Kazufumi Kimoto

Remarks on zeta regularized products

The zeta regularized products play important roles in constructing functions having a certain invariance and a given set of zeros. To deal with wider class of sequences, we introduce an extended version of regularized products. It allows us to treat the case where the attached zeta function has even a log singularity at the origin. In fact, we need the meromorphy of the zeta function for zeta regularizations so far. We discuss examples of the regularized products of sinh(ρ−x)′s especially when ρ runs over the essential zeros of zeta functions.

A variation of multiple

A variation of multiple L-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function can be written in terms of the gamma functions. This result enables us to obtain explicit evaluations of them.

L

For positive integers n and l, we study the cyclic u(gl n )-module generated by the l-th power of the α-determinant det (α) (X). This cyclic module is isomorphic to the n-th tensor space S l (ℂ n ) ⊗n of the symmetric l-th tensor space of ℂ n for all but finitely many exceptional values of α. If α is exceptional, then the cyclic module is equivalent to a proper submodule of S l (ℂ n ) ⊗n , i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in S l (ℂ n ) ⊗n . The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and whose entries are polynomials in α with rational coefficients. In particular, we determine the matrix completely when n = 2. In this case, the matrix becomes a scalar and is essentially given by a classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary. In the Appendix, we consider a variation of the spherical Fourier transformation for (O nl , O n l ) as a main tool for analyzing the same problems, and describe the case where n = 2 by using the zonal spherical functions of the Gelfand pair (O 2l , O 2 l ).

-values arising from the spectral zeta function of the non-commutative harmonic oscillator

As a particular one parameter deformation of the quantum determinant, we introduce a quantum α-determinant detq(α) and study the Uq(gln)-cyclic module generated by it: We show that the multiplicity of each irreducible representation in this cyclic module is determined by a certain polynomial called the q-content discriminant. A part of the present result is a quantum counterpart for the result of Matsumoto and Wakayama [S. Matsumoto, M. Wakayama, Alpha-determinant cyclic modules of gln(C), J. Lie Theory 16 (2006) 393–405], however, a new distinguished feature arises in our situation. Specifically, we determine the degeneration of the multiplicities for ‘classical’ singular points and give a general conjecture for singular points involving semi-classical and quantum singularities. Moreover, we introduce a quantum α-permanent perq(α) and establish another conjecture which describes a ‘reciprocity’ between the multiplicities of the irreducible summands of the cyclic modules generated respectively by detq(α) and perq(α).

ALPHA-DETERMINANT CYCLIC MODULES AND JACOBI POLYNOMIALS

The aim of the present paper is to generalize the notion of the group determinants for finite groups. For a finite group G and its subgroup H, one may define a rectangular matrix of size # H i? # G by X = ( x h g - 1 ) h ? H , g ? G , where { x g | g ? G } are indeterminates indexed by the elements in G. Then, we define an invariant ? ( G , H ) for a given pair ( G , H ) by the k-wreath determinant of the matrix X, where k is the index of H in G. The k-wreath determinant of an n by kn matrix is a relative invariant of the left action by the general linear group of order n and of the right action by the wreath product of two symmetric groups of order k and n. Since the definition of ? ( G , H ) is ordering-sensitive, the representation theory of symmetric groups is naturally involved. When G is abelian, if we specialize the indeterminates to powers of another variable q suitably, then ? ( G , H ) factors into the product of a power of q and polynomials of the form 1 - q r for various positive integers r. We also give examples for non-abelian group-subgroup pairs.

Quantum α-determinant cyclic modules of Uq(gln)

Let be the configuration space of planar -gons having side lengths and modulo isometry group. For generic , the cohomology ring has a form where is the first Stiefel-Whitney class of a certain regular -cover and the ideal is in general big. For generic , we determine the number such that but .

Wreath determinants for group-subgroup pairs

As a particular one parameter deformation of the quantum determinant, we introduce a quantum α-determinant detq(α) and study the Uq(gln)-cyclic module generated by it: We show that the multiplicity of each irreducible representation in this cyclic module is determined by a certain polynomial called the q-content discriminant. A part of the present result is a quantum counterpart for the result of Matsumoto and Wakayama [S. Matsumoto, M. Wakayama, Alpha-determinant cyclic modules of gln(C), J. Lie Theory 16 (2006) 393–405], however, a new distinguished feature arises in our situation. Specifically, we determine the degeneration of the multiplicities for ‘classical’ singular points and give a general conjecture for singular points involving semi-classical and quantum singularities. Moreover, we introduce a quantum α-permanent perq(α) and establish another conjecture which describes a ‘reciprocity’ between the multiplicities of the irreducible summands of the cyclic modules generated respectively by detq(α) and perq(α).

The Height of a Class in the Cohomology Ring of Polygon Spaces

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some

Quantum

q

\alpha

-series identity for proving the zeta function has an Euler product and then, describe the location of zeros. We study further multi-variable and multi-parameter versions of the multiple finite Riemann zeta functions and their infinite counterparts in connection with symmetric polynomials and some arithmetic quantities called powerful numbers.