Spectral Theory of the Nazarov-Sklyanin Lax Operator
In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator 𝓛: 𝐹[𝓌] → 𝐹[𝓌] where 𝐹 is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition 𝐹[𝓌] ≅ ⨁λ 𝑍(𝑗λ, 𝓛) into finite-dimensional 𝓛-cyclic subspaces...
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| description | In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator 𝓛: 𝐹[𝓌] → 𝐹[𝓌] where 𝐹 is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition 𝐹[𝓌] ≅ ⨁λ 𝑍(𝑗λ, 𝓛) into finite-dimensional 𝓛-cyclic subspaces in which Jack polynomials 𝑗λ may be taken as cyclic vectors and (2) prove that the restriction of 𝓛 to each 𝑍(jλ, 𝓛) has simple spectrum given by the anisotropic contents [𝑠] of the addable corners 𝑠 of the Young diagram of λ. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated with 𝓛, both established by Nazarov-Sklyanin. Finally, we conjecture that the 𝓛-eigenfunctions 𝜓ˢλ ∈ 𝐹[𝓌] {with eigenvalue [𝑠] and constant term} 𝜓ˢλ|𝓌₌₀ = 𝑗λ are polynomials in the rescaled power sum basis 𝑉μ𝓌ˡ of 𝐹[𝓌] with integer coefficients.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 063, 22 pages
Spectral Theory of the Nazarov–Sklyanin
Lax Operator
Ryan MICKLER a and Alexander MOLL b
a) Singulariti Research, Melbourne, Victoria, Australia
E-mail: ry.mickler@gmail.com
b) Department of Mathematics and Statistics, Reed College, Portland, Oregon, USA
E-mail: amoll@reed.edu
URL: https://alexander-moll.com
Received March 19, 2023, in final form August 27, 2023; Published online September 10, 2023
https://doi.org/10.3842/SIGMA.2023.063
Abstract. In their study of Jack polynomials, Nazarov–Sklyanin introduced a remark-
able new graded linear operator L : F [w] → F [w] where F is the ring of symmetric
functions and w is a variable. In this paper, we (1) establish a cyclic decomposition
F [w] ∼=
⊕
λ Z(jλ,L) into finite-dimensional L-cyclic subspaces in which Jack polynomi-
als jλ may be taken as cyclic vectors and (2) prove that the restriction of L to each Z(jλ,L)
has simple spectrum given by the anisotropic contents [s] of the addable corners s of the
Young diagram of λ. Our proofs of (1) and (2) rely on the commutativity and spectral
theorem for the integrable hierarchy associated to L, both established by Nazarov–Sklyanin.
Finally, we conjecture that the L-eigenfunctions ψs
λ∈ F [w] with eigenvalue [s] and constant
term ψs
λ|w=0 = jλ are polynomials in the rescaled power sum basis Vµw
l of F [w] with integer
coefficients.
Key words: Jack symmetric functions; Lax operators; anisotropic Young diagrams
2020 Mathematics Subject Classification: 05E05; 33D52; 37K10; 47B35
1 Introduction and statement of results
Nazarov–Sklyanin introduced in [32] a graded linear operator L to the study of Jack polynomi-
als [14, 23, 44]. In Section 1.1, we recall the definition of L : F [w] → F [w] in the polynomial
ring F [w] where F is the ring of symmetric functions with its usual grading and degw = 1.
They proved that if one considers the projection π0 : F [w] → F defined by setting w = 0, then
the operators Tℓ = π0Lℓ pairwise commute in F for all ℓ and are simultaneously diagonalized
on Jack polynomials jλ with explicit eigenvalues. Moreover, as the second author observed
in [27, 29], Nazarov–Sklyanin actually prove in [32] that the eigenvalues of Tℓ at jλ are precisely
the ℓth moments of the transition measure τλ of the anisotropic Young diagram of λ studied
in [2, 5, 12, 16, 18, 20, 39].
In this paper, we determine the spectrum of L in F [w] and identify a distinguished polynomial
basis ψsλ of eigenfunctions of L satisfying π0ψ
s
λ = jλ. We state our result in Theorem 1.4 below.
In subsequent work [24], the first author uses the spectral theorem established in the present
paper to derive a new explicit system of constraints on Jack Littlewood–Richardson coefficients
in terms of a simple new multiplication operation on partitions. Using the results of [24],
Alexandersson–Mickler [1] prove new cases of the strong Stanley conjecture [44]. We hope that
the polynomials ψsλ introduced in this paper may inspire further applications and are also of
independent interest.
mailto:ry.mickler@gmail.com
mailto:amoll@reed.edu
https://alexander-moll.com
https://doi.org/10.3842/SIGMA.2023.063
2 R. Mickler and A. Moll
1.1 The Nazarov–Sklyanin Lax operator
Consider the graded polynomial ring
F = C[V1, V2, . . .] (1.1)
in which deg Vk = k. For any ℏ ∈ C, let V−k be the differential operator in F defined by
V−k = ℏk
∂
∂Vk
. (1.2)
Since [V−k, Vk] = ℏk, (1.1) is the Fock space representation of the Heisenberg algebra at
level ℏ. We will identify F with the ring of symmetric functions in Section 1.2. Introduce a new
variable w with degw = 1 and consider the graded polynomial and Laurent polynomial rings
F [w] = C[w, V1, V2, V3, . . .], (1.3)
F
[
w,w−1
]
= C
[
w,w−1, V1, V2, V3, . . .
]
.
Define the projection π : F
[
w,w−1
]
→ F [w] to be the linear extension of
πwl =
{
wl if l ≥ 0,
0 if l < 0.
(1.4)
We now present the Lax operator from [32] using the conventions from [30, 36].
Definition 1.1. For any ε, ℏ ∈ C and ∂w = ∂
∂w , the Nazarov–Sklyanin Lax operator L [32] is
the graded linear operator L : F [w] → F [w] in the graded polynomial ring F [w] in (1.3) defined
by
L = εw∂w +
∞∑
k=1
V−kw
k +
∞∑
k=1
Vkπw
−k (1.5)
for V−k in (1.2), π in (1.4). Below, we may write L = Lε,ℏ to emphasize that V−k depends on ℏ.
Due to the presence of π in (1.5), L is well defined with codomain F [w]. Moreover, L preserves
total degree in F [w] since w∂w preserves degree, multiplication by wk and Vk raise degree by k,
and V−k and πw−k lower degree by k. The Lax operator Lε,ℏ is a two-parameter perturbation of
a nilpotent linear operator L0,0 by linear differential operators which act as derivations in F [w].
Since
{
wl
}∞
l=0
is a basis of C[w] indexed by N = {0, 1, 2, . . .}, L in (1.5) acts as an N × N
matrix in F [w] = F ⊗C[w] with coefficients in End(F ): using the definition of V−k in (1.2), this
matrix is
L =
0ε V1 V2 V3 V4 · · ·
V−1 1ε V1 V2 V3
. . .
V−2 V−1 2ε V1 V2
. . .
V−3 V−2 V−1 3ε V1
. . .
V−4 V−3 V−2 V−1 4ε
. . .
...
. . .
. . .
. . .
. . .
. . .
. (1.6)
One can see from (1.6) that L0,0 is strictly upper triangular if ε = ℏ = 0, hence nilpotent
in F [w] as we already mentioned. If one sets ε = 0 but keeps ℏ ̸= 0, the diagonal terms
in (1.6) vanish and the entries of L0,ℏ are constant along diagonals. In this special case, one may
regard L0,ℏ as a block Toeplitz operator with infinite blocks from End(F ).
Spectral Theory of the Nazarov–Sklyanin Lax Operator 3
1.2 Jack symmetric functions
Recall that a partition λ of n is a sequence of non-negative integers λi which are weakly decreasing
0 ≤ · · · ≤ λ3 ≤ λ2 ≤ λ1 and satisfy · · ·+ λ3 + λ2 + λ1 = n. For any ε1, ε2 ∈ C, let jλ ∈ F be the
Jack polynomial in the ring F in (1.1) as defined in [22, 30, 36, 37, 40]. For example, the Jack
polynomials in degrees 1 ≤ |λ| ≤ 3 are
j1 = V1,
j2 = V 2
1 + ε1V2,
j1,1 = V 2
1 + ε2V2,
j3 = V 3
1 + 3ε1V1V2 + 2ε21V3,
j1,2 = V 3
1 + (ε1 + ε2)V1V2 + (ε1ε2)V3,
j1,1,1 = V 3
1 + 3ε2V1V2 + 2ε22V3.
For ε2 ̸= 0, these polynomials are equivalent to Jλ(p1, p2, . . . ;α), the integral form Jack sym-
metric function in [14, 23, 44] where pk are the power sum symmetric functions and α is the
Jack parameter. Precisely, for ε2 ̸= 0, if one sets pk = Vk/(−ε2) and α = ε1/(−ε2), then
jλ = (−ε2)|λ|Jλ
(
V1
(−ε2)
,
V2
(−ε2)
, . . . ;
ε1
(−ε2)
)
. (1.7)
At ε1 = α and ε2 = −1, jλ = Jλ.
