TUTORIAL PROBLEMS 1
RATIONAL CHEREDNIK ALGEBRAS
ALGEBRAIC LIE THEORY AND REPRESENTATION THEORY, GLASGOW 2014
1. Reflection groups
Let W be a finite complex reflection group, h its reflection representation, h∗ the dual representation to h, (·, ·) : h∗ ⊗ h → C the natural pairing, S the set of all reflections in W (elements
s ∈ W such that rankh (1 − s) = 1), αs ∈ h∗ and αs∨ ∈ h a choice of basis vectors of Im(1 − s),
normalized so that (αs , αs∨ ) = 2.
(1) a) Prove that any reflection s ∈ W is diagonalizable on h, with eigenvalues 1 of multiplicity dimh − 1, and some λs−1 6= 1 of multilplicity 1.
b) Conclude that s is diagonalizable on h∗ , with eigenvalues 1 of multiplicity dimh − 1,
and λs 6= 1 of multilplicity 1.
c) Prove that the action of s on x ∈ h∗ and y ∈ h can be written as
s(x) = x −
1 − λs
(x, αs∨ )αs
2
1 − λ−1
s
(αs , y)αs∨
2
d) When W is a real reflection group (finite Coxeter group), then any reflection satisfies
s2 = 1, and h∗ ∼
= h. Notice that in that case, c) simplifies to
s(y) = y −
s(x) = x − (x, αs∨ )αs .
(2) For W = Sn the symmetric group on n letters, find h, h∗ , S, conjugacy classes in S, αs ,
αs∨ , and λs .
2. The rational Cherednik algebra and PBW
Let t ∈ C, and let cs ∈ C be a collection of numbers parametrized by reflections s ∈ S, such
that cs = cwsw−1 for any w ∈ W . The rational Cherednik algebra Ht,c (W ) associated to this
data is the quotient of W n T (h ⊕ h∗ ) (the semidirect product of the tensor algebra on h ⊕ h∗
with W ) by the relations: for all x, x0 ∈ h∗ , y, y 0 ∈ h,
[x, x0 ] = 0,
[y, x] = t(x, y) −
[y, y 0 ] = 0
X
cs (αs , y)(x, αs∨ )s
s∈S
(3) Notice that deg(x) = 1, deg(y) = 1, deg(w) = 0 gives a filtration of Ht,c (W ). PBW
theorem states that
gr(Ht,c (W )) ∼
= W n S(h ⊕ h∗ ),
or alternatively, that given bases x1 , . . . xn of h∗ and y1 , . . . yn of h, the set
{xa1 1 . . . xann · w · y1a1 . . . ynan |a1 , . . . bn ∈ Z≥0 , w ∈ W }
is a basis of Ht,c (W ). Using the filtration, prove the easy half of the PBW theorem.
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ALGEBRAIC LIE THEORY AND REPRESENTATION THEORY, GLASGOW 2014
(4) To see that the hard half of the PBW theorem is nontrivial, consider the following algebra
given by generators and relations:
L = C x, y, s|s2 = 1, sxs = −x, sys = −y, [y, x] = 1, (y − s)x = xy + s .
Prove L = 0.
3. The Dunkl embedding
We will fix t = 1 for this part, although most statements work for any t.
Let D(hreg ) be the algebra of differential operators on h, generated by functions x ∈ h∗ and
differential operators ∂y , y ∈ h, with relations
[x, x0 ] = 0,
[∂y , ∂y0 ] = 0,
[∂y , x] = (x, y).
S
Let hreg = h \ ( s {αs = 0}) be the open affine subset obtained by removing the reflection
hyperplanes ker αs = ker(s − 1). Functions on h are C[h] = Sh∗ , and functions on hreg are
obtained from functions on h by inverting all αs . The Dunkl embedding is the map Ht,c (W ) →
W n D(hreg ) given by
x 7→ x
w 7→ w
y
7→ Dy = ∂y −
X 2cs
1−s
(αs , y)
.
1 − λs
αs
s∈S
In the following problems, we will prove this is indeed a homomorphism of algebras, and use it
to define a representation of Ht,c (W ) on C[h].
(5) a) Notice that C[hreg ] is a faithful representation of D(hreg ). (Hint: w ∈ W act
diagonally, f ∈ C[hreg ] act by multiplication, and ∂y act by partial derivations.
There is almost nothing to check.)
b) Show that for any f ∈ C[h] and any reflection s, the polynomial f − s(f ) is divisible
by αs . (Hint: It might be helpful to think about Sn first, or about polynomials of
degree 1).
c) Conclude that Dunkl operators preserve the subspace C[h], and that this is a faithful
representation of the subalgebra of D(hreg ) generated by x, Dy , w.
(6) Show that
wDy w−1 = Dw(y) .
(Hint: start by showing λs = λwsw−1 and wαs w−1 = αwsw−1 ).
(7) Show that
X
[Dy , x] = (x, y) −
cs (αs , y)(x, αs∨ )s.
s∈S
2 1−s
, x] = (x, αs∨ )s, which follows from Problem 1c)).
(Hint: it follows from [ 1−λ
s αs
(8) Show that
[Dy1 , Dy2 ] = 0.
(Hint: it’s possible but pretty hard to show directly. An easier thing is to notice that is
enough to check this in a faithful representation, and then use the representataion from
Problem (5)).
(9) Conclude that the above defined map is really a homomorphism of algebras Ht,c (W ) →
W n Dt (hreg ). Together with Problem (5), notice that we have defined a faithful representation of H1,c (W ) on C[h] = Sh∗ , where x ∈ h∗ act by multiplication, w ∈ W
diagonally, and y ∈ h by Dunkl operators.