# Crystals arising from Stokes phenomenon

Xiaomeng Xu

###### Abstract

In this paper, we give a complex analysis realization of
${\mathfrak{gl}}_{n}$-crystals and their tensor products. Motivated by this
result, we propose that the WKB approximation of the Stokes matrices, of
certain meromorphic linear systems of ordinary differential equations, is
characterized by ${\mathfrak{gl}}_{n}$-crystal structures on the eigenbasis of
shift of argument subalgebras of the universal enveloping algebra
$U({\mathfrak{gl}}_{n})$.

##  1 Introduction

Crystal bases in representation theory were introduced by Kashiwara [15, 16]
and Lusztig [19] in the 1990’s. Since then, they have become ubiquitous in
algebra and geometry. We would like to develop an analytic approach to the
crystal bases in various settings, via the Stokes phenomenon of differential
or difference equations. In this first paper, we give a complex analysis
realization of ${\mathfrak{gl}}_{n}$-crystals. More interestingly, our result
raises a question of characterizing the (quantum) Stokes matrices via
(quantum) algebraic integrable systems.

Let us take the Lie algebra ${\mathfrak{gl}_{n}}$ over the field of complex
numbers, and its universal enveloping algebra $U({\mathfrak{gl}}_{n})$
generated by $\\{E_{ij}\\}_{1\leq i,j\leq n}$ subject to the relation
$[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}$. Let us take the
$n\times n$ matrix $T=(T_{ij})$ with entries valued in
$U({\mathfrak{gl}}_{n})$

$\displaystyle T_{ij}=E_{ij},\ \ \ \ \ \ for\ 1\leq i,j\leq n.$

Let $\mathfrak{h}_{\rm reg}(\mathbb{R})$ denote the set of $n\times n$ real
diagonal matrices with distinct eigenvalues. Given any highest weight
$\lambda$ and the corresponding representation $L(\lambda)$ of
${\mathfrak{gl}}_{n}$, let us consider the linear system of differential
equation

$\displaystyle\frac{dF}{dz}=h\Big{(}\iota
u+\frac{1}{2\pi\iota}\frac{T}{z}\Big{)}\cdot F,$ (1)

for an $n\times n$ matrix function $F(z)$ with entries in ${\rm
End}(L(\lambda))$. Here $\iota=\sqrt{-1}$, $h$ is a real number,
$u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$ is seen as a $n\times n$ matrix with
scalar entries in $U({\mathfrak{gl}}_{n})$, and the action of the coefficient
matrix on $F(z)$ is given by matrix multiplication and the representation of
${\mathfrak{gl}}_{n}$. The equation (1) is a linear system of rank
$n\times{\rm dim}(L(\lambda))$ with a pole of order two at $z=\infty$. In each
Stokes sector ${\rm Sect}_{\pm}=\\{z\in\mathbb{C}~{}|~{}\pm{\rm Re}(z)>0\\}$,
it has a canonical solution specified by a prescribed asymptotics at
$z=\infty$. The quantum Stokes matrices $S_{h\pm}(u)$ are then the transition
matrices between the two canonical solutions, seen as $n\times n$ matrices
$(s_{ij}^{(\pm)}(u))$ with entries in ${\rm End}(L(\lambda))$. Following [25]
(see also Section 3), the matrices $S_{h\pm}(u)$ have canonically regularized
limits as some components $u_{i}$ of $u={\rm diag}(u_{1},...,u_{n})$ collapse
in a comparable speed. The prescription of the possible collapse of the
components, therefore the regularized limits of Stokes matrices, is controlled
by the geometry of the De Concini-Procesi wonderful space
$\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$ of the complement
$\mathfrak{h}_{\rm reg}(\mathbb{R})$ of an arrangement of hyperplanes in
$\mathbb{R}^{n}$, see [8]. Accordingly, we define Stokes matrices at a
boundary point $u_{0}\in\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})\setminus{\mathfrak{h}_{\rm reg}}(\mathbb{R})$ as the
corresponding regularized limits. In particular, the $0$-dimensional stratum
of $\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$ contains the caterpillar
point $u_{\rm cat}$ which corresponds to the limiting point $u_{1}\ll\cdots\ll
u_{n}$, and then the regularized limits $S_{h\pm}(u_{\rm cat})$ of
$S_{h\pm}(u)$ as $u_{1}\ll\cdots\ll u_{n}$ are called the quantum Stokes
matrices at $u_{\rm cat}$.

The WKB method, named after Wentzel, Kramers, and Brillouin, is for
approximating solutions of a differential equation whose highest derivative is
multiplied by a small parameter (other names, including Liouville, Green, and
Jeffreys are sometimes attached to this method). Accordingly, we will study
the asymptotics of $S_{h\pm}(u)$ as $h\rightarrow-\infty$. For that, we fix a
lowest vector $\xi_{0}$, and an inner product on $L(\lambda)$ given by the
conditions $\langle\xi_{0},\xi_{0}\rangle=1$ and $\langle
E_{ij}\eta,\zeta\rangle=\langle\eta,E_{ji}\zeta\rangle$ for any $\eta,\zeta\in
L(\lambda)$. For any $\xi\in L(\lambda)$, we say that a nonzero vector
$\xi^{\prime}$ is the WKB approximation of the vector valued function
$s_{k,k+1}^{(+)}(u)\cdot\xi$ if there exist constants $c$ such that (set
$q=e^{\frac{h}{2}}$)

$\displaystyle q^{c}s_{k,k+1}^{(+)}(u)\cdot\xi\rightarrow\xi^{\prime},\hskip
5.69054ptas\hskip 5.69054ptq\rightarrow 0.$

###  1.1 WKB approximation and crystals

Let us first describe the structure of the WKB approximation of
$S_{h\pm}(u_{\rm cat})$ at the special point $u_{\rm cat}$. Let us take the
Gelfand-Tsetlin orthonormal basis $B(\lambda,u_{\rm cat})$ in $L(\lambda)$,
see Section 3.3. Then in Section 5 we prove

######  Theorem 1.1.

For each $k=1,...,n-1$, there exists canonical operators $\tilde{e}_{k}(u_{\rm
cat})$ and $\tilde{f}_{k}(u_{\rm cat})$ on the finite set $B(\lambda,u_{\rm
cat})$ such that if $\xi\in B(\lambda,u_{\rm cat})$ is a generic element, then
the WKB approximation of $s_{k,k+1}^{(+)}(u_{\rm cat})\cdot\xi$ and
$s_{k+1,k}^{(-)}(u_{\rm cat})\cdot\xi$ are respectively the actions of
$\tilde{e}_{k}(u_{\rm cat})$ and $\tilde{f}_{k}(u_{\rm cat})$ on $\xi$.
Furthermore, the set $B(\lambda,u_{\rm cat})$ equipped with the operators
$\tilde{e}_{k}(u_{\rm cat})$ and $\tilde{f}_{k}(u_{\rm cat})$ is a
${\mathfrak{gl}}_{n}$-crystal.

######  Remark 1.2.

The meaning of "generic" will become clear in Section 4.3. For a none generic
element $\xi$, the $q$ leading term of $s_{k,k+1}^{(+)}(u_{\rm cat})\cdot\xi$
has ambiguity and thus is not well defined. The ambiguity in WKB approximation
along some codimension one walls is an interesting analysis phenomenon, which
is related to cluster mutations in the cluster algebra theory, see our next
paper.

The operators $\tilde{e}_{k}(u_{\rm cat})$ and $\tilde{f}_{k}(u_{\rm cat})$ on
generic elements of $B(\lambda,u_{\rm cat})$ are determined by WKB
approximation, and their extension to none generic elements are uniquely
determined by universal property. That is the operators are canonical.
Besides, the set $B(\lambda,u_{\rm cat})$ can be obtained via the WKB
approximation of Stokes matrices themselves, see Section 4.2. Therefore we
call $(B(\lambda,u_{\rm cat}),\tilde{e}_{k}(u_{\rm cat}),\tilde{f}_{k}(u_{\rm
cat}))$ the WKB datum at $u_{\rm cat}$. Theorem 1.1 states that the WKB datum
at $u_{\rm cat}$ realizes $\mathfrak{gl}_{n}$-crystals. Moreover, the tensor
product of two such $\mathfrak{gl}_{n}$-crystals can also be realized by the
WKB approximation. See Section 5.3. An even stronger statement is that the WKB
datum completely characterizes the WKB approximation (including the power of
$q$ in the approximation), see Section 5.4

Let us formulate the conjectural structure of the WKB approximation of
$S_{h\pm}(u)$ at a generic $u\in\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$.
Let $\mathcal{A}(u)$ be the shift of argument subalgebras of
$U({\mathfrak{gl}}_{n})$, that are maximal commutative subalgebras
parameterized by the de Concini-Procesi space $u\in\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$. See [11, 14] for more details. The action of
$\mathcal{A}(u)$ on $L(\lambda)$ has simple spectrum, and thus decomposes
$L(\lambda)$ into eigenlines. Let $E(\lambda,u)$ denote the set of these
eigenlines. We expect that if $\xi\in L(\lambda)$ is a generic eigenvector of
$\mathcal{A}(u)$, then the WKB approximation of $s_{k,k+1}^{(+)}(u)\cdot\xi$
and $s_{k+1,k}^{(-)}(u)\cdot\xi$ are also eigenvectors. Furthermore, the
induced operators equip $E(\lambda,u)$ with a $\mathfrak{gl}_{n}$-crystal
structure. That is

######  Conjecture 1.3.

For any $u\in\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$, the WKB
approximation of $s_{k,k+1}^{(+)}(u)$ and $s_{k+1,k}^{(-)}(u)\in{\rm
End}(L(\lambda))$, for each $k=1,...,n-1$, produce operators
$\tilde{e}_{k}(u)$ and $\tilde{f}_{k}(u)$ on the set $E(\lambda,u)$ of
eigenlines, such that the WKB datum
$(E(\lambda,u),\tilde{e}_{k}(u),\tilde{f}_{k}(u))$ is a
$\mathfrak{gl}_{n}$-crystal. Moreover, given any two representations
$L(\lambda_{1})$ and $L(\lambda_{2})$, the WKB arrpoximation of

$\displaystyle\Big{(}s_{k,k}^{(+)}\otimes s_{k,k+1}^{(+)}+s_{k,k}^{(+)}\otimes
s_{k,k+1}^{(+)}\Big{)}\hskip 8.53581ptand\hskip
8.53581pt\Big{(}s_{k+1,k}^{(-)}\otimes s_{k,k}^{(-)}+s_{k+1,k+1}^{(-)}\otimes
s_{k+1,k}^{(-)}\Big{)}$ (2)

produce the crystal operators $\tilde{e}_{k}(u)$ and $\tilde{f}_{k}(u)$ on the
tensor product $E(\lambda_{1},u)\otimes E(\lambda_{2},u)$ of
$\mathfrak{gl}_{n}$-crystals.

The shift of argument subalgebra $\mathcal{A}(u_{\rm cat})$ at $u_{\rm cat}$
is the Gelfand-Tsetlin subalgebra, and $E(\lambda,u_{\rm cat})$ is identified
with the set of Gelfand-Tsetlin eigenlines. Thus the conjecture includes
Theorem 1.1 as a special case.

######  Remark 1.4.

The ${\mathfrak{gl}}_{2}$ case can be verified directly, since the quantum
Stokes matrices in this case have closed formula, see the Appendix.

Recall that a ${\mathfrak{gl}}_{n}$-crystal is a finite set which models a
weight basis for a representation of ${\mathfrak{gl}}_{n}$, and crystal
operators indicate the leading order behaviour of the simple root vectors on
the basis under the crystal limit $q\rightarrow 0$ in quantum group
$U_{q}({\mathfrak{gl}}_{n})$. Therefore, from the perspective of quantum
groups, the above conjecture and theorem have the following interpretation: in
[27] we proved that the quantum Stokes matrices $S_{h\pm}(u)$ are the
$L_{\pm}$ operators in the FRT realization of the quantum group
$U_{q}(\mathfrak{gl}_{n})$ [22] (this is why they are called quantum Stokes
matrices). See also the Appendix for more details. Thus the entries
$s_{k,k+1}^{(+)}$, $s_{k+1,k}^{(-)}$ and $s_{k,k}^{(+)}$ are certain
realization of Chevalley generators of $U_{q}({\mathfrak{gl}}_{n})$, and the
operators appearing in (2) are just the images of $s_{k,k+1}^{(+)}$ and
$s_{k+1,k}^{(-)}$ under the coproduct of $U_{q}(\mathfrak{gl}_{n})$.
Conjecture 1.3 and Theorem 1.1 then indicate that the WKB approximation
$h\rightarrow-\infty$ in the differential equation (1) translates to the
crystal limit $q=e^{h/2}\rightarrow 0$ in quantum group.

It seems worth pointing out some approaches to the conjecture for general $u$
cases. First note that the equation (1) is equivalent to the (two variables)
generalized Knizhnik-Zamolodchikov equations (see [10, Theorem 2.1]) taking
values in the tensor product $L(\lambda)\otimes\mathbb{C}^{n}$ of $L(\lambda)$
and the standard representation of ${\mathfrak{gl}}_{n}$. The integral
representations for the solutions of (1), obtained by Felder-Markov-Tarasov-
Varchenko [10, Theorem 3.1], can be rewritten under Bethe vectors/eigenbasis
$B(\lambda,u)$ of the shift of argument subalgebras $\mathcal{A}(u)$. Then the
Stokes matrices $S_{h\pm}(u)$, which measure the difference of the asymptotics
of these solutions as $z\rightarrow\infty$ within different Stokes sectors,
can be expressed under the Bethe vectors $B(\lambda,u)$. The rest is to show
the well definess of the WKB datum
$(E(\lambda,u),\tilde{e}_{k}(u),\tilde{f}_{k}(u))$ applying the method of
steepest descent, and its continuous dependence on
$u\in\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. Then by continuity the WKB
datum at a generic $u$ is equipped with the discrete combinatorial structure
of the WKB datum at $u_{\rm cat}$, and thus is a $\mathfrak{gl}_{n}$-crystal.
Some other more general approach is to use the degeneration of isomonodromy
equation to isospectral equation. We leave it for our next work.

###  1.2 The variation of $u$ in $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$ and cactus group actions on crystals

The local picture. Now let us discuss the variation of $u$ in the closure
$\overline{U_{\rm id}}$ of the connected component $U_{\rm
id}=\\{u\in\mathfrak{h}_{\rm reg}(\mathbb{R})~{}|~{}u_{1}<\cdots<u_{n}\\}$ of
$\mathfrak{h}_{\rm reg}(\mathbb{R})$ in the space $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$. The WKB datum at $u_{\rm cat}$, given in Theorem 1.1,
should provide an applicable model for general $u\in\overline{U_{\rm id}}$.
That is because that the regularized limits $S_{h\pm}(u_{\rm cat})$ simply
encode the leading terms of $S_{h\pm}(u)$ as $u\rightarrow u_{\rm cat}$ from
the connected component $U_{\rm id}$ (see Section 2.4 and 3.4 for more
details), thus we expect the following commutative diagram.

$\begin{CD}S_{h\pm}(u)@>{\text{regularized limits as $u\rightarrow u_{\rm
cat}$ from $U_{\rm id}$}}>{}>S_{h\pm}(u_{\rm cat})\\\ @V{\text{WKB
datum}}V{}V@V{\text{WKB datum}}V{}V\\\
(E(\lambda,u),\tilde{e}_{k}(u),\tilde{f}_{k}(u))@>{\text{continuous as
$u\rightarrow u_{\rm cat}$ from $U_{\rm id}$}}>{}>(E(\lambda,u_{\rm
cat}),\tilde{e}_{k}(u_{\rm cat}),\tilde{f}_{k}(u_{\rm cat}))\end{CD}$

In particular, the discrete combinatorial structure encoded in
$(E(\lambda,u),\tilde{e}_{k}(u),\tilde{f}_{k}(u))$ should be locally
independent of $u\in\overline{U_{\rm id}}$. That is the crystal structure at a
generic $u\in U_{\rm id}$ can be understood from the one at $u_{\rm cat}$. It
particularly implies the following construction of the set $E(\lambda,u)$ from
Stokes matrices themselves.

