Noncommutative complete intersections and matrix integrals

0.5 MB·English

by  Pavel Etingof

NONCOMMUTATIVE COMPLETE INTERSECTIONS
AND MATRIX INTEGRALS
7
0
0
2 PAVEL ETINGOF AND VICTOR GINZBURG

n
a
J To Bob MacPherson on the occasion of his 60th birthday

3
2
Abstract. We introduce a class of noncommutatative algebras called representation

complete intersections (RCI). A graded associative algebra A is said to be RCI pro-
]
G vided there exist arbitrarily large positive integers n such that the scheme Rep A, of
n
n-dimensional representations of A, is a complete intersection. We discuss examples of
A
RCI algebras, includingthose arising from quivers.
.
h Thereisanotherinterestingclassofassociative algebras called noncommutative com-
t plete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also
a
obtainexplicitformulasfortheHilbertseriesofeachnonvanishingcyclicandHochschild
m
homology group of an RCIalgebra. Theproof involvesa noncommutativecyclic Koszul
[ complex, KcqycA,and amatrixintegral similar totheonearising inquivergauge theory.

3 Table of Contents
v
2
7 1. Introduction
2 2. Representation functor and matrix integrals
3 3. Noncommutative complete intersections
0 4. The trace map
6 5. Additional results and examples
0 6. Preprojective algebras and quiver varieties
/
h
t 1. Introduction
a
m 1.1. Inthispaperweworkwithfinitelygeneratedunitalassociative algebrasoverC. Such
: an algebra F is said to be smooth if Ker[F F F], the kernel of the multiplication
v
⊗ →
i map, is a projective F-bimodule.
X
Fix a smooth algebra F, for instance a free algebra F = C x ,...,x . Let J be a
1 d
r h i
a two-sided finitely generated ideal in F.
Definition 1.1.1. The algebra A = F/J (or the pair J F) is called a noncommutative
⊂
complete intersection (NCCI) if J/J2 is a projective A-bimodule. This is equivalent to the
condition that A has a projective A-bimodule resolution of length 2, cf. Theorem 3.1.1.
≤
NCCI algebras have been considered, under various different names, by a number of
authors, see e.g. [AH], [An], [GS], [Go], [Pi]. In the present paper, we are interested in the
case of graded NCCI algebras. Specifically, fix a finite set I and write R for the algebra of
functions I C, with pointwise multiplication. Let Q be a finite quiver with vertex set
→
I, and assign to each edge a Q an arbitrary positive grade degree d(a) > 0. This makes
∈
the path algebra of Q a free graded R-algebra; moreover, any free graded R-algebra F
may be obtained in this way.
1
We study algebras of the form A = F/J, where J is a graded two-sided ideal in the
path algebra F such that J/J2 is a projective finitely generated A-bimodule. Thus, A =
A[r] is a graded algebra such that A[0] = R.
r≥0
The grading on A induces a natural grading on each Hochschild homology group
L
HH (A). Our main results (Theorem 2.2.7 and Theorem 3.7.7) give explicit formulas
k
for the Hilbert series of each Hochschild homology group of A for a class of algebras A
that we call asymptotic representation complete intersections (see below).
For k = 0, for instance, Hochschild homology reduces to the commutator quotient
space, HH (A) = A/[A,A], where [A,A] denotes the C-linear subspace of A spanned by
0
the commutators ab ba, a,b A. The grading on A clearly descends to the commutator
− ∈
quotient. Our result yields a formula, in terms of an infinite product of determinants, for
the Hilbert series of the graded algebra
(A) = Sym A [A,A] . (1.1.2)
C +
O
Here, A+ = r>0A[r], so A+/[A,A] is the p(cid:0)ositi(cid:14)ve degr(cid:1)ee part of A/[A,A]. We remark
that the symmetric algebra in (1.1.2) is equipped with the total grading that comes from
L
the grading on A /[A,A] (this grading should not be confused with the standard grading
+
Sym= Symk, on the Symmetric algebra of a vector space).
k≥0
L
1.2. The main idea of our approach is based on the notion of representation functor and
may be outlined as follows.
