# Delta Operators on Almost Symmetric Functions

Milo Bechtloff Weising

###### Abstract

We construct $\Delta$-operators $F[\Delta]$ on the space of almost symmetric
functions $\operatorname{\mathscr{P}}_{as}^{+}$. These operators extend the
usual $\Delta$-operators on the space of symmetric functions
$\Lambda\subset\operatorname{\mathscr{P}}_{as}^{+}$ central to Macdonald
theory. The $F[\Delta]$ operators are constructed as certain limits of
symmetric functions in the Cherednik operators $Y_{i}$ and act diagonally on
the stable-limit non-symmetric Macdonald functions
$\widetilde{E}_{(\mu|\lambda)}(x_{1},x_{2},\ldots;q,t).$ Using properties of
Ion-Wu limits, we are able to compute commutation relations for the
$\Delta$-operators $F[\Delta]$ and many of the other operators on
$\operatorname{\mathscr{P}}_{as}^{+}$ introduced by Ion-Wu. Using these
relations we show that there is an action of $\mathbb{B}_{q,t}^{\text{ext}}$
on almost symmetric functions which we show is isomorphic to the polynomial
representation of $\mathbb{B}_{q,t}^{\text{ext}}$ constructed by González-
Gorsky-Simental.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Definitions and Conventions
    1. 2.1 Non-symmetric Macdonald Polynomials
    2. 2.2 Stable-Limit DAHA of Ion-Wu
    3. 2.3 Important Limit Operators
  3. 3 Constructing Delta Operators on $\operatorname{\mathscr{P}}_{as}^{+}$
    1. 3.1 Preliminaries
    2. 3.2 Proof of Convergence
    3. 3.3 Commutation Relations
    4. 3.4 $\mathbb{B}_{q,t}^{\text{ext}}$ Action

##  1 Introduction

The symmetric Macdonald functions $P_{\lambda}[X;q,t]$, introduced by
Macdonald [Mac88], are central objects in modern algebraic combinatorics. One
of the many characterizations of the symmetric functions $P_{\lambda}[X;q,t]$
comes from considering the action of the Macdonald operator $\Delta$ on the
ring of symmetric functions $\Lambda.$ The $P_{\lambda}[X;q,t]$ are a
$\Delta$-eigenbasis for the ring of symmetric functions $\Lambda$ with
distinct spectrum.

Through the work of Haiman [Hai99] it is known that the Macdonald operator and
a variant $\widetilde{H}_{\lambda}[X;q,t]$ of the symmetric Macdonald
functions $P_{\lambda}$ carry geometric information about the Hilbert schemes
$\operatorname{\mathrm{Hilb}}^{n}(\mathbb{C}^{2})$ of points in
$\mathbb{C}^{2}.$ This work was extended by Schiffmann-Vasserot in [SV13]
where the authors constructed a representation of the positive elliptic Hall
algebra $\mathscr{E}^{+}$ of Burban-Schiffmann [BS12] on the ring of symmetric
functions $\Lambda$ generated by multiplication operators $p_{r}[X]^{\bullet}$
and the $\Delta$-operators $p_{\ell}[\Delta]$. These $\Delta$-operators act
diagonally on the symmetric Macdonald function basis and satisfy many
interesting commutation relations encoded by the positive elliptic Hall
algebra $\mathscr{E}^{+}$. One may construct these operators directly as
stable limits of certain elements of the spherical double affine Hecke
algebras in type $\operatorname{\mathrm{GL}}.$ In particularly, the operator
$p_{1}[\Delta]=\Delta$ is essentially the stable limit of operators of the
form $Y_{1}^{(n)}+\ldots+Y_{n}^{(n)}$ acting on symmetric polynomials where
$Y_{i}^{(n)}$ denote the Cherednik operators.

Extending the work of Schiffmann-Vasserot, Carlsson-Gorksy-Mellit constructed
an algebra $\mathbb{B}_{q,t}$ which acts on the equivariant K theory of the
parabolic flag Hilbert schemes of points in $\mathbb{C}^{2}$. This algebra is
a non-unital quiver path algebra containing many copies of affine Hecke
algebras in type $\operatorname{\mathrm{GL}}.$ A highly related algebra
$\mathbb{A}_{q,t}$ introduced by Carlsson-Mellit, called the double Dyck path
algebra, was a crucial ingredient in their proof of the Shuffle Theorem
[CM18]. By looking at loops at the vertex labelled $0$ of $\mathbb{B}_{q,t}$
in the polynomial representation $V_{\bullet}$ one may recover many of the
operators coming from the action of $\mathscr{E}^{+}$ on $V_{0}=\Lambda.$
However, one does not find the $\Delta$-operators in this way. More recently,
the authors González-Gorsky-Simental [GGS23] studied a larger algebra
$\mathbb{B}_{q,t}^{\text{ext}}$ which roughly adds in these missing
$\Delta$-elements to $\mathbb{B}_{q,t}$. They construct many representations
for this algebra but in particular they show that there is a unique action of
$\mathbb{B}_{q,t}^{\text{ext}}$ on $V_{\bullet}$ which extends the usual
action of $\mathbb{B}_{q,t}$ and contains the $\Delta$-operators and their
analogues on the equivariant K theory of parabolic flag Hilbert schemes with
flags of non-trivial length.

Through the work of Ion-Wu [IW22] we are able to understand $\mathbb{A}_{q,t}$
and $\mathbb{B}_{q,t}$ through the double affine Hecke algebras in type
$\operatorname{\mathrm{GL}}.$ This relationship comes through the existence of
the $+$-stable limit double affine Hecke algebra $\mathscr{H}$ and its action
on the ring of almost symmetric functions
$\operatorname{\mathscr{P}}_{as}^{+}.$ This action in a sense globalizes the
polynomial representations of $\mathbb{A}_{q,t}$ and $\mathbb{B}_{q,t}$ and is
constructed from the polynomial representations of the double affine Hecke
algebras in type $\operatorname{\mathrm{GL}}.$ Importantly, Ion-Wu construct
limit Cherednik operators $\mathscr{Y}_{i}$ which correspond to the action of
the $z_{i}$ elements in $\mathbb{A}_{q,t}$ and $\mathbb{B}_{q,t}$. The process
used by Ion-Wu for constructing this algebra $\mathscr{H}^{+}$ and its
representation on $\operatorname{\mathscr{P}}_{as}^{+}$ is a not the classical
stable limit procedure but rather a $t$-adically enriched version of
stability. The additional flexibility in the notion of convergence for
sequences of polynomials is enough to make sense of a version of Cherednik
theory for the $+$-stable limit double affine Hecke algebra $\mathscr{H}^{+}$
including the existence of a special basis
$\widetilde{E}_{(\mu|\lambda)}(x_{1},x_{2},\ldots;q,t)$ for
$\operatorname{\mathscr{P}}_{as}^{+}$ of weight vectors for the limit
Cherednik operators $\mathscr{Y}$ [BW23]. However, the spectrum of the limit
Cherednik operators $\mathscr{Y}$ is far from simple which each nonzero weight
space being infinite dimensional. To remedy this issue the author constructed
a new operator $\Psi_{p_{1}}$ on $\operatorname{\mathscr{P}}_{as}^{+}$
extending the usual Macdonald operator on
$\Lambda\subset\operatorname{\mathscr{P}}_{as}^{+}$ which commutes with the
limit Cherednik operators $\mathscr{Y}$ and refines the nonzero
$\mathscr{Y}$-weight spaces to be $1$ dimensional. This operator is
constructed as an Ion-Wu operator limit of the operators
$t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}$ acting on finite variable
polynomials. The author conjectured that a similar operator may be constructed
for any symmetric function $F$ i.e. that the sequence
$F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]$ converges in the sense of Ion-Wu
to an operator on $\operatorname{\mathscr{P}}_{as}^{+}.$

In this paper we confirm the author’s conjecture from [BW23] and construct a
family of operators $F[\Delta]$ on the ring of almost symmetric functions
$\operatorname{\mathscr{P}}_{as}^{+}$ built as Ion-Wu limits of symmetric
polynomials in the Cherednik operators (Theorem 3.10). These operators act
diagonally in the $\widetilde{E}_{(\mu|\lambda)}$ basis and extend the action
of the $\Delta$-operators on
$\Lambda\subset\operatorname{\mathscr{P}}_{as}^{+}.$ The proof that these
sequences of symmetric polynomials in the Cherednik operators converge as
operators is highly technical and involves understanding the action of the
operators $F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]$ on partially symmetric
polynomials in finitely many variables. Note that this is more involved than
understanding the action of symmetric polynomials in the Cherednik operators
on symmetric polynomials which is well understood. Importantly, the
$F[\Delta]$ cannot be expressed entirely in terms of the limit Cherednik
operators $\mathscr{Y}.$ We will obtain an explicit expansion (Proposition
3.9) of the operators $e_{r}[\Delta]$ acting on each of the subspaces
$x_{1}\cdots x_{k}\operatorname{\mathscr{P}}_{as}^{+}$, which depends on $k$,
in terms of the operators $\mathscr{Y}_{i},$ $\epsilon_{k},$ and $\pi.$ Using
properties of Ion-Wu operator limits we compute some commutation relations
(Propositions 3.11) between the $F[\Delta]$ and the operators
$\mathscr{Y}_{i},T_{i},\widetilde{\pi},$ and $\epsilon_{k}.$ At the end of the
paper we will use these commutation relations to show that there exists an
action of $\mathbb{B}_{q,t}^{\text{ext}}$ on $\bigoplus_{k\geq 0}x_{1}\cdots
x_{k}\operatorname{\mathscr{P}}(k)^{+}$ which we show is isomorphic to the
polynomial representation $V_{\bullet}$ of $\mathbb{B}_{q,t}^{\text{ext}}$
constructed by González-Gorsky-Simental (Corollary 3.17). This result shows
that we may consider the $\Delta$-operators on $V_{\bullet}$ in [GGS23] as
limits of symmetric polynomials in the Cherednik operators.

###  1.1 Acknowledgements

The author would like to thank their advisor Monica Vazirani for her continued
support. The author would also like to thank Nicolle González, José Simental,
and Eugene Gorsky for helpful conversations about
$\mathbb{B}_{q,t}^{\text{ext}}$.

##  2 Definitions and Conventions

We present here the conventions that will be used in this paper for the double
affine Hecke algebra of type $GL$. Take note of the quadratic relation
$(T_{i}-1)(T_{i}+t)=0$ which has been chosen to match with the conventions in
[IW22] and the author’s prior work [BW23]. Some of the other notation has been
slightly altered to align with other works of the author.

######  Definition 2.1.

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

  1. (i)

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

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

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

  4. (ii)

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

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

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

  7. (iii)

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

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

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

  10. (iv)

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

  11. (v)

$Y_{1}X_{1}\cdots X_{n}=qX_{1}\cdots X_{n}Y_{1}$

  12.

Further, define the special elements $\pi_{n},\widetilde{\pi}_{n}$ by

  * •

$\pi_{n}:=Y_{1}T_{1}\cdots T_{n-1}$

  * •

$\widetilde{\pi}_{n}:=X_{1}T_{1}^{-1}\cdots T_{n-1}^{-1}.$

This conveniently allows us to write

$Y_{1}=\pi_{n}T_{n-1}^{-1}\cdots T_{1}^{-1}.$

When required we will write $Y_{i}^{(n)}$ for the element $Y_{i}$ in
$\mathscr{H}_{n}$ to differentiate between the element $Y_{i}^{(m)}$ in a
different $\mathscr{H}_{m}$ for $n\neq m$.

