# Double Dyck Path Algebra Representations From DAHA

Milo Bechtloff Weising

###### Abstract

The double Dyck path algebra $\mathbb{A}_{q,t}$ was introduced by Carlsson-
Mellit in their proof of the Shuffle Theorem. A variant of this algebra,
$\mathbb{B}_{q,t}$, was introduced by Carlsson-Gorsky-Mellit in their study of
the parabolic flag Hilbert schemes of points in $\mathbb{C}^{2}$ showing that
$\mathbb{B}_{q,t}$ acts naturally on the equivariant $K$-theory of these
spaces. The algebraic relations defining $\mathbb{B}_{q,t}$ appear
superficially similar to those of the positive double affine Hecke algebras
(DAHA) in type $GL$, $\mathscr{D}_{n}^{+}$, introduced by Cherednik. In this
paper we provide a general method for constructing $\mathbb{B}_{q,t}$
representations from DAHA representations. In particular, every
$\mathscr{D}_{n}^{+}$ module yields a representation of a subalgebra
$\mathbb{B}_{q,t}^{(n)}$ of $\mathbb{B}_{q,t}$ and special families of
compatible DAHA representations give representations of $\mathbb{B}_{q,t}$.
These constructions are functorial. Lastly, we will construct a large family
of $\mathbb{B}_{q,t}$ representations indexed by partitions using this method
related to the Murnaghan-type representations of the positive elliptic Hall
algebra introduced previously by the author.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Notation and Conventions
    1. 2.1 Stable-Limits
    2. 2.2 Positive Double Affine Hecke Algebras
    3. 2.3 The $\mathbb{B}_{q,t}$ Algebra
  3. 3 Main Construction
    1. 3.1 $\mathbb{B}_{q,t}^{(n)}$ Modules From $\operatorname{\mathscr{D}}_{n}^{+}$
    2. 3.2 The Polynomial Case
    3. 3.3 $\mathbb{B}_{q,t}$ Modules From Compatible Sequences
  4. 4 Compatible Sequences From AHA

##  1 Introduction

The algebra $\mathbb{B}_{q,t}$ was introduced by Carlsson-Gorsky-Mellit
[CGM20] as an algebra which has a natural geometric action on the equivariant
$K$-theory of the parabolic flag Hilbert schemes of points in
$\mathbb{C}^{2}.$ This work built on the prior work of Schiffmann-Vasserot
[SV13] who constructed a geometric action of the elliptic Hall algebra
$\mathcal{E}$ on the equivariant $K$-theory of the Hilbert schemes of points
in $\mathbb{C}^{2}.$ These construction are part of a larger story in
Macdonald theory of relating geometric properties of the Hilbert schemes of
points in $\mathbb{C}^{2}$ to the algebraic combinatorics underlying the
modified Macdonald symmetric functions $\widetilde{H}_{\mu}$ and of the
Macdonald operator $\Delta$ (which acts on the space of symmetric functions
$\Lambda$). Importantly, $\mathbb{B}_{q,t}$ is intimately related to the
double Dyck path algebra $\mathbb{A}_{q,t}$ introduced by Carlsson-Mellit in
their proof of the Shuffle Theorem [CM18] regarding the Frobenius character of
the space of diagonal coinvariants and the combinatorics of Dyck paths.

The quiver path algebra $\mathbb{B}_{q,t}$ has relations very similar to the
positive double affine Hecke algebras (DAHA) in type $GL$,
$\operatorname{\mathscr{D}}_{n}^{+}$, introduced by Cherednik [Che03]. In
fact, $\mathbb{B}_{q,t}$ contains many copies of affine Hecke algebras of type
$GL.$ However, there is no direct algebraic relation (no algebra
homomorphisms) between $\mathbb{B}_{q,t}$ (nor $\mathbb{A}_{q,t}$) and DAHAs.
Nevertheless, there are approaches to more indirectly relate these algebras.
Ion-Wu [IW22] defined an algebra called the stable-limit DAHA along with a
polynomial representation on the space of almost symmetric functions
$\operatorname{\mathscr{P}}_{as}^{+}$ which, in a sense, globalizes the
polynomial representation of $\mathbb{A}_{q,t}$ (and as we will see later
$\mathbb{B}_{q,t}$). They used a stable-limit procedure to define this
representation from the polynomial representations of the finite rank DAHAs
$\operatorname{\mathscr{D}}_{n}^{+}$. This representation of the stable-limit
DAHA is much larger than the polynomial representation of $\mathbb{B}_{q,t}$
but the limit Cherednik operators of Ion-Wu, in a sense, behave better on a
certain subspace of $\operatorname{\mathscr{P}}_{as}^{+}$ given by the
following direct sum:

$\bigoplus_{k\geq 0}x_{1}\cdots
x_{k}\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda.$

This subspace aligns with the polynomial representation of $\mathbb{B}_{q,t}.$

Motivated by the construction of Ion-Wu we will in this paper develop a method
for constructing modules for $\mathbb{B}_{q,t}$ directly from the
representation theory of DAHA in type $GL.$ We will use a stable-limit
construction similar to Ion-Wu but will not require any additional non-
archimedean topological considerations as they did. First, we will show
(Proposition 3.4) that given any $\operatorname{\mathscr{D}}_{n}^{+}$ module
$V$ we may construct an action of the subalgebra $\mathbb{B}_{q,t}^{(n)}$ on
the space $L_{\bullet}(V)$ defined by

$L_{\bullet}(V)=\bigoplus_{0\leq k\leq n}L_{k}(V):=\bigoplus_{0\leq k\leq
n}X_{1}\cdots X_{k}\epsilon_{k}(V).$

Here $\epsilon_{k}$ are the partial trivial idempotents of the finite Hecke
algebra. Each space may be considered as a module for the partially
symmetrized positive DAHA,
$\epsilon_{k}\operatorname{\mathscr{D}}_{n}^{+}\epsilon_{k}.$ It will be
immediate to show (Theorem 3.5) that the map $V\rightarrow L_{\bullet}(V)$ is
indeed a functor. We show (Proposition 3.7) that in the case of the polynomial
representations $V_{\operatorname{\mathrm{pol}}}^{(n)}$ of DAHA that
$L_{\bullet}(V_{\operatorname{\mathrm{pol}}}^{(n)})$ is a
$\mathbb{B}_{q,t}^{(n)}$-module quotient of the restriction of the polynomial
representation of $\mathbb{B}_{q,t}$ to $\mathbb{B}_{q,t}^{(n)}$.

Afterwards, we will use stable-limits of the representations $L_{\bullet}(V)$
of $\mathbb{B}_{q,t}^{(n)}$ to build representations of $\mathbb{B}_{q,t}.$
This construction will require the input of an infinite family of
representations of DAHAs, $(V^{(n)})_{n\geq n_{0}}$, along with some
connecting maps, $\Pi^{(n)}:V^{(n+1)}\rightarrow V^{(n)}$, satisfying some
special assumptions. Most interestingly, we require that the following
relations holds:

$\Pi^{(n)}\pi^{(n+1)}T_{n}=\pi^{(n)}\Pi^{(n)}.$

This is the same relation used by Ion-Wu in their construction of the limit
Cherednik operators and is related to certain natural embeddings of the
extended affine symmetric groups
$\widetilde{\mathfrak{S}}_{n}\hookrightarrow\widetilde{\mathfrak{S}}_{n+1}.$
We call such families $C=\left((V^{(n)})_{n\geq n_{1}},(\Pi^{(n)})_{n\geq
n_{1}}\right)$ compatible and construct spaces
$\operatorname{\mathfrak{L}}_{k}(C)$ given by

$\operatorname{\mathfrak{L}}_{k}(C):=\lim_{\leftarrow}L_{k}(V^{(n)}).$

These are the stable-limits of the spaces $L_{k}(V^{(n)})$ with respect to the
maps $\Pi^{(n)}.$ Finally, we package together these spaces to form
$\operatorname{\mathfrak{L}}_{\bullet}(C)$ given as

$\operatorname{\mathfrak{L}}_{\bullet}(C):=\bigoplus_{k\geq
0}\operatorname{\mathfrak{L}}_{k}(C)$

which may be also thought of as the stable-limit of the
$\mathbb{B}_{q,t}^{(n)}$ modules $L_{\bullet}(V^{(n)}).$ We show (Theorem
3.12) that there is a natural action of $\mathbb{B}_{q,t}$ on
$\operatorname{\mathfrak{L}}_{\bullet}(C)$ determined by the
$\mathbb{B}_{q,t}^{(n)}$ module structures on $L_{\bullet}(V^{(n)}).$ This
construction is also functorial.

Lastly, we will use our construction of the functor
$C\rightarrow\operatorname{\mathfrak{L}}_{\bullet}(C)$ to define (Theorem 4.5)
a large family of $\mathbb{B}_{q,t}$ modules,
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda}))$,
indexed by partitions $\lambda.$ These representations in a sense extend the
Murnaghan-type representations of the positive elliptic Hall algebra
previously defined by the author [Wei23]. As such we call these the Murnaghan-
type representations of $\mathbb{B}_{q,t}.$ For $\lambda=\emptyset$,
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\emptyset}))$
recovers the polynomial representation of $\mathbb{B}_{q,t}.$

In a recent paper González-Gorsky-Simental [GGS23] defined an extension
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ of $\mathbb{B}_{q,t}$
containing certain additional $\Delta$-operators as well as a class of
representations of $\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ called
calibrated with special properties. Further, they construct a large class of
calibrated $\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ representations
built from certain posets with exceptional properties. The author conjectures
that the Murnaghan-type representations of $\mathbb{B}_{q,t}$,
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda}))$, have
extended actions by $\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ which are
calibrated. More generally, there should be a special set of conditions on a
compatible sequence $C$ which guarantees that
$\operatorname{\mathfrak{L}}_{\bullet}(C)$ has an extended action by
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ which is calibrated. The
author will return to these questions in the future.

###  1.1 Acknowledgements

The author would like to thank their advisor Monica Vazirani for all of her
help and guidance. The author would also to thank Nicolle González, Eugene
Gorsky, and José Simental for helpful conversations with the author regarding
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ and calibrated
representations. The author would also like to thank Daniel Orr for fruitful
conversations regarding the polynomial representation of $\mathbb{B}_{q,t}.$

##  2 Notation and Conventions

###  2.1 Stable-Limits

In this paper graded will mean $\mathbb{Z}_{\geq 0}$-graded for both vector
spaces and rings. We fix two indeterminants $q,t$ which are algebraically
independent over $\mathbb{Q}$ and consider all vector spaces to be over the
base field $\mathbb{Q}(q,t).$ We will write $\deg(v),\deg(r)$ for the degree
of either a homogeneous vector $v$ in a graded vector space or a homogeneous
element $r$ of a graded ring. For a graded vector space $V$ we will write
$V(d)$ for the degree $d\geq 0$ homogeneous component of $V.$ If $R$ is a
graded ring then we will write $R-\operatorname{\mathrm{Mod}}$ for the
category of consisting of graded left $R$ modules as the objects and with
degree-preserving homomorphisms (homogeneous maps) as the morphisms.

We now review some formalities regarding stable-limits of spaces and modules.

