Sheaves on affine flag manifolds, combinatorics of Bruhat graphs, and modular representations fo algebraic groups <br> Lectures by Professor Pieter Fiebig

Sheaves on affine flag manifolds, combinatorics of Bruhat graphs, and modular representations of algebraic groups
Lectures by Professor Peter Fiebig
7th to 18th October 2013 Institute of Mathematical Sciences (IMSc)

Proposed schedule of the lectures (all 1530--1700 hrs with a break in between for about 10 minutes): Oct 7 (Mon), 9 (Wed), 11 (Fri), 15 (Tue), 16 (Wed), 18 (Fri)

Abstract: We will review recent approaches towards understanding the rational representation theory of reductive groups defined over a field of positive characteristic. A guiding problem for us will be the determination of the irreducible characters of the group. For those, Lusztig stated in 1980 a conjectural formula. It was proven in 1990 in a combined effort of several authors that the formula is "generically true", i.e. in case the characteristic is "big enough". In 2007 a relation between (parity) sheaves on affine flag varieties, Bruhat graphs, and modular representations was used to establish a (huge) upper bound on the exceptional primes. While it was always clear that Lusztig's formula cannot hold true if the characteristic is "small", one was hoping that the exceptions are all smaller than the Coxeter number associated to the underlying root system. Very recently, Geordie Williamson found a counterexample to this.

The lectures will be addressed to graduate students starting to work in areas related to algebraic groups.

Useful references:

Basic monograph by Jens Carsten Jantzen: Representations of Algebraic groups
Notes of lectures by Peter Fiebig which can be accessed from his home page: Shanghai Lectures on Characters of representations and the topology of flag varieties.
Papers by Peter Fiebig:
- An upper bound on the exceptional characteristics for Lusztig's character formula, preprint, 2008, J. Reine Angew. Math. 673 (2012), 1-31.
- Lusztig's conjecture as a moment graph problem, preprint 2007, Bull. London Math. Soc. 42(6) (2010), 957-972.
- Sheaves on affine Schubert varieties, modular representations and Lusztig's conjecture, preprint 2007, J. Amer. Math. Soc. 24 (2011), 133-181.
- The multiplicity one case of Lusztig's conjecture, preprint 2006, Duke Math. J. 153 (2010), 551-571.
Paper by Wolfgang Soergel: On the relation between intersection cohomology and representation theory in positive characteristic, pp. 311-335 in Journal of Pure and Applied Algebra 152 (2000). pdf file

Organizers: Upendra Kulkarni upendra-at-cmi-dot-ac-dot-in and K N Raghavan knr-at-imsc-dot-res-dot-in

Sheaves on affine flag manifolds, combinatorics of Bruhat graphs, and modular representations of algebraic groups Lectures by Professor Peter Fiebig 7th to 18th October 2013 Institute of Mathematical Sciences (IMSc)

Sheaves on affine flag manifolds, combinatorics of Bruhat graphs, and modular representations of algebraic groups
Lectures by Professor Peter Fiebig
7th to 18th October 2013 Institute of Mathematical Sciences (IMSc)