpha\in\Delta^{0}\\}$

The dimension of the root space for $\alpha+n\delta$ is one and the one of
$n\delta$ is the rank of $\mathfrak{g}^{0}$. It follows that the formal
character of the universal Verma module is given by the Weyl denominator, that
is

$\begin{split}\text{ch}[V]&=\frac{\text{ch}[\mathbb{C}_{x}]}{D}\\\
D&=\prod_{n=1}^{\infty}(1-e^{-n\delta})^{r}\prod_{n=1}^{\infty}\prod_{\alpha\in\Delta_{+}^{0}}(1-e^{-\alpha}e^{(1-n)\delta})(1-e^{\alpha}e^{-n\delta}).\end{split}$
(4.1)

This is an equation as formal power series with the short hand notation

$\frac{1}{1-y}=\sum_{n=0}^{\infty}y^{n}.$

Here

$\text{ch}[\mathbb{C}_{x}]=\prod_{i=0}^{r}e^{x_{i}\omega_{i}}$

with $\omega_{i}$ dual to our basis of the Cartan subalgebra, that is
$\omega_{i}(h_{j})=\delta_{i,j}$. The term universal is due to that for a ring
homomorphism $f:R\rightarrow R$ (or $\mathbb{C}$ if desired) one can compose
the action of $\mathfrak{g}$ with $f$ to get a module with character

$\text{ch}[V_{f}]=\prod_{i=0}^{r}e^{f(x_{i})\omega_{i}}\frac{1}{D}.$

in particular any ordinary Verma module over $\mathbb{C}$ can be realized in
this way. If $f$ is an automorphism, then we call $M_{f}$ a universal Verma
module as well. An example is that $f$ maps each $x_{i}$ to
$x_{i}+\lambda_{i}$ for certain constants $\lambda_{i}$.

We note that any ordinary Verma module is simple for generic highest-weight.
Moreover it is also projective and injective in the category of weight Verma
modules, that is semisimple action of the Cartan subalgebra.

####  4.1.1 $\mathfrak{g}=\widehat{\mathfrak{gl}}_{N}$

We identify $\mathfrak{gl}_{N}$ with its defining matrix realization. Let
$E_{i,j}$ be the matrix with entry one at position $(i,j)$ and zero otherwise.
Set $h_{i}=E_{i,i}$ and define $\alpha_{i}$ by
$\alpha_{i}(h_{j})=\delta_{i,j}$. The positive simple roots are then
$\alpha_{i}-\alpha_{i+1}$ for $i=1,\dots,N-1$. The affinization has then basis
$E_{i,j}^{n}$ for $1\leq i,j,\leq N$ and integer $n$. The elements
corresponding to $\mathfrak{g}_{+}$ are the $E_{i,j}^{n}$ for $1\leq i,j,\leq
N$ and either positive integer $n$ or $n=0$ and $i<j$. Similarly the elements
corresponding to $\mathfrak{g}_{-}$ are the $E_{i,j}^{n}$ for $1\leq i,j,\leq
N$ and either negative integer $n$ or $n=0$ and $i>j$. Let $\mathbb{C}_{x}$ be
as above, that is $h_{i}$ acts by multiplication with $x_{i}$ and any element
in $\mathfrak{g}_{+}$ acts trivially. Let $M$ be the induced universal Verma
module. Since $\omega_{i}=\alpha_{i}$ and since

$\Delta_{+}^{0}=\\{\alpha_{i}-\alpha_{j}|1\leq 1<j\leq r\\}$

we have

$\begin{split}\text{ch}[M]&=\frac{\text{ch}[\mathbb{C}_{x}]}{D}\\\
\text{ch}[\mathbb{C}_{x}]&=\prod_{i=0}^{N}e^{x_{i}\alpha_{i}}\\\
D&=\prod_{n=1}^{\infty}(1-e^{-n\delta})^{N}\prod_{n=1}^{\infty}\prod_{1\leq
i<j\leq
r}(1-e^{\alpha_{j}-\alpha_{i}}e^{(1-n)\delta})(1-e^{\alpha_{i}-\alpha_{j}}e^{-n\delta}).\end{split}$
(4.2)

This converges to a meromorphic function on

$\\{|v_{1}|>|v_{2}|>\dots>|v_{N}|\\}\cap\\{|zv_{i}|<|v_{i+1}||i=1,\dots
N-1\\}\subset\mathbb{C}^{N}\times\mathbb{H}$

if we set $v_{i}=e^{\alpha_{i}}$ and $z=e^{-\delta}$. The expression has a
meromorphic continuation to

$\begin{split}\text{ch}[M]&=\frac{\prod\limits_{i=0}^{N}v_{i}^{x_{i}}}{\prod\limits_{n=1}^{\infty}(1-z^{n})^{N}\prod\limits_{n=1}^{\infty}\prod\limits_{1\leq
i<j\leq N}(1-v_{j}v_{i}^{-1}z^{n-1})(1-v_{i}v_{j}^{-1}z^{n})}.\end{split}$
(4.3)

###  4.2 Quantum Hamiltonian reduction

We summarize ${\mathcal{W}}$-algebra basics following [KW04] and using the
summary of [CL22, Section 3]. For this let ${\mathfrak{g}}$ be a Lie algebra,

$(\ \ |\ \ ):{\mathfrak{g}}\times{\mathfrak{g}}\rightarrow\mathbb{C}$ (4.4)

a nondegenerate invariant symmetric bilinear form on ${\mathfrak{g}}$ and
$\\{q^{\alpha}\\}_{\alpha\in S}$ be a basis of ${\mathfrak{g}}$ indexed by the
set $S$. The structure constants are defined by

$[q^{\alpha},q^{\beta}]=\sum_{\gamma\in
S}{f^{\alpha\beta}}_{\gamma}q^{\gamma}.$

The affine vertex algebra $V^{k}({\mathfrak{g}})$ of ${\mathfrak{g}}$ at level
$k\in\mathbb{C}$ is strongly and freely generated by
$\\{X^{\alpha}\\}_{\alpha\in S}$ with the usual operator products

$X^{\alpha}(z)X^{\beta}(w)\sim\frac{k(q^{\alpha}|q^{\beta})}{(z-w)^{2}}+\frac{\sum_{\gamma\in
S}{f^{\alpha\beta}}_{\gamma}X^{\gamma}(w)}{(z-w)}.$

We assume that $k$ is generic. It is more precise to denote
$V^{k}({\mathfrak{g}})$ by $V^{k(\ \ |\ \ )}({\mathfrak{g}})$ to emphazise the
given bilinear form and we will do so later. Set $X_{\alpha}$ to be the field
corresponding to $q_{\alpha}$ where $\\{q_{\alpha}\\}_{\alpha\in S}$ is the
dual basis of ${\mathfrak{g}}$ with respect to $(\ \ |\ \ )$.

Let $f$ be a nilpotent element of ${\mathfrak{g}}$. By the Jacobson-Morozov
theorem, $f$ is part of an ${\mathfrak{s}}{\mathfrak{l}}_{2}$ triple
$\\{f,x,e\\}\subseteq{\mathfrak{g}}$ satisfying the relations
$[x,e]=e,[x,f]=-f,[e,f]=2x$. In particular ${\mathfrak{g}}$ decomposes as an
${\mathfrak{s}}{\mathfrak{l}}_{2}$-module

${\mathfrak{g}}=\bigoplus_{k\in\frac{1}{2}\mathbb{Z}}{\mathfrak{g}}_{k},\qquad{\mathfrak{g}}_{k}=\\{a\in{\mathfrak{g}}|[x,a]=ka\\}.$

Let $S_{k}$ be a basis of ${\mathfrak{g}}_{k}$ and take $S=\bigcup_{k}S_{k}$
as basis of ${\mathfrak{g}}$. Let us also set

${\mathfrak{g}}_{+}=\bigoplus_{k\in\frac{1}{2}\mathbb{Z}_{>0}}{\mathfrak{g}}_{k},\qquad{\mathfrak{g}}_{-}=\bigoplus_{k\in\frac{1}{2}\mathbb{Z}_{<0}}{\mathfrak{g}}_{k}$

with corresponding basis $S_{+}$ of ${\mathfrak{g}}_{+}$; ${\mathfrak{g}}_{-}$
is naturally identified with the dual of ${\mathfrak{g}}_{+}$. On
${\mathfrak{g}}_{\frac{1}{2}}$ one defines the antisymmetric bilinear form

$\langle a,b\rangle:=(f|[a,b]).$

Let $F({\mathfrak{g}}_{+})$ be the vertex superalgebra associated to the
vector superspace ${\mathfrak{g}}_{+}\oplus{\mathfrak{g}}_{+}^{*}$. This
vertex superalgebra is strongly generated by odd fields
$\\{\varphi_{\alpha},\varphi^{\alpha}\\}_{\alpha\in S_{+}}$. The conformal
weight of $\varphi_{\alpha}$ is $j$ if $q^{\alpha}$ is in ${\mathfrak{g}}_{j}$
and the one of $\varphi^{\alpha}$ is $1-j$. The operator products are

$\varphi_{\alpha}(z)\varphi^{\beta}(w)\sim\frac{\delta_{\alpha,\beta}}{(z-w)},\qquad\varphi_{\alpha}(z)\varphi_{\beta}(w)\sim
0\sim\varphi^{\alpha}(z)\varphi^{\beta}(w).$

Let $F({\mathfrak{g}}_{\frac{1}{2}})$ be the neutral vertex algebra associated
to ${\mathfrak{g}}_{\frac{1}{2}}$ with bilinear form $\langle\ \ ,\ \
\rangle$. This has even strong generators $\\{\Phi_{\alpha}\\}_{\alpha\in
S_{\frac{1}{2}}}$ with operator products

$\Phi_{\alpha}(z)\Phi_{\beta}(w)\sim\frac{\langle
q^{\alpha},q^{\beta}\rangle}{(z-w)}\sim\frac{(f|[q^{\alpha},q^{\beta}])}{(z-w)},$
(4.5)

and fields corresponding to the dual basis with respect to $\langle\ \ ,\ \
\rangle$ are denoted by $\Phi^{\alpha}$. Set

$F({\mathfrak{g}},f,k):=F({\mathfrak{g}}_{+})\otimes
F({\mathfrak{g}}_{\frac{1}{2}}).$

The complex is

$C({\mathfrak{g}},f,k):=V^{k}({\mathfrak{g}})\otimes F({\mathfrak{g}},f,k).$

It is $\mathbb{Z}$-graded by ghost number, that is the $\varphi_{\alpha}$ have
charge minus one, the $\varphi^{\alpha}$ charge one and all others charge
zero. One further defines the odd field $d(z)$ of charge minus one by

$\begin{split}d(z)&=\sum_{\alpha\in
S_{+}}:X^{\alpha}\varphi^{\alpha}:-\frac{1}{2}\sum_{\alpha,\beta,\gamma\in
S_{+}}{f^{\alpha\beta}}_{\gamma}:\varphi_{\gamma}\varphi^{\alpha}\varphi^{\beta}:+\sum_{\alpha\in
S_{+}}(f|q^{\alpha})\varphi^{\alpha}+\sum_{\alpha\in
S_{\frac{1}{2}}}:\varphi^{\alpha}\Phi_{\alpha}:.\end{split}$ (4.6)

The zero-mode $d_{0}$ is a differential since $[d(z),d(w)]=0$ by [KRW03, Thm.
2.1]. The $\mathcal{W}$-algebra is defined to be its homology

${\mathcal{W}}^{k}({\mathfrak{g}},f):=H\left(C({\mathfrak{g}},f,k),d_{0}\right).$

The homology is non-trivial only in degree zero [KW04].

