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Middle convolution and non-Schlesinger deformations.

1. Introduction. Various aspects of middle convolution have recently attracted a lot of attention (see, for example, [7-9, 12-14]). Middle convolution is an operation for non-resonant Fuchsian systems of differential equations which preserves deformation equations [9]. In particular, the so-called Hitchin systems [10] obtained from the Schlesinger systems (see below) are invariant under middle convolution. It is known that there exist non-Schlesinger deformations for resonant Fuchsian systems and it is natural to expect that they are also preserved by middle convolution. In this paper we announce that in general non-Schlesinger deformations are not preserved by middle convolution. Moreover, the algorithm of middle convolution gives new explicit examples of non-Schlesinger isomonodromic deformations of the resonant Fuchsian systems of order higher than two and which are different from Bolibruch's example for a Fuchsian system of order 2. This paper announces the main results and the details (including explicit examples) will be published separately [2].

2. Isomonodromic deformations. Let us consider a system of p linear differential equations on the Riemann sphere

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with singularities [a.sup.0.sub.1], ..., [a.sup.0.sub.n] [member of] C, [a.sup.0.sub.n+1] = [infinity]. Here y(z) [member of] [C.sup.p]. System (1) is the Fuchsian system. One can define its monodromy representation [11] by

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

System (1) can be included in the isomonodromic family of Fuchsian systems

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [A.sub.i]([a.sup.0]) = [A.sup.0.sub.i], [a.sup.0] = ([a.sup.0.sub.i], ..., [a.sup.0.sub.n]). The parameters a = ([a.sub.1], ..., [a.sub.n]) of the family (3) are the locations of singular points. Similarly we can define a monodromy representation of system (3) for a [member of]

D([a.sup.0])\[[union].sup.n.sub.i,j=1,i[not equal to]j] {[a.sub.i] = [a.sub.j]} by

(4)[[chi].sub.a] : [[pi].sub.1]([T.sub.a], [z.sub.0]) [right arrow] GL(p, C), [T.sub.a] = [??]\[[union].sup.n+1.sub.i=1] {[a.sub.i]},

where D([a.sup.0]) is a small open disk centered at [a.sup.0].

Definition 2.1 ([3, 4, 5]). The family of Fuchsian systems (3) is isomonodromic if the monodromy representation [[chi].sub.a] coincides with the representation [[chi].sup.0] of system (1) for any a [member of] D([a.sup.0])\[[union].sup.n.sub.i,j=1,i[not equal to]j] {[a.sub.i] = [a.sub.j]}.

The isomonodromuc family (3) is also called the isomonodromic deformation.

The following statement for Definition 2.1 was proved by A. A. Bolibruch.

Theorem 2.1 ([3, 4, 5]). The family of Fuchsian systems (3) is isomonodromic if and only if there exists a matrix-valued differential 1-form [omega] on C x D([a.sup.0])[[union].sup.n.sub.i=1] {z - [a.sub.i] = 0} such that

i) [omega] = [[summation].sup.n.sub.i=1] [A.sub.i](a)/[z-[a.sub.i]] dz for any fixed a [member of] D([a.sup.0]);

ii) d[omega] = [omega] [conjunction] [omega].

Definition 2.1 is the general one. The Schlesinger deformations are the most known in the literature. They are given by the differential form

(5) [[omega].sub.Schl] = [n.summation over (i=1)] [A.sub.i](a)/[z - [a.sub.i]] d(z - [a.sub.i]).

The second condition of Theorem 2.1 is then equivalent to

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is also known as the Schlesinger equation [11]. The fundamental matrix [Y.sub.Schl](z, a) of (3) satisfies

(7) [Y.sub.Schl]([infinity], a) [equivalent to] C,

where C is a constant non-degenerated matrix.

Definition 2.2. Deformations of the Fuchsian systems satisfying (7) are called normalized.

Note that Definition 2.1 does not require any normalization.

An arbitrary isomonodromic deformation is not necessarily the Schlesinger deformation (5). Let us consider a family of Fuchsian systems with the fundamental matrix Y(z, a) = [GAMMA](a)[Y.sub.Schl](z, a), where [GAMMA](a) is a holomorphically invertible matrix. In this case the differential form [omega] = dY(z, a)[Y.sup.-1](z, a) is given by

(8) [omega] = [n.summation over (i=1)] [A.sub.i](a)/[z - [a.sub.i]] d(z - [a.sub.i]) + [n.summation over (k=1)] [[gamma].sub.k](a)d[a.sub.k].