1.3 Addable and removable corners of Young diagrams
Any partition λ determines a set
λ =
∞⋃
r=1
{
(c− 1, r − 1) ∈ N2 : c ∈ {1, 2, . . . , λr}
}
(1.8)
called the Young diagram of λ. We use λ to refer to either the sequence of parts λi or to (1.8).
Definition 1.2. For λ in (1.8), define the addable and removable corner sets by
Aλ =
{
s ∈ N2 : s ̸∈ λ and λ ∪ {s} is also a Young diagram
}
, (1.9)
Rλ =
{
s ∈ N2 : s ∈ λ and λ \ {s} is also a Young diagram
}
. (1.10)
It is also convenient to define the outer corner set R+
λ = Rλ + (1, 1) as shifts of removable
corners.
1.4 The anisotropic content function
We now define the following function [ · ] : Z2 → C.
Definition 1.3. For ε1, ε2 ∈ C, the anisotropic content [s] of s = (N1, N2) ∈ Z2 is defined by
[s] = ε1N1 + ε2N2. (1.11)
For example, the anisotropic contents of the addable corner setAλ of partition λ= (1 ≤ 2 ≤ 6)
in Figure 1 are 3ε2, ε1 + 2ε2, 2ε1 + ε2, and 6ε1. If ε1 = α, ε2 = −1, these are −3, α− 2, 2α− 1,
and 6α, the traditional anisotropic contents of the corresponding α-anisotropic Young diagram
from [16]. For simplicity, we may refer to [s] as the content of s.
4 R. Mickler and A. Moll
Figure 1. The Young diagram of the partition λ = (1 ≤ 2 ≤ 6) of |λ| = 9 with addable corner set Aλ =
{(0, 3), (1, 2), (2, 1), (6, 0)} (depicted above in circles), removable corner set Rλ = {(0, 2), (1, 1), (5, 0)}
(depicted above in diamonds), and outer corner set R+
λ = {(1, 3), (2, 2), (6, 1)}. Note that Rλ ⊂ λ but
R+
λ ∩ λ = ∅.
1.5 Spectral theorem for the Nazarov–Sklyanin Lax operator
Nazarov–Sklyanin proved in [32] that the ingredients in Sections 1.2–1.4 emerge naturally in
the spectral theory of the hierarchy Tℓ : F → F defined by Tℓ = π0(Lε,ℏ)ℓ provided one chooses
ε1, ε2 ∈ C so that
ε = ε1 + ε2, (1.12)
ℏ = −ε1ε2. (1.13)
We will show that one can also discover partitions and Jack polynomials in the spectrum of L
itself.
Theorem 1.4 (main result). Let L be the Nazarov–Sklyanin Lax operator in (1.5). Assume
that ε ∈ R and ℏ > 0. Choose non-zero ε1, ε2 ∈ C parametrizing ε, ℏ by the formulas (1.12)
and (1.13).
(1) The degree n component F [w]n of the graded ring F [w] in (1.3) has a cyclic decomposition
F [w]n =
⊕
|λ|=n
Z(jλ,L) (1.14)
as a direct sum of finite-dimensional L-cyclic subspaces Z(jλ,L) indexed by partitions λ
of size n and generated by Jack polynomials jλ ∈ F defined in (1.7).
(2) The restrictions of L to the subspaces Zλ = Z(jλ,L) in (1.14) all have simple spectrum
spec
(
L
∣∣
Zλ
)
=
{
[s] : s ∈ Aλ
}
(1.15)
given by the anisotropic contents (1.11) of the addable corner set Aλ defined above in (1.9).
In particular, each cyclic subspace Zλ in (1.14) has dimension dimZλ = |Aλ|.
(3) As a distinguished vector space basis ψsλ for the cyclic spaces Z(jλ,L) in (1.14), one can
choose the unique polynomials ψsλ ∈ F [w] in the variables w, V1, V2, . . . indexed by addable
corners s ∈ Aλ which are both eigenfunctions of L with eigenvalues [s] as in (1.15), namely
Lψsλ = [s]ψsλ,
and which project to the Jack polynomial jλ for all s ∈ Aλ upon setting w = 0, namely
π0ψ
s
λ = jλ, (1.16)
Spectral Theory of the Nazarov–Sklyanin Lax Operator 5
where π0 : F [w] → F is evaluation at w = 0. In this normalization,
jλ =
∑
s∈Aλ
τ sλψ
s
λ, (1.17)
where τ sλ > 0 are weights of the anisotropic transition measure [16] defined explicitly by
τ sλ =
∏
r∈R+
λ
[s− r]∏
t∈Aλ\{s}
[s− t]
(1.18)
for Rλ the removable corner set (1.10) and R+
λ = Rλ + (1, 1) the set of outer corners.
Before we proceed, let us illustrate our Theorem 1.4 in degree 1. For λ = (1), j1 = V1 and
Z1 = Z(j1,L) is two-dimensional with basis {V1, w}. In this basis, L acts by the matrix
L|Z1 =
[
0 1
ℏ ε
]
. (1.19)
If ε ∈ R and ℏ > 0, (1.19) has distinct real eigenvalues ε1 and ε2 determined up to permutation
by the equations ε = ε1+ε2 and ℏ = −ε1ε2 characterizing the trace and determinant of (1.19). In
this way, the parametrization (1.12) and (1.13) above is visible already in degree 1. Using (1.11),
these simple eigenvalues ε1 = [(1, 0)], ε2 = [(0, 1)] are the contents of the two addable corners
A1 = {(0, 1), (1, 0)} of the unique partition λ = (1) of size 1. Moreover, as L|Z1-eigenfunctions
we can choose
ψ
(1,0)
1 = V1 + ε1w, ψ
(0,1)
1 = V1 + ε2w
so that at w = 0 one recovers ψs1
∣∣
w=0
= j1 = V1 the Jack polynomial for both s ∈ A1.
1.6 Organization of the paper
In Section 2, we recall two results of Nazarov–Sklyanin [32]. In Appendix A, we collect standard
results for cyclic spaces of self-adjoint operators in a self-contained appendix. In Section 3, we
prove Theorem 1.4 using results from Section 2 and Appendix A. In Section 4, we present the
polynomial eigenfunctions ψsλ of L explicitly in degrees |λ| ≤ 3, state an integrality conjecture
for ψsλ, and prove a principal specialization formula for ψsλ which implies a special case of our
integrality conjecture. In Section 5, we discuss our proof and comment on related results in the
literature.
2 Review of two results of Nazarov–Sklyanin
In this section, we recall two results for the hierarchy Tℓ = π0Lℓ due to Nazarov–Sklyanin [32].
2.1 Definition of the Nazarov–Sklyanin hierarchy
Recall F [w] = C[w, V1, V2, . . .] in (1.3) and the Lax operator L in (1.5). Let π0 be the projection
to F = C[V1, V2, . . .] defined by setting w = 0.
Definition 2.1. The Nazarov–Sklyanin hierarchy is the set of operators Tℓ = π0Lℓ : F → F .
6 R. Mickler and A. Moll
Here Tℓ is the top-left entry of the ℓth power of L in (1.6). Recall V−k = ℏk ∂
∂Vk
in (1.2).
For ℓ = 3,
T3 =
∞∑
k1,k2=1
V−k1−k2Vk1Vk2 +
∞∑
k1,k2=1
V−k1V−k2Vk1+k2 + ε
∞∑
k=1
kVkV−k (2.1)
is the Hamiltonian of the quantum Benjamin–Ono equation on the torus discussed in [32, Sec-
tion 1]. When ε = 0, (2.1) is the well-known cut and join operator discussed, e.g., in [6,
Section 1.2].
2.2 Commutativity and spectral theorem for the hierarchy
Theorem 2.2 (Nazarov–Sklyanin [32]). Choose ε1, ε2 ∈ C which parametrize ε ∈ R and ℏ > 0
by the formulas (1.12) and (1.13). For u ∈ C \ R, consider the bounded transfer operators
T (u) = π0(u− L)−1 : F → F, (2.2)
which encode all unbounded operators in the hierarchy Tℓ = π0Lℓ as coefficients of u−ℓ−1.
(1∨) The transfer operators (2.2) commute in F for distinct values of u and are simultaneously
diagonalized on Jack polynomials jλ in (1.7) with eigenvalues Tλ(u) ∈ C as in
T (u) jλ = Tλ(u) jλ. (2.3)
(2∨) The eigenvalues Tλ(u) ∈ C in (2.3) are determined by the anisotropic contents (1.11) of
the elements of the Young diagram λ in (1.8) and their shifts by the product formula
Tλ(u) = u−1 ·
∏
s∈λ
(u− [s+ (0, 0)])(u− [s+ (1, 1)])
(u− [s+ (1, 0)])(u− [s+ (0, 1)])
.
Equivalently, if Aλ and Rλ are the addable and removable corner sets in (1.9) and (1.10),
Tλ(u) =
∏
r∈R+
λ
(u− [r])∏
s∈Aλ
(u− [s])
, (2.4)
where R+
λ = Rλ + (1, 1) is the set of outer corners.
