The degenerate 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 degenerate affine Hecke algebra

Let $x_1,\dots ,x_n$ be commuting variables and let $ℂ\left[x_1,\dots ,x_n\right]$ be the polynomials in $x_1,\dots ,x_n.$ Let $ℂS_n$ be the group algebra of the symmetric group. The graded Hecke algebra is the vector space $ℂ\left[x_1,\dots ,x_n\right]\otimes S_n$ with multiplication such that

1. $ℂ\left[x_1,\dots ,x_n\right]=ℂ\left[x_1,\dots ,x_n\right]\otimes 1$ is a sublagebra,
2. $ℂS_n=1\otimes ℂS_n$is a subalgebra,
3. $s_i{x}_{i-1}{s}_i=x_i-s_i,$ for $2\le i\le n.$

If $\gamma =\left({\gamma }_1,\dots ,{\gamma }_n\right)$ is an $n$-tuple of nonnegative integers, let $x^{\gamma }=x_1^{{\gamma }_1}\dots x_n^{{\gamma }_n}.$ The elements $$\left\{x^{\gamma }w|\gamma =\left({\gamma }_1,\dots ,{\gamma }_n\right),{\gamma }_i\in {\mathbb{Z}}_{\ge 0},w\in S_n\right\}$$ form a basis of $H_n.$ The map $$\begin{array}{rcl}{H}_n& \to & ℂS_n\\ s_i& \mapsto & s_i\\ x_1& \mapsto & 0\\ x_k& \mapsto & \sum _{i<k}\left(ik\right)\end{array}$$ for $2\le k\le n,$ is a surjective algebra homomorphism.

Lusztig's approach to the passage from the affine Hecke algebra to the graded Hecke algebra is as follows. Let $h:ℂ\left[X\right]\to {ℂ}^{*}$ be $W$-invariant $h\left(X^{\alpha }\right)=1$. Then $I=\ker h$ is the maximal ideal of $ℂ\left[X\right].$Then the associated grading of the filtration $$ℂ\left[X\right]\supseteq I\supseteq I^2\supseteq \dots S\left({𝔥}^{*}\right)=\mathrm{gr}ℂ\left[X\right].$$ The "derivative" of $f$ is the image $d\left(f\right)$ of $f-h\left(f\right)$ in $I/I^2.$ Then $$d\left(f{f}^{\text{'}}\right)=h\left(f\right)d\left(f^{\text{'}}\right)+h\left(f^{\text{'}}\right)d\left(f\right),\text{for }f,f^{\text{'}}\in ℂ\left[X\right].$$ Then $$\tilde{H}$$ is a $ℂ\left[X\right]$-module and we have a filtration $$\tilde{H}\supseteq I\tilde{H}\supseteq I^2\tilde{H}\supseteq \dots$$ such that $\left(I^{k}H\right)\left(I^{l}H\right)\subseteq I^{k+l}H.$ So consider $$\mathrm{gr}\tilde{H}=\stackrel{-}{H}=\underset{k\ge 0}{\oplus }{\stackrel{-}{H}}^k,\text{where }{\stackrel{-}{H}}^k=\left(I^k\tilde{H}\right)/\left(I^{k+1}\tilde{H}\right).$$ Let $w$ be the image of $T_w$ in $\tilde{H}/I\tilde{H}={\stackrel{-}{H}}^0.$$$r=d\left(q\right)\in I/I^2.$$ Then $\stackrel{-}{H}$ is the graded Hecke algebra (Prop 4.4 in [Lu].) Prop 4.5 in [Lu] says that $$Z\left(\stackrel{-}{H}\right)=S{\left({𝔥}^{*}\right)}^W.$$ There should be an analogue for the Pittie-Ram theorem for the graded Hecke algebra.

The degenerate affine Hecke algebra ${\tilde{G}}_n$ is the algebra generated by $ℂ\left[x_1,\dots ,x_n\right]$ and $ℂS_n$ with the relations $$x_i{T}_j=T_j{x}_i\text{if }\left|i-j\right|>1,$$$$s_i{x}_i=x_{i+1}{s}_i-1.$$

There is a surjective evaluation homomorphism $$\begin{array}{rcl}{\tilde{G}}_n& \to & ℂS_n\\ s_i& \mapsto & s_i\\ x_k& \mapsto & \sum _{i<k}\left(i,k\right).\end{array}$$ The degenerate affine Hecke algebra is obtained from the affine Hecke algebra by setting $x_k$ to be the derivative of $X^{{\epsilon }_k}$ at $q=1,$$$x_k=\frac{X^{{\epsilon }_k}-1}{q-1}{|}_{q=1}.$$

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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