# Schedule for: 24w5220 - Branching Problems for Representations of Real, P-Adic and Adelic Groups

Beginning on Sunday, July 7 and ending Friday July 12, 2024

All times in UBC Okanagan, Canada time, PDT (UTC-7).

Sunday, July 7 | |
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16:00 - 23:00 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk Nechako Residence) |

20:00 - 22:00 | Informal gathering (Lounge) |

Monday, July 8 | |
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08:00 - 08:45 |
Breakfast ↓ Please use the meal card and get the breakfast items from Comma(or any food merchants) (UBC Okanagan - Food services) |

08:45 - 09:00 | Introduction and Welcome by BIRS-UBCO Staff (Main Meeting Room - ARTS 110) |

09:00 - 09:30 | Toshiyuki Kobayashi: Overview of branching problems in the real setting (Main Meeting Room - ARTS 110) |

09:45 - 10:25 | Toshihisa Kubo: Symmetry breaking operators and F-method (Main Meeting Room - ARTS 110) |

10:25 - 10:45 | Coffee Break (ARTS 112) |

10:45 - 11:25 | Leticia Barchini: Matching real and p-adic Kazhdan-Lusztig polynomials. (Main Meeting Room - ARTS 110) |

11:30 - 13:00 | Lunch (ARTS 112) |

13:00 - 13:20 | Group Photo (Main Meeting Room - ARTS 110) |

13:20 - 14:00 | Haian HE: On the discretely decomposable restrictions of $(\mathfrak g, K)$-modules (Main Meeting Room - ARTS 110) |

14:20 - 15:00 | Masatoshi Kitagawa: Cartan subalgebras in the branching problem. (Main Meeting Room - ARTS 110) |

15:00 - 15:30 | Coffee Break (ARTS 112) |

15:30 - 15:55 | Víctor Pérez-Valdés: On the differential symmetry breaking operators for principal series representations of $(SO_o(4,1), SO_o(3,1))$. (Main Meeting Room - ARTS 110) |

16:00 - 16:25 |
Baiying Liu: On local Arthur packets and wave front sets of representations ↓ In this talk, I will introduce recent progress on certain problems related to local Arthur packets of classical groups. First, I will introduce the results of Hiraku Atobe, and independently a joint work with Alexander Hazeltine and Chi-Heng Lo on the intersection problem of local Arthur packets for symplectic and split odd special orthogonal groups. Applications of these results include proving the enhanced Shahidi conjecture and producing new families of local Arthur packets such that the GGP pairs are not unique. Then, I will introduce recent results towards studying wavefront sets of individual admissible representations, or those in local Arthur packets or local L-packets, on the upper bound conjecture, the Jiang conjecture, and the generalized Shahidi conjectures. (Main Meeting Room - ARTS 110) |

16:30 - 16:55 |
Yuichiro Tanaka: On the multiplicity-freeness property of cohomology spaces and the visibility of group actions ↓ With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible action for holomorphic actions of Lie groups on complex manifolds. His propagation theorem of the multiplicity-freeness property produces various kinds of multiplicity-free theorems for unitary representations realized in the space of holomorphic sections of an equivariant holomorphic vector bundle whose base space admits a visible action of a Lie group.
Kobayashi has indicated two directions of generalizations of his multiplicity-free theorem. One is a generalization to infinite dimensional manifolds and has been done by Miglioli and Neeb. The other is a generalization to cohomology spaces, which is the main concern of this talk.
I would like to talk about a cohomology version of Kobayashi's theorem and its application to his conjecture on the multiplicity-free restriction of Zuckerman derived functor modules to symmetric subgroups. (Main Meeting Room - ARTS 110) |

17:30 - 18:00 |
Shuttle bus : UBCO Eme building - Parking lot ↓ Please meet up at the parking lot. (Other - See Description) |

18:00 - 20:00 | Dinner (Four Points Hotel) |

Tuesday, July 9 | |
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08:00 - 09:00 |
Breakfast ↓ Please use the meal card and get the breakfast items from Comma(or any food merchants) (UBC Okanagan - Food services) |

09:00 - 09:30 |
Monica Nevins: Overview of branching problems in the $p$-adic setting ↓ We share some p-adic representation theory for a real audience, with focus on aspects that are different from the real case, before describing a variety of branching problems of current interest. (Main Meeting Room - ARTS 110) |

09:45 - 10:25 |
Fiona Murnaghan: Relatively supercuspidal representations ↓ Let $G$ be a connected reductive group defined over a nonarchimedean local field of odd residue characteristic.
Let $H$ be the group of fixed points of an involution of $G$. A representation of $G$ is said to be $H$-distinguished if there exists a nonzero $H$-invariant linear functional on the space of the representation. The $H$-relatively supercuspidal representations of $G$ are the $H$-distinguished representations of $G$ whose generalized matrix coefficients are compactly supported modulo
$HZ$, where $Z$ is the centre of $G$. We will describe results giving a method for constructing $H$-relatively supercuspidal representations of $G$. (Main Meeting Room - ARTS 110) |

