# Stable-Limit Non-symmetric Macdonald Functions

Milo James Bechtloff Weising

(August 31, 2024)

###### Abstract

We construct and study an explicit simultaneous $\mathscr{Y}$-eigenbasis of
Ion and Wu’s standard representation of the +stable-limit double affine Hecke
algebra for the limit Cherednik operators $\mathscr{Y}_{i}$. This basis arises
as a generalization of Cherednik’s non-symmetric Macdonald polynomials of type
$GL$. We utilize links between +stable-limit double affine Hecke algebra
theory of Ion-Wu and the double Dyck path algebra of Carlsson-Mellit that
arose in their proof of the Shuffle Conjecture. As a consequence, the spectral
theory for the limit Cherednik operators is understood. The symmetric
functions comprise the zero weight space. We introduce one extra operator that
commutes with the ${\mathscr{Y}}_{i}$ action and dramatically refines the
weight spaces to now be one-dimensional. This operator, up to a change of
variables, gives an extension of Haiman’s operator $\Delta^{\prime}$ from
$\Lambda$ to $\operatorname{\mathscr{P}}_{as}^{+}.$ Additionally, we develop
another method to build this weight basis using limits of trivial idempotents.

###### Contents

  1. 1 Introduction
  2. 2 Definitions and Notation
    1. 2.1 Double Affine Hecke Algebras in Type GL
      1. 2.1.1 Standard DAHA Representation
      2. 2.1.2 Non-symmetric Macdonald Polynomials
      3. 2.1.3 Combinatorial Formula for Non-symmetric Macdonald Polynomials
    2. 2.2 Symmetric Functions
      1. 2.2.1 Hall-Littlewood Symmetric Functions
    3. 2.3 Stable-Limit DAHA of Ion and Wu
      1. 2.3.1 The Standard +Stable-Limit DAHA Representation
    4. 2.4 Double Dyck Path Algebra
      1. 2.4.1 The Standard $\mathbb{A}_{t,q}$ Representation and the +Stable-Limit DAHA
  3. 3 Stable-Limits of Non-symmetric Macdonald Polynomials
  4. 4 $\mathscr{Y}$-Weight Basis of $\mathscr{P}_{as}^{+}$
    1. 4.1 The $\widetilde{E}_{\mu}$ are $\mathscr{Y}$-Weight Vectors
    2. 4.2 Constructing a Full ${\mathscr{Y}}$-Weight Basis
      1. 4.2.1 Defining the Stable-Limit Non-symmetric Macdonald Functions
    3. 4.3 $\mathcal{A}_{\lambda}$ Basis for $\Lambda$ and Symmetrization via the Trivial Hecke Idempotent
      1. 4.3.1 First Main Theorem and a Full ${\mathscr{Y}}$-Weight Basis of $\operatorname{\mathscr{P}}_{as}^{+}$
  5. 5 Some Recurrence Relations for the $\widetilde{E}_{(\mu|\lambda)}$
  6. 6 Constructing $\widetilde{E}_{(\mu|\lambda)}$-Diagonal Operators from Symmetric Functions

##  1 Introduction

The Shuffle Conjecture [Hag+05], now the Shuffle Theorem [CM18], is a
combinatorial statement regarding the Frobenius character,
$\mathcal{F}_{R_{n}}$, of the diagonal coinvariant algebra $R_{n}$ which
generalizes the coinvariant algebra arising from the geometry of flag
varieties. The conjecture built on the work of many people during the 1990s,
including but not limited to Bergeron, Garsia, Haiman, and Tesler [BG99]
[GH96] [Ber+99]. The following explicit formula is due to Haiman [Hai02]

$\mathcal{F}_{R_{n}}(X;q,t)=(-1)^{n}\nabla e_{n}[X]$

where the operator $\nabla$ is a diagonalizable operator on symmetric
functions prescribed by its action on the modified Macdonald symmetric
functions $\widetilde{H}_{\mu}$ as

$\nabla\widetilde{H}_{\mu}=\widetilde{H}_{\mu}[-1]\cdot\widetilde{H}_{\mu}.$

The original conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov
[Hag+05] states the following:

######  Theorem 1 (Shuffle Theorem).

[CM18]

$(-1)^{n}\nabla
e_{n}[X]=\sum_{\pi}\sum_{w\in\operatorname{WP}_{\pi}}t^{\operatorname{area}(\pi)}q^{\operatorname{dinv}(\pi,w)}x_{w}.$

In the above, $\pi$ ranges over the set of Dyck paths of length $n$ and
$\operatorname{WP}_{\pi}$ is the set of word parking functions corresponding
to $\pi$. The values $\operatorname{area}(\pi)$ and
$\operatorname{dinv}(\pi,w)$ are certain statistics corresponding to $\pi$ and
$w\in\operatorname{WP}_{\pi}$.

In [CM18], Carlsson and Mellit prove the Compositional Shuffle Conjecture of
Haglund, Morse, and Zabrocki [HMZ12], a generalization of the original Shuffle
Conjecture. Carlsson and Mellit construct and investigate a quiver path
algebra called the Double Dyck Path algebra $\mathbb{A}_{q,t}$. They construct
a representation of $\mathbb{A}_{q,t}$, called the standard representation,
built on certain mixed symmetric and non-symmetric polynomial algebras with
actions from Demazure-Lusztig operators, Hall-Littlewood creation operators,
and plethysms. The Compositional Shuffle Theorem falls out after a rich
understanding of the standard representation is developed. Later analysis done
by Carlsson, Gorsky, and Mellit [CGM20] showed that in fact $\mathbb{A}_{q,t}$
occurs naturally in the context of equivariant cohomology of Hilbert schemes.

Recent work by Ion and Wu [IW22] has solidified the links between the work of
Carlsson and Mellit on $\mathbb{A}_{q,t}$ and the representation theory of
double affine Hecke algebras. Ion and Wu introduce the +stable-limit double
affine Hecke algebra $\mathscr{H}^{+}$ along with a representation of
$\mathscr{H}^{+}$ on the space of almost-symmetric functions,
$\mathscr{P}_{as}^{+}$, from which one can recover the standard
$\mathbb{A}_{q,t}$ representation. The main obstruction in making a stable-
limit theory for the double affine Hecke algebras is the lack of an
inverse/directed-limit system of the double affine Hecke algebras in the
traditional sense. Ion and Wu get around this obstruction by introducing a new
notion of convergence (Defn. 14) for sequences of polynomials with increasing
numbers of variables along with limit versions of the standard Cherednik
operators defined by this convergence.

Central to the study of the standard Cherednik operators are the non-symmetric
Macdonald polynomials. The non-symmetric Macdonald polynomials in full
generality were introduced first by Cherednik [Che03] in the context of
proving the Macdonald constant-term conjecture. The introduction of the double
affine Hecke algebra, along with the non-symmetric Macdonald polynomials by
Cherednik, constituted a significant development in representation theory.
They serve as a non-symmetric counterpart to the symmetric Macdonald
polynomials introduced by Macdonald as a $q,t$-analog of Schur functions.
Further, they give an orthogonal basis of the polynomial representation
consisting of weight vectors for the Cherednik operators. The spectral theory
of non-symmetric Macdonald polynomials is well understood using the
combinatorics of affine Weyl groups. The correct choice of symmetrization
applied to a non-symmetric Macdonald polynomial will yield their symmetric
counterpart. The type A symmetric Macdonald polynomials are a remarkable basis
for symmetric polynomials simultaneously generalizing many other well studied
bases which can be recovered by appropriate specializations of values for $q$
and $t$. The aforementioned modified Macdonald functions $\widetilde{H}_{\mu}$
can be obtained via a plethystic transformation from the symmetric Macdonald
polynomials in sufficiently many variables.

It is natural to seek a stable-limit extension for the non-symmetric Macdonald
polynomials following the methods of Ion and Wu. In particular, does the
standard $\mathscr{H}^{+}$ representation $\mathscr{P}_{as}^{+}$ have a basis
of weight vectors for the limit Cherednik operators ${\mathscr{Y}}_{i}$? The
first main theorem of this paper (Theorem 49) answers this question in the
affirmative. In the second main theorem of this paper (Theorem 60) we use a
new operator $\Psi_{p_{1}}$, which commutes with the limit Cherednik
operators, to distinguish between ${\mathscr{Y}}$-weight vectors with the same
${\mathscr{Y}}$-weight. The operator $\Psi_{p_{1}}$ is up to a change of
variables an extension of Haiman’s operator $\Delta^{\prime}$ [Hai99] from
$\Lambda$ to $\operatorname{\mathscr{P}}_{as}^{+}$ (Remark 2). The operator
$\Psi_{p_{1}}$ is a limit of operators from finite variable DAHAs. We
conjecture (Conjecture 1) that for any symmetric function $F\in\Lambda$ there
is an analogous sequence of operators from finite variable DAHAs giving an
analogous operator $\Psi_{F}$ on $\operatorname{\mathscr{P}}_{as}^{+}.$ If
true, this conjecture would yield an action of the elliptic Hall algebra
[BS12] [SV13] on $\operatorname{\mathscr{P}}_{as}^{+}$ (Remark 3).

This paper is the full version of the author’s accepted submission to
FPSAC2023 [BW23].

#### Structure of the paper

Section 2 introduces many of the definitions and notations needed throughout
this paper. In Sections 3 and 4 we construct a basis of weight vectors for the
limit Cherednik operators $\mathscr{Y}_{i}$. Our strategy for this is the
following. First, in Section 3 we show that the non-symmetric Macdonald
polynomials have stable-limits in the sense that if we start with a
composition $\mu$ and consider the compositions $\mu*0^{m}$ for $m\geq 0$ then
the corresponding sequence of non-symmetric Macdonald polynomials
$E_{\mu*0^{m}}$ converges to an element $\widetilde{E}_{\mu}$ of
$\mathscr{P}_{as}^{+}$. Next, in Section 4 we show that these limits of non-
symmetric Macdonald polynomials are $\mathscr{Y}$-weight vectors. Importantly,
the newly constructed set of $\widetilde{E}_{\mu}$ do not span
$\mathscr{P}_{as}^{+}$. To fill in these gaps, the lowering operators $d_{-}$
from $\mathbb{A}_{q,t}$ are used to create enough ${\mathscr{Y}}$-weight
vectors to span $\mathscr{P}_{as}^{+}$. Finally, a symmetrization operator is
used to show that the spanning set obtained from this process is actually a
basis in Theorem 49.

Lemma 27, Corollary 38, and Lemma 37 together give a description of the
weights for the above weight basis of $\mathscr{P}_{as}^{+}$; in other words
we describe the ${\mathscr{Y}}$-spectrum.

In the last two sections of this paper we investigate some applications of
Theorem 49. In Section 5 we derive some recurrence relations for the stable-
limit Macdonald function basis similar to the classical Knop-Sahi relations.
In Section 6 we construct an operator $\Psi_{p_{1}}$ on
$\operatorname{\mathscr{P}}_{as}^{+}$ which is diagonal on the stable-limit
Macdonald function basis, and thus commutes with the limit Cherednik operators
${\mathscr{Y}}$. The action of $\Psi_{p_{1}}$ distinguishes between our basis
elements with identical ${\mathscr{Y}}$-weight. This leads to the second main
theorem of this paper, Theorem 60, where we prove that after adding this new
operator to the algebra of limit Cherednik operators the resulting algebra is
commutative and has one dimensional weight spaces in
$\operatorname{\mathscr{P}}_{as}^{+}.$

#### Acknowledgments

The author would first like to thank their advisor Monica Vazirani for her
tremendous help with the proof checking and editing of this paper and for her
continued guidance of the author through the hurdles of the academic world.
The author would like to thank the FPSAC2023 referees who alerted the author
to an unpublished work of Ion and Wu [IW22a] which independently determines
the ${\mathscr{Y}}$-spectrum on $\mathscr{P}_{as}^{+}$. The author would also
like to thank Daniel Orr and Ben Goodberry for their insights regarding the
symmetrization operators occurring in subsection 4.3.

##  2 Definitions and Notation

###  2.1 Double Affine Hecke Algebras in Type GL

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] but may differ from other authors.

######  Definition 2.

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 element $\omega_{n}$ by

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

This conveniently allows us to write

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

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 $n\neq m$.

We will often use the following basic fact about $\mathscr{H}_{n}$ the proof
of which we will omit.

######  Lemma 3.

Let $f(X_{1},\ldots,X_{n}),g(Y_{1},\ldots,Y_{n})\in\mathscr{H}_{n}$ be
symmetric Laurent polynomials in $X$’s and $Y$’s respectively. Then for all
$1\leq i\leq n-1$,

$[T_{i},f(X_{1},\ldots,X_{n})]=[T_{i},g(Y_{1},\ldots,Y_{n})]=0.$

####  2.1.1 Standard DAHA Representation

######  Definition 4.

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 $\mathscr{P}_{n}$:

  * •

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

  * •

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

  * •

$\omega_{n}f(x_{1},\ldots,x_{n})=f(q^{-1}x_{n},x_{1},\ldots,x_{n-1})$

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 $X$ and $Y$ 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
$\mathscr{P}_{n}$.

######  Definition 5.

For $1\leq i\leq n-1$ define the intertwiners,
$\varphi_{i}^{(n)}\in\mathscr{H}_{n}$, as

$\varphi_{i}^{(n)}:=[T_{i},Y_{i}^{(n)}]=T_{i}Y_{i}^{(n)}-Y_{i}^{(n)}T_{i}.$

The intertwiner elements have the following properties which are readily
verified from the relations of $\mathscr{H}_{n}$:

  * •

$\varphi_{i}^{(n)}=T_{i}(Y_{i}^{(n)}-Y_{i+1}^{(n)})+(1-t)Y_{i+1}^{(n)}$

  * •

$\varphi_{i}^{(n)}Y_{j}^{(n)}=Y_{s_{i}(j)}^{(n)}\varphi_{i}^{(n)}$

  * •

$(\varphi_{i}^{(n)})^{2}=(Y_{i}^{(n)}-tY_{i+1}^{(n)})(Y_{i+1}^{(n)}-tY_{i}^{(n)}).$

####  2.1.2 Non-symmetric Macdonald Polynomials

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

######  Definition 6.

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 in a few instances use the notation $\mathbbm{1}(p)$ to denote the
value $1$ if the statement p is true and $0$ otherwise. In this paper we will
write $\mathfrak{S}_{n}$ for the permutation group on the set
$[n]:=\\{1,\ldots,n\\}.$

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 7.

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 $j.$

As an equivalent definition we say $\alpha<\beta$ if
$\operatorname{sort}(\alpha)\triangleleft\operatorname{sort}(\beta)$ and in
the case that $\lambda=\operatorname{sort}(\alpha)=\operatorname{sort}(\beta)$
then we have $\alpha<\beta$ when $\sigma<\gamma$ in the Bruhat order for
minimal length permutations $\sigma,\gamma$ with $\sigma(\alpha)=\lambda$,
$\gamma(\beta)=\lambda$. It is important to note that with respect to the
Bruhat order any weakly decreasing vector $v\in\mathbb{Z}^{n}$ is the minimal
element in its permutation orbit $\mathfrak{S}_{n}.v$.

######  Definition 8.

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 $\mathscr{P}_{n}^{+}$.

####  2.1.3 Combinatorial Formula for Non-symmetric Macdonald Polynomials

Note that the $q,t$ conventions in [HHL08] differ from those appearing in this
paper. In the below theorem the appropriate translation $q\rightarrow q^{-1}$
has been made.

In [HHL08], Haglund, Haiman, and Loehr give an explicit monomial expansion
formula for the non-symmetric Macdonald polynomials in terms of the
combinatorics of non-attacking labellings of certain box diagrams
corresponding to compositions which we will now review.

######  Definition 9.

[HHL08] For a composition $\mu=(\mu_{1},\dots,\mu_{n})$ define the column
diagram of $\mu$ as

$dg^{\prime}(\mu):=\\{(i,j)\in\mathbb{N}^{2}:1\leq i\leq n,1\leq
j\leq\mu_{i}\\}.$

This is represented by a collection of boxes in positions given by
$dg^{\prime}(\mu)$. The augmented diagram of $\mu$ is given by

$\widehat{dg}(\mu):=dg^{\prime}(\mu)\cup\\{(i,0):1\leq i\leq n\\}.$

Visually, to get $\widehat{dg}(\mu)$ we are adding a bottom row of boxes on
length $n$ below the diagram $dg^{\prime}(\mu)$. Given $u=(i,j)\in
dg^{\prime}(\mu)$ define the following:

  * •

$\operatorname{leg}(u):=\\{(i,j^{\prime})\in dg^{\prime}(\mu):j^{\prime}>j\\}$

  * •

$\operatorname{arm}^{\operatorname{left}}(u):=\\{(i^{\prime},j)\in
dg^{\prime}(\mu):i^{\prime}<i,\mu_{i^{\prime}}\leq\mu_{i}\\}$

  * •

$\operatorname{arm}^{\operatorname{right}}(u):=\\{(i^{\prime},j-1)\in\widehat{dg}(\mu):i^{\prime}>i,\mu_{i^{\prime}}<\mu_{i}\\}$

  * •

$\operatorname{arm}(u):=\operatorname{arm}^{\operatorname{left}}(u)\cup\operatorname{arm}^{\operatorname{right}}(u)$

  * •

$\operatorname{lg}(u):=|\operatorname{leg}(u)|=\mu_{i}-j$

  * •

$a(u):=|\operatorname{arm}(u)|.$

A filling of $\mu$ is a function
$\sigma:dg^{\prime}(\mu)\rightarrow\\{1,...,n\\}$ and given a filling there is
an associated augmented filling
$\widehat{\sigma}:\widehat{dg}(\mu)\rightarrow\\{1,...,n\\}$ extending
$\sigma$ with the additional bottom row boxes filled according to
$\widehat{\sigma}((j,0))=j$ for $j=1,\dots,n$. Distinct lattice squares
$u,v\in\mathbb{N}^{2}$ are said to attack each other if one of the following
is true:

  * •

$u$ and $v$ are in the same row

  * •

$u$ and $v$ are in consecutive rows and the box in the lower row is to the
right of the box in the upper row.

A filling $\sigma:dg^{\prime}(\mu)\rightarrow\\{1,\dots,n\\}$ is non-attacking
if $\widehat{\sigma}(u)\neq\widehat{\sigma}(v)$ for every pair of attacking
boxes $u,v\in\widehat{dg}(\mu).$ For a box $u=(i,j)$ let $d(u)=(i,j-1)$ denote
the box just below $u$. Given a filling
$\sigma:dg^{\prime}(\mu)\rightarrow\\{1,\dots,n\\}$, a descent of $\sigma$ is
a box $u\in dg^{\prime}(\mu)$ such that
$\widehat{\sigma}(u)>\widehat{\sigma}(d(u)).$ Set
$\operatorname{Des}(\widehat{\sigma})$ to be the set of descents of
$\widehat{\sigma}$ and define

$\operatorname{maj}(\widehat{\sigma}):=\sum_{u\in\operatorname{Des}(\widehat{\sigma})}(\operatorname{lg}(u)+1).$

The reading order on the diagram $\widehat{dg}(\mu)$ is the total ordering on
the boxes of $\widehat{dg}(\mu)$ row by row, from top to bottom, and from
right to left within each row. If
$\sigma:dg^{\prime}(\mu)\rightarrow\\{1,\dots,n\\}$ is a filling, an inversion
of $\widehat{\sigma}$ is a pair of attacking boxes $u,v\in\widehat{dg}(\mu)$
such that $u<v$ in reading order and
$\widehat{\sigma}(u)>\widehat{\sigma}(v).$ Set
$\operatorname{Inv}(\widehat{\sigma})$ to be the set of inversions of
$\widehat{\sigma}$. Define the statistics

  * •

$\operatorname{inv}(\widehat{\sigma}):=|\operatorname{Inv}(\widehat{\sigma})|-|\\{i<j:\mu_{i}\leq\mu_{j}\\}|-\sum_{u\in\operatorname{Des}(\widehat{\sigma})}a(u)$

  * •

$\operatorname{coinv}(\widehat{\sigma}):=\left(\sum_{u\in
dg^{\prime}(\mu)}a(u)\right)-\operatorname{inv}(\widehat{\sigma}).$

Lastly, for a filling $\sigma:dg^{\prime}(\mu)\rightarrow\\{1,\dots,n\\}$ set

$x^{\sigma}:=x_{1}^{|\sigma^{-1}(1)|}\cdots x_{n}^{|\sigma^{-1}(n)|}.$

The combinatorial formula for non-symmetric Macdonald polynomials can now be
stated.

