数理解析研究所講究録 1328 巻 2003 年 144-158

144
AN INTRODUCTION TO THE SPECIAL MCKAY

CORRESPONDENCE

東京都立大学

由佳理 (YUKARI ITO )伊藤TOKYO METROPOLITAN UNIVERSITY

1. INTRODUCTION

This note is based on the paper uSpecid McKay correspondence[9] by the author and weshow you more concrete exmaple to see
the structures.
Let us consider an example of 2-dimensional quotient singularity
which is obtained ffom the action of acyclic groupFirst, wethe toric resolution of cyclic quotient singularities
because the quotient spaceis atoric.
Let
be the 2-dimensional real vector space, $\{e^{}|i=1,2\}$ its standard base, the lattice generated by and , $N:=L+ \sum \mathbb{Z}v$ , wheretheruns over an the elements $=1/r(1, a)\in G=C_{r,a}$ ,and$n$$\mathrm{w}\mathrm{i}\mathrm{U}$$\mathrm{r}\mathrm{m}\mathrm{l}$$\mathbb{C}^{2}/G$$\mathrm{v}\mathrm{n}\cdot \mathrm{e}\mathrm{w}$$\mathrm{R}^{2}$$L$$e^{1}$$\mathrm{s}\ovalbox{\tt\small REJECT} \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$e^{2}$$v$$\sigma:=\{.\cdot\sum_{=1}^{2}x:e^{:}\in \mathrm{R}^{2}$,$x\dot{.}\geq 0,\forall i$, $1\leq i\leq 2\}$the naturally defined rational convex polyhedral cone inR.
The corresponding affine torus embedding
is defined as Spae(C[\sigma \check \cap$N$$M$$M])$ , whereis the dual lattice of and the dual cone of in,Vxdefined as{}.
Then $X=\mathbb{C}^{2}/G$ corresponds to the toric variety which is inducedby the cone within the lattice $N$ .
Fact 1We can construct asimplicial decomposition $S$ with theverteces on the Newton Boundary, that is, the convex hull of the lattice
points in except origin.
$N_{\mathrm{B}}=N\otimes_{\mathrm{Z}}$$\mathrm{Y}_{\sigma}$$\check{\sigma}$$\check{\sigma}:=$$\xi\in M_{\mathrm{B}}|\xi(x)\geq 0$$\sigma$$M_{\mathrm{B}}$$\in\sigma$$\sigma$$\sigma$Fact 2Ifis the corresponding torus embedding, then$=\pi_{S}$ :is non-singular. Thus, we obtain the inimal resolution$\tilde{X}=X_{S}-\mathbb{C}^{2}/G=\mathrm{Y}$.
Moreover, each lattice point of the Newton
boundary corresponds to an exceptional divisor.
$\tilde{X}:=X_{S}$$X_{S}$$\pi$

for S(お一迂 k$\mathrm{R}$囮1出Example Let us look at the example of the cyclic quotient singularity of type$\epsilon^{7}=1$$C_{7,3}$

which is generated by the matrix$(\begin{array}{ll}\epsilon 00 \epsilon^{3}\end{array})$

where. The toric resolution of this quotient singularity is given by thetriangulation of alattice
See Figure 1.1.

$N:= \mathrm{Z}^{2}+\frac{1}{7}(1,3)\mathrm{Z}$with the lattice points:

FIGURE 1.1. toric resolution of$\mathrm{C}^{2}/G$

From this Newton polytope, we can see that there are 3exceputional
divisors and the dual graph gives the configuration of the exceptional
components with adeformed coordinate from the original coordinateonas in Figure 1.2.
Therefore we have 4affine pieces in this example and we have 4$\mathrm{C}^{2}$$(\mathrm{x}7\mathrm{y})$coordinate systems corresponding to each affine piece.
In any example of the original McKay correspondence, ffi of nontrivial irreducible representations appear as the corresponding
sentations to the exceptional divisors. However, in this case, there
are only three exceptional curves though there are six non-irreducible
representations. Thereofore we must find the three “special” representations among the six representations, if there exists the generalzed
McKay correspondence.
At the end of this note, we can find the corresponding special
ducible representations, and you will see two ways to find them. The
first original definition will be written in the section 3, and you
beable to see more easy way finffiy.
This paper is organized as follows:
the next section, we
give
ni]l
abrief history of the McKay correspondence and we
discuss the
special representations and the generalized McKay correspondence in$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\triangleright$$\mathrm{i}\mathrm{r}\mathrm{r}*$$\mathrm{w}\mathrm{i}\mathbb{I}$$\mathrm{h}$$\mathrm{w}\mathrm{i}\mathbb{I}$

