# Representation Theory / Number Theory Seminar

### Seminar meets Wednesdays, 3:30-4:30pm MT (unless specified otherwise) in JWB 308. Zoom link will be available for online talks*.

*For security reasons, Zoom links will not be posted here. If you would like to attend a talk, but do not have the link, please contact Gil Moss, Peter Wear, or Petar Bakic (last name at math.utah.edu).

### Spring 2022

#### Friday, January 7, 3:00-4:00pm, LCB 225

(Note the non-standard day/time/room)

Speaker: Spencer Leslie, Duke
Title: Endoscopy for symmetric varieties
Abstract: Relative trace formulas are central tools in the study of relative functoriality. In many cases of interest, basic stability problems have not been addressed. In this talk, I discuss a theory of endoscopy in the context of symmetric varieties with the global goal of stabilizing the associated relative trace formula. I outline how, using the dual group of the symmetric variety, one can give a good notion of endoscopic symmetric variety and conjecture a matching of relative orbital integrals in order to stabilize the relative trace formula. Time permitting, I will explain my proof of these conjectures in the case of unitary Friedberg-Jacquet periods.

#### Wednesday, January 12, 3:30-4:30pm

Speaker: Anna Romanov, University of New South Wales
Title: Costandard Whittaker modules and contravariant pairings
Abstract: I'll discuss recent work with Adam Brown (IST Austria) in which we propose a new definition of costandard Whittaker modules for a complex semisimple Lie algebra using contravariant pairings between Whittaker modules and Verma modules. With these costandard objects, blocks of Milicic-Soergel's Whittaker category have the structure of highest weight categories. This allows us to establish a BGG reciprocity theorem for Whittaker modules. Our costandard objects also give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the flag variety.

#### Friday, January 21, 4:00-5:00pm, LCB 222

(Note the non-standard day/time/room)

Speaker: Rebecca Bellovin, University of Glasgow
Title: Modularity of trianguline Galois representations

#### Wednesday, March 24, 3:00-4:00pm

Speaker: Baiying Liu, Purdue University
Title: On recent progress on Jiang's conjecture on wave front sets of representations in Arthur packets.
Abstract: In this talk, I will introduce some recent progress on Jiang's conjecture on wave front sets of representations in Arthur packets. Jiang's conjecture is a natural generalization of Shahidi's conjecture on tempered L-packets. It shows that there is a strong connection between the structure of Arthur parameters and the wave front sets of representations in the corresponding Arthur packets. This includes some work joint with Dihua Jiang, and joint with Freydoon Shahidi.

#### Wednesday, March 31, 3:00-4:00pm

Speaker: Michael Griffin, BYU
Title: Moonshine
Abstract: In the 1970's, during efforts to completely classify the finite simple groups, several striking apparent coincidences emerged connecting the then-conjectural “Monster group” to the theory of modular functions. Conway and Norton turned these observed 'coincidences' into a precise conjecture known as “Monstrous Moonshine.” Borcherds proved the conjecture in 1992, embedding Monstrous Moonshine in a deeper theory of vertex operator algebras which have important physical interpretations. Fifteen years after Borcherds' proof, Witten conjectured an important role of Monstrous Moonshine in his search for a theory of pure quantum gravity in three dimensions. Under Witten's theory, the irreducible components of the Monster module represent energy states of black holes. The distribution of these energy states can be found using tools from number theory. Moonshine-phenomena have also been observed for other groups besides the Monster. These include the Umbral Moonshine conjectures of Cheng, Duncan, and Harvey which arise from the symmetry groups of each of the 24-dimensional Niemeier lattices. Recently, Moonshine for other sporadic simple groups have been shown to connect arithmetic properties of certain elliptic curves to the class numbers of certain imaginary quadratic fields.

#### Thursday, April 8, time TBD (Colloquium talk)

Speaker: Aaron Pollack, UCSD
Title: TBA
Abstract: TBA

#### Wednesday, April 14, 3:00-4:00pm

Speaker: Martin Weissman, UC Santa Cruz
Title: The compact induction theorem for rank-one p-adic groups
Abstract: A folklore conjecture predicts that when G is a p-adic group, every irreducible supercuspidal representation of G is induced from a compact-mod-center open subgroup. This was proven for GL(n) by Bushnell and Kutzko. For other groups, e.g., classical groups, tame groups, etc., the conjecture is proven for sufficiently large p thanks to hard work by many people. In this talk, I will describe a recent proof of the conjecture which applies to all groups G of relative rank one, with no assumptions about p. The method is to use the work of Schneider and Stuhler to connect supercuspidal representations to sheaves on the Bruhat-Tits tree of G, and "refine" these sheaves until the induction theorem becomes obvious.