For any integers $i,j$ with $1\leq n-i\leq j\leq n-1$, we introduce the
monomial $Q_{ij}=s_{j,j+1}^{(+)}s_{j-1,j}^{(+)}\cdots s_{n-i,n-i+1}^{(+)}$.
For any tuple of nonnegative integers $\underline{d}=(d^{(k)}_{j})_{1\leq
j\leq k\leq n-1}$, we define the operator

$\displaystyle Q(u,h)^{\underline{d}}=\overrightarrow{\prod_{i+j\geq
n}}(Q_{ij})^{d^{(i)}_{j}}\in{\rm End}(L(\lambda)),$ (3)

where the product is taken with $Q_{ij}$ to the right of
$Q_{i^{\prime}j^{\prime}}$ if $j>j^{\prime}$; or $j=j^{\prime}$ and
$i>i^{\prime}$. We say that a nonzero vector $\xi_{\underline{d}}$ is the WKB
approximation of $Q(u,h)^{\underline{d}}\cdot\xi_{0}$ if there exists a
constant $c$ such that

$\displaystyle
q^{c}Q(u,h)^{\underline{d}}\cdot\xi_{0}\rightarrow\xi_{\underline{d}},\hskip
5.69054ptas\hskip 5.69054ptq\rightarrow 0.$

Then for any $u\in U_{\rm id}$, the set of one dimensional subspaces
$[\xi_{\underline{d}}]:=\mathbb{C}\xi_{\underline{d}}$ for all possible
${\underline{d}}$ coincides with $E(\lambda,u)$, and equips $E(\lambda,u)$
with a natural parametrization by the integer points in the Gelfand-Tsetlin
polytope. See Section 4.2 for more details. Furthermore, the WKB operators
$\tilde{e}_{k}(u),\tilde{f}_{k}(u)$ are explicitly given, see the formula in
Theorem 5.3. It gives a realization of the set $E(\lambda,u)$ via the WKB
approximation of polynomial combinations of Stokes matrices themselves.

######  Remark 1.5.

The polynomials $Q(u,h)^{\underline{d}}$ in (3) simply label the elements in
$E(\lambda,u)$ by encoding the ordering of actions of crystal operators on the
lowest vector. However, for $u$ in some other component which is not $U_{\rm
id}$, the WKB approximation of $Q(u,h)^{\underline{d}}\cdot\xi_{0}$ may not
produce a basis anymore. There is a natural candidate resolving this problem.
That is the canonical basis of quantum groups. Since $S_{h+}(u)$ is a
realization of the plus part of $U_{q}(\mathfrak{gl}_{n})$, we can replace the
generators of the plus part of $U_{q}(\mathfrak{gl}_{n})$ by the corresponding
entries of $S_{h+}(u)$, and thus consider the WKB approximation of the
canonical basis. Given the above result, it is natural to expect that the WKB
approximation gives a one-to-one correspondence between the canonical bases
and the integer points in the Gelfand-Tsetlin cone. A more interesting
question is that if the canonical basis can be characterized by the analytic
data.

The global picture. For any $\sigma\in S_{n}$, let $\overline{U_{\sigma}}$ be
the closure of the connected component $U_{\sigma}=\\{u\in\mathfrak{h}_{\rm
reg}(\mathbb{R})~{}|~{}u_{\sigma(1)}<\cdots<u_{\sigma(n)}\\}$ of
$\mathfrak{h}_{\rm reg}(\mathbb{R})$ in the space $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$. The finite set $E(\lambda,u)$ over $u$ in different
chambers can glue together along the intersection
$\overline{U_{\sigma}}\cap\overline{U^{\prime}_{\sigma}}$, according to the
”wall-crossing formula” of the regularized limits of Stokes matrices at a
boundary point, see [25]. In this way, we get a nature cover of the space
$u\in\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$, whose monodromy
representation on the fibre $E(\lambda,u_{\rm cat})$ at $u_{\rm cat}$ can be
computed via the explicit ”wall-crossing formula” of the regularized limits of
Stokes matrices, as $u\rightarrow u_{\rm cat}$ from different chambers
$U_{\tau_{i}}$ with $\tau_{i}\in S_{n}$ being the permutation reversing the
segment $[1,...,i]$ in $[1,...,n]$. In [14] Halacheva, Kamnitzer, Rybnikov and
Weekes (HKRW) defined a covering of the space $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$, whose fibre at $u$ is the set $E(\lambda,u)$ of
eigenlines, and studied its monodromy representation. In particular, the known
${\mathfrak{gl}}_{n}$-crystal structure on the eigenlines $(E(\lambda,u_{\rm
cat}),\tilde{e}_{k}(u_{\rm cat}),\tilde{f}_{k}(u_{\rm cat}))$ of the Gelfand-
Tsetlin subalgebra induces crystal structures on other discrete fibre
$E(\lambda,u)$ by continuity, and globally the monodromy representation gives
rise to the cactus group actions on $\mathfrak{gl}_{n}$-crystals [4]. It is
rather similar to the above consideration. Therefore we expect that the
variation of the WKB datum of quantum Stokes matrices over
$\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$ gives a complex analysis
realization of the HKRW cover, as well as the the crystal structure on each
fibre constructed in [14]. Since the quantum Stokes matrices and their WKB
datum can be seen as the monodromy and spectral datum of the equation (1)
respectively, it is then interesting to realize the cactus group actions on
crystals via an isospectral deformation over $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$.

The construction of [14] works for any simple Lie algebra $\mathfrak{g}$. One
can ask the similar relation between the $\mathfrak{g}$-crystals and the WKB
approximation of quantum Stokes matrices associated to $\mathfrak{g}$.
Furthermore, using the same idea, we expect that various wall crossing type
formula in representation theory can be interpreted via Stokes phenomenon. For
example, motivated by [14, Conjecture 1.14], we expect that the cactus group
action on the Weyl group constructed by Losev [18], via perverse equivalences
coming from wall crossing functors in category $\mathcal{O}$ (a more
elementary definition was given shortly afterwards by Bonnaf$\rm\acute{e}$
[6]), can be realized via the stokes phenomenon of the affine KZ equations
introduced by Cherednik [7].

The organization of the paper is as follows.Section 2 gives the preliminaries
of Stokes matrices of meromorphic linear systems, isomonodromy deformation,
and recalls the regularized limits of Stokes matrices. Section 3 recalls the
quantum Stokes matrices and their regularized limits at $u_{\rm cat}$. Section
4 computes the WKB approximation of the quantum Stokes matrices at $u_{\rm
cat}$, and introduces the notion of WKB datum. The last section shows that the
WKB datum of quantum Stokes matrices are $\mathfrak{gl}_{n}$-crystals.

### Acknowledgements

We would like to thank Anton Alekseev for his valuable suggestions and
comments on the paper.

##  2 Stokes matrices and their regularized limits at caterpillar points

This section gives an introduction to Stokes matrices of meromorphic linear
systems of ordinary differential equations, and gives a brief recall of the
closure of Stokes matrices developed in our previous paper [25]. Section 2.1
recalls the Stokes matrices of certain meromorphic linear systems. Section 2.3
recalls the regularized limits of Stokes matrices and introduces the Stokes
matrices at a caterpillar point. Section 2.4 gives the expression of the
leading terms of Stokes matrices in terms of the Gelfand-Tsetlin systems.

###  2.1 Stokes matrices

Let $\mathfrak{h}(\mathbb{R})$ (resp. $\mathfrak{h}_{\rm reg}(\mathbb{R})$)
denote the set of diagonal matrices with (resp. distinct) real eigenvalues.
Let us consider the meromorphic linear system of differential equation

$\displaystyle\frac{dF}{dz}=\Big{(}iu-\frac{1}{2\pi i}\frac{A}{z}\Big{)}\cdot
F,$ (4)

where $F(z)$ is valued in $\mathbb{C}^{n}$, $u\in\mathfrak{h}(\mathbb{R})$ and
$A\in{{\rm Herm}}(n)$. The system has an order two pole at $\infty$ and (if
$A\neq 0$) a first order pole at $0$.

######  Definition 2.1.

The two Stokes sectors ${\rm Sect}_{\pm}$ of the system are the right/left
half planes ${\rm Sect}_{\pm}=\\{z\in\mathbb{C}~{}|~{}\pm{\rm Re}(z)>0\\}$.

Let us choose the branch of ${\rm log}(z)$, which is real on the positive real
axis, with a cut along the nonnegative imaginary axis $\iota\mathbb{R}_{\geq
0}$. For any two real numbers $a,b$, an open sector and a closed sector with
opening angle $b-a>0$ are respectively denoted by

$S(a,b):=\\{z\in\mathbb{C}~{}|~{}a<{\rm arg}(z)<b\\},\hskip
14.22636pt\overline{S}(a,b):=\\{z\in\mathbb{C}~{}|~{}a\leq{\rm arg}(z)\leq
b\\}.$

The following result are known, see e.g., [2, Chapter 9] or [25].

######  Theorem 2.2.

For any $u\in\mathfrak{h}(\mathbb{R})$, on ${\rm Sect}_{\pm}$ there is a
unique fundamental solution $F_{\pm}:{\rm Sect}_{\pm}\to{\rm
GL}(n,\mathbb{C})$ of equation (4) such that $F_{+}\cdot e^{-\iota uz}\cdot
z^{\frac{\delta(A)}{2\pi\iota}}$ and $F_{-}\cdot e^{-\iota uz}\cdot
z^{\frac{\delta(A)}{2\pi\iota}}$ can be analytically continued to
$S(-\pi,\pi)$ and $S(0,2\pi)$ respectively, and for every small
$\varepsilon>0$,

$\displaystyle\lim_{z\rightarrow\infty}F_{+}(z;u)\cdot e^{-\iota uz}\cdot
z^{\frac{\delta(A)}{2\pi\iota}}$ $\displaystyle=$ $\displaystyle{\rm Id}_{n},\
\ \ as\ \ \ z\in\overline{S}(-\pi+\varepsilon,\pi-\varepsilon),$
$\displaystyle\lim_{z\rightarrow\infty}F_{-}(z;u)\cdot e^{-\iota uz}\cdot
z^{\frac{\delta(A)}{2\pi\iota}}$ $\displaystyle=$ $\displaystyle{\rm Id}_{n},\
\ \ as\ \ \ z\in\overline{S}(\varepsilon,2\pi-\varepsilon),$

Here ${\rm Id}_{n}$ is the rank n identity matrix, and $\delta(A)$ is the
projection of $A$ to the centralizer of $u$ in $\mathfrak{gl}_{n}$. In
particular, if $u$ has distinct eigenvalues, $\delta(A)$ is the diagonal part
of $A$.

######  Definition 2.3.

The Stokes matrices of the system (4) (with respect to ${\rm Sect}_{+}$ and
the branch of ${\rm log}(z)$) are the elements $S_{\pm}(A,u)\in{\rm GL}_{n}$
determined by

$F_{+}(z)=F_{-}(z)\cdot e^{-\frac{\delta(A)}{2}}S_{+}(u,A),\ \ \ \ \
F_{-}(z)=F_{+}(z)\cdot S_{-}(u,A)e^{\frac{\delta(A)}{2}},$

where the first (resp. second) identity is understood to hold in ${\rm
Sect}_{-}$ (resp. ${\rm Sect}_{+}$) after $F_{+}$ (resp. $F_{-}$) has been
analytically continued clockwise.

The following lemma follows from the fact that if $F(z)$ is a solution, so is
$F(-z)^{\dagger}$, see [5].

######  Lemma 2.4.

Let $S_{+}(u,A)^{\dagger}$ denote the conjugation transpose of $S_{+}(u,A)$,
then $S_{-}(u,A)=S_{+}(u,A)^{\dagger}$.

###  2.2 Connection matrices

Since the system (4) is non-resonant, i.e., no two eigenvalues of
$\frac{A}{2\pi\iota}$ for $A\in{\rm Herm}(n)$ are differed by a positive
integer, we have (see e.g [24, Chapter 2]).

######  Lemma 2.5.

There is a unique holomorphic fundamental solution $F_{0}(z)\in{\rm GL}(n)$ of
the system (4) on a neighbourhood of $\infty$ slit along
$\iota\mathbb{R}_{\geq 0}$, such that $F_{0}\cdot
z^{\frac{A}{2\pi\iota}}\rightarrow{\rm Id}_{n}$ as $z\rightarrow 0$.

######  Definition 2.6.

The connection matrix $C(u,A)\in{\rm GL}_{n}(\mathbb{C})$ of the system (4)
(with respect to ${\rm Sect}_{+}$) is determined by $F_{0}(z)=F_{+}(z)\cdot
C(u,A)$, as $F_{0}(z)$ is extended to the domain of definition of $F_{+}(z)$.

Due to the symmetry imposed on the coefficient of the equation (4), the
connection matrix $C(u,A)$ is valued in ${\rm U}(n)$ (see e.g., [5, Lemma
29]).

###  2.3 Regularized limits at caterpillar points and closure of Stokes
matrices

It is well known that for a generic $A\in{\rm Herm}(n)$, the limit of the
Stokes matrices $S_{\pm}(u,A)$, as some components $u_{i}$ of $u={\rm
diag}(u_{1},...,u_{n})\in\mathfrak{h}_{\rm reg}(\mathbb{R})$ collapse, do not
exist. However, in [25] we prove that $S_{\pm}(u,A)$ have canonically
regularized limit as $u_{1},...,u_{n}$ collapse in a comparable speed. In the
following, we will only focus on the special case that all $u_{1},...,u_{n}$
collapse in the speed that $u_{i}$ approaches to $u_{1}$ much faster than
$u_{i+1}$ approaches to $u_{1}$. We refer the reader to [25] for more general
cases.

For any $n\times n$ matrix $A$ and $k=0,...,n-1$, we denote by $\delta_{k}(A)$
the matrix

$\displaystyle\delta_{k}(A)_{ij}=\left\\{\begin{array}[]{lr}A_{ij},&if\ \
1\leq i,j\leq k,\ or\ i=j\\\ 0,&otherwise.\end{array}\right.$ (7)

Then we have

######  Theorem 2.7.

[25] For any fixed $A\in{\rm Herm}(n)$, the limit of the matrix valued
function

$\displaystyle\Big{(}\overrightarrow{\prod_{k=0,...,n-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}\cdot
S_{-}(u,A)S_{+}(u,A)\cdot\Big{(}\overrightarrow{\prod_{k=0,...,n-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}^{-1},$
(8)

as $0<u_{1}\ll u_{2}\ll...\ll u_{n}$, exist. Here $u_{0}:=1$ and ${\rm
log}(\delta_{k}(S_{-})\delta_{k}(S_{+}))$ is the logarithm of the positive
definite Hermitian matrix $\delta_{k}(S_{-}(u,A))\delta_{k}(S_{+}(u,A))$.
Furthermore, if we use the Gauss decomposition to decompose the corresponding
limit of (8) into the product $S_{-}(u_{\rm cat},A)S_{+}(u_{\rm cat},A)$ of
upper and lower triangular matrices $S_{\pm}(u_{\rm cat},A)$ (with the same
diagonal part), then

  * •

the sub-diagonals of $S_{\pm}(u_{\rm cat},A)$ are given by

$\displaystyle(S_{+})_{k,k+1}$ $\displaystyle=$ $\displaystyle
e^{\frac{\small{\lambda^{(k)}_{k+1}-\lambda^{(k-1)}_{k}}}{4}}\sum_{i=1}^{k}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+\frac{\lambda^{(k)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+\frac{\lambda^{(k+1)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+\frac{\lambda^{(k)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+\frac{\lambda^{(k-1)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}\cdot
m^{(k)}_{i},$ (9) $\displaystyle(S_{-})_{k+1,k}$ $\displaystyle=$
$\displaystyle
e^{\frac{\small{\lambda^{(k)}_{k+1}-\lambda^{(k-1)}_{k}}}{4}}\sum_{i=1}^{k}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1-\frac{\lambda^{(k)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1-\frac{\lambda^{(k+1)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1-\frac{\lambda^{(k)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1-\frac{\lambda^{(k-1)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}\cdot\overline{m^{(k)}_{i}}.$
(10)

where $k=1,...,n-1$, $\\{\lambda^{(k)}_{i}\\}_{i=1,...,k}$ are the eigenvalues
of the upper left $k\times k$ submatrix of $A$, $\lambda^{(k)}_{k+1}=A_{kk}$
(k-th diagonal element of $A$), and

$\displaystyle
m^{(k)}_{i}(A)=\sum_{j=1}^{k}\frac{(-1)^{k-j}\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}(\lambda^{(k)}_{i}-A)}{\prod_{l=1,l\neq
i}^{k}(\lambda^{(k)}_{i}-\lambda^{(k)}_{l})}A_{j,k+1}.$ (11)

Here $\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}(\lambda^{(k)}_{i}-A)$ is the
$(k-1)\times(k-1)$ minor of the matrix $\lambda^{(k)}_{i}-A$ (the $\hat{j}$
means that the row index $j$ is omitted).

  * •

the other entries of $S_{\pm}(u_{\rm cat},A)$ are given by explicit algebraic
combinations of the sub-diagonal ones. The algebraic combinations are
determined by the classical RLL formulation, see Section A. The RLL relations
and semiclassical limits for more details.

Through the Gauss decomposition, Theorem 2.7 states that $S_{\pm}(u,A)$ have
properly regularized limits $S_{\pm}(u_{\rm cat},A)$ as $u_{1}\ll\cdots\ll
u_{n}$. In Section 2.4, we will see that the regularized limits completely
encode the leading terms of $S_{\pm}(u,A)$.