######  Theorem 10.

[HHL08] For a composition $\mu$ with $\ell(\mu)=n$ the following holds:

$E_{\mu}=\sum_{\begin{subarray}{c}\sigma:\mu\rightarrow[n]\\\ \text{non-
attacking}\end{subarray}}x^{\sigma}q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu)\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right).$

######  Example.

We finish this subsection with a visual example of a non-attacking filling and
its associated statistics. Below is the augmented filling $\widehat{\sigma}$
of a non-attacking filling $\sigma:(3,2,0,1,0,0)\rightarrow[6]$ pictured as
labels inside the boxes of $\widehat{dg}(3,2,0,1,0,0).$

centertableaux, boxframe= normal, boxsize= 2em 6 &

4 1

1 2 3

1 2 3 4 5 6

Let $u$ be the column 1 box of $\widehat{dg}(3,2,0,1,0,0)$ filled with a $4$
in the above diagram. Notice that $u$ is a descent box of $\widehat{\sigma}$
as $4$ is larger than the label $1$ of the box $d(u)$ just below $u.$ Further,
we see that $a(u)=2$ and $\operatorname{lg}(u)=1$. Considering the diagram as
a whole now we see that $x^{\sigma}=x_{1}^{2}x_{2}x_{3}x_{4}x_{6}$,
$\operatorname{maj}(\widehat{\sigma})=3$,
$|\operatorname{Inv}(\widehat{\sigma})|=21$,
$\operatorname{inv}(\widehat{\sigma})=14$, and
$\operatorname{coinv}(\widehat{\sigma})=1.$ The contribution of this non-
attacking labelling to the HHL formula for
$E_{(3,2,0,1,0,0)}\in\operatorname{\mathscr{P}}_{6}^{+}$ is

$x_{1}^{2}x_{2}x_{3}x_{4}x_{6}q^{-3}t^{1}\left(\frac{1-t}{1-q^{-1}t^{3}}\right)\left(\frac{1-t}{1-q^{-1}t^{2}}\right)\left(\frac{1-t}{1-q^{-2}t^{3}}\right)\left(\frac{1-t}{1-q^{-1}t^{2}}\right).$

###  2.2 Symmetric Functions

######  Definition 11.

Define the ring of symmetric functions $\Lambda$ to be the subalgebra of the
inverse limit of the symmetric polynomial rings
$\mathbb{Q}(q,t)[x_{1},\ldots,x_{n}]^{\mathfrak{S}_{n}}$ with respect to the
quotient maps sending $x_{n}\rightarrow 0$ consisting of those elements with
bounded $x$-degree. For $i\geq 0$ define the $i$-th power sum symmetric
function by

$p_{i}=x_{1}^{i}+x_{2}^{i}+\dots.$

It is a classical result that $\Lambda$ is isomorphic to
$\mathbb{Q}(q,t)[p_{1},p_{2},\dots]$. For any expression
$G=a_{1}g^{\mu_{1}}+a_{2}g^{\mu_{2}}+\dots$ with rational scalars
$a_{i}\in\mathbb{Q}$ and distinct monomials $g^{\mu_{i}}$ in a set of
algebraically independent commuting free variables $\\{g_{1},g_{2},\dots\\}$
the plethsytic evaluation of $p_{i}$ at the expression $G$ is defined to be

$p_{i}[G]:=a_{1}g^{i\mu_{1}}+a_{2}g^{i\mu_{2}}+\dots.$

Note that $g_{i}$ are allowed to be $q$ or $t.$ Here we are using the
convention that $i\mu=(i\mu_{1},\ldots,i\mu_{r})$ for
$\mu=(\mu_{1},\cdots,\mu_{r}).$ The definition of plethystic evaluation on
power sum symmetric functions extends to all symmetric functions $F\in\Lambda$
by requiring $F\rightarrow F[G]$ be a $\mathbb{Q}(q,t)$-algebra homomorphism.
Note that for $F\in\Lambda$, $F=F[x_{1}+x_{2}+\ldots]$ and so we will often
write $F=F[X]$ where $X:=x_{1}+x_{2}+\ldots.$ For a partition $\lambda$ define
the monomial symmetric function $m_{\lambda}$ by

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

where we range over all distinct monomials $x^{\mu}$ such that
$\sigma(\mu)=\lambda$ for some permutation $\sigma$. For $n\geq 0$ define the
complete homogeneous symmetric function $h_{n}$ by

$h_{n}:=\sum_{|\lambda|=n}m_{\lambda}.$

We can extend plethysm to $\mathbb{Q}(q,t)[[p_{1},p_{2},\dots]]$. The
plethystic exponential is defined to be the element of
$\mathbb{Q}(q,t)[[p_{1},p_{2},\dots]]$ given by

$Exp[X]:=\sum_{n\geq 0}h_{n}[X].$

Here we list some notable properties of the plethystic exponential which will
be used later in this paper.

  * •

$Exp[0]=1$

  * •

$Exp[X+Y]=Exp[X]Exp[Y]$

  * •

$Exp[x_{1}+x_{2}+\dots]=\prod_{i=1}^{\infty}\left(\frac{1}{1-x_{i}}\right)$

  * •

$Exp[(1-t)(x_{1}+x_{2}+\dots)]=\prod_{i=1}^{\infty}\left(\frac{1-tx_{i}}{1-x_{i}}\right)$

######  Example.

Here we give a few examples of plethystic evaluation.

  * •

$p_{3}[1+5t+qt^{2}]=1+5t^{3}+q^{3}t^{6}$

  * •

$s_{2}[(1-t)X]=(\frac{p_{2}+p_{1,1}}{2})[(1-t)X]=\frac{(1-t^{2})p_{2}[X]+(1-t)^{2}p_{1,1}[X]}{2}$

  * •

$Exp[\frac{t}{1-t}]=\prod_{n=1}^{\infty}(\frac{1}{1-t^{n}})$

####  2.2.1 Hall-Littlewood Symmetric Functions

For the purposes of this paper we need the following explicit collection of
symmetric functions.

######  Definition 12.

For $n\geq 0$ define the Jing vertex operator $\mathscr{B}_{n}\in
End_{\mathbb{Q}(q,t)}(\Lambda)$ by

$\mathscr{B}_{n}[F]:=\langle z^{n}\rangle F[X-z^{-1}]Exp[(1-t)zX].$

Here $\langle z^{n}\rangle$ denotes the operator which extracts the
coefficient of $z^{n}$ of any formal series in $z$. For a partition
$\lambda=(\lambda_{1},...,\lambda_{r})$ define the Hall-Littlewood symmetric
function, $\mathcal{P}_{\lambda}$, by

$\mathcal{P}_{\lambda}:=\mathscr{B}_{\lambda_{1}}\cdots\mathscr{B}_{\lambda_{r}}(1).$

Note that the operator $\mathscr{B}_{n}$ is graded with degree $n.$ The
definition of the Hall-Littlewood symmetric functions in this paper matches
with [IW22] and [CM18] but differs from that of other authors. As we will see
later in Proposition 46 the $\mathcal{P}_{\lambda}[X]$ are the same as the
dual Hall-Littlewood symmetric functions $Q_{\lambda}[X;t]$ defined by
Macdonald [Mac15]. These symmetric functions have the following useful
properties.

  * •

$\mathcal{P}_{\lambda}$ is homogeneous with degree $|\lambda|$

  * •

$\mathcal{P}_{(n)}[X]=h_{n}[(1-t)X]$

  * •

If $n\geq\lambda_{1}$ then
$\mathscr{B}_{n}(\mathcal{P}_{\lambda})=\mathcal{P}_{n*\lambda}$

  * •

$\mathscr{B}_{0}(\mathcal{P}_{\lambda})=t^{\ell(\lambda)}\mathcal{P}_{\lambda}$

Lastly, it is a classical result that the collection
$\\{\mathcal{P}_{\lambda}\mid\lambda\in\operatorname{\mathrm{Par}}\\}$ is a
basis of $\Lambda$.

###  2.3 Stable-Limit DAHA of Ion and 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 13.

[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 $i\in\mathbb{N}$ satisfy (i) and (ii) of
Defn. 2.

  * •

The generators $T_{i},Y_{i}$ for $i\in\mathbb{N}$ satisfy (i) and (iii) of
Defn. 2.

  * •

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

Importantly, there is no relation of the form $Y_{1}X_{1}\cdots
X_{n}=qX_{1}\cdots X_{n}Y_{1}$ in $\mathscr{H}^{+}$. As such there is no
invertible ’$\omega$’ element in $\mathscr{H}^{+}$ which in $\mathscr{H}_{n}$
normally realizes the cyclic symmetry of the affine type $A$ root systems.

######  Definition 14.

[IW22] Let $\mathscr{P}_{\infty}^{+}$ denote the inverse limit of the rings
$\mathscr{P}_{k}^{+}$ with respect to the homomorphisms
$\pi_{k}:\operatorname{\mathscr{P}}_{k+1}^{+}\rightarrow\operatorname{\mathscr{P}}_{k}^{+}$
which send $x_{k+1}$ to 0 at each step. We can naturally extend $\pi_{k}$ to a
map $\mathscr{P}_{\infty}^{+}\rightarrow\mathscr{P}_{k}$ which will be given
the same name. Let
$\mathscr{P}(k)^{+}:=\mathbb{Q}(q,t)[x_{1},\ldots,x_{k}]\otimes\Lambda[x_{k+1}+x_{k+2}+\ldots]$.
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 $\pi_{n}$.

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 the 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 15.

[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, $\pi_{m}(h_{m+1})=h_{m}$ and
$\pi_{m}(g_{m+1}^{(i)})=g_{m}^{(i)}$ 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
$\lim_{m}f_{m}=h+\sum_{i=1}^{N}a^{(i)}g^{(i)}.$

This definition of convergence is a mix of both the stronger topology arising
from the inverse system given by the maps $\pi_{m}$ and the t-adic topology
arising from the ring $\mathbb{Q}(q,t)$. It is important to note that part of
the above definition requires convergent sequences to always be written as a
finite sum of fixed length with terms that converge independently.

Here we list a few instructive examples of convergent sequences and their
limits:

  * •

$\lim_{m}t^{m}=0$

  * •

$\lim_{m}1+\ldots+t^{m}=\frac{1}{1-t}$

  * •

$\lim_{m}\frac{1}{q^{2}-t^{m}}(x_{3}^{2}+\ldots+x_{m}^{2})=q^{-2}p_{2}[x_{3}+\ldots].$

######  Remark 1.

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 16.

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

There are two important examples of convergent operator sequences which will
be relevant for the rest of this paper. For all $i\geq 1$ and $m\geq 1$ let
$X_{i}^{(m)}$ denote the operator on $\operatorname{\mathscr{P}}_{m}^{+}$
given by $0$ if $m<i$ and by $X_{i}^{(m)}f=x_{i}f$ if $i\leq m$. Similarly for
$i\geq 1$ and $m\geq 1$ let $T_{i}^{(m)}$ denote the operator on
$\operatorname{\mathscr{P}}_{m}^{+}$ given by $0$ if $m-1<i$ and by
$T_{i}f=s_{i}f+(1-t)x_{i}\frac{f-s_{i}f}{x_{i}-x_{i+1}}$ if $i\leq m-1$. Then
for all $i\geq 1$ it is immediate from definition that the sequences
$(X_{i}^{(m)})_{m\geq 1}$ and $(T_{i}^{(m)})_{m\geq 1}$ converge to operators
$X_{i}$ and $T_{i}$ respectively on $\operatorname{\mathscr{P}}_{as}^{+}$.
Further, their corresponding actions are given for
$f\in\operatorname{\mathscr{P}}_{as}^{+}$ simply by

  * •

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

  * •

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

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

######  Proposition 17 (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 $A(f)$.

####  2.3.1 The Standard +Stable-Limit DAHA Representation

Ion and Wu begin their construction of the standard representation of
$\mathscr{H}^{+}$ by noting the following key fact.

######  Proposition 18.

[IW22] For $n\geq 1$

$\pi_{n-1}t^{n}Y_{1}^{(n)}X_{1}=t^{n-1}Y_{1}^{(n-1)}X_{1}\pi_{n-1}.$

In other words, the action of the operators $t^{n}Y_{1}^{(n)}$ and
$t^{n-1}Y_{1}^{(n-1)}$ are compatible on $x_{1}\mathscr{P}_{n}$. As such there
exists a limit operator
$Y_{1}^{(\infty)}:x_{1}\mathscr{P}^{+}_{\infty}\rightarrow
x_{1}\mathscr{P}^{+}_{\infty}$ such that
$\pi_{n}Y_{1}^{(\infty)}=t^{n}Y_{1}^{(n)}$. A crucial idea of Ion and Wu is to
extend the action of the operators $t^{n}Y_{1}^{(n)}$ on
$x_{1}\mathscr{P}_{n}$ to all of $\mathscr{P}_{n}$ using the previously
defined projection $\rho:\mathscr{P}_{n}\rightarrow x_{1}\mathscr{P}_{n}$.

######  Definition 19.

[IW22] Define the operator $\widetilde{Y}_{1}^{(n)}:=\rho\circ
t^{n}Y_{1}^{(n)}$. For $2\leq i\leq n$ define $\widetilde{Y}_{i}^{(n)}$ by
requiring
$\widetilde{Y}_{i}^{(n)}=t^{-1}T_{i-1}\widetilde{Y}_{i-1}^{(n)}T_{i-1}.$

A direct check shows that $\widetilde{Y}_{1}^{(n)}X_{1}=t^{n}Y_{1}^{(n)}X_{1}$
so that $\widetilde{Y}_{1}^{(n)}$ extends the action of $t^{n}Y_{1}^{(n)}$ on
$x_{1}\mathscr{P}_{n}$ as desired. The main utility of this specific choice of
definition is the following theorem.

######  Theorem 20.

[IW22] The sequence $(\widetilde{Y}_{1}^{(m)})_{m\geq 1}$ converges to an
operator $\mathscr{Y}_{1}$ on $\operatorname{\mathscr{P}}_{as}^{+}$. Define
the operators $\mathscr{Y}_{i}$ for $i\geq 2$ by
${\mathscr{Y}}_{i}:=t^{-1}T_{i-1}{\mathscr{Y}}_{i-1}T_{i-1}$. The operators
$\mathscr{Y}_{i}$ along with the Demazure-Lusztig action of the $T_{i}$’s and
multiplication by the $X_{i}$’s generate an $\mathscr{H}^{+}$ action on
$\mathscr{P}_{as}^{+}$.

In particular, the authors Ion and Wu show that despite the fact that for
$1\leq i\neq j\leq n$,
$\widetilde{Y}_{i}^{(n)}\widetilde{Y}_{j}^{(n)}\neq\widetilde{Y}_{j}^{(n)}\widetilde{Y}_{i}^{(n)}$
the limit Cherednik operators commute:

$\mathscr{Y}_{i}\mathscr{Y}_{j}=\mathscr{Y}_{j}\mathscr{Y}_{i}.$

The action of the ${\mathscr{Y}}_{i}$ operators respect the canonical
filtration of $\operatorname{\mathscr{P}}_{as}^{+}=\bigcup_{k\geq
0}\operatorname{\mathscr{P}}(k)^{+}$. For all $n\geq 0$, the operators
$\\{{\mathscr{Y}}_{1},...,{\mathscr{Y}}_{n}\\}$ restrict to operators on the
space $\operatorname{\mathscr{P}}(n)^{+}$ whereas the operators
$\\{{\mathscr{Y}}_{n+1},{\mathscr{Y}}_{n+2},...\\}$ annihilate
$\operatorname{\mathscr{P}}(n)^{+}$. Note that for $n=0$,
$\operatorname{\mathscr{P}}(0)^{+}=\Lambda$ so all of the operators
${\mathscr{Y}}_{i}$ annihilate $\Lambda$.

###  2.4 Double Dyck Path Algebra

The Double Dyck Path Algebra $\mathbb{A}_{q,t}$, introduced by Carlsson and
Mellit [CM18], is a quiver path algebra with vertices indexed by non-negative
integers with the following edge operators:

  * •

$d_{+},d_{+}^{*}:k\rightarrow k+1$

  * •

$T_{1},...,T_{k-1}:k\rightarrow k$

  * •

$d_{-}:k+1\rightarrow k.$

The full set of relations for $\mathbb{A}_{q,t}$ are omitted here but can be
found in [CM18]. In order to match the parameter conventions in Ion and Wu’s
work [IW22] we will consider $\mathbb{A}_{t,q}$ as opposed to
$\mathbb{A}_{q,t}$ formed by simply swapping $q$ and $t$ in the defining
relations of $\mathbb{A}_{q,t}$. Here we highlight a few notable relations of
$\mathbb{A}_{t,q}$ which will be required later:

  * •

The loops $T_{1},...,T_{k-1}$ at vertex $k\geq 2$ generate a type $A$ finite
Hecke algebra

  * •

$d_{-}^{2}T_{k-1}=d_{-}^{2}$ starting at vertex $k\geq 2$

  * •

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

  * •

$z_{i}d_{-}=d_{-}z_{i}$ at vertex k for $1\leq i\leq k-1$ where
$z_{1}:=\frac{t^{k}}{1-t}[d_{+}^{*},d_{-}]T_{k-1}^{-1}\cdots T_{1}^{-1}$ and
$z_{i+1}=t^{-1}T_{i}z_{i}T_{i}.$

####  2.4.1 The Standard $\mathbb{A}_{t,q}$ Representation and the +Stable-
Limit DAHA

Vital to the proof of the Compositional Shuffle Conjecture by Carlsson and
Mellit [CM18] is their construction of a particular representation of
$\mathbb{A}_{t,q}$.

