Fredholm Determinants, Jimbo-Miwa-Ueno τ -Functions, and Representation Theory ALEXEI BORODIN Institute for Advanced Study AND PERCY DEIFT Courant Institute Abstract The au thor s show that a wide class of Fred hol m de terminants aris ing in the re pr e- sentation theory of “big” groups, such as the infinite-dimensional unitary group, solve Painle vé equations . Their methods are based on the theory of inte grable operators and the theory of Riemann-Hilbert problems. c 2002 Wiley Periodi- cals, Inc. Contents Introduction 1161 1. Ha rmon ic An al ys is on th e Infinite- Di me ns ional Uni tary Group 1166 2. Cont inuo us 2 F1 Kernel: Setting of the Problem 1174 3. The Resolvent Kernel and the Corresponding Riemann-Hilber t Prob lem 1183 4. Associated S ystem of L inear Dif ferential Equa tions with Rational Coef ficients 1190 5. General Setting 1197 6. Isomonodromy Deformations: The Jimbo-Miwa-Ue no τ -Function 1200 7. Painlevé VI 1206 8. Other Kernels 1210 9. Differential Equations: A General Approach 1222 App en di x. In te gra ble Ope ra to rs an d Rie mann -Hil be rt Pr oble ms 122 4 Bibliography 1227 Communications on Pure and Applied Mathematics, V ol. L V , 1160–1230 (2002)
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7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
The authors show that a wide class of Fredholm determinants arising in the repre-sentation theory of “big” groups, such as the infinite-dimensional unitary group,solve Painlevé equations. Their methods are based on the theory of integrableoperators and the theory of Riemann-Hilbert problems. c 2002 Wiley Periodi-cals, Inc.
Contents
Introduction 1161
1. Harmonic Analysis on the Infinite-Dimensional Unitary Group 1166
2. Continuous 2 F 1 Kernel: Setting of the Problem 1174
3. The Resolvent Kernel and the Corresponding Riemann-Hilbert Problem 1183
4. Associated System of Linear Differential Equations
with Rational Coefficients 1190
5. General Setting 1197
6. Isomonodromy Deformations: The Jimbo-Miwa-Ueno τ -Function 1200
7. Painlevé VI 1206
8. Other Kernels 1210
9. Differential Equations: A General Approach 1222
Appendix. Integrable Operators and Riemann-Hilbert Problems 1224
Bibliography 1227
Communications on Pure and Applied Mathematics, Vol. LV, 1160–1230 (2002)c 2002 Wiley Periodicals, Inc.
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
a union of a finite number of intervals. They showed that the corresponding Fred-
holm determinant, as a function of the endpoints of the intervals, is a τ -function (in
the sense of [34]) of the corresponding isomonodromy problem. In other words, itcan be expressed through a solution of a “completely integrable” system of partial
differential equations called the Schlesinger equations.
Kernels of the form (0.1) are of great interest in random matrix theory. Indeed,
the Fredholm determinant related to the kernel (0.1) restricted to a domain J , with
A and B being nth and (n−1)th orthogonal polynomials with the weight function ψ ,
measures the probability of having no particles in J for certain n-particle systems
called orthogonal polynomial ensembles. Such systems describe the spectra of
random unitary and Hermitian matrices. We refer the reader to [41] for details.
The results of [33] attracted considerable attention in the random matrix com-
munity. In 1992 M. L. Mehta [42] rederived the Painlevé V equation for the sine
kernel. Approximately at the same time, C. Tracy and H. Widom [58] gave their
own derivation of this result. Moreover, they produced a general algorithm (see
[61]) to obtain a system of partial differential equations for a Fredholm determi-
nant associated with a kernel of type (0.1) restricted to a union of intervals in the
case where the functions ψ , A, and B satisfy a differential equation of the form
(0.2)d
d x √
ψ( x) A( x)
√ ψ ( x) B( x) =R( x)
√ ψ ( x) A( x)
√ ψ( x) B( x) ,
where R( x) is a traceless rational 2 × 2 matrix. Using their method, they derived
different Painlevé equations for a number of kernels relevant to random matrix
theory [58]–[61].
Shortly after, J. Palmer [51] showed that the partial differential equations aris-
ing in the Tracy-Widom method are precisely the Schlesinger equations for an
associated isomonodromy problem.
Among more recent papers, we mention (in no particular order) the works [1, 2],
where a different approach to the kernels arising from matrix models can be found;
the paper [27], where the Painlevé VI equation for the Jacobi kernel was derived;the paper [22], where the theory of Riemann-Hilbert problems was applied to de-
rive the Schlesinger equations for certain kernels and to analyze the asymptotics of
solutions; the paper [28], where a multidimensional analogue of the sine kernel was
treated using the isomonodromy deformation method; and the papers [26, 64, 65],
where, in particular, a two-interval situation was reduced to an ordinary differential
equation in one variable.
Returning to our specific 2 F 1 kernel, we find that our functions ψ , A, and B
satisfy an equation of the form (0.2) (see Remark 4.8).
However, the method in [61] leads in our case to considerable algebraic com-
plexity, and we have not been able to see our way through the calculation. A similar
situation arose in the case of the (simpler) Jacobi kernel, for which the method in
[61] leads to a third-order differential equation. This equation was shown to be
equivalent to the (second-order) Painlevé VI equation only in the later work of
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
solves the RHP (R, v), which is independent of x and t . It follows that ∂∂ x
and ∂∂t
solve the same RHP and hence ∂∂ x
−1 and ∂∂t
−1 have no jump across
=R.
A short calculation then leads to the Lax pair ∂∂ x
= P and ∂∂ x
= L for some
polynomial matrices P = P(ζ ) and L = L(ζ ). Cross-differentiation ∂∂t ∂∂ x
=∂∂ x∂∂ t
then leads to the NLS equation.
As we will see in Section 4, the jump matrix v in steps 1 and 3 is easily con-
jugated to a jump matrix V that is piecewise constant. In the spirit of the above
calculation for NLS, this means that a solution M of the RHP M + = M −V can be
differentiated with respect to the variable ζ on the contour, and also with respect to
s, leading as above to the relations of the form ∂ M ∂ζ
=P M and ∂ M
∂s
=L M , where
P = P(ζ ) and L = L(ζ ) are now rational. Cross-differentiation then leads to aset of differential relations. In order to extract specific equations, such as PVI for
D(s) = det(1 − K |(s,+∞)), we recall the result in [51]. As V is piecewise con-
stant, the above equations ∂ M ∂ζ
= P M and ∂ M ∂s
= L M describe an isomonodromy
deformation, and hence one can construct an associated tau-function τ = τ (s) as
in [34]. A separate calculation (Section 6) shows that in fact D(s) = τ (s), and PVI
follows using calculations similar to those as in [32, appendix C]. The above calcu-
lations generalize immediately to the case where the interval (s, +∞) is replaced
by a union of intervals J .
The idea of reducing the Riemann-Hilbert problem for m s to a problem with a
piecewise constant jump matrix has been recently used in [22, 28, 36, 51]; see also
[29]. However, the method outlined above of performing the reduction seems to be
new.
