Symmetric quotient stacks and Heisenberg actions

0.19 MB·English

by  Andreas Krug

SYMMETRIC QUOTIENT STACKS AND HEISENBERG ACTIONS
ANDREASKRUG
Abstract. ForeverysmoothprojectivevarietyX,weconstructanactionoftheHeisenberg
algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks
[Xn/Sn] which contains the Fock space as a subrepresentation. The action is induced by
functors on the level of the derived categories which form a weak categorification of the
5 action.
1
0
2

n
a 1. Introduction
J
8 A celebrated theorem of Nakajima [Nak97] and Grojnowski [Gro96] identifies the coho-
2 mology of Hilbert schemes of points on a surface with the Fock space representation of the

Heisenberg Lie algebra associated to the cohomology of the surface itself.
]
G One would like to lift the Heisenberg action to the level of the Grothendieck groups or,
A even better, thederived categories of theHilbert schemes. Schiffmann and Vasserot [SV13] as
well as Feigin and Tsymbaliuk [FT11] constructed a Heisenberg action on the Grothendieck
.
h
groups in the case of the affine plane. In [CL12], Cautis and Licata constructed a categorical
t
a Heisenbergaction onthederived categories of theHilbertschemes in thecase thatthesurface
m
is a minimal resolution of a Kleinian singularity. Their construction makes use of the derived
[
McKay correspondence between the derived category of the Hilbert scheme of points on a

1 smooth quasi-projective surface and the derived category of the symmetric quotient stack
v associated to the surface; see [BKR01] and [Hai01].
3
In this paper, we generalise the construction of [CL12] to obtain functors between the
5
2 derived categories of the symmetric quotient stacks of an arbitrary smooth complete variety
7 which descend to a Heisenberg action on the Grothendieck groups of the symmetric quotient
0
stacks. Our construction can also be seen as a global version of a construction of Khovanov
.
1 [Kho14] which deals with the case that the variety is a point. Note, however, that there are
0
deeper categorical structures in [CL12] as well as [Kho14] that we do not generalise; see also
5
1 Section 3.4.
: Our construction is much closer to the construction of [Gro96] (see also [Nak99, Ch. 9])
v
i than to that of [Nak97]. In some sense, what our paper does is to fill in the proof of the
X
claim made in [Gro96, footnote 3], though while working on the higher level of the derived
r
a categories.
1.1. Generators of the Heisenberg algebra. Let V be a vector space over Q (not nec-
essarily of finite dimension) with a bilinear form h , i. Set L = VZ\{0} and denote the
V
image of β ∈ V in the n-th factor of L by a (n) for n ∈ Z\{0}. The Heisenberg algebra H
V β V
associated to (V,h , i) is the unital Q-algebra given by the tensor algebra T(L ) modulo
V
the relations
(1) [a (m),a (n)]= δ nhα,βi.
α β m,−n
1
2 ANDREASKRUG
Note that, with ourdefinition, theHeisenbergalgebrais thequotientof theuniversalenvelop-
ing algebra U(h ) of the Heisenberg Lie algebra associated to (V,h , i) where the central
V
charge is identified with 1. Hence, every module over the Heisenberg algebra H yields a
V
representation of the Heisenberg Lie algebra h . In particular, the Fock space representation
V
is induced from a H -module; see Section 3.1.
V
(n) (n)
We define elements p and q for β ∈ V and n a non-negative integer by the formulae
β β
a (−ℓ) a (ℓ)
(2) p(n)zn = exp β zℓ , q(n)zn = exp β zℓ .
β ℓ β ℓ
nX≥0 Xℓ≥1 nX≥0 Xℓ≥1
   
(n)
As β runs through a basis of V and n through the positive integers, the elements q and
β
p(n) together form a set of generators of H . In order to describetheir relations, we introduce
α V
the following notation.
Definition 1.1. For χ ∈Q and k a non-negative integer, we set
χ+k−1 1
skχ := := (χ+k−1)(χ+k−2)···(χ+1)χ.
(cid:18) k (cid:19) k!