Let A be any graded finitely presented algebra. For each positive dimension d, one has
an affine scheme RepdAof d-dimensional representations of A, cf. 2.1. Heuristically, one
§
expects that the collection of representation schemes RepdA provides, as d , a ‘good
→ ∞
approximation’ of the algebra A (this is not true for nongraded algebras which may have
no finite dimensional representations at all). In particular, one should be able to read off
a lot of information about A from some geometric information about the corresponding
representation schemes.
In more detail, for each d, the general linear group GLd acts naturally on the scheme
RepdAbybasechangetransformationsoftheunderlyingvectorspaceofthed-dimensional
representation. WeconsiderthealgebraC[RepdA]GLd,ofGLd-invariantpolynomialfunc-
tions on RepdA. We show that, in some sense, the commutative algebra (A), in (1.1.2),
is a limit of the algebras C[RepdA]GLd as d goes to infinity. A preciseOversion of this
statement will be proved in 4. The statement implies, in particular, that the Hilbert
series of (A) may be obtain§ed as a limit of Hilbert series of the algebras C[RepdA]GLd
O
as d goes to infinity.
Observe further that the presentation A = F/J makes RepdA a closed subscheme in
RepdF, the latter being a vector space. We introduce a noncommutative cyclic Koszul
complex, KcqycA, such that each of the functors Repd, d ZI, sends KcqycA to the GLd-
∈
fixed part of the standard Koszul complex of the subschemeRepdA RepdF. Moreover,
⊂
we prove that the two complexes become ‘asymptotically isomorphic’ as d .
→ ∞
To be able to exploit usefully the ‘asymptotic isomorphisms’ above we introduce the
notion of representation complete intersection (RCI) algebra, and a weaker (but more use-
ful) notion of asymptotic RCI algebra. The algebra A = F/J is called RCI provided there
exist arbitrarily large positive dimensions d such that the scheme RepdA is a complete
2
intersection in RepdF. We will prove:
RCI asymptotic RCI NCCI with ‘expected’ HH (A) NCCI. (1.2.1)
2
⇒ ⇔ ⇒
It turns out that, for an asymptotic RCI algebra A, there is enough asymptotic infor-
mation about Koszul complexes of the schemes RepdA to deduce formulas for the Hilbert
series of each Hochschild homology group of A.
1.3. Quiver algebras. Our results apply, in particular, in an interesting special case
arising from quivers. Specifically, let Q be a (nonoriented) graph with vertex set I. Let Q
be the quiver obtained by doubling edges of Q so that each nonoriented edge of Q gives
rise to a pair, a,a∗, of oriented edges of Q equipped with opposite orientations. Write
c = c for the adjacency matrix of Q, thus c denotes the number of edges i j in Q.
ij ij
k k →
We take F =F(Q) to be the path algebra of Q. Let
Π = F(Q) [a,a∗]
a∈Q
.(cid:0)X (cid:1)
be the preprojective algebra associated to Q, a quotient of F(Q) by the two-sided ideal
generated by the element (a a∗ a∗ a) F(Q). The grading on the path algebra
a∈Q · − · ∈
induces natural gradings on Π and on Π/[Π,Π].
P
Write 1 for the identity I I-matrix. Thus, 1 tc+t21 is an I I-matrix with entries
in Z[t]. This matrix speciali×zes at t = 1 to the −Car·tan m·atrix of ×Q. The closely related
matrix 1(1 t c+t2 1) = (t+t−1) 1 c, called t-analogue of the Cartan matrix, has
t − · · · −
been considered by Kostant and Lusztig, see [Lu, 3.12-3.13].
§
Applying our general theorem to the algebra Π yields the following result.
Theorem 1.3.1. Let Q be neither Dynkin nor extended Dynkin quiver. Then, Π= Π(Q)
is an RCI algebra, and the Hilbert series of Π, resp. of Π/[Π,Π], is determined from the
formula
1 1
h(Π;t) = (1 t c+t2 1)−1, resp. h (Π);t = .