The following commutation relations may be verified directly:

  * •

$Y_{1}\widetilde{\pi}_{n}=\widetilde{\pi}_{n}qY_{n}$

  * •

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

######  Definition 2.2.

For all $0\leq k<n$ define the element
$\epsilon^{(n)}_{k}\in\operatorname{\mathscr{H}}_{n}$ as

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

Here $\mathfrak{S}_{(1^{k},n-k)}$ is the Young subgroup of $\mathfrak{S}_{n}$
corresponding to the composition $(1^{k},n-k)$,
$T_{\sigma}=T_{s_{i_{1}}}\cdots T_{s_{i_{r}}}$ whenever
$\sigma=s_{i_{1}}\cdots s_{i_{r}}$ is a reduced word representing $\sigma$,
and $[m]_{t}!:=\prod_{i=1}^{m}(\frac{1-t^{i}}{1-t})$ is the $t$-factorial. We
will simply write $\epsilon^{(n)}$ for $\epsilon^{(n)}_{0}$.

######  Definition 2.3.

Let $\mathscr{P}_{n}=\mathbb{Q}(q,t)[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$. The
standard representation of $\mathscr{H}_{n}$ is given by the following action
on $\mathscr{P}_{n}$:

  * •

$T_{i}f(x_{1},\ldots,x_{n})=s_{i}f(x_{1},\ldots,x_{n})+(1-t)x_{i}\frac{1-s_{i}}{x_{i}-x_{i+1}}f(x_{1},\ldots,x_{n})$

  * •

$X_{i}f(x_{1},..,x_{n})=x_{i}f(x_{1},\ldots,x_{n})$

  * •

$\pi_{n}f(x_{1},\ldots,x_{n})=f(x_{2},\ldots,x_{n},qx_{1})$

Here $s_{i}$ denotes the operator that swaps the variables $x_{i}$ and
$x_{i+1}$. Under this action the $T_{i}$ operators are known as the Demazure-
Lusztig operators. For $q,t$ generic $\mathscr{P}_{n}$ is known to be a
faithful representation of $\mathscr{H}_{n}$. The action of the elements
$Y_{1},\ldots,Y_{n}\in\mathscr{H}_{n}$ are called Cherednik operators.

Set $\mathscr{H}_{n}^{+}$ to be the positive part of $\mathscr{H}_{n}$ i.e.
the subalgebra generated by $T_{1},\ldots,T_{n-1}$, $X_{1},\ldots,X_{n}$, and
$Y_{1},\ldots,Y_{n}$ without allowing for inverses in the $X_{i}$ and $Y_{i}$
elements and set $\mathscr{P}_{n}^{+}=\mathbb{Q}(q,t)[x_{1},\ldots,x_{n}]$.
Importantly, $\mathscr{P}_{n}^{+}$ is a $\mathscr{H}_{n}^{+}$-submodule of
$\mathscr{P}_{n}$.

###  2.1 Non-symmetric Macdonald Polynomials

Before discussing non-symmetric Macdonald polynomials we must first review
some basic combinatorial definitions.

######  Definition 2.4.

In this paper, a composition will refer to a finite tuple
$\mu=(\mu_{1},\ldots,\mu_{n})$ of non-negative integers. We allow for the
empty composition $\emptyset$ with no parts. We will let
$\operatorname{\mathrm{Comp}}$ denote the set of all compositions. The length
of a composition $\mu=(\mu_{1},\ldots,\mu_{n})$ is $\ell(\mu)=n$ and the size
of the composition is $|\mu|=\mu_{1}+\ldots+\mu_{n}$. As a convention we will
set $\ell(\emptyset)=0$ and $|\emptyset|=0.$ We say that a composition $\mu$
is reduced if $\mu=\emptyset$ or $\mu_{\ell(\mu)}\neq 0.$ We will let
$\operatorname{\mathrm{Comp}^{\textit{red}}}$ denote the set of all reduced
compositions. Given two compositions $\mu=(\mu_{1},\ldots,\mu_{n})$ and
$\beta=(\beta_{1},\ldots,\beta_{m})$, define
$\mu*\beta=(\mu_{1},\ldots,\mu_{n},\beta_{1},\ldots,\beta_{m})$. A partition
is a composition $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ with
$\lambda_{1}\geq\ldots\geq\lambda_{n}\geq 1$. Note that vacuously we allow for
the empty partition $\emptyset.$ We denote the set of all partitions by
$\operatorname{\mathrm{Par}}$. We denote $\operatorname{sort}(\mu)$ to be the
partition obtained by ordering the nonzero elements of $\mu$ in weakly
decreasing order. The dominance ordering for partitions is defined by
$\lambda\trianglelefteq\nu$ if for all $i\geq 1$,
$\lambda_{1}+\ldots+\lambda_{i}\leq\nu_{1}+\ldots+\nu_{i}$ where we set
$\lambda_{i}=0$ whenever $i>\ell(\lambda)$ and similarly for $\nu$. If
$\lambda\trianglelefteq\nu$ and $\lambda\neq\nu.$ we will write
$\lambda\triangleleft\nu$. We will let $\Phi$ denote the set
$\operatorname{\mathrm{Comp}^{\textit{red}}}\times\operatorname{\mathrm{Par}}$
of all pairs $(\mu|\lambda)$ with
$\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}}$ and
$\lambda\in\operatorname{\mathrm{Par}}.$

We will in a few instances use the notation $\mathbbm{1}(p)$ to denote the
value $1$ if the statement p is true and $0$ otherwise. If $\mu$ is any
composition with size $n$ we let $\mathfrak{S}_{n}/\mathfrak{S}_{\mu}$ denote
the corresponding set of all minimal length left coset representatives.

In line with the conventions in [HHL08] we define the Bruhat order on the type
$GL_{n}$ weight lattice $\mathbb{Z}^{n}$ as follows.

######  Definition 2.5.

Let $e_{1},...,e_{n}$ be the standard basis of $\mathbb{Z}^{n}$ and let
$\alpha\in\mathbb{Z}^{n}$. We define the Bruhat ordering on $\mathbb{Z}^{n}$,
written simply by $<$, by first defining cover relations for the ordering and
then taking their transitive closure. If $i<j$ such that
$\alpha_{i}<\alpha_{j}$ then we say $\alpha>(ij)(\alpha)$ and additionally if
$\alpha_{j}-\alpha_{i}>1$ then $(ij)(\alpha)>\alpha+e_{i}-e_{j}$ where $(ij)$
denotes the transposition swapping $i$ and $j.$

######  Definition 2.6.

The non-symmetric Macdonald polynomials (for $GL_{n}$) are a family of Laurent
polynomials $E_{\mu}\in\mathscr{P}_{n}$ for $\mu\in\mathbb{Z}^{n}$ uniquely
determined by the following:

  * •

Triangularity: Each $E_{\mu}$ has a monomial expansion of the form
$E_{\mu}=x^{\mu}+\sum_{\lambda<\mu}a_{\lambda}x^{\lambda}$

  * •

Weight Vector: Each $E_{\mu}$ is a weight vector for the operators
$Y_{1}^{(n)},\ldots,Y_{n}^{(n)}\in\mathscr{H}_{n}$.

The non-symmetric Macdonald polynomials are a $Y^{(n)}$-weight basis for the
$\mathscr{H}_{n}$ standard representation $\mathscr{P}_{n}$. For
$\mu\in\mathbb{Z}^{n}$, $E_{\mu}$ is homogeneous with degree
$\mu_{1}+\ldots+\mu_{n}$. Further, the set of $E_{\mu}$ corresponding to
$\mu\in\mathbb{Z}_{\geq 0}^{n}$ gives a basis for $\mathscr{P}_{n}^{+}$.

###  2.2 Stable-Limit DAHA of Ion-Wu

As the index $n$ varies, the standard $\mathscr{H}_{n}$ representations,
$\operatorname{\mathscr{P}}_{n}$, fail to form a direct/inverse system of
compatible $\mathscr{H}_{n}$ representations. However, as the authors Ion and
Wu investigate in [IW22], this sequence of representations is compatible
enough to allow for the construction of a limiting representation for a new
algebra resembling a direct limit of the double affine Hecke algebras of type
$GL$. We will start by giving the definition of this algebra.

######  Definition 2.7.

[IW22] The $+$-stable-limit double affine Hecke algebra of Ion and Wu,
$\mathscr{H}^{+}$, is the algebra generated over $\mathbb{Q}(q,t)$ by the
elements $T_{i},X_{i},Y_{i}$ for $i\geq 1$ satisfying the following relations:

  * •

The generators $T_{i},X_{i}$ for $i\geq 1$ satisfy (i) and (ii) of Defn. 2.1.

  * •

The generators $T_{i},Y_{i}$ for $i\geq 1$ satisfy (i) and (iii) of Defn. 2.1.

  * •

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

Note that $X_{i},Y_{i}$ are not invertible in
$\operatorname{\mathscr{H}}^{+}.$

######  Definition 2.8.

Define the ring of symmetric functions $\Lambda$ as the inverse limit of the
graded rings $\mathbb{Q}(q,t)[x_{1},\ldots,x_{n}]^{\mathfrak{S}_{n}}$ with
respect to the homogeneous quotient maps $f(x_{1},\ldots,x_{n+1})\rightarrow
f(x_{1},\ldots,x_{n},0)$ in the category of graded rings. We will write
$X:=x_{1}+x_{2}+\ldots$ and use plethsytic notation
$F(x_{1},x_{2},\ldots)=F[X].$ Define for $\ell\geq 1$ the power sum symmetric
function $p_{\ell}[X]$ by

$p_{\ell}[X]:=\sum_{i\geq 1}x_{i}^{\ell}.$

For $r\geq 1$ define the elementary symmetric function $e_{r}[X]$ as

$e_{r}[X]:=\sum_{i_{1}<\ldots<i_{r}}x_{i_{1}}\cdots x_{i_{r}}.$

As a convention we set $e_{0}[X]:=1.$ If
$\lambda=(\lambda_{1},\ldots,\lambda_{r})$ is a partition define
$e_{\lambda}[X]:=e_{\lambda_{1}}[X]\cdots e_{\lambda_{r}}[X].$ For a partition
$\lambda$ define the monomial symmetric function $m_{\lambda}[X]$ by

$m_{\lambda}[X]:=\sum_{\mu}x^{\mu}$

where we range over all distinct monomials $x^{\mu}$ such that
$\sigma(\mu)=\lambda$ for some permutation $\sigma$.

######  Remark 1.

It is a classical result that the sets
$\\{e_{\lambda}[X]\\}_{\lambda\in\operatorname{\mathrm{Par}}}$ and
$\\{m_{\lambda}[X]\\}_{\lambda\in\operatorname{\mathrm{Par}}}$ are
$\mathbb{Q}(q,t)$-bases for $\Lambda.$ In fact,

$\Lambda\cong\mathbb{Q}(q,t)[e_{1},e_{2},\ldots].$

######  Definition 2.9.

[IW22] Let $\mathscr{P}_{\infty}^{+}$ denote the inverse limit of the rings
$\mathscr{P}_{k}^{+}$ with respect to the homomorphisms
$\Xi_{k}:\operatorname{\mathscr{P}}_{k+1}^{+}\rightarrow\operatorname{\mathscr{P}}_{k}^{+}$
which send $x_{k+1}$ to 0 at each step. We can naturally extend $\Xi_{k}$ to a
map $\mathscr{P}_{\infty}^{+}\rightarrow\mathscr{P}_{k}$ which will be given
the same name. Let
$\mathscr{P}(k)^{+}:=\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda[x_{k+1}+x_{k+2}+\ldots]\subset\mathscr{P}_{\infty}^{+}$.
Define the ring of almost symmetric functions by
$\mathscr{P}_{as}^{+}:=\bigcup_{k\geq 0}\mathscr{P}(k)^{+}$. Note
$\mathscr{P}_{as}^{+}\subset\mathscr{P}_{\infty}^{+}.$ Define
$\rho:\mathscr{P}_{as}^{+}\rightarrow x_{1}\mathscr{P}_{as}^{+}$ to be the
linear map defined by $\rho(x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}F[x_{m}+x_{m+1}+\ldots])=\mathbbm{1}(a_{1}>0)x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}F[x_{m}+x_{m+1}+\ldots]$ for $F\in\Lambda$. Note that $\rho$
restricts to maps $\operatorname{\mathscr{P}}_{n}\rightarrow
x_{1}\operatorname{\mathscr{P}}_{n}$ which are compatible with the quotient
maps $\Xi_{n}$.

