Presentation at the MATRIX conference on Geometric and Categorical Representation Theory

Title: The geometry of local Arthur packets

Speaker: Clifton Cunningham

Abstract: Beginning with a brief overview of the main properties of local Arthur packets and illustrated with examples, this talk explains how an Arthur parameter determines a category of perverse sheaves and how the microlocal perspective on this category reveals an Arthur packet.

Conference on Geometric and Categorical Representation Theory at MATRIX, Creswick Campus, University of Melbourne and Monash University

Program Description: Geometric and categorical representation theory are advancing rapidly, with a growing number of connections to the wider mathematical universe. The goal of this program is to bring international experts in these areas together to facilitate exchange and development of ideas. During the first week, there will be a lecture series by Prof. Luca Migliorini on the arithmetic theory of Higgs bundles.

Organisers: Clifton Cunningham (University of Calgary), Masoud Kamgarpour (University of Queensland), Anthony Licata (Australian National University), Peter McNamara (University of Queensland), Sarah Scherotzke (Bonn University), Oded Yacobi (University of Sydney)

https://www.matrix-inst.org.au/events/geometric-and-categorical-represen...

Speaker: Clifton Cunningham
Room: MS 337
This is a continuation of the talk from last week on admissible representations of $p$-adic G(2) associated to cubic unipotent Arthur parameters.

We have seen how the subregular unipotent orbit in the L-group for split G(2) determines a unipotent Arthur parameter and thus an unramified infinitesimal parameter $\lambda : W_F \to \,^LG(2)$.
Using the Voganish conjectures (\texttt{https://arxiv.org/abs/1705.01885v4}) we find that there are exactly 8 admissible representations with infinitesimal parameter $\lambda$.
Last week Qing Zhang interpreted $\lambda$ as a Langlands parameter for the split torus in $p$-adic G(2) and worked out the corresponding quasi-character $\chi : T(F) \to \mathbb{C}^*$ using the local Langlands correspondence.
We expect that all admissible representations in the composition series of $\mathop{Ind}_{B(F)}^{G(2,F)} \chi$ have infinitesimal parameter $\lambda$; we wonder if not all 8 admissible representations arise in this way.

In this talk I will calculate the multiplicity matrix that describes how these 8 admissible representations are related to 8 standard modules with infinitesimal parameter $\lambda$, assuming the Kazhdan-Lusztig conjecture as in appears in Section 10.2.3 of the preprint above.
To make this calculation I will use the Decomposition Theorem to calculate the stalks of all simple $H_\lambda$-equivariant perverse sheaves on the mini-Vogan variety $V_\lambda$, following the strategy explained in Section 10.3.3 of the preprint.

Speaker: Clifton Cunningham and Qing Zhang,

We consider the Voganish project for the cubic unipotent Arthur parameter for the split exceptional group $G_2$ over a p-adic field, which was first considered by Gan-Gurevich-Jiang. After introducing this parameter $\lambda$, we consider the Vogan variety and its orbits under the action of the natural group $H_\lambda$. We then determine a smooth cover of each orbits which will help to compute the $H_\lambda$-equivariant local systems on each orbit. We also determine the principle series representation of $G_2(F)$ associated with the unramified Langlands parameter.

Location: MS 337, University of Calgary

Speaker: Jerrod Smith

In this talk, we discuss cuspidality of representations of p-adic groups in a relative setting. Let G be a connected reductive group over a p-adic field F and $\theta$ is an involution on G. Let H be the subgroup fixed by $\theta$. One then can define H-relative supercuspidal representations via matrix coefficients. We will discuss the behavior of supercuspidality under induction.

Speaker: Sarah Dijols,

The Generalized Injectivity Conjecture of Casselman-Shahidi states that the unique irreducible generic subquotient of a (generic) standard module is necessarily a subrepresentation. It is related
to L-functions, as studied by Shahidi, hence has some number-theoretical flavor, although our technics lie in the fields of representations of reductive groups over local fields. It was proven for classical groups (SO(2n+1), Sp2n, SO(2n)) by M.Hanzer in 2010. In this talk, I will first explain our interest in this conjecture, and describe its main ingredients. I will further present our proof (under some restrictions) which uses techniques more amenable to prove this conjecture for all quasi-split groups.

Speaker: Qing Zhang

In this talk, we will discuss two integrals involving the exceptional group G2, both discovered by D. Ginzburg. The first integral gives an integral representation of the adjoint L-function of GL(3). Then I will report my recent work joint with J. Hundley on the holomorphy of adjoint L-function of GL(3). The second integral gives an integral representation of standard L-function of G2 itself. Related to this integral, we can consider a Fourier-Jacobi model for G2. I will discuss recent work joint with B. Liu on the uniqueness of such models over finite fields.

http://www.birs.ca/events/2018/focussed-research-groups/18frg237