Table Of ContentINTRODUCTION TO RATIONAL CHEREDNIK ALGEBRAS
YISUN
Abstract. ThesearenotesforatalkintheMIT-NortheasternSpring2015GraduateRepresentationTheory
Seminar. Themainsourceis[BR14].
Contents
1. Definition of rational Cherednik algebra 1
1.1. Complex reflection groups 1
1.2. Invariants of complex reflection groups 1
1.3. The definition of the rational Cherednik algebra 2
1.4. Specialization of Cherednik algebras 2
2. Basic properties of Cherednik algebras 2
2.1. Filtration on H 2
2.2. Dunkl operators and the PBW theorem 2
2.3. The center of H (W) at t(cid:54)=0 3
t,c
2.4. The spherical Cherednik algebra 3
2.5. The Satake isomorphism 4
References 4
1. Definition of rational Cherednik algebra
1.1. Complexreflectiongroups. AgroupW isacomplexreflectiongroupifitisequippedwithareflection
representation V of dimension n and generated by a set of reflections {s ,...,s } for which
1 d
dim(s −id )=1.
i V
Let Ref(W) denote the set of reflections of W, and let ε:W →C× be the composition W →GL(V)d→etC×.
For s∈Ref(W), choose elements α ∈V∗ and α∨ ∈V so that
s s
Im(s−id )=C·α∨ Im(s−id )=C·α .
V s V∗ s
Note that this implies ker(s−id )=ker(α ) and ker(s−id )=ker(α∨).
V s V∗ s
1.2. Invariants of complex reflection groups. TheringoffunctionsC[V]admitsarepresentationofW.
Its W-invariants admit the following description.
Theorem 1.1 (Shephard-Todd,Chevalley). ThealgebraofinvariantsC[V]W isapolynomialringgenerated
by homogeneous elements of degree d ,...,d so that
1 n
(cid:88)
|W|=d ···d and |Ref(W)|= (d −1).
1 n i
i
Date:February4,2015.
1
2 YISUN
1.3. The definition of the rational Cherednik algebra. LetC bethevectorspaceofmapsRef(W)→C
which are constant on conjugacy classes, and let C(cid:101)=C×C. This implies that
C[C(cid:101)]=C[T,(Cs)s∈Ref(W)/W],
where T is projection to C and C evaluation at s∈Ref(W).
s
Definition 1.2. The generic rational Cherednik algebra is the C[C(cid:101)]-algebra H which is the quotient of
C[C(cid:101)]⊗T(V ⊕V∗)(cid:111)C[W]
by the relations
[x,x(cid:48)]=[y,y(cid:48)]=0
[y,x]=T(cid:104)y,x(cid:105)+ (cid:88) (ε(s)−1)C (cid:104)y,αs(cid:105)·(cid:104)αs∨,x(cid:105)s,
s (cid:104)α∨,α (cid:105)
s s
s∈Ref(W)
for x,x(cid:48) ∈V∗ and y,y(cid:48) ∈V.
1.4. Specialization of Cherednik algebras. For (t,c)∈C(cid:101), define the specialization Ht,c of H to be
H :=C ⊗ H
t,c t,c
C[C(cid:101)]
where C[C(cid:101)]→Ct,c is given by evaluation at (t,c). If c=0, we recover the trivial examples
H =C[V ⊕V∗](cid:111)C[W] and H =D (V)(cid:111)C[W],
0,0 t,0 t
where D (V) denotes the ring of differential operators on V, defined as the quotient of C[V ⊕V∗] by
t
[x,x(cid:48)]=0 [y,y(cid:48)]=0 [y,x]=t(cid:104)y,x(cid:105).
Define also D (V) to be the C[T] algebra given as the quotient of C[T]⊗C[V ⊕V∗] by
T
[x,x(cid:48)]=0 [y,y(cid:48)]=0 [y,x]=T(cid:104)y,x(cid:105).
2. Basic properties of Cherednik algebras
2.1. Filtration on H. We define a filtration on H by
• H≤−1 =0;
• H≤0 =C[C(cid:101)]·C[V∗]·C[W];
• H≤1 =H≤0·V +H≤0;
• H≤i =(H≤1)i for i≥2.
2.2. Dunkl operators and the PBW theorem. Let Vreg = V −(cid:83) H = {v ∈ V | Stab (v) = 1}
H W
so that C[Vreg] = C[V][δ−1]. This implies that D (Vreg) = D (V)[δ−1]. Denote by Hreg := H[δ−1]. For
T T
y ∈V, the Dunkl operator Dy is the C[C(cid:101)]-linear endomorphism of C[C(cid:101)]⊗C[V] given by
(cid:88)
D =T∂ − ε(s)C (cid:104)y,α (cid:105)α−1(s−1)∈C[C]⊗D (Vreg)(cid:111)C[W].
y y s s s T
s∈Ref(W)
These operators yield a representation of H on C[C(cid:101)]⊗C[V].
Proposition 2.1. There is a representation of H on C[C(cid:101)]⊗C[V] where V∗ acts by multiplication, V acts
by Dunkl operators, and W acts by the representation action on V.