The ring $\operatorname{\mathscr{P}}_{as}^{+}$ is a free graded
$\Lambda$-module with homogeneous basis given simply by the set of monomials
$x^{\mu}$ with $\mu$ reduced. Therefore, $\operatorname{\mathscr{P}}_{as}^{+}$
has a homogeneous $\mathbb{Q}(q,t)$ basis given by all $x^{\mu}m_{\lambda}[X]$
ranging over all reduced compositions $\mu$ and partitions $\lambda$. Further,
the dimension of the homogeneous degree d part of $\mathscr{P}(k)^{+}$ is
equal to the number of pairs $(\mu|\lambda)$ of reduced compositions $\mu$ and
partitions $\lambda$ with $|\mu|+|\lambda|=d$ and $\ell(\mu)\leq k$.

In order to define the operators required for Ion and Wu’s main construction
we must first review the new definition of convergence introduced in [IW22].

######  Definition 2.10.

[IW22] Let $(f_{m})_{m\geq 1}$ be a sequence of polynomials with
$f_{m}\in\mathscr{P}_{m}^{+}$. Then the sequence $(f_{m})_{m\geq 1}$ is
convergent if there exist some N and auxiliary sequences $(h_{m})_{m\geq 1}$,
$(g^{(i)}_{m})_{m\geq 1}$, and $(a^{(i)}_{m})_{m\geq 1}$ for $1\leq i\leq N$
with $h_{m},g^{(i)}_{m}\in\mathscr{P}_{m}^{+}$,
$a^{(i)}_{m}\in\mathbb{Q}(q,t)$ with the following properties:

  * •

For all $m$, $f_{m}=h_{m}+\sum_{i=1}^{N}a^{(i)}_{m}g^{(i)}_{m}$.

  * •

The sequences $(h_{m})_{m\geq 1}$, $(g^{(i)}_{m})_{m\geq 1}$ for $1\leq i\leq
N$ converge in $\mathscr{P}_{\infty}^{+}$ with limits $h,g^{(i)}$
respectively. That is to say, $\Xi_{m}(h_{m+1})=h_{m}$ and
$\Xi_{m}(g_{m+1}^{(i)})=g_{m}^{(i)}$ for all $1\leq i\leq N$ and $m\geq 1$.
Further, we require $g^{(i)}\in\mathscr{P}_{as}^{+}$.

  * •

The sequences $a^{(i)}_{m}$ for $1\leq i\leq N$ converge with respect to the
t-adic topology on $\mathbb{Q}(q,t)$ with limits $a^{(i)}$ which are required
to be in $\mathbb{Q}(q,t)$.

The sequence is said to have a limit given by
$\lim_{m}f_{m}=h+\sum_{i=1}^{N}a^{(i)}g^{(i)}.$

######  Remark 2.

In this paper we will be entirely concerned with convergent sequences
$(f_{m})_{m\geq 1}$ with almost symmetric limits
$\lim_{m}f_{m}\in\operatorname{\mathscr{P}}_{as}^{+}$. In this case it follows
readily from definition that each of these convergent sequences necessarily
will have the form

$f_{m}(x_{1},\ldots,x_{m})=\sum_{i=1}^{N}c_{i}^{(m)}x^{\mu^{(i)}}F_{i}[x_{1}+\ldots+x_{m}]$

where $N\geq 1$ is fixed, $c_{i}^{(m)}$ are convergent sequences of scalars
with $\lim_{m}c_{i}^{(m)}\in\mathbb{Q}(q,t)$, $F_{i}$ are symmetric functions,
and $\mu^{(i)}$ are compositions. Here we will consider $x^{\mu^{(i)}}=0$ in
$\operatorname{\mathscr{P}}_{m}$ whenever $\ell(\mu^{(i)})>m$.

######  Definition 2.11.

[IW22] For $m\geq 1$ suppose $A_{m}$ is an operator on
$\operatorname{\mathscr{P}}_{m}^{+}$. The sequence $(A_{m})_{m\geq 1}$ of
operators is said to converge if for every
$f\in\operatorname{\mathscr{P}}_{as}^{+}$ the sequence
$(A_{m}(\Xi_{m}(f)))_{m\geq 1}$ converges to an element of
$\operatorname{\mathscr{P}}_{as}^{+}$. From [IW22] the corresponding operator
on $\operatorname{\mathscr{P}}_{as}^{+}$ given by
$A(f):=\lim_{m}A_{m}(\Xi_{m}(f))$ is well defined and said to be the limit of
the sequence $(A_{m})_{m\geq 1}$. In this case we will simply write
$A=\lim_{m}A_{m}$.

The following important technical proposition of Ion and Wu will be used
repeatedly in this paper.

######  Proposition 2.12.

[Prop. 6.21 [IW22]] If $A=\lim_{m}A_{m}$ and $f=\lim_{m}f_{m}$ are limit
operators and limit functions respectively then $A(f)=\lim_{m}A_{m}(f_{m}).$

This is a sort of continuity statement for convergent sequences of operators.
The utility of the above proposition is that for an operator arising as the
limit of finite variable operators, $A=\lim_{m}A_{m}$ say, we can use any
sequence $(f_{m})_{m\geq 1}$ converging to
$f\in\operatorname{\mathscr{P}}_{as}^{+}$ in order to calculate $A(f)$.

It is easy to verify the following proposition using Proposition 2.12.

######  Proposition 2.13.

[IW22] If $A=\lim_{m}A_{m}$ and $B=\lim_{m}$ then $AB=\lim_{m}A_{m}B_{m}.$

###  2.3 Important Limit Operators

In this section we detail the construction of some of the important operators
on $\operatorname{\mathscr{P}}_{as}^{+}$ which will be required later. The
fact that the operators as presented below are in fact well defined is non-
trivial and follows from [IW22] and [BW23].

######  Definition 2.14.

We define the operators $X_{i}$,$T_{i}$,$\mathscr{Y}_{i}$,$\pi$,
$\widetilde{\pi}$ on $\operatorname{\mathscr{P}}_{as}^{+}$ by the following
limits:

  * •

$X_{i}:=\lim_{n}X_{i}$

  * •

$T_{i}:=\lim_{n}T_{i}$

  * •

$\epsilon_{k}:=\lim_{n}\epsilon_{k}^{(n)}$

  * •

$\mathscr{Y}_{i}:=\lim_{n}t^{n-i+1}T_{i-1}\cdots
T_{1}\rho\pi_{n}T_{n-1}^{-1}\cdots T_{i}^{-1}$

  * •

$\pi:=\lim_{n}\pi_{n}$

  * •

$\widetilde{\pi}:=\lim_{n}\widetilde{\pi}_{n}.$

We may explicitly describe the action of some of these operators:

  * •

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

  * •

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

  * •

$\pi(x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}F[X])=x_{2}^{a_{1}}\cdots
x_{k+1}^{a_{k}}F[X+(q-1)x_{1}]$

  * •

$\widetilde{\pi}(x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}F[X])=x_{1}T_{1}^{-1}\cdots
T_{k}^{-1}x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}F[X].$

The operators $\mathscr{Y}_{i}$ were originally constructed by Ion-Wu and are
known as the limit Cherednik operators. They are limits of the deformed
Cherednik operators $\widetilde{Y}_{i}^{(n)}:=t^{n-i+1}T_{i-1}\cdots
T_{1}\rho\pi_{n}T_{n-1}^{-1}\cdots T_{i}^{-1}$ which have the important
property that

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

It is non-trivial that not only are the $\mathscr{Y}_{i}$ operators well
defined, but that they in fact mutually commute i.e.
$[\mathscr{Y}_{i},\mathscr{Y}_{j}]=0.$

It is a result of Ion-Wu that the following representation of
$\operatorname{\mathscr{H}}^{+}$ is well defined.

######  Definition 2.15.

[IW22] The standard representation of $\operatorname{\mathscr{H}}^{+}$
consists of $\operatorname{\mathscr{P}}_{as}^{+}$ with the action generated by
the operators $X_{i},\mathscr{Y}_{i},T_{i}.$

We may give an explicit description of the action of the limit Cherednik
operators $\mathscr{Y}_{i}$ using the stable-limit non-symmetric Macdonald
functions.

######  Definition 2.16.

[BW23] For $(\mu|\lambda)\in\Phi$ define the stable-limit non-symmetric
Macdonald function $\widetilde{E}_{(\mu|\lambda)}$ as

$\widetilde{E}_{(\mu|\lambda)}[x_{1},x_{2},\ldots;q,t]:=\lim_{n}\epsilon_{\ell(\mu)}^{(n)}\left(E_{\mu*\lambda*0^{n-(\ell(\mu)+\ell(\lambda))}}(x_{1},\ldots,x_{n};q,t)\right).$

The following gives an explicit characterization of the action of the
$\mathscr{Y}_{i}$ on $\operatorname{\mathscr{P}}_{as}^{+}.$

######  Theorem 2.17.

[BW23] The $\widetilde{E}_{(\mu|\lambda)}$ are a $\mathscr{Y}$-weight basis
for $\mathscr{P}_{as}^{+}$. Further, we have that

$\mathscr{Y}_{i}(\widetilde{E}_{(\mu|\lambda)})=\widetilde{\alpha}_{(\mu|\lambda)}(i)\widetilde{E}_{(\mu|\lambda)}$

where $\widetilde{\alpha}_{(\mu|\lambda)}(i)$ is given explicitly by

$\widetilde{\alpha}_{(\mu|\lambda)}(i)=\begin{cases}q^{\mu_{i}}t^{\ell(\mu)+\ell(\lambda)+1-\beta_{\mu*\lambda}(i)}&i\leq\ell(\mu),\mu_{i}\neq
0\\\ 0&\text{otherwise}\end{cases}$

and

$\beta_{\nu}(i):=\\#\\{j:1\leq j\leq i\leavevmode\nobreak\
,\nu_{j}\leq\nu_{i}\\}+\\#\\{j:i<j\leq\ell(\nu)\leavevmode\nobreak\
,\nu_{i}>\nu_{j}\\}.$

######  Example.

We can see directly that
$\mathscr{Y}_{1}(\widetilde{E}_{(1|\emptyset)})=qt\widetilde{E}_{(1|\emptyset)}:$

$\displaystyle\mathscr{Y}_{1}(\widetilde{E}_{(1|\emptyset)})$
$\displaystyle=\mathscr{Y}_{1}(x_{1})$
$\displaystyle=\lim_{n}t^{n}\rho\pi_{n}T_{n-1}^{-1}\cdots T_{1}^{-1}(x_{1})$
$\displaystyle=\lim_{n}t^{n}\rho\pi_{n}(t^{-(n-1)}x_{n})$
$\displaystyle=\lim_{n}qt\rho(x_{1})$ $\displaystyle=\lim_{n}qtx_{1}$
$\displaystyle=qtx_{1}.$

##  3 Constructing Delta Operators on $\operatorname{\mathscr{P}}_{as}^{+}$

We begin by recalling the definition of the operator
$\Psi_{p_{1}}:\operatorname{\mathscr{P}}_{as}^{+}\rightarrow\operatorname{\mathscr{P}}_{as}^{+}$
from the author’s prior work. We opt to rename this operator
$\Delta:=\Psi_{p_{1}}$ to better align with the notation used in Macdonald
theory.

