In this chapter, we will study a certain “Hecke algebra” H k (q) that, as we will see, is a deformation of ℂ[S k ]. The ring H k (q) can actually be defined if q is any complex number, but if q is a prime power, it has a representation-theoretic interpretation. We will see that it is the endomorphism ring of the representation of G = GL(k,F q ), where F q is the finite field with q elements, induced from the trivial representation of the Borel subgroup B of upper triangular matrices in G. The fact that it is a deformation of ℂ[S k ] amounts to a parametrization of a certain set of irreducible representations of G — the so-called unipotent ones — by partitions.
KeywordsIrreducible Representation Irreducible Character Borel Subgroup Double Coset Laurent Polynomial
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