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

Last update: 22 February 2013


Fix the following data and notation:

𝔥* is a real vector space of dimensionn, R is a reduced irreducible root system in𝔥*, R+ is a set of positive roots inR, W is the Weyl group ofR, s1,,sn are the simple reflections inW, mij is the order ofsisj inW,ij, R(w)= { αR+ wαR+ } is the inversion set ofwW, (w)=Card (R(w)) is the length ofwW, is the Bruhat-Chevalley order onW, α1,,αn are the simple roots inR+, ω1,,ωn are the fundamental weights, P=i=1n ωi is the weight lattice, P+= i=1n 0ωi is the set of dominant integral weights.

For a brief, easy, introduction to root systems with lots of pictures for visualization see [NRa2003]. By [Bou1981, VI Section 1 no. 6 Corollary 2 to Proposition 17], if w=si1sip is a reduced word for w, then

R(w)= { αip, sip αip-1,, sipsi2 αi1 } . (1.1)

The affine nil-Hecke algebra is the algebra H given by generators T1,,Tn and Xλ, λP, with relations

Ti2=Ti, TiTjTi mijfactors = TjTiTj mijfactors ,XλXμ= Xλ+μ, (1.2)


XλTi=Ti Xsiλ+ xλ-Xsiλ 1-X-αi . (1.3)

Let Tw=Ti1Tip for a reduced word w=si1sip. Then

{ XλTw wW,λP } and { TwXλ wW,λP } (1.4)

are bases of H.

Both the nil-Hecke algebra,

H=-span {TwwW}, and[X]= -span {XλλP} (1.5)

are subalgebras of H. The action of W on [X] is given by defining

wXλ=Xwλ, forwW,λP, (1.6)

and extending linearly. The proof of the following theorem is given in [Ram2003, Theorem 1.13 and Theorem 1.17]. The first statement of the theorem is due to Bernstein, Zelevinsky, and Lusztig [Lus1983, 8.1] and the second statement is due to Steinberg [Ste1975] and is known as the Pittie–Steinberg theorem.

Theorem 1.7 Define

λw=w-1 siw<w ωi,for wW. (1.8)

The center of H is Z(H)= [X]W and each element f[X] has a unique expansion

f=wWfw X-λw, withfw [X]W. (1.9)

Let εi=1-Ti and let εw=εi1εip for a reduced word w=si1sip. Then εw is well defined and independent of the reduced word for w since

εi2=εi, and εiεjεi mijfactors = εjεiεj mijfactors . (1.10)

The second equality is a consequence of the formulas

εw=vw (-1)(v) TvandTw= vw (-1)(v) εv (1.11)

which are straightforward to verify by induction on the length of w.

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and the Schubert calculus authored by Stephen Griffeth and Arun Ram. It was dedicated to Alain Lascoux.

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