Proof. It suffices to check commutation relations involving elements of V. For y ∈ V and x ∈ V∗, notice
that
(cid:104)α∨,x(cid:105)
[α−1s,x]=(ε(s)−1−1) s s
s (cid:104)α∨,α (cid:105)
s s
and therefore
[D ,x]=T(cid:104)y,x(cid:105)+(cid:88)(ε(s)−1)C (cid:104)y,αs(cid:105)(cid:104)αs∨,x(cid:105)s.
y s (cid:104)α∨,α (cid:105)
s s s
By checking directly, we see that wD w−1 =D . Finally, for y,y(cid:48) ∈V, we have that
y w(y)
[[D ,D ],x]=[[D ,x],D ]−[[D ,x],D ],
y y(cid:48) y y(cid:48) y(cid:48) y
INTRODUCTION TO RATIONAL CHEREDNIK ALGEBRAS 3
where we have
[[D ,x],D ]=(cid:88)(ε(s)−1)C (cid:104)y,αs(cid:105)(cid:104)αs∨,x(cid:105)[s,D ]
y y(cid:48) s (cid:104)α∨,α (cid:105) y(cid:48)
s s s
=(cid:88)(ε(s)−1)2C (cid:104)y,αs(cid:105)(cid:104)y(cid:48),αs(cid:105)(cid:104)αs∨,x(cid:105)D s
s (cid:104)α∨,α (cid:105)2 α∨s
s s s
=[[D ,x],D ],
y(cid:48) y
which implies that [D ,D ] commutes with C[V]. On the other hand, [D ,D ] acts by 0 on 1 ∈ C[V],
y y(cid:48) y y(cid:48)
hence by 0 on all of C[V]. Because the action of D (Vreg) on C[V] is faithful, this implies [D ,D ] = 0,
T y y(cid:48)
completing the proof. (cid:3)
Remark. This action is called the polynomial representation of H.
By analyzing the polynomial representation, we are able to obtain a PBW theorem for H.
Proposition 2.2. The linear map
C[C(cid:101)]⊗C[V]⊗C[W]⊗C[V∗]m→ultH
is an isomorphism of C[C(cid:101)]-modules.
Proof. The polynomial representation yields a map
Θ:H→C[C(cid:101)]⊗DT(Vreg).
Denote by Θreg the extension to Hreg →C[C]⊗D (Vreg). Consider the composition
T
η :C[C(cid:101)]⊗C[Vreg]⊗C[W]⊗C[V∗]mu→ltreg Hreg Θ→reg C[C]⊗DT(Vreg)(cid:111)C[W].
Notice that gr(η) is an isomorphism, hence η is an isomorphism. Now, because multreg is surjective by
definition, this implies that it and Θreg are both injections. This implies by restriction that the map
C[C(cid:101)]⊗C[V]⊗C[W]⊗C[V∗]gr(→mult)grH
is injective, hence an isomorphism, which yields the desired. (cid:3)
Corollary 2.3. The polynomial representation is faithful.
Proof. The proof of Proposition 2.2 also shows that Θ is injective by restriction from the isomorphism Θreg.
Faithfulness follows because the map DT(Vreg)(cid:111)C[W] → C[C(cid:101)]⊗Homk(C[V],C[Vreg]) is injective and the
image of the polynomial representation under this identification lands in C[C(cid:101)]⊗Endk(C[V]). (cid:3)
2.3. The center of H (W) at t(cid:54)=0. Let Z denote the center of H and Z its specialization.
t,c t,c
Proposition 2.4. If t(cid:54)=0, then the polynomial representation of H is faithful and Z =C.
t,c t,c
Proof. Faithfulness follows in the same way as in Corollary 2.3, where we note that the polynomial repre-
sentation of D (V)(cid:111)C[W] is faithful only when t(cid:54)=0.
t
Now, by faithfulness, the polynomial representation gives an embedding H (W) (cid:44)→ D (V)(cid:111)C[W] (cid:39)
t,c t
D(V)(cid:111)C[W] for t (cid:54)= 0. Any element of Z must commute with C[V∗] ⊂ D(V)(cid:111)C[W], hence lie in
t,c
C[V∗]. It is easy to check that no non-constant element of C[V∗] commutes with all x ∈ V∗, showing that
Z =C. (cid:3)
t,c
2.4. The spherical Cherednik algebra. The primitive central idempotent of C[W] is
1 (cid:88)
e= w,
|W|
w∈W
and the C[C(cid:101)]-algebra eHe is known as the generic spherical algebra. We denote its specialization by eHt,ce.
By Proposition 2.2, we see that
gr(eHe)=C[C(cid:101)]⊗C[V ⊕V∗]W.
We first examine a few properties of the spherical algebra.
