SMS scnews item created by John Enyang at Mon 16 Apr 2012 1337
Type: Seminar
Distribution: World
Expiry: 21 Apr 2012
Calendar1: 20 Apr 2012 1205-1255
CalLoc1: Carslaw 175
Auth: enyang@penyang.pc (assumed)

# Geometric Satake, Springer correspondence, and small representations

### Henderson

###### 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 $$GL_n(k)$$ 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).

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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

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