######  Definition 2.1.

Let $(V^{(n)})_{n\geq n_{0}}$ be a sequence of graded vector spaces and
suppose $(\Pi^{(n)}:V^{(n+1)}\rightarrow V^{(n)})_{n\geq n_{0}}$ is a family
of degree preserving maps. The stable-limit of the spaces $(V^{(n)})_{n\geq
n_{0}}$ with respect to the maps $(\Pi^{(n)})_{n\geq n_{0}}$ is the graded
vector space $\widetilde{V}:=\lim_{\leftarrow}V^{(n)}$ constructed as follows:
For each $d\geq 0$ we define

$\widetilde{V}(d):=\\{(v_{n})_{n\geq n_{0}}\in\prod_{n\geq
n_{0}}V^{(n)}(d)\leavevmode\nobreak\ |\leavevmode\nobreak\
\Pi^{(n)}(v_{n+1})=v_{n}\\}$

and set

$\widetilde{V}:=\bigoplus_{d\geq 0}\widetilde{V}(d).$

######  Lemma 2.2.

Let $(R^{(n)})_{n\geq n_{0}}$ be a sequence of graded rings with injective
graded ring homomorphisms $(\iota^{(n)}:R^{(n)}\rightarrow R^{(n+1)})_{n\geq
n_{0}}$. We will identify $R^{(n)}$ with its image $\iota_{n}(R^{(n)})\subset
R^{(n+1)}.$ We write $\widetilde{R}=\lim_{\rightarrow}R^{(n)}$ for the direct
limit of the rings $R^{(n)}.$ Suppose $(V^{(n)})_{n\geq n_{0}}$ is a sequence
of graded vector spaces with each $V^{(n)}$ a graded $R^{(n)}$ module and
$(\Pi^{(n)}:V^{(n+1)}\rightarrow V^{(n)})_{n\geq n_{0}}$ a sequence of degree-
preserving maps with each $\Pi^{(n)}$ a graded $R^{(n)}$ module homomorphism.
Then the following defines a graded $\widetilde{R}$ module structure on
$\widetilde{V}:=\lim_{\leftarrow}V^{(n)}$: For $r\in\widetilde{R}$ and
$v\in\widetilde{V}$ with $r\in R_{N}$ and $v=(v_{n})_{n\geq n_{0}}$, define
$r(v)\in\widetilde{V}$ by

$r(v)=\left(\Pi^{(n_{0})}\cdots\Pi^{(N-1)}(r(v_{N})),\ldots,\Pi^{(N-1)}(r(v_{N})),r(v_{N}),r(v_{N+1}),r(v_{N+2}),\ldots\right).$

######  Remark 1.

It is a straightforward exercise to check that the action defined above
actually yields a graded $\widetilde{R}$ module structure on
$\lim_{\leftarrow}V^{(n)}.$ We leave this to the reader. We call
$\lim_{\leftarrow}V^{(n)}$ the stable-limit module corresponding to the
sequence $(V^{(n)})_{n\geq n_{0}}$ and the maps $(\Pi^{(n)})_{n\geq n_{0}}.$
Notice that this construction is functorial. Suppose $(W^{(n)})_{n\geq n_{0}}$
is another sequence of graded vector spaces with each $W^{(n)}$ a graded
$R^{(n)}$ module and $(\Psi^{(n)}:W^{(n+1)}\rightarrow W^{(n)})_{n\geq n_{0}}$
a sequence of degree-preserving maps with each $\Psi^{(n)}$ a graded $R^{(n)}$
module homomorphism and $\phi=(\phi^{(n)})_{n\geq n_{0}}$ is a family of
graded $R^{(n)}$ module maps $\phi^{(n)}:V^{(n)}\rightarrow W^{(n)}$ such that
for all $n\geq n_{0}$

$\phi^{(n)}\Pi^{(n)}=\Psi^{(n)}\phi^{(n+1)}.$

Then $\phi$ determines a graded $\widetilde{R}$ module homomorphism
$\widetilde{\phi}:\lim_{\leftarrow}V^{(n)}\rightarrow\lim_{\leftarrow}W^{(n)}$
given by

$\widetilde{\phi}(v):=(\phi^{(n)}(v_{n}))_{n\geq n_{0}}.$

######  Remark 2.

The stable-limit spaces $\widetilde{V}=\lim_{\leftarrow}V^{(n)}$ may be zero
even if each $V^{(n)}$ is nonzero. However, if each $V^{(n)}$ is nonzero and
the maps $\Pi^{(n)}$ are surjective then $\widetilde{V}$ is nonzero.

###  2.2 Positive Double Affine Hecke Algebras

######  Definition 2.3.

Define the positive double affine Hecke algebra
$\operatorname{\mathscr{D}}_{n}^{+}$ to be the $\mathbb{Q}(q,t)$-algebra
generated by $T_{1},\ldots,T_{n-1}$, $X_{1},\ldots,X_{n}$, and
$Y_{1},\ldots,Y_{n}$ with the following relations:

$(T_{i}-1)(T_{i}+q)=0$,

$T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}$,

$T_{i}T_{j}=T_{j}T_{i}$, $|i-j|>1$,

$T_{i}^{-1}X_{i}T_{i}^{-1}=q^{-1}X_{i+1}$,

$T_{i}X_{j}=X_{j}T_{i}$, $j\notin\\{i,i+1\\}$,

$X_{i}X_{j}=X_{j}X_{i}$,

$T_{i}Y_{i}T_{i}=qY_{i+1}$,

$T_{i}Y_{j}=Y_{j}T_{i}$, $j\notin\\{i,i+1\\}$,

$Y_{i}Y_{j}=Y_{j}Y_{i}$,

$Y_{1}T_{1}X_{1}=X_{2}Y_{1}T_{1}$,

$Y_{1}X_{1}\cdots X_{n}=tX_{1}\cdots X_{n}Y_{1}.$

We will write $\operatorname{\mathscr{A}}_{n}^{X}$ for the subalgebra of
$\operatorname{\mathscr{D}}_{n}^{+}$ generated by
$T_{1},\ldots,T_{n-1},X_{1},\ldots,X_{n}$ and $\operatorname{\mathscr{A}}_{n}$
for the subalgebra generated by $T_{1},\ldots,T_{n-1},Y_{1}\ldots,Y_{n}.$
Further, we write $\operatorname{\mathscr{H}}_{n}$ for the finite Hecke
algebra which is the subalgebra of $\operatorname{\mathscr{D}}_{n}^{+}$
generated by $T_{1},\ldots,T_{n-1}.$

We define the special elements $\pi:=q^{-n}Y_{1}T_{1}\cdots T_{n-1}$ and
$\widetilde{\pi}:=X_{1}T_{1}^{-1}\cdots T_{n-1}^{-1}.$ Further, we define for
$0\leq k\leq n$ the element $\epsilon_{k}$ by

$\epsilon_{k}:=\frac{1}{[n-k]_{q}!}\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}}q^{{n\choose
2}-\ell(\sigma)}T_{\sigma}$

where $T_{\sigma}:=T_{i_{1}}\cdots T_{i_{r}}$ whenever $\sigma=s_{i_{1}}\cdots
s_{i_{r}}$ is a reduced expression for $\sigma$, $\mathfrak{S}_{(1^{k},n-k)}$
is the Young subgroup of the symmetric group $\mathfrak{S}_{n}$ corresponding
to the composition $(1^{k},n-k)$, and
$[m]_{q}!:=\prod_{i=1}^{m}\left(\frac{1-q^{i}}{1-q}\right).$

We will consider $\operatorname{\mathscr{D}}_{n}^{+}$ as a graded algebra with

  * •

$\deg(T_{i})=\deg(Y_{i})=0$

  * •

$\deg(X_{i})=1.$

######  Remark 3.

It is straightforward to check the following additional relations which are
all standard in DAHA theory. We will require all of these relations later in
this paper. For the element $\pi$ we have:

  * •

$\pi X_{i}=X_{i+1}\pi$ for $1\leq i\leq n-1$

  * •

$\pi T_{i}=T_{i+1}\pi$ for $1\leq i\leq n-1$

  * •

$\pi^{2}T_{n-1}=T_{1}\pi^{2}.$

The elements $\epsilon_{k}$ are the partial trivial idempotents. They satisfy
the relations:

  * •

$\epsilon_{k}^{2}=\epsilon_{k}$

  * •

$\epsilon_{k}T_{i}=T_{i}\epsilon_{k}=\epsilon_{k}$ for $k+1\leq i\leq n-1$

  * •

$T_{i}\epsilon_{k}=\epsilon_{k}T_{i}$ for $1\leq i\leq k-1$

  * •

$\epsilon_{k}=\frac{1}{[n-k]_{q}!}\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}}q^{\ell(\sigma)}T_{\sigma}^{-1}$

  * •

$\epsilon_{k}=\left(\frac{1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1}}{1+q+\ldots+q^{n-k-1}}\right)\epsilon_{k+1}$

  * •

$\epsilon_{k}\epsilon_{\ell}=\epsilon_{\min(k,\ell)}.$

We have that the important element $\widetilde{\pi}:=X_{1}T_{1}^{-1}\cdots
T_{n-1}^{-1}$ satisfies the relations:

  * •

$\widetilde{\pi}Y_{i}=Y_{i+1}\widetilde{\pi}$ for $1\leq i\leq n-1$

  * •

$\widetilde{\pi}tY_{n}=Y_{1}\widetilde{\pi}$

  * •

$\widetilde{\pi}T_{i}=T_{i+1}\widetilde{\pi}$ for $1\leq i\leq n-2$

  * •

$\widetilde{\pi}^{2}T_{n-1}=T_{1}\widetilde{\pi}^{2}.$

Note that the last two of these relations only depend of the structure of the
subalgebra $\operatorname{\mathscr{A}}_{n}^{X}$ of
$\operatorname{\mathscr{D}}_{n}^{+}$ and thus hold more generally for all
$2\leq k\leq n$:

  * •

$(X_{1}T_{1}^{-1}\cdots T_{k-1}^{-1})T_{i}=T_{i+1}(X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1})$ for $1\leq i\leq k-2$

  * •

$(X_{1}T_{1}^{-1}\cdots T_{k-1}^{-1})^{2}T_{k-1}=T_{1}(X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1})^{2}.$

Lastly, we have the following expansion of the Jucys-Murphy elements of the
finite Hecke algebra into the standard $T_{\sigma}$ basis:

$q^{n-k}T_{k}^{-1}\cdots T_{n-1}^{-1}T_{n-1}^{-1}\cdots
T_{k}^{-1}=1+(q-1)(T_{k}^{-1}+qT_{k+1}^{-1}T_{k}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k}^{-1}).$

######  Remark 4.

When considering $\operatorname{\mathscr{D}}_{n}^{+}$ and
$\operatorname{\mathscr{D}}_{m}^{+}$ for $n\neq m$ we will write
$Y_{i}^{(n)},\pi^{(n)},\epsilon_{k}^{(n)}$ to help distinguish between the
distinct copies of $Y_{i},\pi,\epsilon_{k}$ in
$\operatorname{\mathscr{D}}_{n}^{+}$ and $\operatorname{\mathscr{D}}_{m}^{+}.$
However, since $\operatorname{\mathscr{A}}_{n}^{X}$ naturally includes into
$\operatorname{\mathscr{A}}_{m}^{X}$ for any $n\leq m$ we will identify the
copies of $X_{i}$ and $T_{i}$ in $\operatorname{\mathscr{A}}_{n}^{X}$ and
$\operatorname{\mathscr{A}}_{m}^{X}.$

######  Definition 2.4.

[Che03] The polynomial representation $V_{\operatorname{\mathrm{pol}}}^{(n)}$
of $\operatorname{\mathscr{D}}_{n}^{+}$ consists of the space
$\mathbb{Q}(q,t)[x_{1},\ldots,x_{n}]$ along with the following operators:

  * •

$T_{i}(f)=s_{i}(f)+(q-1)x_{i}\frac{f-s_{i}(f)}{x_{i}-x_{i+1}}$

  * •

$\pi(f(x_{1},\ldots,x_{n}))=f(x_{2},\ldots,x_{n},tx_{1})$

  * •

$X_{i}(f)=x_{i}f.$

It is a standard exercise in DAHA theory to verify that
$V_{\operatorname{\mathrm{pol}}}^{(n)}$ is a
$\operatorname{\mathscr{D}}_{n}^{+}$ module as defined above (see [Che03]).