More generally, the properly regularized limit of the Stokes matrices
$S_{\pm}(u,A)$, as some components $u_{i}$ of $u={\rm diag}(u_{1},...,u_{n})$
collapse in a comparable speed, was studied in [25]. The prescription of all
the possible collapse of the components $u_{i}$, therefore the properly
regularized limit, is controlled by the geometry of the De Concini-Procesi
space $\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. In [25] we define the
Stokes matrices at the boundary $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})\setminus{\mathfrak{h}_{\rm reg}}(\mathbb{R})$ as the
corresponding regularized limits, and call them the closure of Stokes
matrices. In particular, the "infinite" point $u={\rm diag}(u_{1},...,u_{n})$,
with $u_{1}\ll\cdots\ll u_{n}$, is a point in the $0$-dimensional stratum of
$\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$, called a caterpillar point
$u_{\rm cat}$ (see [23], page 16). Accordingly,

######  Definition 2.8.

[25] For any $A\in{\rm Herm}(n)$, the regularized limits $S_{\pm}(u_{\rm
cat},A)$ of $S_{\pm}(u,A)$, as $0<u_{1}\ll u_{2}\ll...\ll u_{n}$, are called
the Stokes matrices at the caterpillar point $u_{\rm cat}$.

######  Example 2.9 (2 by 2 cases).

Let us consider the rank two case, that is

$\displaystyle\frac{dF}{dz}=\left(\left(\begin{array}[]{cc}\iota u_{1}&0\\\
0&\iota u_{2}\end{array}\right)-\frac{1}{2\pi
iz}{\left(\begin{array}[]{cc}t_{1}&a\\\
\bar{a}&t_{2}\end{array}\right)}\right)\cdot F.\quad$

Following [[3] Proposition 8], the Stokes matrices (with respect to the chosen
branch of ${\rm log}(z)$) are

$S_{-}(u,A)=\left(\begin{array}[]{cc}e^{\frac{t_{1}}{2}}&0\\\
\frac{\bar{a}\cdot
e^{\frac{t_{2}-t_{1}}{4}}({u_{2}-u_{1}})^{\frac{t_{2}-t_{1}}{2\pi\iota}}}{\Gamma(1-\frac{\lambda_{1}-t_{1}}{2\pi\iota})\Gamma(1-\frac{\lambda_{2}-t_{1}}{2\pi\iota})}&e^{\frac{t_{2}}{2}}\end{array}\right),\
\ S_{+}(u,A)=\left(\begin{array}[]{cc}e^{\frac{t_{1}}{2}}&\frac{a\cdot
e^{\frac{t_{2}-t_{1}}{4}}({u_{2}-u_{1}})^{\frac{t_{1}-t_{2}}{2\pi\iota}}}{\Gamma(1+\frac{\lambda_{1}-t_{1}}{2\pi\iota})\Gamma(1+\frac{\lambda_{2}-t_{1}}{2\pi\iota})}\\\
0&e^{\frac{t_{2}}{2}}\end{array}\right).$

By the definition of $\delta_{k}$ given in (7), we have that
$\delta_{0}(S_{\pm}(u,A))=\delta_{1}(S_{\pm}(u,A))$ is the diagonal part of
$S_{\pm}(u,A)$. Then ${\frac{{\rm
log}(\delta_{1}(S_{-})\delta_{1}(S_{+}))}{2\pi\iota}}={{\rm
diag}(\frac{t_{1}}{2\pi\iota},\frac{t_{1}}{2\pi\iota})}$, and we have

$\left(\begin{array}[]{cc}u_{2}^{-\frac{t_{1}}{2\pi\iota}}&0\\\
0&u_{2}^{-\frac{t_{2}}{2\pi\iota}}\end{array}\right)S_{-}(u,A)S_{+}(u,A)\left(\begin{array}[]{cc}u_{2}^{\frac{t_{1}}{2\pi\iota}}&0\\\
0&u_{2}^{\frac{t_{2}}{2\pi\iota}}\end{array}\right)\rightarrow S_{-}(u_{\rm
cat},A)S_{+}(u_{\rm cat},A),\ \ as\ u_{1}\ll u_{2},$

where

$S_{-}(u_{\rm cat},A)^{\dagger}=S_{+}(u_{\rm
cat},A)=\left(\begin{array}[]{cc}e^{\frac{t_{1}}{2}}&\frac{a\cdot
e^{\frac{t_{1}-t_{2}}{4}}}{\Gamma(1+\frac{\lambda_{1}-t_{1}}{2\pi\iota})\Gamma(1+\frac{\lambda_{2}-t_{1}}{2\pi\iota})}\\\
0&e^{\frac{t_{2}}{2}}\end{array}\right).$

###  2.4 The leading terms of Stokes matrices via the Gelfand-Tsetlin systems

Theorem 2.7 was proved in [25] by the method of isomonodromy deformation. In
particular, the asymptotics of solutions of the associated isomonodromy
equations of (4) was studied, using which the regularized limits of Stokes
matrices at $u_{\rm cat}$ in Theorem 2.7 can be rewritten in the following
(stronger) form. For any $A\in{\rm Herm}(n)$, let us introduce the unitary
matrix

$g(u;A)=\overrightarrow{\prod_{k=0,...,n-1}}({u_{k}}/{u_{k+1}})^{\frac{\delta_{k}(A)}{2\pi\iota}}.$

######  Proposition 2.10.

[25] For any fixed $A\in{\rm Herm}(n)$, we have, as $0<u_{1}\ll u_{2}\ll...\ll
u_{n}$,

$\displaystyle S_{\pm}(u,A)=S_{\pm}\Big{(}u_{\rm cat},\hskip
5.69054ptg(u;A)\cdot A\cdot g(u;A)^{-1}+O((u_{2}-u_{1})^{-1})\Big{)}.$ (13)

As a corollary, we can describe explicitly the leading terms of $S_{\pm}(u,A)$
as $u\rightarrow u_{\rm cat}$.

######  Corollary 2.11.

For any fixed $A\in{\rm Herm}(n)$, the subdiagonal entries of Stokes matrices
$S_{\pm}(u,A)$ satisfy

$\displaystyle(S_{+}(u,A))_{k,k+1}={l}^{(\pm)}_{k,k+1}(u,A)+o(l^{(+)}_{k,k+1}(u,A)),\hskip
8.53581ptas\ 0<u_{1}\ll u_{2}\ll...\ll u_{n},$ (14)

where ${l}^{(+)}_{k,k+1}(u)$ coincides with the $(S_{+}(u_{\rm
cat},A))_{k,k+1}$ given by (9), provided we replace the term $m_{i}^{(k)}(A)$
by

$\displaystyle m_{i}^{(k)}(A)\cdot
u_{k}^{-\frac{A_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{\lambda^{(k)}_{i}(A)}{2\pi\iota}}.$
(15)

The corollary is better to be understood in terms of the Gelfand-Tsetlin
integrable systems [13]. Before proving the corollary, let us introduce the
action and angle coordinates of the systems.

Gelfand-Tsetlin maps. For $k\leq n$ let $A^{(k)}\in{\rm Herm}(k)$ denote the
upper left $k$-th submatrix (upper left $k\times k$ corner) of a Hermitian
matrix $A\in{\rm Herm}(n)$, and $\lambda^{(k)}_{i}(A)$-its ordered set of
eigenvalues, $\lambda_{1}^{(k)}(A)\leq\cdots\leq\lambda_{k}^{(k)}(A)$. The map

$\lambda\colon{\rm Herm}(n)\to\mathbb{R}^{\frac{n(n+1)}{2}},$ (16)

taking $A$ to the collection of numbers $\lambda_{i}^{(k)}(A)$ for $1\leq
i\leq k\leq n$, is continuous and is called the Gelfand-Tsetlin map. Its image
$\mathcal{C}(n)$ is the Gelfand-Tsetlin cone, cut out by the following
inequalities,

$\lambda_{i}^{(k+1)}\leq\lambda_{i}^{(k)}\leq\lambda_{i+1}^{(k+1)},\ \ 1\leq
i\leq k\leq n-1.$ (17)

Thimm torus actions. Let $\mathcal{C}_{0}(n)\subset\mathcal{C}(n)$ denote the
subset where all of the eigenvalue inequalities (17) are strict. Let ${\rm
Herm}_{0}(n):=\lambda^{-1}(\mathcal{C}_{0}(n))$ be the corresponding dense
open subset of ${\rm Herm}(n)$. The $k$-torus $T(k)\subset{\rm U}(k)$ of
diagonal matrices acts on ${\rm Herm}_{0}(n)$ as follows,

$t\bullet A=\mathrm{Ad}_{U^{-1}tU}A,\ \ \ \ t\in T(k),\ A\in{\rm
Herm}_{0}(n).$ (18)

Here $U\in{\rm U}(k)\subset{\rm U}(n)$ is a unitary matrix such that
$\mathrm{Ad}_{U}A^{(k)}$ is diagonal, with entries
$\lambda_{1}^{(k)},\ldots,\lambda_{k}^{(k)}$. The action is well-defined since
$U^{-1}tU$ does not depend on the choice of $U$, and preserves the Gelfand-
Tsetlin map (16). The actions of the various $T(k)$’s commute, hence they
define an action of the Gelfand-Tsetlin torus $T(1)\times\cdots\times
T(n-1)\cong{\rm U}(1)^{\frac{(n-1)n}{2}}.$ Here the torus $T(n)$ is excluded,
since the action (18) is trivial for $k=n$.

Action-angle coordinates. Let $A\in\rm Herm_{0}(n)$. There exists a unique
unitary matrix $P_{k}(A)\in{\rm U}(k)\subset{\rm U}(n)$, whose entries in the
$k$-th row are positive and real, such that the upper left $k$-th submatrix of
$P_{k}(A)^{-1}AP_{k}(A)$ is the diagonal matrix ${\rm
diag}(\lambda^{(k)}_{1},...,\lambda^{(k)}_{k})$, i.e.,

$\displaystyle
P_{k}(A)^{-1}AP_{k}(A)=\begin{pmatrix}\lambda^{(k)}_{1}&&&&a^{(k)}_{1}&\cdots\\\
&\ddots&&&\vdots&\cdots\\\ &&\lambda^{(k)}_{k}&&a^{(k)}_{k}&\cdots\\\ &&&&\\\
\overline{a_{1}^{(k)}}&\cdots&\overline{a_{k}^{(k)}}&&\lambda_{k+1}^{(k)}&\cdots\\\
\cdots&\cdots&\cdots&&\cdots&\cdots\end{pmatrix}.$ (19)

The $(i,k+1)$ entry $a^{(k)}_{i}(A)$ are seen as functions on ${\rm
Herm}_{0}(n)$. Then the functions $\\{\lambda^{(k)}_{i}\\}_{1\leq i\leq k\leq
n}$ and $\\{\psi^{(k)}_{i}={\rm Arg}(a^{(k)}_{i})\\}_{1\leq i\leq k\leq n-1}$
on ${\rm Herm}_{0}(n)$ are called the Gelfand-Tsetlin action and angle
coordinates. Under the coordinates, the Thimm action of

${\rm diag}(e^{\iota\theta^{(k)}_{1}},...,e^{\iota\theta^{(k)}_{k}})\in T(k)$

is given by

$\lambda^{(i)}_{j}\mapsto\lambda^{(i)}_{j},\hskip
14.22636pt\psi^{(i)}_{j}=\psi^{(i)}_{j}+\delta_{ki}\delta_{jl}\theta^{(k)}_{l}.$

Proof of Corollary 2.11. First, for any $A$ and $u$, let us introduce an
element in the torus

$\prod_{k=1,...,n}({u_{k-1}}/{u_{k}})^{\frac{\lambda^{(k-1)}(A)}{2\pi\iota}}\in
T(1)\times\cdots\times T(n-1)$

where $\lambda^{(k-1)}(A):={\rm
diag}(\lambda^{(k-1)}_{1},...,\lambda^{(k-1)}_{k-1})$. Let us introduce the
diagonal matrix

$D(u;A)={\rm
diag}(u_{1}^{{A_{11}}/{2\pi\iota}},...,u_{n}^{{A_{nn}}/{2\pi\iota}})\in T(n).$

Then one checks

$\displaystyle
g(u;A)Ag(u;A)^{-1}=D(u;A)^{-1}\cdot\Big{(}\prod_{k=1,...,n}({u_{k-1}}/{u_{k}})^{\frac{\lambda^{(k-1)}(A)}{2\pi\iota}}\bullet
A\Big{)}\cdot D(u;A).$ (20)

Therefore, we have

$\displaystyle\lambda^{(k)}_{i}(g(u;A)Ag(u;A)^{-1})$ $\displaystyle=$
$\displaystyle\lambda^{(k)}_{i}(A),$ $\displaystyle
a^{(k)}_{i}(g(u;A)Ag(u;A)^{-1})$ $\displaystyle=$ $\displaystyle
a^{(k)}_{i}(A)\cdot
u_{i}^{-\frac{A_{ii}}{2\pi\iota}}(u_{k}/u_{k+1})^{\frac{\lambda^{(k)}_{i}(A)}{2\pi\iota}},$

where the $u_{i}^{-{A_{ii}}/{2\pi\iota}}$ term in the second identity comes
from the action of $D(u;A)$ in (20). Furthermore, using the identity of
functions $a^{(k)}_{i}$ and ${m^{(k)}_{i}}$ on ${\rm Herm}_{0}(n)$,

$a^{(k)}_{i}={m^{(k)}_{i}}\cdot\sqrt{\frac{\prod_{v=1,v\neq
i}^{k+1}(\lambda^{(k+1)}_{v}-\lambda^{(k+1)}_{i})}{\prod_{v=1}^{k}(\lambda^{(k)}_{v}-\lambda^{(k+1)}_{i})}},$

we get that the values of the function $m_{i}^{(k)}$ at the two points $A$ and
$g(u;A)Ag(u;A)^{-1}$ are related by

$\displaystyle m_{i}^{(k)}(g(u;A)Ag(u;A)^{-1})=m_{i}^{(k)}(A)\cdot
u_{k}^{-\frac{A_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{\lambda^{(k)}_{i}(A)}{2\pi\iota}},$

that is the identity (15). In the end, the corollary follows from the real
analyticity of $S_{\pm}(u_{\rm cat},A)$ with respect to $A$.

One can prove similar results for the other entries of $S_{\pm}(u,A)$. Since
the functions $m^{(k)}_{i}$ are expressed by
$a^{(k)}_{i}=e^{\iota\psi^{(k)}_{i}}$, $\lambda^{(k)}_{l}$ and
$\lambda^{(k+1)}_{l}$, Corollary 2.11 gives the expression of the leading
terms of Stokes matrices as $u\rightarrow u_{\rm cat}$, in terms of the
Gelfand-Tsetlin action and angle coordinates. Furthermore, we see that the
leading terms include a fast spin on the corresponding Liouville torus of the
integrable systems, which is cancelled out as far as the regularized limit is
considered.

###  2.5 Degenerate irregular terms

The diagonal matrix $\iota u$ is called the irregular term of the meromorphic
linear system of differential equation (4). Although the results in the
previous sections are obtained under the assumption that
$u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, a generalization of the construction
in [25] to the case of degenerate irregular terms, i.e.,
$u\in\mathfrak{h}(\mathbb{R})\setminus\mathfrak{h}_{\rm reg}(\mathbb{R})$ is
direct. For example, given any partition $\underline{d}$ of $n$, i.e., a set
of integers $\\{d_{j}\\}_{j=1,...,m}$ such that $n=d_{1}+\cdots+d_{m}$, let us
consider the subspace of $\mathfrak{h}(\mathbb{R})$

$U_{\underline{d}}=\\{u={\rm
diag}(\underbrace{u_{1},...,u_{1}}_{d_{1}},\underbrace{u_{2},...,u_{2}}_{d_{2}},....,\underbrace{u_{m},...,u_{m}}_{d_{m}})\in\mathfrak{h}(\mathbb{R})~{}|~{}u_{i}\neq
u_{j}\ \ if\ \ i\neq j\\}.$

Any $n\times n$ matrix $A$ can be seen as a blocked matrix $A=(A_{ij})_{1\leq
i,j\leq m}$ according to the tuple $\underline{d}$, with each $A_{ij}$ a
$d_{i}\times d_{j}$ block. Then for each $k=0,...,m-1$, we denote by
$\delta_{k}(A)$ the blocked matrix

$\displaystyle\delta_{k}(A)_{ij}=\left\\{\begin{array}[]{lr}A_{ij},&if\ \
1\leq i,j\leq k,\ or\ i=j\\\ 0,&otherwise.\end{array}\right.$ (23)

######  Remark 2.12.

The identity (23) depends on the partition $d_{1},...,d_{m}$, which
generalizes the definition of $\delta_{k}(A)$ given in (7) corresponding to
the case $d_{1}=\cdots=d_{n}=1$. Although this more general notation can lead
to confusion with (7), the context typically eliminates any ambiguity.