− · · O 1 t2 det(1 tr c+t2r 1)
− r≥1 − · ·
(cid:0) (cid:1) Y
Here, the formula for h(Π,t) has been known for some time, see e.g. [MOV] and
references therein, while the formula for h (Π);t is new.
O
Recallthat,accordingtoVandenBergh[VdB],therearenaturalisomorphismsHHq(Π)
q (cid:0) (cid:1)
= HH2− (Π)relatingHochschildhomology andcohomology. Theseisomorphismsandthe
∼
exact sequence (2.2.8), see Theorem 2.2.7 below, yield formulas for Hilbert series of all
Hochschild cohomology groups of the algebra Π. In particular, this gives an alternate
proof of the result that, for any quiver of neither Dynkin nor extended Dynkin type, the
center of the algebra Π reduces to scalars, cf. [CBEG, Proposition 8.2.2].
Both formulas of Theorem 1.3.1 fail for A,D,E Dynkin quivers.
Now, let Q be an extended Dynkin quiver with extending vertex o I. Then, the
∈
formula for h(Π;t) given in Theorem 1.3.1 still holds, but the formula for h (Π);t has
O
to be modified, see Corollary 6.3.3 in 6.3, and [Su]. In particular, that modified formula
§ (cid:0) (cid:1)
implies a curious identity for the adjacency matrix of Q which we were unable to find in
the literature.
3
The identity involves Chebyshev polynomials of the first kind. Specifically, put ϕ := 1
0
andletϕ C[x], k = 1,2,...,beasequenceofpolynomialsinanindeterminatexdefined
k
∈
as the coefficients in the expansion of the generating function
∞
2 tx
− = 1+ ϕ (x)tk, thus ϕ (2cosz) := 2cos(kz), k 1. (1.3.2)
1 tx+t2 k k ∀ ≥
− k=1
X
For each k 0, we may plug the adjacency matrix c, of Q, into the polynomial ϕ .
k
≥
This way, we obtain an integer valued I I-matrix ϕ (c) = ϕ (c) . The matrix
k k ij
× k k
entries corresponding to the extended vertex i = j = o form a sequence of integers
ϕ (c) , k = 0,1,.... Our identity reads
k oo
det(1 tr c+t2r 1) = (1 tk)ϕk(c)oo. (1.3.3)
− · · −
r≥1 k≥1
Y Y
This identity will be proved in 6.3 by applying the Euler-Poincar´e principle to KcqycΠ,
§
the noncommutative cyclic Koszul complex of the preprojective algebra Π. One can also
verify the identity case by case, for each extended Dynkin quiver of A,D,E types sepa-
rately, see 6.3 for details.
§
1.4. Layoutofthepaper. InSection2,weformulateourfirstimportantresult(Theorem
2.2.7) and carry out crucial matrix integral calculations. The notion of RCI algebra is
introduced in 2.2 and the notion of asymptotic RCI algebra is introduced in 2.4. We
§ §
use the standard Koszul complex associated with a complete intersection to express the
Hilbert series of the coordinate ring C[RepdA] in terms of a matrix integral, an integral
overtheunitarygroupUd. Then,in 2.4,weapplysomeresultsfromthetheoryofrandom
§
matrices to find the asymptotics of our matrix integral as d . For an asymptotic
→ ∞
RCI algebra, that enables us to obtain an asymptotic formula for the Hilbert series of the
algebras C[RepdA] as d goes to infinity.
In 3 we discussNCCI algebras and related homological algebra. Similarly to thefamil-
§
iar case of commutative algebras, to any noncommutative algebra A given by generators
andrelations onecanassociate canonically afreeDGalgebraKqA,called the(noncommu-
tative) Koszul complex of A. The noncommutative Koszul complex is defined in 3.2. It
§
turnsoutthatAisanNCCIalgebraifandonlyiftheDGalgebraKqAisquasi-isomorphic
to A. Main results about NCCI algebras are gathered in Theorems 3.1.1 and 3.2.4.