Lemma 1.2. The relations among the above generators are given by
(3) [q(m),q(n)]= 0 = [p(m),p(n)],
α β α β
min{m,n}
(4) q(m)p(n) = skhα,βi·p(n−k)q(m−k).
α β β α
Xk=0
We will prove Lemma 1.2 in Section 2.1. Note the following simple but important fact.
Lemma 1.3. Let W∗ be a finite-dimensional graded vector space. Then the Euler character-
istic of its symmetric product SkW∗ (formed in the graded sense) is given by
χ(SkW∗) = sk(χ(W∗)).
1.2. Construction and results. Let X be a smooth complete variety over a field k of char-
acteristic zero. For a non-negative integer ℓ, we consider the cartesian product Xℓ together
with the natural action of the symmetric group S given by permuting the factors. The asso-
ℓ
ciated quotient stack [Xℓ/S ] is called theℓ-th symmetric quotient stack. Its derived category
ℓ
can equivalently bedescribedas theS -equivariant derivedcategory of thecartesian product,
ℓ
that means D([Xℓ/S ]) ∼= D (Xℓ). We set
ℓ Sℓ
D := D (Xℓ)
S
ℓ
Mℓ≥0
For 1 ≤ n ≤ N and β ∈ D(X) we define the functor
(5) P(n) : D (XN−n) → D (XN) , E 7→ InfSN β⊠n⊠E .
N,β SN−n SN Sn×SN−n
(cid:0) (cid:1)
Note that for E ∈ D (XN−n) we can consider β⊠n ⊠ E canonically as an object of
SN−n
D (XN). The inflation functor InfSN : D (XN) → D (XN) is the
Sn×SN−n Sn×SN−n Sn×SN−n SN
SYMMETRIC QUOTIENT STACKS AND HEISENBERG ACTIONS 3
adjoint of the forgetful functor; see Section 2.2 for details on the functor Inf and Section 2.4
(n)
for details on the functor P . We set
N,β
(n) (n) (0)
P := P : D → D for n≥ 1, P := id: D → D.
β N,β β
NM≥n
(n) (n)
Finally, we define Q : D → D as the right adjoint of P .
β β
Theorem 1.4. For every α,β ∈D(X) and n,m ∈ N, we have the relations
(6) Q(m)Q(n) ∼= Q(n)Q(m) , P(m)P(n) ∼= P(n)P(m)
α β β α α β β α
min{m,n}
(7) Qα(m)Pβ(n) ∼= SkHom∗(α,β)⊗kPβ(n−k)Qα(m−k).
Mk=0
We denote by K([Xn/S ]) ∼= K (Xn) the equivariant Grothendieck group with coeffi-
n Sn
cients in Q and set K := K (Xn) = K(D). We consider K(X) together with the
ℓ≥0 Sn
bilinear form h , i given bLy the Mukai pairing, i.e.
(8) h[α],[β]i := χ(α,β) := χ Hom∗(α,β)
(cid:0) (cid:1)
where [α] denotes the class of an object α∈ D(X) in the Grothendieck group K(X)
(n) (n)
Corollary 1.5. The descent of the functors P and Q to the level of the Grothendieck
β β
groups makes K into a representation of the Heisenberg algebra H .
K(X)
(n) (n) (n) (n)
Proof. Letp := [P ]: K → Kand q := [Q ]: K → K betheinducedmaps on thelevel
β β β β
of the Grothendieck groups. By Lemma 1.3 and (8), the relations of Theorem 1.4 translate
to the p(n) and q(n) satisfying the relations of Lemma 1.2. (cid:3)
β β
(n) (n)
The category D together with the functors Q and P can be regarded as a categorifi-
β β
cation of the Heisenberg representation K.
Since all the qβ(n) vanish on K0 = KS0(X0) = K(point) ∼= Q, one gets an embedding of the
Fock space representation into K; see Section 3.1 for details.
We will give the proof of Theorem 1.4 in Sections 2.5 and 2.6. One main ingredient is the
following easy fact.