Then the analytic branching rule in [25] states

######  Theorem 2.13.

[25] For any fixed $A\in{\rm Herm}(n)$, the limit of the matrix valued
function of $u\in U_{\underline{d}}$

$\displaystyle\Big{(}\overrightarrow{\prod_{k=0,...,m-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}\cdot
S_{-}(u,A)S_{+}(u,A)\cdot\Big{(}\overrightarrow{\prod_{k=0,...,m-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}^{-1},$
(24)

as $0<u_{1}\ll u_{2}\ll...\ll u_{m}$, equals to (here $u_{0}:=1$)

$\displaystyle\Big{(}\overrightarrow{\prod_{k=1,...,m}}C^{(k)}(A)\Big{)}\cdot
e^{A}\cdot\Big{(}\overrightarrow{\prod_{k=1,...,m}}C^{(k)}(A)\Big{)}^{-1}.$
(25)

Here the product $\overrightarrow{\prod}$ is taken with the index $i$ to the
right of $j$ if $j<i$, and each $C^{(k)}(A)$ denotes the connection matrix of
the system

$\displaystyle\frac{dF}{dz}=\Big{(}\iota
E(\underline{d})_{k}-\frac{1}{2\pi\iota}\frac{\delta_{k}(A)}{z}\Big{)}F,$ (26)

where $\delta_{k}(A)$ is given in (23), and $E(\underline{d})_{k}={\rm
diag}(0,...,0,{\rm Id}_{d_{k}},0,...,0)$ is the blocked diagonal matrix whose
$k-$th blocked digonal entry is the rank $d_{k}$ identity matrix ${\rm
Id}_{d_{k}}$.

######  Remark 2.14.

If the partition $\underline{d}=(1,1,...,1)$, we arrive at the case of
nondegenerate irregular term $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$ studied
in Section 2.3–2.4. In this case, the system (26) is a confluent
hypergeometric system, and the associated connection matrices and Stokes
matrices have closed formula. This is how we derive the explicit formula in
Theorem 2.7.

##  3 Quantum Stokes matrices and their regularized limits at caterpillar
points

This section gives the quantum analog of the results in the previous section,
i.e., the expression of the regularized limits, as well as the leading terms,
of quantum Stokes matrices as $u\rightarrow u_{\rm cat}$, in terms of the
Gelfand-Tsetlin basis. In Section 3.1, we introduce the quantum Stokes
matrices of the linear system (1). In Section 3.2, we recall the regularized
limits of quantum Stokes matrices at a caterpillar point, and their explicit
expressions. In Section 3.3 and 3.4, we we introduce the Gelfand-Tsetlin
basis, and describe the leading terms of quantum Stokes matrices as
$u\rightarrow u_{\rm cat}$ in terms of the basis.

###  3.1 Quantum Stokes matrices

In this subsection, we recall the Stokes matrices of the linear system (1)
associated to a representation $L(\lambda)$

$\displaystyle\frac{dF}{dz}=h\Big{(}\iota
u+\frac{1}{2\pi\iota}\frac{T}{z}\Big{)}\cdot F.$ (27)

Under a choice of basis in $L(\lambda)$, it becomes a special case of the
equation (4) with rank $n\times{\rm dim}(L(\lambda))$. But it is more
conveniently seen as a $n\times n$ system with coefficients valued in a non-
commutative space ${\rm End}(L(\lambda))$. The two Stokes sectors are still
the right/left half planes ${\rm Sect}_{\pm}=\\{z\in\mathbb{C}~{}|~{}\pm{\rm
Re}(z)>0\\}$. And again, let us choose the branch of ${\rm log}(z)$, which is
real on the positive real axis, with a cut along $i\mathbb{R}_{\geq 0}$. As a
special case of Theorem 2.2, we have that

######  Corollary 3.1.

For any $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, on ${\rm Sect}_{\pm}$ there
is a unique (therefore canonical) holomorphic fundamental solution
$F_{h\pm}(z;u)\in{\rm End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ of
(27) such that $F_{h\pm}\cdot e^{-\iota huz}z^{\frac{-h[T]}{2\pi\iota}}$ can
be analytically continued to $S(-\pi,\pi)$ and $S(0,2\pi)$ respectively, and
for every small $\varepsilon>0$,

$\displaystyle\lim_{z\rightarrow\infty}F_{h+}(z;u)\cdot e^{-\iota huz}\cdot
z^{\frac{-h[T]}{2\pi\iota}}$ $\displaystyle=$ $\displaystyle 1,\ \ \ as\ \ \
z\in\overline{S}(-\pi+\varepsilon,\pi-\varepsilon),$
$\displaystyle\lim_{z\rightarrow\infty}F_{h-}(z;u)\cdot e^{-\iota huz}\cdot
z^{\frac{-h[T]}{2\pi\iota}}$ $\displaystyle=$ $\displaystyle 1,\ \ \ as\ \ \
z\in\overline{S}(\varepsilon,2\pi-\varepsilon),$

where $[T]={\rm diag}(E_{11},...,E_{nn})\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$.

######  Definition 3.2.

The quantum Stokes matrices of (27) (with respect to ${\rm Sect}_{+}$ and the
chosen branch of ${\rm log}(z)$) are the elements $S_{h\pm}(u)\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ determined by

$F_{h+}(z)=F_{h-}(z)\cdot e^{\frac{h[T]}{2}}S_{h+}(u),\ \ \ \ \
F_{h-}(z)=F_{h+}(z)\cdot S_{h-}(u)e^{\frac{-h[T]}{2}},$

where the first (resp. second) identity is understood to hold in ${\rm
Sect}_{-}$ (resp. ${\rm Sect}_{+}$) after $F_{h+}$ (resp. $F_{h-}$) has been
analytically continued clockwise.

Let us assume $u\in U_{\rm id}\subset\mathfrak{h}_{\rm reg}(\mathbb{R})$, then
the asymptotics in Theorem 3.1 ensures that $S_{h+}=(s_{ij}^{(+)})$ is a upper
triangular matrix, and $S_{h-}=(s_{ij}^{(-)})$ is lower triangular with
entries $s_{ij}^{(\pm)}$ in ${\rm End}(L(\lambda))$, see e.g., [24, Chapter
9.1].

###  3.2 Quantum Stokes matrices at a caterpillar point

Since (27) can be seen as a special form of the general system (4), we have

$S_{h\pm}(u)=S_{\pm}(hu,-hT),$

where the right hand side are the (classical) Stokes matrices of the equation
(4) with $u$ and $A$ replaced by $hu$ and $-hT$. Here the irregular term
$\iota hu={\rm diag}(\iota hu_{1},...,\iota hu_{n})\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ is degenerate, i.e., has
repeated eigenvalues. One can still study the regularized limits of
$S_{h\pm}(u)$ as the components $u_{i}$ of $u\in\mathfrak{h}_{\rm
reg}(\mathbb{R})$ collapse in a comparable speed, and introduce the quantum
Stokes matrices for any boundary point in $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})\setminus{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. In
particular, the construction in Section 2.5 enables us to define the quantum
Stokes matrices at $u_{\rm cat}$ as follows.

First, the system (27) has rank $n\times{\rm dim}(L(\lambda))$. In terms of
the notation in Section 2.5, let us take the partition $\underline{d}$ of
$n\times{\rm dim}(L(\lambda))$ with $d_{1}=\cdots=d_{n}={\rm
dim}(L(\lambda))$, then the irregular term in the Stokes matrices
$S_{\pm}(hu,-hT)$ of (27) lives in $U_{\underline{d}}$. Following Theorem
2.13, the matrix $S_{-}(hu,-hT)\cdot S_{+}(hu,-hT)$ has a regularized limit.

######  Definition 3.3.

The quantum Stokes matrices at $u_{\rm cat}$ are the upper and lower $n\times
n$ triangular matrices $S_{h\pm}(u_{\rm cat})$ (having the same diagonal
part), with entries valued in ${\rm End}(L(\lambda))$, such that (the blocked
Gauss decomposition)

$S_{h-}(u_{\rm cat})S_{h+}(u_{\rm cat})$

equals to the regularized limit of the function $S_{-}(hu,-hT)\cdot
S_{+}(hu,-hT)$ on $U_{\underline{d}}$, i.e., the limit of

$\displaystyle\Big{(}h^{-\frac{{\delta_{0}}(T)}{2\pi\iota}}\overrightarrow{\prod_{k=0,...,n-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}\cdot
S_{-}S_{+}\cdot\Big{(}h^{-\frac{{\delta_{0}}(T)}{2\pi\iota}}\overrightarrow{\prod_{k=0,...,n-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}(\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}^{-1}$
(28)

as $0<u_{1}\ll u_{2}\ll...\ll u_{n}$, where $\delta_{k}(S_{\pm}(hu,-hT))$
takes the ${\rm End}(\lambda)$ valued entries of the matrices
$S_{\pm}(hu,-hT)$ as in (23) (Here the extra term
$h^{-\frac{{\delta_{0}}(T)}{2\pi\iota}}$ comes from the convention that
$u_{0}=1$ in the product $\overrightarrow{\prod}$).

To write down the explicit expression of $S_{h\pm}(u_{\rm cat})$, as a non-
commutative version of the formula in Theorem 2.7, let us define
$T(\zeta):=\zeta{\rm Id}-T$. Then for any $1\leq m\leq n$, elements
$(a_{1},...,a_{m})$ and $(b_{1},...,b_{m})$ of $\\{1,...,n\\}$, let us
consider the $m$ by $m$ quantum minor

$\displaystyle\Delta^{a_{1},...,a_{m}}_{b_{1},...,b_{m}}(T(\zeta)):=\sum_{\sigma\in
S_{n}}(-1)^{\sigma}T_{a_{\sigma(1)},b_{1}}(\zeta_{1})\cdots
T_{a_{\sigma(m)},b_{m}}(\zeta_{m})\in U({\mathfrak{gl}}_{n})[\zeta],$

where $\zeta_{j}=\zeta+(j-1)$. For any $1\leq k\leq n$, let

$\displaystyle
M_{k}(\zeta):=\Delta^{1,...,k}_{1,...,k}\Big{(}T(\zeta-\frac{1}{2}(k-1))\Big{)}$
(29)

be the upper left $k\times k$ quantum-minor of $T$. It is known that the
subalgebra, generated in $U({\mathfrak{gl}}_{n})$ by the coefficients in all
$M_{k}(\zeta)$ for $1\leq k\leq n$, is commutative and is called the Gelfand-
Tsetlin subalgebra.

Let $\\{\zeta^{(k)}_{i}\\}_{1\leq i\leq k\leq n}$ denote the roots of
$M_{k}(\zeta)=0$ for all $k=1,...,n$ (in an appropriate splitting extension),
that are commutative with each other. Let us consider the equation (1)
associated to $L(\lambda)$, then (see the Appendix for a proof)

######  Theorem 3.4.

For any $1\leq k\leq n-1$, the $(k,k+1)$-entry of $S_{h\pm}(u_{\rm cat})$, as
an element in ${\rm End}(L(\lambda))$ is given by

$\displaystyle s_{k,k+1}^{(+)}=-(\iota
h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1})}{2\pi\iota}}\sum_{i=1}^{k}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi\iota})}\Big{)}\cdot
h\alpha^{(k)}_{i},$ $\displaystyle s_{k+1,k}^{(+)}=-(\iota
h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}+1)}}{2\pi\iota}}\sum_{i=1}^{k}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}+\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}+\frac{1}{2}}{2\pi\iota})}\Big{)}\cdot
h\beta^{(k)}_{i},$

with

$\displaystyle\alpha^{(k)}_{j}=\sum_{j=1}^{k}\frac{(-1)^{k-j}\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}\Big{(}\zeta^{(k)}_{i}-\frac{(k-1)}{2}\Big{)}}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}E_{j,k+1}$ (30)
$\displaystyle\beta^{(k)}_{j}=\sum_{j=1}^{k}\frac{(-1)^{k-j}\Delta_{1,...,\hat{j},...,k}^{1,...,k-1}\Big{(}\zeta^{(k)}_{i}-\frac{(k-3)}{2}\Big{)}}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}E_{k+1,j}.$ (31)

######  Remark 3.5.

A universal version of the formula of quantum Stokes matrices at $u_{\rm cat}$
is given in [27], and is used to derive an explicit Drinfeld isomorphism.

###  3.3 Gelfand-Tsetlin basis for representations of ${\mathfrak{gl}}_{n}$

The expression of quantum Stokes matrices at the caterpillar point is closely
related to the Gelfand-Tsetlin basis. Let us introduce the Gelfand-Tsetlin
orthonormal basis in this subsection.

Recall that $\\{E_{ij}\\}_{i,j=1,...,n}$ is the standard basis of the Lie
algebra ${\mathfrak{gl}}_{n}$. Denote by ${\mathfrak{gl}}_{n-1}$ the
subalgebra spanned by the elements $\\{E_{ij}\\}_{i,j=1,...,n-1}$. Finite
dimensional irreducible representations of ${\mathfrak{gl}}_{n}$ are
parameterized by the highest weight, i.e., $n$-tuples of numbers
$\lambda=(\lambda_{1},...,\lambda_{n})$ with

$\lambda_{i}-\lambda_{i+1}\in\mathbb{Z}_{+},\hskip 14.22636pt,\ \forall\
i=1,...,n-1.$

We denote by $L(\lambda)$ the corresponding representation. It has a lowest
vector $\xi_{0}$ such that $E_{ii}\xi=\lambda_{i}\xi$ for $i=1,...,n$, and
$E_{ij}\xi=0$ for $1\leq j<i\leq n$. Then the simple branching rule for the
reduction from ${\mathfrak{gl}}_{n}$ to ${\mathfrak{gl}}_{n-1}$ states that
the restriction of $L(\lambda)$ to the subalgebra ${\mathfrak{gl}}_{n-1}$ is
isomorphic to the direct sum of pairwise inequivalent irreducible
representations

$L(\lambda)|_{{\mathfrak{gl}}_{n-1}}\cong\underset{\lambda^{(n-1)}}{\bigoplus}L^{\prime}(\lambda^{(n-1)}),$

where the summation is over the highest weights $\lambda^{(n-1)}$ satisfying
the interlacing conditions

$\displaystyle\lambda_{i}^{(n)}-\lambda^{(n-1)}_{i}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\lambda^{(n-1)}_{i}-\lambda^{(n)}_{i+1}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\forall\ i=1,...,n-1.$ (32)

Thus a chain of subalgebras

${\mathfrak{gl}}_{1}\subset\cdots\subset{\mathfrak{gl}}_{n-1}\subset{\mathfrak{gl}}_{n}$

produces a basis in $L(\lambda)$, parameterized by the Gelfand-Tsetlin
patterns. Such a pattern $\Lambda$ (for fixed $\lambda^{(n)}$) is a collection
of numbers $\\{\lambda^{(i)}_{j}\\}_{1\leq i\leq j\leq n-1}$ satisfying the
interlacing conditions

$\displaystyle\lambda_{j}^{(i)}-\lambda^{(i-1)}_{j}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\lambda^{(i-1)}_{j}-\lambda^{(i)}_{j+1}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\forall\ i=1,...,n-1.$

######  Definition 3.6.

We denote by $P_{GZ}(\lambda;\mathbb{Z})$ the set of Gelfand-Tsetlin patterns
in $L(\lambda)$, seen as the set of integer points in the real Gelfand-Tsetlin
polytope $P_{GZ}(\lambda;\mathbb{R})$.

The structure of the basis obtained in this way is summarized in the following
proposition.

######  Proposition 3.7.

There exists an orthonormal basis $\xi_{\Lambda}$ of $L(\lambda)$, called the
Gelfand-Tsetlin basis, parameterized by all patterns $\Lambda\in
P_{GZ}(\lambda;\mathbb{Z})$, such that for any $1\leq i\leq k\leq n$, the
actions of $E_{kk}$, $\zeta^{(k)}_{i}$ and $\alpha^{(k)}_{i}$ on the basis
${\xi_{\Lambda}}$ of $L(\lambda)$ are given by

$\displaystyle E_{kk}\cdot\xi_{\Lambda}=$
$\displaystyle(\sum_{i=1}^{k}\lambda^{(k)}_{i}-\sum_{i=1}^{k-1}\lambda^{(k-1)}_{i})\xi_{\Lambda},$
(33) $\displaystyle\zeta^{(k)}_{i}\cdot\xi_{\Lambda}=$
$\displaystyle(\lambda^{(k)}_{i}(\Lambda)-i+1+(k-1)/2)\xi_{\Lambda},$ (34)
$\displaystyle\alpha^{(k)}_{i}\cdot\xi_{\Lambda}=$
$\displaystyle\sqrt{-\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}{\prod_{l=1}^{k+1}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2})}\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1)}{\prod_{l=1}^{k-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2})}}\cdot\xi_{\Lambda+\delta^{k}_{i}}$
(35)

where the pattern $\Lambda+\delta^{(k)}_{i}$ is obtained from $\Lambda$ by
replacing $\lambda^{(k)}_{i}$ by $\lambda^{(k)}_{i}+1$. It is supposed that
$\xi_{\Lambda}$ is zero if $\Lambda$ is not a pattern.

