there are exact sequences
0→ Kı → OZı → OZı−1 → 0
with Kı a zero dimensional sheaf on C2, for 1 ≤ ı ≤ ℓ. The morphism N (γ) → Sγ(C2) sends a flag of subschemes of the form (5.13) to the cycle classes associated to the zero dimensional sheaves (OZ0, K1, . . . , Kℓ) via the Hilbert-Chow morphism. Then the bottom Cartesian square in (5.12) yields a morphism ργ : N (γ) → eHγ which factors through a morphism ργred:N (γ) → eHredγ . The following generalization of (5.9) will be proven below:
ργred∗ON (γ) =OHeγred. (5.14)
LetUγ = (πγ)−1U be the inverse image of the open subset parameterizing subschemes of C2 supported at r distinct closed points. Then ηγ is an isomorphism over the open subset (ηγ)−1Uγ. Then equation (5.14) follows from the Zariski Main Theorem provided one can prove that N (γ) is connected. This is shown in Appendix C.
For future reference, note that by construction the bundle P = πred∗OHerred is equipped with a fiberwise action of the symmetric groupSr such that its fiber over any point [I] ∈ Hr is isomorphic to the regular representation. By construction,
OHeγ
red ≃(
κred∗OHer
red
)Sγ
where κred : eHrred → eHγred is the morphism of reduced schemes determined by κ in diagram (5.12). Pushing forward this identity to Hr via πredγ , one learns that
πredγ ∗OHeγred ≃(
πred∗OHerred
)Sγ
=PSγ. (5.15)
In particular, since P is locally free, so is Pγ = πredγ ∗OHeγred. Moreover Pγ is equipped with a fiberwise action of Sr such that its fiber at any closed point [I] ∈ Hr is isomorphic to the permutation representation Mγ of Sr with stabilizer Sγ =∏ℓ
ı=0Sγı ⊂ Sr.
bundle F on Hr, let Fµ denote the fiber of F at [Iµ]. An exception will be made for the cotangent bundle T∗Hr, in which case the fiber at [Iµ] will be denoted by Tµ∗Hr. Then equivariant localization yields
χTy˜(Vg,p) = ∑
µ
Ωg,pµ (q1, q2, ˜y), (5.16)
where
Ωg,pµ (q1, q2, ˜y) =
chT(detVg−1+p) chTΛ˜y(Tµ∗Hr⊕gµ )
chTΛ−1(Tµ∗Hr) . (5.17) Then the main formula proven in this section reads
ZK(r)(q1, q2, ˜y, ˜x) = ∑
µ
Ωg,pµ (q1, q2, ˜y) eHµ(q2, q1, ˜x) (5.18)
where eHµ(q2, q1, ˜x) are the modified MacDonald polynomials.
First note that the pushforward formulas (5.14), (5.15) are valid in T-equivariant setting, hence one obtains an identity
χTy˜(N (γ(m)), ηγ∗Vg,p) = χTy˜(Hr, (PSγ ⊗Hr Vg,p)). (5.19) The right hand side of equation (5.19) can be evaluated again by equivariant localization:
χTy˜(Hr, (PSγ ⊗Hr Vg,p)) =∑
µ
Ωg,pµ (q1, q2, ˜y)chT(Pµγ). (5.20)
Now let ˜γ denote the unordered partition of r determined by the sequence γ = (γ0, . . . , γℓ).
Then following formula
chT(Pµγ) =∑
λ
Kλ,˜γKeλ,µ(q1, q2). (5.21) will be proven bellow, where the sum is over all partitions λ of r, Kλ,˜γ are the Kostka numbers, and eKλ,µ(q2, q1) are the modified Kostka–Macdonald coefficients.