###  4.3 Examples for ${\mathfrak{g}}={\mathfrak{g}}{\mathfrak{l}}_{N}$

${\mathcal{W}}$-algebras of ${\mathfrak{g}}{\mathfrak{l}}_{N}$ are
parameterized by partitions of $N$. We introduce them step by step.

######  Example 4.1.

(Principal ${\mathcal{W}}$-algebra: $N=N$)

Let ${\mathfrak{g}}={\mathfrak{g}}{\mathfrak{l}}_{N}$, denote the standard
representation of it by $\mathbb{C}^{N}$ and its conjugate by
$\overline{\mathbb{C}}^{N}$. Let $f$ be the principal nilpotent element. That
is, let
$\rho:{\mathfrak{s}}{\mathfrak{l}}_{2}\rightarrow{\mathfrak{g}}{\mathfrak{l}}_{N}={\mathfrak{g}}{\mathfrak{l}}(\mathbb{C}^{N})$
be the $N$-dimensional irreducible representation of
${\mathfrak{s}}{\mathfrak{l}}_{2}$. We denote this by $\mathbb{C}_{N}$. More
generally, let $\mathbb{C}_{d}$, $d\geq 1$, denote the $d$-dimensional
irreducible representation of ${\mathfrak{s}}{\mathfrak{l}}_{2}$. Then
${\mathfrak{g}}{\mathfrak{l}}_{N}$ decomposes as a module for the
${\mathfrak{s}}{\mathfrak{l}}_{2}$-subalgebra whose basis is the triple
$(e,h,f)$ as

${\mathfrak{g}}{\mathfrak{l}}_{N}\cong_{{\mathfrak{g}}{\mathfrak{l}}_{N}}\mathbb{C}^{N}\otimes\overline{\mathbb{C}}^{N}\cong_{{\mathfrak{s}}{\mathfrak{l}}_{2}}\mathbb{C}_{N}\otimes\mathbb{C}_{N}\cong_{{\mathfrak{s}}{\mathfrak{l}}_{2}}\bigoplus_{J=1}^{N}\mathbb{C}_{2J-1}.$

The free fermion algebra associated to the quantum Hamiltonian reduction is
the free fermion algebra corresponding to the non-zero weight spaces of this
decompositon, whose dimension is $N(N-1)$, i.e. it is $F^{N(N-1)}$ the vertex
superalgebra of $N(N-1)$ free fermions. The complex for the reduction is then
$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})\otimes F^{N(N-1)}$.

######  Example 4.2.

(Rectangular ${\mathcal{W}}$-algebra: $N=s+s+\dots+s$ )

Let ${\mathfrak{g}}={\mathfrak{g}}{\mathfrak{l}}_{N}$ with $N=\ell s$. There
is then an embedding of
${\mathfrak{g}}{\mathfrak{l}}_{\ell}\oplus{\mathfrak{g}}{\mathfrak{l}}_{s}$ in
${\mathfrak{g}}{\mathfrak{l}}_{N}$, such that
$\mathbb{C}^{N}\cong\mathbb{C}^{\ell}\otimes\mathbb{C}^{s}$ is the tensor
product of the standard representations of
${\mathfrak{g}}{\mathfrak{l}}_{\ell}$ and ${\mathfrak{g}}{\mathfrak{l}}_{s}$.
Hence

${\mathfrak{g}}{\mathfrak{l}}_{N}\cong_{{\mathfrak{g}}{\mathfrak{l}}_{N}}\mathbb{C}^{N}\otimes\overline{\mathbb{C}}^{N}\cong(\mathbb{C}^{\ell}\otimes\overline{\mathbb{C}}^{\ell})\otimes(\mathbb{C}^{s}\otimes\overline{\mathbb{C}}^{s})\cong\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{\ell})\otimes\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{s})$

is just the tensor product of the adjoint representations of
${\mathfrak{g}}{\mathfrak{l}}_{\ell}$ and ${\mathfrak{g}}{\mathfrak{l}}_{s}$.
Taking $f$ and ${\mathfrak{s}}{\mathfrak{l}}_{2}$ to be the image in
${\mathfrak{g}}{\mathfrak{l}}_{N}$ of the principal nilpotent element in
${\mathfrak{g}}{\mathfrak{l}}_{s}$ and its associated
${\mathfrak{s}}{\mathfrak{l}}_{2}$ via this embedding we get from the previous
example that

${\mathfrak{g}}{\mathfrak{l}}_{N}\cong\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{\ell})\otimes\left(\bigoplus_{J=1}^{s}\mathbb{C}_{2J-1}\right)$
(4.7)

as a module for
${\mathfrak{g}}{\mathfrak{l}}_{\ell}\otimes{\mathfrak{s}}{\mathfrak{l}}_{2}$.
In particular the complex for this reduction is
$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})\otimes F^{\ell^{2}s(s-1)}$.

######  Example 4.3.

(${\mathcal{W}}$-algebra associated to
$N=s_{1}+s_{1}+\dots+s_{1}+s_{2}+s_{2}+\dots+s_{2}$)

Let $N=N_{1}+N_{2}$ and $N_{1}=m_{1}s_{1},N_{2}=m_{2}s_{2}$, and assume
$0<s_{1}<s_{2}$. We then have an embedding of
${\mathfrak{g}}{\mathfrak{l}}_{N_{1}}\oplus{\mathfrak{g}}{\mathfrak{l}}_{N_{2}}$
in ${\mathfrak{g}}{\mathfrak{l}}_{N}$ and can choose $f=f_{1}+f_{2}$ with
$f_{1}$ and $f_{2}$ the images of the rectangular nilpotent elements in
${\mathfrak{g}}{\mathfrak{l}}_{N_{1}}$ and
${\mathfrak{g}}{\mathfrak{l}}_{N_{2}}$ in ${\mathfrak{g}}{\mathfrak{l}}_{N}$.
Then as a
${\mathfrak{g}}{\mathfrak{l}}_{m_{1}}\otimes{\mathfrak{g}}{\mathfrak{l}}_{m_{2}}\otimes{\mathfrak{s}}{\mathfrak{l}}_{2}$-module
($\mathbf{1}_{v}$ denots the trivial
${\mathfrak{g}}{\mathfrak{l}}_{v}$-module)

$\begin{split}{\mathfrak{g}}{\mathfrak{l}}_{N}\cong\
&\,\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{m_{1}})\otimes\mathbf{1}_{m_{2}}\otimes\left(\bigoplus_{J=1}^{s_{1}}\mathbb{C}_{2J-1}\right)\oplus\mathbf{1}_{m_{1}}\otimes\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{m_{2}})\otimes\left(\bigoplus_{J=1}^{s_{2}}\mathbb{C}_{2J-1}\right)\oplus\\\
&\
\left(\mathbb{C}^{m_{1}}\otimes\overline{\mathbb{C}}^{m_{2}}\oplus\overline{\mathbb{C}}^{m_{1}}\otimes{\mathbb{C}}^{m_{2}}\right)\otimes\left(\bigoplus_{\begin{subarray}{c}d=s_{2}-s_{1}+1\\\
d=s_{2}-s_{1}+1\
(2)\end{subarray}}^{s_{1}+s_{2}-1}\mathbb{C}_{d}\right).\end{split}$ (4.8)

The complex for this reduction is

$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})\otimes
F^{m_{1}^{2}s_{1}(s_{1}-1)}\otimes F^{m_{2}^{2}s_{2}(s_{2}-1)}\otimes
F^{2m_{1}m_{2}s_{1}(s_{2}-1)}$

if $s_{1}=s_{2}\mod 2$ and

$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})\otimes
F^{m_{1}^{2}s_{1}(s_{1}-1)}\otimes F^{m_{2}^{2}s_{2}(s_{2}-1)}\otimes
F^{2m_{1}m_{2}s_{1}s_{2}}\otimes\beta\gamma^{m_{1}m_{2}s_{1}}$

otherwise. Here $\beta\gamma$ denotes the vertex algebra of a pair of
symplectic bosons ($\beta\gamma$-VOA, the neutral free field algebra
associated to ${\mathfrak{g}}_{\frac{1}{2}}$).

######  Example 4.4.

(${\mathcal{W}}$-algebra associated to a general partition of $N$) The general
case is now $N=N_{1}+\dots+N_{L}$, $L\geq 1$, and $N_{i}=m_{i}s_{i}$, $1\leq
i\leq L$, so that ${\mathfrak{g}}{\mathfrak{l}}_{N}$ has
${\mathfrak{g}}{\mathfrak{l}}_{N_{1}}\oplus\dots\oplus{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}$
as subalgebras and we can take $f=f_{1}+\dots+f_{L}$ with $f_{i}$ the
rectangular element in the $i$-th summand. Assume that
$0<s_{1}<s_{2}<\dots<s_{L}$, i.e. the partition is ordered. Then

$\displaystyle{\mathfrak{g}}{\mathfrak{l}}_{N}\cong$ $\displaystyle\ \
\bigoplus_{i=1}^{L}\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{m_{i}})\otimes\left(\bigoplus_{J=1}^{s_{i}}\mathbb{C}_{2J-1}\right)\oplus$
(4.9) $\displaystyle\bigoplus_{1\leq i<j\leq
L}\left(\mathbb{C}^{m_{i}}\otimes\overline{\mathbb{C}}^{m_{j}}\oplus\overline{\mathbb{C}}^{m_{i}}\otimes{\mathbb{C}}^{m_{j}}\right)\otimes\Bigg{(}\bigoplus_{\begin{subarray}{c}d=s_{j}-s_{i}+1\\\
d=s_{j}-s_{i}+1\ (2)\end{subarray}}^{s_{i}+s_{j}-1}\mathbb{C}_{d}\Bigg{)}$

and the free field algebra of the complex is

$F=\bigotimes_{i=1}^{L}F^{m_{i}^{2}s_{i}(s_{i}-1)}\otimes\bigoplus_{1\leq
i<j\leq L}\widetilde{F}^{2m_{i}m_{j}s_{i}(s_{j}-1)}$ (4.10)

with

$\widetilde{F}^{2m_{i}m_{j}s_{i}(s_{j}-1)}:=\begin{cases}F^{2m_{i}m_{j}s_{i}(s_{j}-1)}&s_{i}=s_{j}\mod
2\\\ F^{2m_{i}m_{j}s_{i}s_{j}}\otimes\beta\gamma^{m_{i}m_{j}s_{i}}&s_{i}\neq
s_{j}\mod 2.\end{cases}$ (4.11)

###  4.4 Properties

  1. 1.