Proof. We refer to [20, Sectioin 2] for the existence of the orthogonal
Gelfand-Tsetlin basis, and the norms of the elements in the basis [20,
Proposition 2.4]. We then perform the explicit normalization to get an
orthonormal basis. The actions of $E_{kk}$ and $\zeta^{(k)}_{i}$ on the
orthogonal basis given in [20, Theorem 2.3] imply the identities (33) and
(34).

By the Laplace expansion, the $\alpha^{(k)}_{i}$ in (30) equals to the
normalized quantum minor

$\displaystyle\alpha^{(k)}_{i}=\frac{-1}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}\Delta^{1,...,k}_{1,...,k-1,k+1}\Big{(}T(\zeta-\frac{1}{2}(k-1))\Big{)}.$

Then the identity (35) follows from the action of
$\Delta^{1,...,k}_{1,...,k-1,k+1}$ on the orthogonal basis given in [20,
Section 2.5], provided the normalization, from the orthogonal to the
orthonormal basis, is accounted for.

###  3.4 The leading terms of quantum Stokes matrices via the Gelfand-Tsetlin
basis

Given any $L(\lambda)$, let us think of the associated quantum Stokes matrices
as blocked matrices. Let us take a proper norm on the vector space ${\rm
End}(\lambda)$, then up to a slight modification, Proposition 2.10 and
Corollary 2.11 can be applied to the quantum/blocked cases. That is if we
introduce the unitary matrix

$g(u;T)=\overrightarrow{\prod_{k=0,...,n-1}}({u_{k}}/{u_{k+1}})^{\frac{\delta_{k}(T)}{2\pi\iota}},$

then

######  Corollary 3.8.

For any fixed none zero real number $h$, we have, as $0<u_{1}\ll
u_{2}\ll...\ll u_{n}$,

$\displaystyle S_{h\pm}(u)=S_{\pm}\Big{(}u_{\rm cat},\hskip
5.69054ptg(hu;hT)\cdot hT\cdot g(hu;hT)^{-1}+O((u_{2}-u_{1})^{-1})\Big{)}.$
(36)

Furthermore, the subdiagonal entries of $S_{h\pm}(u)$ satisfy

$\displaystyle
s^{(\pm)}_{k,k+1}(u)={l}^{(\pm)}_{k,k+1}(u)+o(l^{(\pm)}_{k,k+1}(u)),\hskip
8.53581ptas\ 0<u_{1}\ll u_{2}\ll...\ll u_{n},$ (37)

where the leading terms ${l}^{(\pm)}_{k,k+1}(u)$ is given by the expressions
$s^{(\pm)}_{k,k+1}(u_{\rm cat})$ of in Theorem 3.4 provided we replace
respectively $\alpha^{(k)}_{i}$ and $\beta^{(k)}_{i}$ by

$\displaystyle
u_{k}^{-\frac{E_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{h(\zeta^{(k)}_{i}-\frac{k-1}{2})}{2\pi\iota}}\cdot\alpha^{(k)}_{i}\hskip
8.53581ptand\hskip
8.53581ptu_{k}^{-\frac{E_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{h(\zeta^{(k)}_{i}-\frac{k-3}{2})}{2\pi\iota}}\cdot\beta^{(k)}_{i}.$
(38)

Here $\frac{k-1}{2}$ in $(\zeta^{(k)}_{i}-\frac{k-1}{2})$ comes from the
Capelli correction in the quantum minors, see the computation in Appendix.

In particular, by Proposition 3.7 and Corollary 3.8, the leading terms of
entries of quantum Stokes matrices, as elements in ${\rm End}(L(\lambda))$,
can be written down explicitly in terms of the Gelfand-Tsetlin basis.

######  Remark 3.9.

In the next section, we will study the WKB approximation of $S_{h\pm}(u_{\rm
cat})$. By the definition given in the introduction and the replacement of
$\alpha^{(k)}_{i}$ and $\beta^{(k)}_{i}$ given in (38), we see that
${l}^{(\pm)}_{k,k+1}(u)$ and $s^{(\pm)}_{k,k+1}(u_{\rm cat})$ have the same
WKB leading term as $h\rightarrow-\infty.$ Note that the relation in (37) is
for any fixed $h>0$ (the proof of Proposition 2.10 given in [25] makes the
assumption that the norm of $A$ is fixed), therefore, the WKB datum at $u_{\rm
cat}$ encodes the "repeated limit" of $S_{h\pm}(u)$ as $u\rightarrow u_{\rm
cat}$ and then $h\rightarrow-\infty$.

##  4 WKB approximation of quantum Stokes matrices at caterpillar points

In the last two sections, we introduce the relations between the regularized
limits of Stokes matrices at $u_{\rm cat}$ and Gelfand-Tsetlin systems in
classical and quantum settings respectively. In this section, we deepen the
relation between Stokes phenomenon and representation theory by showing that
the WKB approximation of quantum Stokes matrices at a caterpillar point gives
rise to $\mathfrak{gl}_{n}$-crystals. In particular, Section 4.1 computes
explicitly the WKB approximation. Section 4.2 unveils that how to recover the
Gelfand-Tsetlin basis from the WKB approximation. Then Section 4.3 introduces
the notion of WKB datum of quantum Stokes matrices.

###  4.1 WKB approximation of quantum Stokes matrices at caterpillar points

Since the derivative in equation (1) is multiplied by a small parameter $1/h$,
we will call the leading term, as $h\rightarrow-\infty$, of Stokes matrices of
(1) as the WKB approximation. In this subsection, we study the WKB
approximation of Stokes matrices given in (3.4).

First, for any Gelfand-Tsetlin pattern $\Lambda$ of $L(\lambda)$ given in
Section 3.3, set

$\displaystyle
x^{(k)}_{j}(\Lambda):=-\lambda^{(k)}_{j}+\lambda^{(k-1)}_{j-1}-\lambda^{(k)}_{j-1}+\lambda^{(k+1)}_{j},\hskip
8.53581pt1\leq j\leq k+1,$ (39) $\displaystyle
y^{(k)}_{j}(\Lambda):=\lambda^{(k)}_{j}-\lambda^{(k-1)}_{j}+\lambda^{(k)}_{j+1}-\lambda^{(k+1)}_{j+1},\hskip
8.53581pt0\leq j\leq k,$ (40)

and

$\displaystyle X^{(k)}_{j}(\Lambda):=\sum_{i=1}^{j}x^{(k)}_{i}(\Lambda),\hskip
8.53581pt1\leq j\leq k+1,$ (41) $\displaystyle
Y^{(k)}_{j}(\Lambda):=\sum_{i=j}^{k}y^{(k)}_{i}(\Lambda),\hskip 8.53581pt0\leq
j\leq k.$ (42)

Note that $Y^{(k)}_{0}(\Lambda)=Y^{(k)}_{j}(\Lambda)-X^{(k)}_{j}(\Lambda)$ for
any $j=0,...,k+1$. Furthermore, we define

$\displaystyle
wt_{k}(\Lambda)=\sum_{i=1}^{k}\lambda^{(k)}_{i}(\Lambda)-\sum_{i=1}^{k-1}\lambda^{(k-1)}_{i}(\Lambda),$
(43) $\displaystyle\varepsilon_{k}(\Lambda)={\rm
max}\\{X_{1}^{(k)}(\Lambda),X_{2}^{(k)}(\Lambda),...,X_{k}^{(k)}(\Lambda)\\},$
(44) $\displaystyle\phi_{i}(\Lambda)={\rm
max}\\{Y_{1}^{(k)}(\Lambda),Y_{2}^{(k)}(\Lambda),...,Y_{k}^{(k)}(\Lambda)\\},$
(45)

and define the functions $l_{1}(\Lambda)<\cdots<l_{m_{k}}(\Lambda)$ of
$\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ be those ordered labels such that

$\displaystyle X^{(k)}_{l_{j}}(\Lambda)=\varepsilon_{k}(\Lambda).$ (46)

Let us denote by

$P^{k}_{GZ}(\lambda;\mathbb{Z}):=\\{\Lambda\in
P_{GZ}(\lambda;\mathbb{Z})~{}|~{}\Lambda+\delta^{(k)}_{l_{1}}\in
P_{GZ}(\lambda;\mathbb{Z})\ and\ \Lambda+\delta^{(k)}_{l_{i}}\notin
P_{GZ}(\lambda;\mathbb{Z})\ for\ i=2,...,m_{k}\\}.$

In particular, all $\Lambda$ satisfying $m_{k}(\Lambda)=1$ and
$\Lambda+\delta^{(k)}_{l_{i}}\notin P_{GZ}(\lambda;\mathbb{Z})$ are in
$P^{k}_{GZ}(\lambda;\mathbb{Z})$. From (39)–(46), the function
$m_{k}(\Lambda)$ is defined for any point $\Lambda$ in the real polytope
$P_{GZ}(\lambda;\mathbb{R})$. Thus
$P^{k}_{GZ}(\lambda;\mathbb{R}):=P^{k}_{GZ}(\lambda;\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}$
is an open dense part of $P_{GZ}(\lambda;\mathbb{R})$, whose complements are
cut out by various equalities between $\lambda^{(j)}_{i}(\Lambda)$ for
$j=k-1,k,k+1$. By this reason, the elements in the subset
$P^{k}_{GZ}(\lambda;\mathbb{Z})$ are called generic (it gives precise meaning
of the word "generic" in Theorem 1.1).

Let $\xi_{\Lambda}$ be the Gelfand-Tsetlin orthonormal basis given in Section
3.3. Then we have

######  Proposition 4.1.

For $k=1,...,n-1$ and for any $\Lambda\in P^{k}_{GZ}(\lambda;\mathbb{Z})$,
there exists a real number $\theta_{k}\in[0,2\pi)$ independent of $q=e^{h/2}$
such that

$\displaystyle s_{k,k+1}^{(+)}\cdot\xi_{\Lambda}\sim
q^{-\varepsilon_{k}\big{(}\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}\big{)}+\iota\theta_{k}}\xi_{\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}},\hskip
8.53581ptas\ \ q\rightarrow 0.$ (47)

Here note that
$\varepsilon_{k}\big{(}\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}\big{)}=\varepsilon_{k}(\Lambda)+1.$

Proof. By the asymptotics of gamma function

${\rm In}(\Gamma(1+z))\sim z{\rm In}(z)-z+\frac{1}{2}{\rm
In}(z)+\frac{1}{2}{\rm In}(2\pi)+\frac{1}{12z}+o(\frac{1}{z}),\ \ as\
|z|\rightarrow\infty\ at\ constant\ |{\rm arg}(z)|<\pi,$

For $r$ a real number and $h\rightarrow-\infty$, we have

$\displaystyle{\rm
In}(\Gamma(1+\frac{rh}{2\pi\iota}))\sim\left\\{\begin{array}[]{lr}\frac{rh{\rm
In}(-h)}{2\pi\iota}+\frac{rh}{2\pi\iota}{\rm
In}(\frac{r}{2\pi})+\frac{rh}{4}-\frac{rh}{2\pi\iota}+\frac{1}{2}{\rm
In}(\frac{rh}{\iota}),&r>0,\\\ \frac{rh{\rm
In}(-h)}{2\pi\iota}+\frac{rh}{2\pi\iota}{\rm
In}(\frac{-r}{2\pi})-\frac{rh}{4}-\frac{rh}{2\pi\iota}+\frac{1}{2}{\rm
In}(\frac{rh}{\iota}),&r<0.\end{array}\right.$ (50)

Here we use ${\rm In}(\frac{\pm 1}{\iota})=\mp\frac{\pi\iota}{2}$ to separate
the real and imaginary part of the $h$ linear terms. By (34),

$\zeta^{(k)}_{i}\cdot\xi_{\Lambda}=(\lambda^{(k)}_{i}(\Lambda)-i+1+(k-1)/2)\cdot\xi_{\Lambda},$

by abuse of notation, we will take $\zeta^{(k)}_{i}$ as the number
$\lambda^{(k)}_{i}(\Lambda)-i+1+(k-1)/2$ when the vector $\xi_{\Lambda}$ is
specified. Then by (50) and the interlacing inequalities between
$\lambda^{(i)}_{j}$ for $i=k-1,k,k+1$,

$\displaystyle{\rm In}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi\iota})}\Big{)}$
$\displaystyle\sim$ $\displaystyle\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}+\iota
h\theta^{(k)}_{i}+\frac{h}{4}B^{(k)}_{i}+C^{(k)}_{i},\hskip 14.22636ptas\
h\rightarrow-\infty,$

where

$\displaystyle
A^{(k)}_{i}=\sum_{l=1}^{k-1}\zeta^{(k-1)}_{l}+\sum_{l=1}^{k+1}\zeta^{(k+1)}_{l}-2\sum_{l=1}^{k}\zeta^{(k)}_{l}+1,$
$\displaystyle
B^{(k)}_{i}=-2\sum_{l=1}^{i-1}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})+2\sum_{l=i+1}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})+\sum_{l=1}^{i-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l})$
$\displaystyle\hskip
34.1433pt-\sum_{l=i}^{k-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l})+\sum_{l=1}^{i}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l})-\sum_{l=i+1}^{k+1}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}),$
$\displaystyle C^{(k)}_{i}=\frac{1}{2}{\rm
In}\Big{(}-\frac{1}{h^{2}}\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}{\prod_{l=1}^{k+1}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2})}\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1)}{\prod_{l=1}^{k-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2})}\Big{)},$

and

$\displaystyle\theta^{(k)}_{i}=\frac{h}{2\pi}A^{(k)}_{i}-\sum_{l=1}^{i-1}\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi}{\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi})+\sum_{l=i+1}^{k}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}}{2\pi})$ $\displaystyle\hskip
34.1433pt-\sum_{l=1}^{i-1}\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi}{\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi})+\sum_{l=i+1}^{k}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}-1}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}-1}{2\pi})$ $\displaystyle\hskip
34.1433pt+\sum_{l=1}^{i-1}(\frac{\zeta^{(k-1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k-1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi})-\sum_{l=i}^{k-1}(\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi})$
$\displaystyle\hskip
34.1433pt+\sum_{l=1}^{i}(\frac{\zeta^{(k+1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k+1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi})-\sum_{l=i+1}^{k+1}(\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi}).$

Finally, the action of $s^{(+)}_{k,k+1}$ on $\xi_{\Lambda}$ is given by

$\displaystyle(\iota
h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}\sum_{i=1}^{k}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi\iota})}\Big{)}\cdot(-ha^{(k)}_{i})\cdot\xi_{\Lambda}$
$\displaystyle\sim$
$\displaystyle(-h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}(-\iota)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}\sum_{i=1}^{k}e^{\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}}e^{\iota
h\theta^{(k)}_{i}+\frac{h}{4}B^{(k)}_{i}}e^{C^{(k)}_{i}}(-h\alpha^{(k)}_{i})\cdot\xi_{\Lambda}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\Big{(}(-h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}e^{\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}}\Big{)}\Big{(}e^{\frac{h\small{(E_{k+1,k+1}-E_{kk}+1)}}{4}}e^{\iota
h\theta^{(k)}_{i}+\frac{h}{4}B^{(k)}_{i}}\Big{)}\Big{(}-he^{C^{(k)}_{i}}\alpha^{(k)}_{i}\cdot\xi_{\Lambda}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}q^{-X^{(k)}_{i}(\Lambda+\delta^{(k)}_{i})+2\iota
h\theta^{(k)}_{i}}\cdot\xi_{\Lambda+\delta^{(k)}_{i}}.$

Here in the last equality, we use the identities (33), (34), (39) and (41) to
get

$\displaystyle(-h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}e^{\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}}$ $\displaystyle=$ $\displaystyle 1,$
$\displaystyle
e^{\frac{h\small{(E_{k+1,k+1}-E_{kk}+1)}}{4}}e^{\frac{h}{4}B^{(k)}_{i}+\iota
h\theta^{(k)}_{i}}\cdot\xi_{\Lambda+\delta^{(k)}_{i}}$ $\displaystyle=$
$\displaystyle q^{-X^{(k)}_{i}(\Lambda+\delta^{(k)}_{i})+2\iota
h\theta^{(k)}_{i}}\xi_{\Lambda+\delta^{(k)}_{i}},$

and use the expression (35) of $\alpha^{(k)}_{i}\cdot\xi_{\Lambda}$ and the
expression (4.1) of $C^{(k)}_{i}$ to get

$-he^{C^{(k)}_{i}}\alpha^{(k)}_{i}\cdot\xi_{\Lambda}=\xi_{\Lambda+\delta^{(k)}_{i}}.$

Then the proposition follows from the definitions (44) and (46) of
$\varepsilon_{k}(\Lambda)$ and $l_{i}(\Lambda)$ with $i=1,...,m_{k}$, and the
assumption $m_{k}(\Lambda)=1.$ Paricularly the constant $\theta_{k}$ in (47)
is just $2\theta^{(k)}_{l_{1}}$.