Since the fiberwise Sr-action onP is compatible with the T-equivariant structure, there is a direct sum decomposition
Pµ≃⊕
λ
Vµ,λ⊗ Rλ (5.22)
Rλ is the irreducible Sr-representation labelled by the partition λ and Vµ,λ are finite di-mensional representations of T. According to [28, Thm. 3.1, Prop. 3.7.3, Thm. 3.2], the T-character of Vλ,µ is given by the modified Kostka-MacDonald coefficients,
chTVλ,µ = eKλ,µ(q2, q1). (5.23)
The pushforward formula (5.15) shows that the fiberPµγ is theSγ-fixed subspace ofPµ. This yields
chTPµγ = 1
|Sγ|
∑
g=(g0,...,gℓ)∈Sγ
∑
λ
χRλ(g) chTVµ,λ. (5.24) Now recall the branching rule for representations of the symmetric group. Given a sub-groupSr1× Sr2 ⊂ Sr, with r1+ r2 = r, the irreducibleSr-representation Rλ has a direct sum decomposition
Rλ ≃⊕
ν1,ν2
Nν1,ν2,λ(
Rν1 ⊠ Rν2
) (5.25)
where ν1, ν2 are partitions of r1, r2 respectively, and Nν1,ν2,λ are the Littlewood-Richardson coefficients. Applying the rule (5.25) recursively one finds
Rλ ≃ ⊕
ν0,...,νℓ
Nν0,...,νℓ,λ(
Rν0 ⊠ · · · ⊠ Rνℓ
) (5.26)
where νı is a partition of γı for 0≤ ı ≤ ℓ. Substitution in (5.24) yields
chTPµγ =∑
λ
∑
ν0,...,νℓ
Nν0,...,νℓ,λ
∏ℓ ı=0
( 1
|Sγı|
∑
gı∈Sγı
χRνı(gı) )
chTVµ,λ (5.27)
Next note that
1
|Sγı|
∑
gı∈Sγı
χRνı(gı) = dim RSνıγı
is the dimension of theSγı-fixed subspace of Rνı. Since Rνıis an irreducibleSγı-representation, dim RSνıγı = 0
unless Rνı is the trivial representation corresponding to the length one partition νı = (γı).
In the latter case,
dim RS(γγı
ı) = 1.
Then equation (5.27) reduces to
chTPµγ =∑
λ
N(γ0),...,(γℓ),λchTVµ,λ.
formula (5.21) Using equation (5.23), formula (5.21) follows from the identity
N(γ0),...,(γℓ),λ = Kλ,˜γ. (5.28)
The latter is proven in [22, Appendix 9]. More precisely, as shown in loc. cit., the Littlewood-Richardson coefficients occur in the decomposition of the product of two Schur functions:
sν1(x)sν2(x) =∑
λ
Nν1,ν2,λsλ(x). (5.29)
Furthermore, applying formula (5.29) recursively as in [22, Eqn. (A.9) pp. 456] yields s(r1)(x)· · · s(rk)(x) =∑
λ
Kλ,ρsλ(x) (5.30)
where ρ is the partition of r determined by (r1, . . . , rk), and Kλ,ρ are the Kostka numbers.
This implies identity (5.28).
Using equations (5.20), (5.21) the contribution of a fixed point [Iµ] to the partition function (5.6) reduces to:
Ω(g,p)µ (q1, q2, ˜y)∑
ν
∑
λ
Keλ,µ(q2, q1)Kλ,νmν(˜x) =
Ω(g,p)µ (q1, q2, ˜y)∑
λ
Keλ,µ(q2, q1)sλ(˜x) = Ω(g,p)µ (q1, q2, ˜y) eHµ(q2, q1; ˜x).
where mν(˜x) are the monomial symmetric functions, and sλ(˜x) the Schur functions. This concludes the proof of equation (5.18).
6 BPS expansion and a parabolic P = W conjecture
Collecting the results of the previous two sections, here it is shown that geometric engineering yields a conjectural expression for the refined stable pair partition function (1.4), which agrees with the left hand side of the HLRV formula (1.3) by a change of variables. Furthermore, it will be checked that the same change of variables relates the right hand side of equation (1.3) with a refined Gopakumar-Vafa expansion, completing the physical derivation of the HLRV formula.
