The affine Hecke algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 May 2010

The affine Hecke algebra

Let $q \in ℂ^{*} .$ The affine Hecke algebra ${\tilde{H}}_{n}$ is the algebra given by generators $X^{ε_{1}}, T_{2}, T_{3}, \dots, T_{n}$ and relations $\begin{array}{rcl} T_{i} T_{j} & = & T_{j} T_{i} & if |i - j| > 1, \\ T_{i} T_{i + 1} T_{i} & = & T_{i + 1} T_{i} T_{i + 1} & if 2 \leq i \leq n - 1, \\ X^{ε_{1}} T_{j} & = & T_{j} X^{ε_{1}}, & j > 1 \\ X^{ε_{1}} T_{2} X^{ε_{1}} T_{2} & = & T_{2} X^{ε_{1}} T_{2} X^{ε_{1}}, \\ T_{i}^{2} & = & (q - q^{- 1}) T_{i} + 1, & 2 < i < n . \end{array}$ If $q = 1$ then ${\tilde{H}}_{n} = ℂ {\tilde{S}}_{n},$ the group algebra of the affine symmetric group.

Define $X^{ε_{k}} = T_{i} T_{i - 1} \dots T_{2} X^{ε_{1}} T_{2} \dots T_{i - 1} T_{i}, for 1 \leq i \leq n,$ and $X^{λ} = {(X_{1}^{ε})}^{λ_{1}} \dots {(X_{n}^{ε})}^{λ_{n}}, for λ = λ_{1} ε_{1} + \dots λ_{n} ε_{n}, λ_{i} \in ℤ .$ Then $X^{λ} X^{μ} = X^{λ + μ}, for all λ, μ \in ℤ .$ Then $ℂ [X] = ℂ - span \{X^{λ} | λ = λ_{1} ε_{1} + \dots λ_{n} ε_{n}, λ_{i} \in ℤ\}$ is a commutative subalgebra of ${\tilde{H}}_{n}$ isomorphic to the group algebra of $ℤ^{n} .$

The affine Hecke algebra is the unique algebra structure on $ℂ [X] \otimes H$ such that $ℂ [X] = ℂ [X] \otimes 1$ and $H = 1 \otimes H$ are subalgebras and $X^{λ} T_{i} = T_{i} X^{s_{i} λ} + (q - q^{- 1}) \frac{X^{λ} - X^{s_{i} λ}}{1 - X^{ε_{i} - ε_{i - 1}}},$ for all $λ \in ℤ^{n}$ and $1 \leq i \leq n .$

Define $T_{0} = X^{ε_{1} - ε_{n}} T_{n} T_{n - 1} \dots T_{3} T_{2} T_{3} \dots T_{n} and ω = X^{- ε_{1}} T_{2} T_{3} \dots T_{n} .$

The affine Hecke algebra ${\tilde{H}}_{n}$ is presented by generators $T_{0}, T_{1}, \dots, T_{n}, ω$ and relations $\begin{array}{rcl} T_{i} T_{j} & = & T_{j} T_{i} & if |i - j| > 1, \\ T_{i} T_{i + 1} T_{i} & = & T_{i + 1} T_{i} T_{i + 1}, \\ ω T_{i} ω^{- 1} & = & T_{i + 1}, \\ T_{i}^{2} & = & (q - q^{- 1}) T_{i} + 1, & 0 \leq i \leq n - 1, \end{array}$ where the indices are always taken modulo $n .$

If $w \in S_{n}$ let $T_{w} = T_{i_{1}} \dots T_{i_{p}}$ if $w = s_{i_{1}} \dots s_{i_{p}}$ and $p$ is minimal. The affine Hecke algebra ${\tilde{H}}_{n}$ has basis $X^{λ} T_{w}, where w \in S_{n}, λ = λ_{1} ε_{1} + \dots + λ_{n} ε_{n}, λ_{j} \in ℤ .$

The evaluation homomorphism is the surjective homomorphism defined by $\begin{array}{rcl} {\tilde{H}}_{n} & \to & H_{r, 1, n} \\ T_{i} & \mapsto & T_{i}, \\ X^{ε_{k}} & \mapsto & T_{k} \dots T_{2} X^{ε_{1}} T_{2} \dots T_{k} . \end{array}$ Via this homomorphism every irreducible $H_{r, 1, n}$ module $M$ is an irreducible ${\tilde{H}}_{n}$ -module. Conversely, let $M$ be an ${\tilde{H}}_{n}$ -module. Let $u_{0}, \dots, u_{r - 1} \in ℂ$ be such that the minimal polynomial $p (t)$ of the linear transformation of $M$ determined by the action of $X^{ε_{1}}$ dived the polynomial $(t - u_{0}) (t - u_{1}) \dots (t - u_{r - 1}) .$ The ${\tilde{H}}_{n}$ action on $M$ is an $H_{r, 1, n} (u_{0} \dots u_{r - 1}; q)$ action on $M .$

So any $H_{r, 1, n} (u_{0} \dots u_{r - 1}; q)$ -module $M$ is an ${\tilde{H}}_{n}$ and conversely, any ${\tilde{H}}_{n}$ module $M$ is an $H_{r, 1, n} (u_{0} \dots u_{r - 1}; q)$ module for some appropriate choice of $r$ and $u_{0}, \dots, u_{r - 1} \in ℂ .$ So ${\tilde{H}}_{n}$ 's representation theory "contains the representation theory" of all the algebras $H_{r, 1, n} (u_{0} \dots u_{r - 1}; q) .$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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