###  4.2 Gelfand-Tsetlin basis form WKB approximation

Recall that for any tuple ${\underline{d}}=(d^{(k)}_{j})_{1\leq j\leq k\leq
n-1}\in\mathbb{N}^{\frac{n(n-1)}{2}}$, we have defined the operator in the
introduction

$\displaystyle Q(u_{\rm cat})^{\underline{d}}=\overrightarrow{\prod_{i+j\geq
n}}(Q_{ij}(u_{\rm cat}))^{d^{(i)}_{j}}\in{\rm End}(L(\lambda)),$

where $Q_{ij}(u_{\rm cat})=s_{j,j+1}^{(+)}s_{j-1,j}^{(+)}\cdots
s_{n-i,n-i+1}^{(+)}$ are combinations of the entries of Stokes matrix at
$u_{\rm cat}$. Let $\xi_{\Lambda_{0}}$ be the lowest vector in $L(\lambda)$,
which corresponds to the pattern $\Lambda_{0}$ satisfying
$\lambda^{(k)}_{i}(\Lambda_{0})=\lambda^{(k+1)}_{i+1}(\Lambda_{0})$ for all
$i,k$. Following the definition of $Q^{\underline{d}}$ and the proof of
Proposition 4.1, we have

######  Proposition 4.2.

For any $\underline{d}$, there exists a constant $c(\underline{d})$
independent of $q$ such that

$q^{c(\underline{d})}Q(u_{\rm
cat})^{\underline{d}}\cdot\xi_{\Lambda_{0}}\rightarrow\xi_{\Lambda},\hskip
5.69054ptas\hskip 5.69054ptq\rightarrow 0,$

where $\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ is uniquely determined by the
linear equations

$\displaystyle\lambda^{(i)}_{j}(\Lambda)-\lambda^{(i+1)}_{j+1}(\Lambda)=d^{(i)}_{j},\hskip
8.53581ptfor\ all\ 1\leq j\leq i\leq n-1.$ (51)

Proof. The ordering in the definition of $Q^{\underline{d}}$, together with
the formula (35) and the expressions in Theorem 3.4, enforces

$Q(u_{\rm cat})^{\underline{d}}\cdot\xi_{\Lambda_{0}}=f(q)\xi_{\Lambda},$

for certain function of $q$, where $\Lambda$ is determined by (51). Take
$Q^{\underline{d}}=s_{34}(s_{34}s_{23})(s_{34}s_{23}s_{12})^{2}$ as an
example, by (3.4) and (35) we have

$\displaystyle
s_{12}\cdot\xi_{\Lambda_{0}}=f_{1}\xi_{\Lambda_{0}+\delta^{(1)}_{1}},\hskip
8.53581pts_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{2}\xi_{\Lambda_{0}+\delta^{(1)}_{1}+\delta^{(2)}_{2}},\hskip
8.53581pts_{34}s_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{3}\xi_{\Lambda_{0}+\delta^{(1)}_{1}+\delta^{(2)}_{2}+\delta^{(3)}_{3}},$
$\displaystyle
s_{12}s_{34}s_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{4}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+\delta^{(2)}_{2}+\delta^{(3)}_{3}},\hskip
14.22636pts_{23}s_{12}s_{34}s_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{5}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+2\delta^{(2)}_{2}+\delta^{(3)}_{3}},$
$\displaystyle(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{6}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+2\delta^{(2)}_{2}+2\delta^{(3)}_{3}},\hskip
14.22636pts_{23}(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{7}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+3\delta^{(2)}_{2}+2\delta^{(3)}_{3}},$
$\displaystyle
s_{34}s_{23}(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{8}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+3\delta^{(2)}_{2}+3\delta^{(3)}_{3}},\hskip
5.69054pts_{34}s_{34}s_{23}(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{9}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+3\delta^{(2)}_{2}+4\delta^{(3)}_{3}},$

for certain coefficients $\\{f_{i}(q)\\}_{i=1,...,9}$. Thus $Q(u_{\rm
cat})^{\underline{d}}\cdot\xi_{\Lambda_{0}}$ corresponds to a chain of vectors
$f_{i}(q)v_{i}$ as shown in the example, and the ordering in the product
$Q^{\underline{d}}$ guarantees that in each immediate step the assumption in
Proposition 4.1 is satisfied. By Proposition 4.1, for the $i$-th vector in the
chain, there exists a constant $c_{i}$ such that
$q^{c_{i}}\frac{f_{i}(q)}{f_{i-1}(q)}\rightarrow 1$ as $q\rightarrow 0$. Here
we assume $f_{0}(q)=1.$ Since $f(q)=\prod_{i}\frac{f_{i}(q)}{f_{i-1}(q)}$, it
follows from that there exists constants a $c$ such that $q^{c}f(q)\rightarrow
1$ as $q\rightarrow 0$. It proves the proposition.

######  Corollary 4.3.

The set $B(\lambda,u_{\rm cat})$ of the WKB approximation of all nonzero
vectors $Q(u_{\rm cat},h)^{\underline{d}}\cdot\xi_{0}$ is a base of
$L(\lambda)$, with a natural parameterization: the element corresponding to
$Q(u_{\rm cat},h)^{\underline{d}}\cdot\xi_{0}$ is labelled by the integer
point $\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ satisfying (51). Actually
$B(\lambda,u_{\rm cat})$ is isomorphic to the set of Gelfand-Tsetlin basis.

###  4.3 WKB operators

In this subsection, we introduce a combinatorial structure to encode the WKB
leading terms (of entries) of quantum Stokes matrices at a caterpillar point.

As we have seen from the above sections, the $q$ leading term of the action of
monomials $Q^{d}$ for all possible $d$ on the lowest vector $\xi_{0}$ produces
a basis $B(\lambda,u_{\rm cat})$ of $L(\lambda)$, parameterized by
$P_{GZ}(\lambda;\mathbb{Z})$. It coincides with the Gelfand-Tsetlin basis. If
we denote by $B^{k}(\lambda,u_{\rm cat})\subset B(\lambda,u_{\rm cat})$ the
subset consisting of the elements parametrized by
$P_{GZ}^{k}(\lambda;\mathbb{Z})\subset P_{GZ}(\lambda;\mathbb{Z})$, then the
formula (47) shows that taking the $q$ leading term of $s_{k,k+1}^{(+)}$
naturally induces a map

$\displaystyle\tilde{e}_{k}:\ B^{k}(\lambda,u_{\rm cat})\rightarrow
B(\lambda,u_{\rm
cat})~{};~{}\tilde{e}_{k}(\xi_{\Lambda})=\xi_{\Lambda+\delta^{(k)}_{l_{1}}}.$

Equivalently, as we identify $B(\lambda,u_{\rm cat})$ with
$P_{GZ}(\lambda;\mathbb{Z})$ by mapping $\xi_{\Lambda}$ to $\Lambda$,
$\tilde{e}_{k}$ can be seen as a map

$\displaystyle\tilde{e}_{k}:\ P_{GZ}^{k}(\lambda;\mathbb{Z})\rightarrow
P_{GZ}(\lambda;\mathbb{Z})~{};~{}\tilde{e}_{k}({\Lambda})={\Lambda+\delta^{(k)}_{l_{1}}}.$

The map has a canonical extension to the whole $P_{GZ}(\lambda;\mathbb{Z})$ as
follows.

First, the expression of $\tilde{e}_{k}$ is universal, i.e., doesn’t depend on
the choice of $\lambda$. Thus for any positive integer $N$, if we set
$N\lambda=(N\lambda^{(n)}_{1},...,N\lambda^{(n)}_{n})$, then $\tilde{e}_{k}$
can be equivalently seen as a map from
$P^{k}_{GZ}(\lambda;\frac{\mathbb{Z}}{N}):=\\{\Lambda\in
P^{k}_{GZ}(\lambda;\mathbb{R})~{}|~{}\lambda^{(i)}_{j}(\Lambda)\in\frac{\mathbb{Z}}{N}\\}$
to $P_{GZ}(\lambda;\frac{\mathbb{Z}}{N}):=\\{\Lambda\in
P_{GZ}(\lambda;\mathbb{R})~{}|~{}\lambda^{(i)}_{j}(\Lambda)\in\frac{\mathbb{Z}}{N}\\}$

$\displaystyle\tilde{e}_{k}^{\frac{1}{N}}(\Lambda):=\Lambda+\frac{1}{N}\delta^{(k)}_{l},$
(52)

where
$\lambda^{(i)}_{j}(\Lambda+\frac{1}{N}\delta^{(k)}_{l})=\lambda^{(i)}_{j}+\frac{\delta_{ik}\delta_{jl}}{N}$.
Now as $N\rightarrow\infty$, the discrete "dynamical system" (52) approaches
to a unique continuous system in the inner part of the whole real polytope
$P_{GZ}(\lambda;\mathbb{R})$:

$\displaystyle\tilde{e}_{k}^{t}(\Lambda):=\Lambda+t_{1}\delta^{(k)}_{r_{1}}+\cdots+t_{m}\delta^{(k)}_{r_{m}},$
(53)

where the time

$t\in\Big{[}0,\hskip 5.69054pt\sum_{i=1}^{k}({\rm
min}(\lambda^{(k+1)}_{i}(\Lambda),\lambda^{(k-1)}_{i-1}(\Lambda))-\lambda^{(k)}_{i}(\Lambda))\Big{)},$

and $t_{1},...,t_{m}$, $r_{1},...,r_{m}$ are determined by

$\displaystyle{r_{1}}=l_{1}(\Lambda+s\delta^{(k)}_{r_{1}}),\hskip
8.53581pt\text{for all}\ 0\leq s<t_{1},$ $\displaystyle
r_{i}=l_{1}(\Lambda+t_{1}\delta^{(k)}_{r_{1}}+\cdots
t_{i-1}\delta^{(k)}_{r_{i-1}}+s\delta^{(k)}_{r_{i}}),\hskip 8.53581pt\text{for
all}\ 0\leq s<t_{i},\ for\ i=2,...,m,$ $\displaystyle t=t_{1}+\cdots+t_{m}.$

Here the function $l_{1}(\Lambda)$ is defined for any point $\Lambda$ in
$P_{GZ}(\lambda;\mathbb{R})$ just as (39)–(46).

This continuation is unique, thus canonically determines an extension of
$\tilde{e}_{k}$ to the complements of the "generic part"
$P^{k}_{GZ}(\lambda;\mathbb{Z})$ in $P_{GZ}(\lambda;\mathbb{Z})$. It gives
rise to

######  Definition 4.4.

For each $k$, the WKB operator $\tilde{e}_{k}$ from $B(\lambda,u_{\rm
cat})\cong P_{GZ}(\lambda;\mathbb{Z})$ to $B(\lambda,u_{\rm cat})\cup\\{0\\}$
is given by

$\displaystyle\tilde{e}_{k}\cdot\xi_{\Lambda}:=\xi_{\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}},\hskip
14.22636pt\forall\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ (54)

Here recall that $l_{1}(\Lambda)$ is the integer given by (46), and it is
supposed that $\tilde{e}_{k}\cdot\xi_{\Lambda}$ is zero if
$\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}$ doesn’t belong to
$P_{GZ}(\lambda;{\mathbb{Z}})$.

The above computation and discussion carry to the WKB approximation of the
entries $s^{(-)}_{k+1,k}$ of lower triangular Stokes matrix $S_{\hbar-}(u_{\rm
cat})$. In particular, it produces the same set $B(\lambda,u_{\rm cat})$, and
induces operators $\tilde{f}_{k}$ on $B(\lambda,u_{\rm cat})$

$\displaystyle\tilde{f}_{k}\cdot\xi_{\Lambda}=\xi_{\Lambda-\delta^{(k)}_{l_{m_{k}}(\Lambda)}},\hskip
14.22636pt\forall\Lambda\in P_{GZ}(\lambda;\mathbb{Z}).$ (55)

One checks that $\tilde{e}_{k}$ and $\tilde{f}_{k}$ satisfy that for all
$\Lambda,\Lambda^{\prime}\in P_{GZ}(\lambda;{\mathbb{Z}})$,

$\displaystyle\varepsilon_{k}(\Lambda)={\rm
max}\\{j:\tilde{e}_{k}^{j}(\xi_{\Lambda})\neq 0\\},$
$\displaystyle\phi_{k}(\Lambda)={\rm
max}\\{j:\tilde{f}_{k}^{j}(\xi_{\Lambda})\neq 0\\},$

and

$\tilde{e}_{k}\cdot\xi_{\Lambda}=\xi_{\Lambda}^{\prime}\hskip 8.53581ptif\
and\ only\ if\hskip
8.53581pt\tilde{f}_{k}\cdot\xi_{\Lambda}^{\prime}=\xi_{\Lambda}.$

######  Definition 4.5.

We call $(B(\lambda,u_{\rm
cat}),\tilde{e}_{k},\tilde{f}_{k},\varepsilon_{k},\phi_{k})$ the WKB datum of
the Stokes matrices $S_{h\pm}(u_{\rm cat})$ at the caterpillar point
associated to the representation $L(\lambda)$.

##  5 WKB datum are crystals

In Section 5.1 we recall the notion of crystals and their tensor products. In
Section 5.2 we prove that the WKB datum of quantum Stokes matrices at $u_{\rm
cat}$ are $\mathfrak{gl}_{n}$-crystals. Furthermore, in Section 5.3 we also
realize the tensor products of $\mathfrak{gl}_{n}$-crystals by WKB analysis.

###  5.1 Crystals and tensor products

Let $\mathfrak{g}$ be a semisimple Lie algebra with a Cartan datum
$(A,\Delta_{+}=\\{\alpha_{i}\\}_{i\in
I},\Delta^{\vee}_{+}=\\{\alpha_{i}^{\vee}\\}_{i\in I},P,P^{\vee})$ be a Cartan
datum, where $P\subset\mathfrak{h}^{*}$ denotes the weight lattice, $I$
denotes the set of vertices of its Dynkin diagram, $\alpha_{i}\in I$ denote
its simple roots, and $\alpha^{\vee}_{i}$ the simple coroots.

######  Definition 5.1.

A $\mathfrak{g}$-crystal is a finite set $B$ along with maps

$\displaystyle wt$ $\displaystyle:$ $\displaystyle B\rightarrow P,$
$\displaystyle\tilde{e}_{i},\tilde{f}_{i}$ $\displaystyle:$ $\displaystyle
B\rightarrow B\cup\\{0\\},\hskip 5.69054pti\in I,$
$\displaystyle\varepsilon_{i},\phi_{i}$ $\displaystyle:$ $\displaystyle
B\rightarrow\mathbb{Z}\cup\\{-\infty\\},\hskip 5.69054pti\in I,$

satisfying for all $b,b^{\prime}\in B$, and $i\in I$,

  * •

$\tilde{f}_{i}(b)=b^{\prime}$ if and if $b=\tilde{e}_{i}(b^{\prime})$, in
which case

$wt(b^{\prime})=wt(b)-\alpha_{i},\hskip
8.53581pt\varepsilon_{i}(b^{\prime})=\varepsilon_{i}(b)+1,\hskip
8.53581pt\phi_{i}(b^{\prime})=\phi_{i}(b)-1.$

  * •

$\phi_{i}(b)=\varepsilon_{i}(b)+\langle wt(b),\alpha_{i}^{\vee}\rangle$, and
if $\phi_{i}(b)=\varepsilon_{i}(b)=-\infty$, then
$\tilde{e}_{i}(b)=\tilde{f}_{i}(b)=0.$

The map $wt$ is called the weight map, $\tilde{e}_{i}$ and $\tilde{f}_{i}$ are
called Kashiwara operators or crystal operators.