Our main result (see equations (6.4) and (6.5), and Theorems 6.5 and 7.1) can
be stated as follows: Let
J = (a1, a2) (a3, a4) · · · (a2m−1, a2m ) ⊂ R ,
−∞ ≤ a1 < a2 < · · · < a2m ≤ +∞ ,
be a union of disjoint (possibly infinite) intervals inside the real line such that
the closure of J does not contain the points ± 12
. Denote by K J the restriction
of the continuous 2 F 1 kernel K ( x, y) introduced above to J (see Section 2 for the
complete definition of K ( x, y)). Then under suitable restrictions on the parameters
( z, z, w , w), the integral operator defined by K J is trace class, and det(1 − K J )
is a τ -function for the system of Schlesinger equations
∂A
∂ai
= [C i ,A]ai − 1
2
,∂B
∂ai
= [C i ,B ]ai + 1
2
,
∂C j
∂ai
= [C i , C j ]ai − a j
, i = j ; ∂C i
∂ai
= [A, C i ]ai − 1
2
+ [B , C i ]ai + 1
2
+ j=i
[C j , C i ]ai − a j
,
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where A,B , {C i}2mi=1 are nonzero 2 × 2 matrices (if a1 = −∞ or a2m = +∞ then
the corresponding matrices and equations are removed) satisfying
trA = trB = tr C 1 = tr C 2 = · · · = tr C 2m = 0 ,
detA = − z− z
2
2, detB = −
w−w
2
2, detC 1 = detC 2 = · · · = detC 2m = 0 ,
A+ B +2m
i=1
C i =− z+ z+w+w
20
0 z+ z+w+w
2
;
that is,
d ln det(1 − K J
) =2m
i=1
tr(AC i )
ai − 12
+tr(BC i )
ai + 12
+ j=i
tr(C jC i )
ai − a j dai .
If J = (s, +∞), then with the notation
ν1 = ν2 = z + z + w + w
2, ν3 = z − z + w − w
2,
ν4 = z − z − w + w
2,
the function
σ (s) = s − 12s + 1
2d ln det(1 − K |(s,+∞))
ds− ν2
1 s + ν3ν4
2
solves the σ -form of Painlevé VI:
− σ
s − 12
s + 1
2
σ
2 =2(sσ − σ )σ − ν1ν2ν3ν4
2 − σ + ν2
1
σ + ν2
2
σ + ν2
3
σ + ν2
4
.
As noted above, the property of the kernel K that is important for us is the
existence of a simple resolvent kernel L
=K (1
−K )−1. This property seems to
be new and was first observed in the context of the representation theory of theinfinite symmetric group S (∞) in [11]. In random matrix theory the operators K
that arise are projection operators (of Christoffel–Darboux type) or their scaling
limits. All these kernels have norm 1 and hence the operator L = K (1 − K )−1 is
not defined. However, our problem has a different origin that makes it possible not
only to define L but also to express it in an explicit way [11, 16].
The method that we introduce can be used to recover the results in [61] for in-
tegrable operators with entries satisfying equations of type (0.2). We will illustrate
the situation in the specific case of the Airy kernel in Section 9.
In the remainder of the paper we consider a variety of kernels similar to (0.1).
First, we apply our methods to the Jacobi kernel and prove that the determinant
of the identity minus the Jacobi kernel restricted to a finite union of intervals is the
τ -function of the corresponding isomonodromy problem. For the one-interval case
we again get the Painlevé VI equation, re-proving the result of [27].
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
Second, we apply our formalism to the so-called Whittaker kernel and its spe-
cial case, the Laguerre kernel. The Whittaker kernel appeared in works on the
representation theory of the infinite symmetric group [5, 6, 7, 11, 12, 48, 49]. Thecalculations for the 2 F 1 kernel are applicable to (the simpler case of) the Whit-
taker kernel. We prove that the Fredholm determinant of the Whittaker kernel on
a union of intervals is a τ -function of an isomonodromy problem, and we derive
Painlevé V in the one-interval case. This last result was proven in [57], and in [61]
for the special case of the Laguerre kernel.
Finally, we observe that the 2 F 1 kernel degenerates in a certain limit to a kernel
that we call the confluent hypergeometric kernel. This kernel appears in a problem
of decomposing a remarkable family of probability measures on the space of infi-
nite Hermitian matrices on ergodic components; see [14]. It can also be obtained
as a scaling limit of Christoffel-Darboux kernels for the so-called pseudo-Jacobi
orthogonal polynomials; see [14, 64]. We show that the Fredholm determinant
in the one-interval case for this kernel can be expressed in terms of a solution of
the Painlevé V equation. The confluent hypergeometric kernel depends on one
complex parameter r , and for real values of r the last result was proven in [64].
For r = 0 the kernel turns into the sine kernel, which recovers the original result
of [33].
The paper is organized as follows. In Section 1 we describe the representation-
theoretic origin of the problem. In Section 2 we introduce the 2 F 1 kernel and study
its properties. In Section 3 the resolvent kernel L is defined, and the matrix m in
step 1 above is considered. In Section 4 we derive the Lax pair for M as above. In
Section 5 we describe the general setting in which our method is applicable. The
reader interested primarily in the derivation of the differential equations might want
to start reading the paper with this section. In Section 6 we prove that the Fred-
holm determinants of kernels that satisfy the general conditions of Section 5 are
τ -functions of associated isomonodromy problems. In Section 7 we solve our ini-
tial problem: The Painlevé VI equation for det(1−
K |(s,+∞)) is derived. Section 8
deals with the applications of our method to the Jacobi, Whittaker, and confluent
hypergeometric kernels. Section 9 presents a general approach to kernels of the
form (0.1) subject to (0.2), worked out in the case of the Airy kernel. Finally, the
appendix contains a brief description of the formalism of integrable operators and
Riemann-Hilbert problems.
A discrete version of many of the results in this paper is given in [9].
1 Harmonic Analysis on the Infinite-Dimensional Unitary Group
By a character of a (topological) group K (in the sense of von Neumann) we
mean any central (continuous) positive definite function χ on K normalized by
the condition χ (e) = 1. Recall that centrality means χ (gh ) = χ (hg) for any
g, h ∈ K , and positive definiteness means
i, j zi ¯ z j χ (gi g−1 j ) ≥ 0 for any zi ∈ C,
gi ∈ K , i = 1, 2, . . . , n. The characters form a convex set. The extreme points
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
and every character can be written in the form (1.2)
(1.3) χ = λ1≥···≥λ N
P N (λ)χ
λ
,
where χλ is the normalized (as in (1.1)) character of U ( N ) corresponding to λ.
Note that the coordinates of λ may be negative.
Now let K = U (∞) be the infinite-dimensional unitary group defined as the
inductive limit of the finite-dimensional unitary groups U ( N ) with respect to the
natural embeddings U ( N ) → U ( N + 1). Equivalently, U (∞) is the group of
matrices U = [u i j ]∞i, j=1 such that all but finitely many off-diagonal entries are 0,
all but finitely many diagonal entries are equal to 1, and U ∗
=U −1.
