数理解析研究所講究録 1212 巻 2001 年 1-17

1

Witten Laplacian and Twisted de Rham Cohomology
Katsunori Iwasaki (岩崎 克則)
Faculty of Mathematics, Kyushu University
6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581 Japan(九州大学大学院数理学研究院)
Abstract
This paper presents asurvey of arecent work by the author on
an intersection theory for twisted de Rham cohomology and its applications to special function theory. While Witten used atwisted
Laplacian, twisted by aMorse function, to develop his Morse theory
as asuper-symmetric quantum mechanics, we use atwisted Laplacian,twisted by an isolated singularity, in aquite different situation.
Key words: Witten Laplacian, isolated singularity, twisted de Rham
cohomology, duality, super-symmetry, generalized Airy function, Schurpolynomial, skew-Schur polynomial.

2000 Mathematics Subject Classification:$32\mathrm{S}20,35\mathrm{J}10,58\mathrm{A}12$

1Introduction
In his famous paper [24], Witten used atwisted Laplacian, twisted by aMorse function, to develop his Morse theory as asuper-symmetric quantum
mechanics. Afterwards, in their studies on puits multiples en mecanique semiclassique, Helffer and Sj\"ostrand [8] made adetailed investigation of Witten’scomplexes in amathematically rigorous formulation.
Recently, with asimilar philosophy but in aquite different situation,
Iwasaki [9] made use of atwisted Laplacian, twisted by aversal deformationof an isolated singularity (invariant under afinite unitary reflection group),
to construct aduality between apair of polynomial twisted de Rham c0homology groups associated with the isolated singularity. Then Iwasaki andMatsumoto [11] applied this construction to calculate the intersection matrixof ageneralized Airy function in terms of skew-Schur polynomials. The aimof this expository article is to survey the results in the papers $[9, 11]$ .

2
Amotivation for developing atwisted de Rham cohomology theory comesfrom its (possible) applications to the theory of hypergeometric functions intheir broadest sense. This is because, from the standpoint of Euler integral
representations, ahypergeometric function is defined to be an integral ofaclosed twisted differential form along atwisted cycle. See Aomoto andKita [2] and Matsumoto and Yoshida [16] for more information.
Letbe the complex -space with coordinates,$W$
be afinite subgroup of the unitary group $U(n)$ . The case whereand let$W$ is afinite unitary reflection group is of particular interest to us. Consider
a $W$ -invariant polynomialof degree $N=\deg f$ satisfying thefollowing assumption, which will be made throughout the paper.
$Z=\mathbb{C}^{n}$

$n$

$z$

$(z_{1}, \ldots, z_{n})$$=$$f\in \mathbb{C}[z]^{W}$

Assumption 1.1 The top homogeneous component
O $\in Z$ as the only isolated singular point of it.

$f_{0}$of f has the origin

The first aim of this paper is to discuss a $W$ -invariant duality between thepair of twisted polynomial de Rham cohomology groups, where. denotes the space of polynomial differential forms
on $Z$ andare thetwisted exterior differentials defined by$H.(\Omega_{Z}.

, d_{\pm f})$$\Omega_{Z}$$d_{\pm f}$$d_{\pm f}=e^{\mp f}de^{\pm f}=d\pm df\wedge$.

Before entering into the duality, we should describe some elementary properties of the cohomology groupand its $W$-invariant component. Let .be the space of polynomial differential forms on
invariant under the action of $W$ . Standard techniques in finite group actions, such as transfer and the vanishing of the first cohomology of $W$ , imply
that the inclusion of cochain complexes
induces anisomorphism of cohomology groups (the transfer isomorphism):$H^{\cdot}(\Omega_{Z}^{\cdot}, d_{f})$$H^{\cdot}(\Omega_{Z}.

, d_{f})^{W}$$Z$$\Omega_{Z/W}$$(\Omega_{Z/W}., d_{f})\llcornerarrow(\Omega_{Z}.

$H^{\cdot}(\Omega_{Z/W}^{\cdot}, d_{f})\simarrow H^{\cdot}(\Omega_{Z}^{\cdot}, d_{f})^{W}$, d_{f})$.

(1)

When $W$ is afinite unitary reflection group, adetailed description ofis possible. Aclassical theorem of Chevalley [6] and Shephard and Todd [21] tells us that if $W$ is afinite unitary reflection group,
then there exists an -tuple of algebraically independent, homogeneous, W, that generates the invariant algeinvariant polynomials, say, $t=$
. Thus can be thought of as apolynomial of . Letbrabe the complex -space with coordinates . Then the twisted de Rham complex
makes sense, and there exists an inclusion of cochain complexes. Atheorem of Solomon [22] asserts that this inclusion
is an isomorphism. This, together with (1), yields isomorphisms:$H^{\cdot}(\Omega_{Z}.

, d_{f})^{W}$$n$$(t_{1}, \ldots, t_{n})$$\mathbb{C}[z]^{W}=\mathbb{C}[t]$$n$$(\Omega_{T}.

$(\Omega_{T}.

$f$

$t$

$T=\mathbb{C}^{n}$$t$, d_{f})$, d_{f})\epsilonarrow(\Omega_{Z/W}., d_{f})$$H^{\cdot}(\Omega_{T}^{\cdot}, d_{f})\simarrow H^{\cdot}(\Omega_{Z/W}^{\cdot}, d_{f})arrow H^{\cdot}(\sim\Omega_{Z}^{\cdot}, d_{f})^{W}$

.

(2)

3
Assumption 1.1 tells us that , as apolynomial of , has the origin
as the only isolated singular point of it. It is natural to comparewith the cohomology groupof the Koszul complex.
Aclassical result on isolated surface singularities (see e.g., Arnold, GuseinZade and Varchenko [4, Chap. 12] implies that$t$$f_{0}$$\mathrm{O}\in T$$H.(\Omega_{T}.