Set

$I^{\alpha}:=X^{\alpha}+\sum_{\beta,\gamma\in
S_{+}}{f^{\alpha\beta}}_{\gamma}:\varphi_{\gamma}\varphi^{\beta}:+\frac{1}{2}\sum_{\beta\in
S_{\frac{1}{2}}}{f^{\beta\alpha}}_{\gamma}\Phi^{\beta}\Phi_{\gamma}$ (4.12)

for $\alpha\in{\mathfrak{g}}_{0}$. Denote by ${\mathfrak{g}}^{f}$ the
centralizer of $f$ in ${\mathfrak{g}}$, and set
${\mathfrak{b}}:={\mathfrak{g}}^{f}\cap{\mathfrak{g}}_{0}$. It is a Lie
subalgebra of ${\mathfrak{g}}$. [KW04, Thm 2.1] says in particular that
$d_{0}(I^{\alpha})=0$ for $q^{\alpha}\in{\mathfrak{b}}$ and

$I^{\alpha}{}(z)I^{\beta}(w)\sim\frac{B(q^{\alpha}|q^{\beta})}{(z-w)^{2}}+\frac{{f^{\alpha\beta}}_{\gamma}I^{\gamma}}{(z-w)},$

with the bilinear form $B$ on ${\mathfrak{a}}$ given by

$B(q^{\alpha}|q^{\beta})=k(q^{\alpha}|q^{\beta})+\frac{1}{2}\left(\kappa_{\mathfrak{g}}(q^{\alpha},q^{\beta})-\kappa_{{\mathfrak{g}}_{0}}(q^{\alpha},q^{\beta})-\kappa_{\frac{1}{2}}(q^{\alpha},q^{\beta})\right)$

with $\kappa_{\mathfrak{g}}$ and $\kappa_{{\mathfrak{g}}_{0}}$ the usual
Killing form on ${\mathfrak{g}}$ and ${\mathfrak{g}}_{0}$ and
$\kappa_{\frac{1}{2}}$ the trace of ${\mathfrak{g}}_{0}$ on
${\mathfrak{g}}_{\frac{1}{2}}$. Thus ${\mathcal{W}}^{k}({\mathfrak{g}},f)$
contains $V^{B}({\mathfrak{b}})$ as subalgebra for some bilinear form
depending on $({\mathfrak{g}},f,k)$.

  2. 2.

Let ${\mathcal{A}}^{k}$ be some vertex algebra containing
$V^{k}({\mathfrak{g}})$ as subalgebra, then

$H\left({\mathcal{A}}^{k}\otimes F({\mathfrak{g}},f,k),d_{0}\right)$

is a vertex algebra that for generic $k$ contains the ${\mathcal{W}}$-algebra
${\mathcal{W}}^{k}({\mathfrak{g}},f)$ as subalgebra. If ${\mathcal{A}}^{k}$ is
an object in the category of ordinary $V^{k}({\mathfrak{g}})$-modules, usually
denoted by ${\rm{KL}}_{k}({\mathfrak{g}})$ or ${\rm{KL}}_{k(\ \ |\ \
)}({\mathfrak{g}})$, then for generic level the homology is concentrated in
degree zero since Theorem 6.2. of [KW04] applies. In particular the graded
character of the homology is just the Euler-Poincaré character, that is the
supercharacter of the complex.

  3. 3.

Let $M$ be an object in ${\rm{KL}}_{k}({\mathfrak{g}})$, then $M\otimes
F({\mathfrak{g}},f,k)$ is a $V^{B}({\mathfrak{b}})$-module. The zero-modes of
the $I^{\alpha}$ for $q^{\alpha}\in{\mathfrak{b}}$ give rise to an action of
${\mathfrak{b}}$ on $M\otimes F({\mathfrak{g}},f,k)$. This action commutes
with the action of the Virasoro zero-mode of
${\mathcal{W}}^{k}({\mathfrak{g}},f)$ by [KW04, Thm 2.1]. Since conformal
weight spaces of $M\otimes F({\mathfrak{g}},f,k)$ are finite-dimensional (the
conformal weight of the $\Phi^{\alpha}$ is $1/2$) it follows that each
conformal weight space of $M\otimes F({\mathfrak{g}},f,k)$ is an integrable
${\mathfrak{b}}$-module and hence $M\otimes F({\mathfrak{g}},f,k)$ must be an
object in a completion of ${\rm{KL}}_{B}({\mathfrak{b}})$.

  4. 4.

Let $V$ be the universal Verma module as discussed in section 4.1. As
discussed in that section, we can twist the action of $R$ on $V$ by an
automorphism $\varphi$ of $R$, we call the resulting module still a universal
Verma module. Let $\mathcal{V}^{k}$ be a possibly infinite direct sum of
universal Verma modules (all possibly twisted). Then the homology

$H\left(\mathcal{V}^{k}\otimes F({\mathfrak{g}},f,k),d_{0}\right)$

is concentrated in degree zero, since Theorem 6.2. of [KW04] applies as well.

  5. 5.

Let $V$ be the universal Verma module, then $V\otimes F({\mathfrak{g}},f,k)$
is a $V^{B}({\mathfrak{b}})$-module (over $R$). As before, the zero-modes of
the $I^{\alpha}$ for $q^{\alpha}\in{\mathfrak{b}}$ give rise to an action of
${\mathfrak{b}}$ on $V\otimes F({\mathfrak{g}},f,k)$. This action commutes
with the action of the Virasoro zero-mode of
${\mathcal{W}}^{k}({\mathfrak{g}},f)$. Any conformal weight space $V_{n}$ of
$V$ is a direct sum of universal Verma modules for ${\mathfrak{b}}$. Conformal
weight spaces $F({\mathfrak{g}},f,k)_{m}$ of $F({\mathfrak{g}},f,k)$ are
integrable ${\mathfrak{b}}$-modules and the tensor product of an integrable
module with a universal Verma module is a direct sum of universal Verma
modules. As a ${\mathfrak{b}}$-module the weight space of conformal weight $N$
is just the finite direct sum $\bigoplus_{n}V_{n}\otimes
F({\mathfrak{g}},f,k)_{N-n}$ and hence a direct sum of universal Verma modules
for ${\mathfrak{b}}$. The action of the Cartan subalgebra of ${\mathfrak{b}}$
is semisimple on $V$ and $F({\mathfrak{g}},f,k)$ and so it also is on the
tensor product of the two, i.e. $V\otimes F({\mathfrak{g}},f,k)$. is an object
in the category $\mathcal{O}$. Let $U$ be an indecomposable
$V^{B}({\mathfrak{b}})$-submodule of $V\otimes F({\mathfrak{g}},f,k)$. Since
conformal weight of the latter is lower bounded the same must be true for $U$.
The top level of $U$ must be a ${\mathfrak{b}}$-module and hence must contain
a universal ${\mathfrak{b}}$-Verma module as direct summand. It follows that
$U$ must contain the corresponding universal $V^{B}({\mathfrak{b}})$-module,
but universal Verma modules are simple, projective and injective in the
category $\mathcal{O}$, see e.g. [Hum08]. We conclude that $U$ coincides with
the universal $V^{B}({\mathfrak{b}})$-module.

###  4.5 Characters

The Virasoro field $L$ of ${\mathcal{W}}^{k}({\mathfrak{g}},f)$ is

$\begin{split}L_{\text{sug}}&=\frac{1}{2(k+h^{\vee})}\sum_{\alpha\in
S}:X_{\alpha}X^{\alpha}:,\\\ L_{\text{ch}}&=\sum_{\alpha\in
S_{+}}\left(-m_{\alpha}:\varphi^{\alpha}\partial\varphi_{\alpha}:+(1-m_{\alpha}):(\partial\varphi^{\alpha})\varphi_{\alpha}:\right),\\\
L_{\text{ne}}&=\frac{1}{2}\sum_{\alpha\in
S_{\frac{1}{2}}}:(\partial\Phi^{\alpha})\Phi_{\alpha}:,\\\
L&=L_{\text{sug}}+\partial x+L_{\text{ch}}+L_{\text{ne}}.\end{split}$ (4.13)

It has central charge

$c=c({\mathfrak{g}},f,k)=\frac{k\,\text{dim}\,{\mathfrak{g}}}{k+h^{\vee}}-12k(x|x)-\sum_{\alpha\in
S_{+}}(12m_{\alpha}^{2}-12m_{\alpha}+2)-\frac{1}{2}\,\text{dim}\,{\mathfrak{g}}_{\frac{1}{2}}.$
(4.14)

Let $\mathfrak{h}^{\sharp}$ be a Cartan subalgebra of ${\mathfrak{b}}$. Choose
a Cartan subalgebra ${\mathfrak{h}}$ of ${\mathfrak{g}}_{0}$ and hence of
${\mathfrak{g}}$ that contains ${\mathfrak{h}}^{\sharp}$ and $x$. Let
$\Delta_{j}$ be the roots in ${\mathfrak{g}}_{j}$ and fix a set of positive
roots $\Delta_{+}=\Delta_{+}^{0}\cup\bigcup_{j>0}\Delta_{j}$ with
$\Delta_{+}^{0}$ a set of positive roots for ${\mathfrak{g}}_{0}$. Set
$\delta(\alpha)=j$ for $\alpha\in\Delta_{j}$ and set $\epsilon(\alpha)=0$ for
$j\in\mathbb{Z}$ and $\epsilon(\alpha)=\frac{1}{2}$ otherwise. Let
$\Delta_{h.w}$ denote those roots that correspond to a highest-weight vector
for the ${\mathfrak{s}}{\mathfrak{l}}_{2}$-triple $(f,x,e)$. Let
$\Delta_{h.w}^{\sharp}$ be the subset of those roots that act non-trivially on
$h^{\sharp}$ and $\Delta_{h.w.}^{0}$ its complement; set
$\Delta_{h.w}^{\pm,\sharp}=\Delta_{\pm}\cap\Delta_{h.w.}^{\sharp}$.