As in Section 5.4 it will be assumed that there is a single marked point on C. The root stack eC be the root stack has stabilizer µs at the unique orbifold point, for some s ≥ 1. The local threefold eYM is the total space of the rank two bundle ν∗M−1 ⊕ KCe ⊗Ce ν∗M on the root stack eC, with M a degree p line bundle on C. Let Zrefe
YM(q, x, y) be the refined stable pair partition function of Y , where x = (x0, . . . , xs−1, 0, . . .) are the formal counting variables associated to the marked point. Physically, these are chemical potentials for twisted sector
Ramond-Ramond charges at the orbifold point. By analogy with [12, 11], the geometric engineering conjecture reads:
ZYrefe (q, x, y) = 1 +∑
r≥1
ZK(r)(qy−1, q−1y−1, y, (−1)(g−1+p)y−gx). (6.1)
The terms in the right hand side are given by (5.18). Equation (5.17) yields
Ωg,pµ (q1, q2,ey) = ∏
2∈µ
(q1l(2)q2a(2))g−1+p(1− eyq1−l(2)q2a(2)+1)g(1− eyq1l(2)+1q−a(2)2 )g (1− q−l(2)1 qa(2 2)+1)(1− q1l(2)+1q2−a(2))
where a(2), l(2) are the arm and leg length of a box 2 ∈ µ. Making the change of variables in equation (6.1) yields
ZYrefe (q, y, x) = 1 +∑
µ̸=∅
Zµg,p(q, y) eHµ(q−1y−1, qy−1, x), (6.2)
where
Zµg,p(q, y) = (−1)p|µ|∏
2∈µ
(ql(2)−a(2)y1−h(2))p
(qy−1)(2l(2)+1)(g−1)(1− yl(2)−a(2)q−h(2))2g (1− yl(2)−a(2)−1q−h(2))(1− yl(2)−a(2)+1q−h(2))
with h(2) = a(2) + l(2) + 1, and the sum is over all Young diagrams µ. This formula can be also written as
Zµg,p(q, y) = (−1)p|µ|∏
2∈µ
(ql(2)−a(2)y1−h(2))p
(qy)−(2a(2)+1)(g−1)(1− ya(2)−l(2)qh(2))2g
(1− ya(2)−l(2)−1qh(2))(1− ya(2)−l(2)+1q−h(2)) . (6.3) A further change of variables yields
ZYrefe (z−1w, z−1w−1, x) = 1 +∑
µ̸=∅
Hµg,p(z, w) eHµ(z2, w2, x), (6.4)
where
Hg,pµ (z, w) = ∏
2∈µ
(z2a(2)w2l(2))p(z2a(2)+1− w2l(2)+1)2g (z2a(2)+2− w2l(2))(z2a(2)− w2l(2)+2)
For p = 0 this is the left hand side of the HLRV formula evaluated at formal variables x = (x0, . . . , xs−1, 0, 0, . . .).
For the remaining part of this section, let eY := eYOC be the product A1× eS, with eS = tot(KCe). Then it easy to check that any moduli stack of compactly supported Bridgeland stable pure dimension one sheaves on eY with fixed numerical class is isomorphic to a product
A1 × Mβ( eS, γ), where M is a moduli stack of β-stable pure dimension one sheaves on eS with fixed numerical equivalence class γ. The notation used here is the same as in Sections 3.2, equation (3.21), and 3.3.4. Since in this particular case there is a single marked point, and the eigenvalues ξ are trivial, γ will be labelled by integers dl≥ 1, 1 ≤ l ≤ s, and n ∈ Z.
According to Section 3.1, the moduli stack Mβ( eS, γ) is isomorphic to a moduli stack of stable strongly parabolic Higgs bundles on C.
From a string theoretic perspective this chain of isomorphisms identifies parabolic Higgs bundles on C with supersymmetric D2-D0 configurations on the Calabi-Yau threefold eY . Then the HLRV formula is identified with a refined Gopakumar-Vafa expansion [25, 38, 35, 10] provided that one assumes a parabolic variant of the P = W conjecture [14]. Some details are provided below for completeness.
For a precise formulation of the parabolic P = W conjecture, consider a smooth projective curve C with two marked points p,∞ ∈ C and let γp, γ∞∈ π1(C\ {p, ∞}) be the generators associated to the marked points. Let (r, e)∈ Z>0× Z be coprime integers and let Cλ denote the GL(r,C) conjugacy class of a diagonal matrix with (ordered) eigenvalues
λ = (λ1, . . . , λr).
Let also µ = (µ1, . . . , µl) denote the partition of r determined by the multiplicities of the above eigenvalues.
Now let Cλe(C, p,∞) be the character variety with conjugacy classes Cλ, exp(2eπ√
−1/r) at the marked points p,∞. According to [29, Thm. 2.1.5], for sufficiently generic λ, Cλe(C, p,∞) is either empty or a smooth quasi-projective variety of complex dimension
dµ= r2(2g− 2 + 1) −
∑l j=1
(µj)2+ 2. (6.5)
Note that dµ is even; using the identity r =∑l
j=1µj,
dµ= 2bµ, bµ= r2(g− 1) + 1 + ∑
1≤j1,j2≤l j1<j2
µj1µj2. (6.6)
Since the marked curve (C, p,∞) is fixed throughout this section, the character variety will be denoted simply by Cλe in the following.
