The degenerate affine Hecke algebra

The degenerate affine Hecke algebra

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 20 May 2010

The degenerate affine Hecke algebra

Let x1 ,, xn be commuting variables and let x1 xn be the polynomials in x1 ,, xn . Let Sn be the group algebra of the symmetric group. The graded Hecke algebra is the vector space x1 xn Sn with multiplication such that

  1. x1 xn = x1 xn 1 is a sublagebra,
  2. Sn =1 Sn is a subalgebra,
  3. si x i-1 si = xi - si , for 2in.
If γ= γ1 γn is an n -tuple of nonnegative integers, let xγ = x1 γ1 xn γn . The elements xγ w| γ= γ1 γn , γi 0 ,w Sn form a basis of Hn . The map Hn Sn si si x1 0 xk i<k ik for 2kn, 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: X * be W -invariant h Xα =1 . Then I=kerh is the maximal ideal of X . Then the associated grading of the filtration X I I2 S 𝔥 * =gr X . The "derivative" of f is the image d f of f-h f in I/ I2 . Then d f f' =h f d f' +h f' d f ,for  f, f' X . Then H~ is a X -module and we have a filtration H~ I H~ I2 H~ such that Ik H Il H I k+l H. So consider gr H~ = H- = k0 H- k ,where   H- k = Ik H~ / I k+1 H~ . Let w be the image of T w in H~ /I H~ = H- 0 . r=d q I/ I2 . Then H- is the graded Hecke algebra (Prop 4.4 in [Lu].) Prop 4.5 in [Lu] says that Z H- =S 𝔥 * W . There should be an analogue for the Pittie-Ram theorem for the graded Hecke algebra.

The degenerate affine Hecke algebra G~ n is the algebra generated by x1 xn and Sn with the relations xi Tj = Tj xi if   i-j >1, si xi = x i+1 si -1.

There is a surjective evaluation homomorphism G~ n Sn si si xk i<k ik . The degenerate affine Hecke algebra is obtained from the affine Hecke algebra by setting xk to be the derivative of X εk at q=1, xk = X εk -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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