University of Sydney Algebra Seminar

Anthony Henderson (University of Sydney)

Friday 20 April, 12:05-12:55pm, Carslaw 175

Geometric Satake, Springer correspondence, and small representations

Let \(k\) be an algebraically closed field. Consider the irreducible representation of \(GL_n(k)\) with highest weight \((\lambda_1-1,\lambda_2-1,\cdots,\lambda_n-1)\) where \(\lambda\) is a partition of \(n\). The \(0\)-weight space of this representation is a representation of \(S_n\), either irreducible or zero (the latter occurring when \(k\) has characteristic \(p\) and \(\lambda\) is not \(p\)-restricted). This construction produces all the irreducible representations of \(S_n\) over \(k\).

I will explain a generalization where \(GL_n\) is replaced by an arbitrary connected reductive group and \(S_n\) by the Weyl group \(W\). The highest weights to consider are those which are small in the sense of Broer. In the characteristic zero case, the resulting representations of \(W\) were calculated by Reeder: they are sometimes reducible, and not all irreducible representations of \(W\) arise.

In joint work with Pramod Achar, Daniel Juteau, and Simon Riche, we describe these representations of \(W\) in a way which makes sense for any characteristic, using the Springer correspondence (as extended to the modular case by Juteau).