###  2.3 The $\mathbb{B}_{q,t}$ Algebra

######  Definition 2.5.

[CGM20] The algebra $\mathbb{B}_{q,t}$ is generated by a collection of
orthogonal idempotents labelled by $\mathbb{Z}_{\geq 0}$, generators $d_{+}$,
$d_{-}$, $T_{i}$, and $z_{i}$ modulo relations:

$(T_{i}-1)(T_{i}+q)=0$

$T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}$

$T_{i}T_{j}=T_{j}T_{i}$ if $|i-j|>1$

$T_{i}^{-1}z_{i+1}T_{i}^{-1}=q^{-1}z_{i}$ for $1\leq i\leq k-1$

$z_{i}T_{j}=T_{j}z_{i}$ if $i\notin\\{j,j+1\\}$

$z_{i}z_{j}=z_{j}z_{i}$ for $1\leq i,j\leq k$

$d_{-}^{2}T_{k-1}=d_{-}^{2}$ for $k\geq 2$

$d_{-}T_{i}=T_{i}d_{-}$ for $1\leq i\leq k-2$

$T_{1}d_{+}^{2}=d_{+}^{2}$

$d_{+}T_{i}=T_{i+1}d_{+}$ for $1\leq i\leq k-1$

$q\varphi d_{-}=d_{-}\varphi T_{k-1}$ for $k\geq 2$

$T_{1}\varphi d_{+}=qd_{+}\varphi$ for $k\geq 1$

$z_{i}d_{-}=d_{-}z_{i}$

$d_{+}z_{i}=z_{i+1}d_{+}$

$z_{1}(qd_{+}d_{-}-d_{-}d_{+})=qt(d_{+}d_{-}-d_{-}d_{+})z_{k}$ for $k\geq 1$

where $\varphi:=\frac{1}{q-1}[d_{+},d_{-}].$

We will consider $\mathbb{B}_{q,t}$ as a graded algebra with grading
determined by

  * •

$\deg(T_{i})=\deg(z_{i})=\deg(d_{-})=0$

  * •

$\deg(d_{+})=1.$

For $n\geq 0$ define $\mathbb{B}_{q,t}^{(n)}$ to be the subalgebra of
$\mathbb{B}_{q,t}$ given by only considering $T_{i},z_{i},d_{-},d_{+}$ between
the idempotents labelled by $\\{0,\ldots,n\\}.$

######  Remark 5.

The graded algebras $\mathbb{B}_{q,t}^{(n)}$ naturally form a directed system
with

$\operatorname{\mathbb{B}_{q,t}}=\lim_{\rightarrow}\mathbb{B}_{q,t}^{(n)}.$

######  Definition 2.6.

[CGM20] Let $V^{\operatorname{\mathrm{pol}}}_{\bullet}=\bigoplus_{k\geq
0}V^{\operatorname{\mathrm{pol}}}_{k}:=\bigoplus_{k\geq
0}\mathbb{Q}(q,t)[y_{1},\ldots,y_{k}]\otimes\Lambda$ where
$\Lambda=\Lambda[X]$ is the ring of symmetric functions (over
$\mathbb{Q}(q,t)$) in a variable set $X=x_{1}+x_{2}+\ldots.$ Define an action
on $V^{\operatorname{\mathrm{pol}}}_{\bullet}$ by the following operators
given for $F\in V^{\operatorname{\mathrm{pol}}}_{k}$ by

  * •

$T_{i}F:=\frac{(q-1)y_{i+1}F+(y_{i+1}-qy_{i})s_{i}(F)}{y_{i+1}-y_{i}}$ for
$1\leq i\leq k-1$

  * •

$d_{+}F:=-T_{1}\cdots T_{k}F[X+(q-1)y_{k+1}]$

  * •

$d_{-}F:=F[X-(q-1)y_{k}]\operatorname{\mathrm{Exp}}[-y_{k}^{-1}X]|_{y_{k}^{-1}}$

  * •

$z_{k}F:=T_{k-1}\cdots
T_{1}F[X+(1-q)ty_{1}-(q-1)u,y_{2},\ldots,y_{k},u]\operatorname{\mathrm{Exp}}[u^{-1}ty_{1}-u^{-1}X]|_{u^{0}}$

  * •

$z_{i}:=q^{-1}T_{i}^{-1}z_{i+1}T_{i}^{-1}$ for $1\leq i\leq k-1$

where $\operatorname{\mathrm{Exp}}[X]:=\sum_{n\geq 0}h_{n}[X]$ is the
plethystic exponential, $|_{y_{k}^{-1}}$ represents taking the coefficient of
$y_{k}^{-1},$ and $s_{i}$ swaps the variables $y_{i},y_{i+1}$. Here we are
using plethystic notation. This representation
$V^{\operatorname{\mathrm{pol}}}_{\bullet}$ of $\mathbb{B}_{q,t}$ is called
the polynomial representation.

Note that the signs $\pm 1$ of the operators $d_{-},d_{+}$ are reversed in
[CGM20]. This choice is made to align with the conventions in [IW22] and
[Mel21] and does not make a substantial difference in the underlying
representation.

######  Remark 6.

Carlsson-Gorsky-Mellit also construct an action of $\mathbb{B}_{q,t}$ on the
larger space
$W^{\operatorname{\mathrm{pol}}}_{\bullet}:=\bigoplus_{k=0}^{\infty}(y_{1}\cdots
y_{k})^{-1}V^{\operatorname{\mathrm{pol}}}_{k}.$ The space
$V^{\operatorname{\mathrm{pol}}}_{\bullet}$ is isomorphic to the equivariant
$K$-theory of the parabolic flag Hilbert schemes of points in $\mathbb{C}^{2}$
and the larger space $W^{\operatorname{\mathrm{pol}}}_{\bullet}$ is defined in
order to relate the original $\mathbb{A}_{q,t}$ polynomial representation as
defined by Carlsson-Mellit [CM18] to the $\mathbb{A}_{q,t}$ polynomial
representation constructed in [CGM20]. We will use the space
$W^{\operatorname{\mathrm{pol}}}_{\bullet}$ briefly to relate the
$\mathbb{B}_{q,t}$ action on $V^{\operatorname{\mathrm{pol}}}_{\bullet}$ to
the work of Ion-Wu.

##  3 Main Construction

###  3.1 $\mathbb{B}_{q,t}^{(n)}$ Modules From
$\operatorname{\mathscr{D}}_{n}^{+}$

In this section we will take any graded $\operatorname{\mathscr{D}}_{n}^{+}$
module $V$ and construct a corresponding graded $\mathbb{B}_{q,t}^{(n)}$
module $L_{\bullet}(V).$ To do this we will first define the spaces which
constitute $L_{\bullet}(V).$

######  Definition 3.1.

For any graded $\operatorname{\mathscr{D}}_{n}^{+}$ module $V$ and $0\leq
k\leq n$ define the space $L_{k}=L_{k}(V)$ as $L_{k}:=X_{1}\cdots
X_{k}\epsilon_{k}(V).$ We let $L_{\bullet}=L_{\bullet}(V)$ denote the space
$L_{\bullet}:=\bigoplus_{0\leq k\leq n}L_{k}.$

For the remainder of this section let $V$ be a graded module for
$\operatorname{\mathscr{D}}_{n}^{+}.$ We are going to now construct operators
$T_{i},z_{i},d_{+},d_{-}$ on $L_{\bullet}$ which we will show generate a
representation of $\mathbb{B}_{q,t}^{(n)}.$

######  Definition 3.2.

Define the operators

  * •

$T_{i}:L_{k}\rightarrow L_{k}$ for $1\leq i\leq k-1$

  * •

$z_{i}:L_{k}\rightarrow L_{k}$ for $1\leq i\leq k$

  * •

$d_{+}:L_{k}\rightarrow L_{k+1}$ for $0\leq k\leq n-1$

  * •

$d_{-}:L_{k}\rightarrow L_{k-1}$ for $1\leq k\leq n$

as follows:

  * •

$T_{i}(v)$ is defined by the action of $T_{i}$ on $V$

  * •

$z_{i}(v):=Y_{i}(v)$ as defined by the action of $Y_{i}$ on $V$

  * •

$d_{+}(v):=-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}v$

  * •

$d_{-}(v):=(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})(v).$

It is not immediately obvious that these operators are well defined i.e. that
their ranges are correctly specified above. We show this now.

######  Lemma 3.3.

If $v\in L_{k}$ then $T_{i}(v),z_{j}(v)\in L_{k}$ for all $1\leq i\leq k-1$
and $1\leq j\leq k.$ If $k\leq n-1$ then $d_{+}(v)\in L_{k+1}$ and if $1\leq
k$ then $d_{-}(v)\in L_{k-1}.$

###### Proof.

Let $v\in L_{k}$ say, $v=X_{1}\cdots X_{k}\epsilon_{k}(w)$ for $w\in V.$ First
we have,

$\displaystyle T_{i}(v)$ $\displaystyle=T_{i}X_{1}\cdots X_{k}\epsilon_{k}(w)$
$\displaystyle=X_{1}\cdots X_{k}\epsilon_{k}(T_{i}w)\in L_{k}.$

Next we have

$\displaystyle z_{1}(v)$ $\displaystyle=Y_{1}X_{1}\cdots X_{k}\epsilon_{k}(w)$
$\displaystyle=q^{n}\pi T_{n-1}^{-1}\cdots T_{1}^{-1}X_{1}\cdots
X_{k}\epsilon_{k}(w)$ $\displaystyle=q^{n}\pi T_{n-1}^{-1}\cdots
T_{k}^{-1}X_{1}\cdots X_{k}T_{k-1}^{-1}\cdots T_{1}^{-1}\epsilon_{k}(w)$
$\displaystyle=q^{n}\pi X_{1}\cdots X_{k-1}T_{n-1}^{-1}\cdots
T_{k}^{-1}X_{k}T_{k-1}^{-1}\cdots T_{1}^{-1}\epsilon_{k}(w)$
$\displaystyle=q^{n}X_{2}\cdots X_{k}\pi T_{n-1}^{-1}\cdots
T_{k}^{-1}X_{k}T_{k-1}^{-1}\cdots T_{1}^{-1}\epsilon_{k}(w)$
$\displaystyle=q^{n}X_{2}\cdots X_{k}tq^{-(n-k)}X_{1}\pi T_{n-1}\cdots
T_{k}T_{k-1}^{-1}\cdots T_{1}^{-1}\epsilon_{k}(w)$
$\displaystyle=tq^{k}X_{1}\cdots X_{k}\pi T_{n-1}\cdots
T_{k}T_{k-1}^{-1}\cdots T_{1}^{-1}\epsilon_{k}(w)$ $\displaystyle=X_{1}\cdots
X_{k}\pi T_{n-1}\cdots T_{k}\epsilon_{k}(tq^{k}T_{k-1}^{-1}\cdots
T_{1}^{-1}w).$