######  Definition 21.

[CM18] For $k\geq 0$ let
$V_{k}=\mathbb{Q}(q,t)[y_{1},\dots,y_{k}]\otimes\Lambda$ be associated to the
vertex $k$ and denote by $V_{\bullet}$ be the system of spaces $V_{k}$. Let
$\zeta_{k}$ denote the algebra homomorphism

$\zeta_{k}f(y_{1},\dots,.y_{k-1},y_{k})=f(y_{2},\dots,y_{k},qy_{1}).$

If $f$ is a formal series with respect to the variable $y$ with coefficients
in some ring R denote by $\mathfrak{c}_{y}(f)\in R$ the constant term of $f$
i.e. the coefficient of $y^{0}$ in $f$. Note that each $\mathfrak{S}_{k}$ acts
on $V_{k}$ by permuting the variables $y_{1},...,y_{k}.$ Define the following
operators:

  * •

$T_{i}F=s_{i}F+(1-t)y_{i}\frac{F-s_{i}F}{y_{i}-y_{i+1}}$

  * •

$d_{-}F=\mathfrak{c}_{y_{k}}(F[X-(t-1)y_{k}]Exp[-y_{k}^{-1}X])$

  * •

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

  * •

$d_{+}^{*}F=\zeta_{k}F[X+(t-1)y_{k+1}].$

######  Theorem 22.

[CM18] The above operators define a representation of $\mathbb{A}_{t,q}$ on
$V_{\bullet}.$

Ion and Wu use their construction of the standard $\mathscr{H}^{+}$
representation $\operatorname{\mathscr{P}}_{as}^{+}$ to recover the standard
$\mathbb{A}_{t,q}$ representation $V_{\bullet}.$

######  Theorem 23.

[IW22] There exists an $\mathbb{A}_{t,q}$ representation structure on
$\operatorname{\mathscr{P}}_{\bullet}=(\operatorname{\mathscr{P}}(k)^{+})_{k\geq
0}$ isomorphic to the standard representation $V_{\bullet}$ such that at each
vertex $k$, $z_{i}$ acts by ${\mathscr{Y}}_{i}$ and $y_{i}$ acts by $X_{i}$.
Further, according to this isomorphism $\operatorname{\mathscr{P}}(k)^{+}$ is
identified with $V_{k}$ via the map $x_{1}^{a_{1}}\cdots
x_{k}^{a_{k}}F[x_{k+1}+\ldots]\rightarrow y_{1}^{a_{1}}\cdots
y_{k}^{a_{k}}F[\frac{X}{t-1}].$

##  3 Stable-Limits of Non-symmetric Macdonald Polynomials

We start by investigating the properties of certain sequences of non-symmetric
Macdonald polynomials. We will find that if we fix any composition $\mu$ and
consider the sequence of compositions $(\mu*0^{m})_{m\geq 0}$ the
corresponding sequence of non-symmetric Macdonald polynomials
$(E_{\mu*0^{m}})_{m\geq 0}$ will converge in the sense of Definition 15. It is
important to note that in most cases the sequence $(E_{\mu*0^{m}})_{m\geq 0}$
will not converge with respect to the inverse system
$(\pi_{k}:\operatorname{\mathscr{P}}_{k+1}\rightarrow\operatorname{\mathscr{P}}_{k})_{k\geq
1}$. This should be expected because the spectra of the Cherednik operators
acting on $\operatorname{\mathscr{P}}_{k+1}$ are incompatible with the spectra
from the Cherednik operators acting on $\operatorname{\mathscr{P}}_{k}$.
However, by using the HHL explicit combinatorial formula for the non-symmetric
Macdonald polynomials we show that the combinatorics of non-attacking
labellings underlying the sequence $(E_{\mu*0^{m}})_{m\geq 0}$ converge in a
certain sense. The weaker convergence notion introduced by Ion and Wu is
consistent with these combinatorics. For our purposes later in this paper we
will heavily rely on the convergence of these sequences as a bridge between
the limit Cherednik operators ${\mathscr{Y}}_{i}$ and their classical
counterparts.

We now show the convergence of the sequence $(E_{\mu*0^{m}})_{m\geq 0}$.
First, we describe a convenient rearrangement of the monomials in each
$E_{\mu*0^{m}}$.

######  Theorem 24.

Let $\mu$ be a composition with $\ell(\mu)=n$ and $m\geq 0$. Then
$E_{\mu*0^{m}}$ has the explicit expression given by

$E_{\mu*0^{m}}=\sum_{\begin{subarray}{c}\lambda\leavevmode\nobreak\
\text{partition}\\\
|\lambda|\leq|\mu|\end{subarray}}m_{\lambda}[x_{n+1}+\ldots+x_{n+m}]\sum_{\begin{subarray}{c}\sigma:\mu*0^{\ell(\lambda)}\rightarrow[n+\ell(\lambda)]\\\
\text{non-attacking}\\\ \forall i=1,...,\ell(\lambda)\\\
\lambda_{i}=|\sigma^{-1}(n+i)|\end{subarray}}x_{1}^{|\sigma^{-1}(1)|}\cdots
x_{n}^{|\sigma^{-1}(n)|}\Gamma^{(m)}(\widehat{\sigma})$

where

$\Gamma^{(m)}(\widehat{\sigma}):=q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*0^{\ell(\lambda)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{not in row
}1\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right)\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*0^{\ell(\lambda)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{in row
}1\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+m+1)}}\right).\
$

###### Proof.

First, start with directly applying the HHL formula (10):

$E_{\mu*0^{m}}=\sum_{\begin{subarray}{c}\sigma:\mu*0^{m}\rightarrow[n+m]\\\
\text{non-
attacking}\end{subarray}}x^{\sigma}q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*0^{m})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right).$

We know that $E_{\mu*0^{m}}$ is symmetric in the variables
$x_{n+1},...,x_{n+m}$ [Che03] so it follows that the
$\mathbb{Q}(q,t)[x_{1},...,x_{n}]$-coefficient of each monomial in
$x_{n+1},...,x_{n+m}$ is independent of the ordering of the latter variables.
Hence, we find that by grouping these monomials by symmetry

$E_{\mu*0^{m}}=\sum_{\lambda}m_{\lambda}[x_{n+1}+...+x_{n+m}]\sum_{\begin{subarray}{c}\sigma:\mu*0^{m}\rightarrow[n+m]\\\
\text{non-attacking}\\\ \forall i\leavevmode\nobreak\
\lambda_{i}=|\sigma^{-1}(n+i)|\end{subarray}}x_{1}^{|\sigma^{-1}(1)|}\cdots
x_{n}^{|\sigma^{-1}(n)|}q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\leavevmode\nobreak\
\times\\\ \prod_{\begin{subarray}{c}u\in dg^{\prime}(\mu)\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right).$

Note that by degree considerations the only possible partitions $\lambda$ that
have a nonzero contribution to the above sum have $|\lambda|\leq|\mu|$ and
hence we can rewrite the above sums as

$\sum_{\lambda}\sum_{\begin{subarray}{c}\sigma:\mu*0^{m}\rightarrow[n+m]\\\
\text{non-attacking}\\\ \forall i\leavevmode\nobreak\
\lambda_{i}=|\sigma^{-1}(n+i)|\end{subarray}}=\sum_{\begin{subarray}{c}\lambda\leavevmode\nobreak\
\text{partition}\\\
|\lambda|\leq|\mu|\end{subarray}}\sum_{\begin{subarray}{c}\sigma:\mu*0^{m}\rightarrow[n+\ell(\lambda)]\\\
\text{non-attacking}\\\ \forall i\leavevmode\nobreak\
\lambda_{i}=|\sigma^{-1}(n+i)|\end{subarray}}.$

In the latter sum above we have written each $\sigma$ as a non-attacking
labelling $\sigma:\mu*0^{m}\rightarrow[n+\ell(\lambda)]$ to emphasize that the
numbers occurring in this labelling are contained in the set
$[n+\ell(\lambda)]$ which is independent of $m$. However, these are still
considered labellings of the diagram corresponding to $\mu*0^{m}$ and hence we
calculate the corresponding $q,t$ coefficients in the HHL formula accordingly.

We must now understand the dependence on $m$ of the statistics
$\operatorname{maj}$, $\operatorname{coinv}$, $\operatorname{lg}$, and $a$ in
each of the non-attacking labellings
$\sigma:\mu*0^{m}\rightarrow[n+\ell(\lambda)]$ as $m$ varies. Fix a non-
attacking labelling $\sigma:\mu*0^{k}\rightarrow[n+k]$ for some $k\leq m$ and
let $\sigma_{m}$ be the associated labelling of $\mu*0^{m}$. Recall that

$\operatorname{maj}(\widehat{\sigma})=\sum_{u\in\operatorname{Des}(\widehat{\sigma})}(\operatorname{lg}(u)+1)$

and similarly for $\operatorname{maj}(\widehat{\sigma}_{m}).$ The only descent
boxes of $\widehat{\sigma}_{m}$ occur in the diagram $dg^{\prime}(\mu)$ itself
and $\operatorname{lg}(u)$ for these boxes will not depend on m. Therefore,
$\operatorname{maj}(\widehat{\sigma}_{m})=\operatorname{maj}(\widehat{\sigma})$.
For $u\in dg^{\prime}(\mu*0^{m})$ clearly $u\in dg^{\prime}(\mu)$ and by
direct computation we see that when $u$ is not in row 1 then $a(u)$ does not
depend on m. However, for u in row 1 $a(u)$ when calculated in the diagram
$\widehat{dg}(\mu)$ increases to $a(u)+m$ when calculated in the diagram
$\widehat{dg}(\mu*0^{m})$. This comes from counting the extra row 0 boxes for
each box in row 1. Also note that in any non-attacking labelling there cannot
be descent boxes in row 1. Now from careful counting we get the following:

  * •

$|\operatorname{Inv}(\widehat{\sigma}_{m})|=|\operatorname{Inv}(\widehat{\sigma})|+(n+k)(m-k)+\binom{m-k}{2}$

  * •

$|\\{i<j:(\mu*0^{m})_{i}\leq(\mu*0^{m})_{j}\\}|=|\\{i<j:(\mu*0^{k})_{i}\leq(\mu*0^{k})_{j}\\}|+(\\#\\{i:\mu_{i}=0\\}+k)(m-k)+\binom{m-k}{2}$

  * •

$\sum_{u\in\operatorname{Des}(\widehat{\sigma}_{m})}a(u)=\sum_{u\in\operatorname{Des}(\widehat{\sigma})}a(u).$

By using the above calculations and cancelling out terms we get

$\displaystyle\operatorname{inv}(\widehat{\sigma}_{m})$
$\displaystyle=|\operatorname{Inv}(\widehat{\sigma}_{m})|-|\\{i<j:(\mu*0^{m})_{i}\leq(\mu*0^{m})_{j}\\}|-\sum_{u\in\operatorname{Des}(\widehat{\sigma}_{m})}a(u)$
$\displaystyle=|\operatorname{Inv}(\widehat{\sigma})|-|\\{i<j:(\mu*0^{k})_{i}\leq(\mu*0^{k})_{j}\\}|-\sum_{u\in\operatorname{Des}(\widehat{\sigma})}a(u)+(n-\\#\\{i:\mu_{i}=0\\})(m-k)$
$\displaystyle=\operatorname{inv}(\widehat{\sigma})+\\#\\{i:\mu_{i}\neq
0\\}(m-k).$

Further, from the prior observation about how arm, $a(u)$, changes with m we
see that

$\sum_{u\in dg^{\prime}(\mu*0^{m})}a(u)=\\#\\{i:\mu_{i}\neq
0\\}(m-k)+\sum_{u\in dg^{\prime}(\mu*0^{k})}a(u)$

where arm has been calculated in the corresponding diagrams.

We then have

$\displaystyle\operatorname{coinv}(\widehat{\sigma}_{m})$
$\displaystyle=\left(\sum_{u\in
dg^{\prime}(\mu*0^{m})}a(u)\right)-\operatorname{inv}(\widehat{\sigma}_{m})$
$\displaystyle=\left(\\#\\{i:\mu_{i}\neq 0\\}(m-k)+\sum_{u\in
dg^{\prime}(\mu*0^{k})}a(u)\right)-\left(\operatorname{inv}(\widehat{\sigma})+\\#\\{i:\mu_{i}\neq
0\\}(m-k)\right)$ $\displaystyle=\left(\sum_{u\in
dg^{\prime}(\mu*0^{k})}a(u)\right)-\operatorname{inv}(\widehat{\sigma})$
$\displaystyle=\operatorname{coinv}(\widehat{\sigma}).$

Thus
$\operatorname{maj}(\widehat{\sigma}_{m})=\operatorname{maj}(\widehat{\sigma})$
and
$\operatorname{coinv}(\widehat{\sigma}_{m})=\operatorname{coinv}(\widehat{\sigma}).$

Lastly, we return to the expansion of $E_{\mu*0^{m}}$ we found above. For each
partition $\lambda$ with $|\lambda|\leq|\mu|$ we now see that

$\displaystyle\sum_{\begin{subarray}{c}\sigma:\mu*0^{m}\rightarrow[n+\ell(\lambda)]\\\
\text{non-attacking}\\\ \forall i\leavevmode\nobreak\
\lambda_{i}=|\sigma^{-1}(n+i)|\end{subarray}}x_{1}^{|\sigma^{-1}(1)|}\cdots
x_{n}^{|\sigma^{-1}(n)|}q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu)\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}\sigma:\mu*0^{\ell(\lambda)}\rightarrow[n+\ell(\lambda)]\\\
\text{non-attacking}\\\ \forall i\leavevmode\nobreak\
\lambda_{i}=|\sigma^{-1}(n+i)|\end{subarray}}x_{1}^{|\sigma^{-1}(1)|}\cdots
x_{n}^{|\sigma^{-1}(n)|}\Gamma^{(m)}(\widehat{\sigma}).$

where

$\Gamma^{(m)}(\widehat{\sigma}):=q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*0^{\ell(\lambda)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{not in row
}1\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right)\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*0^{\ell(\lambda)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{in row
}1\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+m+1)}}\right).\
$

and we calculate all of the associated statistics in their respective
diagrams.

∎

Now that we have conveniently rearranged the monomial terms of each
$E_{\mu*0^{m}}$ and identified the dependence of the coefficients on the
parameter $m$ we can give a simple proof that the sequence
$(E_{\mu*0^{m}})_{m\geq 0}$ converges.

######  Corollary/Definition 25.

Let $\mu$ be a composition with $\ell(\mu)=n$. The sequence
$(E_{\mu*0^{m}})_{m\geq 1}$ converges to an almost-symmetric function
$\widetilde{E}_{\mu}:=\lim_{m}E_{\mu*0^{m}}\in\operatorname{\mathscr{P}}_{as}^{+}$
given explicitly by

$\widetilde{E}_{\mu}=\sum_{\begin{subarray}{c}\lambda\leavevmode\nobreak\
\text{partition}\\\
|\lambda|\leq|\mu|\end{subarray}}m_{\lambda}[x_{n+1}+\ldots]\sum_{\begin{subarray}{c}\sigma:\mu*0^{\ell(\lambda)}\rightarrow[n+\ell(\lambda)]\\\
\text{non-attacking}\\\ \forall i=1,...,\ell(\lambda)\\\
\lambda_{i}=|\sigma^{-1}(n+i)|\end{subarray}}x_{1}^{|\sigma^{-1}(1)|}\cdots
x_{n}^{|\sigma^{-1}(n)|}\widetilde{\Gamma}(\widehat{\sigma})$

where

$\widetilde{\Gamma}(\widehat{\sigma}):=\lim_{m}\Gamma^{(m)}(\widehat{\sigma})=q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*0^{\ell(\lambda)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{not in row
}1\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right)\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*0^{\ell(\lambda)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{in row }1\end{subarray}}\left(1-t\right).$

###### Proof.

Note that the formula in Theorem 24 is a fixed size finite sum where the only
dependence on m is in the $m_{\lambda}$ symmetric function terms and the
$t^{m}$ occurring in the $\Gamma^{(m)}$ terms. Thus in the sense of Ion and
Wu, see Definition 15, this sequence converges to a well defined element of
$\operatorname{\mathscr{P}}_{as}^{+}$. In particular, each
$m_{\lambda}[x_{n+1}+\ldots+x_{n+m}]$ converges to
$m_{\lambda}[x_{n+1}+\ldots]$ and $t^{m}$ converges to 0 in the
$\widetilde{\Gamma}$-term. Simplifying gives the formula above.

∎

It follows from Corollary 25 that the almost symmetric functions
$\widetilde{E}_{\mu}$ are homogeneous of degree $|\mu|$ and
$\widetilde{E}_{\mu}\in\operatorname{\mathscr{P}}(\ell(\mu))^{+}.$ Note
importantly, that for any composition $\mu$ (not necessarily reduced) and any
$n\geq 0$, by shifting the terms of the sequence $(E_{\mu*0^{m}})_{m\geq 0}$
we see that $\widetilde{E}_{\mu*0^{n}}=\widetilde{E}_{\mu}$.

######  Corollary 26.

Let $\lambda$ be a partition with $\ell(\lambda)=n$ and $|\lambda|=N$. Then
$\widetilde{E}_{\lambda}$ is determined by
$E_{\lambda*0^{N}}\in\operatorname{\mathscr{P}}_{n+N}^{+}$. That is to say, if

$E_{\lambda*0^{N}}(x_{1},...,x_{n+N})=c_{1}x^{\mu^{(1)}}m_{\nu^{(1)}}[x_{n+1}+...+x_{n+N}]+...+c_{k}x^{\mu^{(k)}}m_{\nu^{(k)}}[x_{n+1}+...+x_{n+N}]$

then

$\widetilde{E}_{\lambda}=c_{1}x^{\mu^{(1)}}m_{\nu^{(1)}}[x_{n+1}+...]+...+c_{k}x^{\mu^{(k)}}m_{\nu^{(k)}}[x_{n+1}+...].$

###### Proof.

As $\lambda$ is a partition, row 1 of any non-attacking labelling of $\lambda$
must be 1,2,,…,$\ell(\lambda)$. Thus no boxes of $dg^{\prime}(\lambda)$ in row
1 will have $\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))$ and so there will
be no contributions from any of the terms of the form

$\prod_{\begin{subarray}{c}u\in dg^{\prime}(\lambda)\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{row
}1\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+m+1)}}\right).$

Further, from Corollary 25 it is clear that these are the only coefficients
that depend on m in the limit. Also it follows that each term of the form
$x^{\mu}m_{\nu}[x_{n+1}+\ldots]$ that occurs in the expansion of
$\widetilde{E}_{\lambda}$ appears at least by the $m=N$ step of the limit.
From these two facts it follows that the expansion of
$\widetilde{E}_{\lambda}$ will match that of
$E_{\lambda*0^{N}}(x_{1},...,x_{n+N})$ up to truncating each
$m_{\nu}[x_{n+1}+\ldots]$ to $m_{\nu}[x_{n+1}+\ldots+x_{n+N}]$ using
$\pi_{n+N}.$

∎

##  4 $\mathscr{Y}$-Weight Basis of $\mathscr{P}_{as}^{+}$

Given a family of commuting operators $\\{y_{i}:i\in I\\}$ and a weight vector
v we denote its weight by the function $\alpha:I\rightarrow\mathbb{Q}(q,t)$
such that $y_{i}v=\alpha(i)v.$ We sometimes denote $\alpha$ as
$(\alpha_{1},\alpha_{2},\ldots).$

###  4.1 The $\widetilde{E}_{\mu}$ are $\mathscr{Y}$-Weight Vectors

In what follows, the classical spectral theory for non-symmetric Macdonald
polynomials is used to demonstrate that the limit functions
$\widetilde{E}_{\mu}$ are $\mathscr{Y}$-weight vectors. The below lemma is a
simple application of this classical theory and basic properties of the
$t$-adic topology on $\mathbb{Q}(q,t)$.

######  Lemma 27.

For a composition $\mu$ with $\ell(\mu)=n$ define $\alpha_{\mu}^{(m)}$ to be
the $Y^{(n+m)}$-weight of $E_{\mu*0^{m}}$. Then in the $t$-adic topology on
$\mathbb{Q}(q,t)$ the sequence $(t^{n+m}\alpha_{\mu}^{(m)}(i))_{m\geq 0}$
converges in m to some $\widetilde{\alpha}_{\mu}(i)\in\mathbb{Q}(q,t)$. In
particular, $\widetilde{\alpha}_{\mu}(i)=0$ for $i>n$ and for $1\leq i\leq n$
we have that $\widetilde{\alpha}_{\mu}(i)=0$ exactly when $\mu_{i}=0$.