Our presentation of the results of Nazarov–Sklyanin [32] above differs from what appears
in [32]. For a proof that Theorem 2.2 is equivalent to the original formulation in Nazarov–
Sklyanin [32], see [27, Section 8.2] and the discussion in [29, Section 4.3.3]. We label their two
results (1∨) and (2∨) since we will now derive (1) and (2) in our Theorem 1.4 by proving that
(1∨) ∧ (2∨) ⇒ (1) ∧ (2).
3 Proof of main result
In this section we prove our Theorem 1.4. Throughout, we assume that jλ in (1.7) and L in (1.5)
have parameters ε ∈ R and ℏ > 0 satisfying ε = ε1 + ε2 as in (1.12) and ℏ = −ε1ε2 as in (1.13).
Spectral Theory of the Nazarov–Sklyanin Lax Operator 7
3.1 Jack–Lax cyclic spaces are finite-dimensional
We now recall the definition of certain cyclic spaces first considered in [26, Section 5.2.4].
Definition 3.1. The Jack–Lax cyclic space Zλ is the L-cyclic space generated by jλ
Zλ = Z(jλ,L), (3.1)
i.e., the subspace of F [w] spanned by
{
jλ,Ljλ,L2jλ,L3jλ, . . .
}
.
A priori, we do not know dimZλ for Zλ in (3.1). To apply results from Appendix A, we need
dimZλ < ∞. The fact that Jack–Lax cyclic spaces Zλ are finite-dimensional follows from the
next result.
Lemma 3.2. The graded components F [w]n of F [w] have dimension
dimF [w]n =
n∑
l=0
p(l), (3.2)
where p(l) is the number of partitions of l.
Proof. Write µ =
(
1d12d2 · · ·
)
to denote a partition µ of size |µ| =
∑∞
k=1 kdk with dk parts
of size k. For such µ, let Vµ = V d1
1 V d2
2 · · · . The ring F in (1.1) has graded components Fn
with dimFn = p(n) since {Vµ : |µ| = n} is a basis of Fn. Similarly, F [w] in (1.3) has graded
components F [w]n with basis
{
Vµw
l : |µ|+ l = n
}
so (3.2) holds. ■
For |λ| = n, Jacks are homogeneous jλ ∈ Fn hence jλ ∈ F [w]n. Since L preserves de-
gree, L preserves F [w]n, so Zλ = Z(jλ,L) ⊂ F [w]n and (3.2) implies dimZλ <∞.
3.2 Projections of Jack–Lax cyclic spaces
Next, we show that Jack polynomials diagonalizing the Nazarov–Sklyanin hierarchy in (1∨) of
Theorem 2.2 implies a remarkable property of Zλ.
Lemma 3.3. For any partition λ, the projection of the Jack–Lax cyclic subspace in (3.1) to
F ⊂ F [w] is one-dimensional and spanned by the Jack polynomial jλ:
π0Z(jλ,L) = Cjλ. (3.3)
Proof. For any η ∈ Z(jλ,L), there is a polynomial P so η = P (L)jλ. As a consequence, π0η is
a finite linear combination of π0Lℓjλ indexed by ℓ ≥ 0. Expanding the resolvent (u − L)−1 =∑∞
ℓ=0 u
−ℓ−1Lℓ, part (1∨) of Theorem 2.2 implies all π0Lℓjλ ∈ Cjλ. ■
We emphasize that this short proof does not require the explicit formula (2.4) for the eigen-
values of the Nazarov–Sklyanin hierarchy in (2∨) of their Theorem 2.2, only commutativity
in (1∨).
3.3 The Lax operator in Zλ is self-adjoint for the extended Hall inner product
Let ⟨·, ·⟩ℏ be the inner product on F [w] in which Vµw
l for µ =
(
1d12d2 · · ·
)
are orthogonal with∥∥Vµwl∥∥2ℏ =
∞∏
k=1
(ℏk)dkdk!
By our discussion in Section 1.2, the restriction of ⟨·, ·⟩ℏ to F is the α-Hall inner product
from [23, 44].
Lemma 3.4. If ε ∈ R, ℏ > 0, the restriction of L in (1.5) to any F [w]n or Zλ is self-adjoint.
Proof. Follows from the definition of L in (1.5), the fact that V †
±k = V∓k in (1.2) are mutual
adjoints for ⟨·, ·⟩ℏ, and that the differential operator w∂w is self-adjoint for ⟨·, ·⟩ℏ. ■
8 R. Mickler and A. Moll
3.4 Orthogonality of Jack–Lax cyclic spaces
We now use the projection formula in (3.3) and the orthogonality of Jack polynomials for the
Hall inner product on F [14, 23, 44] to prove that the Jack–Lax cyclic spaces themselves are
orthogonal for the extended Hall inner product.
Lemma 3.5. If λ, γ are distinct partitions of n, Zλ and Zγ are orthogonal in (F [w]n, ⟨·, ·⟩ℏ).
Proof. If η ∈ Zλ = Z(jλ,L), since jλ is cyclic, there is a polynomial P so that η = P (L)jλ.
Similarly, if φ ∈ Zγ = Z(jγ ,L), there is a polynomial Q so that φ = Q(L)jγ . Then
⟨φ, η⟩ℏ = ⟨Q(L)jγ , P (L)jλ⟩ℏ = ⟨jγ , Q(L)P (L)jλ⟩ℏ = 0
since L† = L is self-adjoint in F [w]n, π0Q(L)P (L)jλ ∈ Cjλ by (3.3), and ⟨jγ , jλ⟩ℏ = 0. ■
3.5 Spectrum of the Lax operator in Jack–Lax cyclic spaces
We now show that spectrum of the hierarchy Tℓ = π0Lℓ found by Nazarov–Sklyanin in (2∨) of
their Theorem 2.2 determines that of L in Zλ. Precisely, the eigenvalues of L|Zλ
are simple and
given by the anisotropic contents [s] of addable boxes s ∈ Aλ.
Proof of part (2) of Theorem 1.4. Since dimZλ < ∞ by Lemma 3.2, we may apply the
general linear algebra results in the appendix Section A.2 to the case W = Zλ, J = jλ, and
L = L|Zλ
. From this perspective, formula (2.4) in (2∨) of Nazarov–Sklyanin’s Theorem 2.2 gives
an exact formula for the Titchmarsh–Weyl function T (u) in (A.5), so (A.9) implies (1.15). ■
3.6 Cyclic decomposition of F [w]n into Jack–Lax cyclic spaces Zλ
Proof of part (1) of Theorem 1.4. In any partition λ, the number of addable and removable
corners always differ exactly by 1: (1.9) and (1.10) satisfy
|Aλ| = |Rλ|+ 1.
By induction in n ≥ 0, it is straightforward to prove the combinatorial identities
n∑
l=0
p(l) =
∑
|λ|=n
|Aλ| =
∑
|ν|=n+1
|Rν |, (3.4)
where p(l) is the number of partitions of l. Indeed, (λ, s) 7→ (λ ∪ {s}, s) defines a bijection⋃
|λ|=n
Aλ →
⋃
|ν|=n+1
Rν .
By (3.2) and (3.4), dimF [w]n =
∑
|λ|=n|Aλ|. By formula (1.15) of part (2) of Theorem 1.4 proved
above, since L|Zλ
has simple spectrum indexed by Aλ, dimZλ = |Aλ|. By the orthogonality in
Lemma 3.5,
⊕
|λ|=n Zλ ⊆ F [w]n. However, since we’ve shown dimF [w]n =
∑
|λ|=n dimZλ, we
must have equality
F [w]n =
⊕
|λ|=n
Zλ,
which proves (1.14). ■
Spectral Theory of the Nazarov–Sklyanin Lax Operator 9
3.7 Normalization of eigenfunctions of L
At last, we can complete the proof of Theorem 1.4.
Proof of part (3) of Theorem 1.4. By Lemma 3.4, we may apply the general linear algebra
results in the appendices Appendices A.4 and A.5 to the case W = Zλ, J = jλ, L = L|Zλ
, and
⟨·, ·⟩ = ⟨·, ·⟩ℏ. Throughout Appendix A, dimW = m+1 and we index the L-eigenvector basis ψ(i)
by a superscript i ∈ {0, 1, . . . ,m}. By the formula (1.15) of part (2) of Theorem 1.4 proven above,
dimZλ = |Aλ|, so we may instead index the L-eigenvector basis ψsλ of Zλ by superscripts s ∈ Aλ.
In this way, our desired normalization condition (1.16) on ψsλ(w, V1, V2, . . .) follows immediately
from (A.15). Likewise, the identities (1.17) and (1.18) follow directly from (A.35) and (A.23)
using the aforementioned identification of (2.4) and (1.15). This completes the proof of Theo-
rem 1.4. ■
Remark 3.6. While we prove (1∨) ∧ (2∨) ⇒ (1) ∧ (2), it is not hard to use the same general
framework from Appendix A to prove the converse (1)∧(2) ⇒ (1∨)∧(2∨), so our Theorem 1.4 is
in fact equivalent to results from [32]. That being said, neither the eigenvalues nor eigenfunctions
of L are discussed explicitly in [32]. We give brief comments on the methods of proof in [32] in
Section 5.