145

146

$\mathrm{x}^{7}$

FIGURE 1.2. configuration of$\tilde{X}$the following section.
section four, we treat -Hilbert schemes as a
resolution of singularities, consider the relation with the toric resolution
in the cyclic case, and show how to find the special representations by
combinatorics. In the finfi section, we will compare the difiiculties of
the caiiculations of the special representations between the original way
and this new combinatorial way with an example.
$\mathrm{h}$$G$

2. MCKAY CORRESpONDEN(J147

On the other hand, we have the set of the irreducible representationsof the groupup to isomorphism and let be the natural
sentation in. The tensor product of these representations$\rho_{i}$$G$$\rho$$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\triangleright$$SL(2,\mathbb{C})$$\rho\dot{.}\otimes\rho=\Sigma_{j=0\mathrm{k}j\sqrt j}^{r}‘\cdot$,(1) (ItO-Reid [12]) cohomology groupconjugacyclasses of “age” in $G$}.
(2) (ItO-Nakajima [lO])Grothendie&groupirreducible$G$
representations of }, where $G$ is afinitegroup.
(3) (Bridgeland-King-Reid [3]) Derived category
ducible representations of (?} for any finite subgroups.
$H^{2:}(\tilde{X},\mathbb{Q})rightarrow\{\mathrm{t}\mathrm{h}\mathrm{e}$$i$$K(\tilde{X})rightarrow\{\mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{a}\mathrm{b}\mathrm{e}\underline{\mathrm{l}\mathrm{i}}\mathrm{m}$$D(\tilde{X})rightarrow\{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{n}\triangleright$

Remark 2.1. In (1), the age of $g\in$ (; is defined as follows: After
where is adiagonalzation, if $g^{r}=1$ , we obtain
primitive -th root of unity. Then age(7): $=(a +b +c)/r$ . For theand all ages are integers if
identity element $id$, we define age(id).
The correspondence (2) can be included in (3), but note that the2- imensional numerical McKay correspondence can be explained veryclearly as acorolary of the result in [10].
$g’=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\epsilon^{a},\epsilon^{b}, \epsilon^{\mathrm{c}})$

$r$

$=$

$G\subset SL(3,\mathbb{C})$$0$$\epsilon$148

As acolorful introduction to the McKay correspondence, the author
would like to recommend paper presented at the Bourbaki seminar by
Reid [17] and also on the Web page
, one can find some recent papers related to the McKay$(\mathrm{h}\mathrm{t}\mathrm{t}\mathrm{p}://\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{s}.\mathrm{w}\mathrm{a}r\mathrm{w}\mathrm{i}\mathrm{c}\mathrm{k}.\mathrm{a}\mathrm{c}.\mathrm{u}\mathrm{k}/$$\sim \mathrm{m}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{e}/\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y})$correspondence.

3. SpECIAL REPRESENTATIONSIn this section, we will discuss the special representations. Let beafinite small subgroup of, that is, the action of the group $G$
is free outside the origin, and be arepresentation of $G$ on V. (; actsonand the quotient is avector bundle on
which
can be extended to areflexive sheaf $T$ on $X:=\mathrm{C}^{2}/G$ .
For any reflexive sheaf $T$ on arational surface singularity $X$ and theresolution. We define asheaf.
Definition 3.1. ([5]) The sheaf is called aftdl sheaf on .
Theorem 3.2. ([5]) A sheaf on
is a fidl sheaf if the follo wingconditions are fidffild.
1.
lOcally free,generated by glObal sections,B.
3.
, where means the dual.
$G$$GL(2,\mathbb{C})$$\rho$$\mathbb{C}^{2}\mathrm{x}V$$(\mathbb{C}^{2}\backslash \{0\})/G$$\pi:\tilde{X}\prec X$$\tilde{F}:=\pi^{*}F/\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$$\overline{F}$$\tilde{F}$$\tilde{X}$$\tilde{X}$$\cdot$$F\sim is$$F\sim is$$H^{1}(\tilde{X},\overline{\mathcal{F}}^{\mathrm{v}}\otimes\omega_{\overline{X}})=0$

Note that asheaf$\mathrm{V}$is indecomposable if and only if the corresponding representation is irreducible. Therefore we obtain an indecomposable full sheaf
on for each irreducible representation , but
in general, the number of the irreducible representations is larger thanthat of irreducible exceptional components. Therefore Wunram andRiemenschneider inroduced the notion of aspeciality for ffil sheaves:
Definition 3.3. ([18]) Afiffi sheaf is called special if and only if$\tilde{F}$$\rho$$\overline{F}.\cdot$$\tilde{X}$$\rho.\cdot$$H^{1}(\tilde{X},\tilde{F}^{\vee})=0$.

