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Posts Tagged ‘Hecke algebras’

Let A be an abelian group and T = \{ T_i : A \to A \} be a collection of endomorphisms of A. The commutant T' of T is the set of all endomorphisms of A commuting with every element of T; symbolically,

\displaystyle T' = \{ S \in \text{End}(A) : TS = ST \}.

The commutant of T is equal to the commutant of the subring of \text{End}(A) generated by the T_i, so we may assume without loss of generality that T is already such a subring. In that case, T' is just the ring of endomorphisms of A as a left T-module. The use of the term commutant instead can be thought of as emphasizing the role of A and de-emphasizing the role of T.

The assignment T \mapsto T' is a contravariant Galois connection on the lattice of subsets of \text{End}(A), so the double commutant T \mapsto T'' may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.

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The Hecke algebra attached to a Coxeter system (W, S) is a deformation of the group algebra of W defined as follows. Take the free \mathbb{Z}[q^{ \frac{1}{2} }, q^{ - \frac{1}{2} }]-module \mathcal{H}_W with basis T_w, w \in W, and impose the multiplicative relations

T_w T_s = T_{ws}

if \ell(sw) > \ell(w), and

T_w T_s = q T_{ws} + (q - 1) T_w

otherwise. (For now, ignore the square root of q.) Humphreys proves that these relations describe a unique associative algebra structure on \mathcal{H}_W with T_e as the identity, but the proof is somewhat unenlightening, so I will skip it. (Actually, the only purpose of this post is to motivate the definition of the Kazhdan-Lusztig polynomials, so I’ll be referencing the proofs in Humphreys rather than giving them.)

The motivation behind this definition is a somewhat long story. When W is the Weyl group of an algebraic group G with Borel subgroup B, the above relations describe the algebra of functions on G(\mathbb{F}_q) which are bi-invariant with respect to the left and right actions of B(\mathbb{F}_q) under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group G, and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing G and B with \text{SL}_2(\mathbb{Q}) and \text{SL}_2(\mathbb{Z}) gives the Hecke operators in the theory of modular forms.

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