4 On the eigenfunctions of the Nazarov–Sklyanin Lax operator
In this section, we consider the ψsλ ∈ F [w] of L normalized by the condition π0ψ
s
λ = jλ in part (3)
of our Theorem 1.4. In Section 4.1, we present ψsλ explicitly in degrees |λ| ≤ 3 . In Section 4.2,
we discuss symmetries of ψsλ under ε1 ↔ ε2. In Section 4.3, we state an integrality conjecture
for ψsλ. In Section 4.4, we verify a special case of this conjecture by using Stanley’s principal
specialization formula [44] for ψsλ|w=0 = jλ to derive a similar formula for ψsλ|w=1.
4.1 Examples of eigenfunctions of L
Let jλ be the Jack polynomial in (1.7). Recall
j∅ = 1,
j1 = V1,
j2 = V 2
1 + ε1V2, (4.1)
j1,1 = V 2
1 + ε2V2, (4.2)
j3 = V 3
1 + 3ε1V1V2 + 2ε21V3,
j1,2 = V 3
1 + (ε1 + ε2)V1V2 + (ε1ε2)V3, (4.3)
j1,1,1 = V 3
1 + 3ε2V1V2 + 2ε22V3.
The eigenfunctions ψsλ of L are polynomials in C[w, V1, V2, . . .], normalized by π0ψ
s
λ = jλ, and
in degrees |λ| ≤ 3 are
ψ
(0,0)
∅ = j∅,
ψ
(0,1)
1 = j1 + ε2w,
ψ
(1,0)
1 = j1 + ε1w,
ψ
(0,1)
2 = j2 + ε2V1w + ε1ε2w
2,
ψ
(2,0)
2 = j2 + 2ε1V1w + 2ε21w
2,
10 R. Mickler and A. Moll
ψ
(0,2)
1,1 = j1,1 + 2ε2V1w + 2ε22w
2,
ψ
(1,0)
1,1 = j1,1 + ε1V1w + ε1ε2w
2,
ψ
(0,1)
3 = j3 + ε2V
2
1 w + ε1ε2V2w + 2ε1ε2V1w
2 + 2ε21ε2w
3,
ψ
(3,0)
3 = j3 + 3ε1V
2
1 w + 3ε21V2w + 6ε21V1w
2 + 6ε31w
3,
ψ
(0,2)
1,2 = j1,2 + 2ε2V
2
1 w + ε1ε2V2w + ε2(ε1 + 2ε2)V1w
2 + 2ε1ε
2
2w
3,
ψ
(1,1)
1,2 = j1,2 + (ε1 + ε2)V
2
1 w +
(
ε21 − ε1ε2 + ε22
)
V2w + 3ε1ε2V1w
2 + ε1ε2(ε1 + ε2)w
3,
ψ
(2,0)
1,2 = j1,2 + 2ε1V
2
1 w + ε1ε2V2w + ε1(ε2 + 2ε1)V1w
2 + 2ε21ε2w
3, (4.4)
ψ
(0,3)
1,1,1 = j1,1,1 + 3ε2V
2
1 w + 3ε22V2w + 6ε22V1w
2 + 6ε32w
3,
ψ
(1,0)
1,1,1 = j1,1,1 + ε1V
2
1 w + ε1ε2V2w + 2ε1ε2V1w
2 + 2ε1ε
2
2w
3.
Readers more familiar with the α conventions in [14, 23, 44] may set ε1 = α and ε2 = −1 so
that jλ = Jλ in (1.7) and Vk = pk. For example, if Jλ, mλ, and pµ are the Jack, monomial, and
power sum symmetric functions in [23, 44], J1,1 = 2m1,1 = p21−p2 and J2 = (1+α)m2+2m1,1 =
p21+αp2 gives (4.1) and (4.2) by (1.7). At ε1 = α, ε2 = −1, (4.3) is J1,2 = p31+(α−1)p1p2−αp3
and so (4.4) is ψ
(2,0)
1,2 = p31 + (α− 1)p1p2 − αp3 + 2αp21w − αp2w + α(−1 + 2α)p1w
2 − 2α2w3.
4.2 Symmetries of eigenfunctions of L
As is evident in the examples above, Jack polynomials jλ are invariant under simultaneous
permutation ε1 ↔ ε2 and transposition λ ↔ λ′. This symmetry is easier to see in the modern
conventions [22, 40] in Nekrasov variables ε1, ε2 [36, 37]. Since the Lax operator L itself depends
only on ε = ε1+ε2 and ℏ = −ε1ε2 which are both invariant under ε1 ↔ ε2, its eigenfunctions ψ
s
λ
are invariant under simultaneous permutation ε1 ↔ ε2, transposition λ ↔ λ′, and reflection
s↔ s′, i.e., (N1, N2) ↔ (N2, N1) in N2.
4.3 Integrality conjecture for eigenfunctions of L
Although Jack polynomials Jλ are referred to as the “integral form” Jack polynomials in [23, 44],
the fact that their coefficients in the power sum basis pµ are polynomials in α with integer
coefficients was not proven until [19]. To state this result for jλ as in (1.7), let χλ,µ(ε1, ε2)
denote the coefficient of Vµ = V d1
1 V d2
2 · · · in
jλ(V1, V2, . . . ; ε1, ε2) =
∑
µ
χλ,µ(ε1, ε2)Vµ. (4.5)
These χλ,µ(ε1, ε2) are known as unnormalized Jack characters [5, 20] since they are deformations
of symmetric group characters which appear in the case ε = ε1 + ε2 = 0 of Schur polynomials.
Theorem 4.1 (Lapointe–Vinet [19]). For any λ, µ, in (4.5) one has χλ,µ(ε1, ε2) ∈ Z[ε1, ε2].
Consider the generalized Kostka coefficients κγλ(α) defined by the expansion Jλ=
∑
γ κ
γ
λ(α)m̃γ
of the Jack polynomial Jλ in the basis m̃γ of augmented monomial symmetric functions [23].
Lapointe–Vinet [19] proved κγλ(α) ∈ Z[α] which implies Theorem 4.1 since m̃λ are polynomials
in pµ with integer coefficients [23]. For the same reason, Theorem 4.1 is also a consequence of
the proof of the Macdonald–Stanley conjecture κγλ(α) ∈ Z≥0[α] by Knop–Sahi [17].
We conjecture that an analog of Theorem 4.1 holds for the eigenfunctions ψsλ of the operator L.
Let χs,lλ,µ(ε1, ε2) denote the coefficient of Vµw
l in
ψsλ(w, V1, V2, . . . ; ε1, ε2) =
∑
l
∑
µ
χs,lλ,µ(ε1, ε2)Vµw
l. (4.6)
Spectral Theory of the Nazarov–Sklyanin Lax Operator 11
Conjecture 4.2. For any λ, s, µ, and l in (4.6) one has χs,lλ,µ(ε1, ε2) ∈ Z[ε1, ε2].
This integrality conjecture manifestly holds for the low degree examples of ψsλ presented
in Section 4.1. For example, in the expansion of ψ
(2,0)
1,2 in (4.4), the coefficient of V1w
2 is
χ
(2,0),2
(1,2),(1) = ε1(ε2 + 2ε1). Note that the operator L itself in (1.5) and (1.6) has coefficients
in Z[ε1, ε2]. In addition, by part (2) of our Theorem 1.4, L has eigenvalues in Z≥0[ε1, ε2] since
spec(L) =
{
[s] : s ∈ N2
}
.
We can verify Conjecture 4.2 for the lowest and highest possible powers l of w:
1. If l = 0 so Vµw
l = Vµ, the normalization condition π0ψ
s
λ = jλ implies χs,0λ,µ is the Jack
character χλ,µ in (4.5), so χs,0λ,µ ∈ Z[ε1, ε2] holds as a consequence of Theorem 4.1.
2. If l = |λ| so Vµwl = w|λ|, Proposition 4.3 below implies the content product formula
χ
s,|λ|
λ,0 =
∏
t∈(λ∪s)×
[t], (4.7)
where (λ ∪ s)× = λ ∪ s \ {(0, 0)}. Since [t] are defined by (1.11), χ
s,|λ|
λ,0 ∈ Z≥0[ε1, ε2].
In the next section, we will derive this formula (4.7) for the top degree coefficient of the
eigenfunctions of L. To illustrate this result, consider the partition λ = (1, 2) of size 3 and its
addable corner (2, 0) ∈ A1,2. While the elements of the Young diagram λ = {(0, 0), (0, 1), (1, 0)}
have contents 0, ε2, ε1, if we consider the addable corner s = (2, 0), then the product in (4.7)
is over (λ ∪ s)× = {(0, 1), (1, 0), (2, 0)} have contents ε2, ε1, 2ε1 whose product 2ε21ε2 is the top
degree coefficient of w3 in the formula (4.4) for ψ
(2,0)
1,2 .
4.4 Principal specializations of eigenfunctions of L
We now derive the closed formula (4.7) for the top coefficient of w|λ| in the expansion of ψsλ as
a product of the contents in (λ ∪ s)×. To do so, we derive two principal specialization formulas
for ψsλ.
