Geometric and Categorical Representation Theory

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)

Event Date: 
Monday, December 10, 2018 - 09:00 to Friday, December 21, 2018 - 12:00
Event Type: 

Calculation of the multiplicity matrix associated to the infinitesimal parameter of the cubic unipotent Arthur parameter

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{}) 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.

Event Date: 
Thursday, November 29, 2018 - 10:30 to 11:30

Cubic unipotent Arthur parameter for G2

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

Event Date: 
Thursday, November 22, 2018 - 10:00 to 11:30
Event Type: 
Voganish project

Support of closed orbit relative matrix coefficients

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.

Event Date: 
Thursday, October 11, 2018 - 10:00 to 11:30
Event Type: 

The generalized injectivity conjecture

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.

Event Date: 
Thursday, October 4, 2018 - 10:00 to 11:30

Two integrals involving the exceptional group G2

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.

Event Date: 
Thursday, September 20, 2018 - 10:00 to 11:30
Event Type: 

Calculation of the characteristic cycles of the Kashiwara-Saito singularity

Speaker: Reggie Lybbert

In their 1997 paper, _Geometric construction of crystal bases_ (Duke Math Journal, Vol 89, No. 1), Masaki Kashiwara and Yoshihisa Kashiwa Saito described a singularity in a quiver representation variety of type A_5 with the property that the characteristic cycles of the singularity is reducible, this providing a counterexample to a conjecture of Kazhdan and Lusztig. This singularity is now commonly know at the Kashiwara-Saito singularity. While the 1997 paper showed that the characteristic cycles of the Kashiwara-Saito singularity decomposes into at least two irreducible cycles, they promised, but did not prove, that it decomposes into exactly two irreducible cycles. Using techniques developed in the example part of the Voganish paper, augmented by some computational tools developed this summer, we believe we have a proof of this promise. In this seminar we'll give a sketch of the proof.

Using this calculation and the local Langlands correspondence for GL(16), we should be able to exhibit an irreducible representation \pi of p-adic GL(16) with the property that its ABV-packet (as defined in the Voganish paper) contains exactly one other irreducible representation, \pi', and also describe that representation. We refer to this \pi as the Kashiwara-Saito representation of GL(16) and to \pi' as its coronal representation. This will provide the smallest known example of an irreducible representation of p-adic general linear group with one coronal representation.

Locations: Zoom (, Calgary, Lethbridge

Event Date: 
Tuesday, July 24, 2018 - 10:00 to 11:30
Event Type: 
Voganish Project