Areflexive sheaf $F$ on $X$ is special if is so.
Arepresentation is special if the associated reflexive sheaf $F$ on $X$$\overline{F}$$\rho$

is special.
With these definitions, folowing equivalent conditions for theciality hold:
Theorem 3.4. ([18], [21])1. $F-is$ special
an isomor phism,$F$B.
is spoeial) is reflexive,$\mathrm{s}\mathrm{p}\triangleright$$\Leftrightarrow\tilde{F}\otimes\omega_{\tilde{X}}arrow[(F\otimes\omega_{\overline{X}})^{\mathrm{v}\mathrm{v}}]^{\sim}is$$\Leftrightarrow F\otimes\omega_{\tilde{X}}/torsion$

3.
$V)^{G}$$\rho$

is a special representation

is $su\dot{\eta e}dive$ .

$\Leftrightarrow(\Omega_{\mathrm{C}^{2}}^{2})^{G}\otimes(\mathcal{O}_{\mathrm{C}^{2}}\otimes V)^{G}arrow(\Omega_{G}^{2}\otimes$

149

Then we have following nice generalzed McKay correspondence for
quotient surface singularities:
Theorem 3,5. ([21]) There is a bijection between the set of special
modules F. and the set of irrenon-trivial indecomposable
There
is the first Chern
via
ducible components
class, and also one-tO-One correspondence with the set of special nontrivial imducible representations.
$reff\underline{le}xive$$E_{}$$\cdot$$c_{1}(F.\cdot)E_{j}=\delta_{\dot{\iota}j}$$c_{1}$$a$As acorollary of this theorem, we get the original McKay corresponbadc because in thsi case ffi
dence for finite subgroups inirreducible representations are special.
$SL(2,\mathbb{C})$4.

-HILBERT SCHEMES AND COMBINATORICS$\mathrm{G}$discuss G–Hilbert schemes and anew way
In this section, we
to find the special representations for cyclic quotient singularities bycombinatorics.
can be described as aset of ideals:
Hilbert scheme of -points on$\mathrm{w}\mathrm{i}\mathrm{U}$$\mathbb{C}^{2}$$n$$\mathrm{H}\mathrm{i}1\mathrm{b}^{n}(\mathbb{C}^{2})=$.

{I$\subset \mathbb{C}[x,y]$|I : ideal, m.

$\mathbb{C}[x,y]/I=n$}.

It is a $2\mathrm{r}$ensiond smooth projective variety. The -Hilbert scheme
was introduced in the paper by Nakamura and the author([11]) as follows:$\mathrm{H}\mathrm{i}1\mathrm{b}^{G}(\mathbb{C}^{2})$$\mathrm{H}\mathrm{i}1\mathrm{b}^{G}(\mathbb{C}^{2})=${I$\subset \mathbb{C}[x,y]$|I : -invariant ideal,$G$$\mathbb{C}[x,y]/I\cong \mathbb{C}[G]$

},

150

Theorem 4.1. ([8]) Let be a finite small subgroup of.
(i) -Hilbert scheme
is the minimaI resolution of.
(ii) For, denote by the ideal corresponding to andm\mbox{\boldmath$\alpha$}r\’imal
let $m$ be theideal ofcorresponding to the origin 0. If
is in the exceptional locus, then, as representations of , eve have
$G$

$G$

$GL(2,\mathbb{C})$$Hilb^{G}(\mathbb{C}^{2})$$\mathbb{C}^{2}/G$$y\in Hilb^{G}(\mathbb{C}^{2})$$I_{y}$$y$$\mathcal{O}_{\mathbb{C}^{2}}$

$y$

$G$

(4.2)

where

$I_{y}/mI_{y}\cong\{\begin{array}{l}\rho_{\dot{l}}\oplus hify\in E_{}andy\not\in E_{j}fo\mathrm{r}j\neq i\sqrt\oplus\rho_{j}\oplus mify\in E_{}\cap E_{j}\end{array}$$\rho.\cdot$is the special representation associated uriffi the

ceptional curve$E_{}$$ioedu\dot{\alpha}ble$.