###### Proof.

Take $\mu=(\mu_{1},\ldots,\mu_{n})$. From classical double affine Hecke
algebra theory [Che03] we have
$\alpha_{\mu}^{(0)}(i)=q^{\mu_{i}}t^{1-\beta_{\mu}(i)}$ where

$\beta_{\mu}(i):=\\#\\{j:1\leq j\leq i\leavevmode\nobreak\
,\mu_{j}\leq\mu_{i}\\}+\\#\\{j:i<j\leq n\leavevmode\nobreak\
,\mu_{i}>\mu_{j}\\}.$

If we calculate $\beta_{\mu*0^{m}}(i)$ directly it follows then that

$t^{n+m}\alpha_{\mu}^{(m)}(i)=\begin{cases}q^{\mu_{i}}t^{n+m+1-(\beta_{\mu}(i)+m\mathbbm{1}(\mu_{i}\neq
0))}=t^{n}\alpha_{\mu}^{(0)}(i)&i\leq n,\mu_{i}\neq 0\\\
q^{\mu_{i}}t^{n+m+1-(\beta_{\mu}(i)+m\mathbbm{1}(\mu_{i}\neq
0))}=t^{n+m}\alpha_{\mu}^{(0)}(i)&i\leq n,\mu_{i}=0\\\
t^{n+m+1-(\\#\\{j:\mu_{j}=\leavevmode\nobreak\
0\\}+i-n)}=t^{\\#\\{j:\mu_{j}\neq\leavevmode\nobreak\
0\\}}t^{m+1-(i-n)}&i>n.\end{cases}$

Lastly, by taking the limit $m\rightarrow\infty$ we get the result. ∎

For a composition $\mu$ define the weight $\widetilde{\alpha}_{\mu}$ using the
formula in Lemma 27 for the list of scalars $\widetilde{\alpha}_{\mu}(i)$ for
$i\in\mathbb{N}$.

######  Lemma 28.

For a composition $\mu=(\mu_{1},\ldots,\mu_{n})$ with $\mu_{i}\neq 0$ for
$1\leq i\leq n$, $\widetilde{E}_{\mu}$ is a ${\mathscr{Y}}$-weight vector with
weight $\widetilde{\alpha}_{\mu}$.

###### Proof.

Fix any $r\in\mathbb{N}$. We start by rewriting the operator
${\mathscr{Y}}_{r}$ explicitly in terms of the limit definition of
${\mathscr{Y}}_{1}$.

$\displaystyle{\mathscr{Y}}_{r}$ $\displaystyle=t^{-(r-1)}T_{r-1}\cdots
T_{1}{\mathscr{Y}}_{1}T_{1}\cdots T_{r-1}$
$\displaystyle=t^{-(r-1)}T_{r-1}\cdots
T_{1}\lim_{k}t^{k}\rho\omega_{k}^{-1}T_{k-1}^{-1}\cdots T_{1}^{-1}T_{1}\cdots
T_{r-1}\pi_{k}$ $\displaystyle=\lim_{k}t^{k}T_{r-1}\cdots T_{1}\rho
t^{-(r-1)}\omega_{k}^{-1}T_{k-1}^{-1}\cdots T_{r}^{-1}\pi_{k}$
$\displaystyle=\lim_{k}t^{k}T_{r-1}\cdots T_{1}\rho T_{1}^{-1}\cdots
T_{r-1}^{-1}t^{-(r-1)}T_{r-1}\cdots T_{1}\omega_{k}^{-1}T_{k-1}^{-1}\cdots
T_{r}^{-1}\pi_{k}$ $\displaystyle=\lim_{k}t^{k}T_{r-1}\cdots T_{1}\rho
T_{1}^{-1}\cdots T_{r-1}^{-1}Y_{r}^{(k)}\pi_{k}.$

Applying ${\mathscr{Y}}_{r}$ to $\widetilde{E}_{\mu}$ we see by taking
$k=n+m\geq n$ and shifting the indices that

$\displaystyle{\mathscr{Y}}_{r}(\widetilde{E}_{\mu})$
$\displaystyle=\lim_{m}t^{n+m}T_{r-1}\cdots T_{1}\rho T_{1}^{-1}\cdots
T_{r-1}^{-1}Y_{r}^{(n+m)}(E_{\mu*0^{m}})$ $\displaystyle=\lim_{m}T_{r-1}\cdots
T_{1}\rho T_{1}^{-1}\cdots
T_{r-1}^{-1}t^{n+m}\alpha_{\mu}^{(m)}(r)E_{\mu*0^{m}}$

and by Lemma 27 this converges to

${\mathscr{Y}}_{r}(\widetilde{E}_{\mu})=\widetilde{\alpha}_{\mu}(r)(T_{r-1}\cdots
T_{1}\rho T_{1}^{-1}\cdots T_{r-1}^{-1})\widetilde{E}_{\mu}.$

Importantly, we have implicitly used the fact that both of the sequences
$(E_{\mu*0^{m}})_{m}$ and $(\alpha_{\mu}^{(m)}(r))_{m}$ converge, that the
operator $T_{r-1}\cdots T_{1}\rho T_{1}^{-1}\cdots T_{r-1}^{-1}$ commutes with
the quotient maps
$\pi_{k}:\operatorname{\mathscr{P}}_{k+1}\rightarrow\operatorname{\mathscr{P}}_{k}$
for $k>r$, and Proposition 6.21 in [IW22]. We will show that the right side is
$\widetilde{\alpha}_{\mu}(r)\widetilde{E}_{\mu}$. As
$\widetilde{\alpha}_{\mu}(r)=0$ for $r>n$ by Lemma 27 we reduce to the sub
case $r\leq n$. Fix $r\leq n$. If we could show that $x_{1}$ divides
$T_{1}^{-1}\cdots T_{r-1}^{-1}\widetilde{E}_{\mu}$ then we would have

$\rho(T_{1}^{-1}\cdots T_{r-1}^{-1}\widetilde{E}_{\mu})=T_{1}^{-1}\cdots
T_{r-1}^{-1}\widetilde{E}_{\mu}$

implying that

$\displaystyle{\mathscr{Y}}_{r}(\widetilde{E}_{\mu})$
$\displaystyle=\widetilde{\alpha}_{\mu}(r)(T_{r-1}\cdots T_{1}\rho
T_{1}^{-1}\cdots T_{r-1}^{-1})\widetilde{E}_{\mu}$
$\displaystyle=\widetilde{\alpha}_{\mu}(r)T_{r-1}\cdots T_{1}T_{1}^{-1}\cdots
T_{r-1}^{-1})\widetilde{E}_{\mu}$
$\displaystyle=\widetilde{\alpha}_{\mu}(r)\widetilde{E}_{\mu}$

as desired. To show that $x_{1}|T_{1}^{-1}\cdots
T_{r-1}^{-1}\widetilde{E}_{\mu}$ it suffices to show that for all $m\geq 0$,
$x_{1}|T_{1}^{-1}\cdots T_{r-1}^{-1}E_{\mu*0^{m}}$. To this end fix $m\geq 0$.
We have that

$\displaystyle\alpha_{\mu}^{(m)}(r)E_{\mu*0^{m}}$
$\displaystyle=Y_{r}^{(n+m)}(E_{\mu*0^{m}})$
$\displaystyle=t^{n+m-r+1}T_{r-1}\cdots
T_{1}\omega_{n+m}^{-1}T_{n+m-1}^{-1}\cdots T_{r}^{-1}E_{\mu*0^{m}}.$

Since $\alpha_{\mu}^{(m)}(r)\neq 0$ we can have
$\frac{1}{\alpha_{\mu}^{(m)}(r)}T_{1}^{-1}\cdots T_{r-1}^{-1}$ act on both
sides of the above to get

$\displaystyle T_{1}^{-1}\cdots T_{r-1}^{-1}E_{\mu*0^{m}}$
$\displaystyle=\frac{t^{n+m-r+1}}{\alpha_{\mu}^{(m)}(r)}\omega_{n+m}^{-1}T_{n+m-1}^{-1}\cdots
T_{r}^{-1}E_{\mu*0^{m}}.$

By HHL any non-attacking labelling of $\mu*0^{m}$ will have row 1 diagram
labels given by $\\{1,2,\ldots,n\\}$ so in particular $x_{r}$ divides
$E_{\mu*0^{m}}$ for all $m>0$. Lastly,

$\displaystyle\omega_{n+m}^{-1}T_{n+m-1}^{-1}\cdots T_{r}^{-1}X_{r}$
$\displaystyle=\omega_{n+m}^{-1}t^{-(n+m-r)}X_{n+m}T_{n+m-1}\cdots T_{r}$
$\displaystyle=qt^{-(n+m-r)}X_{1}\omega_{n+m}^{-1}T_{n+m-1}\cdots T_{r}.$

Thus $x_{1}$ divides $T_{1}^{-1}\cdots T_{r-1}^{-1}E_{\mu*0^{m}}$ for all
$m\geq 0$ showing the result.

∎

Now we consider the general situation where the composition $\mu$ can have
some parts which are 0. We can extend the above result, Lemma 28, by a
straight-forward argument using intertwiner theory from the study of affine
Hecke algebras.

######  Theorem 29.

For all compositions $\mu$, $\widetilde{E}_{\mu}$ is a ${\mathscr{Y}}$-weight
vector with weight $\widetilde{\alpha}_{\mu}.$

###### Proof.

Lemma 28 shows that this statement holds for any composition with all parts
nonzero. Fix a composition $\mu$ with length $n$. We know that by sorting in
decreasing order that $\mu$ can be written as a permutation of a composition
of the form $\nu*0^{m}$ for a partition $\nu$ and some $m\geq 0$. From the
definition of Bruhat order it follows that $\nu*0^{m}$ will be the minimal
element out of all of its distinct permutations, including $\mu$. Necessarily,
this finite subposet generated by the permutations of $\nu*0^{m}$ is
isomorphic to the Bruhat ordering on the coset space
$\mathfrak{S}_{n}/\mathfrak{S}_{\kappa}$ where $\mathfrak{S}_{\kappa}$ is the
Young subgroup of $\mathfrak{S}_{n}$ corresponding to the stabilizer of
$\nu*0^{m}$. Hence, it suffices to show inductively that for any composition
$\beta$ with $\nu*0^{m}\leq\beta<s_{i}(\beta)\leq\mu$, if
$\widetilde{E}_{\beta}$ satisfies the theorem then so will
$\widetilde{E}_{s_{i}(\beta)}$. As $\mu$ is finitely many covering elements
away in Bruhat from $\nu*0^{m}$ this induction will indeed terminate after
finitely many steps.

Using the intertwiner operators from affine Hecke algebra theory, given by
$\varphi_{i}=T_{i}{\mathscr{Y}}_{i}-{\mathscr{Y}}_{i}T_{i}$ in this context,
we only need to show that for any composition $\beta$ with
$\nu*0^{m}\leq\beta<s_{i}(\beta)\leq\mu$,

$\varphi_{i}\widetilde{E}_{\beta}=(\widetilde{\alpha}_{\beta}(i)-\widetilde{\alpha}_{\beta}(i+1))\widetilde{E}_{s_{i}(\beta)}.$

Suppose the theorem holds for some $\beta$ with
$\nu*0^{m}\leq\beta<s_{i}(\beta)\leq\mu$. Then we have the following:

$\displaystyle\varphi_{i}\widetilde{E}_{\beta}$
$\displaystyle=(T_{i}({\mathscr{Y}}_{i}-{\mathscr{Y}}_{i+1})+(1-t){\mathscr{Y}}_{i+1})\widetilde{E}_{\beta}$
$\displaystyle=(\widetilde{\alpha}_{\beta}(i)-\widetilde{\alpha}_{\beta}(i+1))T_{i}\widetilde{E}_{\beta}+(1-t)\widetilde{\alpha}_{\beta}(i+1)\widetilde{E}_{\beta}$
$\displaystyle=\lim_{m}(t^{n+m}\alpha_{\beta}^{(m)}(i)-t^{n+m}\alpha_{\beta}^{(m)}(i+1))T_{i}E_{\beta*0^{m}}+(1-t)t^{n+m}\alpha_{\beta}^{(m)}(i+1)E_{\beta*0^{m}}$
$\displaystyle=\lim_{m}(t^{n+m}\alpha_{\beta}^{(m)}(i)-t^{n+m}\alpha_{\beta}^{(m)}(i+1))E_{s_{i}(\beta)*0^{m}}$
$\displaystyle=(\widetilde{\alpha}_{\beta}(i)-\widetilde{\alpha}_{\beta}(i+1))\widetilde{E}_{s_{i}(\beta)}.$

∎

As an immediate consequence of the proof of Theorem 29 we have the following.

######  Corollary 30.

Let $\mu$ be a composition and $i\geq 1$ such that $s_{i}(\mu)>\mu$. Then

$\widetilde{E}_{s_{i}(\mu)}=\left(T_{i}+\frac{(1-t)\widetilde{\alpha}_{\mu}(i+1)}{\widetilde{\alpha}_{\mu}(i)-\widetilde{\alpha}_{\mu}(i+1)}\right)\widetilde{E}_{\mu}.$

We have shown in Theorem 29 there is an explicit collection of
$\mathscr{Y}$-weight vectors $\widetilde{E}_{\mu}$ in $\mathscr{P}_{as}^{+}$
arising as the limits of non-symmetric Macdonald polynomials $E_{\mu*0^{m}}$.
Unfortunately, these $\widetilde{E}_{\mu}$ do not span $\mathscr{P}_{as}^{+}$.
To see this note that one cannot write a non-constant symmetric function as a
linear combination of the $\widetilde{E}_{\mu}$. However, in the below work we
build a full ${\mathscr{Y}}$-weight basis of
$\operatorname{\mathscr{P}}_{as}^{+}$.

###  4.2 Constructing a Full ${\mathscr{Y}}$-Weight Basis

####  4.2.1 Defining the Stable-Limit Non-symmetric Macdonald Functions

To complete our construction of a full weight basis of $\mathscr{P}_{as}^{+}$
we will need the $\partial_{-}^{(k)}$ operators from Ion and Wu. These
operators are, up to a change of variables and plethsym, the $d_{-}$ operators
from Carlsson and Mellit’s standard $\mathbb{A}_{t,q}$ representation.

######  Definition 31.

[IW22] Define the operator
$\partial_{-}^{(k)}:\mathscr{P}(k)^{+}\rightarrow\mathscr{P}(k-1)^{+}$ to be
the $\mathscr{P}_{k-1}^{+}$-linear map which acts on elements of the form
$x_{k}^{n}F[x_{k+1}+x_{k+2}+\ldots]$ for $F\in\Lambda$ and $n\geq 0$ as

$\partial_{-}^{(k)}(x_{k}^{n}F[x_{k+1}+x_{k+2}+\ldots])=\mathscr{B}_{n}(F)[x_{k}+x_{k+1}+\ldots].$

Here the $\mathscr{B}_{n}$ are the Jing operators which serve as creation
operators for Hall-Littlewood symmetric functions $\mathcal{P}_{\lambda}$
given explicitly by the following plethystic formula:

$\mathscr{B}_{n}(F)[X]=\langle z^{n}\rangle F[X-z^{-1}]Exp[(1-t)zX].$

Importantly, the $\partial_{-}^{(k)}$ operators do not come from the
$\mathscr{H}^{+}$ action itself. Note that the $\partial_{-}^{(k)}$ operators
are homogeneous by construction.

We will require the useful alternative expression for the $\partial_{-}^{(k)}$
operators which can be found in [IW22]. Recall the notation $\mathfrak{c}_{y}$
from Definition 21.

######  Lemma 32.

Let $\tau_{k}$ denote the alphabet shift
$\mathfrak{X}_{k}\rightarrow\mathfrak{X}_{k-1}$ acting on symmetric functions
where $\mathfrak{X}_{i}:=x_{i+1}+x_{i+2}+...$. Then for
$f\in\operatorname{\mathscr{P}}_{k}$ and $F\in\Lambda$

$\partial_{-}^{(k)}(f(x_{1},\dots
x_{k})F[\mathfrak{X}_{k}])=\tau_{k}\mathfrak{c}_{x_{k}}f(x_{1},...,x_{k})F[\mathfrak{X}_{k}-x_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}].$

###### Proof.

[IW22]. ∎

As an immediate consequence of this explicit description of the action of the
$\partial_{-}^{(k)}$ operator we get the following required lemmas.

######  Lemma 33.

[IW22] The map
$\partial_{-}^{(k)}:\operatorname{\mathscr{P}}(k)^{+}\rightarrow\operatorname{\mathscr{P}}(k-1)^{+}$
is a projection onto $\operatorname{\mathscr{P}}(k-1)^{+}$ i.e. for
$f\in\operatorname{\mathscr{P}}(k-1)^{+}\subset\operatorname{\mathscr{P}}(k)^{+}$
we have that $\partial_{-}^{(k)}(f)=f$.

