OSHIMA Toshio
Department of Mathematical Sciences, University of Tokyo
381, Komaba, Tokyo 1538914, Japan
Last update is July. 2012.
Japanese version >
Papers and Preprints.

An elementary approach to the Gauss hypergeometric function,
Josai Mathematical Monographs 6(2013), 323.
 abstract:
We give an introduction to the Gauss hypergeometric function, the
hypergeometric equation and their properties in an elementary way.
Moreover we explicitly and uniformly describe the connection
coefficients, the reducibility of the equation and
the monodromy group of the solutions.

Quantization of linear algebra and its application to integral geometry,
(with H. Oda), Geometric Analysis and Integral Geometry, Contemporary Mathematics
598 (2013), 189208.
 abstract:
In order to construct good generating systems of twosided ideals in the universal enveloping
algebra of a complex reductive Lie algebra, we quantize some notions of linear algebra, such
as minors, elementary divisors, and minimal polynomials. The resulting systems are applied to
the integral geometry on various homogeneous spaces of related real Lie groups.

A classification of roots of symmetric KacMoody root systems and its application,
(with K. Hiroe), Symmetries, Integral Systems and Representations,
Springer Proceedings of Mathematics and Statics 40 (2012), 195241.
 abstract:
We study Weyl group orbits in symmetric KacMoody root systems and show a finiteness of orbits of
roots with a fixed index.
We apply this result to the Euler transform of linear ordinary differential equations on the Riemann
sphere whose singular points are regular singular or unramified irregular singular points.

Finite multiplicity theorems
(with T. Kobayashi), Adv. Math. 248 (2013), 912944.
 abstract:
We find upper and lower bounds of the multiplicities of irreducible admissible representations
of a semisimple Lie group occurring in the induced pepresentations from irreducible
represenations of a closed subgroup. We give geometric criteria for the finiteness of the
multiplicities.

Boundary value problems on Riemannian Symmetric Spaces of the noncompact Type
(with N. Shimeno), Lie Groups: Structure, Actions, and Representations, Progress in Mathematics,
307 (2013), 273308, Birkhäuser
 abstract:
We characterize the image of the Poisson transform on each distinguished boundary of
a Riemannian symmetric space of noncompact type by a system of differential equations.

Fractional calculus of Weyl algebra and Fuchsian differential equations,
MSJ Memoirs 28, ixi+203 pages, 2012,
Correction (older version: arXiv:1102.2792v1).
Corrections for UTMS 20115, Feb. 24, 2011.
 abstract:
We give a unified interpretation of confluence, reducibility, contiguity
relation and Katz's middle convolutions for linear ordinary differential
equations with polynomial coefficients and their generalization
to partial differential equations.
The integral representation and series expansion of their solutions
are also within our interpretation.
As applications to Fuchsian differential equations on
the Riemann sphere, we construct a universal model of
Fuchsian differential equations with a given spectral type,
in particular, we construct single ordinary differential
equations without apparent singularities corresponding to
the rigid local systems, whose existence was an open problem
presented by Katz.
Furthermore we obtain an explicit solution of the
connection problem for the rigid Fuchsian differential
equations.
We give many examples calculated by our fractional calculus.

HeckmanOpdam hypergeometric functions and their specializations,
RIMS Kokyuroku Bessatsu B20 (2010), 129162 (with N. Shimeno),
 abstract:
We discuss three topics, confluence, restrictions and real forms for the HeckmanOpdam
hypergeometric functions.

Katz's middle convolution and Yokoyama's extending operation,
arXiv:0812.1135, 18 pages, 2008.
 abstract:
We give a concrete relation between Katz's middle convolution and Yokoyama's
extension and show the equivalence of both algorithms using these operations
for the reduction of Fuchsian systems.

