Research Workshop:
Seminar in Representation Theory
Organizers:
Jan Frahm (Aarhus University)
Bent Ørsted (Aarhus University)
September 8–9, 2021

The purpose of this seminar is to present recent developments in the representation theory of real reductive groups and related areas. The time slots were chosen sothat researchers from both Asia and Europe can attend. We hope that this meeting
also helps to bring together Asian and European colleagues within the AIM ResearchCommunity “Representation Theory & Noncommutative Geometry” on the Sococoplatform.

Zoom details
Zoom link: https://aarhusuniversity.zoom.us/j/68808112524
Meeting-ID: 688 0811 2524

Speakers• Wee Teck Gan (National University of Singapore)• Toshiyuki Kobayashi (The University of Tokyo)• Yoshiki Oshima (Osaka University)• Binyong Sun (Zhejiang University)• Chen-Bo Zhu (National University of Singapore)1

Schedule
WEDNESDAY, SEPTEMBER 8
9:00 – 9:50 CET
15:00 – 15:50 CST

Kobayashi

Tempered representations and limit algebras

16:00 – 16:50 JST
9:50 – 10:40 CET
15:50 – 16:40 CST

Theta correspondence and specialZhu

unipotent representations

16:50 – 17:40 JST

Virtual Coffee/Tea11:00 – 11:50 CET
17:00 – 17:50 CST

GanTwisted GGP problems and conjectures

18:00 – 18:50 JST

THURSDAY, SEPTEMBER 9
9:00 – 9:50 CET
15:00 – 15:50 CSTSun

Archimedean period relations and period relations
for automorphic L-functions

16:00 – 16:50 JST

Virtual Coffee/Tea
10:10 – 11:00 CET
16:10 – 17:00 CST
17:10 – 18:00 JSTOshima

On the asymptotic support of Plancherel
measures for homogeneous spaces

CET = Central European TimeCST = China Standard Time = CET + 6hrs
JST = Japan Standard Time = CET + 7hrs
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Abstracts
Wednesday, September 8
9:00 – 9:50 CET
Toshiyuki Kobayashi (The University of Tokyo)
Tempered representations and limit algebras
I plan to discuss some new connection between the following four (apparently unrelated) topics:
1. (analysis) Tempered unitary representations on homogeneous spaces
2. (combinatorics) Convex polyhedral cones
3. (topology) Limit algebras
4. (symplectic geometry) Quantization of coadjoint orbits,
based on a series of joint papers with Y. Benoist “Tempered homogeneous spaces
I–IV”.
9:50 – 10:40 CET
Chen-Bo Zhu (National University of Singapore)
Theta correspondence and special unipotent representations
The theory of theta correspondence, initiated by Howe, provides a powerful methodof constructing irreducible admissible representations of classical Lie groups. In thistalk, I will discuss a recent work, joint with Barbasch, Ma and Sun, in which weshow that in addition to irreducible unitary parabolic inductions, theta lifts yield
all special unipotent representations of a classical Lie group G. As a consequenceof the construction and the classification, we conclude that all special unipotent
representations of G are unitarizable, as predicted by the Arthur–Barbasch–Voganconjecture.
11:00 – 11:50 CET
Wee Teck Gan (National University of Singapore)
Twisted GGP problems and conjectures
I will discuss some twisted variants of the GGP restriction problems in the setting of
skew-Hermitian spaces. Together with Gross and Prasad, we formulate conjecturalanswers to these twisted GGP problems and provide some evidences in low rank
and for unitary principal series.

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Thursday, September 9
9:00 – 9:50 CET
Binyong Sun (Zhejiang University)
Archimedean period relations and period relations for automorphic L-functions
It was known to Euler that ζ(2k) is a rational multiple of π2k , where ζ is the Euler–Riemann zeta function, and k is a positive integer. Following the pioneering works ofG. Shimura, P. Deligne and etc., D. Blasius proposed a conjecture which asserts thatsimilar rationality results hold for very general automorphic L-functions. We confirm Blasius’s conjecture in two cases: the standard L-functions of symplectic type(joint with Dihua Jiang and Fangyang Tian), and the Rankin-Selberg L-functions for
GL(n) × GL(n − 1) (joint with Jian-Shu Li and Dongwen Liu). The key ingredientis the Archimedean period relations for the modular symbols at infinity. These twocases have already been studied by many authors, including Harris–Lin, Grobner–Raghuram, Harder–Raghuram, Januszewski, Grobner–Lin, and etc.
10:10 – 11:00 CET
Yoshiki Oshima (Osaka University)
On the asymptotic support of Plancherel measures for homogeneous spaces
Let G be a real reductive group and X a homogeneous G-manifold. The Plancherel
measure for X describes how L 2 (X ) breaks up into irreducible unitary representations of G. We discuss asymptotics of the support of Plancherel measure and relate
it with geometry of coadjoint orbits. In particular, we give a sufficient condition for
the existence of discrete series. This is a joint work with Benjamin Harris.

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