This isomonodromic deformation is not normalized. It is clear that this deformation is reduced to the Schlesinger deformation by

(9) [Y.sub.Schl] (z, a) = [[GAMMA].sup.-1](a)Y(z, a).

However, there exist isomonodromic deformations given by differential 1-forms different from (5) and (8). A. A. Bolibruch gave examples of such deformations and obtained a general form of the isomonodromic deformation.

Definition 2.3 ([3, 4]). Let [[lambda].sup.i.sub.1], ..., [[lambda].sup.i.sub.p] be the eigenvalues of the matrix [A.sub.i] of the Fuchsian system (3). A singular point [a.sub.i] is called resonant if there exist at least two non-equal eigenvalues of [A.sub.i] such that their difference is a natural number. A number

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is called a maximal i-resonance of the system.

Theorem 2.2 ([3, 4]). Any matrix-valued differential 1-form [omega] on [??] x D([a.sup.0])\[[union].sup.n.sub.i=1]{z - [a.sub.i] = 0} which defines isomonodromic deformation of the Fuchsian system (3) is given by

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[gamma].sub.m,k,l](a), [[gamma].sub.k](a) are holomorphic in D([a.sup.0]) and [r.sub.l] is a maxmal l-resonance of system (3) for a = [a.sup.0].

We remark that the last terms may be non-zero only if system (3) has some resonant singularities.

The famous example [6, 3, 4] of Bolibruch of the non-Schlesinger normalized isomonodromic deformation is given as follows. The family of Fuchsian systems

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is isomonodromic with the differential 1-form [omega] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here z = -a and z = 0 are resonant singular points. The deformation is normalized (there is no term of the form [gamma](a)da) and it cannot be reduced to the Schlesinger deformation (there is a term [gamma](a)da/(z + a)). In general it is difficult to find explicit examples of the non-Schlesinger deformations (see, for instance, [1,2] for more examples and a discussion).

3. Middle convolution. Middle convolution is an operation on tuples of residue matrices of a Fuchsian system introduced by S. Reiter and M. Dettweiler [7, 8]. For a given parameter [mu] [member of] C one defines residue matrices of dimension pn x pn which are partitioned into blocks and have only one non-zero block consisting of the initial residue matrices and the parameter [mu]. By finding invariant subspaces and reducing the size of the matrices one gets a new Fuchsian system with the same singularities but with new residue matrices. This operation can be realized as an analytic operation for solutions (Euler transformation). Note that the size of matrices in the final system depends on the choice of the parameter [mu]. See [7, 9] for more details and explicit expressions.

The properties of middle convolution for Fuchsian systems, its generalization to systems with irregular singularities and q-difference systems are studied by many Japanese authors (e.g., Y. Haraoka, K. Hiroe, H. Kawakami, T. Oshima, K. Takemura, H. Sakai, M. Yamaguchi, D. Yamakawa and many others). Since middle convolution preserves Schlesinger deformation equations [9], it is natural to ask what happens to Bolibruch's non-Schlesinger isomonodromic deformations under middle convolution.

4. Outline of the main results. In [2] we construct explicit examples that middle convolution does not in general preserve non-Schlesinger deformation. Our examples are based on the modifications of the Bolibruch example. Note that it is very important and of interest to specialists in deformation theory to find new explicit examples of non-Schlesinger isomonodromic deformations because of the difficulty to write down the corresponding differential 1-form [omega] [1]. Our explicit examples show that under middle convoution the resonance condition may appear or disappear. Moreover, the maximal i-resonance of a system may change.

It is easy to show that if we apply middle convolution with [mu] = 0 to Bolibruch's example and get a new isomonodromic (2 x 2)-family, which is non-Schlesinger again and non-normalized. It is an expected result because of the properties of middle convolution and isomonodromic deformations. Therefore, we modify the Bolibruch example.