Now for all $k<i\leq n-1$ we have

$\displaystyle T_{i}\pi T_{n-1}\cdots T_{k}$ $\displaystyle=\pi
T_{i-1}T_{n-1}\cdots T_{k}$ $\displaystyle=\pi T_{n-1}\cdots
T_{i+1}T_{i-1}T_{i}T_{i-1}T_{i-2}\cdots T_{k}$ $\displaystyle=\pi
T_{n-1}\cdots T_{i+1}T_{i}T_{i-1}T_{i}T_{i-2}\cdots T_{k}$ $\displaystyle=\pi
T_{n-1}\cdots T_{k}T_{i}.$

Therefore, we have

$\displaystyle X_{1}\cdots X_{k}\pi T_{n-1}\cdots
T_{k}\epsilon_{k}(tq^{k}T_{k-1}^{-1}\cdots T_{1}^{-1}w)$
$\displaystyle=X_{1}\cdots X_{k}\epsilon_{k}(tq^{k}\pi T_{n-1}\cdots
T_{k}T_{k-1}^{-1}\cdots T_{1}^{-1}w)$

which is clearly in $L_{k}.$

Now for any $1<i\leq k$ since $Y_{i}=q^{-1}T_{i-1}Y_{i-1}T_{i-1}$ we see that

$Y_{i}=q^{-i+1}T_{i-1}\cdots T_{1}Y_{1}T_{1}\cdots T_{i-1}$

and so

$z_{i}=q^{-i+1}T_{i-1}\cdots T_{1}Y_{1}T_{1}\cdots T_{i-1}.$

Since $T_{1}\cdots T_{i-1}v\in L_{k}$ we see that $Y_{1}(T_{1}\cdots
T_{i-1}v)\in L_{k}$ as well and so

$z_{i}(v)=q^{-i+1}T_{i-1}\cdots T_{1}Y_{1}T_{1}\cdots
T_{i-1}(v)=q^{-i+1}T_{i-1}\cdots T_{1}Y_{1}(T_{1}\cdots T_{i-1}v)\in L_{k}.$

We now look at $d_{+}.$ We find that

$\displaystyle d_{+}(v)=$ $\displaystyle=-q^{k}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1}(v)$ $\displaystyle=-q^{k}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1}(X_{1}\cdots X_{k}\epsilon_{k}(w))$ $\displaystyle=-T_{1}\cdots
T_{k}X_{k+1}(X_{1}\cdots X_{k}\epsilon_{k}(w))$ $\displaystyle=-T_{1}\cdots
T_{k}X_{1}\cdots X_{k+1}\epsilon_{k}(w)$ $\displaystyle=-X_{1}\cdots
X_{k+1}(T_{1}\cdots T_{k}\epsilon_{k}(w))$ $\displaystyle=-X_{1}\cdots
X_{k+1}(T_{1}\cdots T_{k}\epsilon_{k+1}\epsilon_{k}(w))$
$\displaystyle=X_{1}\cdots X_{k+1}\epsilon_{k+1}(-T_{1}\cdots
T_{k}\epsilon_{k}(w))\in L_{k+1}.$

Lastly, we look at $d_{-}.$ We suppose $v\in L_{k+1}$ say, $v=X_{1}\cdots
X_{k+1}\epsilon_{k+1}(w)$ for $w\in V.$ We get that

$\displaystyle d_{-}(v)$
$\displaystyle=(1-q)(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1})(v)$
$\displaystyle=(1-q)(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1})(X_{1}\cdots X_{k+1}\epsilon_{k+1}(w))$
$\displaystyle=(1-q)(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1})(\epsilon_{k+1}(X_{1}\cdots X_{k+1}w))$
$\displaystyle=(1-q)(1+q+\ldots+q^{n-k-1})\epsilon_{k}(X_{1}\cdots X_{k+1}w)$
$\displaystyle=X_{1}\cdots X_{k}\epsilon_{k}((1-q^{n-k})X_{k+1}w)\in L_{k}.$

∎

Now we will show that the collection of operators $T_{i},z_{j},d_{-},d_{+}$
acting on the space $L_{\bullet}$ generates an action of $\mathbb{B}_{q,t}.$

######  Proposition 3.4.

$L_{\bullet}$ is a $\mathbb{B}_{q,t}^{(n)}$-module.

###### Proof.

We will show that the operators $T_{i},z_{j},d_{-},d_{+}$ on $L_{\bullet}$
defined in Definition 3.2 satisfy the relations in Definition 2.5. Note first
that the relations involving only $T_{i}$’s and $z_{j}$’s follow immediately
from their definition and the fact that $V$ is a
$\operatorname{\mathscr{D}}^{+}_{n}$-module.

We will start by verifying the relations between $d_{+}$ and the $T_{i}.$ We
will for the remainder of this proof let $v\in L_{k}$ and specify various
conditions on $k$ as needed. Suppose $0\leq k\leq n-1.$ Then for $1\leq i\leq
k-1$ using the braid relations we see

$\displaystyle d_{+}T_{i}(v)$ $\displaystyle=-q^{k}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1}T_{i}(v)$ $\displaystyle=-q^{k}X_{1}T_{i+1}T_{1}^{-1}\cdots
T_{k}^{-1}(v)$ $\displaystyle=T_{i+1}(-q^{k}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1})(v)$ $\displaystyle=T_{i+1}d_{+}(v).$

Now if $0\leq k\leq n-2$ we see from the braid relations and the fact that
$T_{k+1}(v)=v$

$\displaystyle T_{1}^{-1}d_{+}^{2}(v)$
$\displaystyle=T_{1}^{-1}d_{+}(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(v))$
$\displaystyle=T_{1}^{-1}(-q^{k+1}X_{1}T_{1}^{-1}\cdots
T_{k+1}^{-1})(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(v))$
$\displaystyle=q^{2k+1}T_{1}^{-1}X_{1}T_{1}^{-1}T_{2}^{-1}\cdots
T_{k+1}^{-1}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=q^{2k}X_{2}T_{2}^{-1}\cdots T_{k+1}^{-1}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1}(v)$ $\displaystyle=q^{2k}X_{1}X_{2}T_{2}^{-1}\cdots
T_{k+1}^{-1}T_{1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=q^{2k}X_{1}X_{2}T_{2}^{-1}\cdots T_{k+1}^{-1}T_{1}^{-1}\cdots
T_{k}^{-1}T_{k+1}(v)$ $\displaystyle=q^{2k}X_{1}X_{2}T_{1}T_{2}^{-1}\cdots
T_{k+1}^{-1}T_{1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=q^{2k+1}X_{1}T_{1}^{-1}X_{1}T_{2}^{-1}\cdots
T_{k+1}^{-1}T_{1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=q^{2k+1}X_{1}T_{1}^{-1}T_{2}^{-1}\cdots
T_{k+1}^{-1}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=(-q^{k+1}X_{1}T_{1}^{-1}\cdots
T_{k+1}^{-1})(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(v))$
$\displaystyle=d_{+}^{2}(v).$

We will now show that $d_{+}z_{i}=z_{i+1}d_{+}.$ Suppose $1\leq i\leq k\leq
n-1.$ Then we have by using Remark 3

$\displaystyle z_{i+1}d_{+}(v)$
$\displaystyle=-q^{k}Y_{i+1}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=-q^{k}Y_{i+1}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}T_{k+1}^{-1}\cdots
T_{n-1}^{-1}(v)$ $\displaystyle=-q^{k}Y_{i+1}(X_{1}T_{1}^{-1}\cdots
T_{n-1}^{-1})(v)$ $\displaystyle=-q^{k}(X_{1}T_{1}^{-1}\cdots
T_{n-1}^{-1})Y_{i}(v)$ $\displaystyle=-q^{k}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1}Y_{i}T_{k+1}^{-1}\cdots T_{n-1}^{-1}(v)$
$\displaystyle=-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}Y_{i}(v)$
$\displaystyle=d_{+}(z_{i}(v)).$

Next we note that the relations between just $d_{-}$ and the $T_{i}$ follow
trivially from the fact that $d_{-}:L_{k+1}\rightarrow L_{k}$ is a scalar
multiple of $\epsilon_{k}|_{L_{k+1}}$ which follows from the relations (see
Remark 3). Further, the relation $z_{i}d_{-}=d_{-}z_{i}$ also follows easily
from the fact that $Y_{i}T_{j}=T_{j}Y_{i}$ for $i\notin\\{j,j+1\\}.$

Now we are left to show that the relations involving
$\varphi:=\frac{1}{q-1}[d_{+},d_{-}]$ hold. Notice that $\varphi$ may be
computed for $1\leq k\leq n-1$ as

$\displaystyle(q-1)\varphi(v)$ $\displaystyle=[d_{+},d_{-}](v)$
$\displaystyle=(d_{+}d_{-}-d_{-}d_{+})(v)$
$\displaystyle=d_{+}((1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})(v))-d_{-}(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}v)$
$\displaystyle=(1-q)(-q^{k-1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1})(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots T_{k}^{-1})(v)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-(1-q)(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1})(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}v)$
$\displaystyle=(q-1)q^{k-1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}\left((1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})-q(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1})T_{k}^{-1}\right)$
$\displaystyle=(q-1)q^{k-1}X_{1}T_{1}^{-1}\cdots T_{k-1}^{-1}$

so that

$\varphi(v)=q^{k-1}X_{1}T_{1}^{-1}\cdots T_{k-1}^{-1}(v).$

Let $2\leq k\leq n.$ Then

$\displaystyle q\varphi d_{-}(v)$
$\displaystyle=q\varphi(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})v$ $\displaystyle=q(1-q)q^{k-2}X_{1}T_{1}^{-1}\cdots
T_{k-2}^{-1}(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots T_{k}^{-1})v$
$\displaystyle=(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})q^{k-1}X_{1}T_{1}^{-1}\cdots T_{k-2}^{-1}(v)$
$\displaystyle=d_{-}\varphi T_{k-1}(v).$

Let us now show that $T_{1}\varphi d_{+}=qd_{+}\varphi.$ Suppose $1\leq k\leq
n-2.$ Then

$\displaystyle T_{1}\varphi d_{+}(v)$ $\displaystyle
T_{1}\varphi(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1})(v)$
$\displaystyle=T_{1}(q^{k}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1})(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1})(v)$
$\displaystyle=-q^{2k}T_{1}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=-q^{2k}T_{1}(XT_{1}^{-1}\cdots T_{k}^{-1})^{2}(v)$
$\displaystyle=-q^{2k}(X_{1}T_{1}^{-1}\cdots T_{k}^{-1})^{2}T_{k}(v)$
$\displaystyle=-q^{2k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}(v)$ $\displaystyle=q(-q^{k}X_{1}T_{1}^{-1}\cdots
T_{k}^{-1})(q^{k-1}X_{1}T_{1}^{-1}\cdots T_{k-1}^{-1})(v)$
$\displaystyle=qd_{+}\varphi(v).$

Lastly, we show that
$z_{1}(qd_{+}d_{-}-d_{-}d_{+})=qt(d_{+}d_{-}-d_{-}d_{+})z_{k}.$ Take $1\leq
k\leq n-1.$ Then we find