The character of a module is defined to be

$\text{ch}[M](h,q)={\hbox{Tr}}_{M}\left(q^{L_{0}-\frac{c}{24}}e^{h}\right)$
(4.15)

for $h\in{\mathfrak{h}}^{\sharp}$. The character of the
${\mathcal{W}}$-algebra is

$\begin{split}\text{ch}[{\mathcal{W}}^{k}({\mathfrak{g}},f)](h,q)&=q^{-\frac{c}{24}}\prod_{n=1}^{\infty}\prod_{\alpha\in\Delta_{\text{
h.w.}}^{0}}(1-q^{n+\delta(\alpha)})^{-1}\prod_{\alpha\in\Delta_{\text{
h.w.}}^{+,\sharp}}(1-e^{\alpha(h)}q^{n+\delta(\alpha)})^{-1}(1-e^{-\alpha(h)}q^{n+\delta(\alpha)})^{-1},\end{split}$
(4.16)

where the convention is always that

$\frac{1}{1-y}=\sum_{n=0}^{\infty}y^{n}.$

The character of the Verma module $V_{\lambda}$ that is obtained via quantum
Hamiltonian reduction from the Verma module of $V^{k}({\mathfrak{g}})$ of
highest-weight $\lambda$ is

$\text{ch}[V_{\lambda}]=q^{-\frac{c}{24}}q^{\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}-\lambda(x)}\
\prod_{n=1}^{\infty}\prod_{\alpha\in\Delta_{\text{
h.w.}}^{0}}(1-q^{n-\epsilon(\alpha)})^{-1}\prod_{\alpha\in\Delta_{\text{
h.w.}}^{+,\sharp}}(1-e^{\alpha(h)}q^{n-\epsilon(\alpha)})^{-1}(1-e^{-\alpha(h)}q^{n+\epsilon(\alpha)-1})^{-1},$
(4.17)

Consider our main example, that is example 4.4. In this case
${\mathfrak{b}}={\mathfrak{g}}{\mathfrak{l}}_{m_{1}}\oplus\dots\oplus{\mathfrak{g}}{\mathfrak{l}}_{m_{L}}$.
We first explain how to modify the character so that the $\epsilon(\alpha)$
contributions disappear. Note that $\epsilon(\alpha)=0$ for
$\alpha\in\Delta_{\text{ h.w.}}^{0}$. We identify ${\mathfrak{b}}$ with its
natural image in ${\mathfrak{g}}{\mathfrak{l}}_{m_{1}+\dots+m_{L}}$, so that
${\mathfrak{h}}^{\sharp}$ is identified with diagonal matrices whose natural
basis are the $E_{a}$, the diagonal matrices with entry one at position $a$
and all others zero. $\epsilon_{a}$ is then the map from
${\mathfrak{h}}^{\sharp}$ to $\mathbb{C}$ that maps $E_{a}$ to one and all
others to zero. Set $u_{a}=e^{\epsilon_{a}(h)}$ for
$h\in{\mathfrak{h}}^{\sharp}$. Set $x_{i}=\sum_{k=1}^{m_{i}}E_{m_{1}+\dots
m_{i-1}+k}$. Set $t_{j}=0$ if $s_{j}=s_{1}\mod 2$ and $-\frac{1}{2}$ other
wise. Set $y=\sum_{k=1}^{K}t_{k}x_{k}$. For any monomial $\mu$ of $(u_{a}^{\pm
1},y^{\pm 1},z)$, not identically $1$, let

$(\mu)_{\infty}=\prod_{n=1}^{\infty}(1-z^{n-1}\mu).$

Let $I^{y}$ be the field corresponding to $y$ in
${\mathcal{W}}^{k}({\mathfrak{g}},f)$. Then $L+\partial I^{y}$ is another
Virasoro field. The conformal weight of a generator corresponding to $\alpha$
is unchanged if $\epsilon(\alpha)=0$ while otherwise it changes by
$\pm\frac{1}{2}$ for a generator corresponding $\Delta_{h.w}^{\pm,\sharp}$.
Under this procedure the central charge changes to

$c=c({\mathfrak{g}},f,k)-12k(y|y)$

and the Verma character becomes

$\text{ch}[V_{\lambda}]=q^{-\frac{c}{24}}q^{\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}-\lambda(x+y)}\
\prod_{n=1}^{\infty}\prod_{\alpha\in\Delta_{\text{
h.w.}}^{0}}(1-q^{n})^{-1}\prod_{\alpha\in\Delta_{\text{
h.w.}}^{+,\sharp}}(1-e^{\alpha(h)}q^{n})^{-1}(1-e^{-\alpha(h)}q^{n-1})^{-1},$
(4.18)

More explicitly this becomes

$\text{ch}[V_{\lambda}]=q^{-\frac{c}{24}}q^{\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}-\lambda(x+y)}\
\prod_{i=1}^{L}X_{i}\ \prod_{1\leq i<j\leq L}X_{i,j}\ \Big{|}_{y=1}$ (4.19)

where

$\displaystyle X_{i}=$
$\displaystyle\prod_{t=1}^{s_{i}}{(y^{-2t}z)_{\infty}^{-m_{i}}}\prod_{m_{1}+\cdots+m_{i-1}+1\leq
a<c\leq m_{1}+\cdots+m_{i}}\ \ (y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}\
(y^{-2t}u_{a}^{-1}u_{c})^{-1}_{\infty}$ (4.20)

and

$\displaystyle X_{i,j}=$ $\displaystyle\prod_{t=1}^{s_{i}}\ \
\prod_{\begin{subarray}{c}m_{1}+\cdots+m_{i-1}+1\leq a\leq
m_{1}+\cdots+m_{i}\\\ m_{1}+\cdots+m_{j-1}+1\leq c\leq
m_{1}+\cdots+m_{j}\end{subarray}}\ \
(y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}(y^{-2t}u_{a}^{-1}u_{c})^{-1}_{\infty}~{}.$
(4.21)

###  4.6 Iterated reductions

Let ${\mathfrak{g}}$ be a simple Lie algebra and
${\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}$ an ordered set of Lie algebras
such that
${\mathfrak{a}}_{1}\oplus{\mathfrak{a}}_{2}\dots\oplus{\mathfrak{a}}_{L}$ is a
subalgebra of ${\mathfrak{g}}$. Let $f_{1},\dots,f_{L}$ be nilpotent elements
in ${\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}$ and let
$f=f_{1}+f_{2}+\dots+f_{L}$ be their sum in ${\mathfrak{g}}$. Each $f_{i}$ is
also considered to be an element in ${\mathfrak{g}}$ via the embedding of
${\mathfrak{a}}_{i}$ in ${\mathfrak{g}}$. Define Lie algebra
${\mathfrak{b}}_{1},\dots,{\mathfrak{b}}_{L}$ by setting
${\mathfrak{b}}_{1}={\mathfrak{g}}$ and
${\mathfrak{b}}_{i+1}:={\mathfrak{b}}_{i}^{{\mathfrak{a}}_{i}}$ in particular
each ${\mathfrak{b}}_{i}$ has
${\mathfrak{a}}_{i}\oplus\dots\oplus{\mathfrak{a}}_{L}$ as subalgebra and in
particular $f_{i}$ is a nilpotent element in ${\mathfrak{b}}_{i}$
corresponding to an ${\mathfrak{s}}{\mathfrak{l}}_{2}$-triple
$(e_{i},h_{i},f_{i})$ in ${\mathfrak{b}}_{i}$. Let

${\mathfrak{b}}_{i}=\bigoplus_{n\in\frac{1}{2}\mathbb{Z}}{\mathfrak{b}}_{i,n}$

be the corresponding grading by $h_{i}$ eigenvalue. Since
$f_{i}\in{\mathfrak{a}}_{i}$ as well we have that
${\mathfrak{b}}_{i+1}={\mathfrak{b}}_{i}^{{\mathfrak{a}}_{i}}\subset{\mathfrak{b}}_{i}^{f_{i}}\cap{\mathfrak{b}}_{i,0}$.
We define a sequence of vertex algebras
${\mathcal{W}}^{k}_{i}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))$
for $i=1,\dots,h$ with the properties that this algebra contains an affine
vertex algebra $V^{B_{i+1}}({\mathfrak{b}}_{i+1})$ as subalgebra for some
generic level $B_{i+1}$(here set ${\mathfrak{b}}_{s+1}=0$) and such that
${\mathcal{W}}^{k}_{i}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))$
is in a completion of ${\rm{KL}}_{B_{i+1}}({\mathfrak{b}}_{i+1})$. Set

${\mathcal{W}}^{k}_{1}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})):={\mathcal{W}}^{k}({\mathfrak{b}}_{1},f_{1})$

and

${\mathcal{W}}^{k}_{i}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})):=H\left({\mathcal{W}}^{k}_{i-1}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))\otimes
F({\mathfrak{b}}_{i},f_{i},B_{i}),d^{i}_{0}\right).$

with $d_{0}^{i}$ the differential associated to the quantum Hamiltonian
reduction of $V^{B_{i}}({\mathfrak{b}}_{i})$ with respect to the nilpotent
element $f_{i}$. The claimed properties hold by Properties (1, 2, 3) of the
last Section 4.4. In particular each homology is concentrated in degree zero.
By construction each $d_{0}^{i}$ commutes with $d_{0}^{j}$ viewed as operators
on

$V^{k}({\mathfrak{g}})\otimes
F({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))$

with

$F({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})):=\bigotimes_{i=1}^{L}F({\mathfrak{b}}_{i},f_{i},B_{i})$

and so the iterated homology coincides with the total homology, that is

${\mathcal{W}}^{k}_{L}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))=H(V^{k}({\mathfrak{g}})\otimes
F({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})),d_{0})$
(4.22)

and $d=d^{1}+d^{2}+\dots+d^{L}$. It is natural to ask if there is a relation
between $H(V^{k}({\mathfrak{g}})\otimes
F({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})),d_{0})$
and the usual quantum Hamiltonian reduction
${\mathcal{W}}^{k}({\mathfrak{g}},f)$. Two vertex algebra $V_{1},V_{2}$ are
called stable equivalence if there exist a free field algebras $F_{1},F_{2}$
such that

$V_{1}\otimes F_{1}\cong V_{2}\otimes F_{2}$

as vertex algebras. Note that conformal vectors are allowed to differ.

######  Conjecture 4.5.

$H(V^{k}({\mathfrak{g}})\otimes
F({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})),d_{0})$
and ${\mathcal{W}}^{k}({\mathfrak{g}},f)$ are stable equivalent.

The graded characters of these two algebras are given by the supercharacters
of the complexes which agree upon multiplication with suitable free field
algebra characters. Moreover it is a general theme that quantum Hamiltonian
reductions corresponding to the same nilpotent element but to different
complexes and differentials coincide up to some free field algebras, see
[CGL20] and in particular [ACF22] for examples. These examples are motivated
from concatenation of corner vertex algebras [GR19, CG20] and the special
example of our iterated reduction where $f_{1},\dots,f_{L}$ are principal
nilpotent in ${\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}$ is exactly the
gluing of several corner vertex algebras [PR18].