We found an explicit example which shows that applying middle convolution with some [mu], [mu] [not equal to] 0, to a certain non-Schlesinger isomonodromic (2 x 2)-family with five singular points, the resulting family cannot be included in any Bolibruch's non-Schlesinger isomonodromic deformation because of Theorem 2.2. The resulting family does not also satisfy the Schlesinger equations and it cannot be transformed to a Schlesinger isomonodromic deformation by a transformation y = [GAMMA](a)[??] for any holomorphic matrix-value function [GAMMA](a). The initial system has two resonant Fuchsian points with the maximal resonances both equal to 1. The parameter [mu] of middle convolution was chosen to break old resonances without producing new ones, i.e., the resulting system is non-resonant.

Finally, we were able to demonstrate explicitly the opposite case. We can start again with the non-Schlesinger isomonodromic (2 x 2)-family with four singularities. This family has two resonant singularities with the maximal resonances 1 and 4 respectively. After applying middle convolution with [mu] [not member of] 1/2 Z, [mu] [not equal to] 0 we get a new (5 x 5)-family with one resonant point with the maximal resonances equal to 1. This family is the non-Schlesinger isomonodromic deformation and it cannot be reduced to any Schlesinger isomonodromic deformation. To prove this we presented a differential 1-form [omega] for which conditions in Theorem 2.1 are fulfilled. This form contains terms which cannot be elliminated by a holomorphic gauge transformation.

Due to space limitations, the details are published separately [2].

doi: 10.3792/pjaa.91.66

Acknowledgments. Part of this work was carried out while Y. Bibilo was visiting the Univerity of Warsaw in April 2014. The authors acknowledge the support of the Polish NCN Grant 2011/03/B/ST1/00330. Y. Bibilo also acknowledges the support of the Russian Foundation for Basic Research (grant no. RFBR 14-01-00346 A).

References

[1] Y. Bibilo, Inverse monodromy problems for systems with irregular singularities. Ph.D. thesis, Moscow State University, 2012. (in Russian).

[2] Y. Bibilo and G. Filipuk, Non-Schlesinger isomonodromic deformations of Fuchsian systems and middle convolution, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 023, 14 pp.

[3] A. A. Bolibruch, Differential equations with meromorphic coeffcients. Proc. Steklov Inst. Math. 272 (2011), 13-43.

[4] A. A. Bolibruch, On isomonodromic confluences of Fuchsian singularities. Proc. Steklov Inst. Math. 221 (1998), 117-132.

[5] A. A. Bolibruch, Inverse monodromy problems in the analytic theory of differential equations. MCCME, Moscow, 2009. (in Russian).

[6] A. A. Bolibruch, Inverse problems for linear differential equations with meromorphic coefficients, in Isomonodromic deformations and applications in physics (Montreal, QC, 2000), 3 25, CRM Proc. Lecture Notes, 31, Amer. Math. Soc., Providence, RI, 2002.

[7] M. Dettweiler and S. Reiter, An algorithm of Katz and its application to the inverse Galois problem, J. Symbolic Comput. 30 (2000), no. 6, 761-798.

[8] M. Dettweiler and S. Reiter, Middle convolution of Fuchsian systems and the construction of rigid differential systems, J. Algebra 318 (2007), no. 1, 1-24.

[9] Y. Haraoka and G. Filipuk, Middle convolution and deformation for Fuchsian systems, J. Lond. Math. Soc. (2) 76 (2007), no. 2, 438-450.

[10] N. Hitchin, Geometrical aspects of Schlesinger's equation, J. Geom. Phys. 23 (1997), no. 3-4, 287-300.

[11] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painleve, Aspects of Mathematics, E16, Vieweg, Braunschweig, 1991.

[12] H. Kawakami, Generalized Okubo systems and the middle convolution, Int. Math. Res. Not. IMRN 2010, no. 17, 3394-3421.

[13] T. Oshima, Fractional calculus of Weyl algebra and Fuchsian differential equations, MSJ Memoirs, 28, Math. Soc. Japan, Tokyo, 2012.

[14] K. Takemura, Middle convolution and Heun's equation, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 040, 22 pp.

By Yulia Bibilo *) and Galina Filipuk **)

(Communicated by Masaki KASHIWARA, M.J.A., April 13, 2015)

2000 Mathematics Subject Classification. Primary 34M56, 44A15.

*) Department of Theory of Information Transmission and Control, Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny per. 19, Moscow 127994, Russia.

**) Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland.
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Author:Bibilo, Yulia; Filipuk, Galina
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:May 1, 2015
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