A fundamental result of the representation theory of the group U (∞) is a com-plete description of its indecomposable characters. They are naturally parameter-
ized by the points
ω = (α+, β+, α−, β−, γ +, γ −) ∈ R4∞+2
such that
(1.4)
α+1 ≥ α+2 ≥ · · · ≥ 0 , β+1 ≥ β+
2 ≥ · · · ≥ 0 ,
α−1 ≥ α−2 ≥ · · · ≥ 0 , β−1 ≥ β−
2 ≥ · · · ≥ 0 ,
γ +
≥ 0 , γ −
≥ 0 ,∞
i=1
(α+i + β+i + α−i + β−
i ) < ∞ , β+1 + β−
1 ≤ 1 .
The values of extreme characters are provided by Voiculescu’s formulae [63]. This
classification result can be established in two ways: by reduction to a deep theorem
due to Edrei [23] about two-sided, totally positive sequences (see [17, 62]), and by
applying Kerov-Vershik’s asymptotic approach (see [46, 62]).
We denote the set of all points ω satisfying (1.4) by . The coordinates α+i ,
β+i , α−i , β−i , γ +, and γ − are called the Voiculescu parameters.Instead of giving a more detailed description of the indecomposable characters
(which is rather simple and can be found in [63]), we will explain why such param-
eterization is natural. It can be shown that every indecomposable character χ ω of
U (∞) is a limit of indecomposable characters χ λ( N ) of growing finite-dimensional
unitary groups U ( N ) as N → ∞. Here λ( N ) = λ1( N ) ≥ λ2( N ) ≥ · · · ≥ λ N ( N )
is a highest weight of U ( N ). The label ω ∈ of the character χω can be viewed
as a limit of λ( N )’s as N → ∞ in the following way.
We write the set of nonzero coordinates of λ( N ) as a union of two sequences
of positive and negative coordinates:
{λi ( N ) = 0} = λ+( N ) (−λ−( N )) ,
λ+( N ) = λ+1 ( N ) ≥ λ+2 ( N ) ≥ · · · ≥ λ+k ( N )
,
λ−( N ) = λ−1 ( N ) ≥ λ−2 ( N ) ≥ · · · ≥ λ−l ( N )
,
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
where λ+i > 0 and λ−i > 0 for all i , and k and l are the numbers of positive and
negative coordinates in λ( N ), respectively. Note that k
+l
≤N . We now regard
λ+( N ) and λ−( N ) as Young diagrams (of length k and l, respectively), and writethem in the Frobenius notation (see [38, sect. 1] for the definition):
λ+( N ) = p+1 ( N ) > p+2 ( N ) > · · · | q+1 ( N ) > q+2 ( N ) > · · · ,
λ−( N ) = p−1 ( N ) > p−2 ( N ) > · · · | q−1 ( N ) > q−2 ( N ) > · · · .
Then, if χω is a limit of χλ( N ) as N → ∞, we must have
(1.5)
α+i=
lim N →∞
p+i ( N )
N
, β+i
=lim
N →∞
q+i ( N )
N
,
α−i = lim N →∞
p−i ( N )
N , β−
i = lim N →∞
q−i ( N )
N .
for all i = 1, 2, . . . ; see [46, 62]. The parameters γ + and γ − can also be de-
scribed in a similar manner. Since we will not be concerned with them, we refer
the interested reader to [46, 62] for the asymptotic meaning of γ + and γ −.
Observe that the condition β+1 + β−
1 ≤ 1 in (1.4) is now easily explained—it
follows from the relation q+1 + q−1 = k + l − 2 ≤ N .
The next question that we address is how the characters of U (∞) decomposein terms of the indecomposable ones.
THEOREM 1.1 [50] Let χ be a character of U (∞). Then there exists a unique
probability measure P on such that
(1.6) χ =
χωPχ(d ω) ,
where χω is the indecomposable character of U (
∞) corresponding to ω
∈.
The measure Pχ is called the spectral measure of the character χ . The problem
of finding the spectral measure for a given character χ is referred to as the problem
of harmonic analysis for χ .
The decomposition (1.6) is the infinite-dimensional analogue of (1.3).
Since the indecomposable characters χω are limits of the normalized characters
χλ( N ) of U ( N ), it is natural to expect that the measure Pχ from Theorem 1.1 can
be approximated by discrete measures P N from (1.3) as N → ∞. To formulate
the exact result we need more notation.
Define o as the set of points ωo = (α+, β+, α−, β−) ∈ R4∞ satisfying condi-
tions (1.4). There is a natural projection → o that consists of omitting the two
gammas. Denote by P o the push-forward of the measure P under this projection.
Because we will only be concerned with statistical quantities depending on ωo and
not on γ + and γ −, it is enough to consider P o instead of P.
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
For every N = 1, 2, . . . , define a map i N that embeds the set of all highest
weights λ( N ) of U ( N ) into o as follows: For λ( N )
=(λ1( N )
≥λ2( N )
≥ · · · ≥λ N ( N )), using the above notation, we set
i N (λ) =
α+i = p+i ( N )
N , β+
i = q+i ( N )
N , α−i = p−i ( N )
N , β−
i = q−i ( N )
N
∈ o .
THEOREM 1.2 [50] Let χ be a character of U (∞) , χ N be its restriction to U ( N ) ,
and
(1.7) χ |U ( N ) =λ1≥···≥λ N
P N (λ)χλ , P N (λ) ≥ 0 ,λ1≥···≥λ N
P N (λ) = 1 ,
be the decomposition of χ N on indecomposable characters. Then the projection Poχ
of the spectral measure Pχ of χ is the weak limit of push-forwards of the measures
P N under the embeddings i N . In other words, if F is a bounded continuous function
on o , then
lim N →∞
λ1≥···≥λ N
F (i N (λ)) P N (λ) = ω∈o
F (ω) P o(d w) .
Now, following [16], we apply the above general theory to a specific family of
decomposable characters of U (
∞) constructed in [50]. The group U (
∞) does not
carry Haar measure, and hence the naive definition of the regular representationfails. The representations in [50] should be viewed as analogues of the nonexist-
ing regular representation of U (∞). A beautiful geometric construction of these
representations can also be found in [50].
For every N = 1, 2, . . . and a highest weight λ = (λ1 ≥ λ2 ≥ · · · ≥ λ N ), set
P N (λ) = c N · dim2 N (λ) ·
N i=1
f (λi − i ) ,
f ( x) =1
( z − x)( z − x)(w + N + 1 + x)(w + N + 1 + x),
c N = N
i=1
( z + w + i )( z + w + i )( z + w + i )( z + w + i )(i )
( z + z + w + w + i ),
where dim N (λ) is the dimension of the irreducible representation of U ( N ) corre-
sponding to λ,
dim N λ = i≤i< j≤ N
λi − i − λ j + j
j−
i;
see, e.g., [66]. Here z, z , w, and w are complex parameters such that P N (λ) > 0
for all N and λ. This implies that
(1) z = z ∈ C \ Z or k < z, z < k + 1 for some k ∈ Z, and
(2) w = w ∈ C \ Z or l < z, z < l + 1 for some l ∈ Z.
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
and f ( x) was introduced in (1.8).PROPOSITION 1.4 [16] For any highest weight λ = (λ1 ≥ λ2 ≥ · · · ≥ λ N )
P N (λ) = det L( N ) X (λ)
det(1 + L ( N )),
where L( N ) X (λ) denotes the finite submatrix of L ( N ) on X (λ) × X (λ). Moreover, if a
finite-point configuration X ⊂ X( N ) is not of the form X = X (λ) for some highest
weight λ , then det L( N ) X = 0.
