Oberseminar Algebra und Geometrie

Summer term 2026: Irreducible representations of various general linear groups

Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.

Organised by Clotilde Gauthier

All talks take place on Fridays at 14:30 in 25.22.03.73.

If you want get announcements about the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.

Infos für Studierende

Das Oberseminar richtet sich an alle, die einen Einblick in aktuelle Forschung erhalten möchten, ist aber tendenziell eher für fortgeschrittene Studierende geeignet (ab Master). Besonders empfohlen wird die Teilnahme an Oberseminaren, wenn Sie sich vorstellen können zu promovieren. Wenn Sie intessiert sind, können Sie sich einfach (ohne Anmeldung) ins Seminar reinsetzen - gerne auch nur zu einzelnen Vorträgen, die Sie interessieren.

Die Vorträge in diesem Oberseminar sind auf englisch. Üblicherweise nehmen an Oberseminaren auch viele Doktorand:innen, Postdocs und Professor:innen teil. Wahrscheinlich werden Sie nicht alles verstehen; das passiert aber auch den fortgeschritteneren Teilnehmer:innen, und auch, wenn man nicht alles verstanden hat, hat man doch am Ende oft einen interessanten Einblick in ein neues Thema erhalten.

Aims and Content

In this seminar, we are interested in understanding the (complex) representation theory of (finite) general linear groups. We will see that the irreducible representations can be divided in two groups: the principal series and the cuspidal representations. Principal series are obtained as irreducible components of representations induced from representations of specific subgroups, called parabolic subgroups. The cuspidal representations are all the irreducible representations that cannot be obtained in this way. They are much more diffucult to study and come from linear combinations of characters. The main goal of the seminar is to see the machinery involved in the construction of these irreducible representations, and to apply it to small examples to get a better understanding.
You can find a more detailed program here.

Schedule

(The dates of the talks are temporary and might still change.)

