数理解析研究所講究録924 巻 1995 年 198-207

198

July 17, 1995$\mathbb{Z}$-forms of representations of reductive groups
and prehomogeneous vector spaces

AKIHIKO GYOJA

In [1], the representations of reductive group schemes are discussed, andespecially the concept of the ‘split form’ is defined. In the present article, first we

review [1] in a most elementary way, restricting ourselves to the case where the basescheme is$\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{c}}}$Z. Then we discuss how such a general theory can be applied to thetheory of prehomogeneous vector spaces.

1. Reductive group scheme.
1.1. A reductive group scheme overis affine and smooth over$\mathbb{Z}$$\mathbb{Z}$is by definition a group scheme which

whose geometric fibres are connected reductive [5, expos\’e19, 2.7]. (More generally, for any commutative ring$A$or for any scheme

similarly define the concept of reductive group scheme over

$A$

$S$

, we canor $S.$ )1.2. Remark. If the connectedness is not assumed, I do not know how todefine the concept of ‘reductive group scheme’. If the fibre dimension is , then it$0$

would be natural to assume that it is finite \’etale.
1.3.. Remark. In order to consider the bad reduction, it is interesting to

remove the smoothness assumption from the definition of the reductive group scheme.
1.4. Example.
$GL_{n,\mathrm{Z}}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{z}[\{xij\}1\leq i,j\leq n’\det(x_{i}j)^{-1}])$

and

(1)
$SL_{n,\mathrm{Z}}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{z}[\{x_{i}j\}1\leq i,j\leq n]/(\det(x_{ij})-1)\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}1)$

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are reductive group schemes over Z. We shall denote their coordinate rings (i.e., theinside of$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{C}($

$))$

by

$\mathbb{Z}[GLn,\mathrm{Z}]$(2) Let $f:=u_{1}^{2}+\cdots+u_{n}^{2}$ andrespect to

$f$

etc.
$G(\mathrm{C}):=SO_{n}(\mathrm{C})$be the special orthogonal group with(i.e., the group of the usual orthogonal matrices with determinant 1).

Put

$I:=$

{

$\varphi\in \mathbb{Z}[SL_{n},\mathrm{Z}]|\varphi\equiv 0$$G\mathrm{z}:=\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{z}[SL_{n},\mathrm{Z}]/I)$on$so_{n}(\mathrm{C})$},,$G_{A}:=G_{\mathrm{Z}}\otimes_{\mathrm{Z}}A(=G_{\mathrm{Z}}\cross_{\mathrm{S}\mathrm{e}\mathrm{c}}\mathrm{z}\mathrm{p}\mathrm{P}\mathrm{s}\mathrm{e}\mathrm{c}A)$

for any commutative ringand

$G_{\mathrm{Z}}(\mathrm{C})=SO_{n}(\mathrm{C})$$A$. Then] is a reductive group scheme over$G_{\mathrm{Z}[1/2}$

is a reductive algebrai-c group, but

scheme over Z. In fact,$G_{\mathrm{Z}}(\overline{\mathrm{F}}_{2})\subset GL_{n}(\overline{\mathrm{F}}_{2})$$G_{\mathrm{Z}}$is$\mathbb{Z}[1/2]$,

no. a reductive group$t\backslash$is conjugate with$\{\in SL_{n}(\overline{\mathrm{F}}_{2})\}$

in$GL_{n}(\overline{\mathrm{F}^{:_{2}}}),$

of$G_{\mathrm{Z}}$

at

where

$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\overline{\mathrm{F}}2$$\overline{\mathrm{R}}2$

is an algebraic closure of$\mathrm{R}_{2}$, especially the geometric fibre

is not reductive. This phenomenon occurs because$f$becomes adegenerate quadratic form after reduction modulo 2.
It would be worth noting here that

$G_{\mathrm{Q}}$has no model over$\mathbb{Z}$which is reductive

over Z.
(3) Note that, in (2), we can construct the group scheme

quadratic form . If we start from$f$$\sum_{i=1}^{m}x_{i+}x_{m}i$$(n=2m)$$\sum_{i=1^{X}}^{m}iXm+i+x_{2m+1}^{2}$$(n=2m+1)$ ,$f=\{$$G_{\mathrm{Z}}$from any

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then the resulting group scheme is reductive over , whose geometric fibres are always$\mathrm{Z}$special orthogonal groups.
2. Split form of a representation.

2.1. Definition of a split Z-form.

Notation.
$G=G_{\mathrm{C}}=\mathrm{a}$$T=T_{\mathrm{C}}=\mathrm{a}$connected complex reductive group.
maximaI torus.

$M=\mathrm{H}\mathrm{o}\mathrm{m}(T, \mathrm{c}^{\mathrm{x}})$$V=V_{\mathrm{C}}=$.

a finite dimensional multiplicity free rational G-module.

$V=\oplus_{i}V_{i}=\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}$weight space decomposition with respect to$V=\oplus_{\mu\in M}V_{\mu}=\mathrm{t}\mathrm{h}\mathrm{e}$$\mathcal{U}_{\mathrm{Z}}=\mathrm{t}\mathrm{h}\mathrm{e}\mathbb{Z}$where {

$X_{\alpha}|\alpha=$

Consider a triple$G$-subalgebra of the enveloping algebra$U(\mathfrak{g})$generated by, where$(T,\mathcal{U}_{\mathbb{Z}}, V(\mathbb{Z}))$$V(\mathbb{Z})$is a free -submodule of$\mathbb{Z}$(2)$\mathcal{U}_{\mathrm{Z}}\cdot V(\mathbb{Z})\subset V(\mathrm{z})$$X_{\alpha}^{m}/m!$,$V$such that,$V(\mathbb{Z})=\oplus_{\mu\in M}V(\mathrm{Z})\cap V_{\mu}$.

is semisimple, then the condition (3) is redundant [8, p.17, Corollary 1],$(g\tau g^{-1},g\mathcal{U}\mathrm{Z}g^{-}, \sigma gV1(\mathbb{Z}))$lence classes (or
and

.