###  4.7 Iterated Verma modules

We continue with the same set-up as in the previous subsection. Let $M$ be a
module for $V^{k}({\mathfrak{g}})$. Then we can set

$M_{1}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})):=H(V^{k}({\mathfrak{g}})\otimes
F({\mathfrak{g}},f_{1},k),d_{0}^{1})$

and iteratively

$M_{i}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L})):=H\left(M_{i-1}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))\otimes
F({\mathfrak{b}}_{i},f_{i},B_{i}),d^{i}_{0}\right).$

Then
$M_{L}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))$
is automatically a
${\mathcal{W}}^{k}_{L}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))$-module.
If $M$ is the universal Verma module $V$, then all iterated homologies are
concentrated in degree zero by Properties (4) and (5) of Section 4.4. Write
$V^{W}$ for
$V_{L}({\mathfrak{g}},({\mathfrak{a}}_{1},\dots,{\mathfrak{a}}_{L}),(f_{1},\dots,f_{L}))$,
In particular the homology coincides with the total homology and the character
is given by the Euler-Poincaré character of the complex,

$\text{ch}[V^{W}]=\text{sch}[V\otimes F^{\prime}]$ (4.23)

###  4.8
${\mathcal{W}}_{L}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))$:
The iterated ${\mathcal{W}}$-algebras of ${\mathfrak{g}}{\mathfrak{l}}_{N}$

######  Example 4.6.

${\mathcal{W}}_{2}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},{\mathfrak{g}}{\mathfrak{l}}_{N_{2}}),(f_{1},f_{2}))$

We reconsider Example 4.4. Let $N=N_{1}+N_{2}$ and
$M_{1}=m_{1}s_{1},M_{2}=m_{2}s_{2}$, and assume $0<s_{1}<s_{2}$. We then have
an embedding of
${\mathfrak{g}}{\mathfrak{l}}_{M_{1}}\oplus{\mathfrak{g}}{\mathfrak{l}}_{M_{2}}$
in ${\mathfrak{g}}{\mathfrak{l}}_{N}$ and can choose $f=f_{1}+f_{2}$ with
$f_{1}$ and $f_{2}$ the images of the rectangular nilpotent elements in
${\mathfrak{g}}{\mathfrak{l}}_{M_{1}}$ and
${\mathfrak{g}}{\mathfrak{l}}_{M_{2}}$ in ${\mathfrak{g}}{\mathfrak{l}}_{M}$.
Let ${\mathfrak{s}}{\mathfrak{l}}_{2}^{1}$ be the
${\mathfrak{s}}{\mathfrak{l}}_{2}$ corresponding to the rectangular embedding
in ${\mathfrak{g}}{\mathfrak{l}}_{M_{1}}$ and then the latter embedded in
${\mathfrak{g}}{\mathfrak{l}}_{N}$. Then as a
${\mathfrak{g}}{\mathfrak{l}}_{m_{1}}\otimes{\mathfrak{g}}{\mathfrak{l}}_{M_{2}}\otimes{\mathfrak{s}}{\mathfrak{l}}^{1}_{2}$-module
($\mathbf{1}_{m}$ denots the trivial
${\mathfrak{g}}{\mathfrak{l}}_{m}$-module)

$\displaystyle{\mathfrak{g}}{\mathfrak{l}}_{M}\cong$ $\displaystyle\
\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{m_{1}})\otimes\mathbf{1}_{M_{2}}\otimes\left(\bigoplus_{J=1}^{s_{1}}\mathbb{C}_{2J-1}\right)\oplus\mathbf{1}_{m_{1}}\otimes\textup{adj}({\mathfrak{g}}{\mathfrak{l}}_{M_{2}})\otimes\mathbf{1}_{{\mathfrak{s}}{\mathfrak{l}}_{2}^{1}}\oplus$
(4.24) $\displaystyle\
\left(\mathbb{C}^{m_{1}}\otimes\overline{\mathbb{C}}^{M_{2}}\oplus\overline{\mathbb{C}}^{m_{1}}\otimes{\mathbb{C}}^{M_{2}}\right)\otimes\mathbb{C}_{s_{1}}$

The complex for this reduction is

$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{M})\otimes
F^{m_{1}^{2}s_{1}(s_{1}-1)}\otimes F^{2m_{1}m_{2}s_{2}(s_{1}-1)}$

if $s_{1}$ is odd and

$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{M})\otimes
F^{m_{1}^{2}s_{1}(s_{1}-1)}\otimes
F^{2m_{1}m_{2}s_{1}s_{2}}\otimes\beta\gamma^{m_{1}m_{2}s_{2}}$

otherwise. Here $\beta\gamma$ denotes the vertex algebra of a pair of
symplectic bosons ($\beta\gamma$-VOA, the neutral free field algebra
associated to ${\mathfrak{g}}_{\frac{1}{2}}$). The cohomology still has an
affine VOA of type ${\mathfrak{g}}{\mathfrak{l}}_{M_{2}}$ and the iterated
reduction is the rectangular one on this affine subVOA, in particular the
complex for the combined reductions is then

$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})\otimes
F^{m_{1}^{2}s_{1}(s_{1}-1)}\otimes
F^{m_{2}^{2}s_{2}(s_{2}-1)}\otimes\begin{cases}F^{2m_{1}m_{2}s_{2}(s_{1}-1)}&\quad
s_{1}\ \text{odd}\\\
F^{2m_{1}m_{2}s_{1}s_{2}}\otimes\beta\gamma^{m_{1}m_{2}s_{2}}&\quad s_{1}\
\text{even}\end{cases}$

The general case is now $N=N_{1}+\dots+N_{L}$ and $N_{i}=m_{i}s_{i}$ so that
${\mathfrak{g}}{\mathfrak{l}}_{N}$ has
${\mathfrak{g}}{\mathfrak{l}}_{N_{1}}\oplus\dots\oplus{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}$
as subalgebras and we can take $f=f_{1}+\dots+f_{L}$ with $f_{i}$ the
rectangular element in the $i$-th summand. Assume that
$0<s_{1}<s_{2}<\dots<s_{L}$, i.e. the partition is ordered. The first
iteration is as Example 4.6 with

$M^{(1)}=N,\qquad M^{(1)}=M^{(1)}_{1}+M^{(1)}_{2},\qquad
M^{(1)}_{1}=N_{1}~{},\qquad M^{(1)}_{2}=N_{2}+\dots+N_{L}.$

The reduction has an affine subVOA of type
${\mathfrak{g}}{\mathfrak{l}}_{M^{(1)}_{2}}$ and the second iteration is the
reduction on this subVOA again as in Example 4.6 with

$M^{(2)}=M^{(1)}_{2},\qquad M^{(2)}=M^{(2)}_{1}+M^{(2)}_{2},\qquad
M^{(2)}=M^{(1)}_{2},\qquad M^{(2)}_{1}=N_{2}~{},\qquad
M^{(2)}_{2}=N_{3}+\dots+N_{L}.$

The reduction has an affine subVOA of type
${\mathfrak{g}}{\mathfrak{l}}_{M^{(2)}_{2}}$ and one iterates with

$M^{(i)}=M^{(i-1)}_{2},\qquad M^{(i)}=M^{(i)}_{1}+M^{(i)}_{2},\qquad
M^{(i)}_{1}=N_{i}~{},\qquad M^{(i)}_{2}=N_{(i+1)}+\dots+N_{L}~{}.$

Then the complex for all iterations becomes

$V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})\otimes F^{\prime},\qquad
F^{\prime}=\bigotimes_{i=1}^{L}F^{m_{i}^{2}s_{i}(s_{i}-1)}\otimes\bigoplus_{1\leq
i<j\leq L}\widehat{F}^{2m_{i}m_{j}s_{j}(s_{i}-1)}.$

where

$\widehat{F}^{2m_{i}m_{j}s_{j}(s_{i}-1)}:=\begin{cases}F^{2m_{i}m_{j}s_{j}(s_{i}-1)}&\quad
r_{i}\ \text{odd}\\\
F^{2m_{i}m_{j}s_{i}s_{j}}\otimes\beta\gamma^{m_{i}m_{j}s_{j}}&\quad s_{i}\
\text{even}\end{cases}$ (4.25)

for $1\leq i<j\leq L$.

As shown below, this yields the following identity

$\begin{split}\text{ch}[{\mathcal{W}}_{L}^{k}&({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))]={\mathrm{ch}}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}\text{ch}[\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}],\end{split}$ (4.26)

where ${\mathrm{ch}}[{\mathcal{A}}]$ denotes the character of the vertex
algebra ${\mathcal{A}}$. The step by step derivation of this identity proceeds
as follows:

$\begin{split}\text{ch}[{\mathcal{W}}_{L}^{k}&({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))]\\\
&=\text{sch}[V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})\otimes F^{\prime}]\\\
&={\mathrm{ch}}[V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})]\prod_{i=1}^{L}\text{sch}[F^{m_{i}^{2}s_{i}(s_{i}-1)}]\prod_{1\leq
i<j\leq L}\text{sch}[\widehat{F}^{2m_{i}m_{j}s_{j}(s_{i}-1)}]\\\
&={\mathrm{ch}}[V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})]\prod_{i=1}^{L}\text{sch}[F^{m_{i}^{2}s_{i}(s_{i}-1)}]\prod_{1\leq
i<j\leq L}\text{sch}[F^{2m_{i}m_{j}s_{j}(s_{i}-1)}]\\\
&={\mathrm{ch}}[V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})]\prod_{i=1}^{L}\text{sch}[F^{m_{i}^{2}s_{i}(s_{i}-1)}]\prod_{1\leq
i<j\leq
L}\text{sch}[F^{2m_{i}m_{j}s_{i}(s_{j}-1)}]\left(\text{sch}[F^{2m_{i}m_{j}(s_{j}-s_{i})}]\right)^{-1}\\\
&={\mathrm{ch}}[V^{k}({\mathfrak{g}}{\mathfrak{l}}_{N})]\prod_{i=1}^{L}\text{sch}[F^{m_{i}^{2}s_{i}(s_{i}-1)}]\prod_{1\leq
i<j\leq L}\text{sch}[\widetilde{F}^{2m_{i}m_{j}s_{i}(s_{j}-1)}]\prod_{1\leq
i<j\leq L}\text{ch}[\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}]\\\
&={\mathrm{ch}}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}\text{ch}[\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}]\end{split}$ (4.27)

where $\widetilde{F}^{2m_{i}m_{j}s_{i}(s_{j}-1)}$ is defined in equation
(4.11) while $\widehat{F}^{2m_{i}m_{j}s_{j}(s_{i}-1)}$ is defined in equation
(4.25).