Proposition 2.5. The following properties hold:
4 YISUN
(a) eH e is a finitely generated C-algebra without zero divisors;
t,c
(b) H e is a finitely generated right eH e-module;
t,c t,c
(c) left multiplication yields an isomorphism H →End (H e)op;
t,c (eHt,ce)op t,c
Proof. Properties (a) and (b) follow because they hold for gr(eH e). For property (b), let φ : H →
t,c t,c
End (H e)op be the desired morphism. We consider the composition
(eHt,ce)op t,c
ψ :grH gr→(φ)grEnd (H e)op →End (gr(H e))op,
t,c (eHt,ce)op t,c gr(eHt,ce)op t,c
where the first map is by left multiplication and the second is an injection1. Recall from the proof of Propo-
sition 2.2 the isomorphism grH (cid:39)C[V ⊕V∗](cid:111)C[W], under which this map is given by left multiplication
t,c
C[V ⊕V∗](cid:111)C[W]→End(e(C[V⊕V∗](cid:111)C[W])e)op((C[V ⊕V∗](cid:111)C[W])e)op,
henceisanisomorphismbyLemma2.6appliedtoX =V ×V∗ withtheactionofW,wherethecodimension
condition holds because the action of W preserves the pairing between V and V∗. We conclude that ψ is an
isomorphism, hence gr(φ) and φ are, as needed. (cid:3)
Lemma 2.6. Let W act on a smooth affine variety X, let A = C[X], and let R = A (cid:111) C[W]. Let
Xreg ={x∈X |Stab (x)=1}. If codim(X−Xreg)≥2 in each connected component, then the morphism
W
R→End (A)op is an isomorphism.
(AW)op
Proof. We claim that the morphism is injective even if the codimension condition does not hold. Because
Xreg is Zariski dense, we may localize to C[Xreg] to check injectivity, so we may assume that W acts freely
(cid:80) (cid:80)
on X. In this case, choose f ⊗w in the kernel so that f w (f) = 0 for f ∈ A. Because W acts
i i i i i i
freely on X, for any x and z ∈ C we may find some function f ∈ A so that f(w−1·x) = z , meaning that
i i i
(cid:80)
f (x)z =0, whence we conclude f =0, yielding injectivity.
i i i i
If W acts freely on all of X, R and End (A)op are both AW-algebras of rank |W|2, so injectivity
(AW)op
implies surjectivity. For surjectivity in general, for any f ∈End (A)op, cover Xreg by affine open sets
(AW)op
Xj. OneachXj,wemaychoosesome(cid:80) aj·w ∈C[Xreg](cid:111)C[W]withaj ∈C[Xj]andw ∈W whichgives
i i i i i
rise to the restriction of f to Xj. On Xj1 ∩Xj2, the restriction of (cid:80)iaji1 ·wi and (cid:80)iaji2 ·wi gives rise to
the restriction of f to Xj1 ∩Xj2, hence their restrictions are equal. Therefore, the family of functions {aj}
i
glue to a function a on Xreg for which (cid:80) a ·w ∈C[Xreg](cid:111)C[W] gives rise to f| . Each a is regular
i i i i Xreg i
in codimension 2, hence regular by Hartog’s theorem. Thus (cid:80) a ·w lies in R, finishing the proof. (cid:3)
i i i
2.5. The Satake isomorphism. Fortherestofthetalk, weworkinthespecializationt=0. Ourgoalwill
be to prove the Satake isomorphism relating Z and eH e.
0,c 0,c
Theorem 2.7 (Satake isomorphism). The map z (cid:55)→z·e is an isomorphism of algebras Z →eH e.
0,c 0,c
Lemma 2.8. If e is an idempotent of a ring A and left multiplication gives an isomorphism
A→End (Ae)op,
(eAe)op
then the map Z(A)→Z(eAe) given by a(cid:55)→ae is an isomorphism.
Proof. NoticethatwehaveeAe=End (Ae)bydefinition. Therefore,leftmultiplicationonAeyieldsamap
A
α : Z(A) → Z(eAe) so that α(z) = ze implies zm = mα(z) and by the given right multiplication yields a
map β : Z(eAe) → Z(A) so that mz = β(z)m. For z ∈ Z(A), we then have that zm = β(α(z))m, so that
β◦α=id because the left multiplication is faithful. Similarly, we find that α◦β =id. (cid:3)
Proof of Theorem 2.7. By Proposition 2.5(c) and Lemma 2.8, we have Z (cid:39) Z(eH e), so we only need
0,c 0,c
show eH e is commutative. The Dunkl operators at t=0 yield an injection H →C[Vreg⊕V∗](cid:111)C[W]
0,c 0,c
whichrestrictstoaninjectioneH e→C[Vreg⊕V∗]W withthelattercommutative, yieldingtheclaim. (cid:3)
0,c
References
[BR14] C´edricBonnaf´eandRapha¨elRouquier.Calogero-Mosercells.2014.
E-mail address: [email protected]
1IfM andN arefilteredmodules,overafilteredringA,wefilterHomA(M,N)byHomA(M,N)≤i={f ∈HomA(M,N)|
f(M≤j) ⊂ N≤j+i}. There is a map grHomA(M,N) → Homgr(A)(gr(M),gr(N)) which sends [fi] ∈ griHomA(M,N) to
(cid:16) (cid:17)
[mj](cid:55)→[fi(mj)]∈gri+j(N) . WeapplythisconstructionwithA=eHt,ceandM =N =Ht,ce.
