## Spring School on Complete Reducibility

### 9–13 April 2018

This spring school is aimed at introducing graduate students and junior researchers to the notion of G-complete reducibility. Introduced by J.P. Serre in the 1990s, this theory is centrally important in the modern approach to studying linear algebraic groups. The theory generalises the concept of complete reducibility from representation theory, and connects subgroups of linear algebraic groups with tools from geometry and geometric invariant theory.

#### Academic Programme

The principal event of the spring school will be a lecture series by Ben Martin, University of Aberdeen.

Additional talks will be given by Michael Bate (York), Maike Gruchot (Bochum), Alastair Litterick (Bielefeld/Bochum) and David Stewart (Newcastle).

#### Excursion and Dinner

The spring school will also feature a conference dinner, at a fixed cost of €25 per person, and an excursion to the **Zeche Zollverein**, also at €25 per person (including transport). Payment will be collected during registration on Monday morning.

#### Online Registration

Online registration is now **closed**. Please direct further enquiries to: **Frau Claudia Knauber**.

#### Accommodation

We have reserved a number of rooms from 8th-14th April at the **Ibis Bochum Zentrum**, at €66.64 per night. To take advantage of this price, participants must contact the hotel **by phone** (+49 234 33311) or **by email** (**H1440(at)accor.com**), quoting "Spring School on Complete Reducibility".

#### Organising Committee

#### Scientific advisers

#### Schedule

Talks will begin the morning of Monday 9th April and finish 12:30 on Friday 13th

Monday 9th | Tuesday 10th | Wednesday 11th | Thursday 12th | Friday 13th | |
---|---|---|---|---|---|

09:00 - 09:30 | Registration | ||||

09:30 - 10:30 | Martin I | Martin III | Martin V | Litterick | Bate |

10:30 - 11:30 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |

11:30 - 12:30 | Martin II | Martin IV | Martin VI | Gruchot | Martin (additional) |

12:30 - 14:00 | Lunch | Lunch | Lunch | Lunch | Lunch / end |

14:00 - 15:00 | Coffee break | Coffee break | Coffee break | Coffee break | |

15:00 - 16:00 / end | Discussion group | Discussion group | Excursion | Stewart |

##### Dinner

The spring school dinner will take place on Wednesday evening, 19:00 at **Amalfi Pizzeria** (Google Maps), in easy walking distance of the Ibis Hotel Zentrum.

#### Lecture series: Algebraic groups and complete reducibility

Ben Martin, University of Aberdeen

The notion of a *completely reducible* subgroup is important in the study of algebraic groups and their subgroup structure. It generalises the usual idea of complete reducibility from representation theory: a subgroup *H* of a general linear group GL_{n}(*k*) is completely reducible if and only if the inclusion map *i* : H → GL_{n}(*k*) is a completely reducible representation of *H*. In these lectures I will give an introduction to the theory of complete reducibility, and explain an approach to the subject using geometric invariant theory. In particular, I will sketch the Kempf-Hesselink-Rousseau construction of optimal destabilising cocharacters, which has found many applications to algebraic groups and Lie theory.

##### Accompanying notes

Notes to accompany the lecture course can be found below:

Algebraic groups and G-complete reducibility: A geometric approach (422.1 kB)

##### Recommended preparation

I will give a brief review of linear algebraic groups, but it will help if you have met some of the main ideas. The books of Humphreys ("Linear algebraic groups", Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg; corrected printing, 1998) and Springer ("Linear algebraic groups", Modern Birkhäuser Classics, 2nd ed.; 2nd printing, 2008) are a good introduction to the subject. In particular, Chapter I of Humphreys gives a helpful summary of the necessary algebraic geometry. The books of Geck ("An introduction to algebraic geometry and algebraic groups", OUP, Oxford, 2013) and Carter/Segal/Macdonald ("Lectures on Lie groups and Lie algebras", London Mathematical Society Student Texts, Cambridge University Press, 2009) are aimed at undergraduates. Sections 1.1 to 1.3 of Newstead ("Introduction to moduli problems and orbit spaces", Tata Institute of Fundamental Research Publications, Volume 17, 2011) gives a quick and approachable introduction to geometric invariant theory. Part II of Serre's 1998 Moursund lectures ("The notion of complete reducibility in group theory",http://arxiv.org/abs/math/0305257) gives a very readable account of some topics in complete reducibility. The paper of Bate-Martin-Röhrle ("A geometric approach to complete reducibility", Invent. Math. 161 (2005), no. 1, 177-218) describes the approach to complete reducibility via geometric invariant theory.