Next consider the specialization of the HLRV formula (1.3) to the present case taking x∞= (x∞,0, 0, 0, . . .). Since µ∞is the length one partition (r), the variable x∞,0can be scaled
off by a redefiniton of the formal variable x associated to p. Moreover, as observed in [29], the mixed Poincar´e ploynomial Pc(Cλe; u, t) depends only on µ as long as a e is coprime with r. Therefore equation (1.3) yields a formula of the form
∑
µ
Hµg,0(z, w) eHµ(z2, w2; x) = exp (∑∞
k=1
1 k
w−kdµPc,µ(z−2k,−(zw)k)
(1− z2k)(w2k− 1) mµ(xk) )
(6.7)
where Pc,µ(u, t) = Pc(Cλe; u, t).
By analogy with [14], the parabolic P = W conjecture identifies the weight filtration W•Hcpt(Cλe) with the perverse sheaf filtration on the compactly supported cohomology of a moduli space of stable strongly parabolic Higgs bundles. As a first step, note that Conjecture 1.2.1(ii) in [29] yields the identifications
W2pHcpt(Cλe) = W2p+1Hcpt(Cλe) for all p, just as in the unmarked case studied in [30].
Next, let Hem denote the moduli space of rank r ≥ 1, degree e stable parabolic Higgs bundles (E, Φ) on the marked curve (C, p) with parabolic structure of type m at p. The Higgs field Φ : E → E ⊗ KC(p) has nilpotent residue at p with respect to the. Let µ be the partition of r determined by m. For primitive discrete invariants (m, e) and sufficiently generic parabolic weights there are no strictly semistable objects, and the moduli space is a smooth quasi-projective variety of dimension dµ. Furthermore, Hem is diffeomorphic in this case with the character variety Cλe provided the eigenvalues λi are related to the parabolic weights by λi = e2iπαi, 1≤ i ≤ r. There is also a Hitchin map
h :Hem → Bm
with Bm ⊂ ⊕ri=1H0(KC(p)⊗i) a linear subspace of dimension bµ. The generic fibers of h are smooth abelian varieties of dimension bµ, and the total spaceHem is an algebraically complete integrable system. By analogy with [15, 14], this yields a perverse sheaf filtration P•H(Hem).
The parabolic P = W conjecture states that
W2pH(Cλe) = PpH(Hem) (6.8) for all values of p.
Equation (6.8) leads to an identification of the HLRV formula (1.3) with a refined BPS expansion in close analogy with [11, Sect. 4]. Very briefly, using the methods in [15] one
can prove a hard Lefschetz theorem for the parabolic Hitchin map and also choose a (non-canonical) splitting of the perverse sheaf filtration as in [14, Sect 1.4.2, 1.4.3]. This yields an SL(2,C) × C× action on the cohomology H(Hem), which splits as a direct sum
H(Hµe)≃ ⊕bp=0µ R⊕dim(Q(d−p)/2p,0) (6.9) where RjL is the irreducible SL(2,C)-representation of spin jL ∈ 12Z. In the above formula p is the perverse degree and Qp,0 the primitive cohomology of perverse degree p. The coho-mological degree is encoded in the C× action, which scales the quotient GrpPHk(Heµ) with weight l = k− p − bµ. Then specializing x to x = x = (x0, . . . , xs−1, 0, 0, . . .) and making the same change of variables
(z, w) = (
(qy)−1/2, (qy−1)1/2)
as in equation (6.4) converts equation (6.7) into a refined Gopakumar-Vafa expansion. This computation is completely analogous with [11, Sect. 4], hence the details are omitted.
7 Recursion via wallcrossing
The recursion relation conjectured in [12] for the Poincar´e polynomial of the moduli space of Hitchin pairs admits a natural generalization to parabolic Higgs bundles. The derivation of this formula is completely analogous to loc. cit. assuming again all the foundational aspects of motivic Donaldson-Thomas theory [42]. The final result will be recorded below, omitting most intermediary steps.
For simplicity it will be assumed again that the curve C has only one marked point p.
To fix notation, the discrete invariants of a parabolic rank bundle E• on C are the degree e∈ Z and the flag type m = (ma)0≤a≤s−1∈ (Z≥0)×s. Let
|m| =
s−1
∑
a=0
ma, χ(m, e) = e− |m|(g − 1).
For any weights α = (αa)0≤a≤s−1 let
m· α =
s−1
∑
a=0
maαa.