$\displaystyle z_{1}(qd_{+}d_{-}-d_{-}d_{+})$
$\displaystyle=z_{1}(qd_{+}d_{-}(v)-d_{-}d_{+}(v))$
$\displaystyle=z_{1}\left(qd_{+}(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})(v)-d_{-}(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1})(v)\right)$
$\displaystyle=Y_{1}q(1-q)(-q^{k-1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1})(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots T_{k}^{-1})(v)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-Y_{1}(1-q)(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1})(-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1})(v)$
$\displaystyle=(q-1)q^{k}Y_{1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}\left(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1}-(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1}T_{k}^{-1})\right)(v)$
$\displaystyle=(q-1)q^{k}Y_{1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}\left(1+(q-1)(T_{k}^{-1}+qT_{k+1}^{-1}T_{k}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k}^{-1})\right)(v)$ $\displaystyle=(q-1)q^{k}Y_{1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}\left(q^{n-k}T_{k}^{-1}\cdots T_{n-1}^{-1}T_{n-1}^{-1}\cdots
T_{k}^{-1}\right)(v)$
$\displaystyle=q^{n}(q-1)Y_{1}\widetilde{\pi}T_{n-1}^{-1}\cdots T_{k}^{-1}(v)$
$\displaystyle=q^{n}(q-1)\widetilde{\pi}(tY_{n})T_{n-1}^{-1}\cdots
T_{k}^{-1}(v)$ $\displaystyle=tq^{n}(q-1)X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}(T_{k}^{-1}\cdots T_{n-1}^{-1}Y_{n}T_{n-1}^{-1}\cdots
T_{k}^{-1})(v)$ $\displaystyle=tq^{n}(q-1)X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}(q^{-(n-k)}Y_{k})(v)$
$\displaystyle=tq^{k}(q-1)X_{1}T_{1}^{-1}\cdots T_{k-1}^{-1}Y_{k}(v)$
$\displaystyle=qt(q-1)\left(q^{k-1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1}\right)Y_{k}(v)$ $\displaystyle=qt(q-1)\varphi z_{k}(v)$
$\displaystyle=qt[d_{+},d_{-}]z_{k}(v).$

∎

######  Corollary 3.5.

The map $W\rightarrow L_{\bullet}(W)$ is a covariant functor
$\operatorname{\mathscr{D}}_{n}^{+}-\operatorname{\mathrm{Mod}}\rightarrow\mathbb{B}_{q,t}^{(n)}-\operatorname{\mathrm{Mod}}.$

###### Proof.

Suppose $\phi:U\rightarrow W$ is a homogeneous
$\operatorname{\mathscr{D}}_{n}^{+}$-module map. Now for any $0\leq k\leq n$
we see that if $v=X_{1}\cdots X_{k}\epsilon_{k}(u)\in L_{k}(U)$ then

$\phi(v)=\phi(X_{1}\cdots X_{k}\epsilon_{k}(u))=X_{1}\cdots
X_{k}\epsilon_{k}(\phi(u))\in L_{k}(W).$

Thus $\phi$ yields a map $\phi_{\bullet}:L_{\bullet}(U)\rightarrow
L_{\bullet}(W)$ given by restricting $\phi$ to each of the subspaces
$L_{k}(U)\subset U.$ From Definition 3.2 we see that each of the operators
$T_{i},z_{i},d_{-},d_{+}$ is expressed entirely in terms of the action of
$\operatorname{\mathscr{D}}_{n}^{+}$ on $U$ and as such we conclude that
$\phi_{\bullet}$ is a $\mathbb{B}_{q,t}^{(n)}$ module map. ∎

###  3.2 The Polynomial Case

The goal of this section is to relate the $\mathbb{B}_{q,t}^{(n)}$ modules
$L_{\bullet}(W)$ constructed above to the polynomial representation
$V^{\operatorname{\mathrm{pol}}}_{\bullet}$ of $\mathbb{B}_{q,t}$ in the case
when $W=V_{\operatorname{\mathrm{pol}}}^{(n)}.$ We will show that there are
natural maps $x_{1}\cdots
x_{k}\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda\rightarrow x_{1}\cdots
x_{k}\mathbb{Q}(q,t)[x_{1},\ldots,x_{n}]^{\mathfrak{S}_{(1^{k},n-k)}}$ which
are $\mathbb{B}_{q,t}^{(n)}$ module projections. This is nontrivial since the
definitions of $z_{i}$ and $d_{-}$ are quite different in both modules. We
will use the work of Ion-Wu to bridge this gap.

######  Definition 3.6.

[IW22] Let
$\mathscr{P}_{as}^{+}:=\mathbb{Q}(q,t)[x_{1},x_{2},\ldots]\otimes\Lambda$
denote the space of almost symmetric functions with distinguished subspaces
$\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda$ of functions which are
symmetric after k. We will write $\mathfrak{X}_{k}:=x_{k+1}+x_{k+2}+\ldots.$
Let
$\Xi^{(n)}:\mathscr{P}_{as}^{+}\rightarrow\mathbb{Q}(q,t)[x_{1},\ldots,x_{n}]$
be the projection homomorphism defined by $\Xi^{(n)}(x_{n+i})=0$ for all
$i\geq 0.$ By abuse of notation we will also denote by $\Xi^{(n)}$ any
projection map defined on polynomials defined by restricting $\Xi^{(n)}$ to
any subspace of $\mathscr{P}_{as}^{+}$ notably including
$\mathbb{Q}(q,t)[x_{1},\cdots,x_{n},x_{n+1}]$ and $x_{1}\cdots
x_{k}\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda.$ Let
$\mathscr{P}_{\bullet}:=\bigoplus_{k\geq
0}\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda.$ Consider the following
operators given on $f\in\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda$ as
follows:

  * •

$T_{i}(f):=s_{i}(f)+(1-q)x_{i}\frac{f-s_{i}(f)}{x_{i}-x_{i+1}}$

  * •

$d_{+}(f):=-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}(f)$

  * •

$d_{-}(x_{1}^{a_{1}}\cdots
x_{k+1}^{a_{k+1}}F[\mathfrak{X}_{k+1}]):=x_{1}^{a_{1}}\cdots
x_{k}^{a_{k}}F[\mathfrak{X}_{k}-u^{-1}]\operatorname{\mathrm{Exp}}[(1-q)u\mathfrak{X}_{k}]\leavevmode\nobreak\
|_{u^{a_{k+1}}}$

  * •

$z_{i}(f)=\mathscr{Y}_{i}(f):=\lim_{m}\widetilde{Y}_{i}^{(m)}\Xi^{(m)}(f)$

where $\mathscr{Y}_{i}$ are the limit Cherednik operators, $\lim_{m}$ is the
limit as defined by Ion-Wu (with $q$ and $t$ swapped) and
$\widetilde{Y}_{i}^{(m)}$ are the deformed Cherednik operators.

We can use the work of Ion-Wu to relate the above $\mathbb{B}_{q,t}$ module
$\mathscr{P}_{\bullet}$ to the $\mathbb{B}_{q,t}$ module
$W^{\operatorname{\mathrm{pol}}}_{\bullet}$ defined by Carlsson-Gorsky-Mellit
as follows.

######  Theorem.

[IW22] The maps $T_{i},d_{+},d_{-},z_{i}$ on $\mathscr{P}_{\bullet}$ define a
representation of $\mathbb{B}_{q,t}.$ This representation is isomorphic to the
$\mathbb{B}_{q,t}$ representation on
$W^{\operatorname{\mathrm{pol}}}_{\bullet}:=\bigoplus_{k\geq 0}(y_{1}\cdots
y_{k})^{-1}V^{\operatorname{\mathrm{pol}}}_{k}$ defined by Carlsson-Gorsky-
Mellit via the map $\Phi_{\bullet}=\bigoplus_{k\geq 0}\Phi_{k}$ defined by

$\Phi_{k}(x_{1}^{a_{1}}\cdots
x_{k}^{a_{k}}F[\mathfrak{X}_{k}]):=y_{1}^{a_{1}-1}\cdots
y_{k}^{a_{k}-1}F\left[\frac{X}{q-1}\right].$

######  Remark 7.

Ion-Wu in their paper also construct the additional operator $d_{+}^{*}$ on
$\mathscr{P}_{\bullet}$ from which they obtain an action of $\mathbb{A}_{q,t}$
on $\mathscr{P}_{\bullet}$. Further, they show that this $\mathbb{A}_{q,t}$
module is isomorphic to the standard $\mathbb{A}_{q,t}$ representation as
defined by Mellit [Mel21] which is the same as the Carlsson-Gorsky-Mellit
action of $\mathbb{A}_{q,t}$ on $W_{\bullet}^{\operatorname{\mathrm{pol}}}.$
The result as stated above is thus a strictly weaker result than the main
theorem of Ion-Wu but as we are only interested in the subalgebra
$\mathbb{B}_{q,t}$ of $\mathbb{A}_{q,t}$, we will only require the above
result as stated.

By the above theorem of Ion-Wu we find that each of the spaces $y_{1}\cdots
y_{k}W_{k}^{\operatorname{\mathrm{pol}}}=V_{k}^{\operatorname{\mathrm{pol}}}$
gets mapped by $\Phi_{k}^{-1}$ to the space $x_{1}\cdots
x_{k}\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda$ which we will call
$L_{k}^{\operatorname{\mathrm{pol}}}.$ Thus we see that $\mathbb{B}_{q,t}$
acts on the space $L_{\bullet}^{\operatorname{\mathrm{pol}}}:=\bigoplus_{k\geq
0}L_{k}^{\operatorname{\mathrm{pol}}}\subset\operatorname{\mathscr{P}}_{\bullet}.$
For all $n\geq 0$ can relate $L_{\bullet}^{\operatorname{\mathrm{pol}}}$ to
$L_{\bullet}(V_{\operatorname{\mathrm{pol}}}^{(n)})$ in the following way:

######  Proposition 3.7.

The map
$\Xi_{\bullet}^{(n)}:L_{\bullet}^{\operatorname{\mathrm{pol}}}\rightarrow
L_{\bullet}(V_{\operatorname{\mathrm{pol}}}^{(n)})$ defined component-wise by
$\Xi^{(n)}$ is a $\mathbb{B}_{q,t}^{(n)}$ module map.

