Geometric Satake, Springer correspondence, and small representations

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

Speaker:

Anthony Henderson (University of Sydney)

Title:

Geometric Satake, Springer correspondence, and small representations

Abstract:

Let $k$ be an algebraically closed field. Consider the irreducible representation of $G{L}_n(k)$ 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 $G{L}_n$ 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).

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After the seminar we will take the speaker to lunch.

See the Algebra Seminar web page for information about other seminars in the series.

John Enyang John.Enyang@sydney.edu.au