## MEMBERS

### Program Members

For a long time mathematics has been considered a language to describe natural sciences. Similarly, representation theory is a language to describe symmetry, and, as it happens with the whole body of mathematics, has itself a wide range of applications. Representation theory has been extremely useful in diverse areas of mathematics and physics. For instance, if symmetry is considered, then certain cases, which are seemingly complex, reduce to easy-to-handle problems. In these cases representation theory is a powerful tool. Needless to say, there is research on representation theory for its own purpose like other fields of mathematics. For example, classification of “basic representations”, their algebraic and geometric constructions, and issues of decomposing a given representation into such basic representations are particularly important. Although it is difficult to predict when these become useful even within researches in mathematics, there is no doubt about their importance.

My research to date has included applications of representation theory to mathematical physics and number theory, in particular to various analysis related to zeta functions. Below are my recent research projects as well as our research plans as part of the Math-for-Industry.

##### [Spectrum of Non-Commutative Harmonic Oscillators]

A non-commutative harmonic oscillator, which A. Parmeggiani(University of Bologna) and I introduced around 10 years ago, refers to a system of ordinary differential equations with two kinds of non-commutativity inherited from both matrix level and the canonical commutation relations level. In some special cases, the system defines a pair of quantum harmonic oscillators, but generally, the existence of creation and annihilation operators for such a kind of systems is unknown, and actual determination of the spectrum is quite difficult. Since then it has become apparent that this problem is a rich research theme as it relates to a monodromy problem of Heun’s differential equations and to moduli of elliptic curves (through the spectral zeta function). Although the corresponding physical system has yet to be discovered, a certain mechanical engineering model seems to have a similar spectrum (judged from its distribution chart). Hence, this is an interesting topic to pursue.

##### [Representation theory and invariant theory for α-determinants]

Statistics has required the α-determinant to be introduced(Vere-Jones, 1988). An α-determinant interpolates a determinant (α=-1) and permanent (α=1) of a matrix. I started a study about its representation theory several years ago. Actually, I have been working on modules of a general linear group GL_n generated by its (complex) powers in collaboration with Kazufumi Kimoto (University of the Ryukyus) and Sho Matsumoto (Nagoya University). When the power is 1, one has a theory that interpolates the skew-symmetric representation and symmetric tensor representation at the representation level. This study also has a potential to propose a new perspective towards a theory of special functions. With the discovery of a new relative invariant called the wreath-determinant, we are continuing our research from the invariant and combinatorial theoretic points of view. The relationship with zonal polynomials, which are important also in statistics, has been discovered. To date, the α-determinant has multiple connections to statistics as well as a strong connection with probability theory (due to its research on positivity). However, nothing concrete is known about the feedback to statistics from representation theory of α-determinants. Thus, our research objective is to provide such feedback to statistics.

##### [Harmonic analysis on symmetric spaces (particularly symmetric cones) and associated special functions]

In recent years, information geometric approach to optimization theory is becoming more and more used. Typical examples of differential manifolds, which provide the foundation for formulation of problems, include symmetric cones formed by positive definite symmetric matrices. These manifolds can be treated group-theoretically using Lie algebras and Jordan algebras. There is indeed a rich history of research about symmetric spaces, which has accumulated excellent mathematical techniques. Recently, some statisticians have also started to conduct their research from such group-theoretical point of view. Our objective is to study number-theoretical structures of both statistical manifolds compatible which their (flat) connections and applications to statistics.

##### [Multiplicative Radon transform]

My interest in this research topic was triggered when I was studying q-analogues in the Hecke theory for classical automorphic forms. I recently wrote/edited an introductory book of modern mathematics in technology (Iwanami Publ. Tokyo, 2008), which touches upon the Radon transform (theoretically a subject of integrable geometry and representation theory). The Radon transform provides a mathematical foundation for applications such as the computerized tomography. Our study uses the approach from a (multiplicative) discrete transform as a numerical inverse transform of the Radon transform.

One of my Ph.D. students recently published a paper on “q-Hecke theory” (2008), and during the internship in a car manufacturer he made an important contribution to the optimization problem regarding automobile engine control. As part of the internship program, he was studying statistics while receiving the advice of a professor specialized in statistics. This is a good example that even if one’s mathematical foundation is not within a specialty area, it can still be very useful.

I would like to expand my own field and involve students to drive the Math-for-Industry program.