Lemma 1.6 (“Symmetric Ku¨nneth formula”). Let α,β ∈ D(X) and k ∈ N. Then
Hom∗ (β⊠k,α⊠k)∼= SkHom∗(β,α)
DS (Xk)
k
1.3. Conclusion. The reason for the occurrence of Heisenberg actions in the context of
symmetric quotient stacks, and hence also of Hilbert schemes of points on surfaces, can be
summarised in the following simple and, in the author’s opinion, satisfying way:
(i) The Heisenberg algebra has generators whose relations involve the numbers snhα,βi
as coefficients; see Lemma 1.2.
(ii) A “vector-spaceification” of these numbers is given by the graded vector spaces
SnHom∗(α,β); see Lemma 1.3.
(iii) Because of the symmetric Ku¨nneth formula (Lemma 1.6), these graded vector spaces
show up very naturally in the context of derived categories of symmetric quotient
stacks. Inparticular, the relatively simpleconstruction (5)yields acategorical action
of the Heisenberg algebra.
4 ANDREASKRUG
1.4. Organisation of the paper. InSection 2.1weproveLemma1.2. Afterwards, werecall
some basic facts on equivariant derived categories and functors in Section 2.2, introduce some
(n) (n)
notationinSection2.3,anddescribethefunctorsP andQ insomemoredetailinSection
β β
2.4. In Sections 2.5 and 2.6, we give the proof of Theorem 1.4.
WeexplaininSection3.1theembeddingoftheFockspacerepresentationintoK. InSection
3.2 we define further functors which give a categorification of the action on K with respect
to the so-called transposed generators of H . In particular, this recovers the construction
K(X)
of [Kho14]. In Section 3.3 we discuss some generalisations and variants, for example to the
non-complete case and the case where we replace the category D(X) by some equivariant
category. Finally, we point out two open problems in Section 3.4. Conventions With the
exception of Section 3.3, X will be a smooth complete variety over a field k of characteristic
zero. Its derived category is understood to be the bounded derived category of coherent
sheaves, i.e. D(X) := Db(Coh(X)). All functors between derived categories are assumed to
be derived although this will not be reflected in the notation. Acknowledgements. The
author was financially supported by the research grant KR 4541/1-1 of the DFG. He thanks
Daniel Huybrechts, Ciaran Meachan, David Ploog, Miles Reid, and Pawel Sosna for helpful
comments.
2. Proofs
2.1. Proof of Lemma 1.2. Let α,β ∈ V and m,n ∈ N. The vanishing of the commutators
(m) (n) (m) (n)
[q ,q ] and [q ,q ] follows immediately from the fact that [a (r),a (s)] = 0 if r and s
α β α β α β
are both positive or both negative.
For the proof of relation (4) we follow closely the proof of [CL12, Lem. 1]. Set
a (−ℓ) a (ℓ)
A(z) := β zℓ , B(w) := α wℓ
ℓ ℓ
Xℓ≥1 Xℓ≥1
so that q(m)p(n) = [wmzn]exp(B)exp(A). We also set χ := hα,βi. By relation (1), we have
α β
ℓ·χ (wz)ℓ
(9) [B,A] = wℓzℓ = χ = −χlog(1−wz).
ℓ2 ℓ
Xℓ≥1 Xℓ≥1
In particular, [B,A] commutes with A and B. Thus,
(10) exp(B)exp(A) = exp([B,A])exp(A)exp(B);
see [Nak99, Lem. 9.43]. Recall the binomial formula
(11) (1−t)−χ = skχ·tk.
Xk≥0
Together, (9), (10), and (11) give
exp(B)exp(A) = (1−wz)−χexp(A)exp(B)=  skχ·(wz)kexp(A)exp(B).
Xk≥0
 
Comparing the coefficients of wmzn on both sides gives relation (4).
SYMMETRIC QUOTIENT STACKS AND HEISENBERG ACTIONS 5
(n) (n)
In order to show that there are no further relations among the generators q and p we
β β
need to introduce the following notation. Let {β } be a basis of V. Consider maps
i i∈I
(12) ν: I → partitions of n
na≥0(cid:8) (cid:9)
such that ν(i) 6= 0 only for a finite number of i ∈ I. We write the partitions in the form
ν(i) = 1ν(i)12ν(i)2···. We fix a total order on I and set
a(ν) := a (k)ν(i)k , a(−ν):= a (−k)ν(i)k ,
βi βi
Yi∈I(cid:0)kY≥1 (cid:1) Yi∈I(cid:0)kY≥1 (cid:1)
q(ν) := (q(k))ν(i)k , p(ν):= (p(k))ν(i)k .