Proposition 1.4 implies that P N is a determinantal point process (see [10, ap-pendix] and [16, 54] for a general discussion of such processes). In particular, this
implies the following claim:
COROLLARY 1.5 [16] The matrix L ( N ) defines a finite rank (and hence trace class)
operator in 2(X( N )). The correlation functions
ρ( N )k ( x1, x2, . . . , xk ) = P N {λ | { x1, x2, . . . , xk } ⊂ X (λ)}
of the process P N have the determinantal form
ρ( N )k ( x1, x2, . . . , xk ) = det K ( N )( xi , x j )k
i, j=1 , k = 1, 2, . . . ,
where K ( N )( x, y) is the matrix of the operator K ( N ) = L( N )/(1+ L( N )) in 2(X( N )).
Explicit formulae for K ( N ) can be found in [16].
Now we will describe the limit situation as N → ∞. Define the continuous
phase space
X = X(∞) = R \ ± 12
and divide it into two parts:
X = Xin Xout , Xin = − 1
2 ,1
2 , Xout = −∞, −1
2 1
2 , +∞ .
To each point ω ∈ o we associate a point configuration in X as follows:
ω = (α+, β+; α−, β−) → X (ω)
= α+i + 1
2
12
− β+i
−α− j − 12
− 12
+ β− j
,
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
P o{ω | X (ω) intersects each interval ( xi , xi + xi ), i = 1, 2, . . . , k } x1 x2 · · · xk
of the process P have determinantal form
ρk ( x1, x2, . . . , xk ) = det[K ( xi , x j )]k i, j=1 , k = 1, 2, . . . ,
where K ( x, y) is a kernel on X that is the scaling limit of the kernels K ( N )( x, y)
introduced above:
(1.8) K ( x, y) = lim N →∞
N · K ( N )( x N , y N ) , x, y ∈ X .
The kernel K ( x, y) is called the continuous 2 F 1 kernel and is precisely the
kernel in (0.1) for x, y > 12
. Explicit formulae for K ( x, y) can be found in the next
section. This kernel is a real-analytic function of the parameters ( z, z
, w , w
). Wewill use the same notation for its natural analytic continuation.
It is worth noting that the correlation functions ρk ( x1, x2, . . . , xk ) determine the
process P uniquely.
It is a well-known elementary observation that the probability that a determi-
nantal point process with a correlation kernel K does not have particles in a given
part J of the phase space is equal to the Fredholm determinant det(1 − K| J ); see,
e.g., [54, 58].1
In what follows we study determinants of the form det(1
−K
| J ) where K is the
continuous 2 F 1 kernel and J is a union of finitely many (possibly infinite) intervals.
1If the correlation kernel is self-adjoint and this probability is nonzero, then the integral operatordefined by the kernel K is of trace class and the determinant is well-defined; see [54, theorem 4].For kernels that are not self-adjoint, the existence of the determinant, generally speaking, needs to be justified; see, e.g., the end of Section 2.
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
All four functions Rout, S out, Rin, and S in are invariant with respect to the trans-positions z ↔ z and w ↔ w. This follows easily from the above formulae and
the identities
2 F 1
a, b
c
ζ
= (1 − ζ )c−a−b2 F 1
c − a, c − b
c
ζ
,
2 F 1
a, b
c
ζ
= 2 F 1
b, a
c
ζ
.
Since
2 F 1a, b
c
ζ = 2 F 1
a, b
c
ζ
,
where the bar means complex conjugation, the parameters ( z, z ), as well as
(w,w), are either real or complex conjugate, and the functions Rout, S out, Rin,
and S in take real values on Xout and Xin, respectively.
Further, let us denote by C the following change of parameters and independent
variable: ( z, z , w , w, x) ←→ (w,w, z, z , − x). Then
C (ψout) = ψout , C (ψin) = ψin ,C ( Rout) = Rout , C (S out) = −S out , C ( Rin) = Rin , C (S in) = −S in .
For ψout and ψin the claim is obvious from the definition. For Rout and S out, the
symmetry relation follows from the identity
2 F 1
a, b
c
ζ
= (1 − ζ )−a2 F 1
a, c − b
c
ζ
ζ − 1
=(1
−ζ )−b
2F
1c
−a, b
c ζ
ζ − 1 .
For Rin and S in, the symmetry is a corollary of the symmetries of ψin, Rout, S out,
and the branching relation (2.1) below.
2.3 Symmetries of the Kernel
Since the functions Rout, S out, Rin, and S in take real values, the kernel K ( x, y)
is real. Moreover, from the explicit formulae for the kernel, it follows that
K out,out( x, y) = K out,out( y, x) , K in,in( x, y) = K in,in( y, x) ,
K in,out( x, y) = −K out,in( y, x) .
This means that the kernel K ( x, y) is (formally) symmetric with respect to the
indefinite metric id ⊕(− id) on L 2(X, d x) = L 2(Xout, d x) ⊕ L 2(Xin, d x).
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
of these functions. We can view Rout and S out as functions that are analytic and
single-valued on C \ Xin, and Rin and S in as functions that are analytic and single-
valued on C \ Xout. (Recall that the Gauss hypergeometric function can be viewed
as an analytic and single-valued function on C \ [1, +∞).)
For a function F (ζ ) defined onC\R, we will denote by F + and F − its boundary
values:
F +( x) = F ( x + i 0) , F −( x) = F ( x − i 0) .
We will show below that
on Xin
1
ψin
S −out − S +out
2π i= Rin ,
1
ψin
R−out − R+
out
2π i= S in ,(2.1)
on Xout
1
ψout
S −in − S +in
2π i= Rout ,
1
ψout
R−in − R+
in
2πi= S out .(2.2)
We will use the following formula for the analytic continuation of the Gauss hy-
pergeometric function (see [24, 2.1.4(17)]):
2F
1a, b
c ζ =(b − a)(c)
(b)(c − a)(−
ζ )−a
2F
1a, 1
−c
+a
1 − b + a ζ −1+ (a − b)(c)
(a)(c − b)(−ζ )−b
2 F 1
b, 1 − c + b
1 − a + b
ζ −1
.
(2.3)
This formula is valid if b − a /∈ Z, c /∈ {0, −1, −2, . . . }, and ζ /∈ R+.
Both of the formulae in (2.1) are direct consequences of (2.3) and the trivial
relation
on R−(ζ u )− − (ζ u )+
2π i= −sin(πu)
π(−ζ )u , u ∈ C .
To verify the first formula of (2.2), we use the relation (2.3) for both hyperge-ometric functions in the definition of S in. Thus, we get four summands in total.
After computing the jump (S −in − S +in )/2π i , the second and the fourth summands
cancel out. As for the first and the third summands, they produce exactly ψout Rout,
which can be seen from the identities
(s)(1 − s) = π
sin(π s), s ∈ C ,
sin(π( z + w)) sin(π( z + w
))sin(π( z + z + w + w)) sin(π( z − z ))
+sin(π( z + w)) sin(π( z + w))
sin(π( z + z + w + w)) sin(π( z − z))= 1 .
