Betti numbers of holomorphic symplectic quotients
via arithmetic Fourier transform
Tama´s Hausel†
Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom; and Department of Mathematics, University of Texas, Austin, TX 78712 Communicated by Robion C. Kirby, University of California, Berkeley, and approved February 24, 2006 (received for review September 27, 2005) A Fourier transform technique is introduced for counting the
number of solutions of holomorphic moment map equations over a finite field. This technique in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence, simple unified proofs are obtained for formulas of Poincare´ polynomials of toric hyperka¨hler varieties (recovering results of Bielawski–Dancer and Hausel–Sturmfels), Poincare´ polynomials of Hilbert schemes of points and twisted Atiyah–Drinfeld–Hitchin– Manin (ADHM) spaces of instantons on⺓2(recovering results of Nakajima–Yoshioka), and Poincare´ polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable representations of a quiver is announced.
quiver varieties兩 Weyl–Kac character formula
L
et⺛ be a field, which will be either the complex numbers ⺓or the finite field ⺖q in this work. Let G be a reductive
algebraic group over⺛, g its Lie algebra. Consider a
represen-tation : G 3 GL(⺦) of G on a ⺛-vector space ⺦, inducing the
Lie algebra representation | : g 3 gl(⺦). The representation
induces an action : G 3 GL(⺝) on ⺝ ⫽ ⺦ ⫻ ⺦*. The vector
space ⺝ has a natural symplectic structure, defined by the
natural pairing具v, w典 ⫽ w(v), with v 僆 ⺦ and w 僆 ⺦*. With
respect to this symplectic form a moment map : ⺦ ⫻ ⺦* 3 g*,
of is given at X 僆 g by
具共v, w兲, X典 ⫽ 具|共X兲v, w典. [1]
Let now 僆 (g*)Gbe a central element, then the holomorphic
symplectic quotient is defined by the affine geometric invariant theory (GIT) quotient
⺝兾兾兾兾G :⫽ 共⫺1共兲兲兾兾G,
which is the affine algebraic geometric version of the
hyper-ka¨hler quotient construction of ref. 1. In particular our varieties,
in addition to the holomorphic symplectic structure, will carry a
natural hyperka¨hler metric, although the latter will not feature
in what follows.
Our main proposition counts rational points on the varieties
⫺1() over the finite fields ⺖q, where q⫽ pris a prime power.
For convenience we will use the same letters⺦, G, g, ⺝, for
the corresponding vector spaces, groups, Lie algebras, and
matrices over the finite field⺖q. We define the function a|: g 3
⺞ 傺 ⺓ at X 僆 g as
a|共X兲 :⫽ 兩ker共|共X兲兲兩, [2]
where we used the notation兩S兩 for the number of elements in any
set S. In particular a|(X) is always a power of q. For an element
v僆 V of any vector space we define the characteristic function
␦v: V 3⺓ by␦v(x)⫽ 0 unless x ⫽ v when␦v(v)⫽ 1. We can
now formulate our main proposition.
Proposition 1.The number of solutions of the equation(v, w) ⫽
over the finite field ⺖qequals
#兵共v, w兲 僆 ⺝兩共v, w兲 ⫽ 其 ⫽ 兩g兩⫺1/2兩⺦兩F共a |兲共兲 ⫽ 兩g兩⫺1兩⺦兩
冘
X僆g a|共X兲⌿共具X, 典兲.To explain the last two terms in the proposition above, we
need to define Fourier transforms (2) of functions f : g 3⺓ on
the finite Lie algebra g, which here we think of as an abelian
group with its additive structure. To define this fix⌿ : ⺖q3⺓x
a nontrivial additive character, and then we define the Fourier
transform F(f) : g* 3⺓ at a Y 僆 g*
F共 f兲共Y兲 ⫽ 兩g兩⫺1/2
冘
X僆g
f共X兲⌿共具X, Y典兲.