The parabolic slope and the parabolic δ-slope are defined respectively by µ(m, e, α) = χ(m, e) + m· α
|m| , µδ(m, e, α) = χ(m, e) + m· α + δ
|m| ,
and the ordinary slopes are given by
µδ(m, e) = χ(m, e) + δ
|m| , µ(m, e) = χ(m, e)
|m| .
7.1 Generic parabolic weights
The recursion formula will be derived from wallcrossing with respect to variations of the sta-bility parameter δ introduced in Section 4.1. The refined parabolic ADHM invariants will be denoted by Aδ(m, e, α; y) while the refined parabolic Higgs bundle invariants by H(m, e, α; y).
Note that H(m, e, α; y)∈ Q(y) are the rational refined invariants obtained directly from the motivic integration map in [42], not the integral refined invariants H(m, e, α; y)∈ Z[y, y−1].
The relation between the two sets of invariants for sufficiently generic weights is given by the refined multicover formula
H(m, e, α; y) = ∑
k≥1, (m,e)=k(m′,e′)
1
k[k]yH(m′, e′, α; yk). (7.1) For fixed numerical invariants and fixed parabolic weights, there are finitely many critical values δc ∈ R, where strictly semistable objects can exist. Using the formalism of [42], the wallcrossing formula at such a critical value δc̸= 0 is
Aδc+(m, e, α; y)− Aδc−(m, e, α; y) =
∑
l≥2
1 (l− 1)!
∑
∆l(δc,m,e,α)
Aδc−(m1, e1, α; y)
∏l i=2
[χ(mi, ei)]yH(mi, ei, α; y). (7.2)
where
∆l(δc, m, e, α) =
{(m1, . . . , ml), (e1, . . . , el)| mi ∈ (Z≥0)×r, ei ∈ Z, |mi| > 0, 1 ≤ i ≤ l, (m1, e1) +· · · + (ml, el) = (m, e), µ(mi, ei, α) = µδc(m1, e1, α), 2≤ i ≤ l} and
[n]y = yn− y−n y− y−1
for any integer n ∈ Z. This is the same wallcrossing formula as [12, Eqn. 1.3], except the Higgs invariants differ by a sign (−1)χ(m,e) from the used in loc. cit. The present normalization is more natural in this context. There is a similar formula for δc = 0, including an extra term with m1 = 0 as in [12, Eqn. 1.4].
Applying equation (7.2) iteratively from δ >> 0 to δ << 0, and using the duality relations (4.6), one obtains a wallcrossing formula of the form
[χ(m, e)]yH(m, e, α; y) = A+∞(m, e; y)− A+∞( ˇm, ˇe; y)
+∑
l≥2
(−1)l−1 (l− 1)!
∑
∆(>)l (m,e,α)
A+∞(m1, e1; y)
∏l i=2
[χ(mi, ei)]yH(mi, ei, α; y)
−∑
l≥2
(−1)l−1 (l− 1)!
∑
∆(l≥)( ˇm,ˇe,ˇα)
A+∞(m1, e1; y)
∏l i=2
[χ(mi, ei)]yH(mi, ei, α; y)
−∑
l≥2
1 l!
∑
∆(=)l (m,e,α)
∏l i=1
[χ(mi, ei)]yH(mi, ei, α; y)
(7.3)
where
∆(l3)(m, e, α) =
{(m1, . . . , ml), (e1, . . . , el)| mi ∈ (Z≥0)×r, ei ∈ Z, |mi| > 0, 1 ≤ i ≤ l, (m1, e1) +· · · + (ml, el) = (m, e), µ(mi, ei, α) 3 µ(m, e, α), 2 ≤ i ≤ l}
,
(7.4)
the symbol3 taking values >, ≥, = respectively. Note that for any discrete invariants n there exists a lower bound d0 ∈ Z such that A∞(n, d; y) = 0 for all d < d0. This can be proven by standard bounding arguments, or, alternatively, it follows easily from the conjectural formula (6.2). Therefore the number of terms in the right hand side of equation (7.3) is finite and bounded above by a constant independent of the parabolic weights α.
The recursion formula (7.3) together with the geometric engineering conjecture (6.1) completely determines the parabolic refined invariants H(m, e, α; y). Using the arguments employed by Mozgovoy in [49], it will be shown below that the resulting invariants are compatible with those determined by the HRLV formula (6.7).
For simplicity, consider local curves on type (0, 2g− 2) in the following. Using the same notation as [49], the refined partition function (4.7) will be denoted by A∞(q, y, x). Hence
A∞(q, y, x) =∑
m,e
A+∞(m, e; y)qχ(m,e)xm.