βi βi
Yi∈I(cid:0)kY≥1 (cid:1) Yi∈I(cid:0)kY≥1 (cid:1)
Here, the inner product is formed with respect to the usual order of N and the outer product
with respect to the fixed order of I. Since the only relation between the generators a (k) is
β
the commutator relation (1), the elements a(−ν)a(µ) form a basis of H as a vector space.
V
(n) (n)
In the algebra with generators q and p and the relations (3) and (4), one can write every
βi βi
element as a linear combination of the form
(13) λ(ν,µ)p(ν)q(µ) , λ(µ,ν) ∈ Q.
Xν,µ
Thus, it is sufficient to show that the p(ν)q(µ) are linearly independent in H . For two maps
V
ν,ν′ as in (12) we say that ν is coarser than ν′, and write ν ≻ ν′, if |ν(i)| =|ν′(i)| and ν(i) is
a coarser partition than ν′(i) for every i ∈ I. For pairs, we set (ν,µ) ≻ (ν′,µ′) if ν ≻ ν′ and
µ ≻ µ′. Note that by (2), we have
p(n) = a (−n)+ linear combination of a (−1)ν1a (−2)ν2··· for partitions ν of n
β β β β
(cid:0) (cid:1)
(n)
and similarly for q . Thus,
β
(14) p(ν)q(µ) = a(−ν)a(µ)+ linear combination of a(−ν′)a(µ′) for (ν,µ) ≻ (ν′,µ′) .
(cid:0) (cid:1)
Let x = λ(µ,ν)p(ν)q(µ) as in (13) such that not all coefficients vanish. Pick a pair
ν,µ
(ν ,µ ) wPith λ(ν ,µ ) 6= 0 such that for each coarser pair the corresponding coefficient van-
0 0 0 0
ishes. Then, by (14), x is a linear combination of the a(−ν)b(−µ) such that the coefficient of
a(−ν )a(µ ) equals λ(ν ,µ ). Since the a(−ν)b(µ) form a basis, x 6= 0 in H .
0 0 0 0 V
2.2. Basic factsabout equivariant functors. Fordetailsonequivariantderivedcategories
and functors between them, we refer to [BKR01, Sect. 4]. Let G be a finite group acting on a
smooth projective variety M. The equivariant derived category D (M) has as objects pairs
G
(E,λ) where E ∈ D(M) and λ is a G-linearisation of E. A G-linearisation is a family of
isomorphisms (E −→∼= g∗E) in D(M) such that for every pair g,h ∈ G the composition
g∈G
E −λ→g g∗E −g−∗−λ→h g∗h∗E ∼= (hg)∗E
equals λ . The morphisms are morphisms in the ordinary derived category D(X) which are
hg
compatiblewiththelinearisations. Thecategory D (M)isequivalent totheboundedderived
G
category Db(Coh (M)) of G-equivariant coherent sheaves on M; see [Che14] or [Ela14].
G
6 ANDREASKRUG
For a subgroup U ⊂ G there is the functor ResU: D (M) → D (M) given by restricting
G G U
the linearisations. It has the inflation functor InfG: D (M) → D (M) as a left and right
U U G
adjoint. Let U \G be the left cosets and E ∈ D(X). Then
(15) InfG(E) ∼= σ∗E
U
M
σ∈U\G
with the G-linearisation of InfG(E) given by a combination of the U-linearisation of E and
U
permutation of the direct summands.
Let G act trivially on M. Then there is the functor triv: D(M) → D (M) which equips
G
every object with the trivial linearisation. It has the functor of invariants ( )G: D (M) →
G
D(M) as a left and right adjoint.
We use the following principle for the computation of invariants; see e.g. [Kru14, Sect. 3.5].