Proof:Using two basic properties of Fourier transform
F共F共 f兲兲共X兲 ⫽ f共⫺X兲, for X僆 g and
冘
w僆V* ⌿共具v, w典兲 ⫽ 兩V兩␦0共v兲, [3] for v僆 V we get #兵共v, w兲 僆 ⺝兩共v, w兲 ⫽ 其 ⫽冘
v僆⺦w僆⺦*冘
␦共共v, w兲兲 ⫽冘
v僆⺦w冘
僆⺦* F共F共␦兲兲共⫺共v, w兲兲 ⫽冘
v僆⺦w冘
僆⺦*X冘
僆g 兩g兩⫺1/2F共␦ 兲共X兲⌿共具X, ⫺共v, w兲典兲 ⫽冘
v僆⺦X冘
僆g 兩g兩⫺1/2F共␦ 兲共X兲冘
w僆⺦* ⌿共⫺具|共X兲v, w典兲 ⫽冘
v僆⺦X冘
僆g 兩g兩⫺1/2F共␦ 兲共X兲兩⺦兩␦0共|共X兲v兲 ⫽冘
X僆g 兩g兩⫺1/2F共␦ 兲共X兲兩⺦兩a|共X兲 ⫽冘
X僆g 兩g兩⫺1兩⺦兩a |共X兲冘
Y僆g* ␦共Y兲⌿共具X, Y典兲Conflict of interest statement: No conflicts declared. Abbreviation: ADHM, Atiyah–Drinfeld–Hitchin–Manin.
†E-mail: [email protected].
⫽ 兩g兩⫺1兩⺦兩
冘
X僆g
a|共X兲⌿共具X,典兲
⫽ 兩g兩⫺1/2兩⺦兩F共a
|兲共兲.
Affine Toric Hyperka¨hler Varieties
We take G⫽ ⺤d⬵ (⺓⫻)da torus. A vector configuration A⫽
(a1, . . . , an) :⺪n3⺪dgives a representationA:⺤d3⺤n傺
GL(⺦), where ⺦ ⬵ ⺓nis an n-dimensional vector space and⺤n
傺 GL(⺦) is a fixed maximal torus. The corresponding map on
the Lie algebras is |A: td3 tn. The holomorphic moment map
of this actionA:⺦ ⫻ ⺦* 3 (td)* is given by Eq. 1, which in this
case takes the explicit form
A共v, w兲 ⫽
冘
i⫽1
n viwiai.
We take a generic 僆 (td)*. The affine toric hyperka¨hler variety
is then defined as the affine geometric invariant theory quotient:
M(, A) ⫽ A⫺1()兾兾⺤d. To use our main result, we need to
determine a|(X). Note that the natural basis e1, . . . , en僆 (tn)*
gives us a collection of hyperplanes H1, . . . , Hnin td. Now for
X僆 tdwe have that a|(X)⫽ qca(X), where ca(X) is the number
of hyperplanes that contain X. Finally, we take the intersection lattice L(A) of this hyperplane arrangement, i.e., the set of all
subspaces of tdthat arise as the intersection of any collection of
our hyperplanes, with partial ordering given by containment.
The generic choice of will ensure that will not be trivial on
any subspace in the lattice L(A). Thus, for any subspace V僆
L(A), we have from Eq. 3 that兺X僆V⌿ (具X,典) ⫽ 0.
Now we can use Proposition 1. If we perform the sum we get a combinatorial expression #共M共, A兲兲 ⫽ q n⫺d 共q ⫺ 1兲d
冘
X僆td a|共X兲⌿共具X,典兲 ⫽ q n⫺d 共q ⫺ 1兲d冘
V僆L共A兲 L共A兲共V兲qca共V兲,whereL(A)is the Mo¨bius function of the partially ordered set
L(A), while ca(V) is the number of coatoms, i.e., hyperplanes
containing V. Because the count above is polynomial in q and the mixed Hodge structure on M(, A) is pure, we get that for the
Poincare´ polynomial we need to take the opposite of the count
polynomial, i.e., substitute q⫽ 1兾t2and multiply by t4(n⫺d). This
way we get:
Theorem 1.The Poincare´ polynomial of the toric hyperka¨hler variety is given by Pt共M共, A兲兲 ⫽ 1 t2d共1 ⫺ t2兲d
冘
V僆L共A兲 L共A兲共V兲共t2兲n⫺ca共V兲.One can prove by a simple deletion contraction argument (and it also follows from the second proof of proposition 6.3.26 of ref.