###### Proof.

Fix $F\in\Lambda.$ It suffices to show that
$\partial_{-}^{(k)}(F[\mathfrak{X}_{k-1}])=F[\mathfrak{X}_{k-1}].$ By using
the coproduct on $\Lambda$ we can expand
$F[\mathfrak{X}_{k-1}]=F[x_{k}+\mathfrak{X}_{k}]$ in powers of $x_{k}^{i}$
with some coefficients $F_{i}\in\Lambda$ as
$F[x_{k}+\mathfrak{X}_{k}]=\sum_{i\geq 0}x_{k}^{i}F_{i}[\mathfrak{X}_{k}].$
From Lemma 32 we have

$\displaystyle\partial_{-}^{(k)}(F[\mathfrak{X}_{k-1}])$
$\displaystyle=\partial_{-}^{(k)}(F[x_{k}+\mathfrak{X}_{k}])$
$\displaystyle=\partial_{-}^{(k)}(\sum_{i\geq
0}x_{k}^{i}F_{i}[\mathfrak{X}_{k}])$
$\displaystyle=\tau_{k}\mathfrak{c}_{x_{k}}\left(\sum_{i\geq
0}x_{k}^{i}F_{i}[\mathfrak{X}_{k}-x_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]\right)$
$\displaystyle=\tau_{k}\mathfrak{c}_{x_{k}}F[\mathfrak{X}_{k}-x_{k}+x_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]$
$\displaystyle=\tau_{k}\mathfrak{c}_{x_{k}}F[\mathfrak{X}_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]$
$\displaystyle=\tau_{k}F[\mathfrak{X}_{k}]\mathfrak{c}_{x_{k}}Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]$
$\displaystyle=\tau_{k}F[\mathfrak{X}_{k}]$
$\displaystyle=F[\mathfrak{X}_{k-1}].$

∎

######  Lemma 34.

For all $G\in\Lambda$ and $g(x)\in\operatorname{\mathscr{P}}(k)^{+}$

$\partial_{-}^{(k)}(G[x_{k}+x_{k+1}+\dots]g(x))=G[x_{k}+x_{k+1}+\dots]\partial_{-}^{(k)}(g(x)).$

###### Proof.

It suffices to take $g(x)\in\operatorname{\mathscr{P}}(k)^{+}$ to be of the
form $g(x)=f(x_{1},\dots,x_{k})F[\mathfrak{X}_{k}]$ with
$f\in\operatorname{\mathscr{P}}_{k}^{+}$ and $F\in\Lambda$. From Lemma 32 we
get the following:

$\displaystyle\partial_{-}^{(k)}(G[x_{k}+x_{k+1}+\dots]g(x))$
$\displaystyle=\partial_{-}^{(k)}(G[\mathfrak{X}_{k-1}]g(x))$
$\displaystyle=\tau_{k}\mathfrak{c}_{x_{k}}G[\mathfrak{X}_{k-1}-x_{k}]f(x_{1},\dots,x_{k})F[\mathfrak{X}_{k}-x_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]$
$\displaystyle=\tau_{k}\mathfrak{c}_{x_{k}}G[\mathfrak{X}_{k}]f(x_{1},\dots,x_{k})F[\mathfrak{X}_{k}-x_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]$
$\displaystyle=\tau_{k}G[\mathfrak{X}_{k}]\mathfrak{c}_{x_{k}}f(x_{1},\dots,x_{k})F[\mathfrak{X}_{k}-x_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]$
$\displaystyle=G[\mathfrak{X}_{k-1}]\tau_{k}\mathfrak{c}_{x_{k}}f(x_{1},\dots,x_{k})F[\mathfrak{X}_{k}-x_{k}]Exp[-(t-1)x_{k}^{-1}\mathfrak{X}_{k}]$
$\displaystyle=G[\mathfrak{X}_{k-1}]\partial_{-}^{(k)}(f(x_{1},\dots,x_{k})F[\mathfrak{X}_{k}])$
$\displaystyle=G[\mathfrak{X}_{k-1}]\partial_{-}^{(k)}(g(x)).$

∎

######  Corollary 35.

For $G\in\Lambda$ and $g(x)\in\operatorname{\mathscr{P}}(k)^{+}$

$\partial_{-}^{(k)}(G[X]g(x))=G[X]\partial_{-}^{(k)}(g(x)).$

###### Proof.

Take $G\in\Lambda$ and $g(x)\in\operatorname{\mathscr{P}}(k)^{+}$. Expand
$G[X]$ as a finite sum of terms of the form
$f_{i}(x_{1},\dots,x_{k-1})F_{i}[x_{k}+\ldots]$, where
$f_{i}\in\operatorname{\mathscr{P}}_{k-1}$ and $F_{i}\in\Lambda$ so

$G[X]=\sum_{i}f_{i}(x_{1},\dots,x_{k-1})F_{i}[x_{k}+\ldots].$

By Lemma 34 and the fact that $\partial_{-}^{(k)}$ is a
$\operatorname{\mathscr{P}}_{k-1}^{+}$-linear map from Definition 31 we now
see that

$\displaystyle\partial_{-}^{(k)}(G[X]g(x))$
$\displaystyle=\sum_{i}\partial_{-}^{(k)}(f_{i}(x_{1},\dots,x_{k-1})F_{i}[x_{k}+\ldots]g(x))$
$\displaystyle=\sum_{i}f_{i}(x_{1},\dots,x_{k-1})F_{i}[x_{k}+\ldots]\partial_{-}^{(k)}(g(x))$
$\displaystyle=G[X]\partial_{-}^{(k)}(g(x)).$

∎

We can now construct a full ${\mathscr{Y}}$-weight basis of
$\mathscr{P}_{as}^{+}$. We parameterize this basis by pairs $(\mu|\lambda)$
for $\mu$ a reduced composition and $\lambda$ a partition. Combinatorially,
this is reasonable because, as already mentioned, the monomial basis for
$\operatorname{\mathscr{P}}_{as}^{+}$,
$\\{x^{\mu}m_{\lambda}\mid\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}},\lambda\in\operatorname{\mathrm{Par}}\\}$,
is indexed by pairs of reduced compositions and partitions.

######  Definition 36.

For $\mu$ a reduced composition and $\lambda$ a partition define the stable-
limit non-symmetric Macdonald function corresponding to $(\mu|\lambda)$ as

$\widetilde{E}_{(\mu|\lambda)}:=\partial_{-}^{(\ell(\mu)+1)}\cdots\partial_{-}^{(\ell(\mu)+\ell(\lambda))}\widetilde{E}_{\mu*\lambda}.$

For a partition $\lambda$ define

$\mathcal{A}_{\lambda}:=\widetilde{E}_{(\emptyset|\lambda)}\in\Lambda.$ (1)

Later in Theorem 49, we will show that the collection
$\\{\widetilde{E}_{(\mu|\lambda)}\mid\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}},\lambda\in\operatorname{\mathrm{Par}}\\}$
is a ${\mathscr{Y}}$-weight basis for $\operatorname{\mathscr{P}}_{as}^{+}.$

######  Remark.

Note importantly that
$\widetilde{E}_{(\mu|\lambda)}\in\mathscr{P}(\ell(\mu))^{+}$ and
$\widetilde{E}_{(\mu|\lambda)}$ is homogeneous of degree $|\mu|+|\lambda|$.
Further, we have $\widetilde{E}_{(\mu|\emptyset)}=\widetilde{E}_{\mu}$ and
$\widetilde{E}_{(\emptyset|\lambda)}=\mathcal{A}_{\lambda}$. Notice that in
Definition 36 it makes sense to consider $\widetilde{E}_{(\mu|\lambda)}$ when
$\mu$ is not necessarily reduced. However, it is a nontrivial consequence of
Theorem 60 that an analogously defined $\widetilde{E}_{(\mu*0|\lambda)}$ is a
nonzero scalar multiple of $\widetilde{E}_{(\mu|\lambda)}.$ Thus there is no
need to consider the case of $\mu$ non-reduced when building a basis of
$\operatorname{\mathscr{P}}_{as}^{+}.$

There is another basis of $\operatorname{\mathscr{P}}_{as}^{+}$ given by Ion
and Wu in their unpublished work [IW22a] which is equipped with a natural
ordering with respect to which the limit Cherednik operators are triangular.
It follows then that after we show in Corollary 38 that the
$\widetilde{E}_{(\mu|\lambda)}$ are ${\mathscr{Y}}$-weight vectors that each
$\widetilde{E}_{(\mu|\lambda)}$ has a triangular expansion in Ion and Wu’s
basis.

######  Remark.

The stable-limit non-symmetric Macdonald functions
$\widetilde{E}_{(\mu|\lambda)}$ as defined in this paper are distinct from the
stable-limits of non-symmetric Macdonald polynomials occurring in [HHL08]. In
their paper Haglund, Haiman, and Loehr investigate stable-limits of the form
$(E_{0^{m}*\mu})_{m\geq 0}$ where $\mu$ is a composition. Their analysis does
not require the convergence definition of Ion and Wu as the sequences
$(E_{0^{m}*\mu})_{m\geq 0}$ have stable limits in the traditional sense.
Further, the limits of the $(E_{0^{m}*\mu})_{m\geq 0}$ sequences are symmetric
functions whereas, as we will see soon, the $\widetilde{E}_{(\mu|\lambda)}$
are not fully symmetric in general.

The following simple lemma will be used to show that since the
$\widetilde{E}_{\mu*\lambda}$ are ${\mathscr{Y}}$-weight vectors the stable-
limit non-symmetric Macdonald functions $\widetilde{E}_{(\mu|\lambda)}$ are
${\mathscr{Y}}$-weight vectors as well. We describe their weights in Corollary
38.

######  Lemma 37.

Suppose $f\in\mathcal{P}(k)^{+}$ is a ${\mathscr{Y}}$-weight vector with
weight $(\alpha_{1},\ldots,\alpha_{k},0,0,\ldots)$. Then
$\partial_{-}^{(k)}f\in\mathcal{P}(k-1)^{+}$ is a ${\mathscr{Y}}$-weight
vector with weight $(\alpha_{1},\ldots,\alpha_{k-1},0,0,\ldots)$.

###### Proof.

We know that from [IW22] for $g\in\mathcal{P}(k)^{+}$ and $1\leq i\leq k-1$,
${\mathscr{Y}}_{i}\partial_{-}^{(k)}g=\partial_{-}^{(k)}{\mathscr{Y}}_{i}g$ so
${\mathscr{Y}}_{i}\partial_{-}^{(k)}f=\partial_{-}^{(k)}{\mathscr{Y}}_{i}f=\alpha_{i}\partial_{-}^{(k)}f.$
From [IW22] we have that if $i\geq k$ then ${\mathscr{Y}}_{i}$ annihilates
$\mathscr{P}(k-1)$. Since $\partial_{-}^{(k)}f\in\mathcal{P}(k-1)^{+}$ for all
$i\geq k$, ${\mathscr{Y}}_{i}\partial_{-}^{(k)}f=0$.

∎

######  Example.

Here we give a few basic examples of stable-limit non-symmetric Macdonald
functions expanded in the Hall-Littlewood basis $\mathcal{P}_{\lambda}$ and
their corresponding weights.

$\displaystyle\bullet\widetilde{E}_{(\emptyset|2)}$
$\displaystyle=\mathcal{P}_{2}[x_{1}+\ldots]+\frac{q^{-1}}{1-q^{-1}t}\mathcal{P}_{1,1}[x_{1}+\ldots];$
$\displaystyle\text{weight }\widetilde{\alpha}_{(\emptyset|2)}=(0,0,\ldots)$
$\displaystyle\bullet\widetilde{E}_{(2|\emptyset)}$
$\displaystyle=x_{1}^{2}+\frac{q^{-1}}{1-q^{-1}t}x_{1}\mathcal{P}_{1}[x_{2}+\ldots];$
$\displaystyle\text{weight
}\widetilde{\alpha}_{(2|\emptyset)}=(q^{2}t,0,\ldots)$
$\displaystyle\bullet\widetilde{E}_{(1,1,1|\emptyset)}$
$\displaystyle=x_{1}x_{2}x_{3};$ $\displaystyle\text{weight
}\widetilde{\alpha}_{(1,1,1|\emptyset)}=(qt^{3},qt^{2},qt,0,\ldots)$
$\displaystyle\bullet\widetilde{E}_{(1,1|1)}$
$\displaystyle=x_{1}x_{2}\mathcal{P}_{1}[x_{3}+\ldots];$
$\displaystyle\text{weight
}\widetilde{\alpha}_{(1,1|1)}=(qt^{3},qt^{2},0,\ldots)$
$\displaystyle\bullet\widetilde{E}_{(1|1,1)}$
$\displaystyle=x_{1}\mathcal{P}_{1,1}[x_{2}+\cdots];$
$\displaystyle\text{weight }\widetilde{\alpha}_{(1|1,1)}=(qt^{3},0,\ldots)$

As an immediate result of Lemma 37 we have the following:

######  Corollary 38.

For $\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}}$ and
$\lambda\in\operatorname{\mathrm{Par}}$,
$\widetilde{E}_{(\mu|\lambda)}\in\operatorname{\mathscr{P}}_{as}^{+}$ is a
${\mathscr{Y}}$-weight vector with weight $\widetilde{\alpha}_{(\mu|\lambda)}$
given explicitly by

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

###### Proof.

By Definition 36 we have that

$\widetilde{E}_{(\mu|\lambda)}:=\partial_{-}^{(\ell(\mu)+1)}\cdots\partial_{-}^{(\ell(\mu)+\ell(\lambda))}\widetilde{E}_{\mu*\lambda}.$

From Theorem 29 we know that $\widetilde{E}_{\mu*\lambda}$ is a
${\mathscr{Y}}$-weight vector with weight $\widetilde{\alpha}_{\mu*\lambda}.$
Recall that from Lemma 27 that
$\widetilde{\alpha}_{\mu*\lambda}(i)=q^{(\mu*\lambda)_{i}}t^{\ell(\mu*\lambda)+1-\beta_{\mu*\lambda}(i)}$
for $i\leq\ell(\mu*\lambda)$ and equals $0$ for $i>\ell(\mu*\lambda).$ Using
Lemma 37 inductively now shows that $\widetilde{E}_{(\mu|\lambda)}$ is a
${\mathscr{Y}}$-weight vector with weight $\widetilde{\alpha}_{(\mu|\lambda)}$
given by the expression given in the statement of this corollary. ∎

By using the HHL-type formula we proved for the functions
$\widetilde{E}_{\mu}$ in Corollary 25, we readily find a similar formula for
the full set of stable-limit non-symmetric Macdonald functions.

######  Corollary 39.

For a reduced composition $\mu$ and partition $\lambda$ we have that

$\widetilde{E}_{(\mu|\lambda)}=\sum_{\begin{subarray}{c}\nu\leavevmode\nobreak\
\text{partition}\\\
|\nu|\leq|\mu|+|\lambda|\end{subarray}}\sum_{\begin{subarray}{c}\sigma:\mu*\lambda*0^{\ell(\nu)}\rightarrow[\ell(\mu)+\ell(\lambda)+\ell(\nu)]\\\
\text{non-attacking}\\\ \forall i=1,...,\ell(\nu)\\\
\nu_{i}=|\sigma^{-1}(\ell(\mu)+\ell(\lambda)+i)|\end{subarray}}\widetilde{\Gamma}(\widehat{\sigma})x_{1}^{|\sigma^{-1}(1)|}\cdots
x_{\ell(\mu)}^{|\sigma^{-1}(\ell(\mu))|}\leavevmode\nobreak\ \times\\\
\mathscr{B}_{|\sigma^{-1}(\ell(\mu)+1)|}\cdots\mathscr{B}_{|\sigma^{-1}(\ell(\mu)+\ell(\lambda))|}(m_{\nu})[\mathfrak{X}_{\ell(\mu)+\ell(\lambda)}]$

where

$\widetilde{\Gamma}(\widehat{\sigma}):=q^{-\operatorname{maj}(\widehat{\sigma})}t^{\operatorname{coinv}(\widehat{\sigma})}\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*\lambda*0^{\ell(\nu)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{not in row
}1\end{subarray}}\left(\frac{1-t}{1-q^{-(\operatorname{lg}(u)+1)}t^{(a(u)+1)}}\right)\prod_{\begin{subarray}{c}u\in
dg^{\prime}(\mu*\lambda*0^{\ell(\nu)})\\\
\widehat{\sigma}(u)\neq\widehat{\sigma}(d(u))\\\ u\leavevmode\nobreak\
\text{in row }1\end{subarray}}\left(1-t\right).$

Unfortunately, this formula is not nearly as elegant or useful as the HHL
formula (10). The main obstruction comes from not having a full understanding
of the action of the Jing operators $\mathscr{B}_{a}$ on the monomial
symmetric functions. If one were to find an explicit expansion of elements
like $\mathscr{B}_{a_{1}}\cdots\mathscr{B}_{a_{r}}(m_{\lambda})$ into another
suitable basis of $\Lambda$ (possibly the $\mathcal{P}_{\nu}$ basis) one would
be able to give a much more elegant description of these functions. Likely
there is a nice way to do this that has eluded this author.

###  4.3 $\mathcal{A}_{\lambda}$ Basis for $\Lambda$ and Symmetrization via
the Trivial Hecke Idempotent

Lemma 33 shows that the following operator is well defined on
$\operatorname{\mathscr{P}}_{as}^{+}$ i.e. independent of $k$.

######  Definition 40.

For $f\in\mathscr{P}(k)^{+}\subset\mathscr{P}_{as}^{+}$ define

$\widetilde{\sigma}(f):=\partial_{-}^{(1)}\cdots\partial_{-}^{(k)}f.$ (2)

Then $\widetilde{\sigma}$ defines an operator
$\mathscr{P}_{as}^{+}\rightarrow\Lambda$ which we call the stable-limit
symmetrization operator.

######  Remark.

Note that $\widetilde{\sigma}(\widetilde{E}_{\lambda})=\mathcal{A}_{\lambda}$
and
$\widetilde{\sigma}(\widetilde{E}_{(\mu|\lambda)})=\widetilde{\sigma}(\widetilde{E}_{\mu*\lambda}).$

######  Definition 41.

For all $0\leq k<n$ define the operator
$\epsilon^{(n)}_{k}:\operatorname{\mathscr{P}}_{n}^{+}\rightarrow\operatorname{\mathscr{P}}_{n}^{+}$
as

$\epsilon^{(n)}_{k}(f):=\frac{1}{[n-k]_{t}!}\sum_{\sigma\in\mathfrak{S}_{(1^{k},n-k)}}t^{{n-k\choose
2}-\ell(\sigma)}T_{\sigma}(f).$ (3)

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 $\epsilon^{(n)}_{0}$.

For $n\geq 1$ define the rational function

$\Omega_{n}(x)=\Omega_{n}(x_{1},\dots,x_{n};t):=\prod_{1\leq i<j\leq
n}(\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}).$ (4)

We will need the following technical result relating the action of
$\epsilon^{(n)}$ on polynomials to a Weyl character type sum involving
$\Omega_{n}$.

######  Proposition 42.

For $f(x)\in\operatorname{\mathscr{P}}_{n}^{+}$

$\epsilon^{(n)}(f(x))=\frac{1}{[n]_{t}!}\sum_{\sigma\in\mathfrak{S}_{n}}\sigma(f(x)\Omega_{n}(x)).$
(5)

###### Proof.

See Remark 4.17 in [Ram06]. After translating the finite Hecke algebra
quadratic relations in [Ram06] to match those occurring in this paper the
formula matches. ∎

From the formula above in Proposition 42 we can show that the sequence of
trivial idempotents $(\epsilon^{(n)})_{n\geq 1}$ converges in the sense of
[IW22].

######  Proposition 43.

The sequence of operators $(\epsilon^{(n)})_{n\geq 1}$ converges to an
idempotent operator
$\epsilon:\operatorname{\mathscr{P}}_{as}^{+}\rightarrow\Lambda$ such that for
all $i\geq 1$, $\epsilon T_{i}=\epsilon$.

###### Proof.

From [Mac15] in Chapter 3 and Proposition 42 we see that for all partitions
$\lambda$ with $\ell(\lambda)=k$ and $n\geq k$ that

$\epsilon^{(n)}(x^{\lambda})=\frac{[n-k]_{t}!}{[n]_{t}!}v_{\lambda}(t)P_{\lambda}[x_{1}+\ldots+x_{n};t]$
(6)

where $P_{\lambda}[X;t]$ is the Hall-Littlewood symmetric function defined by
Macdonald (not to be confused with $\mathcal{P}_{\lambda}[X]$ seen previously
in this paper) and $v_{\lambda}(t):=\prod_{i\geq 1}([m_{i}(\lambda)]_{t}!)$
where $m_{i}(\lambda)$ is the number of $i$ ’s in
$\lambda=1^{m_{1}(\lambda)}2^{m_{2}(\lambda)}\cdots$. Now we note that with
respect to the $t$-adic topology,

$\lim_{n\rightarrow\infty}\frac{[n-k]_{t}!}{[n]_{t}!}=(1-t)^{k}$

so that

$\lim_{n}\epsilon^{(n)}(x^{\lambda})=v_{\lambda}(t)(1-t)^{\ell(\lambda)}P_{\lambda}[X;t]$

and hence $(\epsilon^{(n)}(x^{\lambda}))_{n\geq 1}$ converges. Note that
following Macdonald’s definitions,

$v_{\lambda}(t)(1-t)^{\ell(\lambda)}P_{\lambda}[X;t]=Q_{\lambda}[X;t].$

Since $\epsilon^{(n)}T_{i}=\epsilon^{(n)}$ for $1\leq i\leq n-1$ it follows
that for all compositions $\mu$, the sequence
$(\epsilon^{(n)}(x^{\mu}))_{n\geq 1}$ is convergent. Clearly from definition
we have that for all symmetric functions $F\in\Lambda$ and
$f(x)\in\operatorname{\mathscr{P}}_{n}^{+}$

$\epsilon^{(n)}(F[x_{1}+\ldots+x_{n}]f(x))=F[x_{1}+\ldots+x_{n}]\epsilon^{(n)}(f(x)).$

It follows now from a straightforward convergence argument using Remark 1 that
for all $g\in\operatorname{\mathscr{P}}_{as}^{+}$ the sequence
$(\epsilon^{(n)}(\pi_{n}(g)))_{n\geq 1}$ converges. The resulting operator
$\epsilon:=\lim_{n}\epsilon^{(n)}\circ\pi_{n}$ is evidently idempotent as its
codomain is $\Lambda$ and certainly $\epsilon$ acts as the identity on
symmetric functions. Further, for all $i\in\mathbb{N}$ we have

$\epsilon T_{i}=\lim_{n}\epsilon^{(n)}\circ\pi_{n}T_{i}$

and since $\pi_{n}$ commutes with $T_{i}$ for $n>i+1$ we see that

$\lim_{n}\epsilon^{(n)}\circ\pi_{n}T_{i}=\lim_{n}\epsilon^{(n)}T_{i}\circ\pi_{n}=\lim_{n}\epsilon^{(n)}\circ\pi_{n}=\epsilon.$

∎

######  Corollary 44.

For all $k\geq 0$ the sequence $(\epsilon^{(n)}_{k})_{n>k}$ converges to an
idempotent operator
$\epsilon_{k}:\operatorname{\mathscr{P}}_{as}^{+}\rightarrow\operatorname{\mathscr{P}}(k)^{+}$
such that for all $i\geq k+1$, $\epsilon_{k}T_{i}=\epsilon_{k}$.

