DUNKL OPERATORS AND QUASI-INVARIANTS 9
Finally, we define the (deformed) Euler vector field
E(k) := E(0) +
H∈A
aH (k)
where E(0) is the usual Euler derivation (i.e. the infinitesimal generator of the
diagonal

action on
K•).
The space
K•
has a bigrading given by
K•
= ⊕m⊕l=0
dim V
C[V ]m ΛlV ∗, and if we let Km l be one of the graded pieces and let ω Km, l then
E(0)ω = (l + m)ω, d(k)ω Km−1,
l+1
∂ω
Km+11−l
The deformations E(k) and d(k) are compatible in the following sense:
E(k) = ∂d(k) + d(k)∂
The following important lemma is used to determine conditions on k which are
sufficient to show E(k) has a small kernel.
Lemma 2.6 ([DO03], Lemma 2.5). Let z(k) =

H∈A
aH (k) CW . Then
(1) The element z(k) is central in CW .
(2) For an irreducible representation τ of CW , let (k) denote the (unique)
eigenvalue for z(k) on τ. Then (k) is a linear function of k with non-
negative integer coefficients.
The following theorem is the main step in the proof of commutativity of the
Dunkl operators.
Theorem 2.7 ([DO03], Theorem 2.9). Assume that k is a parameter such
that −cτ (k) N for all irreducible τ. Then there exists a unique W -invariant
linear isomorphism S(k) : K• K• satisfying the properties
(1) S(k)(Km) l Km,l
(2) The restriction of S(k) to K0 0 is the identity,
(3) S(k)(p ω) = (S(k)(p)) ω,
(4) d(k)S(k) = S(k)d(0).
The intertwining operator S(k) is constructed by induction on m and l. It is not
too difficult to show that the enumerated conditions imply that S(k) is unique and
invertible. The main point is to show the existence of S(k). The proof of existence
is essentially linear algebra using the fact that the assumption on the parameter k
implies (via the previous lemma) that the kernel of E(k) is exactly K0
0.
Corollary 2.8. The map d(k) is a differential on
K•.
Proof. If we assume −cτ (k) N for all irreducible τ, then this corollary
follows immediately from the existence of the intertwining operator (and the fact
that d(0) is a differential). However, the condition
d2(k)
= 0 is a closed condition
(either in Zariski or classical topology), so it must hold on a closed subset of the
space of all parameters k. Since the theorem holds on an open dense set in this
space, it implies the corollary for all parameter values.
As mentioned above, the commutativity of Dunkl operators (Lemma 2.5) fol-
lows directly from this corollary. Indeed, let ei V and xi V

be dual bases, and
write Ti = Tei (k). A straightforward computation shows that for all f C[V ],
d(k)2f
=
ij
(TiTj TjTi)f dxi dxj .
A different proof of commutativity can be found in [EG02b], Section 4.
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