Springer Fibres and Geometric Representation Theory

University of Greenwich

12-16 August

Organisers: Neil Saunders and Lewis Topley

Participants

Sam Gunningham (Edinburgh)

Katerina Hristova (UEA)

Carl Mautner (UC Riverside)

Kayvan Nejabati Zenouz (Greenwich)

Travis Schedler (Imperial College)

David Stewart (Newcastle)

Catharina Stroppel (Bonn)

Ting Xue (Melbourne)

A brief summary of the research program

In 1979 Lusztig and Spaltenstein defined a geometric procedure which allows us to take a nilpotent orbit in a Levi subalgebra of a complex semisimple Lie algebra and induce it to a nilpotent orbit in the Lie algebra itself. Upon closer inspection it becomes apparent that the process of Lusztig--Spaltenstein induction actually induces embeddings of Springer fibres over the respective nilpotent elements. In the subsequent years it has become well understood that the Springer fibres and nilpotent orbits carry a wealth of representation theoretic data:

the Springer correspondence constructs representations of the Weyl group on the cohomology of Springer fibres;
every primitive ideal of an enveloping algebra can be assigned an associated variety, which is the closure of a nilpotent orbit;
in positive characteristics, the work of Bezrukavnikov--Mirkovi´c--Rumynin explains that the derived category of g-modules with generalise central characters is equivalent to the derived category of quasicoherent sheaves on a certain Springer fibre corresponding to the Frobenius central character.

In all three of these situations, there is a well defined notion of representation theoretic induction. The broad goal of this workshop is to study the compatibility between the geometric induction of orbits and Springer fibres, and the algebraic theory of induction of representations and ideals. There are several obvious conjectures which may be formulated and they will form the basis of our research agenda.

Note: We would like to stress that our research agenda is not strictly prescriptive, and if any participants would like to suggest a research question in algebraic or geometric representation theory, then we are open to suggestions.

Workshop Aims

We have invited an array of researchers to discuss questions related to the research program outlined on the home page.

The intention is twofold:

Experienced researchers shall pass on their knowledge of the research topics through a series of informal lectures and discussions;
We shall work together as a group, with the intention of identifying and proving compatibilities between geometric and algebraic induction in various contexts. We hope that new collaborations and research horizons will be bourn forth from our investigations.