$H^{\cdot}(\Omega_{T}^{\cdot}, df_{0}\wedge)$, d_{f})$$(\Omega_{T}^{\cdot}, df_{0}\wedge)$$)$$H^{p}(\Omega_{T}^{\cdot}, df_{0}\wedge)\cong\{J0(p\neq 0)(p=n)$

where
is the Jacobian ring of the isolated
singularity $f_{0}=0$ . Let $d=$
denote the degrees of $W$ , namely,$d_{j}=\deg t_{j}$ .
By aformula of Milnor and Orlik [17] and Arnold [3], thecomplex dimensionof , or the Milnor number of , is given by$J=\mathbb{C}[t]/(\partial f_{0}/\partial t_{1}, \ldots, \partial f_{0}/\partial t_{n})$$(d_{1}, \ldots, d_{n})$$\mu=\dim_{\mathbb{C}}J$$J$$f_{0}$$\mu=\prod_{i=1}^{n}(\frac{N}{d_{i}}-1)<\infty$.

If we provide the complex
with the degree filtration by assigningdegreeto, then the principal term of the exterior differential
and$d_{f}=d+df\wedge=d+df_{0}\wedge+\cdots$ with respect to the filtration is, andthe graduation of the complex
becomes isomorphic to the Koszul
complex. This observation readily leads to the formula:$(\Omega_{T}.

$d_{j}$$t_{j}$, d_{f})$$dt_{j}$$\mathrm{d}/\mathrm{o}\mathrm{A}$$(\Omega_{T}.

$(\Omega_{T}.

, d_{f})$, df_{0}\wedge)$(3)$\dim H^{p}(\Omega_{T}^{\cdot}, d_{f})=\{\begin{array}{l}\mu(p=n)0(p\neq n)\end{array}$

In particular, if $W$ is the trivial group then $d=(1, \ldots, 1)$ and the Milnor
number is given by $\mu=(N-1)^{n}$ . Since $T=Z$ in this case, (3) implies thatunless $p=n$ and $\dim H^{n}(\Omega_{Z}^{\cdot}, d_{f})=(N-1)^{n}$ . So only the-th cohomology group is nontrivial. Note that for any finite subgroup $W$ of$U(n)$ , which may or may not be aunitary reflection group,isfinite-dimensional, as asubspace of the finite-dimensional space.
Now our duality theorems are stated in the following manner.
$H^{p}(\Omega_{Z}^{\cdot}, d_{f})=0$$n$$H^{n}(\Omega_{Z}.

, d_{f})^{W}$$H^{n}(\Omega_{Z}.

, d_{f})$

Theorem 1.2 Let $W$ be a finite subgroup of $U(n)$ . Under Assumption 1.1,
there exists a natural $W$ -invariant duality:$H^{n}(\Omega_{Z}^{\cdot}, d_{f})\cross H^{n}(\Omega_{Z}^{\cdot}, d_{-f})arrow \mathbb{C}$.

(4)

In view of (2), Theorem 1.2 leads to the following corollary.
Corollary 1.3 Let $W$ be a finite unitary reflection group and set $T=Z/W$ .
Under Assumption 1.1, there exists a natural duality:$H^{n}(\Omega_{T}^{\cdot}, d_{f})\cross H^{n}(\Omega_{T}^{\cdot}, d_{-f})arrow \mathbb{C}$

.

(5)

4
Although the statement of the duality does not involve analysis, our construction requires analysis and PDE theory in an essential way. We remarkthat another approach based on algebraic $D$ -module theory, which uses a
result of Sabbah [19], is also possible. In \S 2 we explain how to construct theduality (4). The construction is based on two kinds of comparison theorems
for several types of twisted de Rham cohomology groups. The first comparison theorem (Theorem 2.1) is proved by using Liouville’s theorem in several
complex variables and twisted versions of the Poincare’ lemma (see \S 3). The
second comparison theorem (Theorem 2.2) is based on atwisted version of
Hodge-Kodaira decomposition (see \S 4). It involves apseud0-differential cal. There are natural
culus of Witten’s twisted Laplacian
real structures on the cohomology groups compatible with the duality (Theorem 5.1). We introduce the notion of super-symmetric in \S 6. What we callasuper-symmetry is such atransformation that permutes
and induces symmetric or skew-symmetric self-duality
groups(Theorem 6.2).
on
The construction of Iwasaki [9] was applied to the theory of (generalized)
Airy functions by Iwasaki and Matsumoto [11]. The second aim of this article
is to introduce their result to the reader (see \S 7). Further applications tosingularity theory and special function theory should be made in the future.
$\Delta_{f}=d_{f}d_{f}^{*}+d_{f}^{*}d_{f}$$\mathrm{t}\mathrm{h}\mathrm{e}\pm \mathrm{c}\mathrm{o}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{y}$$H^{n}(\Omega_{Z}.

, d_{\pm f})$$H^{n}(\Omega_{Z}^{\cdot}, d_{f})^{W}$

2Comparison Theorems and Duality
The construction of the duality (4) is based on two kinds of comparison
theorems. The first comparison theorem is stated as follows.
be the space of tempered currents on Z. Then the
Theorem 2.1 Let
induces a $W$ -equivariant isomorinclusion of complexes
phism of cohomology groups:$\mathcal{T}_{Z}^{\cdot}$$(\Omega_{Z}.

, d_{f})\mathrm{e}arrow(\mathcal{T}_{Z}^{\cdot}, d_{f})$$H^{\cdot}(\Omega_{Z}^{\cdot}, d_{f})\simarrow H^{\cdot}(\mathcal{T}_{Z}^{\cdot}, d_{f})$.

(6)

We remark that this theorem holds without Assumption 1.1; the only essential
assumption needed is the condition that is acomplex polynomial. The
second comparison theorem is stated as follows.
$f$be the space of smooth differential forms of Schwartz
Theorem 2.2 Let
inducesclass on Z. Then the inclusion of complexes$W$ -equivariant isomorphismof cohomology groups:$S_{Z}^{\cdot}$$(S_{Z}^{\cdot}, d_{f})\mathrm{c}arrow(\mathcal{T}_{Z}^{\cdot}, df)$

$H^{\cdot}(S_{Z}^{\cdot}, d_{f})\simarrow H^{\cdot}(\mathcal{T}_{Z}^{\cdot}, d_{f})$

.

