Representation theory of p-adic groups

• Petar Bakić: The Langlands Programs: a demonstration 

This will be an “infomercial” talk with the aim of introducing and advertising the Langlands Programs project ( https://langlandsprograms.com/). After a brief introduction to the project, I will demonstrate some of the software that has already been developed.

This is joint work with Elad Zelingher and Thomas Lanard.

• Barbara Bošnjak: On Mœglin parameterization of irreducible tempered representations

J. Arthur classified irreducible tempered representations with tempered A-parameters. C. Mœglin found a procedure how to determine these representations from the parameters. In her construction, one defines representation of the higher rank groups and then uses its derivatives, defined by Jacquet modules, to obtain the desired representation. 

In this talk, we present a work in progress on the classification of the irreducible tempered representations without need to go to higher rank groups. This characterization resembles the parameterization of irreducible square integrable representations by C. Mœglin and M. Tadić. This is joint work with I. Ciganović and M. Tadić.

• Igor Ciganović: Decomposition of some classes of induced representations of p-adic classical groups 

We consider composition series of representations of classical groups over a p-adic field, of characteristic different from 2, induced from two or more irreducible representations, of general linear groups, attached to some classes of segments, and a strongly positive representation of a smaller classical group.

• Iva Kodrnja: Regularity and modularity of number 2025 

Number 2025 is a square of number 45. Number 45 is a triangular number, sum of digits of 10-base number system. So, 2025 is an odd square of a triangular number and thus the sum of cubes of digits by Nichomachus’s theorem.

In this talk we will investigate the regularity of number 2025 as a figurate number connecting it to point lattices, finite group of symmetries and its harmonies through its reduced residue system. 

• Johannes Girsch: Doubling method gamma factors for certain depth zero representation 

Local constants play a central role in the complex representation theory of reductive p-adic groups; for instance, they are essential in the formulation of the local Langlands correspondence. However, for general reductive groups, it can be quite challenging to attach such local constants to irreducible representations. In the case of classical groups, one important approach is the doubling method developed by Piatetski-Shapiro and Rallis. In this talk, I will describe how the doubling gamma factors associated to certain irreducible depth-zero representations of classical groups can be computed explicitly. This is joint work with E. Zelingher. 

• Lovro Greganić: Explicit calculation of the Franke filtration for the general linear group 

The Franke filtration is a descending filtration of spaces of automorphic forms on adelic reductive groups. Its main feature is that the quotients of the filtration can be described in terms of parabolically induced representations. Such a structural description of spaces of automorphic forms has found many applications, but in order to fulfill its full potential, an explicit calculation of the filtration is required. This is a difficult task, not only because of demanding combinatorics, but also because it depends on fine arithmetic information related to Eisenstein series and automorphic L-functions. The goal of this talk is to present an algorithm for explicit calculation of the Franke filtration in the case of the general linear group and its implementation on a computer. This is a joint work with N. Grbac. This work is supported by the Croatian Science Foundation under the project HRZZ-IP-2022-10-4615.  

• Marcela Hanzer: The small vs. the big theta lift 

In this talk, we will give an overview of some results related to the classical and exceptional theta correspondence  for the dual reductive pairs of p-adic groups.

We will recall the notion of the classical theta (Howe) correspondence and discuss the known results of the determination of the full theta lift, especially the situations where the big theta lift is irreducible, thus equal to the small theta lift. The determination of the (full) big theta lift is very important not only in the global considerations, but, recently, in the relative Langlands program, where it is related to some branching problems. We will then  also define  the exceptional theta correspondence and give some examples of the irreducibility of the theta lifts.

• Thomas Lanard: An algorithm for Aubert-Zelevinsky duality à la Mœglin-Waldspurger 

The Aubert-Zelevinsky duality is an involution on the irreducible representations of a p-adic group, playing a central role in representation theory. For GL_n, irreducible representations can be classified by combinatorial objects called multisegments. In this case, an explicit formula to compute the Aubert-Zelevinsky dual was given by Mœglin and Waldspurger. For classical groups such as Sp_{2n} or SO_{2n+1}, irreducible representations can be described in terms of Langlands parameters. In this talk, I will present a combinatorial algorithm, inspired by the Mœglin–Waldspurger approach, to compute the Aubert–Zelevinsky dual in terms of Langlands data. Interestingly, the algorithm was discovered with the help of machine learning tools, which guided us toward patterns leading to its formulation. This is joint work with Alberto Mínguez. 

• Alexander Stadler: Combinatorics of Extended Multi-Segments

Let F be a p-adic field. We consider representations of Arthur type of a classical group G over F. By use of the Endoscopic Classification, Moeglin gave an explicit way to construct Arthur packets when G is a symplectic or special orthogonal group. Further analysis by Xu and Atobe developed the notion of Extended Multi-Segments, which parametrise certain Arthur packets. We review the definition, the construction of these representations in terms of Langlands data and consider some combinatorial problems that arise in this context. 

• Justin Trias: The universal Harish-Chandra j-function 

The Harish–Chandra μ-function plays a central role in the explicit Plancherel formula for a p-adic group G. It arises as the normalising factor for the Plancherel measure on the unitary dual of G, and is defined through the theory of intertwining operators.

In this talk, we show how to extend the construction of the μ-function—or more precisely its inverse, the j-function—to all finitely generated representations, and over general coefficient rings such as Z[1/p]. This leads to a universal j-function with values in the Bernstein centre, which specialises to the classical j-function. Beyond its role in harmonic analysis, the universal j-function also encodes arithmetic information: it reflects aspects of the local Langlands correspondence for classical groups, via criterions of Muić and Moeglin-Tadić connecting it to the reducibility points of parabolically induced representations. Time permitting, we will illustrate how this perspective applies to the study of the local Langlands correspondence in families. This is joint work with Gil Moss.

• Sonja Žunar Kožić: Non-vanishing of Poincaré series on irreducible bounded symmetric domains

Holomorphic automorphic forms on irreducible bounded symmetric domains are a natural and far-reaching generalization of classical modular forms and Siegel modular forms. Every irreducible bounded symmetric domain D can be realized as a quotient G/K, where G is a connected semisimple Lie group with finite center, and K is a maximal compact subgroup of G. 

In this talk, we will consider holomorphic automorphic forms on D given by Poincaré series of polynomial type that lift, in a standard way, to cuspidal automorphic forms on G given by Poincar ́e series of K-finite matrix coefficients of integrable discrete series representations of G. Determining which of these Poincaré series vanish identically is a surprisingly subtle and non-trivial problem. I will explain how it can be answered, in certain cases, by means of Muić's integral non-vanishing criterion for Poincaré  series on locally compact Hausdorff groups.