The Harmonic Analysis Seminar is a meeting of the research team of Prof. Martin Olbrich. Its aim is to present both research works and surveys of mathematical areas of common interest.

### Meetings

The seminar currently meets on Tuesday from 10h30 to 12h00 in room 6A (Campus Belval, Maison du Nombre, 6th floor) unless specified otherwise next to the date.

Occassionally we will have talks by invited guests at other times to be announced here.

### Organizers

• Guenda Palmirotta
• Martin Olbrich

### Upcoming sessions

• Monday, 9.11.2020, 10 a.m. — online
Guenda Palmirotta (Université du Luxembourg), " Solvability of invariant differential equations on the upper half-plane - III"

### Past sessions

• 27.10.2020 — online
Guenda Palmirotta (Université du Luxembourg), " Solvability of invariant differential equations on the upper half-plane - II"
• 20.10.2020 — Room 6A
Guenda Palmirotta (Université du Luxembourg), " Solvability of invariant differential equations on the upper half-plane - I"
• POSTPONED 17.3.2020 - Room 6A
Salah Mehdi (Université de Lorraine, France), "Limit of orbits and representations of Lie groups-II"
• POSTPONED 26.3.2020 - Room MNO 1.040
Lucas Fresse (Université de Lorraine, France), "Limit of orbits and representations of Lie groups-III"
• POSTPONED 07.4.2020 - Room 6A
Polyxeni Spilioti (Aarhus University, Denmark), "tbd"
• 10.3.2020 - Room 6A
Salah Mehdi (Université de Lorraine, France), "Limit of orbits and representations of Lie groups-I"
• 11.2.2020, 10.30 - Room MNO 1.040
Pavle Pandžić (University of Zagreb, Croatia), "On the classification of unitary highest weight modules"
• 8.1.2020 - Room 6A
Spyros Afentoulidis (Université de Lorraine - Metz), "Algebraic Dirac Operators and Representation Theory-III"
• 13.1.2020 - Room 6A
Spyros Afentoulidis (Université de Lorraine - Metz), "Algebraic Dirac Operators and Representation Theory-II"
• 10.12.2019 - Room 6B
Spyros Afentoulidis (Université de Lorraine - Metz), "Algebraic Dirac Operators and Representation Theory-I"
• 14.11.2019 — Room 6A
Guenda Palmirotta (Université du Luxembourg), "Estimates for systems of polynomial equations on $\mathbb C^n$ - II"
• 24.10.2019 — Room 6A
Guenda Palmirotta (Université du Luxembourg), "Estimates for systems of polynomial equations on $\mathbb C^n$"
• 24.09.2019 — Room 6A
Martin Olbrich (Université du Luxembourg), "Hörmander's estimates for Dolbeault operators I"
• 09.07.2019 — Room 6A — 14h00
Martin Olbrich (Université du Luxembourg), "Introductory remarks on sheaves and their cohomology – IV"
• Thursday 20.06.2019 — Room 6A
Martin Olbrich (Université du Luxembourg), "Introductory remarks on sheaves and their cohomology – III"
• 28.05.2019 — Room 6A
Martin Olbrich (Université du Luxembourg), "Introductory remarks on sheaves and their cohomology – II"
• 21.05.2019 — Room 6A — 14h00
Martin Olbrich (Université du Luxembourg), "Introductory remarks on sheaves and their cohomology"
• 14.05.2019 — Room 6A
Guendalina Palmirotta (Université du Luxembourg), "Systems of polynomial equations on $\mathbb C^n$ – II"
• 07.05.2019 — Room 6A
Guendalina Palmirotta (Université du Luxembourg), "Systems of polynomial equations on $\mathbb C^n$"
• 12.03.2019 — Room 6A
Pavle Pandžić (University of Zagreb, Croatia), "Dirac index and twisted characters"
Abstract: Dirac operators have played an important role in representation theory of real reductive Lie groups since the work of Parthasarathy and Atiyah-Schmid on the construction of discrete series representations in the 1970s. One of the important invariants of representations is the Dirac index. An algebraic way to define the Dirac index is as the Euler characteristic of the Dirac cohomology of the associated Harish-Chandra module. The concept of Dirac cohomology was introduced by Vogan and subsequently studied by Huang-Pandzic and others. One of the important properties of the Dirac index of a representation in the equal rank case is its close relationship with the character on the compact Cartan subgroup. In the unequal rank case, the Dirac index of all representations is zero and therefore it is a useless notion. We have however introduced a new invariant, twisted Dirac index, which is a good substitute for the classical notion in the unequal rank cases. In this lecture I will first review some basic facts about representations, Harish-Chandra modules, Dirac cohomology and index. I will then explain the notion of twisted Dirac index and present some examples and applications. This is joint work with Dan Barbasch and Peter Trapa.
• 05.03.2019 — Room 6A
Martin Olbrich (Université du Luxembourg), "The Paley–Wiener theorem for semi-simple Lie groups III"
• 19.02.2019 — Room 6A
Yannick Voglaire (Université du Luxembourg), "Reduction of symplectic symmetric spaces and étale affine representations – III"
