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

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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.

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.

Please meet up at the parking lot.

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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.

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 $G$.

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.

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.

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.

Kelowna Brewing Company 975 Academy Way #121, Kelowna, BC V1V 3A4

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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.

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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.

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.

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.

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.

Please meet up at the parking lot.

Please use the meal card and get the breakfast items from Comma(or any food merchants)

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