### Additional talks

##### Alastair Litterick - Relative Complete Reducibility, I

The notion of *G*-complete reducibility can be expressed in terms of limits of cocharacters. By considering only the cocharacters of some proper reductive subgroup of *G*, we obtain a relative notion of relative complete reducibility, first formulated in 2011 by Bate, Martin, Röhrle and Tange. In this talk we will introduce this notion, explain why it is a natural property to consider, and present a new result which lets us generalise many results from the 'absolute' case.

##### Maike Gruchot - Relative Complete Reducibility, II

Sticking with the theme of the preceding talk, in this talk we again consider the relative notion of complete reducibility, focusing in particular on the many results which generalise from the 'absolute' case. We will also discuss the special case when *G* is the general linear group, and show that the representation-theoretic interpretation of complete reducibility has an analogue in the relative setting.

##### David Stewart - Representations of pseudo-reductive groups

Pseudo-reductive groups are smooth connected linear algebraic groups over a field *k* whose *k*-defined unipotent radical is trivial. If *k* is perfect then all pseudo-reductive groups are reductive, but if *k* is imperfect (hence of characteristic *p*) then one gets a strictly larger collection of groups. They come up in a number of natural situations, not least when one wishes to say something about the simple representations of all smooth connected linear algebraic groups. Recent work by Conrad-Gabber-Prasad has made it possible to reduce the classification of the simple representations of pseudo-reductive groups to the split reductive case. I’ll explain how. This is joint work with Mike Bate.

##### Michael Bate - Complete reducibility and the centre conjecture

In this talk I'll outline the approach to complete reducibility through spherical buildings and show how this gives a third point of view to sit alongside the group theoretic and geometric approaches seen during the course. These relationships are most striking when viewed through the prism of the "Centre Conjecture" due to Tits. I'll talk about this conjecture and some known cases of it, and outline a possible approach to attacking other cases.

##### Ben Martin - Algebraic groups and a question of Külshammer

Let *F* be a finite group and let *G* be a reductive algebraic group over an algebraically closed field of characteristic *p*>0. Burkhard Külshammer asked whether homomorphisms from *F* to *G* are controlled in an appropriate sense by their restrictions to a Sylow *p*-subgroup *F _{p}* of

*F*. Under quite general hypotheses the answer is yes, but a counter-example was recently discovered by Bate, Martin and Röhrle. Külshammer's question makes sense when we replace

*F*with an arbitrary linear algebraic group

*H*and

*F*with a maximal unipotent subgroup of

_{p}*H*. It turns out that the answer is yes if

*H*is connected and semisimple. I will discuss this and some related results. The proofs involve ideas from geometric invariant theory and the theory of complete reducibility. This is joint work with Daniel Lond.

#### Arrival

We expect that most participants will arrive on Sunday, 8th April, and will be staying at the Ibis Hotel Zentrum. In this case, you should head directly to the hotel. If you arrive at Bochum by train, you will notice the hotel as soon as you disembark; it is a tall white building with a large, red "Ibis" sign at the top.

Please be aware that most shops are closed on Sundays in Germany, and the Ibis Hotel Zentrum does not have a restaurant for dinner. However, a number of small cafes, fast-food outlets and bakeries do remain open, especially those located in train stations. **Most shops do not accept credit cards**, so make sure you have Euros on you if you do not own a Eurozone bank card (EC card).