######  Definition 3.1.

For a symmetric function $F$ define the operators
$\Psi_{F}^{(n)}:\operatorname{\mathscr{P}}_{n}^{+}\rightarrow\operatorname{\mathscr{P}}_{n}^{+}$
by

$\Psi_{F}^{(n)}:=F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}].$

For $\nu\in\operatorname{\mathrm{Par}}$ we denote

$\kappa_{\nu}(q,t):=\sum_{i=1}^{\infty}q^{\nu_{i}}t^{i}\in\mathbb{Q}(q,t).$

######  Theorem 3.2.

[BW23] The sequence of operators $(\Psi_{p_{1}}^{(n)})_{n\geq 1}$ converges to
an operator $\Delta$ on $\operatorname{\mathscr{P}}_{as}^{+}$ with the
following properties:

  * •

$\Delta(\widetilde{E}_{(\mu|\lambda)})=\kappa_{\operatorname{sort}(\mu*\lambda)}(q,t)\widetilde{E}_{(\mu|\lambda)}$

  * •

$[\Delta,\mathscr{Y}_{i}]=0$

  * •

$[\Delta,T_{i}]=0$

  * •

The joint $\Delta,\mathscr{Y}_{1},\mathscr{Y}_{2},\ldots$ weight spaces in
$\operatorname{\mathscr{P}}_{as}^{+}$ are all 1-dimensional.

  * •

$\Delta|_{x_{1}\cdots
x_{k}\operatorname{\mathscr{P}}(k)^{+}}=\mathscr{Y}_{1}+\ldots+\mathscr{Y}_{k}+\frac{t}{1-t}\epsilon_{k}T_{k}\cdots
T_{1}\pi.$

######  Remark 3.

The last property of the operator $\Delta$ listed above is not explicitly
stated in [BW23] but follows from the proof of Theorem 3.2 and the fact that
for $1\leq i\leq k-1$

$\displaystyle(\mathscr{Y}_{1}+\ldots+\mathscr{Y}_{k}+\frac{t}{1-t}\epsilon_{k}T_{k}\cdots
T_{1}\pi)T_{i}$
$\displaystyle=(\mathscr{Y}_{1}+\ldots+\mathscr{Y}_{k})T_{i}+\frac{t}{1-t}\epsilon_{k}T_{k}\cdots
T_{1}\pi T_{i}$
$\displaystyle=T_{i}(\mathscr{Y}_{1}+\ldots+\mathscr{Y}_{k})+\frac{t}{1-t}\epsilon_{k}T_{k}\cdots
T_{1}T_{i+1}\pi$
$\displaystyle=T_{i}(\mathscr{Y}_{1}+\ldots+\mathscr{Y}_{k})+\frac{t}{1-t}\epsilon_{k}T_{i}T_{k}\cdots
T_{1}\pi$
$\displaystyle=T_{i}(\mathscr{Y}_{1}+\ldots+\mathscr{Y}_{k})+T_{i}\frac{t}{1-t}\epsilon_{k}T_{k}\cdots
T_{1}\pi$
$\displaystyle=T_{i}(\mathscr{Y}_{1}+\ldots+\mathscr{Y}_{k}+\frac{t}{1-t}\epsilon_{k}T_{k}\cdots
T_{1}\pi).$

This fact will be generalized in Theorem 3.10.

######  Remark 4.

The operator $\Delta$ cannot be expressed entirely in terms of the limit
Cherednik operators $\mathscr{Y}_{i}$ and in particular,

$\Delta\neq\mathscr{Y}_{1}+\mathscr{Y}_{2}+\mathscr{Y}_{3}+\ldots.$

It is easy to see this since each of the operators $\mathscr{Y}_{i}$
annihilates $\Lambda$ whereas $\Delta$ acts diagonally on $\Lambda$ with non-
zero eigenvalues. This is very different than the finite variable situation
where the action of the finite variable Macdonald operator is roughly given by
$Y_{1}^{(n)}+\ldots+Y_{n}^{(n)}.$

At the end of the author’s prior paper [BW23] it is conjectured that for any
symmetric function $F\in\Lambda$ the sequence of operators
$(\Psi_{F}^{(n)})_{n\geq 1}$ converges to an operator on
$\operatorname{\mathscr{P}}_{as}^{+}.$ An affirmation of this conjecture has
direct implications related to the conjectural partially symmetric elliptic
Hall algebras mentioned by Carlsson-Mellit in [CM18] and the extended double
Dyck path algebra $\mathbb{B}_{q,t}^{\text{ext}}$ of González-Gorsky-Simental
in [GGS23].

The main purpose of this section is to give a proof of this conjecture. The
proof involves a detailed computation which will be done in stages. We will
start with some of the required preliminaries.

###  3.1 Preliminaries

There are a few elementary technical results we will need before we are able
to prove the main result of this section Theorem 3.10.

For the remainder of this section we consider $n,k,r$ with $n>k+r\geq 1.$ We
begin by expressing certain partially symmetric polynomials in the Cherednik
elements $Y_{i}$ in terms of products of consecutive Cherednik elements.

######  Lemma 3.3.

$e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]=\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{-\ell(\sigma)}T_{\sigma}t^{rn}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}T_{\sigma^{-1}}$

###### Proof.

Notice that for
$\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}$ the
values $\sigma(k+1),\ldots,\sigma(k+r)$ satisfy
$k+1\leq\sigma(k+1)<\ldots<\sigma(k+r)\leq n$ and uniquely determine $\sigma.$
As such there is a natural bijection
$\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}\rightarrow\\{(i_{1},\ldots,i_{r})|k+1\leq
i_{1}<\ldots<i_{r}\leq n-k\\}$ given by
$\sigma\rightarrow(\sigma(k+1),\ldots,\sigma(k+r)).$ Hence, it suffices to
show that for all such $\sigma$

$Y_{\sigma(k+1)}^{(n)}\cdots
Y_{\sigma(k+r)}^{(n)}=t^{-\ell(\sigma)}T_{\sigma}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}T_{\sigma^{-1}}.$

We proceed by induction on the Bruhat order on
$\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}$. Clearly, this
formula holds for $\sigma=1.$

Suppose $k+1=i_{0}\leq i_{1}<\ldots<i_{r}\leq i_{r+1}=n$ with
$i_{j+1}-i_{j}>1$ for some $0\leq j\leq r.$ Then

$\displaystyle Y_{i_{1}}^{(n)}\cdots
Y_{i_{j-1}}^{(n)}Y_{(i_{j})+1}^{(n)}Y_{i_{j+1}}^{(n)}Y_{i_{j+2}}^{(n)}\cdots
Y_{n}^{(n)}$ $\displaystyle=Y_{i_{1}}^{(n)}\cdots
Y_{i_{j-1}}^{(n)}(t^{-1}T_{i_{j}}Y_{i_{j}}^{(n)}T_{i_{j}})Y_{i_{j+1}}^{(n)}Y_{i_{j+2}}^{(n)}\cdots
Y_{n}^{(n)}$ $\displaystyle=t^{-1}T_{i_{j}}Y_{i_{1}}^{(n)}\cdots
Y_{i_{r}}^{(n)}T_{i_{j}}.$

Now if
$\sigma,\sigma^{\prime}\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}$
are the unique elements with $\sigma(k+\ell)=i_{\ell}$ for $1\leq\ell\leq r$
and $\sigma^{\prime}=s_{i_{j}}\sigma.$ Suppose that
$Y_{\sigma(k+1)}^{(n)}\cdots
Y_{\sigma(k+r)}^{(n)}=t^{-\ell(\sigma)}T_{\sigma}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}T_{\sigma^{-1}}.$ Then from the above we find that

$\displaystyle Y_{\sigma^{\prime}(k+1)}^{(n)}\cdots
Y_{\sigma^{\prime}(k+r)}^{(n)}$ $\displaystyle=Y_{i_{1}}^{(n)}\cdots
Y_{i_{j-1}}^{(n)}Y_{i_{j}+1}^{(n)}Y_{i_{j+1}}^{(n)}Y_{i_{j+2}}^{(n)}\cdots
Y_{n}^{(n)}$ $\displaystyle=t^{-1}T_{i_{j}}Y_{i_{1}}^{(n)}\cdots
Y_{i_{r}}^{(n)}T_{i_{j}}$
$\displaystyle=t^{-1}T_{i_{j}}Y_{\sigma(k+1)}^{(n)}\cdots
Y_{\sigma(k+r)}^{(n)}T_{i_{j}}$
$\displaystyle=t^{-(1+\ell(\sigma))}T_{i_{j}}T_{\sigma}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}T_{\sigma^{-1}}T_{i_{j}}$
$\displaystyle=t^{-\ell(\sigma^{\prime})}T_{\sigma^{\prime}}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}T_{(\sigma^{\prime})^{-1}}.$

∎

Now we may write a product of consecutive Cherednik elements in terms of
$\pi_{n}.$

######  Lemma 3.4.

$t^{rn}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}=t^{(n-k)+\ldots+(n-k-r+1)}(T_{k}\cdots T_{1})(T_{k+1}\cdots
T_{2})\cdots(T_{k+r-1}\cdots T_{r})\pi_{n}^{r}(T_{n-r}^{-1}\cdots
T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots T_{k+r}^{-1})$

###### Proof.

We will show this result by induction. For $r=1$ we see that

$t^{n}Y_{k+1}^{(n)}=t^{n-k}T_{k}\cdots T_{1}\pi_{n}T_{n-1}^{-1}\cdots
T_{k+1}^{-1}.$

Now we find

$\displaystyle t^{(r+1)n}Y_{k+1}^{(n)}\cdots Y_{k+r+1}^{(n)}$
$\displaystyle=t^{rn}Y_{k+1}^{(n)}\cdots Y_{k+r}^{(n)}t^{n}Y_{k+r+1}^{(n)}$
$\displaystyle=t^{(n-k)+\ldots+(n-k-r+1)}(T_{k}\cdots
T_{1})\cdots(T_{k+r-1}\cdots T_{r})\pi_{n}^{r}(T_{n-r}^{-1}\cdots
T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots T_{k+r}^{-1})t^{n}Y_{k+r+1}^{(n)}$
$\displaystyle=t^{(n-k)+\ldots+(n-k-r+1)}(T_{k}\cdots
T_{1})\cdots(T_{k+r-1}\cdots T_{r})\pi_{n}^{r}(T_{n-r}^{-1}\cdots
T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots
T_{k+r}^{-1})\left(t^{n-k-r}T_{k+r}\cdots T_{1}\pi_{n}T_{n-1}^{-1}\cdots
T_{k+r+1}^{-1}\right)$ $\displaystyle=t^{(n-k)+\ldots+(n-k-r)}(T_{k}\cdots
T_{1})\cdots(T_{k+r-1}\cdots T_{r})\pi_{n}^{r}(T_{n-r}^{-1}\cdots
T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots T_{k+r}^{-1})T_{k+r}\cdots
T_{1}\pi_{n}T_{n-1}^{-1}\cdots T_{k+r+1}^{-1}.$

Looking closer we see

$\displaystyle(T_{n-r}^{-1}\cdots T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots
T_{k+r}^{-1})T_{k+r}\cdots T_{1}$ $\displaystyle=(T_{n-r}^{-1}\cdots
T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots T_{k+r+1}^{-1})T_{k+r-1}\cdots T_{1}$
$\displaystyle=(T_{n-r}^{-1}\cdots T_{k+1}^{-1})\cdots(T_{n-2}^{-1}\cdots
T_{k+r-1}^{-1})T_{k+r-1}\cdots T_{1}(T_{n-1}^{-1}\cdots T_{k+r+1}^{-1})$
$\displaystyle=\ldots$ $\displaystyle=(T_{k}\cdots T_{1})(T_{n-r}^{-1}\cdots
T_{k+2}^{-1})\cdots(T_{n-1}^{-1}\cdots T_{k+r+1}^{-1}).$