######  Definition 5.2.

Let $B_{1}$ and $B_{2}$ be two crystals. The tensor product $B_{1}\otimes
B_{2}$ is the crystal with the underlying set $B_{1}\times B_{2}$ (the
Cartesian product) and structure maps

$\displaystyle wt(b_{1},b_{2})$ $\displaystyle=$ $\displaystyle
wt(b_{1})+wt(b_{2}),$ $\displaystyle\tilde{e}_{i}(b_{1},b_{2})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lr}(\tilde{e}_{i}(b_{1}),b_{2}),&if\
\varepsilon_{i}(b_{1})>\phi(b_{2})\\\
(b_{1},\tilde{e}_{i}(b_{2})),&otherwise\end{array}\right.$
$\displaystyle\tilde{f}_{i}(b_{1},b_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lr}(\tilde{f}_{i}(b_{1}),b_{2}),&if\
\varepsilon_{i}(b_{1})\geq\phi(b_{2})\\\
(b_{1},\tilde{f}_{i}(b_{2})),&otherwise.\end{array}\right.$

###  5.2 ${\mathfrak{gl}}_{n}$-crystals from WKB approximation

Let us take the Cartan datum of type $A_{n-1}$, where $I=\\{1,2,...,n-1\\}$,
the weight lattice $P=\mathbb{C}\\{v_{1},...,v_{n}\\}$, and
$\alpha_{i}=v_{i}-v_{i+1}$, $\alpha_{i}^{\vee}$ is given by $\langle
v_{i},\alpha_{j}^{\vee}\rangle=\delta_{ij}-\delta_{i,j+1}.$ Let us consider
the WKB datum, given in Definition 5.1, of the quantum Stokes matrices at the
caterpillar point $u_{\rm cat}$ associated to the representation $L(\lambda)$.
Let us define a weight map

$wt:B(\lambda,u_{\rm cat})\rightarrow
P~{};~{}\xi_{\Lambda}\rightarrow\sum_{k=1}^{n}wt_{k}(\Lambda)v_{k}=\sum_{k=1}^{n}(\sum_{i=1}^{k}\lambda^{(k)}_{i}(\Lambda)-\sum_{i=1}^{k-1}\lambda^{(k-1)}_{i}(\Lambda))v_{k}.$

The following theorem implies Theorem 1.1 in the introduction.

######  Theorem 5.3.

The WKB datum $(B(\lambda,u_{\rm
cat}),\tilde{e}_{k},\tilde{f}_{k},\varepsilon_{k},\phi_{k})$ with the weight
map $wt$ is a ${\mathfrak{gl}}_{n}$-crystal.

Proof. By the formula (54) and (55), the WKB operators on the finite set
$B(\lambda,u_{\rm cat})$ are:

$\displaystyle\tilde{e}_{k}\cdot\xi_{\Lambda}=\left\\{\begin{array}[]{lr}\xi_{\Lambda+\delta^{(k)}_{l}},&if\
\ \varepsilon_{k}(\Lambda)>0,\\\ 0,&if\ \
\varepsilon_{k}(\Lambda)=0,\end{array}\right.where\ l={\rm
min}\\{j=1,...,k~{}|~{}X_{j}^{(k)}(\Lambda)=\varepsilon_{k}(\Lambda)\\},$ (60)
$\displaystyle\tilde{f}_{k}\cdot\xi_{\Lambda}=\left\\{\begin{array}[]{lr}\xi_{\Lambda-\delta^{(k)}_{l}},&if\
\ \phi_{k}(\Lambda)>0,\\\ 0,&if\ \
\phi_{k}(\Lambda)=0,\end{array}\right.where\ l={\rm
max}\\{j=1,...,k~{}|~{}Y_{j}^{(k)}(\Lambda)=\phi_{k}(\Lambda)\\},$ (63)

The explicit realization (60) and (63) of the WKB operators $\tilde{e}_{k}$,
$\tilde{f}_{k}$ coincides with the known crystal operators of
${\mathfrak{gl}}_{n}$-crystal realized on the Gelfand-Tsetlin basis, see e.g.,
[21].

###  5.3 Tensor products from WKB approximation

Recall that given any representation $L(\lambda)$, the quantum Stokes matrices
at $u_{\rm cat}$ produce operators $s_{ij}^{(\pm)}\in{\rm End}(L(\lambda)).$
Now given two representations $L(\lambda_{1})$ and $L(\lambda_{2})$, let us
consider the actions of

$\displaystyle s_{k,k}^{(+)}\otimes s_{k,k+1}^{(+)}+s_{k,k+1}^{(+)}\otimes
s_{k+1,k+1}^{(+)}\ \ and\ \ s_{k+1,k}^{(+)}\otimes
s_{k,k}^{(+)}+s_{k+1,k+1}^{(-)}\otimes s_{k+1,k}^{(-)},\ \ k=1,...,n-1,$ (64)

on the tensor product $L(\lambda_{1})\otimes L(\lambda_{2})$. Let
$\\{\xi_{\Lambda_{1}}\\}$ and $\\{\xi_{\Lambda_{2}}\\}$ be the basis of
$L(\lambda_{1})$ and $L(\lambda_{2})$ respectively. Similar to Proposition
4.1, one can compute the WKB leading term of the operators in (64) under the
basis $\xi_{\Lambda_{1}}\otimes\xi_{\Lambda_{2}}$. (Since the diagonal
elements $s_{k,k}^{(\pm)}$ of quantum Stokes matrices have a rather simple
expression, the computation is direct and is omitted here.) Then following the
same argument as previous sections, one verifies

######  Proposition 5.4.

The WKB approximation of the operators in (64) induces the crystal operators
$\tilde{e}_{k}$ and $\tilde{f}_{k}$ on the tensor product
$E(\lambda_{1},u_{\rm cat})\otimes E(\lambda_{2},u_{\rm cat})$ of
$\mathfrak{gl}_{n}-$crystals.

###  5.4 The structure theorem of the WKB approximation

In a summary, the WKB approximation of quantum Stokes matrices at caterpillar
points naturally produces Gelfand-Tsetlin basis in any $L(\lambda)$: the WKB
approximation of monomials $Q^{d}$ of entries of Stokes matrices produces the
basis $B(\lambda,u_{\rm cat})$ of $L(\lambda)$. Then for any generic $v\in
B(\lambda,u_{\rm cat})$, the WKB approximation of the vector
$s_{k,k+1}^{(+)}\cdot v$ takes the form

$\displaystyle
q^{-\varepsilon_{k}(\tilde{e}_{k}(v))+\iota\theta_{k}(\tilde{e}_{k}(v))}\tilde{e}_{k}(v),$
(65)

where $\tilde{e}_{k}$ is the WKB/crystal operator on $B(\lambda,u_{\rm cat})$.
The expression (65) can be understood as a structure theorem of quantum Stokes
matrices at $u_{\rm cat}$ under the Gelfand-Tsetlin basis. In a next paper, we
would like to use the method of isomonodromy deformation to obtain the same
expression (65) for quantum Stokes matrices at general $u$, with
$B(\lambda,u_{\rm cat})$ replaced by the eigenbasis $B(\lambda,u)$ of the
shift of argument subalgebra $\mathcal{A}(u)$ of $U(\mathfrak{gl}_{n})$.

## Appendix

### A. The RLL relations and semiclassical limits

This appendix explains the reason why the WKB approximation in differential
equations should be related to the theory of crystal bases in quantum groups.
It recalls the quantum and classical RLL relations satisfied by quantum and
classical Stokes matrices respectively.

In [27], we showed that $S_{h\pm}(u)=(s_{ij}^{(\pm)})$ satisfy the RLL
relation, i.e., are $L_{\pm}$-operators, in the Faddeev-Reshetikhin-Takhtajan
realization [22] of the quantum group $U_{q}(\mathfrak{gl}_{n})$ of
${\mathfrak{gl}}_{n}$. To be more precise, let us take the R-matrix $R\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})$ given by

$\displaystyle R=\sum_{i\neq j,i,j=1}^{n}E_{ii}\otimes
E_{jj}+e^{\frac{h}{2}}\sum_{i=1}^{n}E_{ii}\otimes
E_{ii}+(e^{\frac{h}{2}}-e^{-\frac{h}{2}})\sum_{1\leq j<i\leq n}E_{ij}\otimes
E_{ji}.$ (66)

Set $\mathcal{S}_{h+}(u):={S}_{h+}(u)$ and
$\mathcal{S}_{h-}(u):={S}_{h+}(u)^{-1}\in{\rm End}(\mathbb{C}^{n})\otimes{\rm
End}(L(\lambda))$. The following theorem can be found in [26, 27] in various
generalities.

######  Theorem 5.5.

For any $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$ and $L(\lambda)$, the quantum
Stokes matrices $\mathcal{S}_{h\pm}(u)$ satisfy

$\displaystyle
R^{12}\mathcal{S}^{13}_{h\pm}\mathcal{S}^{23}_{h\pm}=\mathcal{S}^{23}_{h\pm}\mathcal{S}^{13}_{h\pm}R^{12}\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})\otimes{\rm
End}(\lambda),$ (67) $\displaystyle
R^{12}\mathcal{S}^{13}_{h+}\mathcal{S}^{23}_{h-}=\mathcal{S}^{23}_{h+}\mathcal{S}^{13}_{h-}R^{12}\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})\otimes{\rm
End}(\lambda),$ (68)

and

$(\mathcal{S}_{h+})_{ii}\cdot(\mathcal{S}_{h-})_{ii}=(\mathcal{S}_{h-})_{ii}\cdot(\mathcal{S}_{h+})_{ii}=1,\
\ \ for\ i=1,...,n.$

Thus the Stokes matrices $S_{h\pm}(u_{\rm cat})$ can be understood as the
$L_{\pm}$ operators in the FRT realization of the quantum group
$U_{q}(\mathfrak{gl}_{n})$ [22] in a representation. The associated coproduct
takes the form

$\Delta(s_{ij}^{(\pm)})=\sum_{k=1}^{n}s_{ik}^{(\pm)}\otimes s_{kj}^{(\pm)},\
i,j=1,...,n.$

Thus the operators in (64) are just the coproduct of the sub-diagonal elements
of quantum Stokes matrices.

One checks that the "gauge transformation" appearing in (28) preserves the RLL
relations, that is if we write the term in (28) as
$\mathcal{G}_{h-}(u)^{-1}\mathcal{G}_{h+}(u)$ via the Gauss decomposition,
then $\mathcal{G}_{\pm}(u)$ satisfy the identities (67) and (68). Since
$S_{h\pm}(u_{\rm cat})$ are the limits of $\mathcal{G}_{h\pm}(u)$, we get

######  Corollary 5.6.

The quantum Stokes matrices $S_{h\pm}(u_{\rm cat})$ at $u_{\rm cat}$ satisfy
the RLL relations (67) and (68).

Similarly, one can show that the RLL relations are preserved under the
regularized limits of quantum Stokes matrices at a general boundary point in
$\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. We refer to [25] for the
classical analog, where the classical RLL relation (69) is shown to be
preserved under the closure of (classical) Stokes matrices. We see that the
Stokes matrices at a caterpillar point share many properties with the generic
ones, thus we can first study these properties using the closed formula at
$u_{\rm cat}$, and then use the method of isomonodromy/isospectral deformation
to understand them at generic $u$.

Semicalssical limits. Let us consider the Lie algebra ${\mathfrak{u}}(n)$ of
the unitary group ${\rm U}(n)$, consisting of skew-Hermitian matrices, and
identify ${\rm Herm}(n)\cong{\mathfrak{u}}(n)^{*}$ via the pairing $\langle
A,\xi\rangle=2{\rm Im}({\rm tr}A\xi)$. Thus ${\rm Herm}(n)$ inherits a Poisson
structure from the canonical linear (Kostant-Kirillov-Souriau) Poisson
structure on ${\mathfrak{u}}(n)^{*}$.

For any fixed $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, by varying $A\in{\rm
Herm}(n)$, we can think of the Stokes matrices $S_{\pm}(u;A)$ of equation (4)
as matrix valued functions $S_{\pm}(u)$ on ${\rm
Herm}(n)\cong\mathfrak{u}(n)^{*}$. Let us introduce the function
$M(u)={S}_{-}(u)^{-1}{S}_{+}(u)$, and let $r_{\pm}\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})$ be the standard
classical r-matrices, that is the semiclassical limit of the $R$-matrix in
(66), see e.g., [1, Formula (212) and (213)]. Then a reformulation of Boalch’s
remarkable theorem states that

######  Theorem 5.7.

[5] For any fixed $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, the matrix valued
function $M(u)$ satisfies

$\displaystyle\\{M^{1},M^{2}\\}=r_{+}M^{1}M^{2}+M^{1}M^{2}r_{+}-M^{1}r_{+}M^{2}-M^{2}r_{-}M^{1}.$
(69)

Here the tensor notation is used: both sides are ${\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})$ valued functions on
${\rm Herm}(n)$, $M^{1}:=M\otimes 1$, $M^{2}:=1\otimes M$, and the $ij,kl$
coefficient of the matrix $\\{M^{1},M^{2}\\}$ is defined as the Poisson
bracket $\\{M_{ij},M_{kl}\\}$ of the functions on ${\rm
Herm}(n)\cong\mathfrak{u}(n)^{*}$.

The right hand side of (69) coincides with the classical RLL formulation of
the Poisson brackets on the standard dual Poisson Lie group (see [1, Formula
(235)]). In particular, the relation (69) can be seen as a semiclassical limit
of the relations (67)-(68). This is why the Stokes matrices of the equation
(27) are called quantum.

### B. The proof of Theorem 3.4

The explicit expressions in Theorem 3.4 can be derived in the same way as the
ones in Theorem 2.7, except that in the non-commutative setting we should use
Proposition 5.8 to exchange the orders of the ${\rm End}(L(\lambda))$ valued
functions $\zeta^{(k)}_{i}$ and $\alpha^{(k)}_{i}$.

Following Definition 3.3 and Theorem 2.13, we have

$\displaystyle S_{h-}(u_{\rm cat})S_{h+}(u_{\rm
cat})=\Big{(}\overrightarrow{\prod_{k=1,...,n}}C^{(k)}_{h}(T)\Big{)}\cdot
e^{hT}\cdot\Big{(}\overrightarrow{\prod_{k=1,...,n}}C^{(k)}_{h}(T)\Big{)}^{-1},$
(70)

where each $C^{(k+1)}_{h}(T)$ denotes the connection matrix of

$\displaystyle\frac{dF}{dz}=h\Big{(}\iota
E_{k+1,k+1}+\frac{1}{2\pi\iota}\frac{\delta_{k+1}(T)}{z}\Big{)}\cdot F.$ (71)

Here $E_{k+1}\in{\rm End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ is the
matrix whose $(k+1,k+1)$ entry is $1\in{\rm End}(L(\lambda))$ and other
entries are zero. Therefore, to get the explicit expressions of
$S_{h\pm}(u_{\rm cat})$, we need to compute each $C^{(k+1)}_{h}(T)$.

Diagonalization in stages. To simplify the equation (71), let us diagonalize
the upper left $k$-th submatrix of its coefficient matrix. Recall that
$\zeta^{(k)}_{1},...,\zeta^{(k)}_{k}$ of $M_{k}(\zeta)$ denote the roots of
the quantum minor
$\Delta^{1,...,k}_{1,...,k}\Big{(}T(\zeta-\frac{h}{2}(k-1))\Big{)}$, which act
diagonally on the Gelfand-Zeitlin basis in $L(\lambda)$.

######  Proposition 5.8.

$(a).$ For any $2\leq k\leq n$, the $n\times n$ matrix $P_{k}$ with entries in
${\rm End}(L(\lambda))$,

$\displaystyle(P_{k})_{ij}:=\frac{(-1)^{k-j}}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}\Big{(}T(\zeta^{(k)}_{i}-\frac{1}{2}(k-1))\Big{)},$
(72) $\displaystyle(P_{k})_{ii}:=1,\ \ if\ i>k,\ \ \ \ \ \ (P_{k})_{ij}:=0,\ \
otherwise.$

diagonalizes the upper left $k$-th principal submatrix $T^{(k)}$ of $T$, and
is such that $T_{k}:=P_{k}\cdot T\cdot P_{k}^{-1}$ takes the form

$T_{k}=\begin{pmatrix}\zeta^{(k)}_{1}+\frac{1}{2}(k-1)&&&&\alpha^{(k)}_{1}&\cdots\\\
&\ddots&&&\vdots&\cdots\\\
&&\zeta^{(k)}_{k}+\frac{1}{2}(k-1)&&\alpha^{(k)}_{k}&\cdots\\\ &&&&\\\
\beta_{1}^{(k)}&\cdots&\beta_{k}^{(k)}&&E_{k+1,k+1}&\cdots\\\
\vdots&\cdots&\vdots&&\vdots&\ddots\end{pmatrix}.$

$(b).$ Furthermore, for any $1\leq k\leq n-1$, and $1\leq i,j\leq k$, we have
the commutators

$\displaystyle[\zeta^{(k)}_{i},\alpha^{(k)}_{j}]=\delta_{ij}\alpha^{(k)}_{j}.$
(73)

Proof. The Laplace expansion, of the quantum minor $M_{k}(\zeta)$ as a
weighted sum of $k$ quantum minors of size $k-1$, shows that for any tuples
$(a_{1},...,a_{k})$ and $(b_{1},...,b_{k})$,

$\Delta^{a_{1},...,a_{k}}_{b_{1},...,b_{k}}(T)(\zeta)=\sum_{j=1}^{k}(-1)^{k-j}\Delta^{a_{1},...,\widehat{a_{j}},...,a_{k}}_{b_{1},...,b_{k-1}}(T)(\zeta)\cdot
T_{a_{j}b_{k}}(\zeta+(k-1)).$

In the following, let us fix $(a_{1},...,a_{k})=(1,...,k)$ and
$(b_{1},...,b_{k-1})=(1,...,k-1)$. If we take $b_{k}=k$ and for each
$i=1,...,k$ plug in $\zeta=\zeta_{i}^{(k)}-\frac{1}{2}(k-1)$, we get the
identity

$\displaystyle
0=\Delta^{1,...,k}_{1,...,k}(T)\Big{(}\zeta^{(k)}_{i}-\frac{1}{2}(k-1)\Big{)}=\sum_{j=1}^{k}(-1)^{k-j}\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}(T)\Big{(}\zeta^{(k)}_{i}-\frac{1}{2}(k-1)\Big{)}\cdot
T_{jk}\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}.$

Similarly, taking $b_{k}=1,...,k-1$ and
$\zeta=\zeta_{i}^{(k)}-\frac{1}{2}(k-1)$ eventually implies that for each $i$,
the row vector

$v_{i}=\frac{1}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}\Big{(}(P_{k})_{i1},...,(P_{k})_{ik}\Big{)}$

satisfies

$v_{i}\cdot T^{(k)}=\Big{(}\zeta^{(k)}_{i}+\frac{h}{2}(k-1)\Big{)}v_{i}.$

Here we denote by $T^{(k)}$ the upper left $k$-th principal submatrix of $T$.
It verifies $(a)$.