Since there is a single marked point, the formal variable x = (x0, x1, . . .) does not carry an extra index.
As shown in Section 6, geometric engineering predicts that A∞(q, y, x) is determined by equation (6.1)
A∞(q, y, x) = 1 +∑
µ̸=∅
Zµg,0(q, y) eHµ(q−1y−1, qy−1; x).
Following [49], let ePm(q, y) be defined by the formula A∞(q, y, x) = exp[ ∑
k≥1
∑
m
xkm
k f (qk, yk) ePm(qk, yk) ]
(7.5)
where
f (q, y) = q
(1− qy)(y − q).
Note that ePm(q, y) is related to the mixed Poincar´e polynomial of the character variety Cλe
defined in Section 6, where λ is the partition of r =|m| determined by m. Using the change of variables (z2, w2) = (q−1y−1, qy−1) in equation (6.7), one obtains
Pem(q, y) = ybλ+2q−bλPc(Cλe, qy,−y−1)
where bλ = dλ/2 is half the complex dimension of the character variety.
Now let
Ω(m, e; y) = yH(m, e, α; y)
for any discrete invariants (m, e). Assuming that the invariants Ω(m, e; y) are independent of the degree e∈ Z for any m, it will be shown below that
Ω(m, e; y) = ePm(1, y) (7.6)
for all (m, e). The proof is entirely analogous to the proof of [49, Thm. 4.6], some details being presented below for completeness. Note that the assumption that Ω(m, e; y) are inde-pendent of e is a standard conjecture [36] for Donaldson-Thomas invariants of pure dimension one sheaves on Calabi-Yau threefolds.
Following [49], for any series I = ∑
(m,e)
I(m, e)qχ(m,e)xm ∈ Q(y)[[q±1, x]]
and any µ∈ R, let
I3µ = ∑
µ(m,e,α)3µ
I(m, e)qχ(m,e)xm
where 3 ∈ {=, ≥, >, ≤, <}. Furthermore, for any µ ∈ R define Cµ(q, y, x) = exp[ ∑
k≥1
∑
µ(m,e,α)=µ
xkm
k(y2k− 1)Ω(m, e; yk)(
(qy)kχ(m,e) − (qy−1)kχ(m,e))]
and
C3µ(q, y, x) =∏
η3µ
Cη(q, y, x)
Then the recursion relation (7.3) can be recast in the form Cµ(q, y, x) =(
A∞(q, y, x)C>µ−1(q, y, x))
µ− (A∞(q−1, y, x)C≥−µ−1 (q−1, y, x))
−µ (7.7)
by analogy with [49, Remark. 4.5], where
A∞(q, y, x) = A∞(q, y, x)− 1.
In order to prove equation (7.6) it suffices to show that the statement of [49, Thm 4.7]
holds in the present context. Namely, it suffices to prove the identity
A∞(q, y, x)C>µ−1(q, y, x) = A∞(q−1, y, x)C≥−µ−1 (q−1, y, x) (7.8) inQ(y)[[q±1, x]]. The proof given in [49, Sect 5] is based on several essential facts.
First note that independence of Ω(m, e; y) of degree yields a factorization of the form
∑
(m,e)
Ω(m, e; y)qχ(m,e)(
yχ(m,e)− y−χ(m,e)) xm = ( ∑
m
Ω(m; y)xm) ∑
n∈Z
((qy)n− (qy−1)n)) ,
where Ω(m; y) denotes the common value of Ω(m, e; y). Moreover, note that the function f (q, y) defined below equation (7.5) satisfies
f (q, y) = f (q−1, y).
These two facts imply that completely analogous statements to [49, Lemma 5.1], [49, Lemma 5.4] and [49, Prop. 5.5] hold in the present context.
The next important observation is that
A∞(q, y, x) = A∞(q−1, y, x). (7.9)
In the present context, this follows from equation (6.3), which shows that Zλg,0(q−1, y) = Zλg,0t (q, y),
and the standard property of MacDonald polynomials Heλ(t, s; x) = eHλt(s, t; x),
which yields
Heλ(qy−1, q−1y−1; x) = eHλt(q−1y−1, qy−1; x).
Since f (q, y) is invariant under q 7→ q−1, equation (7.9) implies that Pem(q, y) = ePm(q−1, y)
which is analogous to [49, Lemma 5.7].
From this point on, the proof of identity (7.6) is identical with the proof of [49, Thm.
4.7] given in Section 5 of loc. cit.