Therefore,

$\displaystyle(T_{k}\cdots T_{1})\cdots(T_{k+r-1}\cdots
T_{r})\pi_{n}^{r}(T_{n-r}^{-1}\cdots T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots
T_{k+r}^{-1})T_{k+r}\cdots T_{1}\pi_{n}T_{n-1}^{-1}\cdots T_{k+r+1}^{-1}$
$\displaystyle=(T_{k}\cdots T_{1})\cdots(T_{k+r-1}\cdots
T_{r})\pi_{n}^{r}(T_{k}\cdots T_{1})(T_{n-r}^{-1}\cdots
T_{k+2}^{-1})\cdots(T_{n-1}^{-1}\cdots
T_{k+r+1}^{-1})\pi_{n}T_{n-1}^{-1}\cdots T_{k+r+1}^{-1}$
$\displaystyle=(T_{k}\cdots T_{1})\cdots(T_{k+r-1}\cdots T_{r})(T_{k+r}\cdots
T_{r+1})\pi_{n}^{r}(T_{n-r}^{-1}\cdots T_{k+2}^{-1})\cdots(T_{n-1}^{-1}\cdots
T_{k+r+1}^{-1})\pi_{n}T_{n-1}^{-1}\cdots T_{k+r+1}^{-1}$
$\displaystyle=(T_{k}\cdots T_{1})\cdots(T_{k+r-1}\cdots T_{r})(T_{k+r}\cdots
T_{r+1})\pi_{n}^{r+1}(T_{n-r-1}^{-1}\cdots
T_{k+1}^{-1})\cdots(T_{n-2}^{-1}\cdots T_{k+r}^{-1})(T_{n-1}^{-1}\cdots
T_{k+r+1}^{-1})$

so that

$\displaystyle t^{(r+1)n}Y_{k+1}^{(n)}\cdots Y_{k+r+1}^{(n)}$
$\displaystyle=t^{(n-k)+\ldots+(n-k-r)}(T_{k}\cdots T_{1})\cdots(T_{k+r}\cdots
T_{r+1})\pi_{n}^{r+1}(T_{n-r-1}^{-1}\cdots
T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots T_{k+r+1}^{-1}).$

∎

We need the following standard result.

######  Lemma 3.5.

$\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{{n-k\choose
2}-{n-k-r\choose 2}-{r\choose
2}-\ell(\sigma)}=\frac{[n-k]_{t}!}{[n-k-r]_{t}![r]_{t}!}$

###### Proof.

We see the following:

$\displaystyle[n-k]_{t}!$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}}t^{{n-k\choose
2}-\ell(\sigma)}$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}\sum_{\gamma\in\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{{n-k\choose
2}-\ell(\sigma\gamma)}$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}\sum_{\gamma\in\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{{n-k\choose
2}-\ell(\sigma)-\ell(\gamma)}$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{{n-k\choose
2}-{n-k-r\choose 2}-{r\choose
2}-\ell(\sigma)}\sum_{\gamma\in\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{{n-k-r\choose
2}+{r\choose 2}-\ell(\gamma)}$
$\displaystyle=[n-k-r]_{t}![r]_{t}!\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{{n-k\choose
2}-{n-k-r\choose 2}-{r\choose 2}-\ell(\sigma)}.$

∎

Using the prior lemmas in this section now shows the following:

######  Lemma 3.6.

$e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]\epsilon_{k}^{(n)}=t^{{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots
T_{r})\pi_{n}^{r}\epsilon_{k}^{(n)}.$

###### Proof.

Putting together Lemmas 3.3, 3.4, and 3.5 we get the following computation:

$\displaystyle
e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]\epsilon_{k}^{(n)}$
$\displaystyle=e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right](\epsilon_{k}^{(n)})^{2}$
$\displaystyle=\epsilon_{k}^{(n)}e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]\epsilon_{k}^{(n)}$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{-\ell(\sigma)}\epsilon_{k}^{(n)}T_{\sigma}t^{rn}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}T_{\sigma^{-1}}\epsilon_{k}^{(n)}$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{-\ell(\sigma)}\epsilon_{k}^{(n)}t^{rn}Y_{k+1}^{(n)}\cdots
Y_{k+r}^{(n)}\epsilon_{k}^{(n)}$
$\displaystyle=\left(\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{-\ell(\sigma)}\right)\epsilon_{k}^{(n)}t^{(n-k)+\ldots+(n-k-r+1)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots T_{r})\pi_{n}^{r}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times(T_{n-r}^{-1}\cdots T_{k+1}^{-1})\cdots(T_{n-1}^{-1}\cdots
T_{k+r}^{-1})\epsilon_{k}^{(n)}$
$\displaystyle=\left(\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{-\ell(\sigma)}\right)\epsilon_{k}^{(n)}t^{(n-k)+\ldots+(n-k-r+1)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots
T_{r})\pi_{n}^{r}\epsilon_{k}^{(n)}$ $\displaystyle=t^{{r\choose
2}+r}\left(\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}/\mathfrak{S}_{(1^{k},r,n-k-r)}}t^{{n-k\choose
2}-{n-k-r\choose 2}-{r\choose
2}-\ell(\sigma)}\right)\epsilon_{k}^{(n)}(T_{k}\cdots T_{1})(T_{k+1}\cdots
T_{2})\cdots(T_{k+r-1}\cdots T_{r})\pi_{n}^{r}\epsilon_{k}^{(n)}$
$\displaystyle=t^{{r+1\choose
2}}\frac{[n-k]_{t}!}{[n-k-r]_{t}![r]_{t}!}\epsilon_{k}^{(n)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots
T_{r})\pi_{n}^{r}\epsilon_{k}^{(n)}$ $\displaystyle=t^{{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots
T_{r})\pi_{n}^{r}\epsilon_{k}^{(n)}.$

∎

######  Remark 5.

Note that Lemma 3.6 shows that in any $\operatorname{\mathscr{H}}_{n}^{+}$
module V, the operators
$e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]$ preserve the
subspace $\epsilon_{k}^{(n)}V.$

The next result will be important for the proof of Theorem 3.9 where we will
need to argue that the operator
$e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]$ preserves the
space $x_{1}\ldots
x_{k}\mathbb{Q}(q,t)[x_{1},\ldots,x_{n}]^{\mathfrak{S}_{(1^{k},n-k)}}$ in the
polynomial representation of $\operatorname{\mathscr{H}}_{n}^{+}.$

######  Lemma 3.7.

$\displaystyle
e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]\epsilon_{k}^{(n)}X_{1}\cdots
X_{k}$ $\displaystyle=X_{1}\cdots X_{k}t^{rk+{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}^{-1}\cdots
T_{1}^{-1})(T_{k+1}^{-1}\cdots T_{2}^{-1})\cdots(T_{k+r-1}^{-1}\cdots
T_{r}^{-1})\pi_{n}^{r}\epsilon_{k}^{(n)}$

###### Proof.

$\displaystyle
e_{r}\left[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}\right]\epsilon_{k}^{(n)}X_{1}\cdots
X_{k}$ $\displaystyle=t^{{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots
T_{r})\pi_{n}^{r}\epsilon_{k}^{(n)}X_{1}\cdots X_{k}$
$\displaystyle=t^{{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots T_{r})\pi_{n}^{r}X_{1}\cdots
X_{k}\epsilon_{k}^{(n)}$ $\displaystyle=t^{{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots T_{r})X_{r+1}\cdots
X_{k+r}\pi_{n}^{r}\epsilon_{k}^{(n)}.$

Further,

$\displaystyle T_{k+r-1}\cdots T_{r}X_{r+1}\cdots X_{r+k}$
$\displaystyle=T_{k+r-1}\cdots T_{r+1}(T_{r}X_{r+1})X_{r+2}\cdots X_{r+k}$
$\displaystyle=T_{k+r-1}\cdots T_{r+1}(tX_{r}T_{r}^{-1})X_{r+2}\cdots X_{r+k}$
$\displaystyle=tT_{k+r-1}\cdots T_{r+1}X_{r}X_{r+2}\cdots X_{r+k}T_{r}^{-1}$
$\displaystyle=tX_{r}T_{k+r-1}\cdots T_{r+1}X_{r+2}\cdots X_{r+k}T_{r}^{-1}$
$\displaystyle=tX_{r}T_{k+r-1}\cdots T_{r+2}tX_{r+1}T_{r+1}^{-1}X_{r+3}\cdots
X_{r+k}T_{r}^{-1}$ $\displaystyle=t^{2}X_{r}X_{r+1}T_{k+r-1}\cdots
T_{r+2}X_{r+3}\cdots X_{r+k}T_{r+1}^{-1}T_{r}^{-1}$ $\displaystyle=\ldots$
$\displaystyle=t^{k}X_{r}\cdots X_{r+k-1}T_{k+r-1}^{-1}\cdots T_{r}^{-1}.$

By applying this argument repeatedly we find that

$\displaystyle(T_{k}\cdots T_{1})\cdots(T_{k+r-2}\cdots
T_{r-1})(T_{k+r-1}\cdots T_{r})X_{r+1}\cdots X_{k+r}$
$\displaystyle=t^{k}(T_{k}\cdots T_{1})\cdots(T_{k+r-2}\cdots
T_{r-1})X_{r}\cdots X_{k+r-1}(T_{k+r-1}^{-1}\cdots T_{r}^{-1})$
$\displaystyle=\ldots$ $\displaystyle=t^{rk}X_{1}\cdots X_{k}(T_{k}^{-1}\cdots
T_{1}^{-1})\cdots(T_{k+r-1}^{-1}\cdots T_{r}^{-1})$

so therefore,

$\displaystyle t^{{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+r-1}\cdots T_{r})X_{r+1}\cdots
X_{k+r}\pi_{n}^{r}\epsilon_{k}^{(n)}$ $\displaystyle=t^{{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}t^{rk}X_{1}\cdots
X_{k}(T_{k}^{-1}\cdots T_{1}^{-1})\cdots(T_{k+r-1}^{-1}\cdots
T_{r}^{-1})\pi_{n}^{r}\epsilon_{k}^{(n)}$ $\displaystyle=X_{1}\cdots
X_{k}t^{rk+{r+1\choose
2}}\left(\frac{1-t^{n-k-r+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{r}}\right)\epsilon_{k}^{(n)}(T_{k}^{-1}\cdots
T_{1}^{-1})\cdots(T_{k+r-1}^{-1}\cdots
T_{r}^{-1})\pi_{n}^{r}\epsilon_{k}^{(n)}.$

∎

Lastly, we have the standard coproduct formula for the elementary symmetric
functions.

######  Lemma 3.8.

$e_{r}[X+Y]=\sum_{s=0}^{r}e_{s}[X]e_{r-s}[Y].$

###### Proof.

This result is standard. ∎

###  3.2 Proof of Convergence

We will now use all of the lemmas proven above to show the following result:

######  Proposition 3.9.

For $r\geq 1$ the sequence of operators $(\Psi_{e_{r}}^{(n)})_{n\geq 1}$
converges to an operator $e_{r}[\Delta]$ on
$\operatorname{\mathscr{P}}_{as}^{+}.$ The operators $e_{r}[\Delta]$ satisfy
the following properties:

  * •

$e_{r}[\Delta](\widetilde{E}_{(\mu|\lambda)})=e_{r}[\kappa_{\operatorname{sort}(\mu*\lambda)}(q,t)]\widetilde{E}_{(\mu|\lambda)}$

  * •

$[e_{r}[\Delta],\mathscr{Y}_{i}]=0$

  * •

$[e_{r}[\Delta],T_{i}]=0$

  * •

$[e_{r}[\Delta],e_{s}[\Delta]]=0$

  * •

$e_{r}[\Delta]|_{x_{1}\cdots
x_{k}\operatorname{\mathscr{P}}(k)^{+}}=\sum_{s=0}^{r}\left[\prod_{i=1}^{s}\left(\frac{t^{i}}{1-t^{i}}\right)\right]e_{r-s}(\mathscr{Y}_{1},\ldots,\mathscr{Y}_{k})\epsilon_{k}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+s-1}\cdots T_{s})\pi^{s}.$

###### Proof.

The structure of the following argument is similar to the proof of Theorem 59
in [BW23].