The first equality in (4.27) is that the character coincides with the
supercharacter of the complex as all higher homologies vanish. The third
equality follows as the supercharacters of
$\widehat{F}^{2m_{i}m_{j}s_{j}(s_{i}-1)}$ and $F^{2m_{i}m_{j}s_{j}(s_{i}-1)}$
coincide. The fifth equality uses that the supercharacters of
$\widetilde{F}^{2m_{i}m_{j}s_{i}(s_{j}-1)}$ and
$F^{2m_{i}m_{j}s_{i}(s_{j}-1)}$ coincide as well as that the $\beta\gamma$
character is the inverse of the supercharacter of two free fermions. Finally
the last equality is the coincidence of the supercharacter of the complex with
the ${\mathcal{W}}$-algebra character.

This coincidence of characters is expected to also hold on the level of actual
VOAs, that is

######  Conjecture 4.7.

${\mathcal{W}}_{L}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))\cong{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)\otimes\bigotimes\limits_{1\leq
i<j\leq L}\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}$.

In order to conclude this section, note that a completely analogous identity
holds for characters of universal Verma modules i.e.

$\begin{split}\text{ch}^{V}[{\mathcal{W}}_{L}^{k}&({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))]={\mathrm{ch}}^{V}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}\text{ch}^{V}[\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}].\end{split}$
(4.28)

###  4.9 Conformal weights and refined characters

Let $V$ be a vertex algebra that is $\frac{1}{2}\mathbb{Z}$-graded by
conformal weight. Let $\\{F_{p}V\subset V|p\in\frac{1}{2}\mathbb{Z}\\}$ be an
increasing, exhaustive and separated filtration, that is

  1. 1.

$F_{p}V\subset F_{p+\frac{1}{2}}V$

  2. 2.

$V=\bigcup\limits_{p}F_{p}V$

  3. 3.

$0=\bigcap\limits_{p}F_{p}V$

Let $L_{0}$ be the Virasoro zero-mode of a conformal field of $V$. The
filtration is compatible with the vertex algebra structure and the conformal
structure if

  1. 1.

$|\,0\,\rangle\in F_{0}V\setminus F_{-\frac{1}{2}}V$

  2. 2.

$TF_{p}V\subset F_{p}V$ ($T$ the derivation of $V$)

  3. 3.

$L_{0}F_{p}V\subset F_{p}V$

  4. 4.

$v_{n}F_{q}V\subset F_{p+q}V$. for all $p,q\in\frac{1}{2}\mathbb{Z}$, $v\in
F_{p}V$ and $n\in\frac{1}{2}\mathbb{Z}$.

The associated graded

$\text{gr}^{F}V=\bigoplus_{p\in\frac{1}{2}\mathbb{Z}}F_{p}V/F_{p-\frac{1}{2}}V$

is then a vertex algebra, the associated graded of $V$ with respect to $F$.
Assume that $V$ is graded by some Cartan subalgebra ${\mathfrak{h}}$, such
that

$hF_{p}V\subset F_{p}V$

for $h$ in ${\mathfrak{h}}$ then ${\mathfrak{h}}$ and $L_{0}$ induce an action
on $\text{gr}^{F}V$ and we can add an additional grading $g$ acting on
$F_{p}V/F_{p-\frac{1}{2}}V$ by multiplication with $p$. The refined character
of $V$ is then

$\text{ch}_{\rm
ref}[V](y,z,q)=\text{tr}_{\text{gr}^{F}V}\left(q^{L_{0}-\frac{c}{24}}z^{h}y^{-2g}\right).$
(4.29)

######  Example 4.8.

Let $V$ be the vertex algebra of $m_{i}m_{j}(s_{j}-s_{i})$ $\beta\gamma$-VOAs.
It is strongly generated by $\\{\beta^{t}_{a,b},\gamma^{t}_{a,b}|1\leq a\leq
m_{i},1\leq b\leq m_{j},1\leq t\leq s_{j}-s_{i}\\}$. The OPE is

$\beta^{t}_{a,b}(z)\gamma^{t}_{a,b}(w)=(z-w)^{-1}$

and all others are zero. Assigning the field $\beta^{t}_{a,b}(z)$ (and hence
any of its modes) the degree $t$ and the field $\gamma^{t}_{a,b}(z)$ the
degree $1-t$ clearly gives a filtration such that the associated graded is
commutative. Using the same variables as in equations (4.20) and (4.21), the
refined character is

$\displaystyle{\rm ch}_{\rm ref}[\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}]=$
$\displaystyle\prod_{t=1}^{s_{j}-s_{i}}\ \ \ \
\prod_{m_{1}+\cdots+m_{i-1}+1\leq a\leq m_{1}+\cdots+m_{i}}\ \
\prod_{m_{1}+\cdots+m_{j-1}+1\leq b\leq
m_{1}+\cdots+m_{j}}(y^{-2t}zu_{a}u_{b}^{-1})^{-1}_{\infty}\
(y^{2-2t}u_{a}^{-1}u_{b})^{-1}_{\infty}~{}.$

The algebra of modes satisfies

$[\beta^{t}_{a,b,n}\,,\,\gamma^{t}_{a,b,m}]=\delta_{n+m,0}$

and all other commutators vanish. Thus the $\beta^{t}_{a,b}$ form a
commutative subalgebra and the submodule $B(m_{i},m_{j},s_{j}-s_{i})$ of the
$\beta\gamma$-VOA generated by this subalgebra acting on the vacuum has
refined character

$\displaystyle{\rm ch}_{\rm ref}[B(m_{i},m_{j},s_{j}-s_{i})]=$ (4.30)
$\displaystyle\prod_{t=1}^{s_{j}-s_{i}}\ \ \ \
\prod_{m_{1}+\cdots+m_{i-1}+1\leq a\leq m_{1}+\cdots+m_{i}}\ \
\prod_{m_{1}+\cdots+m_{j-1}+1\leq b\leq
m_{1}+\cdots+m_{j}}(y^{-2t}zu_{a}u_{b}^{-1})^{-1}_{\infty}~{}.$

The standard Filtration for a vertex algebra is given by fixing a minimal set
of strong generators and given $x_{n}$ degree $h$ if the strong generator
$x(z)$ has conformal weight $h$ [Li04]. This depends on a choice of conformal
vector. Recall that the strong generators of
${\mathcal{W}}^{k}({\mathfrak{g}},f)$ are labelled by
$\alpha\in\Delta_{h.w.}$. If $\delta(\alpha)=j$, then the strong generator
corresponds to a highest-weight vector of highest-weight $j$ with respect to
the $\mathfrak{sl}_{2}$-triple for the quantum Hamiltonian reduction. It thus
corresponds to the $u=2j+1$ dimensional representation. The conformal weigth
of the generator is $j+1$ or in terms of the dimension $(u+1)/2$. From this we
can read of that the refined character with respect to the standard filtration
of Example 4.4 is

$\text{ch}_{\rm
ref}[V_{\lambda}]=q^{-\frac{c}{24}}q^{\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}-\lambda(x+y)}\
\prod_{i=1}^{L}X_{i}\ \prod_{1\leq i<j\leq L}X_{i,j}$ (4.31)

where

$\displaystyle X_{i}=$
$\displaystyle\prod_{t=1}^{s_{i}}(y^{-2t}z)_{\infty}^{-m_{i}}\prod_{m_{1}+\cdots+m_{i-1}+1\leq
a<c\leq m_{1}+\cdots+m_{i}}\ \ (y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}\
(y^{-2t}u_{a}^{-1}u_{c})^{-1}_{\infty}$ (4.32)

and

$\displaystyle X_{i,j}=$ $\displaystyle\prod_{t=1}^{s_{i}}\ \
\prod_{\begin{subarray}{c}m_{1}+\cdots+m_{i-1}+1\leq a\leq
m_{1}+\cdots+m_{i}\\\ m_{1}+\cdots+m_{j-1}+1\leq c\leq
m_{1}+\cdots+m_{j}\end{subarray}}\ \
(y^{-2t}y^{s_{i}-s_{j}}zu_{a}u_{c}^{-1})^{-1}_{\infty}(y^{-2t}y^{s_{i}-s_{j}}u_{a}^{-1}u_{c})^{-1}_{\infty}~{}.$
(4.33)

Let $w=\frac{1}{2}\sum_{k=1}^{L}s_{k}x_{k}$ and define a new conformal vector
$L+\partial w$. The refined character for this filtration is

$\text{ch}_{\rm
ref}[V_{\lambda}]=q^{-\frac{c}{24}}q^{\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}-\lambda(x+y)}\
\prod_{i=1}^{L}X_{i}\ \prod_{1\leq i<j\leq L}X_{i,j}$ (4.34)

where

$\displaystyle X_{i}=$
$\displaystyle\prod_{t=1}^{s_{i}}(y^{-2t}z)_{\infty}^{-m_{i}}\prod_{m_{1}+\cdots+m_{i-1}+1\leq
a<c\leq m_{1}+\cdots+m_{i}}\ \ (y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}\
(y^{-2t}u_{a}^{-1}u_{c})^{-1}_{\infty}$ (4.35)

and

$\displaystyle X_{i,j}=$ $\displaystyle\prod_{t=1}^{s_{i}}\ \
\prod_{\begin{subarray}{c}m_{1}+\cdots+m_{i-1}+1\leq a\leq
m_{1}+\cdots+m_{i}\\\ m_{1}+\cdots+m_{j-1}+1\leq c\leq
m_{1}+\cdots+m_{j}\end{subarray}}\ \
(y^{-2t}y^{2(s_{i}-s_{j})}zu_{a}u_{c}^{-1})^{-1}_{\infty}(y^{-2t}u_{a}^{-1}u_{c})^{-1}_{\infty}~{}.$
(4.36)

In conclusion, note that the conformal weight refinement of identity (4.28)
reads

$\displaystyle{\rm ch}^{V}_{\rm ref}[{\mathcal{W}}_{L}^{k}$
$\displaystyle({\mathfrak{g}}{\mathfrak{l}}_{N},({\mathfrak{g}}{\mathfrak{l}}_{N_{1}},\dots,{\mathfrak{g}}{\mathfrak{l}}_{N_{L}}),(f_{1},\dots,f_{L}))]={\rm
ch}_{\rm
ref}^{V}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}{\rm ch}_{\rm ref}[\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}].$ (4.37)

where

${\rm ch}_{\rm
ref}^{V}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]=\prod_{i=1}^{L}X_{i}\prod_{1\leq
i<j\leq L}X_{i,j}~{}.$ (4.38)

##  5 Comparison to partition functions of affine Laumon spaces

The goal of this section is to prove equation (1.7), which relates the
partition (1.3) to the refined universal Verma character associated to the
$\beta$-truncation of the vertex algebra

${\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)\otimes\bigotimes\limits_{1\leq
i<j\leq L}\beta\gamma^{m_{i}m_{j}(s_{j}-s_{i})}.$