3) that for any matroid MᏭand its dual MᏮ
1 qd共1 ⫺ q兲d
冘
V僆L共A兲 L共A兲共V兲共q兲n⫺ca共V兲⫽ h共MᏮ兲, where h共MᏮ兲 ⫽冘
i⫽0 n⫺d hi共MᏮ兲qi,is the h-polynomial of the dual matroid MᏮ. This way we recover
a result of refs. 4 and 5; for a more recent arithmetic proof see (6):
Corollary 1.The Poincare´ polynomial of the toric hyperka¨hler variety is given by
Pt共M共, A兲兲 ⫽ h共MB兲共t2兲,
where B is a Gale dual vector configuration of A.
Hilbert Scheme ofn-Points on⺓2and Atiyah–Drinfeld–Hitchin– Manin (ADHM) Spaces
Here G⫽ GL(V), where V is an n-dimensional ⺛ vector space.
We need three types of basic representations of G. The adjoint
representationad: GL(V) 3 GL(gl(V)), the defining
repre-sentationdef⫽ Id : G 3 GL(V), and the trivial representations
trivk ⫽ 1 : G 3 GL(⺛k). Fix k and n. Define⺦ ⫽ gl(V) ⫻ V R
⺛k,⺝ ⫽ ⺦ ⫻ ⺦* and : G 3 GL(⺦) by ⫽
ad⫻defRtrivk .
Then we take the central element ⫽ IdV僆 gl(V) and define
the twisted ADHM space as
M共n, k兲 ⫽ ⺝兾兾兾兾G⫽⫺1共兲兾兾G,
where
共A, B, I, J兲 ⫽ 关A, B兴 ⫹ IJ,
with A, B僆 gl(V), I 僆 Hom(⺛k, V) and J僆 Hom(V, ⺛k).
The space M(n, k) is empty when k ⫽ 0 (the trace of a
commutator is always zero), diffeomorphic with the Hilbert
scheme of n-points on⺓2, when k⫽ 1, and is the twisted version
of the ADHM space (7) of U(k) Yang–Mills instantons of charge
n on⺢4(cf. ref. 8). By our main Proposition 1 the number of
solutions over⺛ ⫽ ⺖qof the equation
关A, B兴 ⫹ IJ ⫽ IdV,
is the Fourier transform on g of the function a|(X) ⫽
兩ker(|(X))兩. First we determine a|(X) for X僆 g ⫽ gl(V). By the
definition of | we have
ker共|共X兲兲 ⫽ ker共|ad共X兲兲 ⫻ ker共|def兲 丢 ⺛k,
and so if a|ad(X)⫽ 兩ker(|ad(X))兩 and a|def⫽ 兩ker(|def)兩, then we have
a|共X兲 ⫽ a|ad共X兲a|defk 共X兲.
Now Proposition 1 gives us
#共M共n, k兲兲 ⫽ 1 兩G兩 #兵共v, w兲 僆 ⺝兩共v, w兲 ⫽ 其 ⫽兩g储G兩兩⺦兩
冘
X僆g a|共X兲⌿共具X, 典兲 ⫽兩g储G兩兩⺦兩冘
X僆ga|ad共X兲a|defk 共X兲⌿共具X, 典兲.
We will perform the sum adjoint orbit by adjoint orbit. The adjoint orbits of gl(n), according to their Jordan normal forms, fall into types, labeled by T(n), which stands for the set of all possible
Jordan normal forms of elements in gl(n). We denote by Treg(t)
the types of the regular (i.e., nonsingular) adjoint orbits, while
Tnil(s)⫽ P(s) denotes the types of the nilpotent adjoint orbits,
which are just given by partitions of s. First we do the k⫽ 0 case
where we know a priori, that the count should be 0, because the commutator of any two matrix is always trace-free and thus
cannot equal (for almost all q). Additionally, if we separate the nilpotent and regular parts of our adjoint orbits we get
0⫽ 1 兩G兩 X
冘
僆ga|ad共X兲⌿共具X, 典兲 ⫽冘
n⫽s⫹t僆T冘
nil共s兲 兩ᑝ兩 兩C兩 僆T冘
reg共t兲 兩ᑝ兩 兩C兩 ⌿共具X, 典兲,where C and, respectively, ᑝ denotes the centralizer of an
element X of g of type in the adjoint representation of G,
respectively, g on g.