$a$

(7)

5
We remark that Assumption 1.1 is essential in this theorem.
Once Theorems 2.1 and 2.2 are established, the construction of the duality (4) proceeds along the same line as in the classical counterparts: thePoincare duality for de Rham cohomology, and the Serre duality for Dolbeault cohomology on acompact (complex in the latter case) manifold. To
explain this, consider the diagram:$H^{n}(\Omega_{Z}.

, d_{f})arrow(\mathrm{a})H^{n}(\tau_{1^{Z}(\mathrm{d})}.,d_{f})$$H^{n}(S_{Z}.

$H^{n}(\Omega_{Z}^{\cdot},d_{-f})\downarrow(\mathrm{b})$,, d_{-f})arrow(\mathrm{c})H^{n}(\mathcal{T}_{Z}^{\cdot}, d_{-f})$

where the arrows (a), (b), (c) are $W$ -equivariant isomorphisms; (a) and (b)come from Theorem 2.1, and (c) comes from Theorem 2.2, respectively. The
arrow (d) indicates anatural $W$ -invariant duality between
andinduced from the topological duality between the space of tempered distributions and that of smooth functions of Schwartz class. The
reduction to the former situation from the latter is made by astandard
functional-analytical method as in Serre [20], based on the finite dimensionality of, which in turn follows from the finite
anddimensionality ofthrough the isomorphisms (6) and (7). In
this manner, establishing the duality (4) has been reduced to proving thecomparison theorems, Theorems 2.1 and 2.2.
Putting Theorems 2.1 and 2.2 together yields natural $W$-equivariant is0morphisms of cohomology groups:$H^{n}(\mathcal{T}_{Z}^{\cdot}, d_{f})$$H^{n}(S_{Z}^{\cdot}, d_{-f})$$H^{n}(\mathcal{T}_{Z}^{\cdot}, d_{f})$$H^{n}(S_{Z}^{\cdot}, d_{-f})$$H^{n}(\Omega_{Z}^{\cdot}, d_{\pm f})$$H^{n}(\Omega_{Z}^{\cdot}, d_{\pm f})\simarrow H^{n}(S_{Z}^{\cdot}, d_{\pm f})$.

(8)

Therefore every cohomology classes
can be represented bysame-closecl smooth -forms of Schwartz class on $Z$ , say,
Then theduality (4) is expressed by the integrals:$\emptyset\pm\in H^{n}(\Omega_{Z}.

$d_{\pm f}$, d_{\pm f})$$n$$\emptyset\pm\cdot$$\langle\phi_{+}, \phi_{-}\rangle=\int_{Z}\psi_{+}\wedge\phi_{-}=\int_{Z}\phi_{+}\wedge\psi_{-}=\int_{Z}\psi_{+}\wedge\psi_{-}$

Note that these integrals are convergent, sinceandare rapidly decreasing as $|z|arrow\infty$ .

$\emptyset\pm \mathrm{a}\mathrm{r}\mathrm{e}$.

of polynomial growth$\psi_{\pm}$3Twisted Poincar\’e Lemmas
We present an outline of the proof of Theorem 2.1. Amain idea is to represent
the polynomial de Rham complex,
in terms of the double complex$(\Omega_{Z}.$ $d_{f})$

6
0

0

0

0

111
111

1
1

$0 arrow\Omega_{Z}^{0}arrow\Omega_{Z}^{1}\frac{d_{f}}{r}d_{f}\Omega_{Z}^{2}arrow\cdots\frac{d_{f}}{r}d_{f}\Omega_{Z}^{n}arrow 0$$F_{z_{\overline{\partial}}}^{0} \frac{\partial_{f_{\mathrm{t}}}}{r}\tau_{Z}’\frac{\partial_{f_{1}}}{r}\tau_{z_{\overline{\partial}}}^{2,0}\cdots\underline{\backslash },\tau_{Z}^{n,0}arrow 0\iota’\iota^{10}-\overline{\partial}\iota^{\frac{\partial_{f_{1}}}{r}}\downarrow\pm\overline{\partial}\partial_{f}$

. . .

.

,

$\tau z^{0,1^{\partial_{f}}}arrow\tau_{z_{\overline{\partial}}}^{1,1}\frac{\partial_{f1}}{r}\mathcal{T}_{Z}\underline{\iota},\ldots\underline{\mathrm{c}},$$7_{Z}^{\mathrm{m},1_{arrow 0}}\downarrow-\partial\downarrow\iota^{2,1}-\overline{\partial}\iota_{\mp}\partial\partial_{f}\partial$$\mathcal{T}_{Z}^{0,n}\downarrow\pm\overline{\partial}\underline{\partial_{f1}}$$\mathcal{T}_{Z}^{1,n}\frac{\partial_{f_{1}}}{r}\mathcal{T}_{Z}^{2,n}\frac{\partial_{f_{1}}}{\sim}\cdots\frac{\partial_{f_{\iota}}}{\neg}\mathcal{T}_{Z}^{\mathrm{n},n}\downarrow\mp\overline{\partial}\downarrow\pm\overline{\partial}\downarrow\overline{\partial}arrow 0$,$\downarrow$$\downarrow$

0

0

1

1

0

0

Figure 1: Double Complex

in amanner indicated in Figure 1, whereis the space of
$Z$
$(p,
q)$
tempered-currents on , andis the holomorphic twisted exteriordifferential fined by$(\mathcal{T}_{Z’}^{\cdot}\cdot, \partial_{f},\overline{\partial})$$\mathcal{T}_{Z}^{p,q}$$\partial_{f}$$\mathrm{d}_{1}\mathrm{e}$$\partial_{f}=e^{-f}\partial e^{f}=\partial+df\wedge$.