The second part of (2.2) is proved similarly.
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The restriction b − a /∈ Z for (2.3) in our situation means that our proof works
when z
=z. For z
=z the result is obtained by the limit transition z
→z in
(2.1) and (2.2).
2.5 Differential Equations (due to G. Olshanski)
We use Riemann’s notation
P
t 1 t 2 t 3a b c ζ
a b c
to denote the two-dimensional space of solutions to the second-order Fuchs’ equa-
tion with singular points t 1, t 2, and t 3 and exponents a and a , b and b, and c andc; see, e.g., [24, 2.6]. If a − a /∈ Z, then this means that about t 1, there are two
solutions of the form
(ζ − t 1)a × {a holomorphic function} , (ζ − t 1)a × {a holomorphic function} .
If a = a, then the basis of the space of solutions near t 1 has the form
(ζ − t 1)a × {a holomorphic function} ,
ln(ζ − t 1)(ζ − t 1)a × {a holomorphic function} .
The holomorphic functions above must take nonzero values at t 1. For t 2 and t 3 thepicture is similar.
We always have a + a + b + b + c + c = 1.
The Gauss hypergeometric function 2 F 1
a,bc | ζ
belongs to the space
P
0 ∞ 1
0 a 0 ζ
1 − c b c − a − b
,
and, since it is holomorphic around the origin, it corresponds to the exponent 0 at
the origin.
Riemann showed (see [24, 2.6.1]) that
(2.4)
ζ − t 1
ζ − t 2
κ ζ − t 3
ζ − t 2
µP
t 1 t 2 t 3a b c ζ
a b c
=
t 1 t 2 t 3a + κ b − κ − µ c + µ ζ
a
+κ b
−κ
−µ c
+µ
,
where if t n = ∞, then the factor ζ − t n should be replaced by 1, and
P
t 1 t 2 t 3a b c ζ
a b c
= P
s1 s2 s3
a b c η
a b c
,
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PROPOSITION 2.1 The kernels K J out,out( x, y) and K J
in,in( x, y) define positive trace
class operators in L 2( J out, d x) and L 2( J in, d x) , respectively.
PROOF: Sections 2.1 and 2.3 above imply that the kernels K J out,out( x, y) and
K J in,in( x, y) are smooth, real-valued, and symmetric. Moreover, the principal mi-
nors of these kernels are always nonnegative, because the kernel K was obtained
as a limit of matrices with nonnegative principal minors; see Section 1. Thus, it
remains to prove that the integrals J out
K J out,out( x, x)d x and
J in
K J in,in( x, x)d x
converge. For the second integral the claim is obvious, since J in ⊂ (− 12
, 12
), and the
integrand is bounded on J in. For the first integral we need to control the behavior
of the integrand near infinity (if J out is not bounded). Since ψout( x) = O( x−S) as x → ∞, by Section 2.1 and (2.8) we see that
K ( x, x) = O( x−2−S) , x → ∞ .
AsS > −1, the integral converges.
We will assume that K J out,in( x, y) = 0 and K J
in,out( x, y) = 0 if ( x, y) does not
belong to the domain of definition of the corresponding kernel ( J out × J in for the
first kernel and J in
×J out for the second one).
PROPOSITION 2.2 The kernel K 0( x, y) = K J out,in( x, y) + K J
in,out( x, y) defines a
trace class operator in L 2( J , d x).
PROOF: Consider the operator − d 2
d x2 acting, respectively, on
(i) C ∞0 (R) , (ii) C ∞0 ( J ) , (iii) C ∞0 (R \ J ) .
In all three cases the operator is essentially self-adjoint, giving rise to the pos-
itive self-adjoint operators H , H J , and H R\ J in L2(R), L 2( J ), and L 2(R \ J ),
respectively. It is well-known (see e.g., [52, theorem XI.21]) that the operatorT = (1 + x 2)−1(1 + H )−1 is trace class in L 2(R). A direct proof can be given
as follows. Let p denote the (self-adjoint) closure of −i d d x
acting on C ∞0 ; then
H = p2. Commuting (1 − i x)−1 and (1 + i p)−1 in the representation
T = (1 + i x)−1(1 − i x)−1(1 + i p)−1(1 − i p)−1 ,
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
vanishes at a j for j = i . Clearly K 2( x, y) = 0 for x ∈ ∂ J , which implies that
K 2(
·, y)
∈dom H J for all y
∈Xout. By using the decay conditions (2.8) (each
differentiation with respect to x gives an extra power of decay), it follows that(1 + H J )(1 + x 2)K 2( x, y) gives rise to a bounded operator on L 2( J ), and hence
K 2 = (1 + x 2)−1(1 + H J )
−1
(1 + H J )(1 + x 2)K 2
is trace class. But clearly K 2 is a finite-rank perturbation of K out,in. A similar
computation is true for K in,out, and we conclude that K 0 is trace class on L 2( J ).
Propositions 2.7 and 2.8 prove that the operator
K J = K J
out,out
K J
out,inK J in,out K J
in,in
is trace class. This shows that the determinant det(1 − K J ) is well-defined.
3 The Resolvent Kernel and
the Corresponding Riemann-Hilbert Problem
Starting from this point we assume that the reader is familiar with the material
in the appendix.
As was explained in Section 1 (see Theorem 1.6 et seq.), the 2 F 1 kernel K
is a limit of certain discrete kernels that we denoted as K ( N ). Moreover, these
discrete kernels have rather simple resolvent kernels L ( N ) = K ( N )/(1 − K ( N )); see
Corollary 1.5. The kernels L ( N ) are integrable, and thus the kernels K ( N ) can be
found through solving (discrete) Riemann-Hilbert problems; see [8].
Our first observation is that the kernel L ( N ) admits a scaling limit as N → ∞.
Recall that for x ∈ X, we denote by x N the point of the lattice X( N ) that is closest
to x N .
The proof of the following proposition is straightforward.
PROPOSITION 3.1 [16] The limit
L( x, y) = lim N →∞
N · L( N )( x N , y N ) , x, y ∈ X ,
exists. In the block form corresponding to the splitting X = Xout Xin , the kernel
L( x, y) has the following representation:
L =
0 A
− A∗ 0
,
where A is a kernel on Xout ×Xin of the form
A( x, y) =√
ψout( x)ψin( y)
x − y,
where the functions ψout and ψin were introduced at the beginning of Section 2.
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Our conditions on the parameters ( z, z , w , w) imply that |( z− z )| < 1. Then
the asymptotic formulae of Section 2.6 imply that m is locally square integrable
near ζ = 12
, and so are
m0 and
m−1
0 , as follows from the formula above. Since
m
and m0 locally solve the same RHP, we obtain that mm−10 has no jump on R near
ζ = 12 , and it is locally integrable as a product of two locally square integrablefunctions. Hence, this ratio is a locally holomorphic function. We denote this
holomorphic function by H 1/2(ζ ), and set
U 1 = U , U 2 = U
1 −2πi C ( z, z )
0 1
.
Because v( x) in (3.2) has determinant 1, it follows that det m+( x) = det m−( x).