ex-

Remark 4.3. In dimension two, we can say that -Hilbert scheme isthe same as a2-dimensional irreducible component of the (Afixed setof. Asimilar statement holds forin dimen. ensiond irreduciblesion three, that is, the -Hilbert scheme is acomponent of the -fixexi set ofand acrepant resolution of
the quotient singularity. In this case note thatis not$G$$\mathrm{H}\mathrm{i}1\mathrm{b}^{\mathrm{n}}(\mathrm{C}^{2})$$G\subset SL(3,\mathbb{C})$$\mathrm{r}$$\mathrm{G}$$G$$\mathrm{H}\mathrm{i}1\mathrm{b}^{n}(\mathbb{C}^{3})$$\mathbb{C}^{3}/G$$\mathrm{H}\mathrm{i}1\mathrm{b}^{n}(\mathbb{C}^{3})$smooth.
Moreover, Haiman proved that
crepant resolution of’-Hilbert scheme
symmetric product of$\mathrm{G}$$\mathrm{S}$$\mathbb{C}^{2n}/\mathrm{S}_{\mathrm{n}}=7\mathrm{k}\cdot \mathrm{t}\mathrm{h}$$\mathrm{H}\mathrm{i}1\mathrm{b}^{\mathrm{S}_{n}}(\mathbb{C}^{2n})$”$\mathrm{H}\mathrm{i}1\mathrm{b}^{*}(\mathbb{C}^{2\iota}’)\cong \mathrm{H}\mathrm{i}1\mathrm{b}$$\mathbb{C}^{2}$is a, i.e.,$(\mathbb{C}^{2})$in process of the proof of $n!$ conjecture, (cf. [13])rom now on, we restrict our considerations to$G\subset GL(2,\mathrm{C})$cyclic.

Wunram constructed the generalized McKay correspondence for cyclic
surface singularities in the paper [20] and we have to consider the corresponding geometrical informations (the inimal resolution, reflexive sheaves and so on) to obtain the special representations. Here wewould like to give anew characterization of the special representationsin terms of combinatorics. It is much easier to find the special
sentation because we d.on’t need any geometrical objects, but based on
the result of -Hilbert schemes.
Let us discuss the new characterization of the special representationsin terms of combinatorics. Let be acyclic groupwhich is generated by amatrix
where
and
and consider$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\triangleright$$\mathrm{G}$$G$$(\begin{array}{ll}\epsilon 00 \epsilon^{a}\end{array})$$C_{r,a}$$\epsilon^{r}=1$$\mathrm{g}\mathrm{c}\mathrm{d}(r,a)=1$

acharacter map
asand $y\vdash*f$ , then we haveacorresponding characters for each monomials in.
(Afixed(AElbert
Let be the ideal of thepoint in thescheme,then we can define the following sets$\mathbb{C}[x,y]arrow \mathbb{C}[t]/t^{r}$$x\vdash\succ t$$\mathbb{C}[x,y]$$I_{\mathrm{p}}$

$p$

151

for which
Consider a -invariant subscheme
is the regular representation of . Then the G- ilbert scheme
can be regarded as amodul space of such .
$Z_{p}\subset \mathbb{C}^{2}$$\mathrm{G}$$H^{0}(Z_{p}, \mathcal{O}_{Z_{\mathrm{p}}})=$$G$$\mathcal{O}_{\mathrm{C}^{2}}/I_{p}$$Z_{p}$

Definition 4.4. The set of monomials inare not incluster if all monomials on
aYoung diagram of $|G|$ boxes.

$\mathbb{C}[x,y]$$\mathrm{Y}(Z_{p})$$I_{p}$

is cWed G-$\mathrm{Y}(Z_{p})$and it can be drawn as

Definition 4.5. For any small cyclic group , let $B(G)$ be the set ofmonomials which are not divisible, by any G-.nvariant monomial andcffi it G-basis.
$G$, then let $L(G)$ be $\{1, x, \cdots, x^{r-1},y, \cdots,y^{r-1}\}$ ,Definition 4.6. If
or $xy$. We
i.e., the set of monomials which cannot be devided by ,cffi it -space for $G$ because the shape of this diagram looks as the$|G|=\mathrm{r}$$x^{r}$

chapital$y^{r}$”$\mathrm{a}\mathrm{L}.$Definition 4.7. The monomial$x^{m}y^{n}$is of weight k if m $+an$ $=k$ .