In 3.7, we introduce the cyclic Koszul complex KcqycA that computes cyclic homology
§
of the NCCI algebra A. We prove that these vanish in all degrees > 1. We formulate the
mainresultofthepaper,Theorem3.7.7,whichclaims,amongotherthings,theequivalence
in the middle of (1.2.1).
The proofs of the two main theorems are completed in 4. In that section, we compare
the algebras A and C[RepdA] via natural evaluation ma§ps. In particular, we construct
an evaluation morphism from the cyclic Koszul complex KcqycA to the ordinary Koszul
complex of the representation scheme RepdA, that commutes with the respective Koszul
differentials. We prove that, in the d limit, evaluation morphisms become iso-
→ ∞
morphisms. Thus, homology vanishing for the noncommutative Koszul complex KcqycA is
4
equivalent to an asymptotic homology vanishingfor the ordinaryKoszulcomplexes of rep-
resentation schemes RepdA. Furthermore theHilbert series of KcqycA may beexpressed in
terms of Hilbert series of the corresponding Koszul complexes of representation schemes.
In 5, we consider various examples of RCI and asymptotic RCI algebras. We also
§
provide examples of non RCI algebras which, nevertheless, are asymptotic RCI, see 5.4.
§
Specifically, we show that if the relations of A are pairwise non-overlapping monomials,
then A is an asymptotic RCI but not necessarily RCI. We use a deformation argument
to deduce a similar result for algebras with generic homogeneous relations of degrees
equal to the lengths of these non-overlapping monomials. This way, one obtains many
nontrivial examples of asymptotic RCI algebras with explicit infinite productformulas for
the corresponding Hilbert series.
In 6, we show that, for connected quivers of neither Dynkin nor extended Dynkin
§
type, the corresponding preprojective algebra is RCI. Also, partial preprojective algebras
of any connected quiver (in the sense of [EE]), are RCI algebras. In 6.2, we give an
§
interesting asymptotic formula for Hilbert series of Nakajima quiver varieties. Extended
Dynkin quivers are discussed in 6.3, where we prove the identity in (1.3.3).
§
Remark 1.4.1. In the present paper, the ground field is the field C of complex numbers.
All the results of the paper can be routinely generalized to the setting of an arbitrary
algebraically closed ground field of characteristic zero. Characteristic zero assumption is
essential for the geometric results involving representation schemes. It is also essential
for the cyclic/Hochschild homology computation in Lemma 3.6.1 and Proposition 3.7.1,
which involves Karoubi-de Rham complex. However, the rest of 3 applies verbatim in
§
the case of ground fields of arbitrary characteristic.
1.5. Ackmowledgments. TheauthorsaregratefultoM.Artin,L.Hesselholt,G.Lusztig,H.Nakajima,
D.Piontkovsky,E.Rains,andR.Stanleyforusefulcommentsanddiscussions. WealsothankE.Rainsfor
explainingtoustheresultsofDiaconisandShahshahanionmatrixintegrals,andforprovidingaMAGMA
code that we used at all stages of this work. The work of P.E. and V.G. was partially supported by the
NSFgrants DMS-0504847 and DMS-0303465, respectively, and by theCRDFgrant RM1-2545-MO-03.
2. Representation functor and matrix integrals
2.1. Basic definitions. Throughout, we fix a finite set I and let R be the algebra of
functions I C, with pointwise multiplication. Thus, R is a commutative semisimple
C-algebra. S→imple R-modules are 1-dimensional and are parametrized by elements of the
set I; an R-module is the same thing as an I-graded vector space, M = M . If M is
i∈I i
⊕
finite-dimensional and d = dimM , we say that d = d , an I-tuple of nonnegative
i i i i∈I
{ }
integers, is the dimension vector for M.
Similarly, R-bimodules are I I-graded vector spaces, M = M .
i,j∈I ij
Let M = M[r] be a Z×-graded R-bimodule such that⊕each graded component
r≥0 +
M[r] is a finite dimensional R-bimodule, that is we have M[r] = M [r], where
L i,j∈I ij
dimM [r] < . The Hilbert series of M is a Z[[t]]-valued I I-matrix defined as
ij
∞ × L
h(M) = h (M;t) , h (M;t) := dimM [r] tr.
ij ij ij
k k r≥0 ·
5 X
It is clear that h(M M′)= h(M) h(M′) for any Z -graded R-bimodules M and M′.