Abstract: We introduce a notion of symplectic reduction for symplectic symmetric spaces as a means to the study of their structure theory. We give a rough classification in terms of their so-called core, or symplectically irreducible base. Underlying symplectic reduction is a notion of so-called pre-Lie triple system. We will explain how these are related to étale affine representations of Lie triple systems, how any symplectic symmetric space and any Jordan triple system yield such a structure, and how they allow to build "double extensions" of symplectic symmetric spaces.
This (series of) talk(s) will be a more detailed version of my talk at the last SL2R conference. I will start from scratch but skip most of the motivation to focus more on the details and proofs of the results. In this second talk, we will finish the proof of the decomposition theorem and introduce étale affine representations.
• 15.01.2019 — Room 6A
Yannick Voglaire (Université du Luxembourg), "Reduction of symplectic symmetric spaces and étale affine representations – II"
Abstract: We introduce a notion of symplectic reduction for symplectic symmetric spaces as a means to the study of their structure theory. We give a rough classification in terms of their so-called core, or symplectically irreducible base. Underlying symplectic reduction is a notion of so-called pre-Lie triple system. We will explain how these are related to étale affine representations of Lie triple systems, how any symplectic symmetric space and any Jordan triple system yield such a structure, and how they allow to build "double extensions" of symplectic symmetric spaces.
This (series of) talk(s) will be a more detailed version of my talk at the last SL2R conference. I will start from scratch but skip most of the motivation to focus more on the details and proofs of the results. In this second talk, we will finish the proof of the decomposition theorem and introduce étale affine representations.
• 11.12.2018 — Room MNO 1.010
Guendalina Palmirotta (Université du Luxembourg), "Solvability of systems of linear partial differential equations with constant coefficients - Part 2"
• 04.12.2018 — Room MNO 1.010
Martin Olbrich (Université du Luxembourg), "The Paley–Wiener theorem for semi-simple Lie groups II"
• 27.11.2018 — Room MNO 1.010
Guendalina Palmirotta (Université du Luxembourg), "Solvability of systems of linear partial differential equations with constant coefficients - Part 1"
• 20.11.2018 — Room 5A
Yannick Voglaire (Université du Luxembourg), "Reduction of symplectic symmetric spaces and étale affine representations – I"
Abstract: We introduce a notion of symplectic reduction for symplectic symmetric spaces as a means to the study of their structure theory. We give a rough classification in terms of their so-called core, or symplectically irreducible base. Underlying symplectic reduction is a notion of so-called pre-Lie triple system. We will explain how these are related to étale affine representations of Lie triple systems, how any symplectic symmetric space and any Jordan triple system yield such a structure, and how they allow to build "double extensions" of symplectic symmetric spaces.
This (series of) talk(s) will be a more detailed version of my talk at the last SL2R conference. I will start from scratch but skip most of the motivation to focus more on the details and proofs of the results. In this first talk, we will concentrate on the proof of the symplectic Levi decomposition.
• 06.11.2018 — Room MNO 1.020
Martin Olbrich (Université du Luxembourg), "The Paley–Wiener theorem for semi-simple Lie groups I"
• 04.07.2018 Wednesday
Laurent La Fuente-Gravy (Université du Luxembourg), "Symplectic Dirac operators: Construction and kernels"
• 26.06.2018
Yannick Voglaire (Université du Luxembourg), "Differential operators on sections of homogeneous bundles over (homogeneous spaces of) Lie groupoids"
• 12.06.2018
Gang Liu (Université de Lorraine), "A geometric interpretation of Kirillov's conjecture for tempered representations"
• 22.05.2018
Roger Zierau (Oklahoma State University, Université de Lorraine), "Dirac index and characters of discrete series representations"
• 15.05.2018
Salah Mehdi (Université de Lorraine, France), "Orbital parameters for invariant differential operators on nilpotent homogeneous spaces - Part 2"
• 08.05.2018
Salah Mehdi (Université de Lorraine, France), "Orbital parameters for invariant differential operators on nilpotent homogeneous spaces"
• 24.04.2018
Martin Olbrich (Université du Luxembourg), "Differential operators on sections of homogeneous bundles - Part 3"
• 10.04.2018 — special time 10:00
Martin Olbrich (Université du Luxembourg), "Differential operators on sections of homogeneous bundles - Part 2"
• 27.03.2018 — Room 6A
Martin Olbrich (Université du Luxembourg), "Differential operators on sections of homogeneous bundles"