Notice that every element of $\operatorname{\mathscr{P}}_{as}^{+}$ is a finite
$\mathbb{Q}(q,t)$-linear combination of terms of the form
$T_{\sigma}x^{\lambda}F[X]$ where $\sigma$ is a permutation, $\lambda$ is a
partition, and $F\in\Lambda$. Therefore, to show that the sequence of
operators $(\Psi_{e_{r}}^{(n)})_{n\geq 1}$ converges it suffices to show that
sequences of the form

$(\Psi_{e_{r}}^{(n)}(T_{\sigma}x^{\lambda}F[x_{1}+\ldots+x_{n}]))_{n\geq 1}$

converge. For $n$ sufficiently large, $T_{\sigma}$ commutes with
$\Psi_{e_{r}}^{(n)}=e_{r}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]$ so it
suffices to consider only sequences of the form

$(\Psi_{e_{r}}^{(n)}(x^{\lambda}F[x_{1}+\ldots+x_{n}]))_{n\geq 1}.$

Let $\lambda$ be a partition, $k:=\ell(\lambda)$, $F\in\Lambda$, and take
$n>k+r$. Set $\lambda^{\prime}:=(\lambda_{1}-1,\ldots,\lambda_{k}-1).$ Recall
that $\widetilde{Y}_{1}^{(n)}X_{1}=t^{n}Y_{1}^{(n)}X_{1}$ from which it
follows directly that $\widetilde{Y}_{i}^{(n)}X_{i}=t^{n}Y_{i}^{(n)}X_{i}$ for
all $1\leq i\leq n$. Then for all $1\leq i\leq k$ we have that

$t^{n}Y_{i}^{(n)}X_{1}\cdots X_{k}=\widetilde{Y}_{i}^{(n)}X_{1}\cdots X_{k}.$

This means that as operators on $x_{1}\ldots
x_{k}\operatorname{\mathscr{P}}_{k}^{+}$,
$t^{n}Y_{i}^{(n)}=\widetilde{Y}_{i}^{(n)}.$ Note that these operators preserve
the subspace $x_{1}\ldots x_{k}\operatorname{\mathscr{P}}_{k}^{+}$. Further,
we may naturally extend this argument to show that as operators on
$x_{1}\cdots x_{k}\operatorname{\mathscr{P}}_{k}^{+}$ for any
$a_{1},\ldots,a_{k}\geq 0$, and any permutation $\gamma\in\mathfrak{S}_{k}$,

$(t^{n}Y_{\gamma(1)}^{(n)})^{a_{1}}\cdots(t^{n}Y_{\gamma(k)}^{(n)})^{a_{k}}|_{x_{1}\cdots
x_{k}\operatorname{\mathscr{P}}_{k}^{+}}=(\widetilde{Y}_{\gamma(1)}^{(n)})^{a_{1}}\cdots(\widetilde{Y}_{\gamma(k)}^{(n)})^{a_{k}}|_{x_{1}\cdots
x_{k}\operatorname{\mathscr{P}}_{k}^{+}}.$

This is notable because, unlike the Cherednik operators $Y_{i}^{(n)},$ the
deformed Cherednik operators $\widetilde{Y}_{i}^{(n)}$ do not mutually
commute. Therefore again as operators on $\operatorname{\mathscr{P}}_{k}^{+}$,
for all $0\leq s\leq r$

$e_{r-s}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{k}^{(n)}]X_{1}\cdots
X_{k}=e_{r-s}(\widetilde{Y}_{1}^{(n)},\ldots,\widetilde{Y}_{k}^{(n)})X_{1}\cdots
X_{k}.$

Using Lemma 3.8 we now find the following:

$\displaystyle
e_{r}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}](x^{\lambda}F[x_{1}+\ldots+x_{n}])$
$\displaystyle=e_{r}[(t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{k}^{(n)})+(t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)})](x^{\lambda}F[x_{1}+\ldots+x_{n}])$
$\displaystyle=\sum_{s=0}^{r}e_{r-s}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{k}^{(n)}]e_{s}[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}](x^{\lambda}F[x_{1}+\ldots+x_{n}]).$

Importantly, since $x^{\lambda}F[x_{1}+\ldots+x_{n}]$ is symmetric in
$k+1,\ldots,n$,

$\epsilon_{k}^{(n)}(x^{\lambda}F[x_{1}+\ldots+x_{n}])=x^{\lambda}F[x_{1}+\ldots+x_{n}]$

so that by using Lemmas 3.6 and 3.7 we find

$\displaystyle\sum_{s=0}^{r}e_{r-s}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{k}^{(n)}]e_{s}[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}](x^{\lambda}F[x_{1}+\ldots+x_{n}])$
$\displaystyle=\sum_{s=0}^{r}e_{r-s}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{k}^{(n)}]e_{s}[t^{n}Y_{k+1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]\epsilon_{k}^{(n)}X_{1}\cdots
X_{k}(x^{\lambda^{\prime}}F[x_{1}+\ldots+x_{n}])$
$\displaystyle=\sum_{s=0}^{r}e_{r-s}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{k}^{(n)}]X_{1}\cdots
X_{k}t^{sk+{s+1\choose
2}}\left(\frac{1-t^{n-k-s+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{s}}\right)\epsilon_{k}^{(n)}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times(T_{k}^{-1}\cdots T_{1}^{-1})(T_{k+1}^{-1}\cdots
T_{2}^{-1})\cdots(T_{k+s-1}^{-1}\cdots
T_{s}^{-1})\pi_{n}^{r}\epsilon_{k}^{(n)}(x^{\lambda^{\prime}}F[x_{1}+\ldots+x_{n}])$
$\displaystyle=\sum_{s=0}^{r}e_{r-s}(t^{n}\widetilde{Y}_{1}^{(n)},\ldots,t^{n}\widetilde{Y}_{k}^{(n)})X_{1}\cdots
X_{k}t^{sk+{s+1\choose
2}}\left(\frac{1-t^{n-k-s+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{s}}\right)\epsilon_{k}^{(n)}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times(T_{k}^{-1}\cdots T_{1}^{-1})(T_{k+1}^{-1}\cdots
T_{2}^{-1})\cdots(T_{k+s-1}^{-1}\cdots
T_{s}^{-1})\pi_{n}^{r}\epsilon_{k}^{(n)}(x^{\lambda^{\prime}}F[x_{1}+\ldots+x_{n}])$
$\displaystyle=\sum_{s=0}^{r}e_{r-s}(t^{n}\widetilde{Y}_{1}^{(n)},\ldots,t^{n}\widetilde{Y}_{k}^{(n)})t^{{s+1\choose
2}}\left(\frac{1-t^{n-k-s+1}}{1-t}\right)\cdots\left(\frac{1-t^{n-k}}{1-t^{s}}\right)\epsilon_{k}^{(n)}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times(T_{k}\cdots T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+s-1}\cdots
T_{s})\pi_{n}^{s}(x^{\lambda}F[x_{1}+\ldots+x_{n}]).$

From here it is clear that

$\displaystyle\lim_{n}\Psi_{e_{r}}^{(n)}(x^{\lambda}F[x_{1}+\ldots+x_{n}])$
$\displaystyle=\sum_{s=0}^{r}\left[\prod_{i=1}^{s}\left(\frac{t^{i}}{1-t^{i}}\right)\right]e_{r-s}(\mathscr{Y}_{1},\ldots,\mathscr{Y}_{k})\epsilon_{k}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+s-1}\cdots
T_{s})\pi^{s}(x^{\lambda}F[X])$

which is evidently an element of
$\operatorname{\mathscr{P}}(k)^{+}\subset\operatorname{\mathscr{P}}_{as}^{+}.$
Therefore, the sequence of operators $(\Psi_{e_{r}}^{(n)})_{n\geq 1}$
converges to an operator on $\operatorname{\mathscr{P}}_{as}^{+}$ which we
will call $e_{r}[\Delta].$

We will now prove various properties of $e_{r}[\Delta].$ For all $1\leq i\leq
k-1$ and $0\leq s\leq r$

$\displaystyle\epsilon_{k}(T_{k}\cdots T_{1})(T_{k+1}\cdots
T_{2})\cdots(T_{k+s-1}\cdots T_{s})\pi^{s}T_{i}$
$\displaystyle=\epsilon_{k}(T_{k}\cdots T_{1})(T_{k+1}\cdots
T_{2})\cdots(T_{k+s-1}\cdots T_{s})T_{i+s}\pi^{s}$
$\displaystyle=\epsilon_{k}(T_{k}\cdots T_{1})(T_{k+1}\cdots
T_{2})\cdots(T_{k+s-2}\cdots T_{s-2})T_{i+s-1}(T_{k+s-1}\cdots T_{s})\pi^{s}$
$\displaystyle=\ldots$ $\displaystyle=\epsilon_{k}T_{i}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+s-1}\cdots T_{s})\pi^{s}$
$\displaystyle=T_{i}\epsilon_{k}(T_{k}\cdots T_{1})(T_{k+1}\cdots
T_{2})\cdots(T_{k+s-1}\cdots T_{s})\pi^{s}.$

Therefore, for any $f\in x_{1}\cdots x_{k}\operatorname{\mathscr{P}}(k)^{+}$,
by expanding $f$ into a sum of terms of the form $T_{\sigma}x^{\lambda}F[X]$
where $\sigma\in\mathfrak{S}_{k}$ and $\lambda$ is a partition with
$\ell(\lambda)=k$, we find that

$e_{r}[\Delta](f)=\sum_{s=0}^{r}\prod_{i=1}^{s}\left(\frac{t^{i}}{1-t^{i}}\right)e_{r-s}(\mathscr{Y}_{1},\ldots,\mathscr{Y}_{k})\epsilon_{k}(T_{k}\cdots
T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+s-1}\cdots T_{s})\pi^{s}(f).$

Now let $(\mu|\lambda)\in\Phi.$ Using Corollary 47 in [BW23] (see Definition
2.16) and Proposition 2.12 we have that

$\displaystyle e_{r}[\Delta](\widetilde{E}_{(\mu|\lambda)})$
$\displaystyle=\lim_{n}e_{r}[t^{n}Y_{1}^{(n)}+\cdots+t^{n}Y_{n}^{(n)}](\epsilon_{\ell(\mu)}^{(n)}(E_{\mu*\lambda*0^{n-(\ell(\mu)+\ell(\lambda))}}))$
$\displaystyle=\lim_{n}\epsilon_{\ell(\mu)}^{(n)}e_{r}[t^{n}Y_{1}^{(n)}+\cdots+t^{n}Y_{n}^{(n)}](E_{\mu*\lambda*0^{n-(\ell(\mu)+\ell(\lambda))}})$
$\displaystyle=\lim_{n}\epsilon_{\ell(\mu)}^{(n)}\left(e_{r}\left[\sum_{i=1}^{n}q^{\operatorname{sort}(\mu*\lambda)_{i}}t^{i}\right]E_{\mu*\lambda*0^{n-(\ell(\mu)+\ell(\lambda))}}\right)$
$\displaystyle=\lim_{n}e_{r}\left[\sum_{i=1}^{n}q^{\operatorname{sort}(\mu*\lambda)_{i}}t^{i}\right]\epsilon_{\ell(\mu)}^{(n)}(E_{\mu*\lambda*0^{n-(\ell(\mu)+\ell(\lambda))}})$
$\displaystyle=e_{r}[\kappa_{\operatorname{sort}(\mu*\lambda)}(q,t)]\widetilde{E}_{(\mu|\lambda)}.$

To see that $[e_{r}[\Delta],T_{i}]=0$ we may check directly:

$\displaystyle e_{r}[\Delta]T_{i}$
$\displaystyle=\lim_{n}e_{r}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]T_{i}$
$\displaystyle=\lim_{n}T_{i}e_{r}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]$
$\displaystyle=T_{i}e_{r}[\Delta].$

Lastly, since the $\widetilde{E}_{(\mu|\lambda)}$ are a basis of
$\operatorname{\mathscr{P}}_{as}^{+}$ (Theorem 2.17) it follows that for all
$i,r,s\geq 1$,

  * •

$[e_{r}[\Delta],\mathscr{Y}_{i}]=0$

  * •

$[e_{r}[\Delta],e_{s}[\Delta]]=0.$

∎

######  Remark 6.

Note that

$(T_{k}\cdots T_{1})(T_{k+1}\cdots T_{2})\cdots(T_{k+s-1}\cdots
T_{s})\pi^{s}=(T_{k}\cdots T_{1}\pi)^{s}$

so we have that

$e_{r}[\Delta]|_{x_{1}\cdots
x_{k}\operatorname{\mathscr{P}}(k)^{+}}=\sum_{s=0}^{r}\left[\prod_{i=1}^{s}\left(\frac{t^{i}}{1-t^{i}}\right)\right]e_{r-s}(\mathscr{Y}_{1},\ldots,\mathscr{Y}_{k})\epsilon_{k}(T_{k}\cdots
T_{1}\pi)^{s}.$

As an immediate consequence we have the following result confirming the
conjecture posed in [BW23].