###### Proof.

This follows immediately from Proposition 43 after shifting indices and noting
that the operators $\epsilon^{(n)}_{k}$ commute with multiplication by
$x_{1},\dots,x_{k}$. ∎

Now we will extend our definition of the stable-limit symmetrization operator
$\widetilde{\sigma}$ to partial symmetrization operators in the natural way.

######  Definition 45.

For $k\geq 0$ let
$\widetilde{\sigma}_{k}:\operatorname{\mathscr{P}}_{as}^{+}\rightarrow\operatorname{\mathscr{P}}(k)^{+}$
be defined on $g\in\operatorname{\mathscr{P}}(n)^{+}$ for $n\geq k$ by

$\widetilde{\sigma}_{k}(g):=\partial_{-}^{(k+1)}\cdots\partial_{-}^{(n)}(g).$
(7)

######  Remark.

The operators $\widetilde{\sigma}_{k}$ are well defined by Lemma 33. In
particular, if $g\in\operatorname{\mathscr{P}}(\ell)^{+}$ for $0\leq\ell\leq
k$ then
$\operatorname{\mathscr{P}}(\ell)^{+}\subset\operatorname{\mathscr{P}}(k)^{+}$
and there is no ambiguity in defining
$\widetilde{\sigma}_{k}(g)=\partial_{-}^{(k+1)}\cdots\partial_{-}^{(n)}(g)$ as
above. Note that $\widetilde{\sigma}_{0}=\widetilde{\sigma}$. Further, for all
$\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}}$ and
$\lambda\in\operatorname{\mathrm{Par}}$ we see that in this new terminology

$\widetilde{E}_{(\mu|\lambda)}=\widetilde{\sigma}_{\ell(\mu)}(\widetilde{E}_{\mu*\lambda}).$

Further, if $k\leq\ell$ then
$\widetilde{\sigma}_{k}\widetilde{\sigma}_{\ell}=\widetilde{\sigma}_{k}.$

We will now show that as operators on $\operatorname{\mathscr{P}}_{as}^{+}$,
$\epsilon_{\ell}=\widetilde{\sigma}_{\ell}$ for all ${\ell}\geq 0$.

######  Proposition 46.

For all $\ell\geq 0$, $\epsilon_{\ell}=\widetilde{\sigma}_{\ell}.$

###### Proof.

By shifting indices it suffices to just prove that
$\epsilon=\widetilde{\sigma}$, i.e., the $\ell=0$ case. Further, since both
maps are $T_{i}$-equivariant $\Lambda$-module maps (see Corollary 35) it
suffices to show that for all partitions $\lambda$,
$\epsilon(x^{\lambda})=\widetilde{\sigma}(x^{\lambda}).$ From the proof of
Proposition 43 we saw that $\epsilon(x^{\lambda})=Q_{\lambda}[X;t]$ whereas it
follows from the definition of the Jing vertex operators that
$\widetilde{\sigma}(x^{\lambda})=\mathcal{P}_{\lambda}[X]$. Therefore, it
suffices to argue that $Q_{\lambda}[X;t]=\mathcal{P}_{\lambda}[X].$ To this
end we will prove that

$\mathcal{P}_{\lambda}[X]=\langle z_{1}^{\lambda_{1}}\cdots
z_{r}^{\lambda_{r}}\rangle Exp[(1-t)(z_{1}+\ldots+z_{r})X]Exp[(t-1)\sum_{1\leq
i<j\leq r}\frac{z_{j}}{z_{i}}]$ (8)

which by 2.15 in Macdonald Chapter 3 [Mac15] is an alternative definition for
$Q_{\lambda}[X;t].$

Suppose $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ is a partition. Note first
that by definition
$\mathcal{P}_{\lambda}[X]=\mathscr{B}_{\lambda_{1}}\cdots\mathscr{B}_{\lambda_{r}}(1).$
We will now induct on the number of operators $\mathscr{B}$ acting on $1$ in
the expression $\mathscr{B}_{\lambda_{1}}\cdots\mathscr{B}_{\lambda_{r}}(1).$
As a base case

$\mathscr{B}_{\lambda_{r}}(1)=\langle z_{r}^{\lambda_{r}}\rangle
1[X-z_{r}^{-1}]Exp[(1-t)z_{r}X]=\langle z_{r}^{\lambda_{r}}\rangle
Exp[(1-t)z_{r}X].$

We claim that for all $1\leq k\leq r$

$\mathscr{B}_{\lambda_{k}}\cdots\mathscr{B}_{\lambda_{r}}(1)=\langle
z_{k}^{\lambda_{k}}\cdots z_{r}^{\lambda_{r}}\rangle
Exp[(1-t)(z_{k}+\ldots+z_{r})X]Exp[(t-1)\sum_{k\leq i<j\leq
r}\frac{z_{j}}{z_{i}}].$ (9)

Suppose the above is true for some $1<k\leq r$. Then

$\displaystyle\mathscr{B}_{\lambda_{k-1}}\mathscr{B}_{\lambda_{k}}\cdots\mathscr{B}_{\lambda_{r}}(1)$
$\displaystyle=\mathscr{B}_{\lambda_{k-1}}\left(\langle
z_{k}^{\lambda_{k}}\cdots z_{r}^{\lambda_{r}}\rangle
Exp[(1-t)(z_{k}+\ldots+z_{r})X]Exp[(t-1)\sum_{k\leq i<j\leq
r}\frac{z_{j}}{z_{i}}]\right)$ $\displaystyle=\langle
z_{k-1}^{\lambda_{k-1}}\rangle\langle z_{k}^{\lambda_{k}}\cdots
z_{r}^{\lambda_{r}}\rangle
Exp[(1-t)(z_{k}+\ldots+z_{r})(X-z_{k-1}^{-1})]Exp[(t-1)\sum_{k\leq i<j\leq
r}\frac{z_{j}}{z_{i}}]Exp[(1-t)z_{k-1}X].$

Now we use the additive property of the plethystic exponential namely,
$Exp[A+B]=Exp[A]Exp[B]$, to rearrange terms and get

$\langle z_{k-1}^{\lambda_{k-1}}\cdots z_{r}^{\lambda_{r}}\rangle
Exp[(1-t)(z_{k}+\ldots+z_{r})X]Exp[(1-t)z_{k-1}X]Exp[(t-1)\sum_{k\leq i<j\leq
r}\frac{z_{j}}{z_{i}}]Exp[(t-1)(\frac{z_{k}}{z_{k-1}}+\ldots+\frac{z_{r}}{z_{k-1}})]$

which simplifies to

$\langle z_{k-1}^{\lambda_{k-1}}\cdots z_{r}^{\lambda_{r}}\rangle
Exp[(1-t)(z_{k-1}+z_{k}+\ldots+z_{r})X]Exp[(t-1)\sum_{k-1\leq i<j\leq
r}\frac{z_{j}}{z_{i}}]$

showing that the formula (9) holds for all $1\leq k\leq r$. Taking $k=1$ shows
equation (8) holds. ∎

As an immediate consequence of Proposition 46 we find the following
enlightening description for the $\widetilde{E}_{(\mu|\lambda)}$ functions.

######  Corollary 47.

For all $(\mu|\lambda)$ with $\mu$ a reduced composition and $\lambda$ a
partition,

$\widetilde{E}_{(\mu|\lambda)}=\lim_{n}\epsilon_{\ell(\mu)}^{(n)}(E_{\mu*\lambda*0^{n-(\ell(\mu)+\ell(\lambda))}}).$
(10)

In particular, for partitions $\lambda$,
$\mathcal{A}_{\lambda}[X]=(1-t)^{\ell(\lambda)}v_{\lambda}(t)P_{\lambda}[X;q^{-1},t]$
where $P_{\lambda}[X;q^{-1},t]$ is the symmetric Macdonald function. As a
consequence the set
$\\{\mathcal{A}_{\lambda}:\lambda\in\operatorname{\mathrm{Par}}\\}$ is a basis
of $\Lambda$.

######  Remark.

The $P_{\lambda}[X;q,t]$ are the symmetric Macdonald functions as defined by
Macdonald in [Mac15] and seen in Cherednik’s work [Che03] not to be confused
with the modified symmetric Macdonald functions $\widetilde{H}_{\mu}$ seen in
many places but in particular in the work of Haiman [Hai02]. Further,
Corollary 47 gives an interpretation of the $\widetilde{E}_{(\mu|\lambda)}$ as
limits of partially symmetrized non-symmetric Macdonald polynomials. Goodberry
in [Goo22] and Lapointe in [Lap22] have investigated similar families of
partially symmetric Macdonald polynomials. Up to a change of variables and
limiting these different notions are likely directly related.

In order to prove the first main theorem in this paper, Theorem 49, we will
require the following straightforward lemma.

######  Lemma 48.

For any composition $\mu$ there is some nonzero scalar
$\gamma_{\mu}\in\mathbb{Q}(q,t)$ such that

$\widetilde{\sigma}(\widetilde{E}_{\mu})=\gamma_{\mu}\mathcal{A}_{\operatorname{sort}(\mu)}$

where $\gamma_{\mu}=1$ when $\mu$ is a partition.

###### Proof.

We know that for all partitions $\lambda$,
$\widetilde{\sigma}(\widetilde{E}_{\lambda})=\mathcal{A}_{\lambda}$ so this
lemma holds trivially for partitions. Now we proceed by induction on Bruhat
order similarly to the argument in the proof of Theorem 29. To show the lemma
holds it suffices to show that if $\mu$ is a composition and $k$ such that
$s_{k}(\mu)>\mu$ in Bruhat order and
$\widetilde{\sigma}(\widetilde{E}_{\mu})=\gamma_{\mu}\mathcal{A}_{\operatorname{sort}(\mu)}$
for $\gamma_{\mu}\neq 0$ then
$\widetilde{\sigma}(\widetilde{E}_{s_{k}(\mu)})=\gamma_{s_{k}(\mu)}\mathcal{A}_{\operatorname{sort}(\mu)}$
for $\gamma_{s_{k}(\mu)}\neq 0$. To this end fix such $\mu$ and $k$. Then by
Corollary 30

$\widetilde{E}_{s_{k}(\mu)}=\left(T_{k}+\frac{(1-t)\widetilde{\alpha}_{\mu}(k+1)}{\widetilde{\alpha}_{\mu}(k)-\widetilde{\alpha}_{\mu}(k+1)}\right)\widetilde{E}_{\mu}.$

From Proposition 46 $\widetilde{\sigma}=\lim_{m}\epsilon^{(m)}$ so that
$\widetilde{\sigma}T_{k}=\widetilde{\sigma}$. Therefore,

$\displaystyle\widetilde{\sigma}(\widetilde{E}_{s_{k}(\mu)})$
$\displaystyle=\widetilde{\sigma}\left(\left(T_{k}+\frac{(1-t)\widetilde{\alpha}_{\mu}(k+1)}{\widetilde{\alpha}_{\mu}(k)-\widetilde{\alpha}_{\mu}(k+1)}\right)\widetilde{E}_{\mu}\right)$
$\displaystyle=\left(1+\frac{(1-t)\widetilde{\alpha}_{\mu}(k+1)}{\widetilde{\alpha}_{\mu}(k)-\widetilde{\alpha}_{\mu}(k+1)}\right)\widetilde{\sigma}(\widetilde{E}_{\mu})$
$\displaystyle=\left(\frac{\widetilde{\alpha}_{\mu}(k)-t\widetilde{\alpha}_{\mu}(k+1)}{\widetilde{\alpha}_{\mu}(k)-\widetilde{\alpha}_{\mu}(k+1)}\right)\gamma_{\mu}\mathcal{A}_{\operatorname{sort}(\mu)}.$

By Lemma 27 we see that since $s_{k}(\mu)>\mu$ it follows that
$\widetilde{\alpha}_{\mu}(k)\neq t\widetilde{\alpha}_{\mu}(k+1).$ Hence,
$\gamma_{s_{k}(\mu)}:=\left(\frac{\widetilde{\alpha}_{\mu}(k)-t\widetilde{\alpha}_{\mu}(k+1)}{\widetilde{\alpha}_{\mu}(k)-\widetilde{\alpha}_{\mu}(k+1)}\right)\gamma_{\mu}\neq
0$ so the result follows. ∎

######  Remark.

Note that using the recursive formula
$\gamma_{s_{k}(\mu)}=\left(\frac{\widetilde{\alpha}_{\mu}(k)-t\widetilde{\alpha}_{\mu}(k+1)}{\widetilde{\alpha}_{\mu}(k)-\widetilde{\alpha}_{\mu}(k+1)}\right)\gamma_{\mu}$
in the proof of Lemma 48, the formula for the eigenvalues
$\widetilde{\alpha}_{\mu}(k)$ in Lemma 27, and the base condition
$\gamma_{\mu}=1$ for $\mu$ a partition, it is possible to give an explicit
expression for $\gamma_{\mu}$ for any composition $\mu.$ However, all we need
for the purposes of this paper is that $\gamma_{\mu}\neq 0$ so we will not
find such an explicit expression for $\gamma_{\mu}.$

####  4.3.1 First Main Theorem and a Full ${\mathscr{Y}}$-Weight Basis of
$\operatorname{\mathscr{P}}_{as}^{+}$

Finally, we prove that the stable-limit non-symmetric Macdonald functions are
a basis for $\mathscr{P}_{as}^{+}$. To do this we will use the stable-limit
symmetrization operator to help distinguish between stable-limit non-symmetric
Macdonald functions with the same ${\mathscr{Y}}$-weight.

######  Theorem 49.

(First Main Theorem) The $\widetilde{E}_{(\mu|\lambda)}$ are a
${\mathscr{Y}}$-weight basis for $\mathscr{P}_{as}^{+}$.

###### Proof.

As there are sufficiently many $\widetilde{E}_{(\mu|\lambda)}$ in each graded
component of every $\mathscr{P}(k)^{+}$ it suffices to show that these
functions are linearly independent. Certainly, weight vectors in distinct
weight spaces are linearly independent. Using Lemmas 27 and 38, we deduce that
if $\widetilde{E}_{(\mu^{(1)}|\lambda^{(1)})}$ and
$\widetilde{E}_{(\mu^{(2)}|\lambda^{(2)})}$ have the same weight then
necessarily $\mu^{(1)}=\mu^{(2)}$. Hence, we can restrict to the case where we
have a dependence relation

$c_{1}\widetilde{E}_{(\mu|\lambda^{(1)})}+\ldots+c_{N}\widetilde{E}_{(\mu|\lambda^{(N)})}=0$

for $\lambda^{(1)},\ldots,\lambda^{(N)}$ distinct partitions. By applying the
stable-limit symmetrization operator we see that

$\widetilde{\sigma}(c_{1}\widetilde{E}_{(\mu|\lambda^{(1)})}+\ldots+c_{N}\widetilde{E}_{(\mu|\lambda^{(N)})})=\widetilde{\sigma}(c_{1}\widetilde{E}_{\mu*\lambda^{(1)}}+\ldots+c_{N}\widetilde{E}_{\mu*\lambda^{(N)}})=0.$

Now by Lemma 48,
$\widetilde{\sigma}(\widetilde{E}_{\mu*\lambda^{(i)}})=\gamma_{\mu*\lambda^{(i)}}\mathcal{A}_{\operatorname{sort}(\mu*\lambda^{(i)})}$
with nonzero scalars $\gamma_{\mu*\lambda^{(i)}}$ yielding

$0=c^{\prime}_{1}\mathcal{A}_{\operatorname{sort}(\mu*\lambda^{(1)})}+\ldots+c^{\prime}_{n}\mathcal{A}_{\operatorname{sort}(\mu*\lambda^{(N)})}.$

The partitions $\lambda^{(i)}$ are distinct so we know that the partitions
$\operatorname{sort}(\mu*\lambda^{(i)})$ are distinct as well. By Corollary 47
the symmetric functions $\mathcal{A}_{\operatorname{sort}(\mu*\lambda^{(i)})}$
are linearly independent. Thus $c^{\prime}_{i}=0$ implying $c_{i}=0$ for all
$1\leq i\leq N$ as desired. ∎

##  5 Some Recurrence Relations for the $\widetilde{E}_{(\mu|\lambda)}$

In this section we will discuss a few recurrence relations for the stable-
limit non-symmetric Macdonald functions. We start by looking at the action of
the Demazure-Lusztig operators $T_{i}$ and the lowering operators
$\partial_{-}$.