######  Proposition 5.1.

Suppose the numerical invariants ${\bf r}=(r_{0},\ldots,r_{\ell-1})$ are given
by

$(r_{0},\ldots,r_{\ell-1})=\big{(}\underbrace{s_{L},\ldots,s_{L}}_{m_{L}}\,,\,\underbrace{s_{L-1},\ldots,s_{L-1}}_{m_{L-1}},\ldots,\underbrace{s_{1},\ldots,s_{1}}_{m_{1}}\big{)}$
(5.1)

for some $m_{1},\ldots,m_{L}>0$ and $0<s_{1}<\cdots<s_{L}$, $L\geq 1$. Then

$\displaystyle Z_{\bf r}={\rm ch}_{\rm
ref}^{V}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}\ {\rm ch}_{\rm ref}[B(m_{i},m_{j},s_{j}-s_{i})].$ (5.2)

Proof. As shown in Proposition 3.4, the partition function $Z_{\bf r}$ is
given by

$\displaystyle Z_{\bf r}=$
$\displaystyle\prod_{a=0}^{\ell-1}\bigg{(}\prod_{t=1}^{r_{a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}}\bigg{)}$ $\displaystyle\prod_{1\leq a<c\leq\ell}\
\bigg{(}\prod_{t=1}^{r_{\ell-c}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a}}\prod_{k=1}^{\infty}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)}.$

The data $(m_{1},\ldots,m_{L})$ determines a partition

$\\{1,\ldots,\ell\\}\simeq\bigcup_{i=1}^{L}\\{m_{1}+\cdots+m_{i-1}+1,\ldots,m_{1}+\cdots+m_{i}\\}$

where the sets in the right hand side are pairwise disjoint. By convention,
$m_{0}=0$. Moreover, given $1\leq c\leq\ell$, one has

$r_{\ell-c}=s_{i}$

if and only if $m_{1}+\cdots+m_{i-1}+1\leq c\leq m_{1}+\cdots+m_{i},\ 1\leq
i\leq L$. Then the above expression for $Z_{\bf r}$ is equivalent to

$Z_{\bf r}=\prod_{i=1}^{L}Z_{i}\ \prod_{1\leq i<j\leq L}Z_{i,j}$ (5.3)

where

$\displaystyle Z_{i}=$
$\displaystyle\prod_{t=1}^{s_{i}}{(y^{-2t}z)_{\infty}^{m_{i}}}\prod_{m_{1}+\cdots+m_{i-1}+1\leq
a<c\leq m_{1}+\cdots+m_{i}}\ \ (y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}\
(y^{-2t}u_{a}^{-1}u_{c})^{-1}_{\infty}$ (5.4)

and

$\displaystyle Z_{i,j}=$ $\displaystyle\prod_{t=1}^{s_{j}}\prod_{v=1}^{s_{i}}\
\ \prod_{m_{1}+\cdots+m_{i-1}+1\leq a\leq m_{1}+\cdots+m_{i}}\ \
\prod_{m_{1}+\cdots+m_{j-1}+1\leq c\leq
m_{1}+\cdots+m_{j}}(y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}(y^{-2v}u_{a}^{-1}u_{c})^{-1}_{\infty}~{}.$
(5.5)

Note that the $Z_{i}$ coincide with the factors $X_{i}$ defined in equation
(4.35). Furthermore, note that

$\displaystyle Z_{i,j}$
$\displaystyle=\prod_{(a,c)}\prod_{t=1}^{s_{j}}\prod_{v=1}^{s_{i}}\prod_{(a,c)}(y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}(y^{-2v}u_{a}^{-1}u_{c})^{-1}_{\infty}$
(5.6)
$\displaystyle=\prod_{(a,c)}\prod_{t=1}^{s_{j}-s_{i}}(y^{-2t}zu_{a}u_{c}^{-1})^{-1}_{\infty}$
$\displaystyle\ \ \
\prod_{(a,c)}\prod_{t=1}^{s_{i}}\prod_{v=1}^{s_{i}}(y^{-2(t+s_{j}-s_{i})}zu_{a}u_{c}^{-1})^{-1}_{\infty}(y^{-2v}u_{a}^{-1}u_{c})^{-1}_{\infty}~{}.$

where $\prod_{(a,c)}$ stands for

$\prod_{m_{1}+\cdots+m_{i-1}+1\leq a\leq m_{1}+\cdots+m_{i}}\ \
\prod_{m_{1}+\cdots+m_{j-1}+1\leq c\leq m_{1}+\cdots+m_{j}}~{}.$

By comparison to equations (4.30) and (4.36) one obtains

$Z_{i,j}=X_{i,j}\,{\rm ch}_{\rm ref}[B(m_{i},m_{j},s_{j}-s_{i})].$

In conclusion, using equation (4.38), one obtains,

$\displaystyle Z$ $\displaystyle=\prod_{i=1}^{L}X_{i}\prod_{1\leq i<j\leq L}\
X_{i,j}\ {\rm ch}_{\rm ref}[B(m_{i},m_{j},s_{j}-s_{i})]$ (5.7)
$\displaystyle={\rm ch}_{\rm
ref}^{V}[{\mathcal{W}}^{k}({\mathfrak{g}}{\mathfrak{l}}_{N},f)]\prod_{1\leq
i<j\leq L}\ {\rm ch}_{\rm ref}[B(m_{i},m_{j},s_{j}-s_{i})].$

$\Box$

##  Appendix A Some combinatorial identities

This section proves equations (3.4) and (3.5).

Given a Young diagram $\mu$ and an integer $-\ell+1\leq c\leq\ell-1$, let

$\displaystyle{\sf N}^{\geq 0}_{1}(\mu;c)$
$\displaystyle=\\{(i,j)\in\mu\,|\,\mu^{\prime}_{i}-j\equiv c\ {\rm mod}\
\ell\,\\}$ $\displaystyle{\sf N}^{>0}_{1}(\mu;c)$
$\displaystyle=\\{(i,j)\in\mu\,|\,\mu^{\prime}_{i}-j>0,\
\mu^{\prime}_{i}-j\equiv c\ {\rm mod}\ \ell\,\\}$ $\displaystyle{\sf N}^{\geq
0}_{2}(\mu;c)$ $\displaystyle=\\{(i,j)\in\mu\,|\,j-1\equiv c\ {\rm mod}\
\ell\,\\}$

Moreover, for fixed $1\leq i\leq{\rm col}(\mu)$ let

$\displaystyle{\sf N}^{\geq 0}_{1}(i;c)$ $\displaystyle=\\{1\leq
j\leq\mu_{i}^{\prime}\,|\,\mu^{\prime}_{i}-j\equiv c\ {\rm mod}\ \ell\,\\}$
$\displaystyle{\sf N}^{>0}_{1}(i;c)$ $\displaystyle=\\{1\leq
j\leq\mu_{i}^{\prime}-1\,|\,\mu^{\prime}_{i}-j\equiv c\ {\rm mod}\ \ell\,\\}$
$\displaystyle{\sf N}^{\geq 0}_{2}(i;c)$ $\displaystyle=\\{1\leq
j\leq\mu_{i}^{\prime}\,|\,j-1\equiv c\ {\rm mod}\ \ell\,\\}$

where ${\rm col}(\mu)$ denotes the number of columns of $\mu$ and
$\mu_{i}^{\prime}$ denotes the height of the $i$-th column.

As in equation (2.3) a partition $\mu$ will be said to be $a$-colored if and
only if the box $(i,j)\in\mu$ is marked by $a-j+1$ mod $\ell$.

######  Lemma A.1.

Let $n_{a-c}$ denote the number of boxes of $\mu$ marked by $a-c$ mod $\ell$.
Then

$n_{a-c}=|{\sf N}_{1}^{\geq 0}(\mu;c)|$ (A.1)

for any $-\ell+1\leq c\leq\ell-1$.

Proof. By definition, $n_{a-c}=|{\sf N}_{2}^{\geq 0}(\mu;c)|$. Hence it
suffices to show that

$|{\sf N}_{1}^{\geq 0}(\mu;c)|=|{\sf N}_{2}^{\geq 0}(\mu;c)|$ (A.2)

for any $-\ell+1\leq c\leq\ell-1$. Setting $j^{\prime}=\mu_{i}^{\prime}-j$,
note that

$\displaystyle{\sf N}^{\geq 0}_{1}(i;c)$ $\displaystyle=\\{1\leq
j\leq\mu_{i}^{\prime}\,|\,\mu^{\prime}_{i}-j\equiv c\ {\rm mod}\ \ell\,\\}$
$\displaystyle=\\{0\leq j^{\prime}\leq\mu_{i}^{\prime}-1\,|\,j^{\prime}\equiv
c\ {\rm mod}\ \ell\,\\},$

which coincides with ${\sf N}^{\geq 0}_{2}(i;c)$ by definition. In conclusion,
one obtains

$|{\sf N}_{1}^{\geq 0}(i;c)|=|{\sf N}_{2}^{\geq 0}(i;c)|$

for any $1\leq i\leq{\rm col}(\mu)$, hence equation (A.5) holds.

$\Box$

######  Lemma A.2.

The following identity holds for any partition $\mu$:

$|{\sf N}_{1}^{>0}(\mu;c)|=\left\\{\begin{array}[]{ll}|{\sf N}_{2}^{\geq
0}(\mu;c)|&{\rm for}\ -\ell+1\leq c\leq\ell-1,\ c\neq 0\\\ &\\\ |{\sf
N}_{2}^{\geq 0}(\mu;c)|-{\rm col}(\mu)&{\rm for}\ c=0,\\\ \end{array}\right.$
(A.3)

where ${\rm col}(\mu)$ denotes the number of columns of $\mu$.

