# 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).