###### Proof.

We need to show that $\Xi^{(n)}$ commutes with the operators
$T_{i},z_{i},d_{+},d_{-}$ as defined on both of the spaces
$L_{\bullet}^{\operatorname{\mathrm{pol}}}$ and
$L_{\bullet}(V_{\operatorname{\mathrm{pol}}}^{(n)})$ respectively. For $T_{i}$
and $d_{+}$ this is immediate. For $z_{i}$ we note that from the construction
of the deformed Cherednik operators $\widetilde{Y}_{i}^{(n)}$ [IW22] we have
that if $1\leq i\leq k$ then

$\widetilde{Y}_{i}^{(n)}X_{i}=Y_{i}^{(n)}X_{i}.$

Further, from Ion-Wu we also know that (using this paper’s conventions) for
all $n\geq i$

$Y_{i}^{(n)}X_{i}\leavevmode\nobreak\ \Xi^{(n)}=\Xi^{(n)}Y_{i}^{(n+1)}X_{i}.$

Thus for $f\in L_{k}^{\operatorname{\mathrm{pol}}}$ and $1\leq i\leq k$ we
find

$Y_{i}^{(n)}\Xi^{(n)}(f)=\Xi^{(n)}\mathscr{Y}_{i}(f)$

and so

$z_{i}\Xi^{(n)}=\Xi^{(n)}z_{i}.$

Let $0\leq k\leq n-1$ and $f=x_{1}\cdots x_{k+1}g$ for
$g\in\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}].$ From the author’s prior paper
[BW23], we know that

$\displaystyle d_{-}(f)$
$\displaystyle=\lim_{m}\left(\frac{1+qT_{k+1}^{-1}+\ldots+q^{m-k-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}}{1+q+\ldots+q^{m-k-1}}\right)\Xi^{(m)}(f)$
$\displaystyle=\left(\frac{1}{1+q+q^{2}+\ldots}\right)\lim_{m}\left(1+qT_{k+1}^{-1}+\ldots+q^{m-k-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}\right)\Xi^{(m)}(f)$
$\displaystyle=(1-q)\lim_{m}\left(1+qT_{k+1}^{-1}+\ldots+q^{m-k-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}\right)\Xi^{(m)}(f)$
$\displaystyle=(1-q)\lim_{m}\left(1+qT_{k+1}^{-1}+\ldots+q^{m-k-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}\right)X_{1}\cdots X_{k+1}\Xi^{(m)}(g).$

Now if $m\geq k$ then

$\displaystyle\Xi^{(m)}\left(1+qT_{k+1}^{-1}+\ldots+q^{m-k}T_{m}^{-1}\cdots
T_{k+1}^{-1}\right)X_{1}\cdots X_{k+1}$
$\displaystyle\Xi^{(m)}\left(1+qT_{k+1}^{-1}+\ldots+q^{m-k-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}\right)X_{1}\cdots X_{k+1}+\Xi^{(m)}q^{m-k}T_{m}^{-1}\cdots
T_{k+1}^{-1}X_{1}\cdots X_{k+1}$
$\displaystyle=\left(1+qT_{k+1}^{-1}+\ldots+q^{m-k-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}\right)X_{1}\cdots X_{k+1}\Xi^{(m)}+\Xi^{(m)}X_{m+1}T_{m}\cdots
T_{k+1}X_{1}\cdots X_{k}$
$\displaystyle=\left(1+qT_{k+1}^{-1}+\ldots+q^{m-k-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}\right)X_{1}\cdots X_{k+1}\Xi^{(m)}.$

Therefore,

$\displaystyle\Xi^{(n)}(d_{-}(f))$
$\displaystyle=(1-q)\left(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1}\right)X_{1}\cdots X_{k+1}\Xi^{(n)}(g)$
$\displaystyle=(1-q)\left(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1}\right)\Xi^{(n)}(X_{1}\cdots X_{k+1}g)$
$\displaystyle=(1-q)\left(1+qT_{k+1}^{-1}+\ldots+q^{n-k-1}T_{n-1}^{-1}\cdots
T_{k+1}^{-1}\right)\Xi^{(n)}(f)$ $\displaystyle=d_{-}\Xi^{(n)}(f).$

Thus $\Xi^{(n)}d_{-}=d_{-}\Xi^{(n)}$ and so $\Xi_{\bullet}^{(n)}$ is a
$\mathbb{B}_{q,t}^{(n)}$ module map. ∎

######  Remark 8.

Since $L_{\bullet}^{\operatorname{\mathrm{pol}}}$ is isomorphic as a
$\mathbb{B}_{q,t}$ module to $V_{\bullet}^{\operatorname{\mathrm{pol}}}$ via
the map $\Phi_{\bullet}$ and from Proposition 3.7 we know that
$\Xi^{(n)}:L_{\bullet}^{\operatorname{\mathrm{pol}}}\rightarrow
L_{\bullet}(V_{\operatorname{\mathrm{pol}}}^{(n)})$ is a
$\mathbb{B}_{q,t}^{(n)}$ module quotient, it follows that
$L_{\bullet}(V_{\operatorname{\mathrm{pol}}}^{(n)})$ is a
$\mathbb{B}_{q,t}^{(n)}$ module quotient of
$\operatorname{Res}^{\mathbb{B}_{q,t}}_{\mathbb{B}_{q,t}^{(n)}}V_{\bullet}^{\operatorname{\mathrm{pol}}}.$

###  3.3 $\mathbb{B}_{q,t}$ Modules From Compatible Sequences

We will now build representations for the full $\mathbb{B}_{q,t}$ algebra
given certain special families of DAHA representations.

######  Definition 3.8.

Let $C=\left((V^{(n)})_{n\geq n_{1}},(\Pi^{(n)})_{n\geq n_{1}}\right)$ be a
collection of $\mathbb{Q}(q,t)$-vector spaces and maps
$\Pi^{(n)}:V^{(n+1)}\rightarrow V^{(n)}$ with $n_{1}\geq 1.$ We call $C$ a
compatible sequence if the following conditions hold:

  * •

Each $V^{(n)}$ is a graded $\operatorname{\mathscr{D}}_{n}^{+}$-module

  * •

The maps $\Pi^{(n)}:V^{(n+1)}\rightarrow V^{(n)}$ are degree-preserving.

  * •

Each map $\Pi^{(n)}$ is a $\operatorname{\mathscr{A}}_{n}^{X}$ module map.

  * •

$\Pi^{(n)}X_{n+1}=0$

  * •

$\Pi^{(n)}\pi^{(n+1)}T_{n}=\pi^{(n)}\Pi^{(n)}.$

Given compatible sequences $C=\left((V^{(n)})_{n\geq n_{1}},(\Pi^{(n)})_{n\geq
n_{1}}\right)$ and $D=\left((W^{(n)})_{n\geq n_{2}},(\Psi^{(n)})_{n\geq
n_{2}}\right)$ a homomorphism $\phi:C\rightarrow D$ is a collection of maps
$\phi=(\phi^{(n)})_{n\geq\max(n_{1},n_{2})}$ with
$\phi^{(n)}:V^{(n)}\rightarrow W^{(n)}$ such that

  * •

$\phi^{(n)}$ are degree-preserving $\operatorname{\mathscr{D}}_{n}^{+}$ module
maps.

  * •

$\phi^{(n)}\Pi^{(n)}=\Psi^{(n)}\phi^{(n+1)}.$

We will write $\mathfrak{C}$ for the category of compatible sequences.

######  Remark 9.

The importance of the relation $\Pi^{(n)}\pi^{(n+1)}T_{n}=\pi^{(n)}\Pi^{(n)}$
can be traced back to at least the work of Ion-Wu [IW22] on their stable-limit
DAHA. This relation allowed Ion-Wu to construct the limit Cherednik operators
on the space of almost symmetric functions utilizing a remarkable stability
relation for the classical Cherednik operators. We will be following a similar
idea in a different setting in this section of the paper.

This relation may be interpreted as relating to the natural inclusion map on
extended affine symmetric groups
$\widehat{\mathfrak{S}}_{n}\rightarrow\widehat{\mathfrak{S}}_{n+1}$ given by
$s_{i}\rightarrow s_{i}$ for $1\leq i\leq n-1$ and $\pi\rightarrow\pi s_{n}.$
Diagrammatically, this map sends the crossing diagram for some
$\sigma\in\widehat{\mathfrak{S}}_{n}$ on $n$-strands to the corresponding
crossing diagram on $(n+1)$-strands where we send $n+1$ to itself.

For the remainder of this section we fix a compatible sequence
$C=\left((V^{(n)})_{n\geq n_{0}},(\Pi^{(n)})_{n\geq n_{0}}\right).$ It is easy
to check that for $0\leq k\leq n,$ $\Pi^{(n)}(L_{k}(V^{(n+1)}))\subset
L_{k}(V^{(n)})$ so that the following definition makes sense.

######  Definition 3.9.

For $k\geq 0$ define
$\operatorname{\mathfrak{L}}_{k}=\operatorname{\mathfrak{L}}_{k}(C)$ to be the
stable-limit
$\operatorname{\mathfrak{L}}_{k}:=\lim_{\leftarrow}L_{k}(V^{(n)})$ with
respect to the maps $\Pi^{(n)}.$ We define
$\operatorname{\mathfrak{L}}_{\bullet}=\operatorname{\mathfrak{L}}_{\bullet}(C)$
as $\operatorname{\mathfrak{L}}_{\bullet}=\bigoplus_{k\geq
0}\operatorname{\mathfrak{L}}_{k}.$ We will write
$\Pi^{(n)}_{\bullet}:L_{\bullet}(V^{(n+1)})\rightarrow L_{\bullet}(V^{(n)})$
for the map obtained by restricting $\Pi^{(n)}$ to each component
$L_{k}(V^{(n+1)}).$

If we let $\widetilde{V}$ denote the stable-limit of the spaces $V^{(n)}$ with
respect to the maps $\Pi^{(n)}$ then we can reinterpret the spaces
$\operatorname{\mathfrak{L}}_{k}$ as

$\operatorname{\mathfrak{L}}_{k}=\\{v\in X_{1}\cdots
X_{k}\widetilde{V}|\leavevmode\nobreak\ T_{i}(v)=v\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
i>k\\}.$

######  Lemma 3.10.

For $n\geq n_{0}$ the map
$\Pi^{(n)}_{\bullet}:L_{\bullet}(V^{(n+1)})\rightarrow L_{\bullet}(V^{(n)})$
is a $\mathbb{B}_{q,t}^{(n)}$-module map.

###### Proof.

By definition $\Pi^{(n)}$ is a $\operatorname{\mathscr{A}}_{n}^{X}$-module map
so for $1\leq i\leq k-1,$ $\Pi^{(n)}T_{i}=T_{i}\Pi^{(n)}.$ Further, we also
know that if $k\leq n-1$ then $\Pi^{(n)}d_{+}=d_{+}\Pi^{(n)}$ since on
$L_{k},$ $d_{+}=-q^{k}X_{1}T_{1}^{-1}\cdots T_{k}^{-1}.$

Now let $1\leq k\leq n$ and $v\in L_{k}$ say, $v=X_{1}\cdots
X_{k}\epsilon_{k}(w).$ We see that from a nearly identical calculation to one
seen in the proof of Proposition 3.7

$\displaystyle\Pi^{(n)}d_{-}(v)$
$\displaystyle=\Pi^{(n)}(1-q)(1+qT_{k}^{-1}+\ldots+q^{n+1-k}T_{n}^{-1}\cdots
T_{k}^{-1})(v)$
$\displaystyle=\Pi^{(n)}(1-q)(1+qT_{k}^{-1}+\ldots+q^{n+1-k}T_{n}^{-1}\cdots
T_{k}^{-1})(X_{1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=\Pi^{(n)}(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})(X_{1}\cdots
X_{k}\epsilon_{k}(w))+\Pi^{(n)}(1-q)q^{n+1-k}T_{n}^{-1}\cdots
T_{k}^{-1}X_{1}\cdots X_{k}\epsilon_{k}(w)$
$\displaystyle=(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})\Pi^{(n)}(X_{1}\cdots
X_{k}\epsilon_{k}(w))+\Pi^{(n)}(1-q)X_{n+1}T_{n}\cdots T_{k}X_{1}\cdots
X_{k-1}\epsilon_{k}(w)$
$\displaystyle=(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})\Pi^{(n)}(X_{1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=(1-q)(1+qT_{k}^{-1}+\ldots+q^{n-k}T_{n-1}^{-1}\cdots
T_{k}^{-1})\Pi^{(n)}(v)$ $\displaystyle=d_{-}\Pi^{(n)}(v).$

Lastly, let $1\leq i\leq k.$ Using the relation
$\Pi^{(n)}\pi^{(n+1)}T_{n}=\pi^{(n)}\Pi^{(n)}$ we find