So if we define the generating series
⌽nil0 共T兲 ⫽ 1 ⫹
冘
s⫽1 ⬁冘
僆Tnil共s兲 兩ᑝ兩 兩C兩 T s, and ⌽reg共T兲 ⫽ 1 ⫹冘
t⫽1 ⬁冘
僆Treg共t兲 兩ᑝ兩 兩C兩 ⌿共具X, 典兲T t, then we have ⌽nil0 共T兲⌽reg共T兲 ⫽ 1.However,⌽nil0 is easy to calculate (9)
⌽nil 0 共T兲 ⫽
写
i⫽1 ⬁写
j⫽1 ⬁ 1 共1 ⫺ Tiq1⫺j兲, [4] thus we get ⌽reg共T兲 ⫽写
i⫽1 ⬁写
j⫽1 ⬁ 共1 ⫺ Tiq1⫺j兲. [5]Now the general case is easy to deal with:
#共M共n, k兲兲 qnk ⫽ 1 兩G兩 X
冘
僆ga|ad共X兲⌿共具X, 典兲 ⫽冘
n⫽s⫹t僆T冘
nil共s兲 兩ᑝ兩a|defk 共X兲 兩C兩 僆T冘
reg共t兲 兩ᑝ兩 兩C兩 ⌿共具X, 典兲.Thus, if we define the grand generating function by
⌽k共T兲 ⫽ 1 ⫹
冘
n⫽1 ⬁ #共M共n, k兲兲T n qkn, [6] and ⌽nil k 共T兲 ⫽ 1 ⫹冘
s⫽1 ⬁冘
僆Tnil共s兲 兩ᑝ储ker共X兲兩k 兩C兩 T s,then for the latter we get similarly to the argument for Eq. 4 in ref. 9 that ⌽nilk ⫽ ⌽nilk 共T兲 ⫽
写
i⫽1 ⬁写
j⫽1 ⬁ 1 共1 ⫺ Tiqk⫹1⫺j兲.For the grand generating function then we get
⌽k共T兲 ⫽ ⌽ nil k 共T兲⌽ reg共T兲 ⫽
写
i⫽1 ⬁写
l⫽1 k 1 共1 ⫺ Tiql兲.Because the mixed Hodge structure is pure, and this count is
polynomial, we get the compactly supported Poincare´
polyno-mial. To get the ordinary Poincare´ polynomial, we need to
replace q⫽ 1兾t2and multiply the nth term in Eq. 6 by t4kn. This
way we get Theorem 2.
Theorem 2.The generating function of the Poincare´ polynomials of the twisted ADHM spaces are given by
冘
n⫽0 ⬁ Pt共M共k, n兲兲Tn⫽写
i⫽1 ⬁写
b⫽1 k 1 共1 ⫺ t2共k共i⫺1兲⫹b⫺1兲Ti兲.This result appeared as corollary 3.10 in ref. 10.
Quiver Varieties of Nakajima
Here we recall the definition of the affine version of Nakajima’s
quiver varieties (11). Let Q⫽ (V, E) be a quiver, i.e., an oriented
graph on a finite set V⫽ {1, . . . , n} with E 傺 V ⫻ V a finite
set of oriented (perhaps multiple and loop) edges. To each vertex
i of the graph, we associate two finite dimensional ⺛ vector
spaces Viand Wi. We call (v1, . . . , vn, w1, . . . , wn)⫽ (v, w) the
dimension vector, where vi⫽ dim(Vi) and wi ⫽ dim(Wi). To
these data we associate the grand vector space
⺦v,w⫽
丣
共i, j兲僆E Hom共 Vi, Vj兲 丣
丣
i僆VHom共 Vi, Wi兲,
the group and its Lie algebra
Gv⫽
⫻
i僆V GL共 Vi兲 gv⫽丣
i僆V gl共 Vi兲,and the natural representation
v,w: Gv 3 GL(⺦v,w),
with derivative
|v,w: gv 3 gl共⺦v,w兲.