Proposition 4.3 (principal specializations). For λ with |λ| = n, let ψsλ(w, V1, V2, . . . , Vn) be the
polynomial eigenfunction of L with eigenvalue [s] normalized by π0ψ
s
λ = jλ. Fix z ∈ C. Then
ψsλ(0, z, z, . . . , z) =
∏
t∈λ
(z + [t]), (4.8)
ψsλ(1, z, z, . . . , z) =
∏
t∈(λ∪s)×
(z + [t]) (4.9)
are two content product formulae for principal specializations at w = 0 and w = 1, respectively.
Proof. Since π0ψ
s
λ = jλ, the w = 0 principal specialization formula (4.8) is an immediate
consequence of the well-known principal specialization result for Jλ due to Stanley [44]. To
prove (4.9), let L ≡ Lε,ℏ be the Lax operator in (1.5), with the choices (1.12) and (1.13). The
eigenvalue equation is
Lε,ℏψsλ = [s]ψsλ. (4.10)
Taking π0 of both sides of (4.10), one can use π0ψ
s
λ = jλ on the right side, then observe that we
can replace π0Lε,ℏ by π0L0,0 on the left side. Indeed, by inspecting the terms in (1.5) with ℏ
and ε, one has π0Lε,ℏη = π0L0,0η for any η ∈ F [w]. As a consequence,
π0L0,0ψ
s
λ = [s]jλ. (4.11)
12 R. Mickler and A. Moll
Since π0L0,0 =
∑∞
k=1 Vkπ0w
−k, if we expand our eigenfunction as
ψsλ =
n∑
l=0
(πlψ
s
λ)w
l (4.12)
with each πlψ
s
λ ∈ F = C[V1, V2, . . .] homogeneous of degree n− l, (4.11) reads
V1π1ψ
s
λ + V2π2ψ
s
λ + · · ·+ Vnπnψ
s
λ = [s]jλ(V1, V2, . . . , Vn). (4.13)
On the other hand, the difference ψsλ(1)− ψsλ(0) between ψ
s
λ at w = 1 and w = 0 is
π1ψ
s
λ + π2ψ
s
λ + · · ·+ πnψ
s
λ = ψsλ(1, V1, V2, . . . , Vn)− jλ(V1, V2, . . . , Vn). (4.14)
Equation (4.13) is an equality of polynomials in Vk which are homogeneous of degree n, whereas
equation (4.14) is an equality of polynomials in Vk which are not homogeneous. If we evalu-
ate Vk = z for all 1 ≤ k ≤ n in both (4.13) and (4.14) and assume z ̸= 0, the left sides of each
differ by an overall factor of z, so we can conclude that
ψsλ(1, z, z, . . . , z) =
(
1 +
[s]
z
)
· jλ(z, z, . . . , z). (4.15)
Substituting Stanley’s result (4.8) into (4.15) yields (4.9). ■
Using the notation πlψ
s
λ for the coefficient of wl in ψsλ as in (4.12), setting z = 0 in (4.9)
yields the content product formula for the non-vanishing top degree coefficient πnψ
s
λ presented
in (4.7).
5 Comments and comparison with previous results
In this section, we discuss the larger context of our Theorem 1.4. In Section 5.1, we comment on
the classical Lax operator L of Nakamura [31] and Bock–Kruskal [3] which served as the main
inspiration for the construction of the Lax operator L by Nazarov–Sklyanin [32]. In Section 5.2,
we compare the appearance of Jack polynomials at w = 0 in π0ψ
s
λ = jλ of our Theorem 1.4
to a recent result of Gérard–Kappeler [9] for π0Φl of the classical L eigenfunctions Φl. In
Section 5.3, we comment on developments since [32]. Finally, in Section 5.4 we compare the
eigenvalue equation Lψsλ = [s]ψsλ in our Theorem 1.4 indexed by pairs (λ, λ∪s) of partitions which
differ by s to two other instances of this equation in the literature: (i) in type A representation
theory [25, 38] at ε = 0 and (ii) in the equivariant cohomology of nested Hilbert schemes of
points in C2 [22, 30, 37, 40].
5.1 Comments on the Nakamura–Bock–Kruskal classical Lax operator
In [32, Sections 1 and 2], Nazarov–Sklyanin discuss how they thought to introduce their Lax
operator L in (1.5) and (1.6) to the study of Jack polynomials jλ. They did so as a natural
consequence of two observations. On the one hand, Jack polynomials have been long known to be
eigenfunctions of the Hamiltonian (2.1) of the quantum Benjamin–Ono equation on the torus –
see [32, Section 2] and [27] and references therein. On the other hand, the classical Benjamin–
Ono equation admits a Lax pair due to Nakamura [31] and Bock–Kruskal [3]. When the spatial
geometry is a torus, the Lax operator L from [3, 31] takes the form in (5.1) below. Assume
u ∈ L2(T,R) is a real-valued distribution with Fourier modes uk ∈ C satisfying
∑∞
k=1 |uk|2 <∞,
u−k = uk, and u0 = 0. Then with π as in (1.4) and ∂w = ∂
∂w ,
L = ε0w∂w +
∞∑
k=1
ukw
k +
∞∑
k=1
ukπw
−k (5.1)
Spectral Theory of the Nazarov–Sklyanin Lax Operator 13
is the Nakamura–Bock–Kruskal Lax operator for the classical Benjamin–Ono equation on the
torus with dispersion coefficient ε0 ∈ R [3, 31]. This L is partially-defined on C[w] and essentially
self-adjoint with respect to the inner product ⟨·, ·⟩ on C[w] in which wl are an orthonormal basis.
Just like (1.5), only the first terms involve the projection π in (1.4). However, unlike (1.5), the
second infinite sum over ukw
k applied to a polynomial in C[w] will not yield a polynomial if
infinitely-many uk = u−k ̸= 0. As in Section 1.1, we can write L as a N× N matrix
L =
0ε0 u1 u2 u3 u4 · · ·
u1 1ε0 u1 u2 u3
. . .
u2 u1 2ε0 u1 u2
. . .
u3 u2 u1 3ε0 u1
. . .
u4 u3 u2 u1 4ε0
. . .
...
. . .
. . .
. . .
. . .
. . .
. (5.2)
With these two observations in mind, Nazarov–Sklyanin [32] realized that canonical quantization
via (1.2)
(uk, u−k) −→ (Vk, V−k) (5.3)
performed directly in the matrix elements of L in (5.2) yields a well-defined L : F [w] → F [w]
in (1.6) whose powers are well defined without normal ordering. Equivalently, Nazarov–Sklyanin
realized that the classical field u(x) can be directly replaced in L by the affine ĝl1-current at
level ℏ to get L. For a discussion of (5.3) from the point of view of geometric quantization,
see [27].
5.2 Comparison to Gérard–Kappeler’s action-angle coordinates
This paper was inspired by recent spectral analysis [7, 8, 9, 10, 11, 28] of the Nakamura–Bock–
Kruskal Lax operator L in (5.2). At ε0 = 0, (5.2) is a Toeplitz operator whose spectrum has
been studied for over a century [4, 43]. At ε0 ̸= 0, the spectrum of L is simple [9, 28]. In [27, 28],
the second author proved that Gérard–Kappeler [9] independently found the classical limit of
the quantum hierarchy of Nazarov–Sklaynin [32]. We can now make a second comparison to [9].
On the one hand, a main result of [9] is that the constant terms of the classical L eigenfunctions
determine the action-angle coordinates of the classical Benjamin–Ono equation on the torus. On
the other hand, in (3) of our Theorem 1.4, we proved that the constant terms of the quantum L
eigenfunctions are Jack polynomials. We hope that the many structural results in Gérard–
Kappeler [9] admit explicit quantizations which can shed further light on the objects in this
paper.
5.3 Comments on developments since the work of Nazarov–Sklyanin
The study of Tℓ = π0Lℓ and intricate proof of (1∨) and (2∨) in Theorem 2.2 by Nazarov–
Sklyanin [32] draws on their prior work [33] on the N → ∞ limits of the Sekiguchi–Debiard
operators A
(1)
N , A
(2)
N , . . . for Jack polynomials [23]. In [34, 35], Nazarov–Sklyanin generalized
their results to the case of Macdonald polynomials. In [35], they mention that their Theorem 2.2
was independently discovered in a different form by Sergeev–Veselov [41, 42]. The relationship
between the eigenfunctions ψsλ considered in this paper, the framework in [41, 42], and the
quantum Baker–Achiever function in [32] deserves further study.
14 R. Mickler and A. Moll
5.4 Comparison to spectral theorems in representation theory and geometry
Our Theorem 1.4 is not the only appearance of contents of addable corners as eigenvalues of
operators. In the case ε1 = −ε2 = ε so that ε = 0, ℏ = ε2, and α = 1, our eigenvalue equation
degenerates to
L0,ℏψ
s
λ = [s]ψsλ,
where L0,ℏ is a block Toeplitz operator, ψsλ are polynomials which recover Schur polynomials
at w = 0, and the anisotropic content is ε times the usual content. In this case, the content
of a single addable corner is well known to describe the spectral theory at the heart of the
representation theory of glN and the symmetric group S(n), see, e.g., discussions in Molev–
Nazarov–Olshanski [25] and Okounkov–Vershik [38].