(9)

Since is acomplex polynomial, the exterior differential
is decomposed
$(0,
1)$
$(1,
0)$
into the-and-components as. The key ingredients ofthe proof consist of the Liouville theorem in several complex variables andtwo kinds of Poincare’ lemmas.
$f$$d_{f}$$d_{f}=\partial_{f}+\overline{\partial}$

Lemma 3.1 (Liouville Theorem) There following sequence is exact:$0arrow\Omega_{Z}^{p,0}arrow \mathcal{T}_{Z}^{p,0}arrow^{\overline{\partial}}\mathcal{T}_{Z}^{p,1}$

7
Lemma 3.2 ( -Poincare Lemma) The following sequence is exact$\mathrm{d}$$\mathcal{T}_{Z}^{p,0}arrow\overline{\partial}\mathcal{T}_{Z}^{p,1}arrow\overline{\partial}$

Lemma 3.3 ($\partial_{f^{-}}\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{a}\mathrm{r}\acute{\mathrm{e}}$

.

(10)

.

$..arrow\overline{\partial}\mathcal{T}_{Z}^{p,n}arrow 0$

Lemma) The following sequence is exact:$\mathcal{T}_{Z}^{0,q}arrow\partial_{f}\mathcal{T}_{Z}^{1,q}arrow\partial_{f}$...

$arrow\partial_{f}\mathcal{T}_{Z}^{n,q}arrow 0$

(11)

.

These lemmas are incorporated into the commutative diagram in Figure 1in which all the horizontal sequences except the top one, as well as all
the vertical sequences, are exact. In view of these exactness, ageneral theorem in homological algebra implies that there is anatural isomorphism of
cohomology groups, since the total complex of the
double complex
is given by. One can easily check that
this isomorphism is actually induced from the inclusion.
Hence Theorem 2.1 is proved if Lemmas 3.1, 3.2 and 3.2 are established.
Lemma 3.1 is quite standard in several complex variables. The proofs ofLemmas 3.2 and 3.3 are reduced to showing the exactness of the sequence:$H^{\cdot}(\Omega_{Z}^{\cdot}, d_{f})arrow H^{\cdot}(\sim \mathcal{T}_{Z}^{\cdot}, d_{f})$$(\mathcal{T}_{Z’}^{\cdot}\cdot, \partial_{f},\overline{\partial})$$(\mathcal{T}_{Z}^{\cdot}, d_{f})$$(\Omega_{Z}^{\cdot}, d_{f})\mapsto(\mathcal{T}_{Z}^{\cdot}, d_{f})$$\mathcal{T}_{Z}^{0,q}arrow\partial \mathcal{T}_{Z}^{1,q}arrow^{\partial}$...

$arrow\partial \mathcal{T}_{Z}^{n,q}arrow 0$

(12)

.

Indeed, (10) is transformed into (12) by just taking complex conjugate. As
for (11), we notice that there is acommutative diagram:...
$arrow\partial_{f}\mathcal{T}_{Z,\downarrow e^{2i1\mathrm{m}f}}^{n,q},arrow 0$$\tau_{Z,\iota^{0q^{\partial_{f}}}}’arrow e^{2i1\mathrm{m}f}\mathcal{T}_{Z,\downarrow e^{2i1\mathrm{m}f}}^{1,q^{\partial}},arrow^{f}$(13)$\mathcal{T}_{Z}^{0,q}arrow\partial \mathcal{T}_{Z}^{1,q}arrow^{\partial}$a...

$\mathcal{T}_{Z}^{n,q}arrow 0$,

where the vertical arrows are the multiplications by the function. Iview of (9), they should be the multiplication by the function . However,since is acomplex polynomial, the functionis aconstant with respect
to the differential , namely, it is an anti-holomorphic function, and s
can be replaced by. This replacement is acrucial
trick in our argument. Since the function
and all its derivatives areat most of polynomial growth as, each vertical arrow in (13) is an
isomorphism, yielding an isomorphism between (11) and (12). In this manner(11) has been untwisted and transformed into (12). Finally the exactness of(12) is obtained from Lemma 3.4 below by dualizing the sequence (14).
$e^{2i{\rm Im} f}$$\mathrm{n}$$e^{f}$$e^{-\overline{f}}$$f$$e^{f}$$\partial$$\mathrm{o}$$e^{f}e^{-\overline{f}}=e^{f-\overline{f}}=e^{2i{\rm Im} f}$$e^{2i{\rm Im} f}$$|z|$$arrow\infty$

Lemma 3.4 Let
denote the space of smooth $(p, q)$ -forms of Schwartz
class on Z. The following sequence is exact:$S_{Z}^{p,q}$$0arrow S_{Z}^{0,q}arrow^{\partial}S_{Z}^{1,q}arrow\partial$...

$arrow^{\partial}S_{Z}^{n,q}$

.

(14)

8
This lemma is established by using the Fourier transform, the Borel-Litttheorem, adivision theorem for smooth functions, the flatness of the formal
power series ring over the polynomial ring and the exactness of an algebraic
Koszul complex.

4Twisted Hodge-Kodaira Decomposition
The strategy for proving Theorem 2.2 is to develop atwisted version of
Hodge-Kodaira theory and acrucial role is played by the twisted Laplacian:$\Delta_{f}=d_{f}d_{f}^{*}+d_{f}^{*}d_{f}$,is the formal Hermitian adjoint of . Asimple inspection showswhereis unitarily equivalent to the real twisted Laplacianthat
associated with the real part of , namely,$d_{f}$$d_{f}^{*}$$\Delta_{g}=d_{g}d_{g}^{*}+d_{g}^{*}d_{g}$$\Delta_{f}$

$g$

$f$

wwiitthh g$\Delta_{f}=e^{:{\rm Im} f}\Delta_{g}e^{-i{\rm Im} f}$f.

