Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicit...

Full description

Saved in:

Bibliographic Details
Published in:	Indagationes mathematicae 2022-01, Vol.33 (1), p.218-237
Main Authors:	Juteau, Daniel, Lecouvey, Cédric, Sorlin, Karine
Format:	Article
Language:	eng
Subjects:	Classical types Mathematics Modular representation theory Reductive algebraic groups Representation Theory Springer correspondence
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Description
Summary:	We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic p, and with coefficients in a field of odd characteristic ℓ≠p): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We provide a quick proof, by sorting characters according to the dimension of the corresponding Springer fibre, an invariant which is directly computable from symbols.
ISSN:	0019-3577 1872-6100