17.04.26: tba: Setting the stage
The goal of the talk is to give the representation theory background that is required for the upcoming talks.
Give the basic definition of representation theory (representation, subrepresentation, irreducible representation, dimension, morphism of representations). State Schur's Lemma and Maschke's theorem. Give the definition of the character of a representation and state Frobenius' Theorem. Give the orthogonality relations between irreducible characters and draw the character table of $\mathfrak{S}_3$ as an example. Mention the representation ring and define the notion of virtual representation. Then, define induced representations and state Frobenius' Reciprocity and Mackey's induction-restriction formula. Also give the formula for the character of an induced representation. To finish the talk, move away from representation theory and describe the conjugacy classes of $\mathrm{Gl}_2(\mathbb{F}_q)$ as this is needed in the three next talks.
References: [FH04], [BH06] and Serre
24.04.26: tba: Principal series of $\mathrm{GL}_2(\mathbb{F}_q)$
The goal of this talk is to separate the irreducible representations of $\mathrm{GL}_2(\mathbb{F}_q)$ into two categories and study the representations that falls under the category called principal series.
Define all the groups involved and state the Bruhat decomposition for $\mathrm{GL}_2(\mathbb{F}_q)$ [BH06, §5.2]. Follow [BH06, §6.2] to describe the character of the subgroup $N$ of unipotent upper triangular matrices and deduce from it that any irreducible representation $\pi$ of $\mathrm{GL}_2(\mathbb{F}_q)$ is such that $\mathrm{Res}_N^G\pi$ either contains all character of $N$ or all characters of $N$ but the trivial one. Describe the linear characters of $\mathrm{GL}_2(\mathbb{F}_q)$ (see [FH04,p.69]). Follow [BH06, §6.3] to completely describe the principal series of $\mathrm{GL}_2(\mathbb{F}_q)$.
References: [FH04] and [BH06]
01.05.26: No seminar
08.05.26: tba: Cuspidal representations of $\mathrm{GL}_2(\mathbb{F_q})$
The goal of this talk is to build and study the irreducible representations of $\mathrm{GL}_2(\mathbb{F}_q)$ that does not qualify as principal series. They are called cuspidal representations and cannot directly be obtained with induction.
Follow [BH06, §6.4] to get the classification of cuspidal representations (sketch the proof of the theorem). Draw a recap table of the four kind of irreducible representations of $\mathrm{GL}_2(\mathbb{F}_q)$ (linear characters, Steinberg, irreducible principal series and cuspidal representations). Give their repspective number and dimension. Finally, draw the character table of $\mathrm{GL}_2(\mathbb{F_q})$.
Reference: [BH06]
15.05.26: No seminar at the moment
22.05.26: tba: What about $\mathrm{SL}_2(\mathbb{F}_q)$?
The goal of this talk is to see how the classification of irreducible representations obtained for $\mathrm{GL}_2(\mathbb{F}_q)$ enables to get the irreducible representations of $\mathrm{SL}_2(\mathbb{F}_q)$.
Start by describing the $q+4$ conjugacy classes of $\mathrm{SL}_2(\mathbb{F}_q)$ and briefly explain how to get them from the conjugacy classes of $\mathrm{GL}_2(\mathbb{F}_q)$. Explain what happens after restricting the different types of irreducible representations of $\mathrm{GL}_2(\mathbb{F}_q)$ and see that this process already gives $q$ irreducible representations of $\mathrm{SL}_2(\mathbb{F}_q)$. Explain how to get the four remaining irreducible representations.
Reference: [FH04, p71-72]
29.05.26: tba: Conjugacy classes of $\mathrm{GL}_n(\mathbb{F}_q)$
The goal of this talk is to describe the conjugacy classes of $\mathrm{GL}_n(\mathbb{F}_q)$. We want to understand how to classify those conjugacy classe and see which can be grouped together to form a type of conjugacy class. This is the first step towards the construction of the irreducible characters.
Follow the first section of [Gre55] (pages 405-408) to describe the conjugacy classes of $\mathrm{GL}_n(\mathbb{F}_q)$ in terms of partitions and irreducible polynomials. Define the type of a conjugacy classe, and give the number of types of conjugacy classes in terms of $n$. Give the number of conjugacy classes of $\mathrm{GL}_n(\mathbb{F}_q)$ in terms of $n$ and $q$. Finally, as an example, compute the (types of) conjugacy classes of $\mathrm{GL}_3(\mathbb{F}_q)$.
Reference: [Gre55]
05.06.26: No seminar at the moment
12.06.26: No seminar
19.06.26: tba: Parabolic induction
The goal of this talk is to define the concept of parabolic induction and to understand the formula for the character of representations obtained through this process.
Follow [Gre55, p403] to define parabolic subgroups and parabolic induction. State and explain Theorem 2 of [Gre55]. State Lemmas 2.6, 2.7 and 2.8 as consequences and as an example, compute the value of the character $\mathrm{Ind}_{P_{(2\:1)}}^{\mathrm{GL}_3(\mathbb{F}_q)}\chi\otimes\phi$ where $\chi$ is an arbitrary character of $\mathrm{GL}_2(\mathbb{F}_q)$ and $\phi$ is an arbitrary character of $\mathbb{F}_q^\times$.
Reference: [Gre55]
26.06.26: tba: Uniform functions
This talk is a bit more of a stand alone talk, even though the examples will be used again in the next talk. The goal is to define and study uniform functions which are of great help to define the irreducible characters of $\mathrm{GL}_n(\mathbb{F}_q)$.
Follow [Gre55, p420-423] in order to define uniforms functions, and present the example defined in the program. State theorems 6,7 and 11, which are crucial in proving that the characters we will define are irreducible.
Reference: [Gre55]
03.07.26: tba: Irreducible characters of $\mathrm{GL}_n(\mathbb{F}_q)$
The goal of this talk is to define the irreducible characters of $\mathrm{GL}_n(\mathbb{F}_q)$ with uniform functions, and to explicitly compute the character table of $\mathrm{GL}_3(\mathbb{F}_q)$.
Give definition 3.1 of [Gre55]. Give the definition of an $s$-simplex [Gre55, p438-439] and state Lemma 7.7. State Theorem 12, recall the example of the previous talk and see that the uniform functions defined there where the primary irreducible characters. State Theorem 13 and using the examples of Talks 6 and 7, draw the character table of $\mathrm{GL}_3(\mathbb{F}_q)$.
Reference: [Gre55]
10.07.26: tba: The irreducible representations of $\mathrm{GL}_2(\mathbb{Q}_p)$: an overview
The goal of this talk is to give the classification of the irreducible representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ and see the similarities and differences with the finite case.
Define the notions of smooth representations and smooth induction [BH06, §2.1 and §2.4]. Give the definition of Jacquet module and sketch the proof of the proposition [BH06, §9.1]. Explain the difference between $\mathrm{GL}_2(\mathbb{Q}_p)$ and the finite case, and state the classification of principal series.
24.07.26:Program discussion

Oberseminar Algebra und Geometrie

Summer term 2026: Irreducible representations of various general linear groups

Infos für Studierende

Aims and Content

Schedule

References:

Archive