For generic ε = ε1 + ε2, it is also well known that contents of addable corners arise in the
equivariant cohomology of nested Hilbert schemes of points in the affine plane [22, 30, 37, 40].
Let X [n] be the Hilbert scheme of n points in C2, i.e., all ideals I ⊂ C[x1, x2] so C[x1, x2]/I is
a vector space of dimension n. Then
X [n,n+1] =
{(
I, Ĩ
)
∈ X [n] ×X [n+1] : I ⊂ Ĩ
}
is the nested Hilbert scheme of points in C2. The tautological line bundle L → X [n,n+1] has
fibers
L|
(I,Ĩ)
= Ĩ/I. (5.4)
The action of the torus T = C× × C× on C2 induces an action on both X [n] and X [n,n+1]. The
T-fixed points Iλ ∈ X [n] are monomial ideals indexed by partitions λ with |λ| = n. The T-fixed
points ηsλ ∈ X [n,n+1] are
ηsλ = (Iλ, Iλ∪s)
indexed by pairs (λ, s) with |λ| = n and s an addable corner in Aλ. For this reason, the
classes [Iλ] and [ηsλ] must be a basis in the T-equivariant cohomology rings of X [n] and X [n,n+1],
respectively. Moreover, if s = (N1, N2) ∈ N2, the torus action with characters ε1, ε2 on the
monomial xN1
1 xN2
2 ∈ C[x1, x2] is determined by the content [s] = ε1N1 + ε2N2 in (1.11). As
a consequence, the operator c1(L)∪− of cup product with the first Chern class of L in (5.4) must
act diagonally in the basis [ηsλ] and satisfy the same eigenvalue equation as the Nazarov–Sklyanin
Lax operator in our Theorem 1.4. This observation suggests an extension of the isomorphism
identifying Jack polynomials jλ with [Iλ] [22, 30, 40] and T3 = π0L3 in (2.1) with the operator
in Lehn [21] to an identification of ψsλ ∈ F [w] introduced in this paper and [ηsλ].
A Cyclic spaces of self-adjoint operators
In this appendix, we recall several standard results for cyclic spaces of self-adjoint operators
following the treatment of Jacobi operators by Kerov [15, Section 6]. To streamline our proof in
Section 3, we present these results without choosing an orthogonal basis in which our self-adjoint
operators are tridiagonal.
A.1 Cyclic spaces W , W̃ associated to an operator L with cyclic vector J
Let W be a C-vector space with dimCW = m + 1 < ∞. Let L : W → W be a linear operator
(not necessarily self-adjoint) and J ∈ W a non-zero vector. Recall that W is a L-cyclic space
Spectral Theory of the Nazarov–Sklyanin Lax Operator 15
generated by J if the set of m+ 1 vectors
{
J, LJ, L2J, . . . , LmJ
}
is a basis for W . In this case,
we say that J is a cyclic vector for the operator L and write W = Z(J, L). In Proposition A.1
below, we recall a recipe which produces new cyclic spaces W̃ = Z
(
J̃ , L̃
)
from a given cyclic
spaceW = Z(J, L). Let CJ denote the span of J . Choose any complementary subspace W̃ ⊂W
of codimension 1 so that
W = CJ + W̃ . (A.1)
Let ΠJ and Π̃ be the canonical projections onto CJ and W̃ , respectively, with kerΠJ = W̃ and
ker Π̃ = CJ . For these canonical projections, one has IdW = ΠJ + Π̃. Let a ∈ C and J̃ ∈ W̃ be
uniquely determined by the expansion of the vector LJ with respect to (A.1) as in
LJ = aJ + J̃ , (A.2)
so that aJ = ΠJLJ and J̃ = Π̃LJ . Let L̃ : W̃ → W̃ be the linear operator with domain W̃
defined by
L̃ = Π̃L|
W̃
, (A.3)
the restriction of Π̃L : W →W to W̃ .
Proposition A.1. Consider W̃ , J̃ , L̃ in (A.1)–(A.3). If W = Z(J, L) is L-cyclic with cyclic
vector J , then the codimension 1 subspace W̃ is L̃-cyclic with cyclic vector J̃ , i.e., W̃ = Z
(
J̃ , L̃
)
.
Proof. For any η̃ ∈ W̃ , we need to find a polynomial P̃ so that
η̃ = P̃
(
L̃
)
J̃ .
Choose η ∈W such that Π̃η = η̃. Since W = Z(J, L), there is a polynomial P so that
η = P (L)J. (A.4)
Apply Π̃ to both sides of (A.4). One can then use the identity IdW = ΠJ + Π̃, (A.2), and (A.3)
to prove that Π̃LℓJ for ℓ = 0, 1, 2, . . . are in the span of J̃ , L̃J̃ , L̃2J̃ , . . . by induction on ℓ, thus
defining P̃ from P . ■
A.2 Two rational functions T , T̃ defined by L and J
For u ∈ C with both u − L and u − L̃ invertible, namely u ̸∈
(
spec(L) ∪ spec(L̃)
)
, let T (u)
and T̃ (u) be the unique scalars for which
ΠJ
1
u− L
J = T (u)J, (A.5)
ΠJL
1
u− L̃
J̃ = T̃ (u)J. (A.6)
Formulas (A.5) and (A.6) define two meromorphic functions T , T̃ of u ∈ C away from spec(L),
spec
(
L̃
)
, respectively.
Proposition A.2. Assume for simplicity that a = 0 in (A.2) so that LJ = J̃ . For all u ∈ C at
which T (u) ̸= 0 and T̃ (u) is well defined, the two functions T and T̃ in (A.5) and (A.6) satisfy
1
T (u)
+ T̃ (u) = u. (A.7)
In particular, the left-hand side of (A.7) is analytic in u ∈ C.
16 R. Mickler and A. Moll
Proof. The following argument is standard both in the theory of Jacobi matrices and in the
study of lattice paths, see, e.g., [15, Lemma 6.3.2]. However, this result is usually presented
in the context of a choice of a tridiagonal matrix representation of L and assuming that L is
self-adjoint, neither of which is necessary for the argument below. Since a = 0, u−2ΠJLJ = 0.
Expanding the resolvent using a geometric series, (A.5) becomes
T (u)J = u−1J +
∞∑
ℓ=2
u−ℓ−1ΠJL
ℓJ. (A.8)
For terms in (A.8) with ℓ ≥ 2, use (A.1) to insert ℓ − 1 copies of IdW = ΠJ + Π̃ between
the ℓ copies of L which appear in the expansion of Lℓ. This produces 2ℓ−1 terms of the form
ΠJΠ1LΠ2LΠ3 · · ·LΠℓ−1LJ with each Πi ∈
{
ΠJ , Π̃
}
. Group these terms by the minimum value
of i so that Πi = ΠJ . Since a = 0, we can assume i ≥ 2. After relabeling indices, one checks (A.8)
becomes T (u) = u−1 + u−1T̃ (u)T (u) with T̃ (u) as defined in (A.6). ■
The first function T in (A.5) is the ratio of the characteristic polynomials of L̃ and L.
Proposition A.3. The complex function T (u) defined for u ̸∈ spec(L) by (A.5) is
T (u) =
det
(
u− L̃
)
det(u− L)
(A.9)
a rational function with poles at spec(L) and zeroes at spec
(
L̃
)
.
Proof. Apply Cramer’s rule for ΠJ of solutions Φ(u) of the linear system (u− L)Φ = J . ■
As a consequence of (A.7) and (A.9), the second function T̃ is also a rational function.
Corollary A.4. If a = 0 in (A.2), the function T̃ (u) defined for u ̸∈ spec
(
L̃
)
by (A.6) is
T̃ (u) = u− det(u− L)
det
(
u− L̃
)
a rational function with poles at spec
(
L̃
)
.
A.3 The case of self-adjoint L with cyclic vector J
Introduce a Hermitian inner product ⟨·, ·⟩ on the finite-dimensional complex vector space W .
Adopt the convention ⟨η, αξ⟩ = α⟨η, ξ⟩ that the inner product is C-linear in the right-most entry.
We now revisit the results in Appendices A.1 and A.2 for L, L̃ and T , T̃ under the assumption
that L is a self-adjoint operator in the space (W, ⟨·, ·⟩).
Given a non-zero vector J ∈ W , let W̃ = (CJ)⊥ be the orthogonal complement of CJ .
Let ΠJ and Π̃ = Π⊥
J be the orthogonal projections so that (A.1) is an orthogonal decomposition
W = CJ ⊕W̃ = CJ ⊕ (CJ)⊥. By restriction, ⟨·, ·⟩ defines an inner product on W̃ = (CJ)⊥. If L
is self-adjoint in (W, ⟨·, ·⟩), L̃ = Π̃LΠ̃ in (A.3) is self-adjoint in (W̃ , ⟨·, ·⟩) since Π̃ is self-adjoint.