10:25 - 10:45 | Coffee Break (ARTS 112) |

10:45 - 11:25 |
Monica Nevins: Relating $p$-adic types between a group and a fixed-point subgroup. ↓ Let $G$ be a connected reductive group over a local nonarchimedean field of residual characteristic $p$ and set $H=(G^\Gamma)^\circ$, where $\Gamma \subset Aut(G)$ is a finite group such that $p\not| \Gamma$. The restriction of an Adler--Yu type $(J,\lambda)$ to its pro-$p$ subgroup $J_+$ is called a \emph{semisimple character} in the setting of Bushnell--Kutzko--Stevens types. In this talk we show that the restriction of any $\Gamma$-stable datum defining a semisimple character for $G$ gives that of a semisimple character for $H$ and sketch an argument that all semisimple characters for $H$ arise in this way. Part of this is joint work with Peter Latham. (Main Meeting Room - ARTS 110) |

11:30 - 13:00 | Lunch (ARTS 112) |

13:00 - 13:40 | Taito Tauchi: Relationship between multiplicities of induced representations and orbit decomposition on real and complex flag varieties. (Main Meeting Room - ARTS 110) |

14:00 - 14:40 | Hiroyuki Ochiai: A $D$-module approach to invariant distributions with finitely many orbits. (Main Meeting Room - ARTS 110) |

14:45 - 15:15 | Coffee Break (ARTS 112) |

15:15 - 15:55 | Yoshiki Oshima: Discrete branching laws of derived functor modules. (Main Meeting Room - ARTS 110) |

16:15 - 16:40 |
Zhiyu Zhang: Arithmetic fundamental lemmas and non-reductive subgroups. ↓ I will discuss the (arithmetic) twisted Fourier-Jacobi Gan-Gross-Prasad conjecture, which studies certain period integrals and branching laws involved with Weil representations of unitary groups. Via relative trace methods, we are led to the (arithmetic) fundamental lemmas on (arithmetic) orbital integrals. (Main Meeting Room - ARTS 110) |

16:45 - 17:10 |
Ekta Tiwari: Branching rules for irreducible supercuspidal representations of unramified $U(1,1)$. ↓ The restriction of a supercuspidal representation of unramified $p$-adic $U (1, 1)$ to a maximal compact subgroup decomposes as a multiplicity-free direct sum of irreducible representations. Moreover, we show that when restricted to a small enough neighbourhood, such a representation decomposes as a direct sum of irreducible representations constructed from nilpotent elements of the Lie algebra. In this talk, we present an explicit description of this decomposition. (Main Meeting Room - ARTS 110) |

17:30 - 19:00 |
Dinner ↓ Kelowna Brewing Company
975 Academy Way #121, Kelowna, BC V1V 3A4 (Other - See Description) |

Wednesday, July 10 | |
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08:00 - 08:45 |
Breakfast ↓ Please use the meal card and get the breakfast items from Comma(or any food merchants) (UBC Okanagan - Food services) |

09:00 - 09:40 |
Dipendra Prasad: What does the restriction really looks like in the GGP problems for $p$-adics - Online talk. ↓ Branching laws for complex representations of groups describe how a representation of a group decomposes when restricted to a subgroup which in the case of compact groups is a direct sum of irreducible representations.
When dealing with infinite dimensional representations of a $p$-adic group $G$ restricted to a subgroup $H$, the branching laws describe irreducible representations of $H$ which arise as a quotient representation. It is not clear if this allows one to describe how an irreducible representation of $G$ restricted to $H$ looks like. The lecture is an attempt to discuss this question. (Main Meeting Room - ARTS 110) |

09:50 - 10:20 | Dihua Jiang: Overview of branching problems in the automorphic setting (Main Meeting Room - ARTS 110) |

10:20 - 10:45 | Coffee Break (ARTS 112) |

10:45 - 11:25 | Genkai Zhang: Multiplicities of $SU(2)$-representations for quaternionic symmetric pair $(G, K)$. Online talk (Main Meeting Room - ARTS 110) |

11:30 - 13:00 | Lunch (ARTS 112) |

13:00 - 17:30 | Free Afternoon (Kelowna) |

17:30 - 19:00 | Dinner (ARTS 112) |

Thursday, July 11 | |
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08:00 - 09:00 |
Breakfast ↓ Please use the meal card and get the breakfast items from Comma(or any food merchants) (UBC Okanagan - Food services) |

09:00 - 09:40 | Michael Harris: Restriction problems and ∂-cohomology: Work in progress with Kobayashi and Speh. Online talk (Main Meeting Room - ARTS 110) |