Classification of Fuchsian systems and their connection problem,
arXiv:0811.2916, 29 pages, 2008, to appear in RIMS Kokyuroku Bessatsu.
 abstract:
We review the DeligneSimpson problem, a combinatorial structure of middle convolutions
and their relation to a KacMoody root system discovered by CrawleyBoevey.
We show with examples that middle convolutions transform the Fuchsian systems with a
fixed number of accessory parameters into fundamental systems whose spectral type is in
a finite set and we give an explicit connection formula for solutions of Fuchsian
differential equations without moduli.
 HeckmanOpdam hypergeometric functions and their specializations,
Harmonische Analysis und Darstellungstheorie Topologischer Gruppen,
Mathematisches Forschungsinstitut Oberwolfach,
Report No. 49/2007, 3840.
 A classification of subsystems of a root system,
preprint, 47pp, 2006, math.RT/0611904, submitted.
 abstract:
We classify isomorphic classes of the homomorphisms of a root system
$\Xi$ to a root system $\Sigma$ which keep Cartan integers invariant.
We examine different types of isomorphic classes defined by the Weyl group of
$\Sigma$, that of $\Xi$ and the automorphisms of $\Sigma$ or $\Xi$ etc.
We also distinguish the subsystem generated by a subset of a fundamental system.
We introduce the concept of the dual pair for root systems which helps
the study of the action of the outer automorphism of $\Xi$ on the homomorphisms.
 Commuting differential operators with regular singurarities,
(preprint, 30pp, 2006, math.AP/0611899),
Algebraic Analysis of Differential Equations
 from Microlocal Analysis to Exponential Asymptotics 
Festschrift in Honor of Takahiro Kawai SpringerVerlag, Tokyo, 2007. 195224.
 abstract:
We study a system of partial differential equations defined by
commuting family of differential operators with regular singularities.
We construct ideally analytic solutions depending on a holomorphic parameter.
We give some explicit examples of differential operators related
to $SL(n,\mathbb R)$ and completely integrable quantum systems.
 Completely integrable systems associated with
classical root systems, (2005, preprint),
mathph/0502028, SIGMA 3(2007), 061, 50pages,
 abstract:
We explicitly construct sufficient integrals of completely integrable quantum and classical systems associated with classical root systems, which include CalogeroMoserSutherland models, Inozemtsev models and Toda finite lattices with boundary conditions. We also discuss the classification of the completely integrable systems.
 A class of completely integrable quantum systems associated with
classical root systems, (19pp, preprint,
2004), Indag. Mathem. 16(2005), 655677.
 abstract: We classify the completely integrable systems associated
with classical root systems whose potential functions are meromorphic at
an infinite point.
 Minimal polynomials and annihilators of generalized Verma modules of the
scalar type, (56pp, UTMS 20043, DVI file,
PDF file), Journal of Lie Theory 16(2006), 155219
(with H. Oda).
 abstract: We calculate the minimal polynomial associated to the pair of
any finite dimensional representation of any semisimple Lie algebra and a
generarized Verma module of the scalar type of the Lie algebra.
Using this minimal polynomial, we give a generator system of a generalized Verma
module of the scalar type. In the classical limit, it gives the generator system
of the defining ideal of a coadjoint orbit.
 A calculation of cfunctions for semisimple symmetric spaces,
Lie Groups and Symmetric Spaces, in the memory of F. I. Karpelevich,
edited by Gindikin, AMS Translation Series, 210(2003), 315339
 Fatou's theorems and Hardytype spaces for eigenfunctions of the invariant differential operators on symmetric spaces, International Mathematics Research Notices 16(2003), 915931 (with S. B. Said and N. Shimeno).
 Harmonic analysis on semisimple symmetric spaces, Sugaku Expositions, 15(2002), 151170, AMS, translated from Sugaku 37(1985), 97112.
 Annihilators of generalized Verma modules of the scalar type
for classical Lie algebras, (29pp, preprint, UTMS, 200129, DVI file),
"Harmonic Analysis, Group Representations, Automorphic forms and Invariant Theory",
in honor of Roger Howe, Vol. 12 (2007), Lecture Notes Series,
Institute of Mathematical Sciences, National University of Singapore, 277319.
abstract:
We construct a generator system of the annihilator of a Verma mudule of a classical
reductive Lie algebra induced from a character of a parabolic subalgebra as an analogue
of the minimal polynomial of a matrix. In a classical limit it gives a generator system
of the defining ideal of any semisimple coadjoint orbit of the Lie algerba.  A quantization of conjugacy classes of matrices, 16pp, preprint, UTMS, 200038,
DVI file, PDF file. Adv. in Math. 196(2005), 124146.
 abstract: We construct a generator system of the annihilator of a generalized
Verma module of $\mathfrak {gl}(n,\mathbb C)$ induced from any character of any parabolic
subalgebra as an analogue of minors and elementary divisors. The generator system has a
quantization parameter $\epsilon$ and it generates the defining ideal of the conjugacy
class of square matrices at the classical limit $\epsilon=0$.
 Dimensions of spaces of generalized spherical functions,
Amer. J. Math., 118(1996), 637652, (with J. Huang and N. Wallach).
 View Top
 Generalized Capelli identities and boundary value problems for GL(n) ,
Structure of Solutions of Differential Equations, Katata/Kyoto 1995, 307335, World
Scientific, 1996.
 abstract: The Capelli identity is extended to the case of minors. The operators
appearing in the generalized identities give the annihilator of the degenerate principal
series for $GL(n)$ and characterizes the image of the Poisson transform of the
hyperfunctions on several boundaries of $GL(n)$. Hypergeometric functions are defined
through realizations of some special sections of the degenerate principal series and the
realizations on boundaries of $GL(n)$ generalize Gelfand's hypergeometric functions.
Related Radon transforms for Grassmannians are discussed.
 View Top
 Commuting families of differential operators invariant under the action of a Weyl
group, J. Math. Sci. Univ. Tokyo, 2(1995), 175 (with H. Sekiguchi).
 abstract: We study the family of commuting differential operators invariant under
the natural action of an irreducible classical Weyl group. When their highest order terms
generate the invariant differential operators with constant coefficients, we determine the
potential function of the Schrodinger operator in the family.
 View Top  Commuting differential operators of type B2, UTMS
9465, Dept. of Mathematical Sciences, Univ. of Tokyo, 1994, 31pp (with H. Ochiai), Funkcialaj Ekvacioj 46(2003), 297336.