Let us describe the method to find the special representations ofwith these diagrams:

$G$

, the
Theorem 4.8. For a small finite cyclic subgroup ofis special if and only if the $correspondrightarrow$
irreducible representation$ing$ monomial in $B(G)$ are not contained in the set of monomials$GL(2,\mathbb{C})$$\rho\dot{.}$$B(G)\backslash L(G)$.

Proof In Theorem 3.4 (3), we have the definition of the special$\mathrm{r}\mathrm{e}\triangleright$

resentation, and it is not easy to compute all special representations.
under theHowever look at the behavior of the monomials inmap
for each representation :First, let us consider the monomial bases of each set. Let
:
.
.
if
is in
. An element $f(x, y)dx$ A
andand only if$\mathbb{C}[x,y]$$\Phi_{:}(\Omega_{G}^{2})^{G}\otimes(\mathcal{O}_{\mathrm{C}^{2}},\otimes V.\cdot)^{G}arrow(\Omega_{\mathrm{C}^{2}}^{2}\otimes V_{})^{G}$$\rho\dot{.}$$V_{i}=\mathbb{C}e$$\rho(g)\mathfrak{g}$$=\epsilon^{-}\dot{.}$$dy\otimes\rho_{1}$$(\Omega_{\mathrm{C}^{2}}^{2}\otimes V\dot{.})^{G}$,$g^{*}f(x,y)dx\Lambda dy\cdot\epsilon^{1+a}\otimes\epsilon^{-:}=f(x, y)dx\wedge dy$that is,($f$ ( ,y)&A $dy$) $=\epsilon^{-(a+1)}\dot{.}(f(x,y)dx$ A $dy$).
Therefore the monomial base foris aset of monomiffici$f(x,y)$ such that$g:f(x,y)$ $|arrow\epsilon^{-(a+1)}f(x,y)$under the action of $G$, that is, monomials of weight $i-(a+1)$ .
Similarly, we have the monomial bases foras the set of monomials $f(x.y)$ of weight $r-(a+1)$ .
$g^{*}$$x$$(\Omega_{\mathrm{C}^{2}}^{2}$$V_{})^{G}$$(\Omega_{\mathrm{C}^{2}}^{2})^{G}$

152

The monomial bases foris given as aset of monomials$f(x,y)$ of weight .
Let us check the surjectivity of the map . If
is surjective, then
all the monomial bases incan be obtained as aproduct ofthe monomial basis of two other sets. Therefore the degree of the mon0means inmust be higher than the degree of the monomials.
inNow look at the map. The vector spaceis$x^{a+1},xy$,, wheregenerated by the monomials of weight $a+1$ , i.e.,. On the other hand,is generated by
the degree 0monomial 1. Then the map
is not surjective.
, where $mn\neq 0$ , is abase of
By this, if amonomial of type, then there exists amonomial $x^{m-1}y^{n-1}$ inandthe degree become smaller under tha map . This means
is notsurjective.
Moreover, if the bases ofis generated only by and, then the degrees of the monomials in
where
is bigger andis surjective. Thus we have the assertion.
$(\mathcal{O}_{\mathrm{C}^{2}}\otimes V.\cdot)^{G}$$i$$\Phi_{:}$$\Phi_{i}$$(\Omega_{\mathrm{C}^{2}}^{2}\otimes V\dot{.})^{G}$$(\Omega_{\mathrm{C}^{2}}^{2}\otimes V_{i})^{G}$$(\mathcal{O}_{\mathrm{C}^{2}}\otimes V.\cdot)^{G}$$(\mathcal{O}_{\mathbb{C}^{2}}\otimes V_{a+1})^{G}$$\Phi_{a+1}$$\cdots$$ab=a+1\mathrm{m}\mathrm{o}\mathrm{d} r$$y^{b}$$(\Omega_{\sigma}^{2}\otimes V_{a+1})^{G}$$\Phi_{a+1}$$x^{m}y^{\iota}.$$(\mathcal{O}_{C^{2}}\otimes V_{})^{G}$$(\Omega_{\mathrm{C}^{2}}^{2}\otimes V.\cdot)^{G}$$\Phi_{:}$$\Phi_{:}$$x^{:}$$(\mathcal{O}_{\mathrm{C}^{2}}\otimes V_{})^{G}$$y^{j}$$aj=i\mathrm{m}\mathrm{o}\mathrm{d} r$$(\Omega_{C^{2}}^{2}\otimes V_{})^{G}\square$$\Phi_{:}$

Remark 4.9. From this theorem, we can also say that arepresentationis special if and only if the number of the generators of the spaceis 2.