######  Proposition 50.

For a reduced composition $\mu=(\mu_{1},\dots,\mu_{r})$ and partition
$\lambda=(\lambda_{1},\dots,\lambda_{k})$ if $\mu_{r}\geq\lambda_{1}$ and
$\mu_{r-1}\neq 0$ then

$\partial_{-}^{(r)}\left(\widetilde{E}_{(\mu_{1},\dots,\mu_{r}|\lambda_{1},\dots,\lambda_{k})}\right)=\widetilde{E}_{(\mu_{1},\dots,\mu_{r-1}|\mu_{r},\lambda_{1},\dots,\lambda_{k})}.$

###### Proof.

This follows immediately from the definitions of
$\widetilde{E}_{(\mu|\lambda)}$ and $\partial_{-}^{(r)}.$ ∎

######  Proposition 51.

Take $\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}}$ and
$\lambda\in\operatorname{\mathrm{Par}}$ and suppose $1\leq i\leq\ell(\mu)-1$
such that $s_{i}(\mu)>\mu$ and
$s_{i}(\mu)\in\operatorname{\mathrm{Comp}^{\textit{red}}}.$ Then

$\widetilde{E}_{(s_{i}(\mu)|\lambda)}=\left(T_{i}+\frac{(1-t)\widetilde{\alpha}_{\mu*\lambda}(i+1)}{\widetilde{\alpha}_{\mu*\lambda}(i)-\widetilde{\alpha}_{\mu*\lambda}(i+1)}\right)\widetilde{E}_{(\mu|\lambda)}.$

###### Proof.

Since $s_{i}(\mu)>\mu$ we know that $s_{i}(\mu*\lambda)>\mu*\lambda$ so by
Corollary 30

$\widetilde{E}_{s_{i}(\mu*\lambda)}=\left(T_{i}+\frac{(1-t)\widetilde{\alpha}_{\mu*\lambda}(i+1)}{\widetilde{\alpha}_{\mu*\lambda}(i)-\widetilde{\alpha}_{\mu*\lambda}(i+1)}\right)\widetilde{E}_{\mu*\lambda}.$

Now we know $T_{i}$ commutes with the operators
$\partial_{-}^{(\ell(\mu)+1)},\dots,\partial_{-}^{(\ell(\mu)+\ell(\lambda))}$
and thus we see that

$\displaystyle\widetilde{E}_{(s_{i}(\mu)|\lambda)}$
$\displaystyle=\partial_{-}^{(\ell(\mu)+1)}\cdots\partial_{-}^{(\ell(\mu)+\ell(\lambda))}(\widetilde{E}_{s_{i}(\mu*\lambda)})$
$\displaystyle=\partial_{-}^{(\ell(\mu)+1)}\cdots\partial_{-}^{(\ell(\mu)+\ell(\lambda))}\left(\left(T_{i}+\frac{(1-t)\widetilde{\alpha}_{\mu*\lambda}(i+1)}{\widetilde{\alpha}_{\mu*\lambda}(i)-\widetilde{\alpha}_{\mu*\lambda}(i+1)}\right)\widetilde{E}_{\mu*\lambda}\right)$
$\displaystyle=\left(T_{i}+\frac{(1-t)\widetilde{\alpha}_{\mu*\lambda}(i+1)}{\widetilde{\alpha}_{\mu*\lambda}(i)-\widetilde{\alpha}_{\mu*\lambda}(i+1)}\right)\partial_{-}^{(\ell(\mu)+1)}\cdots\partial_{-}^{(\ell(\mu)+\ell(\lambda))}(\widetilde{E}_{\mu*\lambda})$
$\displaystyle=\left(T_{i}+\frac{(1-t)\widetilde{\alpha}_{\mu*\lambda}(i+1)}{\widetilde{\alpha}_{\mu*\lambda}(i)-\widetilde{\alpha}_{\mu*\lambda}(i+1)}\right)\widetilde{E}_{(\mu|\lambda)}.$

∎

######  Proposition 52.

For
$\mu=(\mu_{1},\dots,\mu_{r})\in\operatorname{\mathrm{Comp}^{\textit{red}}}$
and $\lambda\in\operatorname{\mathrm{Par}}$ we have that

$T_{r}\widetilde{E}_{(\mu|\lambda)}=\frac{\gamma_{\mu*\lambda}}{\gamma_{(\mu_{1},\ldots,\mu_{r-1},0,\mu_{r})*\lambda}}\widetilde{E}_{(\mu_{1},\dots,\mu_{r-1},0,\mu_{r}|\lambda)}.$

###### Proof.

First note that by Corollary 38

$\varphi_{r}(\widetilde{E}_{(\mu|\lambda)})=(T_{r}({\mathscr{Y}}_{r}-{\mathscr{Y}}_{r+1})+(1-t){\mathscr{Y}}_{r+1})\widetilde{E}_{(\mu|\lambda)}=(\widetilde{\alpha}_{\mu*\lambda}(r)-0)T_{r}\widetilde{E}_{(\mu|\lambda)}+(1-t)(0)\widetilde{E}_{(\mu|\lambda)}=\widetilde{\alpha}_{\mu*\lambda}(r)T_{r}\widetilde{E}_{(\mu|\lambda)}$

and by Lemma 27 $\widetilde{\alpha}_{\mu*\lambda}(r)\neq 0$ since $\mu_{r}\neq
0$. Therefore, $\varphi_{r}(\widetilde{E}_{(\mu|\lambda)})$ is nonzero and
therefore must be a ${\mathscr{Y}}$-weight vector with weight
$(\widetilde{\alpha}_{\mu*\lambda}(1),\dots,\widetilde{\alpha}_{\mu*\lambda}(r-1),0,\widetilde{\alpha}_{\mu*\lambda}(r),0,\dots)$.
By using the explicit formula for the eigenvalues
$\widetilde{\alpha}_{\mu*\lambda}(i)$ from Lemma 27 we see that for $1\leq
i\leq r$, $\widetilde{\alpha}_{\mu*\lambda}(i)=0$ exactly when $\mu_{i}=0$ and
further, for all $1\leq i\leq r$ with $\mu_{i}\neq 0$,
$\widetilde{\alpha}_{\mu*\lambda}(i)=q^{\mu_{i}}t^{b_{i}}$ for some $b_{i}$.
Hence by Theorem 49 and Corollary 38,
$\varphi_{r}(\widetilde{E}_{(\mu|\lambda)})$ is of the form

$\varphi_{r}(\widetilde{E}_{(\mu|\lambda)})=\sum_{\nu}a_{\nu}\widetilde{E}_{(\mu_{1},\dots,\mu_{r-1},0,\mu_{r}|\nu)}$

$\nu$ ranges over a finite set of partitions $\nu$ and $a_{\nu}$ are some
scalars. Note that we have

$\widetilde{\sigma}(\varphi_{r}(\widetilde{E}_{(\mu|\lambda)}))=\widetilde{\sigma}(\widetilde{\alpha}_{\mu*\lambda}(r)T_{r}\widetilde{E}_{(\mu|\lambda)})$

and since $\widetilde{\sigma}T_{r}=\widetilde{\sigma}$

$\widetilde{\sigma}(\varphi_{r}(\widetilde{E}_{(\mu|\lambda)}))=\widetilde{\alpha}_{\mu*\lambda}(r)\widetilde{\sigma}(\widetilde{E}_{(\mu|\lambda)})=\widetilde{\alpha}_{\mu*\lambda}(r)\gamma_{\mu*\lambda}\mathcal{A}_{\operatorname{sort}(\mu*\lambda)}$

using Lemma 48. Similarly, we see that

$\widetilde{\sigma}\left(\sum_{\nu}a_{\nu}\widetilde{E}_{(\mu_{1},\dots,\mu_{r-1},0,\mu_{r}|\nu)}\right)=\sum_{\nu}a_{\nu}\gamma_{(\mu_{1},\dots,\mu_{r-1},0,\mu_{r})*\nu}\mathcal{A}_{\operatorname{sort}(\mu*\nu)}$

since
$\operatorname{sort}((\mu_{1},\ldots,\mu_{r-1},0,\mu_{r})*\nu)=\operatorname{sort}(\mu*\nu)$
for all $\nu.$

Thus

$\mathcal{A}_{\operatorname{sort}(\mu*\lambda)}=\sum_{\nu}a_{\nu}^{\prime}\mathcal{A}_{\operatorname{sort}(\mu*\nu)}$

where

$a^{\prime}_{\nu}:=\frac{a_{\nu}\gamma_{(\mu_{1},\dots,\mu_{r-1},0,\mu_{r})*\nu}}{\widetilde{\alpha}_{\mu*\lambda}(r)\gamma_{\mu*\lambda}}.$

By Corollary 47 we know that the $\mathcal{A}_{\beta}$ are a basis for
$\Lambda$ and so we see that the only possible partition $\nu$ that can
contribute a nonzero term in the above expansion is $\nu=\lambda$. Further,
$a^{\prime}_{\lambda}=1$ and thus
$a_{\lambda}=\frac{\widetilde{\alpha}_{\mu*\lambda}(r)\gamma_{\mu*\lambda}}{\gamma_{(\mu_{1},\ldots,\mu_{r-1},0,\mu_{r})*\lambda}}.$

Therefore,

$\varphi_{r}(\widetilde{E}_{(\mu|\lambda)})=\widetilde{\alpha}_{\mu*\lambda}(r)T_{r}\widetilde{E}_{(\mu|\lambda)}=\frac{\widetilde{\alpha}_{\mu*\lambda}(r)\gamma_{\mu*\lambda}}{\gamma_{(\mu_{1},\ldots,\mu_{r-1},0,\mu_{r})*\lambda}}\widetilde{E}_{(\mu_{1},\dots,\mu_{r-1},0,\mu_{r}|\lambda)}$

which yields

$T_{r}\widetilde{E}_{(\mu|\lambda)}=\frac{\gamma_{\mu*\lambda}}{\gamma_{(\mu_{1},\ldots,\mu_{r-1},0,\mu_{r})*\lambda}}\widetilde{E}_{(\mu_{1},\dots,\mu_{r-1},0,\mu_{r}|\lambda)}.$

∎

######  Definition 53.

Define $\widetilde{\omega}_{m}:=X_{1}T_{1}^{-1}\cdots T_{m-1}^{-1}$ considered
as an operator on $\operatorname{\mathscr{P}}_{m}^{+}$.

######  Remark.

These operators are the same as the corresponding operators of the same name
defined by Ion and Wu up to inversion and some scalars. We have defined the
operators as above for convenience. The operators $\omega_{m}$ and
$\widetilde{\omega}_{m}$ are used by Ion and Wu [IW22] to give operators
analogous to the $d_{+},d_{+}^{*}$ operators in $\mathbb{A}_{t,q}$.

######  Lemma 54.

The sequences of operators $(\widetilde{\omega}_{m})_{m\geq 1}$ and
$(\omega_{m}^{-1})_{m\geq 1}$ converge to operators
$\widetilde{\omega},\omega^{*}:\operatorname{\mathscr{P}}_{as}^{+}\rightarrow\operatorname{\mathscr{P}}_{as}^{+}$
respectively with actions given by

  * •

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

  * •

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

###### Proof.

Let $(f_{m})_{m\geq 1}$ be a convergent sequence with limit
$f\in\operatorname{\mathscr{P}}(k)^{+}$. We start by showing the sequence
$(\widetilde{\omega}_{m}(f_{m}))_{m\geq 1}$ converges to an element of
$\operatorname{\mathscr{P}}_{as}^{+}$. It follows directly by the definition
of convergence that there exists some $M>k$ such that for all $i$ and $m$ with
$m\geq M$ and $k+1\leq i\leq m-1$, $T_{i}f_{m}=f_{m}$. Therefore, for all
$m\geq M$

$\widetilde{\omega}_{m}(f_{m})=x_{1}T_{1}^{-1}\cdots T_{k}^{-1}f_{m}$

which clearly converges to $x_{1}T_{1}^{-1}\cdots T_{k}^{-1}f$. It follows
then that the sequence of operators $(\widetilde{\omega}_{m})_{m\geq 1}$
converges to an operator which we call $\widetilde{\omega}.$ By considering
$f=x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}F[X]$ with $F\in\Lambda$ we get the first
formula in the lemma statement above.

Next we will show the sequence $(\omega^{-1}_{m}(\pi_{m}(f)))_{m\geq 1}$
converges. Expand $f$ as

$f=\sum_{i=1}^{N}c_{i}x^{\mu^{(i)}}F_{i}[X]$

for $c_{i}\in\mathbb{Q}(q,t)$, compositions $\mu^{(i)}$, and $F_{i}\in\Lambda$
where we may assume each composition $\mu^{(i)}$ has length $k$ so that for
all $m\geq k$

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

Applying $\omega_{m}^{-1}$ to $\pi_{m}(f)$ gives for $m\geq k$

$\omega_{m}^{-1}(\pi_{m}(f))=\sum_{i=1}^{N}c_{i}x_{2}^{\mu^{(i)}_{1}}\cdots
x_{k+1}^{\mu^{(i)}_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}]$

so therefore we get

$\lim_{m}\omega_{m}^{-1}(\pi_{m}(f))=\sum_{i=1}^{N}c_{i}x_{2}^{\mu^{(i)}_{1}}\cdots
x_{k+1}^{\mu^{(i)}_{k}}F[X+(q-1)x_{1}].$

Thus the sequence of operators $(\omega_{m}^{-1})_{m\geq 1}$ converges to an
operator which we call $\omega^{*}$. Lastly, by applying this formula to
$f=x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}F[X]$ with $F\in\Lambda$ to see the second
formula given in the lemma statement. ∎

In line with the above results in this section we will now give a partial
generalization of the classical Knop-Sahi relation regarding the action of the
$\omega$ operators on Macdonald polynomials.

######  Proposition 55.

For all compositions $\mu$

$t^{\\#\\{j:\mu_{j}\neq
0\\}}\widetilde{\omega}(\widetilde{E}_{\mu})=x_{1}\omega^{*}(\widetilde{E}_{\mu})=\widetilde{E}_{1*\mu}.$

###### Proof.

Suppose $\ell(\mu)=n$. Recall that for all $m\geq 1$

$(Y^{(n+m)}_{n+m})^{-1}=t^{n+m-1}\omega_{n+m}T_{1}^{-1}\cdots T_{n+m-1}^{-1}.$

Therefore, by recalling the eigenvalue notation in Lemma 27 we have

$t^{n+m-1}\omega_{n+m}T_{1}^{-1}\cdots
T_{n+m-1}^{-1}E_{\mu*0^{m}}=(Y^{(n+m)}_{n+m})^{-1}E_{\mu*0^{m}}=\alpha_{\mu}^{(m)}(n+m)^{-1}E_{\mu*0^{m}}$

so that

$t^{n+m-1}\alpha_{\mu}^{(m)}(n+m)x_{1}T_{1}^{-1}\cdots
T_{n+m-1}^{-1}E_{\mu*0^{m}}=x_{1}\omega_{n+m}^{-1}E_{\mu*0^{m}}.$

From Lemma 27 we see that

$t^{n+m-1}\alpha_{\mu}^{(m)}(n+m)=t^{\\#\\{j:\mu_{j}\neq 0\\}}$

which gives

$t^{\\#\\{j:\mu_{j}\neq 0\\}}x_{1}T_{1}^{-1}\cdots
T_{n+m-1}^{-1}E_{\mu*0^{m}}=t^{\\#\\{j:\mu_{j}\neq
0\\}}\widetilde{\omega}_{n+m}(E_{\mu*0^{m}})=x_{1}\omega_{n+m}^{-1}E_{\mu*0^{m}}.$

From the classical Knop-Sahi relations (see [HHL08]) applied to
$E_{\mu*0^{m}}$ we get
$x_{1}\omega_{n+m}^{-1}E_{\mu*0^{m}}=E_{1*\mu*0^{m-1}}.$ Applying Corollary 25
and Lemma 54 as $m\rightarrow\infty$ now gives

$t^{\\#\\{j:\mu_{j}\neq
0\\}}\widetilde{\omega}(\widetilde{E}_{\mu})=x_{1}\omega^{*}(\widetilde{E}_{\mu})=\widetilde{E}_{1*\mu}.$

∎

##  6 Constructing $\widetilde{E}_{(\mu|\lambda)}$-Diagonal Operators from
Symmetric Functions

The main goal of the following section of this paper is to construct an
operator on $\operatorname{\mathscr{P}}_{as}^{+}$ which is diagonal in the
stable-limit Macdonald function basis, commutes with the limit Cherednik
operators ${\mathscr{Y}}_{i}$, but does not annihilate $\Lambda$. This
operator will be constructed from a limit of operators arising from the action
of $t^{m}Y^{(m)}_{1}+\ldots+t^{m}Y^{(m)}_{m}$ on
$\operatorname{\mathscr{P}}_{m}^{+}$. After finding the eigenvalues of this
new operator we will show that the addition of this operator to the algebra
generated by the limit Cherednik operators has simple spectrum on
$\operatorname{\mathscr{P}}_{as}^{+}$.

We begin with the following natural definition.

######  Definition 56.

For $F\in\Lambda$ define the operator
$\Psi_{F}^{(m)}:\operatorname{\mathscr{P}}_{m}^{+}\rightarrow\operatorname{\mathscr{P}}_{m}^{+}$
by

$\Psi_{F}^{(m)}:=F[t^{m}Y_{1}^{(m)}+\ldots+t^{m}Y_{m}^{(m)}].$ (11)

Further, for a composition $\mu$ with $\ell(\mu)=n$ and $m\geq 0$ define the
scalar $\kappa_{\mu}^{(m)}(q,t)$ as

$\kappa_{\mu}^{(m)}(q,t):=\sum_{i=1}^{n+m}t^{n+m}\alpha_{\mu}^{(m)}(i).$

Recall from Lemma 27 that $\alpha_{\mu}^{(m)}(i)$ is given by
$Y_{i}^{(n+m)}E_{\mu*0^{m}}=\alpha_{\mu}^{(m)}(i)E_{\mu*0^{m}}$.

######  Lemma 57.

For all compositions $\mu$ the sequence $(\kappa_{\mu}^{(m)}(q,t))_{m\geq 0}$
converges to some $\kappa_{\mu}(q,t)\in\mathbb{Q}(q,t).$ Further,
$\kappa_{\mu}(q,t)=\kappa_{\mu*0^{k}}(q,t)$ for all $k\geq 0$ and
$\kappa_{\mu}(q,t)=\kappa_{s_{i}(\mu)}(q,t)$ for all $1\leq i\leq\ell(\mu)-1$.

###### Proof.

Using Lemma 27 we get the following:

$\displaystyle\kappa_{\mu}^{(m)}(q,t)$
$\displaystyle=\sum_{i=1}^{n+m}t^{n+m}\alpha_{\mu}^{(m)}(i)$
$\displaystyle=\sum_{i=1}^{n}t^{n+m}\alpha_{\mu}^{(m)}(i)+\sum_{i=n+1}^{n+m}t^{\\#\\{j:\mu_{j}\neq
0\\}}t^{m+1-(i-n)}$
$\displaystyle=\sum_{i=1}^{n}t^{n}\alpha_{\mu}^{(0)}(i)t^{m\mathbbm{1}(\mu_{i}=0)}+t^{\\#\\{j:\mu_{j}\neq
0\\}}\sum_{i=1}^{m}t^{m-i+1}$ $\displaystyle=\sum_{\mu_{i}\neq
0}t^{n}\alpha_{\mu}^{(0)}(i)+t^{m}\sum_{\mu_{i}=0}t^{n}\alpha_{\mu}^{(0)}(i)+t^{\\#\\{j:\mu_{j}\neq
0\\}}\sum_{i=1}^{m}t^{i}.$

Therefore,

$\kappa_{\mu}(q,t):=\lim_{m}\kappa_{\mu}^{(m)}(q,t)=\left(\sum_{i:\mu_{i}\neq
0}t^{n}\alpha_{\mu}^{(0)}(i)\right)+\frac{t^{1+\\#\\{j:\mu_{j}\neq
0\\}}}{1-t}\in\mathbb{Q}(q,t).$ (12)

The last statement regarding $\kappa_{\mu*0^{k}}(q,t)$ and
$\kappa_{s_{i}(\mu)}(q,t)$ follows now directly from Lemma 27 and classical
DAHA intertwiner theory. ∎

######  Remark.

Recall from the proof of Lemma 27 that

$t^{n}\alpha_{\mu}^{(0)}=q^{\mu_{i}}t^{n+1-\beta_{\mu}(i)}.$

Applying this to the Lemma 57 gives the combinatorial formula

$\kappa_{\mu}(q,t)=\frac{t^{1+\\#\\{j:\mu_{j}\neq
0\\}}}{1-t}+\sum_{\mu_{i}\neq 0}q^{\mu_{i}}t^{n+1-\beta_{\mu}(i)}.$

If we consider the partition $\lambda$ to have an infinite string of $0$’s
attached to its tail then

$\kappa_{\lambda}(q,t)=\sum_{i=1}^{\infty}q^{\lambda_{i}}t^{i}.$

Notice that this is exactly equal to

$\frac{t}{1-t}\left(1-(1-t)(1-q)B_{\lambda}(q,t)\right)$

where $B_{\lambda}(q,t)$ is the diagram generator of $\lambda$ in [Hai99].

######  Corollary 58.

Let $\lambda$ and $\nu$ be partitions. Then
$\kappa_{\lambda}(q,t)=\kappa_{\nu}(q,t)$ if and only if $\lambda=\nu$.