##### Local registration

Registration and talks will take place at **Wasserstrasse 221** on Monday morning. **Note: This is not on the main university campus**. If you wish to attend the excursion and/or dinner, please bring payment with you in cash.

##### Getting to Bochum by air

The nearest international airports to Bochum are Düsseldorf (DUS) and Dortmund (DTM). Cologne/Bonn Airport (CGN) is further away, but is on a direct railway line to Bochum. **Be warned** that Düsseldorf Weeze (NRN) is **not** easy to reach from Bochum.

**From Düsseldorf International (DUS)** you must take the 'SkyTrain' to Düsseldorf Flughafen railway station and change onto a main-line train. SkyTrain tickets can be purchased from machines at the station, and a separate ticket is required for the main-line train.

**From Dortmund airport (DTM)** the easiest route is to take the AirportExpress bus to Dortmund Main Station (Dortmund Hbf), and from there take a train to Bochum. Tickets for the AirportExpress can be purchased on the bus. A rail ticket must be purchased separately. **Warning: do not take the 'AirportShuttle'** as this goes to a small local train station that is not useful for Bochum.

##### Getting to Bochum by rail

Bochum is very well-connected by rail. Frequent trains run from the main station (Bochum Hbf) to many other major cities. The main station is located in the city centre, and is immediately next to the Ibis Hotel Zentrum.

The basic type of ticket is a 'VRR' ticket, which can be purchased from Deutsche Bahn ticket machines. Once stamped, this ticket will let you use all buses, U-Bahn, S-Bahn and local trains (RE/RB) for at least the next 90 minutes. The price of the ticket depends on how far you want to go. To get to and from Dortmund, you require a Zone B ticket (currently (€5.90). To get to and from Düsseldorf, you require a Zone D ticket (currently €15.30). Note that these tickets do **not** let you use the IC or ICE train services.

### Local Information

##### Travel between the hotel and talk venue (Off-campus)

All talks will take place at **Wasserstrasse 221** (Google Maps), away from the University Campus. From the hotel, one can either follow the 30-minute walk shown on the map, or take the underground (U-Bahn U35) from Bochum main station to Wasserstrasse. For the U-Bahn, blue signs and downward escalators to the U-Bahn are at both ends of Bochum main station. You require a 'VRR Zone A ticket' for this journey; these currently cost €2.80 each, or €10.20 for a ticket which can be stamped four times. Tickets can be stamped on board the U-Bahn.

##### Food and drink

For lunch, we recommend following a local. For dinner, there are many good places to eat in Bochum city centre. We recommend heading to the **Bermuda Dreieck** which is a ten-minute walk from the main train station, and features an array of restaurants, bars and pubs.

#### List of Participants

- Chris Attenborough (University of York)
- Falk Bannuscher (Ruhr-University Bochum)
- Leon Barth (Ruhr-University Bochum)
- Michael Bate (University of York)
- Alexandre Borovik (University of Manchester)
- Melvin Dauter (TU Kaiserslautern)
- Ögmundur Eiriksson (Bielefeld University)
- Lukas Gohla (TU Dresden)
- Maike Gruchot (Ruhr-University Bochum)
- Ulrike Hansper (Bielefeld University)
- Mikko Korhonen (École Polytechnique Fédérale de Lausanne)
- Denis Kuz (Ruhr-University Bochum)
- Alastair Litterick (Bielefeld University / Ruhr-University Bochum)
- Ben Martin (University of Aberdeen)
- Tilman Möller (Ruhr-University Bochum)
- Paul Mücksch (Ruhr-University Bochum)
- Gerhard Röhrle (Ruhr-University Bochum)
- Nithi Rungtanapirom (Goethe University Frankfurt)
- Yuri Santos (Bielefeld University)
- Anne Schauenburg (Ruhr-University Bochum)
- David Stewart (Newcastle University)
- Michele Zordan (Katholieke Universiteit Leuven)