abstract: We determine the completely integrable quantum systems with $B_2$ symmetry.
We also study the reducibility of the systems of differential equations defined by these
quantum systems.
 View Top  Completely integrable systems with a symmetry in coordinates,
UTMS 946, Dept. of Mathematical Sciences, Univ. of Tokyo, 1994, 22pp.
Asian J. Math., 2(1998), 935955.
 abstract: We explicitly construct the integrals of completely integrable quantum
or classical systems whose potential functions are invariant under the action of a
classical Weyl group. Our potential functions and integrals are expressed by the
Weierstrass elliptic function.
 Commuting families of symmetric differential operators,
Proc. Japan. Acad. 70(1994), 6266 (with H. Ochiai and H. Sekiguchi).
 abstract: This paper is a resume of main results in the following three papers.
 View All
 Multiplicities of representations on homogeneous spaces,
Abstracts from the conference `Harmonic Analysis on Lie Groups held at Sandbjerg Gods',
August 2630, 1991, 2pp, edited by N. V. Pedersen, 1991, Mathematical Institute,
Copenhagen University.
 Paley Wiener theorems on a symmetric space and its
applications, Differential Geometry and Its Applications, 1(1991), 247278 (with
Y. Saburi and M. Wakayama).
 Embeddings of discrete series into principal series, The Orbit Method in
Representation Theory Proceeding of a Conference Held in Copenhagen, August to September
1988, 1990, Birkhauser, 147175 (with T. Matsuki).
 Asymptotic behavior of FlenstedJensen's spherical trace functions with respect to
spectral parameters, Algebraic Analysis, Geometry and Number Theory, Proceedings of
JAMI Inaugural Conference, Supplement to Amer. J. Math., The Johns Hopkins University
Press, 1989, 313323.
 A note on Ehrenpreis' fundamental principle on a symmetric space, Algebraic
Analysis, edited by T. Kawai and M. Kashiwara, Academic Press, 1988, 681697 (with Y.
Saburi and M. Wakayama).
 A method of harmonic analysis on semisimple symmetric spaces, Algebraic
Analysis, edited by T. Kawai and M. Kashiwara, Academic Press 1988, 667680.
 Boundedness of certain unitarizable HarishChandra modules, Advanced Studies in
Pure Math., 14(1988), 651660 (with M. FlenstedJensen and H. Schlichtkrull).
 A realization of semisimple symmetric spaces and construction of boundary value maps,
Advanced Studies in Pure Math., 14(1988), 603650.
 Asymptotic behavior of spherical functions on semisimple symmetric spaces,
Advanced Studies in Pure Math., 14(1988), 561601.
 Open problems suggested by T. Oshima, Open Problems in Representation Theory of
Lie Groups, Proceeding of Eighteenth International Symposium, Division of Mathematics, The
Taniguchi Foundation, edited by T. Oshima, 1987, 2123.
 Discrete series for semisimple symmetric spaces,
Proceedings of the International Congress of Mathematicians, 1984, 901904
 The restricted root systems of semisimple symmetric pairs,
Advanced Studies in Pure Math., 4(1984), 433497 (with J. Sekiguchi).
 Boundary value problems for systems of linear partial differential equations with regular singularities, Advanced Studies in Pure Math., 4(1984), 391432.
 A description of discrete series for semisimple symmetric spaces,
Advanced Studies in Pure Math., 4(1984), 331390 (with T. Matsuki).
 A definition of boundary values of solutions of partial differential equations with regular singularities, Publ. RIMS Kyoto Univ., 22(1983), 12031230.
 Fourier analysis on semisimple symmetric spaces
,NonCommutative Harmonic Analysis, Proceedings, 1980, Lect. Notes in Math.,
880(1981), 357369, DVI file, PDF file.
 Microlocal analysis of prehomogeneous vector spaces,
Invent. Math., 62(1980), 117179 (with M. Sato, M. Kashiwara and T. Kimura).