$\rho.\cdot$$(\mathrm{O}\mathrm{q}2\otimes V\dot{.})^{G}$

Theorem 4.10. Let be a fied point by -action, then we can defineby the -cluster and the configuration of the exceptional
an ideal
locus can be described by these data.
$G$

$p$

$G$

$I_{p}$$P$

roof. The defining equation of the ideal

is given by$I_{\mathrm{p}}$$\{\begin{array}{l}x^{a}=\alpha y^{\mathrm{c}}y^{b}=\sqrt x^{d}x^{a-d}y^{b-\mathrm{c}}=\alpha\beta\end{array}$andwhere and are complex numbers and bothand (resp.
correspond the same representation (or character).
is alocal affine coorinate near the fixed point and
The pair
it is also obtained from the calculation with toric geometry. Moreover
each axis of the affine chart is just aexceptional curve or the original
and the points
axis of . The exceptional curve is isomorphic to a:
) which ison it is written by the ratio lke $[? : y^{b}]$ (resp.
). The fixed(resp.
corresponding to aspecial representationand .
point is the intersection point of 2exceptional curves
Thus we can get the whole space of exceptional locus by deformationClof the point and patching the affine pieces.
$\alpha$$x^{a}$$\beta$$y^{b}$$y^{\mathrm{c}}$$x^{d})$$(\alpha,\beta)$$p$$\mathrm{P}^{1}$$\mathbb{C}^{2}$$[x^{d}$$\rho_{a}$$y^{\mathrm{c}}]$$\rho_{d}$$E_{a}$$p$$p$$E_{d}$

153

We will see aconcrete example in the folowing section. Here wewould like to make one remark as acorollary:
Corollary 4.11. For -type simple singularities, all $n+1$ affine charts
can be described by $n+1$ Young diagrams of type $(1, \cdots, 1, k)$ .
$A_{n}$

case, $xy$ is always -invariant, hence $B(G)=L(G)$ .
Therefore we have $n+1G$-clusters and each of them corresponds tothe monomial ideal $(x^{k},y^{n-k+2},xy)$ .

Proof In$\mathrm{G}$$A_{\mathrm{n}}$$\square$

5. EXAMPLE
Now let us go badc to our example in the first section: the cyclic qu0tient singularity of type

$C_{7,3}$

which is generated by the matrix$(\begin{array}{ll}\epsilon 00 \epsilon^{3}\end{array})$.
where
By the Theorem 3.4, we have away to find the three special
sentations corresponding to the exceptional curves, i.e., let us consider
is surjective.
when the map, we have the following:
As abasis of the$\epsilon^{7}=1$$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\triangleright$$(\Omega_{\mathrm{C}^{2}}^{2})^{G}\otimes(\mathcal{O}_{\mathrm{C}^{2}}\otimes V)^{G}arrow(\Omega_{\mathrm{C}^{2}}^{2}\otimes V)^{G}$$(\mathcal{O}_{\mathbb{C}^{2}}\otimes V)^{G}$

$k=1$
$k=2$
$k=3$
$k=4$
$k=5$
$k=6$
$k=7$

$\{x^{4},xy\}$$\{x^{5},x^{2}y\}$$\{x^{6},x^{3}y,y^{2}\}${1}$\{x\}$$\{x^{2}\}$$\{x^{3},y\}$The basis of the space$(\Omega_{\mathrm{C}^{2}}^{2})^{G}$is$\{x^{3},y\}$.

Moreover, we have the following basis of the space
$k=1$
$k=2$
$k=3$
$k=4$
$k=5$
$k=6$
$k=7$

$(\Omega_{\mathrm{C}^{2}}^{2}\otimes V)^{G}$:$\{x\}$$\{x^{2}\}$$\{x^{3},y\}$$\{x^{4},xy\}$$\{x^{5},x^{2}y\}$$\{x^{6},x^{3}y,y^{2}\}$$\{x^{7},x^{4}y,xy^{2}\}$Thus the map wiU be surjective when k $=1,2$ or 3, and we have
md .
three special representations ,$\rho_{1}$$\rho_{2}$$\rho_{3}$Now we have achance to use our new way to find the special
sentations! Let us draw the diagram which corresponds to the (Abasis$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\triangleright$154

and -space. First we have the following -basis $B(G)$ and the corresponding characters in asame diagram. In Figure 5.1 we draw the-space as shaded part in $B(G)$ .
$G$$\mathrm{L}$FIGURE 5.1.

