## Preliminaries

Last update: 22 February 2013

## Preliminaries

Fix the following data and notation:

$𝔥* is a real vector space of dimension n, R is a reduced irreducible root system in 𝔥*, R+ is a set of positive roots in R, W is the Weyl group of R, s1,…,sn are the simple reflections in W, mij is the order of sisj in W,i≠j, R(w)= { α∈R+ ∣ wα∉R+ } is the inversion set of w∈W, ℓ(w)=Card (R(w)) is the length of w∈W, ≤ is the Bruhat-Chevalley order on W, α1,…,αn are the simple roots in R+, ω1,…,ωn are the fundamental weights, P=∑i=1n ℤωi is the weight lattice, P+= ∑i=1n ℤ≥0ωi is the set of dominant integral weights.$

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 $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ is a reduced word for $w,$ then

$R(w)= { αip, sip αip-1,…, sip…si2 αi1 } . (1.1)$

The affine nil-Hecke algebra is the algebra $\stackrel{\sim }{H}$ given by generators ${T}_{1},\dots ,{T}_{n}$ and ${X}^{\lambda },$ $\lambda \in P,$ with relations

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

and

$XλTi=Ti Xsiλ+ xλ-Xsiλ 1-X-αi . (1.3)$

Let ${T}_{w}={T}_{{i}_{1}}\dots {T}_{{i}_{p}}$ for a reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}\text{.}$ Then

${ XλTw ∣ w∈W,λ∈P } and { TwXλ ∣ w∈W,λ∈P } (1.4)$

are bases of $\stackrel{\sim }{H}\text{.}$

Both the nil-Hecke algebra,

$H=ℤ-span {Tw ∣ w∈W}, andℤ[X]=ℤ -span {Xλ ∣ λ∈P} (1.5)$

are subalgebras of $\stackrel{\sim }{H}\text{.}$ The action of $W$ on $ℤ\left[X\right]$ is given by defining

$wXλ=Xwλ, for w∈W,λ∈P, (1.6)$

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

The center of $\stackrel{\sim }{H}$ is $Z\left(\stackrel{\sim }{H}\right)=ℤ{\left[X\right]}^{W}$ and each element $f\in ℤ\left[X\right]$ has a unique expansion

$f=∑w∈Wfw X-λw, with fw∈ℤ [X]W. (1.9)$

Let ${\epsilon }_{i}=1-{T}_{i}$ and let ${\epsilon }_{w}={\epsilon }_{{i}_{1}}\dots {\epsilon }_{{i}_{p}}$ for a reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}\text{.}$ Then ${\epsilon }_{w}$ 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

$εw=∑v≤w (-1)ℓ(v) TvandTw= ∑v≤w (-1)ℓ(v) εv (1.11)$

which are straightforward to verify by induction on the length of $w\text{.}$

## 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.