Let E = (E,λ) be a G-equivariant sheaf such that E = ⊕ E for some finite index set K.
i∈K i
Assume that there is an action of G on K such that λ (E ) = E for all i ∈ K and g ∈ G.
g i g·i
We say that the linearisation λ induces the action on the index set K.
Lemma 2.1. Let R ⊂ K be a set of representatives of the G-orbits in K under the action
induced by the linearisation. Then EG ∼= EGi where G ⊂ G is the stabiliser subgroup.
i∈R i i
L
2.3. Combinatorial notations. For a finite set I we set XI := X ∼= X|I|. For J ⊂ I
I
we denote by prI : XI → XJ the projection. We often drop the indQex I in the notation and
J
simply write pr when the source of the projection should be clear from the context.
J
For n ∈ N, we set n := [1,n] = {1,2,...,n}. By convention, 0 := ∅. Later, there will be a
fixed number N ∈ N such that all occurring finite sets will be subsets of N. Hence, we will
often write N instead of N to ease the notation. Furthermore, for a subset J ⊂ N we set
J¯:= N \J. For 0 ≤ n ≤ N we set n := n = N \n = [n+1,N].
(n) (n)
2.4. The functors P and Q . For every object β ∈ D(X) and N ≥ n, the functor
β α
P(n) : D (XN−n) → D (XN) is given by the composition
N,β SN−n SN
triv pr∗
D (XN−n)−−→ D (XN−n) −−→n D (XN)
SN−n Sn×SN−n Sn×SN−n
−⊗−p−r−∗n−β−⊠→n D (XN) −I−nf−SS−Nn−×−S→n D (XN)
Sn×SN−n SN
where we use the identifications Xn ∼= Xn and Xn = X[n+1,N] ∼= XN−n. Hence, the right-
adjoint Q(n) : D (XN) → D (XN−n) is the composition
N,β SN SN−n
D (XN−n)←(−−)S−−n D (XN−n) ←p−r−n−∗ D (XN)
SN−n Sn×SN−n Sn×SN−n
←⊗−p−r−∗n−β−∨⊠−−n D (XN)←R−es−SS−Nn−×−S−n D (XN).
Sn×SN−n SN
Note that (β∨)⊠n ∼= (β⊠n)∨ so that we simply write β∨⊠n without ambiguity. Since
σ∗ pr∗ β⊠⊗pr∗ E ∼= pr∗ β⊠ ⊗pr∗ E
n n σ−1(n) σ−1(n)
(cid:0) (cid:1) (cid:1)
for every E ∈ D (XN−n) and σ ∈ S , we get by (15)
SN−n n
(16) P(n)(E) ∼= pr∗β⊠n⊗pr∗E
N,β I I¯
M
I⊂N,|I|=n
SYMMETRIC QUOTIENT STACKS AND HEISENBERG ACTIONS 7
Furthermore, the functor Q(m) is given on objects F ∈ D (XN) by
α SN
(17) Q(m)(F) ∼= pr (pr∗ α∨⊠m⊗F)Sm
α m∗ m
Remark 2.2. Let I be any set of cardinality N and J ⊂ I be any subset of cardinality m.
Making the identifications XI ∼= XN, XJ ∼= Xm, and XI\J ∼= XN−m we can also write
Q(m)(F) ∼= prI (prI∗α∨⊠m⊗F)SJ .
N,α I\J∗ J
2.5. Proof of relation (6). In order to verify the relations (6) of Theorem 1.4 it is sufficient
to show the first relation, i.e. Qα(m)Q(βn) ∼= Q(βn)Qα(m) for α,β ∈ D(X) and m,n ∈ N. The
relation Pα(m)Pβ(n) ∼= Pβ(n)Pα(m) follows by adjunction. This means that we have to show that
Q(m) Q(n) ∼= Q(n) Q(m) : D (XN) → D (XN−m−n) for every N ≥ n+m.