$G$-baeis $B(G)$ and the charactersin $B(G)\backslash L(G)$ andand
Now we have three monomials $xy$ ,they correspond to the characters (resp. representations) 4, 5and 6,and ). Therefore we can find aset of special repre(resp.
, and fiad the corresponding G-clusters,
sentations, that is,representing the origin of the affine charts of the resolution, can be
drawn as 4young diagrams and get the corresponding special representations in this case. See Figure 5.2.
Let us see the meanings of the corresponding -clusters in this case.
for (2), we obtain an ideal $I_{2}=(y^{5},x^{2},xy^{2})$ for the origin ofFrom
the affine chart (2) in Figure 1.2, and the corresponding representations
corresponding
are , and . If we take the maximal ideal $m$ ofto the origin 0, then we have$x^{2}y$$\rho_{4}$$x^{3}y$$\rho_{6}$$\rho_{5}$$\{\rho_{1}, \rho_{2}, \rho_{3}\}$$G$$\mathrm{Y}(Z_{p})$$\#\mathrm{c}2$$\rho_{1}$$\rho_{2}$$\rho_{0}$$I_{2}/mI_{2}\cong\rho_{1}\oplus\rho_{2}\oplus\rho_{0}$

.

155

156

(1)

(2)

$t4)$

$C3)$

FIGURE 5.2,

$G$-cluster$\mathrm{Y}(Z_{p})$Similarly we have the ideal $I_{3}=(y^{3}, x^{3}, xy^{2})$ and$I_{3}/mI_{3}\underline{\simeq}\rho_{2}\oplus\rho_{3}$$\oplus\rho_{0}$.

These descriptions coincide with the results of Theorem 4.1 for an.
intersecting point at
For any other points on the exceptional component , we musthave$E_{1}\cap E_{2}$$p$$E_{\dot{1}}$$I_{\mathrm{p}}/mI_{\mathrm{p}}\underline{\simeq}\rho_{\dot{1}}$$\oplus\rho_{0}$

.

$(*)$

In fact, we can see that on the exceptional divisor
in this example
was determined by the ratio: , that is, the corresponing ideal of
apoint on
can be described as. Therefore
the ratio
gives the coordinate of the exceptional curve
and we also have $(*)$ .
$E_{2}$$x^{2}$$I_{p}=(\alpha x^{2}-\beta y^{3},xy^{2}-\gamma)$$E_{2}$$(\alpha :$y^{3}$\beta)$$(^{\underline{\simeq}}\mathrm{P}^{1})$We discussed special McKay correspondence in 2-dimensional case inthis paper. In dimension three, it is convenient to consider crepant resolutions as aminimal resolution and we have much more complicated, we havesituation. Even in the casein general. Of course we can use the same definition for the special
representations in the higher dimensional case, but all non-trivial irreducible representations ofare special. On the other
hand, the number of the exceptional divisors is less than that of the
non-trivial irreducible representations. Therefore, it looks very difficult
to generalize this special McKay correspondence. That is, we shouldmake adifference, say akind of the grading, in the set of the special(or non-trivial) representations like “age” of the conjugacy classes$G\subset 5\mathrm{L}(3, \mathbb{C})$$G\subset SL(3, \mathbb{C})$$H^{4}(\tilde{X},\mathrm{Q})$$\neq 0$

157

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[21]

DBPARPRNr OF MATHEMATICS, TOKYO, TOKYO 192-0397JAPAN$E$-rrgail
address: yDarlMth.metro-u.ac.jp$\mathrm{M}\mathrm{E}\mathrm{T}\mathrm{R}\mathrm{O}\mathrm{P}\mathrm{o}\iota \mathrm{r}\mathrm{r}\mathrm{A}\mathrm{N}$$\mathrm{m}\mathrm{o}\sqrt{1}$$\mathrm{U}\ovalbox{\tt\small REJECT} \mathrm{r}\mathrm{r}\mathrm{v}$, HA-