Also, as above, det m(ζ ) = det
m(ζ ) is locally integrable. Thus, det m(ζ ) is entire.
If z + z
+ w + w
> 0, then as noted in Proposition 3.2, det m(ζ ) → 1 as ζ → ∞,and hence, by Liouville’s theorem, det m(ζ ) ≡ 1. Analytic continuation in the
parameters z, z, w, and w ensures that the same is true for all (allowable) values
of the parameters. The fact that H 1/2( 12
) is invertible now follows from the fact that
det m(ζ ) = det C (ζ ) ≡ 1, and det U 1 and det U 2 are nonzero. The proof of (i) is
complete.
Assume now that z = z . Then there exists a nondegenerate matrix V such that
1 −2πi C ( z, z )
0 1 e−iπ( z+ z) 0
2π i eiπ( z+ z) =
V −1 1 1
0 1 V ,
and the local solution of the RHP with the jump matrix v has the form
m0(ζ ) =
1 ln
ζ − 12
0 1
V .
Repeating word for word the argument above, we get (ii) with
V 1
=V , V 2
=V
1 −2π i C ( z, z )
0 1 .
Similarly to Proposition 3.3 we have the following:
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
Since L∗ = − L , we know that if L is bounded, then (1 + L) is invertible. It
seems very plausible that whenever the operator L is bounded, the relation K
= L(1 + L)−1 should hold. We are able to prove this under the additional restriction z + z + w + w > 0.
PROPOSITION 3.6 Assume that z + z +w+w > 0 , | z + z| < 1 , and |w+w| < 1.
Then K = L(1 + L)−1.
PROOF: Since L is bounded and L = − L∗, L has a pure imaginary spectrum,
and 1 + L is invertible. Hence, it is enough to show that K + K L = L. The
restrictions on the parameters and the asymptotics of the functions Rout, S out, Rin,
and S in from Section 2.6 imply that the relation (2.1) and (2.2) can be rewritten inthe integral form:
(3.7) Xout
ψout( x) Rout( x)
x − yd x = −S in( y) ,
Xout
ψout( x)S out( x)
x − yd x = 1 − Rin( y) ,
Xin
ψin( x)S in( x)
x − yd x = 1 − Rout( y) ,
Xin
ψin( x) Rin( x)
x − yd x = −S out( y) .
The 1’s on the right-hand side appear because Rout(ζ ) ∼ 1 and Rin(ζ ) ∼ 1 asζ → ∞. The restriction z + z + w + w > 0 is needed to ensure the convergence
of the first integral at infinity. Indeed, ψout( x) Rout( x) ∼ x− z− z−w−w as x → ∞.
The identity
(3.8) K ( x, y) + X
L( x, α) K (α, y)d α = L( x, y)
for all x, y
∈X follows directly from the relations (3.7); see [13, theorem 3.3] for
a similar computation. On the other hand, by (2.8) we see that for any g ∈ C ∞0 (X),
G(α) = X
K (α, y)g( y)d y = K g(α)
lies in L 2(X, d α). Integrating (3.8) against g( y), we see that (1 + L)G = Lg
and hence K g = (1 + L)−1 Lg in L 2(X). It follows that K extends to a bounded
operator (1 + L)−1 L = L(1 + L)−1 in L2(X). Conversely, we see that the bounded
operator L(1
+L)−1 has a kernel action given by the 2 F 1 kernel K ( x, y).
Proposition 3.6 has the following corollary, which will be important for us later.
COROLLARY 3.7 Assume that z + z +w +w > 0 , | z + z | < 1 , and |w +w| < 1.
Then, in the notation of Section 2 , the operator 1 − K J is invertible.
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
Hence, K J is an integrable kernel. Since J is bounded away from the points ± 12
,
it is easy to see that the functions F i and G i (which are, in fact, the functions
√ ψout Rout, √ ψout S out, √ ψin Rin, and √ ψin S in rearranged in a certain way) belongto L p( J , d x) ∩ L∞( J , d x) for any p > 2S−1. This follows from (2.8) and (2.9).
Set
v J = I − 2π i F G t =
1 − 2π i F 1G1 −2πi F 1G2
−2πi F 2G1 1 − 2π i F 2G2
.
Note that F t ( x)G( x) = F 1( x)G1( x) + F 2( x)G2( x) = 0.
PROPOSITION 4.1 Assume that the operator 1− K J is invertible. Then there exists
a solution m J of the normalized RHP ( J , v J ) such that the kernel of the operator
R
J
= K
J
(1 − K
J
)
−1
has the form
R J ( x, y) = F 1( x)G 1( y) + F 2( x)G 2( y)
x − y,
F = m J +F = m J −F , G = m J +G = m J −G .
The matrix m J is locally square integrable near the endpoints of J .
PROOF: See Proposition A.2 and the succeeding comment.
Concerning the invertibility of (1 − K J ), see Corollary 3.7 and Remark 3.8.
Later on we will need the following property of the decay of m J at infinity:
PROPOSITION 4.2 As ζ → ∞ , ζ ∈ C \ R , we have m J (ζ )m−1
J (ζ ) = o(|ζ |−1).
PROOF: We will give the proof for J = (s, +∞), s > 12
. The proof for general
J is similar.
Observe that det v J ≡ 1. Then det m J has no jump on J . Since m J is square
integrable near t , det m J is locally integrable. Moreover, det m J (ζ ) → 1 as ζ →∞, because m J (ζ ) → I . Again by Liouville’s theorem, det m J ≡ 1, and m−1
J is
bounded near ζ = ∞. Therefore, it suffices to show that m J (ζ ) = o(|ζ |−1).
The proof of proposition A.2 given in [20] implies that for ζ ∈ C \ R
m J (ζ ) = I − +∞
s
m J +(t )F (t )G t (t )
t − ζ dt = I −
+∞s
m J −(t )F (t )G t (t )
t − ζ dt ;
therefore,
m J (ζ ) = −
+∞s
m J +(t )F (t )G t (t )
(t − ζ )2dt = −
+∞s
m J −(t )F (t )G t (t )
(t − ζ )2dt .
If ζ > const |ζ |, then |t − ζ | > const |ζ |. That is, the distance of the point ζ to
the contour of integration is of order |ζ |. Since m J (t ) is bounded and F (t )G t (t )
decays at infinity as a positive power of t , we see that m J (ζ ) = o(|ζ |
−1
).If the point ζ is closer to the real line and, say, ζ < 0, we can deform the line
of integration up to the line s + t eiθ , 0 < θ < π2
. In other words,
m J (ζ ) = −
+∞0
m J (s + t eiθ )F (s + t eiθ )G t (s + teiθ )
(s + t eiθ − ζ )2eiθ dt .
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
PROOF: Since M satisfies the jump condition with a piecewise constant jumpmatrix V (Lemma 4.5), M satisfies the jump condition with exactly the same jump
matrix. Therefore, M M −1 has no jump across X. Note that
det M = det m J det m(det C )−1 ≡ 1 ,
and hence M −1 exists.
Thus, we know that M M −1 is a holomorphic function away from the points
{±12} ∪ {ai}2m
i=1. We now investigate the behavior of M near these points.