The relation in $(b)$ follows from Proposition 3.7.

One can define $P_{k}$ recursively. That is for any $1\leq k\leq n-1$, we
introduce the $n\times n$ matrix $L^{(k+1)}$ with entries in ${\rm
End}(L(\lambda))$

$\displaystyle
L^{(k+1)}_{ij}=\alpha^{(k)}_{i}\cdot\frac{1}{\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2}},\
\ \ 1\leq i\leq k,\ \ 1\leq j\leq k+1,$ (74) $\displaystyle
L^{(k+1)}_{k+1,j}=1,\ \ \ \ \ 1\leq j\leq k+1.$ (75) $\displaystyle
L^{(k+1)}_{ii}=1,\ \ if\ i>k+1,\ \ \ L^{(k+1)}_{ij}=0,\ otherwise.$

Then ${L^{(k+1)}}^{-1}\cdot T_{k}\cdot{L^{(k+1)}}=T_{k+1}$, and
$P_{k}=L^{(k+1)}P_{k+1}$.

Confluent hypergeometric functions in representation spaces. Diagonalizing the
upper left $k$-th submatrix of the coefficient of (71), we are left with the
equation

$\displaystyle\frac{dF}{dz}=h\Big{(}\iota
E_{k+1,k+1}+\frac{1}{2\pi\iota}\frac{\delta_{k+1}(T_{k})}{z}\Big{)}\cdot F.$
(76)

A solution of this equation, as a function valued in ${\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$, is described using the
generalized confluent hypergeometric functions as follows. First recall that
the confluent hypergeometric functions associated to any
$\alpha_{j}\in\mathbb{C}$ and
$\beta_{j}\in\mathbb{C}\setminus\\{0,-1,-2,...\\},1\leq j\leq m$, are

${}_{k}F_{k}(\alpha_{1},...,\alpha_{k},\beta_{1},...,\beta_{k};z)=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}\cdots(\alpha_{k})_{n}}{(\beta_{1})_{n}\cdots(\beta_{k})_{n}}\frac{z^{n}}{n!},$
(77)

where $(\alpha)_{0}=1$, $(\alpha)_{n}=\alpha\cdots(\alpha+n-1),$ $n\geq 1$.
Since $\\{\zeta^{(k)}_{i}\\}_{1\leq i\leq k\leq n}$ are commutative elements
in ${\rm End}(L(\lambda))$, we can introduce the following confluent
hypergeometric functions valued in ${\rm End}(L(\lambda))$,

$\displaystyle
H(z)_{ij}=\scriptsize{\frac{1}{h(\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2})}}\cdot{{}_{k}F_{k}}(\alpha_{ij,1},...,\alpha_{ij,k},\beta_{ij,1},...,\widehat{\beta_{ij,j}},...,\beta_{ij,k+1};\iota
z),\ \ \ 1\leq i\leq k,1\leq j\leq k+1,$ $\displaystyle
H(z)_{k+1,j}={{}_{k}F_{k}}(\alpha_{k+1j,1},...,\alpha_{k+1j,k},\beta_{k+1j,1},...,\widehat{\beta_{k+1j,j}},...,\beta_{k+1j,k+1};\iota
z),\ \ i=k+1,\ 1\leq j\leq k+1,$

with the variables $\\{\alpha_{ij,l}\\}$ and $\\{\beta_{ij,l}\\}$ are given by

$\displaystyle\alpha_{ij,i}=\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2}\Big{)},\
\ \ 1\leq i\leq k,\ 1\leq j\leq k+1,$
$\displaystyle\alpha_{(k+1,j),i}=1+\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2}\Big{)},\
\ \ 1\leq j\leq k+1,$
$\displaystyle\alpha_{ij,l}=1-\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k)}_{l}-\frac{1}{2}\Big{)},\
\ \ \ l\neq i,\ 1\leq l\leq k,\ 1\leq i,j\leq k+1,$
$\displaystyle\beta_{ij,l}=1+\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k+1)}_{l}\Big{)},\
\ \ \ l\neq j,\ 1\leq l\leq k+1,\ \ 1\leq i,j\leq k+1.$

######  Proposition 5.9.

The equation (76) has a fundamental solution $F(z)\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ taking the form

$\displaystyle F(z)=\left(\begin{array}[]{cc}{\rm
diag}(ha^{(k)}_{1},...,ha^{(k)}_{k},1)&0\\\ 0&{\rm
Id}_{n-k-1}\end{array}\right)\left(\begin{array}[]{cc}\big{(}H_{ij}(z)\big{)}_{i,j=1}^{k+1}&0\\\
0&{\rm Id}_{n-k-1}\end{array}\right)\cdot z^{\frac{h[T_{k+1}]}{2\pi\iota}},$
(82)

where $[T_{k+1}]$ is the diagonal part of $T_{k+1}$, i.e.,

${[T_{k+1}]}=\left(\begin{array}[]{cc}{\rm
diag}({\zeta^{(k+1)}_{1}+\frac{k}{2}},...,{\zeta^{(k+1)}_{k+1}+\frac{k}{2}})&0\\\
0&{\rm diag}({E_{k+2,k+2}},...,{E_{n,n}})\end{array}\right).$

Proof. We only need to verify the identities

$\displaystyle
z\frac{d}{dz}(F(z)_{ij})=\frac{h}{2\pi\iota}\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}F(z)_{ij}+\frac{h}{2\pi\iota}\alpha^{(k)}_{i}F(z)_{k+1,j},\
\ \ for\ 1\leq i\leq k,$ (83) $\displaystyle
z\frac{d}{dz}(F(z)_{k+1,j})=h\iota
zF(z)_{k+1,j}+\frac{h}{2\pi\iota}\sum_{i=1}^{k}\beta^{(k)}_{i}F(z)_{ij}+\frac{h}{2\pi\iota}\zeta^{(k)}_{k+1}F(z)_{k+1,j},$
(84)

for the functions

$\displaystyle F(z)_{ij}=h\alpha^{(k)}_{i}\cdot H_{ij}(z)\cdot
z^{\frac{h}{2\pi\iota}(\zeta^{(k+1)}_{j}+\frac{k}{2})},\ \ \ for\ 1\leq i\leq
k,\ 1\leq j\leq k+1$ $\displaystyle F(z)_{k+1,j}=H_{k+1,j}(z)\cdot
z^{\frac{h}{2\pi\iota}(\zeta^{(k+1)}_{j}+\frac{k}{2})},\ \ \ for\ 1\leq j\leq
k+1.$

Since $\alpha^{(k)}_{i}$ and $\zeta^{(k)}_{i}$ do not commute, to compare the
left and right hand sides of (83), we need to keep the terms
$\alpha^{(k)}_{i}$ in the front. That is to pull the term $\alpha^{(k)}_{i}$
in

$\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}F(z)_{ij}=\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}\cdot\alpha^{(k)}_{i}H_{ij}(z)z^{\frac{1}{2\pi\iota}(\zeta^{(k+1)}_{j}+\frac{k}{2})}$

in front, via the commutator relation

$\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}\cdot\alpha^{(k)}_{i}=\alpha^{(k)}_{i}\cdot\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k+1)\Big{)}.$

Then the equation (83) is equivalent to the differential equation

$\displaystyle
z\frac{dH_{ij}}{dz}=\frac{h({\small\zeta^{(k)}_{i}-\zeta^{(k+1)}_{j}}+\frac{1}{2})}{2\pi\iota}H_{ij}-\frac{1}{2\pi\iota}H_{k+1,j},\
\ \ for\ 1\leq i\leq k,$ (85)

which in turn can be verified by the definition (77) of the confluent
hypergeometric functions. Using the similar differential equation of confluent
hypergeometric functions, as well as the characteristic polynomials of
$\delta_{k+1}(T)$, one can verify (83).

We denote by $C^{(k+1)}_{h}(T_{k})$ the connection matrix of the equation
(76). Then the Stokes matrices $S_{h+}^{(k+1)}(T)$ and the renormalized
connection matrix $\widetilde{C}_{h}^{(k+1)}(T):=C^{(k+1)}_{h}(T_{k})\cdot
L^{(k+1)}$ of the linear system (76) is described by

######  Proposition 5.10.

$(1)$ The entries $s_{ij}$ of $S_{h+}^{(k+1)}(T)$ are

$\displaystyle s_{j,k+1}=-\frac{(\iota
h)^{\frac{h(\zeta^{(k)}_{j}-E_{k+1,k+1}-1)}{2\pi\iota}}e^{\frac{-h\zeta^{(k)}_{j}}{2}}\prod_{l=1,l\neq
j}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{j}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{j}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\cdot
h\alpha^{(k)}_{j},\ \ \ for\ j=1,...,k;$

and

$s_{ii}=e^{\frac{E_{ii}}{2}},\ \ \ i=1,...,n;\hskip 14.22636pts_{ij}=0,\ \
otherwise.$

$(2)$. The entries $c_{ij}$ of the matrix $\widetilde{C}_{h}^{(k+1)}(T)$ are

$\displaystyle c_{ij}=-\frac{(\iota
h)^{\frac{h(\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i})}{2\pi\iota}}\prod\limits_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k+1)}_{l}}{2\pi\iota})\prod\limits_{l=1}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod\limits_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k)}_{l}}{2\pi\iota})\prod\limits_{l=1,l\neq
j}^{k+1}\Gamma(1+h\frac{\zeta^{(k+1)}_{l}-\zeta^{(k)}_{i}+\frac{1}{2}}{2\pi\iota})}\cdot
h\alpha^{(k)}_{i},$

for $1\leq j\leq k+1,1\leq i\leq k$;

$\displaystyle c_{k+1,j}=\frac{(\iota
h)^{\frac{hE_{k+1,k+1}-h\zeta^{(k+1)}_{j}}{2\pi\iota}}\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k+1)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k)}_{l}-\frac{1}{2}}{2\pi\iota})},\
\ \ for\ 1\leq j\leq k+1;$

and

$\ c_{ii}=1\ \ for\ \ k+2\leq i\leq n;\hskip 14.22636ptc_{ij}=0\ \ otherwise.$

Proof. It follows from Proposition 5.9 and the known asymptotics of the
functions ${}_{k}F_{k}(\alpha_{1},...,\alpha_{k},\beta_{1},...,\beta_{k};z)$:

$\displaystyle\frac{\prod_{l=1}^{k}\Gamma(\alpha_{l})}{\prod_{l=1}^{k}\Gamma(\beta_{l})}\cdot{{}_{k}F_{k}}(\alpha_{1},...,\alpha_{k},\beta_{1},...,\beta_{k};z)$
(86) $\displaystyle\sim$
$\displaystyle\sum_{m=1}^{k}\Gamma(\alpha_{m})\frac{\prod_{l=1,l\neq
m}^{k}\Gamma(\alpha_{l}-\alpha_{m})}{\prod_{l=1}^{k}\Gamma(\beta_{l}-\alpha_{m})}(\mp
z)^{-\alpha_{m}}(1+O(\frac{1}{z}))+e^{z}z^{\sum_{l=1}^{k}(\alpha_{l}-\beta_{l})}(1+O(\frac{1}{z})).$

where upper or lower signs are chosen according as $z$ lies in the upper or
lower half-plane. To be more precise, if we take the solution $F(z)$ in
Proposition 5.9, then

$\displaystyle F(z)\sim\hat{F}(z)\cdot Y\cdot U_{+},\ \ \ as\ z\rightarrow\
\infty\ in\ \overline{S}(-\pi+\varepsilon,\pi-\varepsilon)$ $\displaystyle
F(z)\sim\hat{F}(z)\cdot Y\cdot U_{-},\ \ \ as\ z\rightarrow\ \infty\ in\
\overline{S}(\varepsilon,2\pi-\varepsilon),$

for small $\varepsilon$, where

$\hat{F}(z)=(1+O(z^{-1}))e^{\iota
hzE_{k+1,k+1}}z^{\frac{h}{2\pi\iota}[T_{k}]}$

is the unique formal solution, and $U_{\pm}$ are explicit matrices determined
by the asymptotics (86). Note that the matrices $U_{\pm}$ are certain
functions of the commutative variables $\\{\zeta^{(k)}_{j}\\}_{j=1,...,k}$ and
$\\{\zeta^{(k+1)}_{j}\\}_{j=1,...,k+1}$. Then by the uniqueness in Theorem
2.2, we know that the canonical solutions

$\displaystyle F(z)=F_{+}(z)YU_{+}\hskip 8.53581ptin\ \ {\rm Sect}_{+},$
$\displaystyle F(z)=F_{-}(z)YU_{-}\hskip 8.53581ptin\ \ {\rm Sect}_{-}.$

Then by definition, the Stokes matrices are given by (here to derive the
second formula, the change in choice of ${\rm log}(z)$ is accounted for)

$\displaystyle S^{(k+1)}_{h+}(T)=YU_{-}U_{+}^{-1}Y^{-1},\ \ \ \ \ \
S^{(k+1)}_{h-}(T)=YU_{-}e^{h[T_{k+1}]}U_{+}^{-1}Y^{-1}.$

Then to get the expression of $b^{(k+1)}_{j}$ in the proposition, we only need
to use the commutative relation (73) to commute $a^{(k)}_{j}$ in the first
factor of $Y$ in $YU_{-}U_{+}^{-1}Y^{-1}$ with the arguments $\zeta^{(k)}_{j}$
in $U_{-}U_{+}^{-1}$. Similarly, one can get the expression of the
renormalized connection matrix $\widetilde{C}^{(k+1)}(T)$. The involved
computation using the asymptotics of ${}_{k}F_{k}$ is straight-forward but
lengthy. Here we only outline the computation, and refer the reader to [17]
for the full details including the explicit expressions of $U_{\pm}$.

Subdiagonal entries of quantum Stokes matrices. The connection matrices of the
equations (71) and (76) are related by

$C_{h}^{(k+1)}(T)=P_{k}^{-1}\cdot C_{h}^{(k+1)}(T_{k})\cdot P_{k},$

where $P_{k}$ is given in Proposition 5.8. By the relation
$\widetilde{C}_{h}^{(k+1)}(T)=C^{(k+1)}_{h}(T_{k})\cdot L^{(k+1)}$ and
$P_{k}=L^{(k+1)}P_{k+1}$, we get

$\displaystyle
C_{h}^{(k+1)}(T)=P_{k}^{-1}\cdot\widetilde{C}_{h}^{(k+1)}(T_{k})\cdot
P_{k+1}.$ (87)

Plugging (87) for all $k=1,...,n-1$ into (70) gives

$\displaystyle S_{h-}(u_{\rm cat})S_{h+}(u_{\rm
cat})=\Big{(}\overrightarrow{\prod_{k=1,...,n}}\widetilde{C}_{h}^{(k)}(T)\Big{)}\cdot
e^{h[T]}\cdot\Big{(}\overrightarrow{\prod_{k=1,...,n}}\widetilde{C}_{h}^{(k)}(T)\Big{)}^{-1}.$

A manipulation of (blocked) Gauss decomposition, as well as the explicit
formula of $\widetilde{C}_{h}^{(k+1)}(T)$ and $S_{h+}^{(k+1)}(T)$ given in
Proposition 5.10, gives rise to the formula of $s^{(+)}_{k,k+1}$ in Theorem
3.4. We refer the reader to [17] for the full details.

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School of Mathematical Sciences & Beijing International Center for
Mathematical Research, Peking University, Beijing 100871, China

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