$={\rm Re}$(15)

It should be mentioned here that Witten [24] used the twisted Laplaon aRiemannian manifold to develop his Morse theory as asupercian
symmetric quantum mechanics, where is aMorse function and is alarge
parameter. Afterwards, Helffer and Sj\"ostrand [8] worked out to justify someof Witten’s speculations mathematically. On the other hand, we use thetwisted Laplacian rather differently; we are not concerned with Morse theory but with Hodge-Kodaira theory. Moreover our function $g={\rm Re} f$ is notnecessarily aMorse function.
. To state it
We need an explicit formula for the twisted Laplacian
we introduce some notation. Setting $m=2n$ and $z_{j}=x_{2j-1}+ix_{2j}(j=$, ), we think ofwith real,as the real Euclidean space. For an ordered subset
coordinates $x=$$\{1, \ldots, m\}$ , we write
and$=p$ . We employ aformula of Witten [24, formula (13)]. In our case this formula reads:$\Delta_{tg}$

$t$

$g$

$\Delta_{g}$$1$$\ldots$$Z=\mathbb{C}^{n}$$n$$\mathbb{R}^{m}$$I$$(x_{1}, \ldots, x_{m})$$dxi=dx_{i_{1}}\wedge\cdots\wedge dx_{\dot{\iota}_{\mathrm{p}}}$

Proposition 4.1 The twisted Laplacian$\Delta_{g}$

:

$I$

$|I|$

acting on$p$-forrms is written$\sum_{|I|=p}\phi_{I}dx_{I}\vdash\neq\sum_{|I|=p}\sum_{|J|=p}A_{IJ}\phi_{J}dx_{I}$where the matrix elements(i) Diagonal: if$\Delta_{g}$$=J$ ,$A_{IJ}$

are described as follows:

then$A_{II}= \Delta+|dg|^{2}+\sum_{i=1}^{m}\epsilon_{i}(I)$$=(i_{1}, \ldots, i_{p})\subset$$\frac{\partial^{2}g}{\partial x_{i}^{2}}$

,

,

9

a

is the usual Laplacian andwhere
is a sign function defined by$=-(\partial^{2}/\partial x_{1}^{2}+\cdots+\partial^{2}/\partial x_{m}^{2})$$\epsilon_{i}(I)$(ii) First off-diagonals: if

$|I$

$=\{$

$+1$

$(i\in I)$,-1$(i\not\in I)$.

$\cap J|=p-1$ , then$A_{IJ}=2(-1)^{\ell_{i}(I)+\ell_{j}(J)} \frac{\partial^{2}g}{\partial x_{i}\partial x_{j}}$whereelement of .

$i\in I\backslash J$

and$\epsilon_{i}(I)$$j\in J\backslash I$, and$\ell_{i}(I)$

$=k$

if

,

$i\in I$is the -th smallest
$k$

$I$

(iii) Remaining entries: if$|I$$\cap J|<p-1$ , then $A_{IJ}=-0$ .

may be thought ofThis proposition shows that the twisted Laplacian
as aSchrodinger operator with matrix-valued potential: $\Delta_{g}=\Delta+|dg|^{2}+V$ ,and the tri-diagonal
where the potential term consists of the scalar part$V$
matrix part . Since is acomplex polynomial having as its real part, theCauchy-Riemann equation for yields $|dg|^{2}=|df|^{2}$ . Moreover, since is a
as its top homogeneous component,
polynomial of degree $N$ havingof degree
is expressed as the sum of the real homogeneous polynomial
2 $(N-1)$ with areal polynomial of degree less than 2 $(N-1)$ . On the otheris areal polynomial ofhand, Proposition 4.1 implies that each entry ofis written in the form:
degree less than $N-1$ . Hence$\Delta_{g}$$|dg|^{2}$

$f$

$g$

$f$

$f$

$|dg|^{2}$$f_{0}$$|df_{0}|^{2}$$V$$\Delta_{g}$,$\Delta_{g}=\Delta+|df_{0}|^{2}+U$

where is a -diagonal symmetric matrix whose entries are real polynomialsof degree less than $2(N-1)$ . Assumption 1.1 then assures that the principalof the potential has auniform polynomial growth of degree $2(N-$term$1)\geq 2$ as $|x|arrow\infty$ . This fact leads us to atwisted version of Hodge-Kodaira.
decomposition on the open manifold$U$$\mathrm{t}\mathrm{r}\mathrm{i}$$|df_{0}|^{2}$$Z=\mathbb{R}^{m}$

Theorem 4.2 (Twisted Hodge-Kodaira Decomposition) Every-harmonic smooth differential
harmonic tempered current is necessarily aform of Schwartz class, and vice versa, namely,$\Delta_{g^{-}}$$\Delta_{g}$$\{\phi\in \mathcal{T}_{Z}^{\cdot} :\Delta_{g}\phi=0\}=\{\phi \in S_{Z}^{\cdot} :\Delta_{g}\phi=0\}$

.

(16)

The linear space
defined by the both sides of (16) is finite dimensional:
There exists a $W$ -equivariant, contin tous, linear operator,:{Green operator), which restricts to a continuous operator
such that the following conditions are satisfied$\mathcal{H}_{g}^{\cdot}$$G_{g}$$G_{g}$$\mathcal{T}_{Z}^{\cdot}arrow \mathcal{T}_{Z}^{\cdot}$$S_{Z}^{\cdot}arrow S_{Z}^{\cdot}$10(i) There exist $W$ -equivariant direct sum decompositions:$\mathcal{T}_{Z}^{\cdot}$$S_{Z}^{\cdot}$$=\mathcal{H}_{g}^{\cdot}\oplus d_{g}d_{g}^{*}G_{g}\mathcal{T}_{Z}^{\cdot}\oplus d_{g}^{*}d_{g}G_{g}\mathcal{T}_{Z}.$.

$=\mathcal{H}_{g}^{\cdot}\oplus d_{g}d_{g}^{*}G_{g}S_{Z}$$\oplus d_{g}^{*}d_{g}G_{g}S_{Z}^{\cdot}$

,

(17)
(18)

.