R +
Given a dimension⊗vector d, let Cd·:= Cdi. This is an I-graded vector space, and
i∈I
we consider an associative algebra
L
Ed := HomC(Cd,Cd) = Eij, where Eij = HomC(Cdi,Cdj).
i,j∈I
M
The I-grading makes the vector space Cd a left R-module, and the action map for this
module gives a natural algebra homomorphism R → i∈I Eii ⊂ Ed.
Notation 2.1.1. We use unadorned symbols like Sym,LΛ,Hom, , etc., for symmetric alge-
bra, exterior algebra, Hom-space, tensor product, etc., all take⊗n over C (not over R).
Given a finite dimensional R-bimodule M, we reserve the corresponding blackboard
font notation for the vector space
Md := HomR-bimod(M,Ed). (2.1.2)
The exceptions to this convention are: N,Z ,Z, and C, the sets of natural numbers,
+
nonnegative integers, integers, and complex numbers, respectively. ♦
An N-grading on an R-bimodule M gives rise to a linear C×-action on the vector space
Md. This action contracts the vector space to its origin, the only C×-fixed point.
LetF beanR-algebra, thatis,afinitely presentedassociativeunitalC-algebraequipped
with an algebra imbedding R ֒ F. Given a dimension vector d, let RepdF be the set of
all algebra maps F Ed := H→omC(Cd,Cd) which restrict to the natural map R Ed.
→ →
The set RepdF has the natural structure of a (not necessarily reduced) affine scheme
of finite type, called the representation scheme of d-dimensional representations of the
algebra F. Let C[RepdF] denote the coordinate ring of this scheme.
2.2. Main result. Let T V = R V V V ..., denote the tensor algebra of an
R R
R-bimodule V, and write T+V = V V ⊗V ..., for the augmentation ideal of T V.
R L L⊗R L R
Definition 2.2.1. A pair of finite dLimensionalLN-graded R-bimodules V and L, together
with a graded R-bimodule imbedding j :L ֒ T+V, will be referred to as (V,L)-datum.
→ R
Fix a (V,L)-datum, where V = V[r], and L = L[r]. We define
r≥1 r≥1
L◦ :=Lj−1 [T V,T V] LL. (2.2.2)
R R
⊂
Thus, L◦ = L◦[r] is a graded subsp(cid:0)ace in L. W(cid:1)e put m = dimL◦[r], and let
r≥1 r
L λ(L◦) := (1 tr)mr Z[t]. (2.2.3)
r>0 − ∈
Y
We also introduce the following generating ζ-function
∞
1
ζ(V,L) := Z[[t]]. (2.2.4)
det 1 h(V;ts)+h(L;ts) ∈
s=1 −
Y
The gradings on V,L, and L◦, give(cid:0)rise to total gradings on(cid:1)various objects like exterior
algebra ΛL◦, tensor algebra T V, etc. These total gradings will becalled weight gradings.
R
6
For the corresponding bigraded algebras, we write e.g., ΛL◦ = (ΛpL◦)[r]. With
p,r≥0
this notation, for the polynomial in (2.2.3), one has
L
λ(L◦) = ( 1)p tr dim((ΛpL◦)[r]).
p,r − · ·
Associated with a (V,L)-datuXm, is the two-sided ideal J := (L) T V generated
R
⊂
by j(L), and the corresponding quotient algebra A := T V/J = T V/(L). The weight
R R
grading on T V makes J a graded ideal and A a graded algebra, A = A[r]. For
R r≥0
each r 0, the homogeneous component A[r] is a finite dimensional R-bimodule; we have
≥ L
A[0] = R.