N−n,α N,β N−m,β N,α SN SN−m−n
Indeed, using Remark 2.2, the projection formula along prN, the projection formula along
n
prN , and finally Remark 2.2 again, we get isomorphisms
[n+1,n+m]
(m) (n)
Q Q (F)
N−n,α N,β
S
∼= prn prn∗ α∨⊠m⊗ prN (prN∗β∨⊠n⊗F) Sn [n+1,n+m]
n+m∗ [n+1,n+m] n∗ n
h (cid:16) (cid:17)i
(cid:2) (cid:3)
∼= prN prN∗β∨⊠n⊗prN∗ α∨⊠m⊗F Sn×S[n+1,n+m]
n+m∗ n [n+1,n+m]
h i
(cid:0) (cid:1)
∼= pr[n+1,n+m]∗ pr[n+1,n+m]∗β∨⊠n⊗ prN (prN∗ α∨⊠m⊗F) S[n+1,n+m] Sn
n+m∗ n [n+1,n+m]∗ [n+1,n+m]
h (cid:16) (cid:17)i
(cid:2) (cid:3)
∼=Q(n) Q(m)(F)
N−m,β N,α
which are functorial in F ∈ D (XN).
SN
2.6. Proof of relation (7). Let N ≥ max{m,n} and E ∈ D (XN−n). Combining (16)
SN−n
and (17) we get
Q(m)P(n)(E) ∼= T Sm
α β I
(cid:2)MI∈K (cid:3)
where
K = {I ⊂ N , |I| = n} and T = pr pr∗ α∨⊠m⊗pr∗β⊠n⊗pr∗E .
I m∗ m I I¯
(cid:0) (cid:1)
The S -linearisation of ⊗ T induces on the index set K the action σ·I = σ(I). A set of
m I∈K I
representatives of the S -orbits in K is given by
m
R= {I = k∪J | n+m−N ≤ k ≤ min{m,n}, J ⊂ m, |J| = n−k}.
Thus, Lemma 2.1 yields
(18) Q(m)P(n)(E) ∼= TSk×S[k+1,m].
α β I
I=kM∪J∈R
8 ANDREASKRUG
Let I = k∪J ∈ R and consider the diagram
prm
(19) Xm J // Xm∩I = XJ
vv♠p♠r♠mm♠\♠J♠♠♠♠♠♠♠p♠rkm OO ee▲▲▲▲▲▲p▲r▲mN▲▲▲ prN
XI∩m = Xm\J Xk = Xm∪I oo XN k // Xm∩I = Xk
hh◗◗◗◗p◗r◗Im◗\◗J◗◗◗◗◗◗◗p◗rkIX(cid:15)(cid:15)I yyssssssppsrrsNkNIssssprI[k+1,m] // Xm∩I = X[k+1,m].
The two triangles in the middle of the diagram are commutative and the square on the left
is cartesian. We can rewrite T = T as
I k∪J
T ∼= prk prN prN∗ prk∗prm∗β⊠n−k ⊗prk∗ E ⊗prI∗ α∨⊠m−k ⊗prN∗(α∨⊗β)⊠k .
I m∗ k∗ k m J I [k+1,m] k
h (cid:16) (cid:17) i
(cid:0) (cid:1)
By the projection formula along prN, we get
k
T ∼= prk prk∗prm∗β⊠n−k ⊗prk∗ E⊗prI∗ α∨⊠m−k ⊗prN prN∗(α∨ ⊗β)⊠k .
I m∗ m J I [k+1,m] k∗ k
h i
(cid:0) (cid:1)
By the fact that XN = Xk ×Xk and Ku¨nneth formula
prkN∗prNk ∗(α∨⊗β)⊠k ∼= OXk ⊗kΓ (α∨⊗β)⊠k ∼= OXk ⊗kHom∗(α⊠k,β⊠k)
(cid:0) (cid:1) ∼= O ⊗kHom∗(α,β)⊗k.