Under these assumptions, we can give alternative formulas for the rational functions T and T̃ :
⟨·, ·⟩
T (u) =
〈
J, (u− L)−1J
〉
⟨J, J⟩
(A.10)
T̃ (u) =
∥∥J̃∥∥2
∥J∥2
·
〈
J̃ , (u− L̃)−1J̃
〉〈
J̃ , J̃
〉 , (A.11)
Spectral Theory of the Nazarov–Sklyanin Lax Operator 17
using (A.5) and (A.6), and ΠJξ = ⟨J,ξ⟩
⟨J,J⟩J . Here (A.11) is T̃ (u) = AT∨(u) where A = ∥J̃∥2
∥J∥2 > 0
and
T∨(u) =
〈
J̃ , (u− L̃)−1J̃
〉〈
J̃ , J̃
〉 . (A.12)
As we discussed in [28], we write (A.11) without simplifying ∥J̃∥2
⟨J̃ ,J̃⟩
= 1 to clarify that T∨(u)
in (A.12), not T̃ (u) in (A.11), is the Stieltjes transform of a cotransition measure in Kerov [15, 16].
Note that T and T̃ are often called the Titchmarsh–Weyl functions of L and L̃ – see [15,
Section 6].
A.4 Spectral theorem for self-adjoint operators L, L̃ with cyclic vectors J , J̃
Assume that L and L̃ are self-adjoint in (W, ⟨·, ·⟩) and W̃ = (CJ)⊥ with cyclic vectors J and J̃
as in Appendix A.3.
Theorem A.5 (spectral theorem for cyclic spaces of self-adjoint operators). Let L, L̃ be the self-
adjoint operators in W , W̃ of dimensions m+1, m with cyclic vectors J , J̃ as in Appendix A.3.
Then L, L̃ have real eigenvalues σ(i), σ̃(i) which are simple and strictly interlacing
σ(m) < σ̃(m) < σ(m−1) < · · · < σ̃(2) < σ(1) < σ̃(1) < σ(0).
Since the spectrum is simple, there exist bases
{
ψ(i)
}m
i=0
,
{
ψ̃(i)
}m
i=1
of the cyclic spaces W , W̃ ,
which are L, L̃ eigenvectors
Lψ(i) = σ(i)ψ(i), (A.13)
L̃ψ̃(i) = σ̃(i)ψ̃(i) (A.14)
defined uniquely up to complex rescaling of each eigenvector. In fact, the normalizations of each
eigenvector can be fixed by the canonical constraints
ΠJψ
(i) = J, (A.15)
ΠJLψ̃
(i) = J. (A.16)
Proof. By the Cauchy interlacing theorem – e.g., [13, p. 242] or [15, Section 6] – it remains to
prove (A.15) and (A.16). For any basis of L eigenvectors ψ(i), since ΠJ projects onto CJ ,
ΠJψ
(i) = γ(i)J (A.17)
for some γ(i) ∈ C for all i ∈ {0, 1, . . . ,m}. We first argue that γ(i) = 0 is not possible in (A.17),
i.e., ΠJψ
(i) ̸= 0. Pairing (A.17) with J , using the inner product and the fact that the orthogonal
projection ΠJ is self-adjoint, one arrives at the identity
⟨J, ψ(i)⟩ = γ(i)∥J∥2. (A.18)
As a consequence, if we expand J in the ψs basis, formula (A.18) determines the coefficients
J =
m∑
i=0
(
γ(i)
∥J∥2∥∥ψ(i)
∥∥2
)
ψ(i). (A.19)
Since J is cyclic for L inW , its coefficients in the ψ(i) basis are non-zero for all i ∈ {0, 1, . . . ,m},
thus (A.19) guarantees γ(i) ̸= 0 for all i ∈ {0, 1, . . . ,m}. Since the eigenvalues σ(i) of L in W
18 R. Mickler and A. Moll
are distinct, the ψ(i) are uniquely determined up to overall factors in C×, so we can choose
these factors uniquely so γ(i) = 1 in (A.17), proving (A.15). By Proposition A.1, L̃ is cyclic in
W̃ = (CJ)⊥. Since L̃ = L̃†, the same argument above repeated in W̃ instead of W , this time
choosing γ̃(i) = ∥J∥2/∥J̃∥2 instead of γ(i) = 1, guarantees that if we consider J̃ ∈ W̃ in (A.2),
there is a unique basis ψ̃(i) of L̃-eigenvectors of W̃ with
Π
J̃
ψ̃(i) =
∥J∥2∥∥J̃∥∥2 J̃ . (A.20)
Pairing both sides of (A.20) with J̃ and using the fact that the orthogonal projection Π
J̃
to CJ̃
is self-adjoint, (A.20) implies〈
J̃ , ψ̃(i)
〉
= ∥J∥2. (A.21)
At the same time, since J̃ = Π̃LJ holds by (A.2), L† = L, Π̃ = Π̃†, and Π̃ψ̃(i) = ψ̃(i), we also
have 〈
J̃ , ψ̃(i)
〉
=
〈
Π̃LJ, ψ̃(i)
〉
=
〈
J, LΠ̃ψ̃(i)
〉
=
〈
J, Lψ̃(i)
〉
. (A.22)
Equating (A.21) and (A.22) implies ⟨J,Lψ̃(i)⟩
⟨J,J⟩ = 1 which is equivalent to (A.16). ■
A.5 Residues of T , T̃ and eigenvectors of self-adjoint L, L̃
Let T and T̃ be the rational functions associated to generic linear operators L with cyclic vector J
in Appendix A.2. Under the assumption that L and L̃ are self-adjoint as in Appendix A.3,
consider the residues
τ (i) = Resu=σ(i)T (u), (A.23)
τ̃ (i) = Resu=σ̃(i) T̃ (u) (A.24)
of T and T̃ at the simple eigenvalues σ(i) ∈ spec(L) and σ̃(i) ∈ spec
(
L̃
)
from Appendix A.4. We
first recall how these residues appear in the calculation of the squared norms
∥∥ψ(i)
∥∥2, ∥∥ψ̃(i)
∥∥2 of
the eigenvectors of L and L̃.
Proposition A.6. Let ψ(i), ψ̃(i) be the sets of eigenvectors of the self-adjoint operators L, L̃
normalized with respect to the cyclic vector J by the conditions (A.15) and (A.16). Let τ (i), τ̃ (i)
be the residues in (A.23) and (A.24) of the rational functions T (u), T̃ (u) in (A.10) and (A.11).
Then
τ (i)∥ψ(i)∥2 = ∥J∥2, (A.25)
τ̃ (i)∥ψ̃(i)∥2 = ∥J∥2. (A.26)
In particular, the residues τ (i) and τ̃ (i) in (A.23) and (A.24) are all positive.
Proof. For non-zero η ∈W , let Πη be the orthogonal projection onto Cη defined by Πηξ=
⟨η,ξ⟩
⟨η,η⟩η.
Since the orthogonal bases ψ(i), ψ̃(i) of W , W̃ determine resolutions of identities IdW , Id
W̃
, we
have
J =
m∑
i=0
Πψ(i)J =
m∑
i=0
〈
ψ(i), J
〉∥∥ψ(i)
∥∥2 ψ(i), (A.27)
Spectral Theory of the Nazarov–Sklyanin Lax Operator 19
J̃ =
m∑
i=1
Π
ψ̃(i) J̃ =
m∑
i=1
〈
ψ̃(i), J̃
〉∥∥ψ̃(i)
∥∥2 ψ̃(i). (A.28)
Substituting (A.27) and (A.28) into the inner product formulas (A.10) and (A.11) for T , T̃ ,
one may use the eigenvalue equations (A.13) and (A.14) and the orthogonality of eigenfunctions
to get
T (u) =
1
∥J∥2
m∑
i=0
1
u− σ(i)
·
∣∣〈J, ψ(i)
〉∣∣2∥∥ψ(i)
∥∥2 , (A.29)
T̃ (u) =
1
∥J∥2
m∑
i=0
1
u− σ̃(i)
·
∣∣〈J̃ , ψ̃(i)
〉∣∣2∥∥ψ̃(i)
∥∥2 . (A.30)
Taking residues of (A.29) and (A.30) at eigenvalues σ(i), σ̃(i) gives
τ (i) =
1
∥J∥2
·
∣∣〈J, ψ(i)
〉∣∣2∥∥ψ(i)
∥∥2 , (A.31)
τ̃ (i) =
1
∥J∥2
·
∣∣〈J̃ , ψ̃(i)
〉∣∣2∥∥ψ̃(i)
∥∥2 . (A.32)
Finally, since the normalization conditions (A.15) and (A.16) and L = L† imply〈
J, ψ(i)
〉
= ∥J∥2, (A.33)〈
J̃ , ψ̃(i)
〉
= ∥J∥2, (A.34)
substituting (A.33) and (A.34) into (A.31) and (A.32) completes the proof. ■
In fact, these residues relate the eigenvectors and cyclic vectors themselves, not just their
norms.