###### Proof.

This follows readily from the identity

$\kappa_{\lambda}(q,t)=\sum_{i=1}^{\infty}q^{\lambda_{i}}t^{i}$

given in the prior remark. ∎

In this next result we will show that the sequence of operators
$(\Psi_{p_{1}}^{(m)})_{m\geq 1}$ converges to a well defined map on
$\operatorname{\mathscr{P}}_{as}^{+}$. As expected these operators are well-
behaved on sequences of the form
$\epsilon_{\ell(\mu)}^{(m)}(E_{\mu*\lambda*0^{m-(\ell(\mu)+\ell(\lambda))}}).$
In fact it is not hard to show that $(\Psi_{p_{1}}^{(m)})_{m\geq 1}$ converges
on the former sequences. However, this is not a sufficient argument to show
the convergence of the $(\Psi_{p_{1}}^{(m)})_{m\geq 1}$. In order to obtain a
well-defined operator on $\operatorname{\mathscr{P}}_{as}^{+}$ from the
sequence of operators $(\Psi_{p_{1}}^{(m)})_{m\geq 1}$ one needs to show that
given an arbitrary convergent sequence $(f_{m})_{m\geq 1}$ the corresponding
sequence $(\Psi_{p_{1}}^{(m)}(f_{m}))_{m\geq 1}$ converges. Therefore, the
difficulty in the following proof is to show that the $\Psi_{p_{1}}^{(m)}$ are
well behaved in general.

######  Theorem 59.

The sequence of operators $(\Psi_{p_{1}}^{(m)})_{m\geq 1}$ converges to an
operator
$\Psi_{p_{1}}:\operatorname{\mathscr{P}}_{as}^{+}\rightarrow\operatorname{\mathscr{P}}_{as}^{+}$
which is diagonal in the $\widetilde{E}_{(\mu|\lambda)}$ basis with

$\Psi_{p_{1}}(\widetilde{E}_{(\mu|\lambda)})=\kappa_{\mu*\lambda}(q,t)\widetilde{E}_{(\mu|\lambda)}.$

###### Proof.

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 $(\Psi_{p_{1}}^{(m)})_{m\geq 1}$ converges it suffices using Remark
1 to show that sequences of the form

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

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

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

Let $\lambda$ be a partition, $k:=\ell(\lambda)$, $F\in\Lambda$, and take
$m>k$. Recall that $\widetilde{Y}_{1}^{(m)}X_{1}=t^{m}Y_{1}^{(m)}X_{1}$ (19)
from which it follows directly that
$\widetilde{Y}_{i}^{(m)}X_{i}=t^{m}Y_{i}^{(m)}X_{i}$ for all $1\leq i\leq m$.
Then for all $1\leq i\leq k$ we have that since $\lambda_{i}\neq 0$,

$t^{m}Y_{i}^{(m)}(x^{\lambda}F[x_{1}+\ldots+x_{m}])=\widetilde{Y}_{i}^{(m)}(x^{\lambda}F[x_{1}+\ldots+x_{m}]).$

Therefore,

$t^{m}(Y_{1}^{(m)}+\ldots+Y_{k}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}])=(\widetilde{Y}_{1}^{(m)}+\ldots+\widetilde{Y}_{k}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}]).$

Now since $x^{\lambda}F[x_{1}+\ldots+x_{m}]$ is symmetric in the variables
$\\{k+1,\dots,m\\}$ we see that

$\displaystyle
t^{m}(Y_{k+1}^{(m)}+\ldots+Y_{m}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}])$
$\displaystyle=(t^{m-k}T_{k}\cdots T_{1}\omega_{m}^{-1}T_{m-1}^{-1}\cdots
T_{k+1}^{-1}+t^{m-k-1}T_{k+1}\cdots T_{1}\omega^{-1}_{m}T_{m-1}^{-1}\cdots
T_{k+2}^{-1}+\ldots+tT_{m-1}\cdots
T_{1}\omega_{m}^{-1})(x^{\lambda}F[x_{1}+\ldots+x_{m}])$
$\displaystyle=(t^{m-k}T_{k}\cdots T_{1}+t^{m-k-1}T_{k+1}\cdots
T_{1}+\ldots+tT_{m-1}\cdots
T_{1})\omega_{m}^{-1}(x^{\lambda}F[x_{1}+\ldots+x_{m}])$
$\displaystyle=(t^{m-k}T_{k}\cdots T_{1}+t^{m-k-1}T_{k+1}\cdots
T_{1}+\ldots+tT_{m-1}\cdots T_{1})(x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}])$
$\displaystyle=(t^{m-k}+t^{m-k-1}T_{k+1}+\ldots+tT_{m-1}\cdots
T_{k+1})\left(T_{k}\cdots T_{1}x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}]\right).$

Notice that since $T_{k}\cdots T_{1}x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}]$ is symmetric in the
variables $\\{k+2,\ldots,m\\}$

$\epsilon_{k+1}^{(m)}(T_{k}\cdots T_{1}x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}])=T_{k}\cdots
T_{1}x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}].$

Therefore,

$\displaystyle
t^{m}(Y_{k+1}^{(m)}+\ldots+Y_{m}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}])$
$\displaystyle=(t^{m-k}+\ldots+tT_{m-1}\cdots
T_{k+1})\epsilon_{k+1}^{(m)}(T_{k}\cdots T_{1}x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}])$
$\displaystyle=t(t^{m-k-1}+\ldots+1)\epsilon_{k}^{(m)}(T_{k}\cdots
T_{1}x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}])$

where the last equality follows from

$\left(\frac{t^{m-k-1}+t^{m-k-2}T_{k+1}+\ldots+T_{m-1}\cdots
T_{k+1}}{t^{m-k-1}+t^{m-k-2}+\ldots+1}\right)\epsilon_{k+1}^{(m)}=\epsilon_{k}^{(m)}.$

Putting it all together we see that

$\displaystyle\Psi_{p_{1}}^{(m)}(x^{\lambda}F[x_{1}+\ldots+x_{m}])$
$\displaystyle=t^{m}(Y_{1}^{(m)}+\ldots+Y_{m}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}])$
$\displaystyle=t^{m}(Y_{1}^{(m)}+\ldots+Y_{k}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}])+t^{m}(Y_{k+1}^{(m)}+\ldots+Y_{m}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}])$
$\displaystyle=(\widetilde{Y}_{1}^{(m)}+\ldots+\widetilde{Y}_{k}^{(m)})(x^{\lambda}F[x_{1}+\ldots+x_{m}])+t(t^{m-k-1}+\ldots+1)\epsilon_{k}^{(m)}(T_{k}\cdots
T_{1}x_{2}^{\lambda_{1}}\cdots
x_{k+1}^{\lambda_{k}}F[qx_{1}+x_{2}+\ldots+x_{m}])$

which by Theorem 20 and Corollary 44 converges to

$({\mathscr{Y}}_{1}+\ldots+{\mathscr{Y}}_{k})(x^{\lambda}F[X])+\frac{t}{1-t}\epsilon_{k}(T_{k}\cdots
T_{1}x_{2}^{\lambda_{1}}\cdots x_{k+1}^{\lambda_{k}}F[X+(q-1)x_{1}]).$

Therefore, the limit operator $\Psi_{p_{1}}:=\lim_{m}\Psi_{p_{1}}^{(m)}$ is
well defined.

We will now show that the $\widetilde{E}_{(\mu|\lambda)}$ are weight vectors
of $\Psi_{p_{1}}$ and compute their corresponding weight values. Let
$\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}}$ and
$\lambda\in\operatorname{\mathrm{Par}}$. By Corollary 47 we have that

$\widetilde{E}_{(\mu|\lambda)}=\lim_{m}\epsilon_{\ell(\mu)}^{(m)}(E_{\mu*\lambda*0^{m-(\ell(\mu)+\ell(\lambda))}}).$

Therefore, by Proposition 6.21 from [IW22], Lemma 57, and Lemma 3 it follows
that

$\displaystyle\Psi_{p_{1}}(\widetilde{E}_{(\mu|\lambda)})$
$\displaystyle=\lim_{m}\Psi_{p_{1}}^{(m)}(\epsilon_{\ell(\mu)}^{(m)}(E_{\mu*\lambda*0^{m-(\ell(\mu)+\ell(\lambda))}}))$
$\displaystyle=\lim_{m}t^{m}(Y_{1}^{(m)}+\ldots+Y_{m}^{(m)})\epsilon_{\ell(\mu)}^{(m)}(E_{\mu*\lambda*0^{m-(\ell(\mu)+\ell(\lambda))}})$
$\displaystyle=\lim_{m}\epsilon_{\ell(\mu)}^{(m)}(t^{m}(Y_{1}^{(m)}+\ldots+Y_{m}^{(m)})E_{\mu*\lambda*0^{m-(\ell(\mu)+\ell(\lambda))}})$
$\displaystyle=\lim_{m}\kappa_{\mu*\lambda}^{(m-(\ell(\mu)+\ell(\lambda)))}(q,t)\epsilon_{\ell(\mu)}^{(m)}(E_{\mu*\lambda*0^{m-(\ell(\mu)+\ell(\lambda))}})$
$\displaystyle=\kappa_{\mu*\lambda}(q,t)\widetilde{E}_{(\mu|\lambda)}.$

∎

######  Remark 2.

From the proof of Theorem 59 we see that in particular, for partitions
$\lambda$ we have that

$\Psi_{p_{1}}(\mathcal{A}_{\lambda}[X])=\frac{t}{1-t}(1-(1-t)(1-q)B_{\lambda}(q,t))\mathcal{A}_{\lambda}[X].$

We saw that in Corollary 47
$\mathcal{A}_{\lambda}[X]=(1-t)^{\ell(\lambda)}v_{\lambda}(t)P_{\lambda}[X;q^{-1},t]$
so, following the argument of Haiman in [Hai99], the operator
$t^{-1}(1-t)\Psi_{p_{1}}$ is up to a change of variables equal to
$\Delta^{\prime}$. Therefore, we can view $t^{-1}(1-t)\Psi_{p_{1}}$ in a
certain sense (after changing variables) as extending the operator
$\Delta^{\prime}$ from $\Lambda$ to $\operatorname{\mathscr{P}}_{as}^{+}$.
Further, Theorem 59 does not follow immediately from the work of Ion and Wu in
[IW22] and in particular,

$\Psi_{p_{1}}\neq{\mathscr{Y}}_{1}+{\mathscr{Y}}_{2}+\dots$

although the latter operator is certainly well defined in a weak sense as a
diagonal operator in the $\widetilde{E}_{(\mu|\lambda)}$ basis. The easiest
way to see this is to note that ${\mathscr{Y}}_{1}+{\mathscr{Y}}_{2}+\dots$
will annihilate $\Lambda$ whereas $\Psi_{p_{1}}$ acting on the basis
$\mathcal{A}_{\lambda}$ of $\Lambda$ has nonzero eigenvalues
$\kappa_{\lambda}(q,t)\neq 0$.

######  Theorem 60 (Second Main Theorem).

Let $\widetilde{Y}$ denote the $\mathbb{Q}(q,t)$-subalgebra of
$End_{\mathbb{Q}(q,t)}(\operatorname{\mathscr{P}}_{as}^{+})$ generated by
$\Psi_{p_{1}}$ and ${\mathscr{Y}}_{i}$ for $i\geq 1$.
$\operatorname{\mathscr{P}}_{as}^{+}$ has a basis of $\widetilde{Y}$-weight
vectors and every $\widetilde{Y}$-weight space of
$\operatorname{\mathscr{P}}_{as}^{+}$ is 1-dimensional.

###### Proof.

Since $\Psi_{p_{1}}$ is diagonal in the $\widetilde{E}_{(\mu|\lambda)}$ basis,
see Theorem 49, it commutes with each ${\mathscr{Y}}_{i}$. Therefore,
$\widetilde{Y}$ is a commutative algebra of operators on
$\operatorname{\mathscr{P}}_{as}^{+}$ so it makes sense to ask about its
weights in $\operatorname{\mathscr{P}}_{as}^{+}$. To show that the
$\widetilde{Y}$-weight spaces of $\operatorname{\mathscr{P}}_{as}^{+}$ are
1-dimensional it suffices to show that if
$(\mu^{(1)}|\lambda^{(1)})\neq(\mu^{(2)}|\lambda^{(2)})$ for
$\mu^{(1)},\mu^{(2)}\in\operatorname{\mathrm{Comp}^{\textit{red}}}$ and
$\lambda^{(1)},\lambda^{(2)}\in\operatorname{\mathrm{Par}}$ with
$\widetilde{E}_{(\mu^{(1)}|\lambda^{(1)})}$ and
$\widetilde{E}_{(\mu^{(2)}|\lambda^{(2)})}$ having the same
${\mathscr{Y}}$-weight then the $\Psi_{p_{1}}$ eigenvalues for
$\widetilde{E}_{(\mu^{(1)}|\lambda^{(1)})}$ and
$\widetilde{E}_{(\mu^{(2)}|\lambda^{(2)})}$ are distinct. Necessarily, from
the proof of Theorem 49, if $\widetilde{E}_{(\mu^{(1)}|\lambda^{(1)})}$ and
$\widetilde{E}_{(\mu^{(2)}|\lambda^{(2)})}$ have the same
${\mathscr{Y}}$-weight then $\mu^{(1)}=\mu^{(2)}=\mu$. Since
$(\mu|\lambda^{(1)})\neq(\mu|\lambda^{(2)})$ it follows that
$\lambda^{(1)}\neq\lambda^{(2)}$ so that
$\operatorname{sort}(\mu*\lambda^{(1)})\neq\operatorname{sort}(\mu*\lambda^{(2)})$.
From Lemma 57 we then know that
$\kappa_{\mu*\lambda^{(1)}}\neq\kappa_{\mu*\lambda^{(2)}}$ so lastly by
Theorem 59 we see that the $\Psi_{p_{1}}$ eigenvalues for
$\widetilde{E}_{(\mu|\lambda^{(1)})}$ and
$\widetilde{E}_{(\mu|\lambda^{(2)})}$ are distinct. Hence, the
$\widetilde{Y}$-weight spaces of $\operatorname{\mathscr{P}}_{as}^{+}$ are
1-dimensional.

∎

Theorem 59 motivates the following definition.

######  Definition 61.

For $F\in\Lambda$ let
$\Psi_{F}:\operatorname{\mathscr{P}}_{as}^{+}\rightarrow\operatorname{\mathscr{P}}_{as}^{+}$
be the diagonal operator in
$End_{\mathbb{Q}(q,t)}(\operatorname{\mathscr{P}}_{as}^{+})$ in the
$\\{\widetilde{E}_{(\mu|\lambda)}:\mu\in\operatorname{\mathrm{Comp}^{\textit{red}}},\lambda\in\operatorname{\mathrm{Par}}\\}$
basis given by

$\Psi_{F}(\widetilde{E}_{(\mu|\lambda)}):=F[\kappa_{\mu*\lambda}(q,t)]\widetilde{E}_{(\mu|\lambda)}.$

Notice that by construction every operator $\Psi_{F}$ commutes with the image
of every ${\mathscr{Y}}_{i}\in
End_{\mathbb{Q}(q,t)}(\operatorname{\mathscr{P}}_{as}^{+})$ since from
Corollary 49 we know that the $\widetilde{E}_{(\mu|\lambda)}$ are a basis of
$\operatorname{\mathscr{P}}_{as}^{+}$.

######  Conjecture 1.

For all $F\in\Lambda$ we have that

$\Psi_{F}=\lim_{m}\Psi_{F}^{(m)}.$

######  Remark 3.

Trivially, this conjecture holds for $F=1\in\Lambda$ and Theorem 59 shows that
this conjecture holds for $F=p_{1}$. Thus we see that the conjecture holds for
$F\in\mathbb{Q}(q,t)[p_{1}]$. It is easy to extend part of the argument in the
proof of Theorem 59 to show that

$\lim_{m}\Psi_{F}^{(m)}(\epsilon_{\ell(\mu)}^{(m)}(E_{\mu*\lambda*0^{m-(\ell(\mu)+\ell(\lambda))}}))=F[\kappa_{\mu*\lambda}(q,t)]\widetilde{E}_{(\mu|\lambda)}=\Psi_{F}(\widetilde{E}_{(\mu|\lambda)}).$

However, this is not sufficient to prove the conjecture. Similarly, to the
proof of Theorem 59 one needs to know that the sequence of operators
$(\Psi_{F}^{(m)})_{m\geq 1}$ is well behaved on arbitrary convergent sequences
in order to prove convergence to an operator in
$End_{\mathbb{Q}(q,t)}(\operatorname{\mathscr{P}}_{as}^{+})$. It would be
sufficient to show that for every $r\geq 2$ the sequence of operators
$(\Psi_{p_{r}}^{(m)})_{m\geq 1}$ converges since if this sequence converges
its limit operator will necessarily agree with $\Psi_{p_{r}}$ on the
$\widetilde{E}_{(\mu|\lambda)}$ basis.

This conjecture would imply the existence of an action of the elliptic Hall
algebra [BS12] [SV13] on the space of almost symmetric functions. In
independent work, according to private communications, Dongyu Wu has
constructed an elliptic Hall algebra on $\operatorname{\mathscr{P}}_{as}^{+}$.
Further, in Wu’s action $P_{(0,1)}$ acts identically to $\Psi_{p_{1}}$ up to a
scalar.

## References

  * [Ber+99] F. Bergeron, A. M. Garsia, M. Haiman and G. Tesler  “Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions” Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part III  In _Methods Appl. Anal._ 6.3, 1999, pp. 363–420  DOI: 10.4310/MAA.1999.v6.n3.a7
  * [BG99] F. Bergeron and A. M. Garsia  “Science fiction and Macdonald’s polynomials”  In _Algebraic methods and $q$-special functions (Montréal, QC, 1996)_ 22, CRM Proc. Lecture Notes  Amer. Math. Soc., Providence, RI, 1999, pp. 1–52  DOI: 10.1090/crmp/022/01
  * [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 in Type A”  In _Sém. Lothar. Combin._ , 2023
  * [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
  * [Che03] Ivan Cherednik  “Double affine Hecke algebras and difference Fourier transforms”  In _Invent. Math._ 152.2, 2003, pp. 213–303  DOI: 10.1007/s00222-002-0240-0
  * [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
  * [GH96] A. M. Garsia and M. Haiman  “A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion”  In _J. Algebraic Combin._ 5.3, 1996, pp. 191–244  DOI: 10.1023/A:1022476211638
  * [Goo22] Ben Goodberry  “Partially-Symmetric Macdonald Polynomials”, 2022
  * [Hag+05] J. Haglund et al.  “A combinatorial formula for the character of the diagonal coinvariants”  In _Duke Math. J._ 126.2, 2005, pp. 195–232  DOI: 10.1215/S0012-7094-04-12621-1
  * [Hai02] Mark Haiman  “Vanishing theorems and character formulas for the Hilbert scheme of points in the plane”  In _Invent. Math._ 149.2, 2002, pp. 371–407  DOI: 10.1007/s002220200219
  * [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
  * [HMZ12] J. Haglund, J. Morse and M. Zabrocki  “A compositional shuffle conjecture specifying touch points of the Dyck path”  In _Canad. J. Math._ 64.4, 2012, pp. 822–844  DOI: 10.4153/CJM-2011-078-4
  * [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
  * [IW22a] Bogdan Ion and Dongyu Wu  “Unpublished Notes”, 2022
  * [Lap22] Luc Lapointe  “$m$-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures”, 2022  arXiv:2206.05177 [math.CO]
  * [Mac15] I. G. Macdonald  “Symmetric functions and Hall polynomials” With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [ MR1354144], Oxford Classic Texts in the Physical Sciences  The Clarendon Press, Oxford University Press, New York, 2015, pp. xii+475
  * [Ram06] Arun Ram  “Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux”  In _Pure Appl. Math. Q._ 2.4, Special Issue: In honor of Robert D. MacPherson. Part 2, 2006, pp. 963–1013  DOI: 10.4310/PAMQ.2006.v2.n4.a4
  * [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