Given an integer N and a dimension vector d, we write d N if d > N for all i I.
i
≻ ∈
Definition 2.2.5. A (V,L)-datum is called representation complete intersection (RCI) if
for any integer N there exists a dimension vector d N such that, cf. (2.1.2):
≻
dimRepdA= dimVd dimLd+dimL◦. (2.2.6)
−
The geometric meaning of equation (2.2.6) will be explained in 2.3 below.
§
Abusing terminology, we often say that the algebra A = T V/(L) arising from the
R
(V,L)-datum is an RCI algebra.
One of our main results reads
Theorem 2.2.7. Let (V,L) be an RCI datum and A = T V/(L). Then, we have
R
HH (A) = 0 for all k > 2 (i.e., A is a NCCI algebra, cf. 3.1), and HH (A) = L◦.
k § 2 ∼
Furthermore, there is a natural exact sequence
0 R HH (A) HH (A) HH (A) 0. (2.2.8)
0 1 2
−→ −→ −→ −→ −→
The Hilbert series of A, resp. of HH (A) = A/[A,A], is determined by the formula
0
(i) h(A) = 1 h(V)+h(L) −1, resp., (ii) h( (A)) = ζ(V,L)/λ(L◦). (2.2.9)
− O
Remark 2.2.10. Onemay call p(V,L;t) := 1 h(V;t)+h(L;t), a Z[t]-valued I I-matrix,
(cid:0) (cid:1)
− ×
the Cartan polynomial of the (V,L)-datum. Thus, formulas (2.2.9) read
∞
1 1
h(A) = p(V,L;t)−1, resp., h( (A)) = .
O λ(L◦)· detp(V,L;ts)
s=1
Y
Remark 2.2.11. It follows from the isomorphism L◦ = HH (A) and from (2.2.8) that one
∼ 2
has h(HH (A)) = h A [A,A] +h(L◦). Therefore, the Theorem implies also that the
1 +
Hilbert series of HH (A) may be found from the formula
1
(cid:0) (cid:14) (cid:1)
h(SymHH (A)) = ζ(V,L)/λ(L◦)2.
1
The proof of Theorem 2.2.7 will be completed in 4. We provide, in effect, two different
§
proofs of the Theorem. The first proof is based on a matrix integral calculation, to be
carried out in 2.3-2.4 below. Thesecond, purely algebraic proof, is based on the formula
§§
for h( (A)) in the special case of a free R-algebra A. Such a formula is known, it can
O
be obtained by a combinatorial argument that involves counting cyclic paths on a graph,
cf. [St]. The general case of Theorem 2.2.7 for an arbitrary RCI algebra A can then be
deduced from the special case of free R-algebras by applying the formula proved in the
special case to the noncommutative Koszul complex KqA, which is free as an R-algebra,
cf. 3.2.
§
7
2.3. Hilbert series of representation schemes. Fixadimensionvector danda(V,L)-
datum. Clearly, one has a natural isomorphism Repd(TRV) ∼= HomR-bimod(V,Ed) = Vd,
cf. (2.1.2).
Now, let A = T V/(L). The algebra projection T V ։ A induces a natural closed
R R
imbedding of representation schemes RepdA ֒ Repd(TRV). Dually, the R-bimodule
→
imbedding j : L ֒ T V induces a natural restriction morphism
R
→
jd∗ : Vd = HomR-bimod(V,Ed)∼= Repd(TRV) −→ Ld. (2.3.1)
It is clear that the scheme RepdA is the scheme-theoretic zero fiber of the morphism jd∗.
In general, the map j∗ is not surjective. Indeed, for any algebra homomorphism ρ :
d
TRV Ed and any x [TRV,TRV], we have Trρ(x) = 0. In particular, the composite
→ ∈
map
L◦ ֒j [TRV,TRV] ֒ TRV ρ Ed Tr C
→ → −→ −→
must vanish. Therefore, the image of the map j∗ in (2.3.1) is contained in a proper
d
subspace L♯d := Kerψ Ld, the kernel of the following composite map
⊂
L♯d := Ker ψ : Ld resLL◦// // Hom(L◦,Ed) Tr|Ed// // Hom(L◦,C) = (L◦)∗ . (2.3.2)
Write pd : Rep(cid:2)d(TRV) L♯d for the resulting map and let o denote th(cid:3)e origin of the
→
vector space L♯. This way, we get the following commutative diagram
d
Repo(cid:15)(cid:15)d(cid:31)A(cid:127) (cid:31)(cid:127) // Vd = R// Lep(cid:15)(cid:15)♯dpd(cid:31)d(cid:127)(TRRRVRR)RRRRRjdR∗RRRRRRR//)) Ld (2.3.3)
{ }
The left square in the diagram is cartesian, so RepdA is the scheme-theoretic zero fiber
of the morphism pd. The following result provides a geometric meaning for the notion of
representation complete intersection datum.