Near ζ
=12
, m J (ζ ) is holomorphic, and the behavior of m(ζ )C −1(ζ ) is de-
scribed by Proposition 3.3. This implies, in the notation of Proposition 3.3, that for
z = z
M (ζ ) M −1(ζ ) = 1
ζ − 12
m J
12
H 1/2
12
z− z
20
0 z− z2
H −1
1/2
12
m−1
J
12
+ O(1) ,
and for z = z
M (ζ ) M −1(ζ ) = 1
ζ
−12
m J
12
H 1/2
12
0 1
0 0 H −1
1/212
m−1 J
12
+ O(1) .
Similarly, near ζ = − 12
we have the following: For w = w,
M (ζ ) M −1(ζ ) =1
ζ + 12
m J
− 12
H −1/2
− 12
w−w2
0
0 w−w2
H −1−1/2
−12
m−1
J
− 12
+ O(1) ,
and for w = w,
M (ζ ) M −1(ζ ) =1
ζ + 12
m J
− 12
H −1/2
−12
0 1
0 0
H −1−1/2
−12
m−1
J
− 12
+ O(1) .
As for the points {a j}2m j=1, we will prove the following claim:
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
PROOF OF LEMMA 4.7: Let us give a proof for an odd value of j . The proof
for the even j ’s is obtained by changing the sign of ζ . We will omit the subscript
j in a j and C j .
Near the point a the jump matrix for M has the form (Lemma 4.5)
( M −( x))−1 M +( x) =
C o + 2π i f o(go)t , x < a ,
C o , x > a ,
where f o = C − f and go = C −t + g are locally constant vectors, and C o = C −C −1
+
is a locally constant matrix. Note that ((C o)−1 f o)t go = f t g = 0.
Set
M (ζ ) = M ( ζ ) , ζ > 0 ,
M (ζ )C o , ζ < 0 .
Then
( M −( x))−1 M +( x) =
I + 2π i (C o)−1 f o(go)t , x < a .
I , x > a ,
and we note that
M o = exp 1
2π i ln( I + 2πi (C o)−1 f o(go)t ) ln(ζ − a)= exp
(C o)−1 f o(go)t ln(ζ − a)
= I + (C o)−1 f o(go)t ln(ζ − a)
is also a solution of this local RHP. (Here we use the fact that ((C o)−1 f o(go)t )2 =0.) Hence,
M o(ζ ) = M o( ζ ) , ζ > 0 ,
M o(ζ)(C o)−1 , ζ < 0 ,
is a local solution of the RHP for M near ζ = a.Since M = m J mC −1, m and C −1 are bounded near a, and m J is square inte-
grable near a (Proposition 4.1), we conclude that M is square integrable near a.
Clearly, M o is also locally square integrable, and det M o ≡ 1. Hence, H a (ζ ) ≡ M (ζ)( M o(ζ))−1 is locally integrable and does not have any jump across R near a.
Therefore, H a(ζ ) is holomorphic near a. Since det M ≡ 1, H a is nonsingular. We
obtain
(4.3) M (ζ ) = H a (ζ ) M o( ζ ) .
Computing M M −1 explicitly, we arrive at the desired claim with
h = H a H −1a and C = H a(a)(C o)−1 f ogo H −1
a (a) .
Since (C o)−1 f ogo is nilpotent and nonzero, the proof of Lemma 4.7 and Theo-
rem 4.6 is complete.
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Thus, the X γ ’s provide a “monodromy representation” of the fundamental group
of C
\ {b1, b2, . . . , bn
}:
X : π1(C \ {b1, b2, . . . , bn}) → G L( N ,C) , [γ ] → X γ .
Now view the singular points {b1, b2, . . . , bn} as variables. It may happen that
moving these points a little and changing the rational matrix B(ζ ) in an appropriate
way, we do not change the monodromy representation. In such a case we say that
we have an isomonodromy deformation of the initial differential equation.
For general information on isomonodromy deformations, we refer the reader to
[31, 34].
Without loss of generality, we can assume that, in the notation of Section 5, the
first k ≤ n points {b1, b2, . . . , bk } of the set {b j}n j=1 are exactly those endpoints of J that are not the endpoints of X. Clearly, {b j}k
j=1 ⊂ {a j }2m j=1.
The following statement is immediate:
PROPOSITION 6.1 Under the assumptions of Theorem 5.1 (or Theorem 5.3) , there
exists > 0 with the property that moving the points b1, b2, . . . , bk within their
-neighborhoods inside R provides an isomonodromy deformation of the equation
(5.1) (or of the equation (5.3) , respectively).
Note that the matrices {B j}n j=1 are now functions of b1, b2, . . . , bk .
PROOF: Choose > 0 so that the points b1, b2, . . . , bk cannot collide between
themselves or with the other endpoints bk +1, bk +2, . . . , bn . Since the matrix M =m J mC −1 has a nonzero determinant, this matrix can be viewed as a fundamental
solution of (5.1). The monodromy of this solution, as we go along any closed
curve that avoids the singular points, is equal to the product of the values of the
jump matrix V or their inverses at the points where the curve meets X. Since V
does not depend on b1, b2, . . . , bk , the proof is complete.
In 1912, Schlesinger realized that if the matrix B(ζ ) has simple poles, then a
deformation of the b j ’s preserves monodromy if and only if the residues {B j} of Bat the singular points, as functions of the b j ’s, satisfy a certain system of nonlinear
partial differential equations. These equations are called the Schlesinger equations.
The analogues of the Schlesinger equations in the case when B has higher-order
poles were derived in [34].
In what follows we will use the Schlesinger equations arising from the isomon-
odromy deformation described in Proposition 6.1. Since our situation is simpler
than the general case in [34], it is more instructive to rederive the equations that we
need rather than to refer to the general theory.
PROPOSITION 6.2 (Schlesinger Equations) (i) The matrices {B j}n j=1 from for-mula (5.1) , as functions in b1, b2, . . . , bk , satisfy the equations
(6.1)∂B l
∂b j
= [B j ,B l]b j − bl
,∂B j
∂b j
=
1≤l≤nl= j
[B j ,B l]bl − b j
,
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where j = 1, 2, . . . , k and l = 1, 2, . . . , n.
(ii) The matrices
{B j
}n
j=1
from (5.3) , as functions in b1, b2, . . . , bk , satisfy the
equations
(6.2)∂B l
∂b j
= [B j ,B l]b j − bl
,∂B j
∂b j
=
1≤l≤nl= j
[B j ,B l]bl − b j
− [B j , D] ,
where j = 1, 2, . . . , k and l = 1, 2, . . . , n.
SKETCH OF THE PROOF: Since M satisfies an RHP with a constant jump ma-
trix V , the derivative M b j = ∂ M ∂b j
satisfies the same jump condition, j = 1, 2, . . . , k .
Hence, the matrix M b j M −1 has no jump across X. Thus, it is holomorphic inC \ {b j }. As was shown in the proof of Lemma 4.7, locally near ζ = b j we have
M (ζ ) = H (ζ ) exp
(C o)−1 f o(go)t ln(ζ − b j )
= H (ζ ) I + (C o)−1 f o(go)t ln(ζ − b j )
,
where H is holomorphic. With some additional effort, one can show that H is
differentiable with respect to b j , and differentiating with respect to b j , we see that
M b j
(ζ ) M −1(ζ )= −
H (b j )(C o)−1 f o go H −1(b j )
ζ − b j +O(1)
= −B j
z − b j +O(1) .