$\displaystyle\Pi^{(n)}z_{i}(v)$ $\displaystyle=\Pi^{(n)}Y_{i}(v)$
$\displaystyle=\Pi^{(n)}q^{n-i+2}T_{i-1}\cdots
T_{1}\pi^{(n+1)}T_{n}^{-1}\cdots T_{i}^{-1}(v)$
$\displaystyle=\Pi^{(n)}q^{n-i+2}T_{i-1}\cdots
T_{1}\pi^{(n+1)}T_{n}^{-1}\cdots T_{i}^{-1}(X_{1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=\Pi^{(n)}q^{n-i+2}T_{i-1}\cdots
T_{1}\pi^{(n+1)}T_{n}^{-1}\cdots T_{i}^{-1}X_{i}(X_{1}\cdots
X_{i-1}X_{i+1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=\Pi^{(n)}q^{2}T_{i-1}\cdots T_{1}\pi^{(n+1)}X_{n+1}T_{n}\cdots
T_{i}(X_{1}\cdots X_{i-1}X_{i+1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=\Pi^{(n)}q^{2}tT_{i-1}\cdots T_{1}X_{1}\pi^{(n+1)}T_{n}\cdots
T_{i}(X_{1}\cdots X_{i-1}X_{i+1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=q^{2}tT_{i-1}\cdots T_{1}X_{1}\Pi^{(n)}\pi^{(n+1)}T_{n}\cdots
T_{i}(X_{1}\cdots X_{i-1}X_{i+1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=q^{2}tT_{i-1}\cdots T_{1}X_{1}\pi^{(n)}\Pi^{(n)}T_{n-1}\cdots
T_{i}(X_{1}\cdots X_{i-1}X_{i+1}\cdots X_{k}\epsilon_{k}(w))$
$\displaystyle=q^{2}T_{i-1}\cdots T_{1}\pi^{(n)}X_{n}T_{n-1}\cdots
T_{i}(X_{1}\cdots X_{i-1}X_{i+1}\cdots X_{k}\Pi^{(n)}(\epsilon_{k}(w))$
$\displaystyle=q^{n-i+1}T_{i-1}\cdots T_{1}\pi^{(n)}T_{n-1}^{-1}\cdots
T_{i}^{-1}X_{i}(X_{1}\cdots X_{i-1}X_{i+1}\cdots
X_{k}\Pi^{(n)}(\epsilon_{k}(w))$ $\displaystyle=q^{n-i+1}T_{i-1}\cdots
T_{1}\pi^{(n)}T_{n-1}^{-1}\cdots T_{i}^{-1}\Pi^{(n)}(X_{1}\cdots
X_{k}\epsilon_{k}(w))$ $\displaystyle=z_{i}\Pi^{(n)}(v).$

∎

As an immediate consequence of Lemma 3.10 and Lemma 2.2 we may make the
following definition.

######  Definition 3.11.

We define the graded $\mathbb{B}_{q,t}$ module structure on
$\operatorname{\mathfrak{L}}_{\bullet}$ given by the stable-limit of the
graded $\mathbb{B}_{q,t}^{(n)}$ modules
$\operatorname{\mathfrak{L}}_{\bullet}^{(n)}$ with respect to the maps
$\Pi^{(n)}_{\bullet}:\operatorname{\mathfrak{L}}_{\bullet}^{(n+1)}\rightarrow\operatorname{\mathfrak{L}}_{\bullet}^{(n)}.$

######  Example 1.

In the case of the polynomial representations of
$\operatorname{\mathscr{D}}_{n}^{+}$, $V_{\operatorname{\mathrm{pol}}}^{(n)}$,
we see using Proposition 3.7 that
$\operatorname{\mathfrak{L}}_{\bullet}(C_{\operatorname{\mathrm{pol}}})\cong
V^{\operatorname{\mathrm{pol}}}_{\bullet}$ where

$C_{\operatorname{\mathrm{pol}}}:=\left((V_{\operatorname{\mathrm{pol}}}^{(n)})_{n\geq
1},(\Xi_{\bullet}^{(n)})_{n\geq 1}\right).$

The construction in Definition 3.11 associates to any compatible sequence $C$
a graded module $\operatorname{\mathfrak{L}}_{\bullet}(C)$ of
$\mathbb{B}_{q,t}.$ We can easily see that this construction is functorial.

######  Theorem 3.12.

(Main Theorem) The map $C\rightarrow\operatorname{\mathfrak{L}}_{\bullet}(C)$
is a covariant functor
$\mathfrak{C}\rightarrow\mathbb{B}_{q,t}-\operatorname{\mathrm{Mod}}.$

###### Proof.

This follows immediately using the functoriality described in Remark 1 and
from the fact that the operators on $\operatorname{\mathfrak{L}}_{\bullet}(C)$
are described entirely in terms of the action of each
$\operatorname{\mathscr{D}}_{n}^{+}$ on $V^{(n)}.$ ∎

######  Remark 10.

Recently, González-Gorsky-Simental [GGS23] introduced the extended algebra
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ and the notion of calibrated
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ modules. The extended algebra
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ contains additional
$\Delta$-operators with specific relations motivated by the $\Delta$-operators
in Macdonald theory. Calibrated
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ modules are those modules
with a basis of joint eigenvectors for the $z_{i}$’s and the additional
operators $\Delta_{p_{m}}$ with simple nonzero spectrum.

In the case of the polynomial representations of DAHAs, the $\mathbb{B}_{q,t}$
representation
$\operatorname{\mathfrak{L}}_{\bullet}(C_{\operatorname{\mathrm{pol}}})$ has
an extended action by $\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ using
$\Delta$-operators and this representation is calibrated. It is an interesting
question to figure out exactly which properties of the family of DAHA modules
$C_{\operatorname{\mathrm{pol}}}$ allow for this extended calibrated action by
$\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$.

##  4 Compatible Sequences From AHA

In this section we give a method for defining compatible sequences. We will
consider families of representations for the affine Hecke algebras
$\operatorname{\mathscr{A}}_{n}$ in type $GL$ with special properties which we
call pre-compatible. These families of representations for
$\operatorname{\mathscr{A}}_{n}$ can then be induced to give representations
of the corresponding $\operatorname{\mathscr{D}}_{n}^{+}$ which can be shown
to be compatible after carefully defining the correct connecting maps.

######  Definition 4.1.

Let $C=\left((U^{(n)})_{n\geq n_{0}},(\kappa^{(n)})_{n\geq n_{0}}\right)$ be a
collection of $\mathbb{Q}(q,t)$-vector spaces and maps
$\kappa^{(n)}:U^{(n+1)}\rightarrow U^{(n)}$ with $n_{1}\geq 1.$ We call $C$ a
pre-compatible sequence if the following hold:

  * •

Each $U^{(n)}$ is a graded $\operatorname{\mathscr{A}}_{n}$ module

  * •

The maps $\kappa^{(n)}:U^{(n+1)}\rightarrow U^{(n)}$ are degree preserving
$\operatorname{\mathscr{H}}_{n}$ module maps

  * •

$\kappa^{(n)}\pi^{(n+1)}T_{n}=\pi^{(n)}\kappa^{(n)}.$

Given any pre-compatible sequence $C$ we define the spaces
$(V_{C}^{(n)})_{n\geq n_{0}}$ by

$V_{C}^{(n)}:=\operatorname{Ind}_{\operatorname{\mathscr{A}}^{(n)}}^{\operatorname{\mathscr{D}}_{n}^{+}}U^{(n)}$

which we endow with the grading inherited by
$\mathbb{Q}(q,t)[X_{1},\ldots,X_{n}]\otimes U^{(n)}$ (which is isomorphic as a
vector space). Define the maps $(\Pi_{C}^{(n)}:V_{C}^{(n+1)}\rightarrow
V_{C}^{(n)})_{n\geq n_{0}}$ by

$\Pi_{C}^{(n)}(X_{1}^{\alpha_{1}}\cdots X_{n+1}^{\alpha_{n+1}}\otimes
v):=\mathbbm{1}(\alpha_{n+1}=0)\otimes\kappa^{(n)}(v).$

We will write $\operatorname{Ind}(C)$ for the family

$\operatorname{Ind}(C):=\left((V_{C}^{(n)})_{n\geq
n_{0}},(\Pi_{C}^{(n)})_{n\geq n_{0}}\right).$

######  Proposition 4.2.

If $C$ is pre-compatible then $\operatorname{Ind}(C)$ is compatible.

###### Proof.

By construction each space $V_{C}^{(n)}$ is a graded
$\operatorname{\mathscr{D}}_{n}^{+}$ module and the maps $\Pi_{C}^{(n)}$ are
degree preserving $\operatorname{\mathscr{A}}_{n}^{X}$ maps with
$\Pi_{C}^{(n)}X_{n+1}=0.$ Thus we only need to show that
$\Pi_{C}^{(n)}\pi^{n+1}T_{n}=\pi^{(n)}\Pi_{C}^{(n)}.$ To see this we have the
following:

$\displaystyle\Pi_{C}^{(n)}\pi^{(n+1)}T_{n}(X_{1}^{\alpha_{1}}\cdots
X_{n+1}^{\alpha_{n+1}}\otimes v)$ $\displaystyle=X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}\pi^{(n+1)}T_{n}(X_{n}^{\alpha_{n}}X_{n+1}^{\alpha_{n+1}}\otimes
v)$ $\displaystyle=X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}\pi^{(n+1)}\left(\left(X_{n}^{\alpha_{n+1}}X_{n+1}^{\alpha_{n}}T_{n}+(1-q)X_{n}\frac{X_{n}^{\alpha_{n}}X_{n+1}^{\alpha_{n+1}}-X_{n}^{\alpha_{n+1}}X_{n+1}^{\alpha_{n}}}{X_{n}-X_{n+1}}\right)\otimes
v\right)$ $\displaystyle=X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}\pi^{(n+1)}\left(X_{n}^{\alpha_{n+1}}X_{n+1}^{\alpha_{n}}\otimes
T_{n}v\right)+(1-q)X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}\pi^{(n+1)}X_{n}\frac{X_{n}^{\alpha_{n}}X_{n+1}^{\alpha_{n+1}}-X_{n}^{\alpha_{n+1}}X_{n+1}^{\alpha_{n}}}{X_{n}-X_{n+1}}(1\otimes
v)$ $\displaystyle=X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}X_{n+1}^{\alpha_{n+1}}(tX_{1})^{\alpha_{n}}\pi^{(n+1)}\left(1\otimes
T_{n}v\right)$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +(1-q)X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}X_{n+1}\pi^{(n+1)}\frac{X_{n}^{\alpha_{n}}X_{n+1}^{\alpha_{n+1}}-X_{n}^{\alpha_{n+1}}X_{n+1}^{\alpha_{n}}}{X_{n}-X_{n+1}}(1\otimes
v)$ $\displaystyle=X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}X_{n+1}^{\alpha_{n+1}}(tX_{1})^{\alpha_{n}}\pi^{(n+1)}\left(1\otimes
T_{n}v\right)$
$\displaystyle=\mathbbm{1}(\alpha_{n+1}=0)(tX_{1})^{\alpha_{n}}X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\Pi_{C}^{(n)}(1\otimes\pi^{(n+1)}T_{n}(v))$
$\displaystyle=\mathbbm{1}(\alpha_{n+1}=0)(tX_{1})^{\alpha_{n}}X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\otimes\kappa^{(n)}(\pi^{(n+1)}T_{n}(v))$
$\displaystyle=\mathbbm{1}(\alpha_{n+1}=0)(tX_{1})^{\alpha_{n}}X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\otimes\pi^{(n)}\kappa^{(n)}(v)$
$\displaystyle=\mathbbm{1}(\alpha_{n+1}=0)(tX_{1})^{\alpha_{n}}X_{2}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n-1}}\pi^{(n)}\otimes\kappa^{(n)}(v)$
$\displaystyle=\pi^{(n)}\left(\mathbbm{1}(\alpha_{n+1}=0)X_{1}^{\alpha_{1}}\cdots
X_{n}^{\alpha_{n}}\otimes\kappa^{(n)}(v)\right)$
$\displaystyle=\pi^{(n)}\Pi_{C}^{(n)}(X_{1}^{\alpha_{1}}\cdots
X_{n+1}^{\alpha_{n+1}}\otimes v).$

