RIMS Workshop: Developments in Representation
Theory and Related Topics
Date: 9–12, July, 2019
Location: Room 420, RIMS, Kyoto University
Organizer: Yoshiki Oshima (Osaka University)

Schedule9(Tue)
13:20 – 14:20 Toshiyuki Kobayashi (University of Tokyo)
Regular Representations on Homogeneous Spaces
14:35 – 15:35 Taito Tauchi (University of Tokyo)
A generalized uniformly bounded multiplicity theorem
15:50 – 16:50 Yosuke Morita (Kyoto University)
Cartan projections of abelian horospherical subgroups and proper actions
on homogeneous spaces
10(Wed)9:50 – 10:50 Ryotaro Kawago (Okayama University of Science)
Multiplicities of points on Schubert varieties in the Symplectic flag variety
11:05 – 12:05 Yuichiro Tanaka (University of Tokyo)
Visible actions of compact Lie groups on complex spherical varieties
13:20 – 14:20 Takahiro Nagaoka (Kyoto University)
Strong Lefschetz property and prehomogeneous vector space
14:35 – 15:35 Kazuki Ishimoto (Kobe University)
Fourier transform for prehomogeneous vector spaces over finite field
15:50 – 16:50 Hideto Nakashima (Nagoya University)
An algebraic proof for explicit formulas of basic relative invariants ofhomogeneous cones111(Thu)
9:50 – 10:50 Masaki Hidaka, Minoru Itoh (Kagoshima University)
The Schur polynomials in all nth primitive roots of unity
11:05 – 12:05 Genki Shibukawa (Kobe University)
Pieri type formulas and diﬀerence relations for Jack interpolation polynomials
13:20 – 14:20 Perez Victor (University of Tokyo)
Normality and Duality on Topological Groups
14:35 – 15:35 Masaru Sugawara (Tohoku University)
Convex bases for aﬃne quantum groups and its application for quantum
dilogarithm identities
15:50 – 16:50 Noriyuki Abe (University of Tokyo)
On Soergel bimodules
12(Fri)
9:10 – 10:10 Takashi Satomi (University of Tokyo)
A refinement of Young’s inequality for the convolution product on groups
10:20 – 11:20 Ryosuke Nakahama (University of Tokyo)
Weighted Bergman inner products on subspaces of bounded symmetricdomains
11:30 – 12:30 Takuma Hayashi (University of Tokyo)
Half-integrality of the closed SO(3)-orbit on the flag variety of SL32

Abstracts
Toshiyuki Kobayashi (University of Tokyo)
Title: Regular Representations on Homogeneous Spaces
I plan to discuss some basic questions about regular representations on
X acted algebraically by real reductive groups G.
1. (function spaces) Does the group G have a “good control” on the space
C(X) of functions on X?
2. (L2 theory) What can we say about “spectrum” for L2 (X)?
We highlight “multiplicities and “temperdness” for these questions, andgive their geometric criteria.
If time permits, I will mention some applications to branching problems
for restriction of infinite-dimensional representations.
Masaki Hidaka, Minoru Itoh (Kagoshima University)
Title: The Schur polynomials in all nth primitive roots of unity
We show that the values of the Schur polynomials in all nth primitiveroots of unity are in {1, 0, −1}, if n has at most two distinct odd prime factors.
This result can be regarded as a generalization of properties of the cyclotomicpolynomial. Indeed, when the partition is (1k ), our result is equivalent to
the well-known fact that the coeﬃcients of the nth cyclotomic polynomial
are in {1, 0, −1}, if n has at most two distinct odd prime factors. Moreover,when the partition is (k), our result is equivalent to a similar property of thecoeﬃcients of the multiplicative inverse of the nth cyclotomic polynomialwhich was proved by P. Moree about 10 years ago.
Genki Shibukawa (Kobe University)
Title: Pieri type formulas and diﬀerence relations for Jack interpolation
polynomials.
We obtain new Pieri type formulas for Jack polynomials. From these newPieri formulas, we give Pieri type formulas and diﬀerence relations for Jack
interpolation polynomials, and their applications.

3Perez Victor (University of Tokyo)
Title: Normality and Duality on Topological Groups
This talk can be divided essentially in two parts. In the first one, we introduce basic concepts and classical results of the class of topological groups.
Concretely, we deal with the Pontryagin Duality Theorem for locally compact abelian groups. In the second one, we present a result of normality
using techniques of duality theory of topological groups. A group topology
ν in a topological group (G, τ ) is said to be a compatible topology withthe duality (G, G∧ ) if the dual groups of G induced by τ and ν coincide,i.e. (G, ν)∧ = (G, τ )∧ .
The result we show states that for a concrete group (namely, the productof R-copies of Z), there do not exist normal topologies compatible with theduality (G, G∧ ) that belongs to a particular family; namely, the family of thelocally quasi-convex topologies.
Ryosuke Nakahama (University of Tokyo)
Title: Weighted Bergman inner products on subspaces of bounded symmetric domains
Let Hr (F) (F = R, C, H) be the space of Hermitian matrices, and weconsider the bounded symmetric domain D ⊂ Hr (F)C . In this talk we present
a result on the computation of the weighted Bergman inner product on D of apolynomial on the subspace Hr′ (F) ⊕ Hr′′ (F) and an exponential function on
Hr (F). Also, as an application, we present a result on explicit constructionof intertwining operators from representations of Sp(r, R) to those of thesubgroup U (r′ , r′′ ).

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Takuma Hayashi (University of Tokyo)
Title: Half-integrality of the closed SO(3)-orbit on the flag variety of SL3
In representation theory over C of a real reductive group GR , the geometryof K-orbits on the flag variety of G over C plays an important role. For
example, we can attach fundamental series representations of GR from closed
K- orbits.
Recently, several people have started to work on rational and integral
structures of Harish-Chandra modules. In this talk, I will show that theclosed SO(3)-orbit on the flag variety of SL3 is defined over Z [1/2], motivated by Michael Harris’ work on a construction of rational structures ofHarish-Chandra modules and Fabian Januszewski’s descent theorem for certain Harish-Chandra modules. This is a joint work with Fabian Januszewski.

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