The action is from both left and right on the first term and from the left on the second.
We now have Gvacting on⺝v,w⫽ ⺦v,w⫻ ⺦*v,wpreserving the
symplectic form with moment mapv,w:⺦v,w⫻ ⺦*v,w3 g*vgiven
by Eq. 1. We take now Ev⫽ (IdV1, . . . , IdVn)僆 (g*v)
Gv, and define
the affine Nakajima quiver variety (11) as
M共v, w兲 ⫽v⫺1,w共v兲兾兾Gv.
Here we determine the Betti numbers of M(v, w) using our main
Proposition 1, by calculating the Fourier transform of the
func-tion a|
v,wgiven in Eq. 2.
First, we introduce, for a dimension vector w 僆 V⺞, the
⌽nil共w兲 ⫽
冘
v⫽共v1, . . . , vn兲僆V⺞ i写
僆V Ti vi冘
1僆Tnil共v1兲 · · ·冘
n僆Tnil共vn兲 a| v,w共X1, . . . , Xn兲 兩C1兩· · ·兩Cn兩 ,where Tnil(s) is the set of types of nilpotent s⫻ s matrices, where
a type is given by a partition 僆 P(s) of s, Xdenotes the typical
s⫻ s nilpotent matrix in gl(s) in Jordan form of type, Cis the
centralizer of Xunder the adjoint action of GL(s) on gl(s). We
also introduce the generating function
⌽reg⫽
冘
v⫽共v1, . . . , vn兲僆V⺞写
i僆V Ti vi冘
1僆Treg共v1兲 · · ·冘
n僆Treg共vn兲 a| v,0共X1, . . . , Xn兲 兩C1兩· · ·兩Cn兩 ⌿共具X, v典兲,where Treg(t) is the set of types, i.e., Jordan normal forms, of
a regular t⫻ t matrix Xin gl(t), C傺 GL(t) its centralizer under
the adjoint action. Note also that for a regular element X僆 gv,
a|
v,w(X) ⫽ a|v,0(X) does not depend on w僆 V
⺞.
Now we introduce for w僆 V⺞the grand generating function
⌽共w兲 ⫽
冘
v僆V⺞
#共M共v, w兲兲 兩gv兩
兩⺦v,w兩
Tv. [7]
As in the previous section, our main Proposition 1 implies
⌽共w兲 ⫽ ⌽nil共w兲⌽reg. [8]
Finally, we note that when w⫽ 0 we have |v,0(*v)⫽ 0, where*v⫽
(IdV1, . . . , IdVn)僆 g, thus by Eq. 1 具v,0(v, 0),*v典 ⫽ 0. Because
具v,*v典 ⫽ 兺 vi, the equationv,0(v, w)⫽vhas no solutions (for
almost all q). This way we get that⌽(0) ⫽ 1, and so Eq. 8 yields
⌽reg⫽ 1兾⌽nil(0), giving the result
⌽共w兲 ⫽⌽nil共w兲
⌽nil共0兲
.