Proposition A.7. In the self-adjoint case, the residues τ (i) and τ̃ (i) of T and T̃ arise as
coefficients in the expression of the cyclic vectors J and J̃ as linear combinations of eigenvectors
of L and L̃
J =
m∑
i=0
τ (i)ψ(i), (A.35)
J̃ =
m∑
i=1
τ̃ (i)ψ̃(i) (A.36)
assuming that ψ(i) and ψ̃(i) are normalized by the conditions (A.15) and (A.16).
Proof. Formulas (A.25) and (A.33) imply τ (i) = ⟨ψ(i),J⟩
⟨ψ(i),ψ(i)⟩ . Hence (A.35), since
Πψ(i)ξ =
〈
ψ(i), ξ
〉〈
ψ(i), ψ(i)
〉ψ(i).
Similarly, combining formulas (A.26) and (A.34) implies (A.36). ■
We conclude with a ‘converse’ of Proposition A.7: the eigenvectors of L and L̃ themselves
can be determined directly from the cyclic vectors J and J̃ and the resolvents of L and L̃.
20 R. Mickler and A. Moll
Proposition A.8. Let ψ(i) and ψ̃(i) be eigenvectors of self-adjoint operators L and L̃ normal-
ized with respect to the cyclic vector J by the conditions (A.15) and (A.16) with corresponding
eigenvalues σ(i) and σ̃(i). Then, assuming a = 0 in (A.2) so that LJ = J̃ , the eigenvectors of L
and L̃ satisfy
ψ̃(i) =
1
L− σ̃(i)
J, (A.37)
ψ(i) = J +
1
σ(i) − L̃
J̃ . (A.38)
Proof. We first prove (A.37). To begin, we observe that
ΠJ
1
L− σ̃
J = 0 (A.39)
since (A.5) implies that the left-hand side of (A.39) is −T (σ̃t)J , but we know that T (σ̃t) = 0
since σ̃(i) ∈ spec
(
L̃
)
is necessarily a zero of the rational function T by Proposition A.3. As
a consequence of (A.39), Π̃ 1
L−σ̃(i)J = 1
L−σ̃(i)J which implies the eigenvalue relation
L̃
1
L− σ̃(i)
J = Π̃LΠ̃
1
L− σ̃(i)
J = Π̃
(
L− σ̃(i) + σ̃(i)
) 1
L− σ̃(i)
J = σ̃(i)
1
L− σ̃(i)
J.
Finally, since formula (A.39) also implies that ΠJL
1
L−σ̃(i)J = ΠJJ + σ̃(i)ΠJ
1
L−σ̃(i)J = J , we have
shown that 1
L−σ̃(i)J is an L̃-eigenvector with eigenvalue σ̃(i) satisfying (A.16). Since these prop-
erties uniquely characterize ψ̃(i), we have proven (A.37). Next, we prove (A.38) assuming a = 0
so that LJ = J̃ in (A.2). To begin, the resolvent of L̃ is well defined at u = σ(i) since in
Theorem A.5 we have seen that spec(L) ∩ spec
(
L̃
)
= ∅. As a consequence, J̃ ∈ W̃ implies that
1
σ(i)−L̃
J̃ ∈ W̃ , hence the vector on the right-hand side of (A.38) satisfies
ΠJ
(
J +
1
σ(i) − L̃
J̃
)
= J
the normalization condition (A.15). To verify that this vector is indeed an L-eigenvector with
eigenvalue σ(i), multiply by L then calculate its projections onto CJ and W̃ = (CJ)⊥. In the CJ
projection, we can simplify using our assumption a = 0 in (A.2), the definition (A.6) of T̃ (u),
and T̃
(
σ(i)
)
= σ(i) at the special value u = σ(i) due to Corollary A.4:
ΠJL
(
J +
1
σ(i) − L̃
J̃
)
= ΠJLJ +ΠJL
1
σ(i) − L̃
J̃ = aJ + T
(
σ(i)
)
J = σsJ. (A.40)
In the W̃ = (CJ)⊥ projection, using 1
σ(i)−L̃
J̃ ∈ W̃ , Π̃LΠ̃ = L̃, and L̃ = −
(
σ(i) − L̃
)
+ σ(i),
Π̃L
(
J +
1
σ(i) − L̃
J̃
)
= Π̃LJ + L̃
1
σ(i) − L̃
J̃ = J̃ − J̃ + σ(i)
1
σ(i) − L̃
J̃
= σ(i)
1
σ(i) − L̃
J̃ . (A.41)
Since (A.40) and (A.41) are σ(i) times the same projections of J + 1
σ(i)−L̃
J̃ , (A.38) follows. ■
Acknowledgements
The authors would like to thank the referees for many helpful comments and suggestions. We
would also like to express our sincere thanks to the staff at Darwin’s Ltd. coffee and sandwich
shop on Cambridge Street in Cambridge, MA for supporting our collaboration during the years
2014–2019.
Spectral Theory of the Nazarov–Sklyanin Lax Operator 21
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1 Introduction and statement of results
1.1 The Nazarov–Sklyanin Lax operator
1.2 Jack symmetric functions
1.3 Addable and removable corners of Young diagrams
1.4 The anisotropic content function
1.5 Spectral theorem for the Nazarov–Sklyanin Lax operator
1.6 Organization of the paper
2 Review of two results of Nazarov–Sklyanin
2.1 Definition of the Nazarov–Sklyanin hierarchy
2.2 Commutativity and spectral theorem for the hierarchy
3 Proof of main result
3.1 Jack–Lax cyclic spaces are finite-dimensional
3.2 Projections of Jack–Lax cyclic spaces
3.3 The Lax operator in Z_lambda is self-adjoint for the extended Hall inner product
3.4 Orthogonality of Jack–Lax cyclic spaces
3.5 Spectrum of the Lax operator in Jack–Lax cyclic spaces
3.6 Cyclic decomposition of F[w]_n into Jack–Lax cyclic spaces Z_lambda
3.7 Normalization of eigenfunctions of L
4 On the eigenfunctions of the Nazarov–Sklyanin Lax operator
4.1 Examples of eigenfunctions of L
4.2 Symmetries of eigenfunctions of L
4.3 Integrality conjecture for eigenfunctions of L
4.4 Principal specializations of eigenfunctions of L
5 Comments and comparison with previous results
5.1 Comments on the Nakamura–Bock–Kruskal classical Lax operator
5.2 Comparison to Gérard–Kappeler's action-angle coordinates
5.3 Comments on developments since the work of Nazarov–Sklyanin
5.4 Comparison to spectral theorems in representation theory and geometry
A Cyclic spaces of self-adjoint operators
A.1 Cyclic spaces W, tilde W associated to an operator L with cyclic vector J
A.2 Two rational functions T, tilde T defined by L and J
A.3 The case of self-adjoint L with cyclic vector J
A.4 Spectral theorem for self-adjoint operators L, tilde L with cyclic vectors J, tilde J
A.5 Residues of T, tilde T and eigenvectors of self-adjoint L, tilde L
References
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| id | nasplib_isofts_kiev_ua-123456789-212021 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T12:36:10Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mickler, Ryan Moll, Alexander 2026-01-22T09:21:20Z 2023 Spectral Theory of the Nazarov-Sklyanin Lax Operator. Ryan Mickler and Alexander Moll. SIGMA 19 (2023), 063, 22 pages 1815-0659 2020 Mathematics Subject Classification: 05E05; 33D52; 37K10; 47B35 arXiv:2211.01586 https://nasplib.isofts.kiev.ua/handle/123456789/212021 https://doi.org/10.3842/SIGMA.2023.063 In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator 𝓛: 𝐹[𝓌] → 𝐹[𝓌] where 𝐹 is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition 𝐹[𝓌] ≅ ⨁λ 𝑍(𝑗λ, 𝓛) into finite-dimensional 𝓛-cyclic subspaces in which Jack polynomials 𝑗λ may be taken as cyclic vectors and (2) prove that the restriction of 𝓛 to each 𝑍(jλ, 𝓛) has simple spectrum given by the anisotropic contents [𝑠] of the addable corners 𝑠 of the Young diagram of λ. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated with 𝓛, both established by Nazarov-Sklyanin. Finally, we conjecture that the 𝓛-eigenfunctions 𝜓ˢλ ∈ 𝐹[𝓌] {with eigenvalue [𝑠] and constant term} 𝜓ˢλ|𝓌₌₀ = 𝑗λ are polynomials in the rescaled power sum basis 𝑉μ𝓌ˡ of 𝐹[𝓌] with integer coefficients. The authors would like to thank the referees for many helpful comments and suggestions. We would also like to express our sincere thanks to the staff at Darwin’s Ltd. coffee and sandwich shop on Cambridge Street in Cambridge, MA, for supporting our collaboration during the years 2014–2019. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spectral Theory of the Nazarov-Sklyanin Lax Operator Article published earlier |
| spellingShingle | Spectral Theory of the Nazarov-Sklyanin Lax Operator Mickler, Ryan Moll, Alexander |
| title | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_full | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_fullStr | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_full_unstemmed | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_short | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_sort | spectral theory of the nazarov-sklyanin lax operator |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212021 |
| work_keys_str_mv | AT micklerryan spectraltheoryofthenazarovsklyaninlaxoperator AT mollalexander spectraltheoryofthenazarovsklyaninlaxoperator |