Xk
This gives
TI ∼= Hom∗(α,β)⊗k ⊗kT˜I , T˜I = prkm∗ prkm∗prmJ∗β⊠n−k ⊗prkI∗ E ⊗prI[k∗+1,m]α∨⊠m−k .
h i
(cid:0) (cid:1)
By the projection formula along prk and base change along the cartesian square in (19),
m
T˜ ∼= prm∗β⊠n−k ⊗prk prk∗ E ⊗prI∗ α∨⊠m−k
I J m∗ I [k+1,m]
∼= prm∗β⊠n−k ⊗prm∗ prI(cid:0) E⊗prI∗ α∨⊠m(cid:1)−k
J m\J m\J∗ [k+1,m]
(cid:0) (cid:1)
In summary,
TI ∼= Hom∗(α,β)⊗k ⊗kprmJ∗β⊠n−k ⊗prmm∗\J prIm\J∗ E⊗prI[k∗+1,m]α∨⊠m−k .
(cid:0) (cid:1)
Taking S ×S -invariants yields
k [k+1,m]
TISk×S[k+1,m] ∼= SkHom∗(α,β)⊗kprmJ∗β⊠n−k ⊗prmm∗\J prIm\J∗ E⊗prI[k∗+1,m]α∨⊠m−k S[k+1,m].
h i
(cid:0) (cid:1)
Plugging this into (18) gives
min{m,n}
Q(m)P(n)(E) ∼= SkHom∗(α,β)⊗T
α β k
k=nM+m−N
b
where
T := prm∗β⊠n−k ⊗prm∗ prI (E ⊗prI∗ α∨⊠m−k) S[k+1,m].
k J m\J m\J∗ [k+1,m]
J⊂m,M|J|=n−k (cid:2) (cid:3)
b
SYMMETRIC QUOTIENT STACKS AND HEISENBERG ACTIONS 9
Using Remark 2.2, we see that T ∼=P(n−k) Q(m−k)(E). Hence,
k N−m,β N−n,α
mbin{m,n}
Qα(m)Pβ(n)(E) ∼= SkHom∗(α,β)⊗kPN(n−−mk),βQN(m−−nk,α)(E).
k=nM+m−N
Since all the isomorphisms used above are functorial, this shows the isomorphism (7).
3. Further remarks
3.1. The Fock space. TheFock space representation of the Heisenberg algebra is defined as
the quotient F := H /I where I is the left ideal generated by all the a (n) with n > 0. Let
V V β
1 ∈F be the class of 1 ∈ H . The Fock space is an irreducible representation. Hence, given
V V
any representation M of H together with an element 0 6= x ∈ M such that a (n)·x = 0 for
V β
all β ∈ V and n > 0, the map
F → M , 17→ x
V
gives an embedding of H -representations.
V
Let K = K (Xℓ) be the representation described in Section 1.2. Then every non-
ℓ≥0 Sℓ
zero x ∈ K L(X0) ∼= K(point) ∼= Q is annihilated by all the q(n) and hence by the a (n) for
S0 β β
n > 0. Thus, the Fock space can be realised as a subrepresentation of K.
In general, it is a proper subrepresentation, i.e. F ( K. However, if K(X) is of finite
K(X)
dimension and the exterior product K(X)⊗n → K(Xn) is an isomorphism, for example if X
has a cellular decomposition, we do have the equality F = K. In this case
K(X)
K := K (Xℓ)∼= Sν1K(X) ⊗ Sν2K(X) ⊗...
ℓ Sℓ
ν parMtitionofℓ(cid:0) (cid:1) (cid:0) (cid:1)
as follows from the general decomposition of equivariant K-theory describedin [Vis91]. Thus,
the dimensions of F and K agree in every degree ℓ.
K(X)
3.2. Transposed generators. There is yet another set of generators of the Heisenberg al-
(1n) (1n)
gebra, namely p , q defined by
β β
p(1n)zn = exp− aβ(−ℓ)zℓ , q(1n)zn = exp− aβ(ℓ)zℓ ;
β ℓ β ℓ
nX≥0 Xℓ≥1 nX≥0 Xℓ≥1
   
compare [CL12, Sect. 2.2.2]. The relations among these generators are exactly the same as
(n) (n)
those between the p and q . Moreover, one can mix these two sets of generators to get
β β
(n) (1n) (1n) (n)
the set of generators consisting of p and q (or, alternatively, p and q ). These are
β β β β
the generators used by Khovanov in [Kho14] for his construction of a categorification of the
Heisenberg algebra H in the case that V = Q. The relations between these generators are
V
min{m,n}
[q(1m),q(1n)] = 0= [p(m),p(n)] , q(m)p(n) = sk(−hα,βi)·p(n−k)q(m−k).
