Presenting the Affine Hecke algebra: Iwahori and Bernstein Presentations and the Path model

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 05 April 2012

Initial data

The inital data is

  1. a finite -reflection group, (W1,𝔥).
This means
  1. 𝔥 is a free -module (has -basis {ω1,...,ωl})
  2. W0 is a finite subgroup of GL(𝔥) generated by reflections.
A reflection is a matrix conjugate (in GL(𝔥)) to ξ 1 0 0 1 with ξ1.

HW: Show that if s is a reflection in a finite subgroup of GLn() then ξ is a root of unity?

Our favourite example: Type SL3(). 𝔥 = span{ω1,ω2} W0 = dihedral group of order 6 generated by   s1,s2 ω1 ω2 𝔥θ 𝔥α1 𝔥α2 W0 = {1,s1, s2, s1s2, s2s1, s1s2s2} contains 3 reflection, s1, s2 and s1s2s1 = sθ, the reflections in 𝔥α1,  𝔥α2,  𝔥θ, respectively.

The affine Weyl group (semidirect product presentation)

W = {twXλ  |  wW0,  λ𝔥} with tutv = tuv, XλXμ = Xλ+μ and twXλ = Xwλtw for u,v,wW0 and λ,μ𝔥. Then W acts on 𝔥 = 𝔥Λ0 ( 𝔥 = 𝔥 = -span{ω1,...,ωl} ) by twXλ (μ+mΛ0) = tw λ 0 0 1 μ m = twμ+mλ m = twμ + mλ + mΛ0.

In our SL3() example: 𝔥0 = 𝔥0Λ0 Xλ   acts trivially twμ=wμ Xλμ=μ 𝔥1 = 𝔥+Λ0

Coxeter generators

Cm are fundamental regions for W acting on 𝔥m such that

  1. C0_ C2_ C1_ 0.
  2. 𝔥α1 ,..., 𝔥αl are the walls of C0.
  3. 𝔥α0 ,..., 𝔥αl are the walls of C1,
  4. s0,...,sl the corresponding reflections.
  5. Ω = {gW  |  gC1=C1}.

W is presented by generators s0,s1,...,sl and Ω such that Ω is a subgroup, si2 = 1, for   i=0,1,...,l, sisjsi mij   factors = sjsisj mij   factors for   ij,  where   πmij = 𝔥αi 𝔥αj gsig-1 = sg(i), where   g𝔥αi = 𝔥αg(i).

The Dynkin, or Coxeter diagram of W is the graph with

  1. vertices α0 ,..., αl,
  2. and edges αi  mij  αj,
(the graph of the "1-skeleton of C1").

The affine Hecke algebra (Bernstein presentation)

H is generated by T1,...,Tl and Xλ, λ𝔥 with Ti2 = 1   for   i=1,2,...,l, XλXμ = Xλ+μ, TiTjTi mij   factors = TjTiTj mij   factors for   ij, TiXλ = Xsiλ + (q-q-1) Xλ-Xsiλ 1-X-αi , where siλ = λ- λ,αi αi determines αi𝔥.

The affine Hecke algebra (Coxeterish presentation)

H is presented by generators T0,T1,...,Tl and Ω with Ti2 = (q-q-1)Ti+1, for   i=0,1,...,l, TiTjTi mij   factors = TjTiTj mij   factors for   ij, gh = hg and gTig-1 = Tg(i) for g,hΩ.

Bases of H

Let wW. A reduced word for w, w = gsi1 sil, gΩ, i1,...,il {0,1,...,l} is a minimal length sequence w = ( g, C1 𝔥β1 si1C1 𝔥β2 si1 si2C1 𝔥βl si1 sil C1 ). The periodic orientation is:

  1. If 0𝔥α then C0 is on the positive side of 𝔥α.
  2. Parallel hyperplanes have parallel orientation.
For a reduced work w=gsi1sil define Tw = gTi1 Til and Xw = gTi1ϵ1 Tilϵl where ϵj = { +1, if the   jth   step of w   is 𝔥βj + - -1, if the   jth   step of   w   is 𝔥βj - + } Then {Tw  |  wW}, {Xw  |  wW} {TvXμ  |  vW0,  λ𝔥}, {XλTw  |  λ𝔥,  wW0} are bases of H.

Note: Xλ = XXλ and Xw =? TvXμ   if   w=vtμ.

The path model (the algebra of paths)

Four kinds of steps j v vsj - + j v vsj - + j v vsj - + j v vsj - + An alcove walk is a sequence of steps such that

  1. the tail of the first step is in C1
  2. at every step, the head of each arrow is in the same alcove as the tail of the next.
Use the relations - + =  - + +  - + and - + =  - + +  - + to straighten any sequence of steps to a linear combination of alcove walks.

  1. Fix λ𝔥 and wW0.
  2. Fix a minimal length walk ci1- cir- from C1 to wC1
  3. and a minimal length walk cj1ϵ1 cjsϵs from C1 to λ+C1.
Then Tw-1-1 Xλ = p (-1)# neg folds of  p (q-q-1)# of folds of  p Xwt(p) Tφ(p)-1-1 where the sum is over all alcove walks ci1- cir- pj1 pjs where   pjk   is   cjk+   or   cjk-   or   fjkϵk and end(p) = wt(p) + φ(p)C1 is the ending alcove of p.

Notes and References

These notes are from lecture notes of Arun Ram (CBMS Lecture 1).



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