(ii) Let:be the projection relative to the decomposition (17).
Then it restricts to the projection:
relative to the decomposition (18). Moreover, as operators acting on
or on . ’ thefollowing commutation relations are valid:$H_{g}$$\mathcal{T}_{Z}^{\cdot}arrow \mathcal{H}_{g}^{\cdot}$$H_{g}$$S_{Z}^{\cdot}arrow \mathcal{H}_{g}^{\cdot}$$\mathcal{T}_{Z}^{\cdot}$$S_{Z}$(a) I $=H_{g}+\Delta_{g}G_{g}=H_{g}+G_{g}\Delta_{g}$ ,(b) $H_{g}G_{g}=G_{g}H_{g}=H_{g}\Delta_{g}=\Delta_{g}H_{\mathit{9}}=0$,(c) $d_{g}G_{g}=Ggdg$ ,$d_{g}^{*}G_{g}=G_{g}d_{g}^{*}$.

Theorem 4.2 implies that both of the cohomology groups
and$W$
are -equivariantly isomorphic to the harmonic space . Inview of (15), Theorem 4.2 clearly leads to the same type of decomposition
theorem for the twisted Laplacian . Thus we obtain the following corollary.
$H.(\mathcal{T}_{Z}., d_{g})$$H^{\cdot}(S_{Z}.

, d_{g})$$\mathcal{H}_{g}^{\cdot}$$\Delta_{f}$Corollary 4.3 There exist $W$ -equivariant isomorphisms:$H^{\cdot}(S_{Z}^{\cdot}, d_{f})\simarrow \mathcal{H}_{f}^{\cdot}arrow H^{\cdot}(\sim \mathcal{T}_{Z}^{\cdot}, d_{f})$

,

where
is the space of 1 -harmonic tempered -currents on $Z$ , or equivalently the space of -harmonic smooth forms of Schwartz class on .
$\mathcal{H}_{f}^{\cdot}$

$p$

$f$

$Z$

$\Delta_{f}$Once Theorem 4.2 is established, Theorem 2.2 readily follows from Corollary 4.3. Basically the proof of Theorem 4.2 follows the standard arguments
as in Wells [23, Chap. 4]. An essential difference here is to make use of thepseud0-differential calculus of KumanO-go and Taniguchi [14]. They developed aglobal calculus which allows for weight functions of polynomial growthasand permits aglobal treatment of operators
its inverse (see also Beals [5]).
$|x|arrow\infty$$1\mathrm{i}\mathrm{k}\mathrm{e}-\Delta+|x|^{2k}$and5Real Structure
There are natural real structures oncompatible with the duality.
Let us recall some terminology. Areal structure on acomplex vector space,say, , is an anti-CMinear automorphism : $Varrow V$ such that $J^{2}=1$ .
Given areal structure on , let $V_{\mathbb{R}}=\{v\in V : Jv=v\}$ . Thenis a
$H^{n}(\Omega_{Z}.

$V$, d_{\pm f})$

$J$

$J$

$V$

$V_{\mathbb{R}}$11real vector space such that
. We refer to 6 as the real partof V relative to J. When agroup W acts on V, areal structure J on V is
said to be $W$ -equivariant if J commutes with the action of W on V. If J is
-equivariant, then 6 is acted on by W in anatural manner.
$\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}\mathrm{H}(3_{1}\mathrm{C}\ovalbox{\tt\small REJECT}$$\mathrm{I}\mathrm{I}^{\ovalbox{\tt\small REJECT}}$

Theorem 5.1 The cohomology groups
admit W-equivariant
real structures:
such that the W-invariant
complex duality (4) restricts to a $W$ -invariant real duality:$H^{n}(\Omega_{Z}.

$J_{\pm f}$$H^{n}(\Omega_{Z}.

, d_{\pm f})arrow H^{n}(\Omega_{Z}.

, d_{\pm f})$, d_{\pm f})$,$H_{\mathbb{R}}^{n}(\Omega_{Z}^{\cdot}, d_{f})\cross H_{\mathbb{R}}^{n}(\Omega_{Z}^{\cdot}, d_{-f})arrow \mathbb{R}$

where

,

$H_{\mathbb{R}}^{n}(\Omega_{Z}.$$d_{\pm f})$denote the real par ts of(19)$H^{n}(\Omega_{Z}^{\cdot}, d_{\pm f})$relative to$J_{\pm f}$.

The real structures mentioned in Theorem 5.1 are constructed as follows.
There exists (well-defined) anti-C linear automorphisms:$J_{\pm f}$:$S_{Z}^{p}arrow S_{Z}^{p}$,$\psi$}$arrow e^{\mp 2i{\rm Im} f}\overline{\psi}$.

It is easy to see thatgive real structures on . Asimple check shows that
commute with the twisted exterior differentials
and sodefinereal structures on the twisted de Rham complexesof Schwartz
class. Passing to cohomology, they induce $W$ -equivariant real structureson. Then, through the $W$-equivariant isomorphism (8), one
obtains $W$ -equivariant real structureson. The complex
duality (4) induces the real duality (19), when restricted to the real parts.
$J_{\pm f}$$S_{Z}^{p}$$J_{\pm f}$$d_{\pm f}$$(S_{Z}.

$J_{\pm f}$, d_{\pm f})$$J_{\pm f}$$H^{n}(S_{Z}^{\cdot}, d_{\pm f})$$J_{\pm f}$$H^{n}(\Omega_{Z}.

, d_{\pm f})$Corollary 5.2 Let $W$ be a finite unitary reflection group and set $T=Z/W$ .
Then there exist natural real structures onsuch that the complex
duality in Corollary 1.3 restricts to a real duality:$H^{n}(\Omega_{T}.

, d_{\pm f})$$H_{\mathbb{R}}^{n}(\Omega_{T}^{\cdot}, d_{f})\cross H_{\mathbb{R}}^{n}(\Omega_{T}^{\cdot}, d_{-f})arrow \mathrm{R}$.

6Super-symmetry
It is interesting to consider atransformation that permutes thegroups,under which the duality behaves naturally.

$\pm \mathrm{c}\mathrm{o}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{y}$$H^{n}(\Omega_{Z}.$ $d_{\pm f})$Definition 6.1 Given afinite subgroup $W$ of $U(n)$ and a $W$-invariant polynomial , aunitary transformation $\tau\in U(n)$ is called asuper-symmetry
relative to $(W, f)$ , if the following conditions are satisfied:$f$$\tau^{2}\in W$

and$\tau^{*}f=-f$.