 Eigenspaces of invariant differential operators on an affine symmetric space
, Invent. Math., 57(1980), 181 (with J. Sekiguchi).
 Orbits on affine symmetric spaces under the action of the isotropy subgroups
, J. Math. Soc. Japan, 32(1980), 399414 (with T. Matsuki).
 Poisson transformations on affine symmetric spaces,
Proc. Japan Acad. 55A(1979), 323327.
 A study of Feynman integrals by microdifferential equations,
Comm. Math. Phys., 60(1978), 97130 (with M. Kashiwara and T. Kawai).
 On analytic equivalence of glancing hypersurfaces,
Sci. Papers College Gen. Ed. Univ. Tokyo, 28(1978), 5157.
 A realization of Riemannian symmetric spaces, J. Math. Soc. Japan,
30(1978), 117132.
 Eigenfunctions of invariant differential operators on a symmetric space,
Ann. of Math., 107(1978), 139 (with M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto and M. Tanaka).
 Systems of differential equations with regular singularities and their boundary value problems, 106(1977), 145200 (with M. Kashiwara).
 Boundary value problem on symmetric homogeneous spaces,
Proc. Japan Acad., 53A(1977), 8183 (with J. Sekiguchi).
 Holonomy structure of Landau singularities and Feynman integrals,
Publ. RIMS Kyoto Univ., 12 Suppl(1977), 387438 (with M. Sato, T. Miwa and M. Jimbo)
 Introduction to microlocal analysis,
Publ. RIMS Kyoto Univ., 12 Suppl(1977), 267300 (with T. Miwa and M. Jimbo).
 Boundary value problem with regular singularity and HelgasonOkamoto conjecture, Publ. RIMS Kyoto Univ., 12 Suppl(1977), 257265 (with K. Minemura).
 Structure of a single pseudo differential equation in a real domain,
J. Math. Soc. Japan, 28(1976), 8085 (with T. Kawai and M. Kashiwara).
 Structure of cohomology groups whose coefficients are microfunction solution sheaves of systems of pseudodifferential equations with multiple characteristics
II, Proc. Japan Acad., 50(1974), 549550 (with T. Kawai and M. Kashiwara).
 Structure of cohomology groups whose coefficients are microfunction solution sheaves of systems of pseudodifferential equations with multiple characteristics
I, Proc. Japan Acad., 50(1974), 420425 (with T. Kawai and M. Kashiwara).
 On the global existence of solutions of systems of linear differential equations with constant coefficients, J. Math. Soc. Japan, 26(1974), 575586.
 A proof of Ehrenpreis' fundamental principle in hyperfunctions,
Proc. Japan Acad., 50(1974), 1418.
 Singularities in contact geometry and degenerate pseudodifferential equations
, J. Fac. Sci. Univ. Tokyo Sec. IA 21(1974), 4387.
 On the theorem of CauchyKowalevsky for first order linear differential equations with degenerate principal symbols, Proc. Japan Acad., 49(1973), 8387.
Others
 Fractional calculus of Weyl algebra and its application to differential equations on the Riemann sphere, Mathematics  String theory theminar, IPMU, Univ. of Tokyo, September 8, 2009
 Fractional calculus of Weyl algebra and
its applications,
Representation Theory of Real reductive Lie groups,
held at University of Utah, July 30, 2009.
 Classification of Fuchsian systems and their connection problem, Differential equations and symmetric spaces,
held at the Univ. of Tokyo, January 15, 2009
 Completely integrable systems with coordinate symmetries,
"Non Commutative Harmonic Analysis" held at RIMS, Kyoto University, August 1994,
20pp(in Japanese)
 Capelli identities, degenerate series and hypergeometric
functions, Okinawa Symposium on Representation Theory of Lie Groups, December,
1995, 19pp.
 Twisted Radon transformation on Grassmanians, "Integral Geometry in Representation
Theory" held at MSRI, Berkeley, October 8, 2001. DVI
files, PS files.(copy of slides: 12in x 9 in, 14pp)
to Oshima Laboratory Home Page
oshima@ms.utokyo.ac.jp