######  Theorem 3.10.

For any symmetric function $F\in\Lambda$ the sequence of operators
$(\Psi_{F}^{(n)})_{n\geq 1}$ converges to an operator on
$\operatorname{\mathscr{P}}_{as}^{+}$ which we may call $F[\Delta].$ These
operators satisfy the following properties:

  * •

$F[\Delta](\widetilde{E}_{(\mu|\lambda)})=F[\kappa_{\operatorname{sort}(\mu*\lambda)}(q,t)]\widetilde{E}_{(\mu|\lambda)}$

  * •

$[F[\Delta],\mathscr{Y}_{i}]=0$

  * •

$[F[\Delta],T_{i}]=0$

  * •

$[F[\Delta],G[\Delta]]=0.$

###### Proof.

Recall that the ring of symmetric functions $\Lambda$ is generated
algebraically by the elementary symmetric polynomials $e_{1},e_{2},\ldots.$
For any $F\in\Lambda$ we may write $F=f(e_{1},e_{2},\ldots,e_{r})$ so that for
all $n\geq 1$

$\Psi_{F}^{(n)}=f(\Psi_{e_{1}}^{(n)},\ldots,\Psi_{e_{r}}^{(n)}).$

By applying Propositions 2.13 and 3.9 we find that $(\Psi_{F}^{(n)})_{n\geq
1}$ converges and that

$F[\Delta]:=\lim_{n}\Psi_{F}^{(n)}=f(\lim_{n}\Psi_{e_{1}}^{(n)},\ldots,\lim_{n}\Psi_{e_{r}}^{(n)})=f(e_{1}[\Delta],\ldots,e_{r}[\Delta]).$

For $(\mu|\lambda)\in\Phi$ we see that

$\displaystyle F[\Delta](\widetilde{E}_{(\mu|\lambda)})$
$\displaystyle=f(e_{1}[\Delta],\ldots,e_{r}[\Delta])(\widetilde{E}_{(\mu|\lambda)})$
$\displaystyle=f(e_{1}[\kappa_{\operatorname{sort}(\mu*\lambda)}(q,t)],\ldots,e_{r}[\kappa_{\operatorname{sort}(\mu*\lambda)}(q,t)])(\widetilde{E}_{(\mu|\lambda)})$
$\displaystyle=F[\kappa_{\operatorname{sort}(\mu*\lambda)}(q,t)]\widetilde{E}_{(\mu|\lambda)}.$

The other properties follow directly from Theorem 2.17 and Proposition 3.9.

∎

######  Example.

The operator $e_{2}[\Delta]$ acts on
$x_{1}x_{2}x_{3}\operatorname{\mathscr{P}}(3)^{+}$ as

$(\mathscr{Y}_{1}\mathscr{Y}_{2}+\mathscr{Y}_{1}\mathscr{Y}_{3}+\mathscr{Y}_{2}\mathscr{Y}_{3})+\frac{t}{1-t}(\mathscr{Y}_{1}+\mathscr{Y}_{2}+\mathscr{Y}_{3})\epsilon_{3}T_{3}T_{2}T_{1}\pi+\frac{t^{3}}{(1-t)(1-t^{2})}\epsilon_{3}T_{3}T_{2}T_{1}T_{4}T_{3}T_{2}\pi^{2}.$

If we instead consider $e_{4}[\Delta]$ acting on
$\operatorname{\mathscr{P}}(0)^{+}=\Lambda[X]$ then we get

$\frac{t^{10}}{(1-t)(1-t^{2})(1-t^{3})(1-t^{4})}\epsilon\pi^{4}.$

As an example computation we have that

$p_{2}[\Delta](\widetilde{E}_{(4,1,2|5,4,2,2,1)})=\left(q^{10}t^{2}+q^{8}t^{4}+q^{8}t^{6}+q^{4}t^{8}+q^{4}t^{10}+q^{4}t^{12}+q^{2}t^{14}+q^{2}t^{16}+t^{18}+t^{20}+\ldots\right)\widetilde{E}_{(4,1,2|5,4,2,2,1)}.$

######  Remark 7.

It is easy to see that for all $F\in\Lambda$,
$\epsilon_{k}F[\Delta]=F[\Delta]\epsilon_{k}$ by direct calculation:

$\displaystyle\epsilon_{k}F[\Delta]$
$\displaystyle=\lim_{n}\epsilon_{k}^{(n)}\Psi_{F}^{(n)}$
$\displaystyle=\lim_{n}\frac{1}{[n-k]_{t}!}\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}}t^{{n-k\choose
2}-\ell(\sigma)}T_{\sigma}F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]$
$\displaystyle=\lim_{n}\frac{1}{[n-k]_{t}!}\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}}t^{{n-k\choose
2}-\ell(\sigma)}F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]T_{\sigma}$
$\displaystyle=\lim_{n}F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]\frac{1}{[n-k]_{t}!}\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}}t^{{n-k\choose
2}-\ell(\sigma)}T_{\sigma}$ $\displaystyle=F[\Delta]\epsilon_{k}.$

###  3.3 Commutation Relations

In this section we compute some of the commutation relations between the
$\Delta$-operators $F[\Delta]$ and the operator $\widetilde{\pi}$ on
$\operatorname{\mathscr{P}}_{as}^{+}$. These relations are conceptually
important because in the case of the finite rank DAHA in type
$\operatorname{\mathrm{GL}}_{n}$ the analogous commutation relations allow for
one to develop a theory of intertwiners and the Knop-Sahi relations for non-
symmetric Macdonald polynomials.

We start with the following result which will follow easily using the
properties of Ion-Wu limits and particularly Proposition 2.13.

######  Proposition 3.11.

For $F\in\Lambda$

$\widetilde{\pi}F[\Delta]=F[\Delta+(q^{-1}-1)\mathscr{Y}_{1}]\widetilde{\pi}.$

###### Proof.

Let $F\in\Lambda$ and $G_{i},H_{i}\in\Lambda$ such that

$F[X+Y]=\sum_{i}G_{i}[X]H_{i}[Y].$

We may compute directly:

$\displaystyle\widetilde{\pi}F[\Delta]$
$\displaystyle=\lim_{n}\widetilde{\pi}_{n}F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]$
$\displaystyle=\lim_{n}F[t^{n}Y_{2}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}+q^{-1}t^{n}Y_{1}^{(n)}]\widetilde{\pi}_{n}$
$\displaystyle=\lim_{n}F[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}+(q^{-1}-1)t^{n}Y_{1}^{(n)}]\widetilde{\pi}_{n}$
$\displaystyle=\lim_{n}\sum_{i}G_{i}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]H_{i}[(q^{-1}-1)t^{n}Y_{1}^{(n)}]\widetilde{\pi}_{n}$
$\displaystyle=\lim_{n}\sum_{i}G_{i}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]H_{i}[(q^{-1}-1)t^{n}Y_{1}^{(n)}]X_{1}T_{1}^{-1}\cdots
T_{n-1}^{-1}$
$\displaystyle=\lim_{n}\sum_{i}G_{i}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]H_{i}[(q^{-1}-1)\widetilde{Y}_{1}^{(n)}]X_{1}T_{1}^{-1}\cdots
T_{n-1}^{-1}$
$\displaystyle=\lim_{n}\sum_{i}G_{i}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]H_{i}[(q^{-1}-1)\widetilde{Y}_{1}^{(n)}]\widetilde{\pi}_{n}$
$\displaystyle=\sum_{i}\left(\lim_{n}G_{i}[t^{n}Y_{1}^{(n)}+\ldots+t^{n}Y_{n}^{(n)}]\right)\left(\lim_{n}H_{i}[(q^{-1}-1)\widetilde{Y}_{1}^{(n)}]\right)\left(\lim_{n}\widetilde{\pi}_{n}\right)$
$\displaystyle=\sum_{i}G_{i}[\Delta]H_{i}[(q^{-1}-1)\mathscr{Y}_{1}]\widetilde{\pi}$
$\displaystyle=F[\Delta+(q^{-1}-1)\mathscr{Y}_{1}]\widetilde{\pi}.$

∎

As an immediate consequence we find the following.

######  Corollary 3.12.

For every $r\geq 1$

$[\widetilde{\pi},p_{r}[\Delta]]=(q^{-r}-1)\mathscr{Y}_{1}^{r}\widetilde{\pi}.$

###### Proof.

Using Proposition 3.11 applied to $F[X]=p_{r}[X]$ gives

$\widetilde{\pi}p_{r}[\Delta]=p_{r}[\Delta+(q^{-1}-1)\mathscr{Y}_{1}]\widetilde{\pi}=(p_{r}[\Delta]+(q^{-r}-1)\mathscr{Y}_{1}^{r})\widetilde{\pi}.$

∎

######  Remark 8.

More generally, we may write for any $F\in\Lambda$

$[\widetilde{\pi},F[\Delta]]=\left(F[\Delta+(q^{-1}-1)\mathscr{Y}_{1}]-F[\Delta]\right)\widetilde{\pi}.$

Lastly, we compute the full commutation relations between the limit Cherednik
operators $\mathscr{Y}_{i}$ and $\widetilde{\pi}.$ Interestingly, most of
these relations mimic the standard finite rank DAHA situation except for
$\mathscr{Y}_{1}\widetilde{\pi}$ which now involves $\Delta.$

######  Proposition 3.13.

$\mathscr{Y}_{i}\widetilde{\pi}=\begin{cases}\widetilde{\pi}\mathscr{Y}_{i-1}&i>1\\\
\frac{\widetilde{\pi}\Delta-\Delta\widetilde{\pi}}{q^{-1}-1}&i=1.\\\
\end{cases}$

###### Proof.

For $i=1$ we have that from Proposition 3.11

$\widetilde{\pi}\Delta-\Delta\widetilde{\pi}=(q^{-1}-1)\mathscr{Y}_{1}\widetilde{\pi}$

and hence

$\mathscr{Y}_{1}\widetilde{\pi}=\frac{\widetilde{\pi}\Delta-\Delta\widetilde{\pi}}{q^{-1}-1}.$