Proof. The defining conditions of ${\sf N}_{1}^{>0}(i;c)$ are equivalent to

$1\leq j\leq\mu^{\prime}_{i}-1,\qquad\mu^{\prime}_{i}-j=k\ell+c,\
k\in\mathbb{Z}.$

Suppose $\mu^{\prime}_{i}=q_{i}\ell+a_{i}$, $0\leq a_{i}\leq\ell-1$. Then
these conditions are equivalent to

$1\leq(q_{i}-k)\ell+a_{i}-c\leq q_{i}\ell+a_{i}-1~{},$

hence also to

${1-c\over\ell}\leq k\leq q_{i}+{a_{i}-c-1\over\ell}~{}.$

Since $\ell\geq 2$, and $k,c,q_{i},a_{i}$ are integers so that

$-\ell+1\leq c\leq\ell-1,\qquad 0\leq a_{i}\leq\ell-1~{},$

these conditions are further equivalent to

$\displaystyle 0\leq k\leq q_{i}-1$ $\displaystyle{\rm if}\ 1\leq c\leq\ell-1\
{\rm and}\ 0\leq a_{i}\leq c$ (A.4) $\displaystyle 0\leq k\leq q_{i}$
$\displaystyle{\rm if}\ 1\leq c\leq\ell-2\ {\rm and}\ c+1\leq a_{i}\leq\ell-1$
$\displaystyle 1\leq k\leq q_{i}-1$ $\displaystyle{\rm if}\ c=0\ {\rm and}\
a_{i}=0$ $\displaystyle 1\leq k\leq q_{i}$ $\displaystyle{\rm if}\ c=0\ {\rm
and}\ 1\leq a_{i}\leq\ell-1$ $\displaystyle 1\leq k\leq q_{i}$
$\displaystyle{\rm if}\ -\ell+1\leq c\leq-1\ {\rm and}\ 0\leq a_{i}\leq\ell+c$
$\displaystyle 1\leq k\leq q_{i}+1$ $\displaystyle{\rm if}\ -\ell+1\leq
c\leq-2\ {\rm and}\ \ell+c+1\leq a_{i}\leq\ell-1$

On the other hand, as shown in the proof of Lemma A.1, the defining conditions
of ${\sf N}_{2}^{\geq 0}(i;c)$ are

$-{c\over\ell}\leq k\leq q_{i}+{a_{i}-c-1\over\ell}~{}.$ (A.5)

These conditions are similarly equivalent to

$\displaystyle 0\leq k\leq q_{i}-1$ $\displaystyle{\rm if}\ 0\leq c\leq\ell-1\
{\rm and}\ 0\leq a_{i}\leq c$ (A.6) $\displaystyle 0\leq k\leq q_{i}$
$\displaystyle{\rm if}\ 0\leq c\leq\ell-2\ {\rm and}\ c+1\leq a_{i}\leq\ell-1$
$\displaystyle 1\leq k\leq q_{i}$ $\displaystyle{\rm if}\ -\ell+1\leq c\leq-1\
{\rm and}\ 0\leq a_{i}\leq\ell+c$ $\displaystyle 1\leq k\leq q_{i}+1$
$\displaystyle{\rm if}\ -\ell+1\leq c\leq-2\ {\rm and}\ \ell+c+1\leq
a_{i}\leq\ell-1.$

By comparison with (A.4), one obtains the following table

$\begin{array}[]{cccc}&&|{\sf N}_{1}^{>0}(i;c)|&|{\sf N}_{2}^{\geq 0}(i;c)|\\\
&&\\\ 1\leq c\leq\ell-1,\ 0\leq a_{i}\leq c&&q_{i}&q_{i}\\\ &&\\\ 1\leq
c\leq\ell-2,\ c+1\leq a_{i}\leq\ell-1&&q_{i}+1&q_{i}+1\\\ &&\\\ c=0,\
a_{i}=0&&q_{i}-1&q_{i}\\\ &&\\\ c=0,\ 1\leq a_{i}\leq\ell-1&&q_{i}&q_{i}+1\\\
&&\\\ -\ell-1\leq c\leq-1,\ 0\leq a_{i}\leq\ell+c&&q_{i}&q_{i}\\\ &&\\\
-\ell-1\leq c\leq-2,\ \ell+c+1\leq a_{i}\leq\ell-1&&q_{i}+1&q_{i}+1\\\ &&\\\
\end{array}$

In conclusion,

$|{\sf N}_{1}^{>0}(i;c)|=\left\\{\begin{array}[]{ll}|{\sf N}_{2}^{\geq
0}(i;c)|&{\rm for}\ -\ell+1\leq c\leq\ell-1,\ c\neq 0\\\ &\\\ |{\sf
N}_{2}^{\geq 0}(i;c)|-1&{\rm for}\ c=0.\\\ \end{array}\right.$

This yields equation (A.3).

$\Box$

##  Appendix B Some product identities

This section proves equation (3.19).

######  Lemma B.1.

The following identity holds:

$\displaystyle\prod_{\begin{subarray}{c}1\leq a,c\leq\ell-1\\\ a\neq
c\end{subarray}}\prod_{t=1}^{r_{a}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}=$
(B.1) $\displaystyle\prod_{1\leq
a<c\leq\ell-1}\bigg{(}\prod_{t=1}^{r_{a}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}$
$\displaystyle\qquad\qquad\ \prod_{t=1}^{r_{a-c+\ell}}{1\over
1-y^{-2t}z^{k-1}{\widetilde{q}}_{a-c}{\widetilde{q}}_{a-c-1}\cdots{\widetilde{q}}_{a-\ell+1}}\bigg{)}.$

Proof. Note the set isomorphism

$\\{1\leq c<a\leq\ell-1\\}\xrightarrow{\sim}\\{1\leq a<c\leq\ell-1\\}$
$(a,c)\mapsto(a-c,\ell-c)$

Let $a^{\prime}=a-c$ and $c^{\prime}=\ell-c$. Since

$z={\widetilde{q}}_{a}{\widetilde{q}}_{a-1}\cdots{\widetilde{q}}_{a-\ell+1}$

for any $a\in\mathbb{Z}$, one has

$\displaystyle\prod_{1\leq c<a\leq\ell-1}\prod_{t=1}^{r_{a}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}=$
$\displaystyle\prod_{1\leq c<a\leq\ell-1}\prod_{t=1}^{r_{a}}{1\over
1-y^{-2t}z^{k-1}{\widetilde{q}}_{a}{\widetilde{q}}_{a-c-1}\cdots{\widetilde{q}}_{a-c+1}}=$
$\displaystyle\prod_{1\leq c<a\leq\ell-1}\prod_{t=1}^{r_{a}}{1\over
1-y^{-2t}z^{k-1}{\widetilde{q}}_{a-\ell}{\widetilde{q}}_{a-c-1-\ell}\cdots{\widetilde{q}}_{a-c+1-\ell}}=$
$\displaystyle\prod_{1\leq
a^{\prime}<c^{\prime}\leq\ell-1}\prod_{t=1}^{r_{a^{\prime}-c^{\prime}+\ell}}{1\over
1-y^{-2t}z^{k-1}{\widetilde{q}}_{a^{\prime}-c^{\prime}}{\widetilde{q}}_{a^{\prime}-c^{\prime}-1}\cdots{\widetilde{q}}_{a^{\prime}-\ell+1}}=$
$\displaystyle\prod_{1\leq a<c\leq\ell-1}\prod_{t=1}^{r_{a-c+\ell}}{1\over
1-y^{-2t}z^{k-1}{\widetilde{q}}_{a-c}{\widetilde{q}}_{a-c-1}\cdots{\widetilde{q}}_{a-\ell+1}}$

$\Box$

Recall that the variables (3.13) are given by

$u_{a}=({\widetilde{q}}_{-a}\cdots{\widetilde{q}}{\widetilde{q}}_{-\ell+1})^{-1},\
1\leq a\leq\ell-1.$

Then one has:

######  Lemma B.2.

Identity (B.1) can be further written as

$\displaystyle\prod_{\begin{subarray}{c}1\leq a,c\leq\ell-1\\\ a\neq
c\end{subarray}}\prod_{t=1}^{r_{a}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}=$
(B.2) $\displaystyle\prod_{1\leq a<c\leq\ell-1}\
\bigg{(}\prod_{t=1}^{r_{\ell-c}}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a}}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)}$

Proof. Again, note the set isomorphism

$\\{1\leq a<c\leq\ell-1\\}\xrightarrow{\sim}\\{1\leq a<c\leq\ell-1\\}$
$(a,c)\mapsto(c-a,\ell-a)$

Let $a^{\prime}=c-a$ and $c^{\prime}=\ell-a$. This yields

$\displaystyle\prod_{1\leq a<c\leq\ell-1}\bigg{(}\prod_{t=1}^{r_{a}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1}}$
$\displaystyle\qquad\qquad\ \ \prod_{t=1}^{r_{a-c+\ell}}{1\over
1-y^{-2t}z^{k-1}{\widetilde{q}}_{a-c}{\widetilde{q}}_{a-c-1}\cdots{\widetilde{q}}_{a-\ell+1}}\bigg{)}=$
$\displaystyle\prod_{1\leq
a^{\prime}<c^{\prime}\leq\ell-1}\bigg{(}\prod_{t=1}^{r_{\ell-c^{\prime}}}{1\over
1-y^{-2t}z^{k}({\widetilde{q}}_{-a^{\prime}})^{-1}({\widetilde{q}}_{-a^{\prime}-1})^{-1}\cdots({\widetilde{q}}_{-c^{\prime}+1})^{-1}}$
$\displaystyle\qquad\qquad\qquad\ \prod_{t=1}^{r_{\ell-a^{\prime}}}{1\over
1-y^{-2t}z^{k-1}{\widetilde{q}}_{-a^{\prime}}{\widetilde{q}}_{-a^{\prime}-1}\cdots{\widetilde{q}}_{-c^{\prime}+1}}\bigg{)}=$
$\displaystyle\prod_{1\leq
a^{\prime}<c^{\prime}\leq\ell-1}\bigg{(}\prod_{t=1}^{r_{\ell-c^{\prime}}}{1\over
1-y^{-2t}z^{k}u_{a^{\prime}}u_{c^{\prime}}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a^{\prime}}}{1\over
1-y^{-2t}z^{k-1}u_{a^{\prime}}^{-1}u_{c^{\prime}}}\bigg{)}$

as claimed in identity (B.2).

$\Box$

In conclusion, Lemmas B.1 and B.2 yield identity (3.19) i.e.

$\displaystyle\prod_{a=1}^{\ell-1}\prod_{t=1}^{r_{a}}\prod_{\begin{subarray}{c}c=1\\\
c\neq
a\end{subarray}}^{\ell-1}{1\over(1-y^{-2t}z^{k}({\widetilde{q}}_{a-c})^{-1}({\widetilde{q}}_{a-c-1})^{-1}\cdots({\widetilde{q}}_{a-\ell+1})^{-1})}=$
$\displaystyle\prod_{1\leq a<c\leq\ell-1}\
\bigg{(}\prod_{t=1}^{r_{\ell-c}}{1\over
1-y^{-2t}z^{k}u_{a}u_{c}^{-1}}\bigg{)}\
\bigg{(}\prod_{t=1}^{r_{\ell-a}}{1\over
1-y^{-2t}z^{k-1}u_{a}^{-1}u_{c}}\bigg{)}.$

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Thomas Creutzig, Department of Mathematics and Statistics, University of
Alberta, Canada

E-mail address<EMAIL_ADDRESS>
Duiliu-Emanuel Diaconescu, New High Energy Theory Center, Rutgers University,
USA

E-mail address<EMAIL_ADDRESS>
Mingyang Ma, Department of Physics and Astronomy, Rutgers University, USA

E-mail address<EMAIL_ADDRESS>