09:50 - 10:30 |
Dihua Jiang: Branching Problem, Wavefront Set, and Langlands Functoriality ↓ We discuss the branching problem in the automorphic setting, which is formulated as understanding of periods of automorphic forms. With certain information from the wavefront sets as input, we discuss conjectures and results that relate the non-vanishing of periods to the Langlands functoriality. (Main Meeting Room - ARTS 110) |

10:30 - 10:50 | Coffee Break (ARTS 112) |

10:50 - 11:30 |
Andre Reznikov: Mass distribution of automorphic Frobenius functionals. Online talk. ↓ Automorphic representation comes equipped with the natural Frobenius functional. In the classical picture Frobenius functional is given by evaluation at a point.
As such, it is highly relevant for problems dealing with periods. I will discuss how the quantitative version of the Orbit method developed by P. Nelson and A. Venkatesh could be used in order to formulate questions and obtain non-trivial bounds on mass distribution of Frobenius functional associated to a cuspidal representation. (Main Meeting Room - ARTS 110) |

11:30 - 13:00 | Lunch (ARTS 112) |

13:00 - 13:40 | Ryosuke Nakahama: Holographic and symmetry breaking operators of holomorphic discrete series representations for $(SU(3,3),SO^*(6))$. (Main Meeting Room - ARTS 110) |

14:00 - 14:40 | Hideko Sekiguchi: Isomorphisms between certain cohomologies over indefinite Grassmannian manifolds. (Main Meeting Room - ARTS 110) |

14:45 - 15:15 | Coffee Break (ARTS 112) |

15:15 - 15:55 | Paul-Emile Paradan: Multiplicities of discrete series representations with respect to a symmetric pair and index theory. (Main Meeting Room - ARTS 110) |

16:15 - 16:40 |
Danielle Wang: Twisted GGP conjecture in the unramified case ↓ We present the relative trace formula approach to the global twisted Gan-Gross-Prasad conjecture. In particular, we explain how the relevant fundamental lemma can be reduced to the Jacquet-Rallis fundamental lemma, which allows us to prove the conjecture in the unramified case under some local conditions. (Main Meeting Room - ARTS 110) |

16:45 - 17:10 |
Huajie Li: On the Guo-Jacquet trace formula ↓ In this talk, I shall report on joint work in progress with Pierre-Henri Chaudouard which establishes the coarse Guo-Jacquet trace formula. It serves as an example of relative trace formulae for symmetric spaces and should be useful in generalisation of Waldspurger’s theorem on central L-values for $GL(2)$. In order to understand geometric terms in this trace formula, we also obtain a formula of semi-simple descent. In particular, regular semi-simple terms are written as explicit weighted orbital integrals. (Main Meeting Room - ARTS 110) |

17:30 - 18:00 |
Shuttle bus : UBCO Eme building - Parking lot ↓ Please meet up at the parking lot. (Other - See Description) |

18:00 - 20:00 | Dinner (Four Points Hotel) |

Friday, July 12 | |
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08:00 - 09:00 |
Breakfast ↓ Please use the meal card and get the breakfast items from Comma(or any food merchants) (UBC Okanagan - Food services) |

08:50 - 09:00 | Checkout by 11AM (Front Desk Nechako) |

09:00 - 09:40 |
Dmitry Gourevitch: Orthogonal families of hypergeometric polynomials. ↓ Motivated by the theory of hypergeometric orthogonal polynomials, we consider quasi-orthogonal polynomial families - those that are orthogonal with respect to a non-degenerate bilinear form defined by a linear functional - in which the ratio of successive coefficients is given by a rational function $f(u,s)$ which is polynomial in $u$. We call this a family of Jacobi type.
Our main result is that, up to rescaling and renormalization, there are only five families of Jacobi type. These are the classical families of Jacobi, Laguerre and Bessel polynomials, and two more one parameter families $E^c, F^c$. Each family arises as a specialization of some hypergeometric series. The last two families can also be expressed through Lommel polynomials, and they are orthogonal with respect to a positive measure on the real line for $c>0$ and $c>-1$ respectively.
We also consider the more general rational families, i.e. quasi-orthogonal families in which the ratio $f(u,s)$ of successive coefficients is allowed to be rational in u as well. I will formulate the two main theorems, one on Jacobi families and one on rational families, as well as the main ideas of the proofs. This is a joint work with J. Bernstein and S. Sahi: https://arxiv.org/abs/2401.14715 (Main Meeting Room - ARTS 110) |

09:45 - 10:15 | Quentin Labriet: Differential symmetry breaking operators for the pair $(GL(n+1,\mathbb R),GL(n,\mathbb R))$. Online talk. (Main Meeting Room - ARTS 110) |

10:15 - 10:45 | Coffee Break (ARTS 112) |

10:45 - 11:25 | Chengbo Zhu: On the Arthur-Barbasch-Vogan conjecture. (Main Meeting Room - ARTS 110) |

11:30 - 13:00 | Lunch (ARTS 112) |