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European Mathematical Society Publishing House
2024-03-28 23:52:29
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=PRIMS&vol=47&iss=2&update_since=2024-03-28
Publications of the Research Institute for Mathematical Sciences
Publ. Res. Inst. Math. Sci.
PRIMS
0034-5318
1663-4926
General
10.4171/PRIMS
http://www.ems-ph.org/doi/10.4171/PRIMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Research Institute for Mathematical Sciences, Kyoto University
47
2011
2
On M. Sato's Classification of Some Reductive Prehomogeneous Vector Spaces
Tatsuo
Kimura
University of Tsukuba, IBARAKI, JAPAN
Yukimi
Ishii
University of Tsukuba, IBARAKI, JAPAN
Inhyun
Ryu
University of Tsukuba, IBARAKI, JAPAN
Michio
Hamada
University of Tsukuba, IBARAKI, JAPAN
Yoshiteru
Kurosawa
University of Tsukuba, IBARAKI, JAPAN
Masaya
Ouchi
University of Tsukuba, IBARAKI, JAPAN
Tomohiro
Kamiyoshi
University of Tsukuba, IBARAKI, JAPAN
prehomogeneous vector spaces, classification
Under some condition, M. Sato classified reductive prehomogeneous vector spaces of the form $(G_{0}\times G,\Lambda _{1} \otimes \rho ,V(n)\otimes V)$. In this paper, under the other condition, we classify the prehomogeneous vector spaces of the same form. We consider everything over the complex number field $\mathbb{C}$.
Number theory
Group theory and generalizations
General
397
418
10.2977/PRIMS/40
http://www.ems-ph.org/doi/10.2977/PRIMS/40
Asymptotic Behaviour of Variation of Pure Polarized TERP Structure
Takuro
Mochizuki
Kyoto University, KYOTO, JAPAN
harmonic bundle, TERP structure, new supersymmetric index
The purpose of this paper is twofold. One is to give a survey of our study on reductions of harmonic bundles, and the other is to explain a simple application in the study of TERP structures. In particular, we investigate the asymptotic behaviour of the "new supersymmetric index" for variation of pure polarized TERP structure.
Several complex variables and analytic spaces
Algebraic geometry
General
419
534
10.2977/PRIMS/41
http://www.ems-ph.org/doi/10.2977/PRIMS/41
Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace
Boris
Feigin
Independent University of Moscow, MOSCOW, RUSSIAN FEDERATION
Michio
Jimbo
Rikkyo University, TOKYO, JAPAN
Tetsuji
Miwa
Kyoto University, KYOTO, JAPAN
difference Toda Hamiltonian, quantum groups, fermionic formulas, bosonic formulas
We derive a bosonic formula for the character of the principal space in the level $k$ vacuum module for $\widehat{\mathfrak{sl}}_{n+1}$, starting from a known fermionic formula for it. In our previous work, the latter was written as a sum consisting of Shapovalov scalar products of the Whittaker vectors for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$. In this paper we compute these scalar products in the bosonic form, using the decomposition of the Whittaker vectors in the Gelfand-Zetlin basis. We show further that the bosonic formula obtained in this way is the quasi-classical decomposition of the fermionic formula. %for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$ % modules computing it %by using the decomposition %of the Whittaker vectors in the Gelfand-Zetlin basis. %We show that the bosonic formula obtained in this way %is the quasi-classical decomposition of the fermionic formula.
Associative rings and algebras
General
535
551
10.2977/PRIMS/42
http://www.ems-ph.org/doi/10.2977/PRIMS/42
Baker–Akhiezer Modules on the Intersections of Shifted Theta Divisors
Koji
Cho
Kyushu University, FUKUOKA, JAPAN
Andrey
Mironov
Siberian Branch of the Russian Academy of Sciences, NOVOSIBIRSK, RUSSIAN FEDERATION
Atsushi
Nakayashiki
Tsuda College, TOKYO, JAPAN
abelian variety, theta function, D-module, commuting differential operators
The restriction, on the spectral variables, of the Baker–Akhiezer (BA) module of a g-dimensional principally polarized abelian variety with the non-singular theta divisor to an intersection of shifted theta divisors is studied. It is shown that the restriction to a k-dimensional variety becomes a free module over the ring of di erential operators in k variables. The remaining g–k derivations de ne evolution equations for generators of the BA-module. As a corollary new examples of commutative rings of partial di fferential operators with matrix coe cients and their non-trivial evolution equations are obtained.
Algebraic geometry
General
553
567
10.2977/PRIMS/43
http://www.ems-ph.org/doi/10.2977/PRIMS/43
Wall Crossing and M-Theory
Mina
Aganagic
University of California, BERKELEY, UNITED STATES
Hirosi
Ooguri
California Institute of Technology, PASADENA, UNITED STATES
Cumrun
Vafa
Harvard University, Cambridge, UNITED STATES
Masahito
Yamazaki
California Institute of Technology, PASADENA, UNITED STATES
string theory, bound state, Fock space, Calabi–Yau manifold
We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a Calabi-Yau 3-fold $X$. When $X$ has no compact 4-cyles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in M-theory compactified down to 5 dimensions by a Calabi-Yau 3-fold $X$. The generating function of the D-brane bound states is expressed as a reduction of the square of the topological string partition function, in all chambers of the Kähler moduli space.
Algebraic geometry
General
569
584
10.2977/PRIMS/44
http://www.ems-ph.org/doi/10.2977/PRIMS/44
Algebraic Analysis of Minimal Representations
Toshiyuki
Kobayashi
University of Tokyo, TOKYO, JAPAN
minimal representations, hyperfunction, branching law, reductive group, generalized Fourier transform, holomorphic semigroup, conservative quantity, ${\mathcal{D}}$-module
Small representations of a group yield large symmetries in the representation space. Analysis of minimal representations utilizes large symmetries in their geometric models, and serves as a driving force in creating new interesting problems that interact with other branches of mathematics. This article discusses the following three topics that arise from minimal representations of the inde nite orthogonal group: 1. construction of conservative quantities for ultra-hyperbolic equations, 2. quantitative discrete branching laws, 3. deformation of the Fourier transform, with emphasis on the prominent role of Sato's ideas in algebraic analysis.
Topological groups, Lie groups
Functional analysis
Differential geometry
Global analysis, analysis on manifolds
585
611
10.2977/PRIMS/45
http://www.ems-ph.org/doi/10.2977/PRIMS/45
Microlocal Geometry and Valued Fields
François
Loeser
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
non-archimedean geometry, valued fields, Milnor fiber, tangent cones, microlocal geometry
This is an overview of some recent connections between non-archimedean geometry and microlocal analysis, with some emphasis on the motivic Milnor fiber and the properties of tangent cones.
Mathematical logic and foundations
Number theory
Algebraic geometry
General
613
627
10.2977/PRIMS/46
http://www.ems-ph.org/doi/10.2977/PRIMS/46
The Laplace Transform of the Cut-and-Join Equation and the Bouchard–Mariño Conjecture on Hurwitz Numbers
Bertrand
Eynard
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
Motohico
Mulase
University of California at Davis, DAVIS, UNITED STATES
Bradley
Safnuk
Central Michigan University, MOUNT PLEASANT, UNITED STATES
Hurwitz numbers, cut-and-join equation, linear Hodge integral, topological recursion, Lambert curve
We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil–Petersson volume of the moduli space of bordered hyperbolic surfaces. We fi nd that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Mariño using topological string theory.
Algebraic geometry
Combinatorics
Quantum theory
General
629
670
10.2977/PRIMS/47
http://www.ems-ph.org/doi/10.2977/PRIMS/47