Let $i>1.$ We see that

$\displaystyle\mathscr{Y}_{i}\widetilde{\pi}$
$\displaystyle=\lim_{n}\widetilde{Y}_{i}^{(n)}\widetilde{\pi}_{n}$
$\displaystyle=\lim_{n}\left(t^{n-i+1}T_{i-1}\cdots
T_{1}\rho\pi_{n}T_{n-1}^{-1}\cdots
T_{i}^{-1}\right)\left(X_{1}T_{1}^{-1}\cdots T_{n-1}^{-1}\right)$
$\displaystyle=\lim_{n}t^{n-i+1}T_{i-1}\cdots
T_{1}\rho\pi_{n}X_{1}T_{n-1}^{-1}\cdots T_{i}^{-1}T_{1}^{-1}\cdots
T_{n-1}^{-1}$ $\displaystyle=\lim_{n}t^{n-i+1}T_{i-1}\cdots T_{1}\rho
X_{2}\pi_{n}T_{n-1}^{-1}\cdots T_{i}^{-1}T_{1}^{-1}\cdots T_{n-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+1}T_{i-1}\cdots
T_{1}X_{2}\rho\pi_{n}T_{n-1}^{-1}\cdots T_{i}^{-1}T_{1}^{-1}\cdots
T_{n-1}^{-1}$ $\displaystyle=\lim_{n}t^{n-i+1}T_{i-1}\cdots
T_{2}(tX_{1}T_{1}^{-1})\rho\pi_{n}T_{n-1}^{-1}\cdots
T_{i}^{-1}T_{1}^{-1}\cdots T_{n-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+2}X_{1}T_{i-1}\cdots
T_{2}T_{1}^{-1}\rho\pi_{n}T_{1}^{-1}\cdots T_{i-2}^{-1}T_{n-1}^{-1}\cdots
T_{i}^{-1}T_{i-1}^{-1}\cdots T_{n-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+2}X_{1}T_{i-1}\cdots T_{2}T_{1}^{-1}\rho
T_{2}^{-1}\cdots T_{i-1}^{-1}\pi_{n}T_{n-1}^{-1}\cdots
T_{i}^{-1}T_{i-1}^{-1}\cdots T_{n-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+2}X_{1}T_{i-1}\cdots
T_{2}T_{1}^{-1}T_{2}^{-1}\cdots T_{i-1}^{-1}\rho\pi_{n}T_{n-1}^{-1}\cdots
T_{i}^{-1}T_{i-1}^{-1}\cdots T_{n-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+2}X_{1}T_{1}^{-1}\cdots
T_{i-1}^{-1}T_{i-2}\cdots T_{1}\rho\pi_{n}T_{i-1}^{-1}\cdots
T_{n-2}^{-1}T_{n-1}^{-1}\cdots T_{i-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+2}X_{1}T_{1}^{-1}\cdots
T_{i-1}^{-1}T_{i-2}\cdots T_{1}\rho T_{i}^{-1}\cdots
T_{n-1}^{-1}\pi_{n}T_{n-1}^{-1}\cdots T_{i-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+2}X_{1}T_{1}^{-1}\cdots
T_{i-1}^{-1}T_{i-2}\cdots T_{1}T_{i}^{-1}\cdots
T_{n-1}^{-1}\rho\pi_{n}T_{n-1}^{-1}\cdots T_{i-1}^{-1}$
$\displaystyle=\lim_{n}t^{n-i+2}X_{1}T_{1}^{-1}\cdots
T_{i-1}^{-1}T_{i}^{-1}\cdots T_{n-1}^{-1}T_{i-2}\cdots
T_{1}\rho\pi_{n}T_{n-1}^{-1}\cdots T_{i-1}^{-1}$
$\displaystyle=\lim_{n}\left(X_{1}T_{1}^{-1}\cdots
T_{n-1}^{-1}\right)\left(t^{n-i+2}T_{i-2}\cdots
T_{1}\rho\pi_{n}T_{n-1}^{-1}\cdots T_{i-1}^{-1}\right)$
$\displaystyle=\lim_{n}\widetilde{\pi}_{n}\widetilde{Y}_{i-1}^{(n)}$
$\displaystyle=\widetilde{\pi}\mathscr{Y}_{i-1}.$

∎

###  3.4 $\mathbb{B}_{q,t}^{\text{ext}}$ Action

We refer the reader to [CGM20] and [GGS23] for relevant background surrounding
the algebra $\mathbb{B}_{q,t}.$ Importantly, in their conventions $q,t$ have
swapped roles. We will show in this short section how to relate the
$\Delta$-operators $F[\Delta]$ constructed in this paper to those defined by
González-Gorsky-Simental.

Let us begin by recalling the definition of the extended algebra
$\mathbb{B}_{q,t}^{\text{ext}}$.

######  Definition 3.14.

The algebra $\mathbb{B}_{q,t}^{\text{ext}}$ is generated by $\mathbb{B}_{q,t}$
along with loops at each vertex $k\geq 0$ labelled by $\Delta_{p_{m}}$ for all
$m\geq 1$ such that

  * •

$[\Delta_{p_{m}},\Delta_{p_{\ell}}]=0$

  * •

$[\Delta_{p_{m}},T_{i}]=[\Delta_{p_{m}},z_{i}]=[\Delta_{p_{m}},d_{-}]=0$

  * •

$[\Delta_{p_{m}},d_{+}]=z_{1}^{m}d_{+}.$

We begin by noting that the commutation relation in Corollary 3.12 is very
similar to the defining relations for $\mathbb{B}_{q,t}^{\text{ext}}$ but not
exactly identical. To better match conventions we may consider for $m\geq 1$
the normalized operators

$\Delta_{p_{m}}:=\frac{q^{m}}{q^{m}-1}\left(p_{m}[\Delta]-\frac{t^{m}}{1-t^{m}}\right).$

Note now that for any $(\mu|\lambda)\in\Phi$ and $m\geq 1$

$\Delta_{p_{m}}(\widetilde{E}_{(\mu|\lambda)})=(qt)^{m}\left(\sum_{i\geq
1}\left(\frac{q^{m\operatorname{sort}(\mu*\lambda)_{i}}-1}{q^{m}-1}\right)t^{m(i-1)}\right)\widetilde{E}_{(\mu|\lambda)}.$

In particular, $\Delta_{p_{m}}(1)=0$ for all $m\geq 1.$ Using Theorem 3.10,
Remark 7, and Corollary 3.12 the usual action of $\mathbb{B}_{q,t}$ (see the
author’s prior paper for more details [Wei24]) along with these new
$\Delta$-operators $\Delta_{p_{m}}$ define a representation of
$\mathbb{B}_{q,t}^{\text{ext}}$ on $L_{\bullet}:=\bigoplus_{k\geq
0}x_{1}\cdots x_{k}\operatorname{\mathscr{P}}(k)^{+}$. In particular, since

$[\widetilde{\pi},p_{m}[\Delta]]=(q^{-m}-1)\mathscr{Y}_{1}^{m}\widetilde{\pi}$

we see that

$[\Delta_{p_{m}},\widetilde{\pi}]=\mathscr{Y}_{1}^{m}\widetilde{\pi}.$

From the work of Ion-Wu [IW22] we know there exists a $\mathbb{B}_{q,t}$
module isomorphism $L_{\bullet}\rightarrow V_{\bullet}$ given by

$x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}F[x_{k+1}+x_{k+2}+\ldots]\rightarrow
y_{1}^{a_{1}-1}\cdots y_{k}^{a_{k}-1}F\left[\frac{X}{t-1}\right]$

where

$V_{\bullet}:=\bigoplus_{k\geq
0}\mathbb{Q}(q,t)[y_{1},\ldots,y_{k}]\otimes\Lambda[X]$

is the $\mathbb{B}_{q,t}$ module originally considered by Carlsson-Gorsky-
Mellit [CGM20]. Now we may use the following lemma:

######  Lemma 3.15.

[GGS23] The $\mathbb{B}_{q,t}$ polynomial representation $V_{\bullet}$ is
generated by $1\in V_{0}.$

From this González-Gorsky-Simental find that the
$\mathbb{B}_{q,t}^{\text{ext}}$ module structure on $V_{\bullet}$ is uniquely
determined by the commutation relations for the $\Delta_{p_{m}}$ operators in
the following way:

######  Corollary 3.16.

[GGS23] We can uniquely define the operators $\Delta_{p_{m}}$ on the
polynomial representation $V_{\bullet}$ by setting $\Delta_{p_{m}}(1)=0$ and
extending via Lemma 3.15 using the relations in Definition 3.14.

Putting together all of the work in this section thus far we get the following
result.

######  Corollary 3.17.

The Ion-Wu map $L_{\bullet}\rightarrow V_{\bullet}$ is an isomorphism of
$\mathbb{B}_{q,t}^{\text{ext}}$ modules.

######  Remark 9.

In particular, the $\Delta_{p_{m}}$ operators on $V_{\bullet}$ of González-
Gorsky-Simental correspond through the Ion-Wu map to limits of sums of
symmetric polynomials in the Cherednik operators as in Theorem 3.10. It would
interesting to know if the formula from Proposition 3.9

$e_{r}[\Delta]|_{x_{1}\cdots
x_{k}\operatorname{\mathscr{P}}(k)^{+}}=\sum_{s=0}^{r}\left[\prod_{i=1}^{s}\left(\frac{t^{i}}{1-t^{i}}\right)\right]e_{r-s}(\mathscr{Y}_{1},\ldots,\mathscr{Y}_{k})\epsilon_{k}(T_{k}\cdots
T_{1}\pi)^{s}$

corresponds to something geometrically meaningful on the González-Gorsky-
Simental $\mathbb{B}_{q,t}^{\text{ext}}$ module $V_{\bullet}.$

## References

  * [BS12] Igor Burban and Olivier Schiffmann  “On the Hall algebra of an elliptic curve, I”  In _Duke Math. J._ 161.7, 2012, pp. 1171–1231  DOI: 10.1215/00127094-1593263
  * [BW23] Milo Bechtloff Weising  “Stable-Limit Non-symmetric Macdonald Functions”, 2023  arXiv: https://arxiv.org/abs/2307.05864
  * [CGM20] Erik Carlsson, Eugene Gorsky and Anton Mellit  “The $\mathbb{A}_{q,t}$ algebra and parabolic flag Hilbert schemes”  In _Math. Ann._ 376.3-4, 2020, pp. 1303–1336  DOI: 10.1007/s00208-019-01898-1
  * [CM18] Erik Carlsson and Anton Mellit  “A proof of the shuffle conjecture”  In _J. Amer. Math. Soc._ 31.3, 2018, pp. 661–697  DOI: 10.1090/jams/893
  * [GGS23] Nicolle González, Eugene Gorsky and José Simental  “Calibrated representations of the double Dyck path algebra”, 2023  arXiv:2311.17653 [math.RT]
  * [Hai99] Mark Haiman  “Macdonald polynomials and geometry”  In _New perspectives in algebraic combinatorics_ volume 38 of MSRI Publ., 1999, pp. pp. 207–254
  * [HHL08] J. Haglund, M. Haiman and N. Loehr  “A combinatorial formula for nonsymmetric Macdonald polynomials”  In _Amer. J. Math._ 130.2, 2008, pp. 359–383  DOI: 10.1353/ajm.2008.0015
  * [IW22] Bogdan Ion and Dongyu Wu  “THE STABLE LIMIT DAHA AND THE DOUBLE DYCK PATH ALGEBRA”  In _Journal of the Institute of Mathematics of Jussieu_ Cambridge University Press, 2022, pp. 1–46  DOI: 10.1017/S1474748022000445
  * [Mac88] Ian Macdonald  “A New Class of Symmetric Functions”  In _Sém. Lothar. Combin._ , 1988
  * [SV13] Olivier Schiffmann and Eric Vasserot  “The elliptic Hall algebra and the $K$-theory of the Hilbert scheme of $\mathbb{A}^{2}$”  In _Duke Math. J._ 162.2, 2013, pp. 279–366  DOI: 10.1215/00127094-1961849
  * [Wei24] Milo Bechtloff Weising  “Double Dyck Path Algebra Representations From DAHA”, 2024  arXiv:2402.02843 [math.RT]