12
acting on
Here is asimple example. Let $W$ be the symmetric groupby permuting the coordinates, and be asymmetric polynomial having
only odd homogeneous components. Then $\tau=-1$ is asuper-symmetry.
Asuper-symmetry induces $W$ -equivariant isomorphisms of de Rham.
, and then,, wherecomplexes :’$W$ -equivariant isomorphisms of de Rham cohomology groups:$S_{n}$$f$$\mathbb{C}^{1}$$\tau$$\tau$$F_{Z}^{\cdot}=\Omega_{Z}$$(F_{Z}^{\cdot}, d_{\pm f})arrow\sim(\mathcal{F}_{Z}^{\cdot}, d_{\mp f})$$\tau$:$H^{n}(F_{Z}^{\cdot}, d_{\pm f})\simarrow H^{n}(F_{Z}^{\cdot}, d_{\mp f})$$S_{Z}^{\cdot}$$\mathcal{T}_{Z}^{\cdot}$$(\mathcal{F}_{Z}^{\cdot}=\Omega_{Z}^{\cdot}, S_{Z}^{\cdot}, \mathcal{T}_{Z}^{\cdot})$

.

(20)

Note that they are compatible with the isomorphisms (8). Thus asupergroups. The duality (4), combined
symmetry permutes thewith the super symmetry (20), induces a $W$ -invariant self-duality:$\pm \mathrm{c}\mathrm{o}\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{y}$$H^{n}(\Omega_{Z}^{\cdot}, d_{f})\cross H^{n}(\Omega_{Z}^{\cdot}, d_{f})arrow \mathbb{C}$,$(\phi_{1}, \phi_{2})-\rangle\langle\langle\phi_{1}, \phi_{2}\rangle\rangle:=\langle\phi_{1}, \tau\phi_{2}\rangle$. (21)

Note that (21) is compatible with the real structure . The self-duality (21)has abeautiful property, when restricted to the $W$ -invariant components.
$J_{f}$

Theorem 6.2 For a super-symmetry relative to $(W, f)$ , the associated selfduality (21) induces a nondegenerate bilinear $fom$ :$\tau$$H^{n}(\Omega_{Z}^{\cdot}, d_{f})^{W}\cross H^{n}(\Omega_{Z}^{\cdot}, d_{f})^{W}arrow \mathbb{C}$which is symmetric when$n$,

is even, and skew-symmetric when$n$

is odd.

denote the $W$ -invariant component of the real cohomology, or equivalently the real part of the $W$ -invariant complex
group. The compatibility of the self-duality (21)cohomology groupleads to arefinement of Theorem 6.2.
with the real structureLet$H_{\mathbb{R}}^{n}(\Omega_{Z}.

, d_{f})^{W}$$H_{\mathbb{R}}^{n}(\Omega_{Z}.

, d_{f})$$H^{n}(\Omega_{Z}^{\cdot}, d_{f})^{W}$$J_{f}$

Theorem 6.3 For a super-symmetry relative to $(W, f)$ , the self-duality(21) induces a real nondegenerate bilinear form:$\tau$,$H_{\mathbb{R}}^{n}(\Omega_{Z}^{\cdot}, d_{f})^{W}\cross H_{\mathbb{R}}^{n}(\Omega_{Z}^{\cdot}, d_{f})^{W}arrow \mathrm{R}$which is symmetric when$n$

is even, and skew-symmetric when$n$

is odd.

When $W$ is afinite unitary reflection group, Theorem 6.3 yields:
Corollary 6.4 Let $W$ be a finite unitary reflection group and set $T=Z/W$ .
For a super-symmetry relative to $(W, f)$ , there exists a real nondegeneratebilinear $fom$ :(22).
$\tau$$H_{\mathbb{R}}^{n}(\Omega_{T}^{\cdot}, d_{f})\cross H_{\mathbb{R}}^{n}(\Omega_{T}^{\cdot}, d_{f})arrow \mathrm{R}$which is symmetric when n is even, and skew-symmetric when n is odd.

13
We pose the problem of constructing abasis ofand computing
the intersection matrix of the self-duality (22) relative to that basis for various
finite unitary reflection groups $W$ and various $W$ -invariant polynomialssatisfying Assumption 1.1. It is also an interesting problem to discuss theconnection of our construction with the classical Picard-Lefschetz theory for
isolated surface singularities, vanishing cycles, Hodge structures, and so on.
$H_{\mathbb{R}}^{n}(\Omega_{T}.

, d_{f})$$f$

7Generalized Airy functions
The classical Airy function in single-variable is defined by an integral:Ai(a)$= \int_{c}e^{\frac{1}{3}t^{3}+at}dt$(a$\in \mathbb{C})$,where is acycle chosen in such away that the integrand is exponentially decreasing at infinity along (see Figure 2). The Airy function is an important
special function arising in mathematical optics (see Airy [1]). Ageneralization of the Airy function into several variables was introduced by Gel’fand,Retakh and Serganova [7] and was studied in some depth by Kimura $[12, 13]$ .
$c$

$c$

Figure 2: Cycles for the Airy integral
Following [7, 12], we recall the definition of ageneralized Airy function.
First, let
be the -th coefficient of agenerating series:$\theta_{k}(t)$$k$$\log(1+t_{1}X+\cdots+t_{n}X^{n})=\sum_{k=1}^{\infty}\theta_{k}(t)X^{k}$

Then
t
$=$

$\theta_{k}(t)$$(t_{1},$is aweighted homogeneous polynomial of degree k in variables, where is assumed to be of degree j. For small values of k,$t_{n})$\ldots ,.