Therefore, it is enough to understand⌽nil(w), which reduces to
a simple linear algebra problem of determining a|
v,w(X1, . . . ,
Xn). Putting together everything yields the following:
Theorem 3.Let Q⫽ (V, E) be a quiver, with V ⫽ {1, . . . , n} and
E傺 V ⫻ V, with possibly multiple edges and loops. Fix a dimension
vector w 僆 ⺞V. The Poincare´ polynomials P
t(M(v, w)) of the
corresponding Nakajima quiver varieties are given by the generating function
冘
v僆⺞V Pt共M共v, w兲兲t⫺d共v,w兲Tv ⫽冘
v僆⺞V Tv冘
1僆P共v1兲 · · · n僆P共vn兲冉
共i, j兲僆E写
t⫺2n共i,j兲冊冉
写
i僆V t⫺2n共i,共1wi兲兲冊
写
i僆V冉
t⫺2n共i,i兲写
k写
j⫽1 mk共i兲 共1 ⫺ t2j兲冊
冘
v僆⺞V Tv冘
1僆P共v1兲 · · · n僆P共vn兲写
共i, j兲僆E t⫺2n共i,j兲写
i僆V冉
t⫺2n共i,i兲写
k写
j⫽1 mk共i兲 共1 ⫺ t2j兲冊
, [9]where d(v, w)⫽ 2兺(i, j)僆Evivj⫹ 2兺i僆Vvi(wi⫺ vi) is the dimension of M(v, w), and Tv⫽ 兿
i僆VTi vi
. P(s) stands for the set of partitions¶
of s僆 ⺞. For two partitions ⫽ (1, . . . ,l)僆 P(s) and ⫽
(1, . . . ,m)僆 P(s) we define n(, ) ⫽ 兺i, jmin(i, j), and
if we write ⫽ (1m1(), 2m2(), . . .) 僆 P(s), then we can define
l() ⫽ 兺 mi() ⫽ l the number of parts in . With this notation
n(i, (1wi))⫽ w
il(i) in the above formula.
Remark:This single formula encompasses a surprising amount
of combinatorics and representation theory. When v⫽ (1, . . . ,
1), the Nakajima quiver variety is a toric hyperka¨hler variety,
thus Eq. 9 gives a new formula for its Poincare´ polynomial, which
was given in Corollary 1. If additionally w⫽ (1, 0, . . . , 0) then
M(v, w) is the toric quiver variety of ref. 5. Therefore, its
Poincare´ polynomial, which is the reliability polynomial㛳of the
graph underlying the quiver (5) also can be read off from the above formula 9.
When the quiver is just a single loop on one vertex, our
formula 9 reproduces Theorem 2. When the quiver is of type An
Nakajima (14) showed, that the Poincare´ polynomials of the
quiver variety are related to the combinatorics of Young-tableaux, while in the general ADE case, Lusztig (15)
conjec-tured a formula for the Poincare´ polynomial, in terms of
formulae arising in the representation theory of quantum groups. When the quiver is star-shaped recent work (T.H., unpublished work and T.H., E. Letellier, and F. Rodriguez–
Villegas, unpublished work) calculates these Poincare´
polyno-mials using the character theory of reductive Lie algebras over finite fields (2) and arrives at formulas determined by the Hall–Littlewood symmetric functions (12), which arose as the pure part of Macdonald symmetric polynomials (12). In the case when the quiver has no loops, Nakajima (16) gives a combina-torial algorithm for all Betti numbers of quiver varieties, moti-vated by the representation theory of quantum loop algebras. Finally, through the paper (17) of Crawley-Boevey and Van den
Bergh, Poincare´ polynomials of quiver varieties are related to the
number of absolutely indecomposable representations of quivers in the work of Kac (18), which were eventually completely determined by Hua (19).
In particular, formula 9, when combined with results in refs. 19 and 11 and the Weyl–Kac character formula in the representation theory of Kac–Moody algebras (20), yields a simple proof of conjecture 1 of Kac (18). Consequently, formula 9 can be viewed as a q-deformation of the Weyl–Kac character formula (20).
A detailed study of the above generating function 9, its relationship to the wide variety of examples mentioned above, and details of the proofs of the results of this work will appear elsewhere.
¶The notation for partitions is that of ref. 12.
㛳Incidentally, the reliability polynomial measures the probability of the graph remaining connected when each edge has the same probability of failure, a concept heavily used in the study of reliability of computer networks (13).
I thank Ed Swartz for ref. 3, Fernando Rodriguez-Villegas for ref. 9, Bala´zs Szendro¨i for ref. 10, and Hiraku Nakajima for suggestions regarding Eq. 9. This work was supported by a Royal Society Univer-sity Research Fellowship, National Science Foundation Grant DMS-0305505, an Alfred P. Sloan Research Fellowship, and a Summer Research Assignment of the University of Texas at Austin.
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