Thus $\Pi_{C}^{(n)}\pi^{(n+1)}T_{n}=\pi^{(n)}\Pi_{C}^{(n)}$ and so
$\operatorname{Ind}(C)$ is compatible. ∎

We will now give a large family of pre-compatible sequences built from Young
diagrams. The modules in these sequences are the same (up to changing
conventions) as the modules in [DL11] and the author’s prior paper [Wei23]

######  Definition 4.3.

Define the $\mathbb{Q}(q,t)$-algebra homomorphism
$\rho_{n}:\operatorname{\mathscr{A}}_{n}\rightarrow\operatorname{\mathscr{H}}_{n}$
by

  * •

$\rho_{n}(T_{i})=T_{i}$ for $1\leq i\leq n-1$

  * •

$\rho_{n}(\pi^{(n)})=T_{1}^{-1}\cdots T_{n-1}^{-1}.$

For a $\operatorname{\mathscr{H}}_{n}$-module $V$ we will denote by
$\rho_{n}^{*}(V)$ the $\operatorname{\mathscr{A}}_{n}$-module with action
defined for $v\in V$ and $X\in\operatorname{\mathscr{A}}_{n}$ by
$X(v):=\rho_{n}(X)(v).$

######  Definition 4.4.

Let $\lambda$ be a Young diagram. We will write
$\operatorname{\mathrm{SYT}}(\lambda)$ for the set of Young tableaux of shape
$\lambda.$ Let $n_{\lambda}:=|\lambda|+\lambda_{1}$ and for $n\geq
n_{\lambda}$ define $\lambda^{(n)}:=(n-|\lambda|,\lambda).$ Given a box,
$\square$, in a Young diagram $\lambda$ we define the content of $\square$ as
$c(\square):=a-b$ where $\square=(a,b)$ as drawn in the
$\mathbb{N}\times\mathbb{N}$ grid (in English notation). We will write
$c_{\tau}(i)=c(\square)$ where $\tau(\square)=i.$ We order the set
$\operatorname{\mathrm{SYT}}(\lambda)$ by defining the cover relation
$s_{i}(\tau)>\tau$ if $c_{\tau}(i)-c_{\tau}(i+1)>1.$ Define the
$\operatorname{\mathscr{H}}_{n}$-module $S_{\lambda}$ as the space with basis
$e_{\tau}$ labelled by Young tableaux
$\tau\in\operatorname{\mathrm{SYT}}(\lambda)$ and
$\operatorname{\mathscr{H}}_{n}$ action given by:

  * •

$T_{i}(e_{\tau})=e_{\tau}$ if $i,i+1$ are in the same row of $\tau$

  * •

$T_{i}(e_{\tau})=-qe_{\tau}$ if $i,i+1$ are in the same column of $\tau$

  * •

$T_{i}(e_{\tau})=e_{s_{i}(\tau)}+\frac{(1-q)q^{c_{\tau}(i)}}{q^{c_{\tau}(i)}-q^{c_{\tau}(i+1)}}e_{\tau}$
if $s_{i}(\tau)>\tau$

  * •

$T_{i}(e_{\tau})=-\frac{(q^{c_{\tau}(i+1)+1}-q^{c_{\tau}(i)})(q^{c_{\tau}(i)+1}-q^{c_{\tau}(i+1)})}{(q^{c_{\tau}(i+1)}-q^{c_{\tau}(i)})^{2}}e_{s_{i}(\tau)}+\frac{(1-q)q^{c_{\tau}(i)}}{q^{c_{\tau}(i)}-q^{c_{\tau}(i+1)}}e_{\tau}$
if $s_{i}(\tau)<\tau.$

For $n\geq n_{\lambda}$ define the $\operatorname{\mathscr{A}}_{n}$ modules
$U_{\lambda}^{(n)}:=\rho_{n}^{*}(S_{\lambda^{(n)}})$ and maps
$\kappa_{\lambda}^{(n)}:U_{\lambda}^{(n+1)}\rightarrow U_{\lambda}^{(n)}$
given for $\tau\in\operatorname{\mathrm{SYT}}(\lambda^{(n+1)})$ as

$\kappa_{\lambda}^{(n)}(e_{\tau}):=\begin{cases}e_{\tau|_{\lambda^{(n)}}}&\tau(\square_{0})=n+1\\\
0&\tau(\square_{0})\neq n+1.\end{cases}$

where $\square_{0}$ is the unique square in $\lambda^{(n+1)}/\lambda^{(n)}.$

We consider the $\operatorname{\mathscr{A}}_{n}$ modules $U_{\lambda}^{(n)}$
as graded with the trivial grading i.e.
$U_{\lambda}^{(n)}=U_{\lambda}^{(n)}(0).$ We will write $C_{\lambda}$ for the
family

$C_{\lambda}:=\left((U_{\lambda}^{(n)})_{n\geq
n_{\lambda}},(\kappa_{\lambda}^{(n)})_{n\geq n_{\lambda}}\right).$

######  Remark 11.

It is a straightforward (but tedious) exercise to show that the
$\operatorname{\mathscr{H}}_{n}$ module defined on $S_{\lambda}$ as above is
actually a $\operatorname{\mathscr{H}}_{n}$ module. We leave this to the
reader. In fact this representation is the unique irreducible
$\operatorname{\mathscr{H}}_{n}$ module corresponding to the shape $\lambda$.
Definition 4.4 follows the conventions of the author’s paper [Wei23] which are
slightly different than the definitions of the spaces of vector valued
polynomials considered by Dunkl-Luque in [DL11]. As constructed, the elements
$e_{\tau}$ of the $\operatorname{\mathscr{A}}_{n}$ module $U_{\lambda}^{(n)}$
are not weight vectors for the Cherednik elements $Y_{i}$ but rather for the
reversed orientation Cherednik elements $\theta_{i}$ given by
$\theta_{i}=q^{i-1}T_{i-1}^{-1}\cdots T_{1}^{-1}\pi T_{n-1}\cdots T_{i}.$
Explicitly, we have that for $\tau\in\operatorname{\mathrm{SYT}}(\lambda)$ and
$1\leq i\leq n$, $\theta_{i}(e_{\tau})=q^{c_{\tau}(i)}e_{\tau}.$

######  Theorem 4.5.

For any $\lambda,$ $\operatorname{Ind}(C_{\lambda})$ is a compatible sequence
with $\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda}))$
a nonzero graded $\mathbb{B}_{q,t}$ module.

###### Proof.

It is easy from the explicit $T_{i}$ relations given in Definition 4.4 to
verify that for every $n\geq n_{\lambda}$ the map
$\kappa_{\lambda}^{(n)}:U_{\lambda}^{(n+1)}\rightarrow U_{\lambda}^{(n)}$ is a
$\operatorname{\mathscr{H}}_{n}$ module map. We therefore also have that

$\displaystyle\kappa_{\lambda}^{(n)}\pi^{(n+1)}T_{n}$
$\displaystyle=\kappa_{\lambda}^{(n)}\rho_{n+1}(\pi^{(n+1)})T_{n}$
$\displaystyle=\kappa_{\lambda}^{(n)}T_{1}^{-1}\cdots T_{n}^{-1}T_{n}$
$\displaystyle=\kappa_{\lambda}^{(n)}T_{1}^{-1}\cdots T_{n-1}^{-1}$
$\displaystyle=T_{1}^{-1}\cdots T_{n-1}^{-1}\kappa_{\lambda}^{(n)}$
$\displaystyle=\pi^{(n)}\kappa_{\lambda}^{(n)}.$

Hence, $C_{\lambda}$ is a pre-compatible sequence and so by Proposition 4.2 it
follows that $\operatorname{Ind}(C_{\lambda})$ is a compatible sequence. Thus
we may consider the graded $\mathbb{B}_{q,t}$ module
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda})).$

To show that
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda}))$ is
nonzero it suffices to show that
$\operatorname{\mathfrak{L}}_{0}(\operatorname{Ind}(C_{\lambda}))$ is nonzero.
This is space is the stable-limit of the symmetrized spaces
$\epsilon_{0}^{(n)}(\operatorname{Ind}_{\operatorname{\mathscr{A}}_{n}}^{\operatorname{\mathscr{D}}_{n}^{+}}U_{\lambda}^{(n)})$
with respect to the maps $\kappa_{\lambda}^{(n)}.$ However, this space is the
Murnaghan-type representation $\widetilde{W}_{\lambda}$ of the positive
elliptic Hall algebra of shape $\lambda$ from the author’s prior paper
[Wei23]. This space is infinite dimensional for any $\lambda$ and so clearly
$\operatorname{\mathfrak{L}}_{0}(\operatorname{Ind}(C_{\lambda}))$ is nonzero.
∎

We can show further that for all $k\geq 0,$
$\operatorname{\mathfrak{L}}_{k}(\operatorname{Ind}(C_{\lambda}))$ is infinite
dimensional. To see this note that
$d_{+}^{k}:\operatorname{\mathfrak{L}}_{0}(\operatorname{Ind}(C_{\lambda}))\rightarrow\operatorname{\mathfrak{L}}_{k}(\operatorname{Ind}(C_{\lambda}))$
is given by

$(-q^{k-1}X_{1}T_{1}^{-1}\cdots
T_{k-1}^{-1})\cdots(-q^{2}X_{1}T_{1}^{-1}T_{2}^{-1})(-qX_{1}T_{1}^{-1})(-X_{1})$

which is clearly injective. Thus since
$\operatorname{\mathfrak{L}}_{0}(\operatorname{Ind}(C_{\lambda}))=\widetilde{W}_{\lambda}$
is infinite-dimensional the same is true for
$\operatorname{\mathfrak{L}}_{k}(\operatorname{Ind}(C_{\lambda}))$.

As the $\mathbb{B}_{q,t}$ modules
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda}))$
contain the Murnaghan-type representation $\widetilde{W}_{\lambda}$ of EHA we
will refer to these modules as the $\mathbb{B}_{q,t}$ modules of Murnaghan-
type.

######  Remark 12.

The author conjectures that each of the Murnaghan-type $\mathbb{B}_{q,t}$
modules,
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda}))$, has
an extended action by $\mathbb{B}_{q,t}^{\operatorname{\mathrm{ext}}}$ and
that these extended modules are calibrated. Evidence for this conjecture comes
from the author’s prior paper [Wei23] where the author constructs
$\Delta$-operators on the space
$\widetilde{W}_{\lambda}=\operatorname{\mathfrak{L}}_{0}(\operatorname{Ind}(C_{\lambda}))$
which have distinct nonzero spectrum. Extending these $\Delta$-operators to
the whole space
$\operatorname{\mathfrak{L}}_{\bullet}(\operatorname{Ind}(C_{\lambda}))$ is
nontrivial. The author will investigate this in a subsequent paper.

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