$t_{j}$

14$\theta_{1}(t)=t_{1}$$\theta_{2}(t)=t_{2}-\frac{1}{2}t_{1}^{2}$03$(t)=t_{3}-t_{2}t_{1}+ \frac{1}{3}t_{1}^{3}$$\theta_{4}(t)=t_{4}-t_{3}t_{1}-\frac{1}{2}t_{2}^{2}+t_{2}t_{1}^{2}-\frac{1}{4}t_{1}^{4}$$\theta_{5}(t)=t_{5}-t_{4}t_{1}-t_{3}t_{2}+t_{3}t_{1}^{2}+t_{2}^{2}t_{1}-t_{2}t_{1}^{3}+\frac{1}{5}t_{1}^{5}$

Table 1: Polynomials
polynomials$\theta_{k}(t)$$\theta_{k}(t)$are illustrated in Table 1. Set$f=f(a, t)= \sum_{k=0}^{N}(-1)^{k}e_{k}(a)\theta_{N-k+1}(t)$,is the -th elementary symmetric polynomial of $a=(a_{1}, \ldots, a_{N})$where
be the complex -space with coordinates $t=$with $N\geq n$ . Let. Next we define afamily of supports in as follows: an element, quicker thanof is aclosed subset of such that(see Pham [18]). A
for some $q>0$ as
generalized Airy function is now defined by an integral:$k$$e_{k}(a)$$T=\mathbb{C}^{n}$

$n$

$T$

$\Phi$$(t_{1}, \ldots, t_{n})$$\Phi$$c$$-||t||^{q}$$T$${\rm Re}\theta_{N+1}(t)|_{c}arrow-\infty$$||t||:= \sum_{j=1}^{n}|t_{j}|^{1/j}arrow \mathrm{o}\mathrm{o}$$A(a)= \int_{\mathrm{c}}e^{f(a,t)}\omega$,

is a -closed polynomial -form and is an -cycle with. This integral depends only on the cohomology classdenote theand the homology class $[c]\in H_{n}^{\Phi}(T)$ , where
integral -th homology group of with supports in .
such that
If we introduce new variables $z=$

where
support in$\omega$$H^{n}(\Omega_{T}.

$\in\Omega_{T}^{n}$$n$$d_{f}$$n$$c$$[\omega]\in$$\Phi$$H_{n}^{\Phi}(T)$, d_{f})$

$T$

$n$

$\Phi$$(z_{1}, \ldots, z_{n})$$(j=1, \ldots, n)$$t_{j}=(-1)^{j}e_{j}(z)$,is the -th elementary symmetric polynomial of , then we canwhereeasily see that , as apolynomial of , satisfies Assumption 1.1. In this caseby permuting theacting on
the group $W$ is the symmetric group, $n)$ . Thus we havecoordinates, having degrees $d=$ 1, 2,$e_{j}(z)$

$z$

$j$

$f$

$z$

$Z=\mathbb{C}^{n}$$S_{n}$$($$\dim H^{n}(\Omega_{T}^{\cdot}, d_{\pm f})=\mu$$\ldots$,where$\mu=(\begin{array}{l}Nn\end{array})$.

15
There are bases of the cohomology groups
written in termsof Schur polynomials. Given aYoung diagram , letdenote the Schurpolynomial in attached to the diagram A(see Macdonald [15]). Let $R(p, q)$
be the rectangular Young diagram with rows and columns, and let
be the set of all Young subdiagrams of $R(p, q)$ .
$H^{n}(\Omega_{T}.

$\lambda$, d_{\pm f})$$s_{\lambda}(z)$

$z$

$p$

$\mathcal{Y}(p, q)$$q$

Theorem 7.1 Denote by
the cohomology classes in
reprersented by the polynomial differential -form. Then the sets$N-n)$ } form bases of the cohomology groups.
{ : A$H^{n}(\Omega_{T}.

$\phi_{\lambda}^{\pm}$$s_{\lambda}(z)dt_{1}\wedge\cdots\wedge dt_{n}$$n$$\phi_{\lambda}^{\pm}$, d_{\pm f})$$\in \mathcal{Y}(n,$$H^{n}(\Omega_{T}^{\cdot}, d_{\pm f})$

We are interested in the intersection matrix of the duality (5) relative tothe bases in Theorem 7.1. Iwasaki and Matsumoto [11] were able to calculate
it explicitly in terms of skew-Schur polynomials. Let
denote theskew-Schur polynomial of $a=$
attached to apair
of Young
diagrams (see Macdonald [15]). To state the result, we need the concept ofcomplementary diagrams. Given aYoung diagram, its complementary diagram A
is defined by$s_{\lambda/\mu}(a)$$(a_{1}, \ldots, a_{N})$$(\lambda, \mu)$$\lambda$$=$$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{p})\in$$\in \mathcal{Y}(p, q)$$\mathcal{Y}(p, q)$$\check{\lambda}=(q-\lambda_{p}, q-\lambda_{p-1}, \ldots, q-\lambda_{1})$.

Pictorially, is obtained by rotating the rectangle $R(p, q)$ , together with ,around its center by 180 and then deleting Afrom $R(p, q)$ . For instance, aYoung diagram $\lambda=(5,3,3,3,2,1,1)$ has the complementary diagram$(4,4,3,2,2,2,0)$ in(see Figure 3). Taking complementary diagrams
defines an involution on the set.
$\check{\lambda}$$\lambda$$\lambda=$$\mathcal{Y}(7,5)$$\lambda\vdasharrow\check{\lambda}$$\mathcal{Y}(p, q)$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$rightarrow$

A $=(5,$ 3,3,3,2,1,1)

A $=(4,$ 4,3,2,2,2,0)

Figure 3: Complementary Diagram$\mathrm{s}$16Theorem 7.2 With respect to the bases
the intersection pairing$H^{n}(0\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}, d_{f})\mathrm{x}H^{n}(*\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT},$$\langle\phi_{\lambda}^{+}, \phi_{\mu}^{-}\rangle=(-1)^{\frac{1}{2}n(n-1)}n!s_{\lambda/\overline{\mu}}(a)$constructed in Theorem 7.1,
dis represented as$\{+\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\}$for$f)-+\mathrm{C}$$\lambda$, $\mu\in \mathcal{Y}(n, N-n)$ .

This theorem provides us with acohomological interpretation of skew-Schurpolynomials by means of atwisted intersection theory.

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