Lemma 2.3.4. The dimension equality (2.2.6) holds for a dimension vector d if and only
if the map pd, in diagram (3.3.4), is flat.
To prove the Lemma, we observe that the gradings on V and L make each of the spaces
in diagram (2.3.3) a C×-variety, such that all maps in the diagram become C×-equivariant
morphisms. Moreover, the C×-action on L♯ being a contraction, we deduce that the
d
dimension of any fiber of the map pd is less than or equal to the dimension of the zero
fiber, that is to dimRepdA. Therefore, the morphism pd is flat if and only if one has
dimRepdA= dimVd dimL♯d.
−
Observe next that the map ψ in (2.3.2) is surjective. Hence, one has an exact sequence
0 L♯d Ld ψ (L◦)∗ 0. (2.3.5)
−→ −→ −→ −→
We deduce: dimL♯d = dimLd dimL◦, hence, dimVd dimL♯d = dimVd dimLd +
− − −
dimL◦, and the Lemma follows. (cid:3)
8
Given i I and an element g GL(Cdi), we write g∨ = (g⊤)−1 : (Cdi)∗ (Cdi)∗,
∈ i ∈ i i →
whereg⊤ denotesthedualendomorphismofthedualvectorspace. Foreachpairofvertices
i
i,j I, we consider endomorphisms of the vector space (Cdi)∗ Cdj = Hom(Cdi,Cdj) =
E ∈of the form g∨ g . Thus, given g GL(Cdi) and g ⊗GL(Cdj), for any integer
ij i ⊗ j i ∈ j ∈
r 0, there is a well defined polynomial
≥
det(1Eij −tr ·gi∨⊗gj)∈ C[t].
We putGd = GL(Cdi). This is a reductive group that acts naturally on the vector
i∈I
space Cd = Cdi, hence, also on the algebra Ed, by conjugation.
i∈IQ
Next, for each d, we fix a maximal compact subgroup U(d) GL(Cd). The group
L ⊂
Ud := i∈IU(di) is a maximal compact subgroup of Gd. Let dg be the Haar measure on
Ud normalized so that the total volume of Ud equals 1.
Q
We introduce the following matrix integral
dg
I(V,L,d) := . (2.3.6)
ZUd rY>0iY,j∈I det(1Eij −tr ·gi∨⊗gj)crij(V,L)
Here, cr (V,L) := dimV [r] dimL [r], and the above expression is viewed as a formal
ij ij − ij
power series I(V,L,d) = I (V,L,d)tm C[[t]] given by the power series expansion
m≥0 m · ∈
of the RHS of (2.3.6).
P
Conjugation-action of the group Gd on Ed induces linear Gd-actions on the vector
spaces Vd,Ld, etc. Furthermore, all the maps in diagram (3.3.4) are Gd-equivariant
morphisms. In particular, the scheme RepdA is a Gd-stable subscheme of Vd, so the
group Gd acts on the coordinate ring C[RepdA] by algebra automorphisms.
We also have the C×-action on the scheme RepdA Vd that commutes with the Gd-
action. The induced C×-action on the subalgebra of G⊂d-invariants gives rise to a grading
C[RepdA]Gd = C[RepdA]Gd[r], (2.3.7)
r≥0
to be referred to as weight grading. M
Proposition 2.3.8. Let A = T V be the graded algebra associated to a (V,L)-datum.