α β α β α β β α
Xk=0
(1n) (1n)
For β ∈D(X) and n > 0, we set P := P with
β N≥n N,β
L
P(1n): D (XN−n)→ D (XN) , E 7→ InfSN (β⊠n⊗a )⊠E .
N,β SN−n SN Sn×SN−n n
(cid:0) (cid:1)
10 ANDREASKRUG
Here, a denotes the non-trivial character of S . Again, Q(1n) is defined as the right-adjoint
n n β
(1n)
of P . Using the “anti-symmetric Ku¨nneth formula”
β
Hom∗ (β⊠k ⊗a ,α⊠k) ∼= ∧kHom∗(β,α),
DS (Xk) k
k
one can prove in complete analogy to Sections 2.5 and 2.6 the relations
(20) Q(1m)Q(1n) ∼= Q(1n)Q(1m) , P(m)P(n) ∼= P(n)P(m)
α β β α α β β α
min{m,n}
(21) Q(β1m)Pα(n) ∼= ∧kHom∗(β,α)⊗kPα(n−k)Qβ(1m−k).
Mk=0
One can reformulate Lemma 1.3 as the formula χ(∧kW∗) = sk(−χ(W∗)) for W∗ a graded
(n) (1n)
vector space. Thus, relations (20) and (21) show that the P and Q give again a cat-
β β
(n) (1n)
egorical Heisenberg action, this time lifting the generators p and q . The special case
β β
that X is a point is exactly the construction of [Kho14].
(n)
3.3. Generalisations and variants. Since the construction (5) of the functors P is very
β
simple and formal, it generalises well to other settings beside smooth complete varieties.
Let X be a non-complete smooth variety. Then Theorem 1.4 continues to hold if we
assume that the objects α and β have complete support, i.e. α,β ∈ Dc(X), which ensures
that Hom∗(α,β) is a finite dimensional graded vector space. Thus, we get an induced action
of the Heisenberg algebra associated to Kc(X) := K(Dc(X)) on K (note that K needs not
necessarily be replaced by Grothendieck groups with finite support).
Similarly, if one wants to also drop the smoothness assumption one needs α and β to be
perfect objects with complete support.
Furthermore, one can replace the variety X by a Deligne–Mumford stack X. The case that
X = [X/C∗] for X the affine plane or a minimal resolution of a Kleinian singularity gives a
Heisenberg action on the equivariant Grothendieck groups K (Hilbℓ(X)) as in [SV13],
ℓ≥0 C∗
[FT11], and [CL12]. L
3.4. Open problems. In [GK14], the symmetric product SnT of a category T is defined in
such a way that Sn(D(X)) ∼= D (Xn) for a smooth projective variety X. Thus, one may
Sn
hope that our construction can be generalised to a setting where D(X) is replaced by any
(Hom-finite and symmetric monoidal) dg-enhanced triangulated category T.
Note that in [Kho14] and [CL12] a categorification of the Heisenberg action is given in
a much stronger sense than in this paper. In particular, 2-categories whose Grothendieck
groups are isomorphic to the Heisenberg algebra are constructed. Clearly, one would like to
do the same in our more general setting too.
References
[BKR01] Tom Bridgeland, Alastair King, and Miles Reid. The McKay correspondence as an equivalence of
derived categories. J. Amer. Math. Soc., 14(3):535–554 (electronic), 2001.
[Che14] X. Chen. A note on separable functors and monads. arXiv:1403.1332, 2014.
[CL12] Sabin Cautis and AnthonyLicata. Heisenberg categorification and Hilbert schemes. Duke Math. J.,
161(13):2469–2547, 2012.
[Ela14] A. D. Elagin. On equivariant triangulated categories. arXiv:1403.7027, 2014.
[FT11] B. L. Feigin and A. I. Tsymbaliuk. Equivariant K-theory of Hilbert schemes via shuffle algebra.
Kyoto J. Math., 51(4):831–854, 2011.