Since M ∼ I at ζ = ∞, one can show that M b j M −1 → 0 as ζ → ∞. By
Liouville’s theorem, M b j M −1 + B j /( z − b j ) ≡ 0, and
(6.3) M b j = − B j
ζ − b j
M .
The linear equations (5.1) and (6.3) form a Lax pair for (6.1).
Differentiating (5.1) with respect to b j and (6.3) with respect to ζ , subtracting
the results, and multiplying the difference by M −1
on the right, we obtainn
l=1
∂B l
∂b j
1
ζ − bl
= 1
ζ − b j
1≤l≤n
l= j
[B l ,B j ]ζ − bl
.
The equality of residues at the points {bl}nl=1 on both sides of this identity gives
(6.1). The equations (6.2) are proved in exactly the same way.
COROLLARY 6.3 In the notation of Theorem 4.6 ,
∂A
∂a j= [
C j ,A
]a j − 1
2
, ∂B
∂a j= [
C j ,B
]a j + 1
2
,(6.4)
∂C l
∂a j
= [C j , C l]a j − al
,∂C j
∂a j
= −[C j ,A]a j − 1
2
− [C j ,B ]a j + 1
2
−
1≤l≤2ml= j
[C j , C l]a j − al
.(6.5)
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PROOF: The proof of (1) is very similar to the case of the 2 F 1 kernel. The
kernel L( x, y) has the form
L = 0 A
− A∗ 0
,
where A is a kernel on R+ × R− of the form
A( x, y) =√
ψ+( x)ψ−( y)
x − y= C ( z, z ) x−
z+ z
2 e− x (− y) z+ z
2 e y
x − y.
The jump condition is verified using the formulae [24, 6.5(7), 6.8(15), 6.9(4)].
(2) can be either verified in the same way as for the 2 F 1 kernel or deduced from
the fact that the Whittaker kernel is the correlation kernel of a determinantal pointprocess that has finitely many particles in J almost surely (see [54, theorem 4] for
the general theorem about determinantal point processes, and [11, 7] for the needed
property of the Whittaker kernel).
(3) If | z + z| < 1, then the kernel L introduced above defines a skew, bounded
operator in L 2(X, d x) and K = L(1 + L)−1; see [11, 49]. Then, similarly to
Corollary 3.7, we can prove that 1 − K J is invertible.
However, for the restricted operator K J , we can prove the invertibility of 1−K J
for all admissible values of ( z, z ). The following argument is due to G. Olshanski.
Write K J in the block form
K J =
K J +,+ K J
+,−
K J −,+ K J
−,−
corresponding to the splitting J = ( J ∩R+)( J ∩R−). Since K is a correlation ker-
nel, K J +,+ and K J
−,− are positive definite. Moreover, K J −,+( x, y) = −K J
+,−( y, x)
by definition of the Whittaker kernel. Thus, it is enough to prove the invertibility
of 1 − K J +,+ and 1 − K J
−,− (see the proof of Corollary 3.7).
We consider K J +,+; the proof for K J
−,− is similar. By [54, theorem 3], K +,+
≤1
and K J +,+ = K +,+| J ≤ 1. The only way K +,+ can have norm 1 (remember that
K J +,+ is of trace class and hence compact) is that K +,+ has an eigenfunction with
eigenvalue 1 that is supported on J ∩ R+. By [49, prop. 3.1] (see also [11]),
K +,+ = K +,+( x, y) commutes with a Sturm-Liouville operator
D x = − d
d x x 2 d
d x+ ( z + z + x)2
4
in the sense that
K +,+( x, y) D y
=D x K +,+( x, y)
for all x, y > 0. Suppose f ∈ L 2(R+) is an eigenfunction of K +,+ with eigenvalue
1 and supported in J ∩R+, i.e., R+
K +,+( x, y) f ( y)d y =
J ∩R+
K +,+( x, y) f ( y)d y = f ( x) , x > 0 .
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Then using the decay and smoothness properties of K +,+( x, y), which follow eas-
ily from the known properties of the Whittaker function, one sees that D x f also
belongs to L 2(R+) and R+
K +,+( x, y) D y f ( y)d y = D x f ( x) ;
thus
V = Span
Dk x f : k ≥ 0
⊂ Ker
1 − K J +,+
⊂ L 2(R+) .
But because K J +,+ is compact, dim V < ∞, and hence V is a finite-dimensional
invariant subspace for D x . It follows that there exists a nonzero v ∈ V such that
D x v = λv for some scalar λ. But since v ∈ V , it must vanish in a neighborhoodof x = 0, which is not possible for nontrivial solutions v( x) of the differential
equation D x v = λv. Thus, we obtain a contradiction, and hence K J +,+ < 1 and
1 − K J +,+ is invertible. The proof of (3) is now complete.
(4) and (5a–d), (5e) are easily verified. The proofs of (6) and (7) are similar to
the case of the 2 F 1 kernel, and we do not reproduce them here.
By Theorem 5.3, the matrix M for the Whittaker kernel satisfies the differential
equation
M (ζ ) = Aζ
+2m
j=1
C jζ − a j
− σ 32 M ( ζ ) .
The matrices {C j}2m j=1 are nilpotent (if a1 = −∞ or a2m = +∞, then C 1 = 0 or
C 2m = 0, respectively), and an analogue of Proposition 3.3 shows that
trA = 0, detA = −
z − z
2
2
.
THEOREM 8.9 The Fredholm determinant det(1 − K | J ) , where K is the Whittaker
kernel, is the τ -function for the system of Schlesinger equations
(8.1)∂A
∂a j
=2m
j=1
[C j ,A]a j − 1
2
,∂C l
∂a j
= [A, C j ]a j
−
1≤l≤2ml= j
[C j , C l]a j − al
+ [C j , σ 3]2
.
The matrices {C j}2m j=1 are nilpotent (if a1 = −∞ or a2m = +∞ , then C 1 = 0 or
C 2m = 0 , respectively) , and
trA = 0 , detA = −
z − z
2 2
.
PROOF: These results follow from Theorem 6.5.
The next step is to consider J = (s, +∞), s > 0. It turns out that in this case
the Schlesinger equations reduce to the σ -form of the Painlevé V equation. This
reduction can be performed in the spirit of Section 7, following the corresponding
7/29/2019 Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory
Acknowledgments. The authors would like to thank A. Kitaev for important
discussions about this work and O. Costin and R. D. Costin for informing us of
their calculations on Painlevé VI. The authors would also like to thank A. Its and
G. Olshanski for many useful discussions. This research was partially conductedduring the period the first author served as a Clay Mathematics Institute Long-Term
Prize Fellow. The work of the first author was also supported in part by NSF Grant
DMS-9729992, and the work of the second author was supported in part by NSF
Grant DMS-0003268.
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ALEXEI BORODIN PERCY DEIFT
Institute for Advanced Study Courant Institute
School of Mathematics 251 Mercer Street
Einstein Drive New York, NY 10012-1185Princeton, NJ 08540 E-mail: [email protected]