R
Assume that the map pd in (2.3.3) is flat, so the scheme RepdA is a complete intersection
in Vd. Then, the Hilbert series of the graded algebra C[RepdA]Gd is given by
h(C[RepdA]Gd) = I(V,L,d)/λ(L◦).
Proof. Associated with the zero fiber of the map pd, see (2.3.3), is the standard Koszul
complex with differential d of degree 1:
K
−
q
Kq(RepdA) = C[Vd] Λ (L♯d)∗, such that (2.3.9)
⊗
C[RepdA] = Coker[dK :K1(RepdA) K0(RepdA)].
→
For each p 0, the C×-action on Vd and on L♯d gives an additional weight grading
≥
on the vector space Kp(RepdA). The Koszul differential respects the weight grading.
Writing [r] to denote weight r homogeneous component, we get a bigraded direct sum
decomposition
Kq(RepdA)= Kp(RepdA)[r]. (2.3.10)
p,r≥0
M9
Observe next that all the maps in (2.3.5) are Gd-equivariant, provided one treats (L◦)∗
as a vector space with the trivial Gd-action. Therefore, dualizing (2.3.5) yields a Gd-
equivariant short exact sequence
0 L◦ ψ⊤ L∗ (L♯)∗ 0. (2.3.11)
d d
−→ −→ −→ −→
Thus, one can rewrite the Koszul complex in the form
q
Kq(RepdA) = C[Vd] Λ (L∗d/L◦). (2.3.12)
⊗
Assume now that the map pd in (3.3.4) is flat so that the scheme RepdA is a complete
intersection. Then, the Koszul complex provides a graded, Gd-equivariant DG algebra
resolution of the coordinate ring C[RepdA]. It follows, since the group Gd is reductive,
that the Gd-fixed part of the Koszul complex provides a graded DG algebra resolution of
C[RepdA]Gd, the subalgebra of Gd-invariants in C[RepdA]. Thus, by the Euler-Poincar´e
principle, the Hilbert series of C[RepdA]Gd may be expressed in terms of Hilbert series of
the graded spaces Kp(RepdA)Gd = r≥0Kp(RepdA)Gd[r] as follows
h(C[RepdA]Gd) = L ( 1)p h Kp(RepdA)Gd (2.3.13)
p≥0 − ·
= X ( 1)p h(cid:0) C[Vd] Λp(L(cid:1)∗d/L◦) Gd .
p≥0 − · ⊗
To compute the last expression,Xrecall that for(cid:16)a(cid:0)ny two linear maps,(cid:1)u :(cid:17)M M and
→
v :N N, of finite dimensional vector spaces, one has
→
det(1 t u)
( 1)p tm Tr(v uSymmN⊗ΛpM) = − · .
− · · ⊗ | det(1 t v)
p,m≥0 − ·
X
Now, for any p 0, we have a direct sum decomposition with respect to the weight
≥
grading
C[Vd] Λp(L∗d/L◦) = C[Vd] Λp(L∗d/L◦) [r]. (2.3.14)
⊗ r≥0 ⊗
Thus, using the above formula and theMnotati(cid:0)on cr = dimV [r](cid:1) dimL [r], for any
ij ij − ij
g Gd, we find
∈
( 1)p tr Tr g (2.3.15)
− · · |(C[Vd]⊗Λp(L∗d/L◦))[r]
r,p≥0
(cid:16) (cid:17)
P
= 1 det(1−trg|Eij)dimLij[r] = 1 1 .
r>0 (1−tr)mr · i,j∈I det(1−trg|Eij)dimVij[r]! λ(L◦)iY,j∈I det(1−trg|Eij)crij
Q Q
Recall that, for any finite dimensional Gd-module M, the dimension of the Gd-fixed
point subspace is given by the integral dimMGd = Tr(g )dg. We apply this to
Ud |M
each direct summand of the graded Gd-module in the right-hand side of (2.3.14). We
R
conclude that the Hilbert series of the subcomplex of Gd-invariants from the last line
in (2.3.13) is obtained by averaging the expression in (2.3.15) over Ud, which gives the
desired formula. (cid:3)
10