,

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}_{\mathrm{Z}}V(\mathrm{Z})=\dim_{\mathrm{C}}V$

but it is necessary in general.) Consider the equivalence relation

$T$

$T$

root} is a Chevalley system. See [3, \S 1] for Chevalley system.

(1)

(3)
(If

decomposition.

$\mathcal{U}_{\mathrm{Z}}$for$g\in G(\mathrm{C})$$(T,u_{\mathrm{Z}},$ $V(\mathbb{Z}))$anditself) a split$\sigma\in \mathrm{A}\mathrm{u}\mathrm{t}_{G}V$$\mathbb{Z}$$(T,\mathcal{U}_{\mathrm{Z}}, V(\mathbb{Z}))\sim$. We call each of the equiva--form of $(G, V)$ . (If we can understandfrom the context, we sometimes call abusively$V(\mathbb{Z})$a split Z-form.)

2.2. Dual. Let $(G, V^{\vee})=(G, \oplus_{i}V_{i}^{})$ be the dual of $(G, V)$ . Put$\{v^{\vee}\in V^{}|\langle v^{}, V(\mathrm{z})\rangle\subset \mathbb{Z}\}$$(G, V^{\vee})$($=$the dual lattice). Then

$V^{\vee}(\mathrm{Z})$is a split$V^{\vee}(\mathbb{Z})$$\mathbb{Z}$$:=$-form of.
2.3. Minimal split$\mathrm{Z}$-form. Let$\mu_{i}$be a weight of $(G, V_{i})$ which is highest

201with respect to some fixed Borel subgroup $B$ containing . Take
$T$

put

$V_{\min}(\mathbb{Z}):=\oplus_{i}\mathcal{U}\mathrm{z}\cdot v_{i}$$\oplus_{i}\mathbb{Z}v_{i}$. Thenis a split -form such that$\mathbb{Z}$and$V_{\min}(\mathbb{Z})\cap\oplus iV_{\mu}i=$.
2.4. Maximal split

Put$V_{\min}(\mathbb{Z})$$0\neq v_{i}\in V_{i}\cap V_{\mu}.\cdot$

and$V_{\min}^{}(\mathbb{Z}):=\oplus_{i}\mathcal{U}_{\mathrm{Z}}\cdot v^{\mathrm{v}}i$

is a split -form such that$\mathbb{Z}$$\mathbb{Z}$-form. Take$v_{i}^{}\in V_{i}^{}\cap V_{\mu:}^{}$

so that

$V_{\max}(\mathbb{Z}):=\{v\in V|\langle v, V_{\min(}^{}\mathbb{Z}))\subset \mathbb{Z}\}$$V_{\max}(\mathbb{Z})\cap\oplus_{i}V_{\mu:}=\oplus_{i}\mathbb{Z}v_{i}$2.5. (1) Every split -form$\mathbb{Z}$$V(\mathbb{Z})$$(v_{i}^{\vee}, v_{i})=1$. Then.

$V_{\max}(\mathbb{Z})$.

normalized so that$\oplus_{i}V(\mathrm{z})\cap V_{\mu}.\cdot=\oplus_{i}\mathbb{Z}v_{i}$satisfies$V_{\min}(\mathbb{Z})\subset V(\mathrm{z})\subset V\mathrm{m}\mathrm{a}\mathrm{x}(\mathbb{Z})$Conversely, any -submodule$\mathbb{Z}$$V(\mathbb{Z})$of$V(\mathrm{C})$.

such that$V_{\min}(\mathbb{Z})\subset V(\mathbb{Z})\subset V_{\max}(\mathbb{Z})$, and$\mathcal{U}\mathrm{z}V(\mathbb{Z})\subset V(\mathbb{Z})$$V( \mathbb{Z})=\bigoplus_{\mu\in M}V(\mathrm{z})\cap V_{\mu}(\mathrm{c})$gives a split Z-form.
Problem A. When two such

$V(\mathbb{Z})’ s$are?

$equi_{\mathrm{V}}\mathrm{a}lent.$

If the answer of the following problem is affirmative,$\mathrm{t}\mathrm{h}^{\backslash }\mathrm{e}\mathrm{n}$different$V(\mathbb{Z})’ \mathrm{s}$are never equivalent.

Problem B.

?

$V(\mathbb{Z})=\oplus_{i}(V_{i}(\mathrm{C})\cap V(\mathbb{Z})).$(2) If $(G, V)$ is a (not necessarily reduced) saturated [3, Introduction], irre-

ducible, regular, prehomogeneous vector space, then$V_{\max}(\mathbb{Z})/V_{\min}(\mathbb{Z})=0$(The proof uses the classification of M.Sato and T.Kimura. Note thator$=\mathbb{Z}/2\mathbb{Z}$$V_{\min}(\mathbb{Z})$

.

and

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depends only on$V_{\max}(\mathbb{Z})$$\mathcal{U}_{\mathrm{Z}}$, and that they behave well under the castling transfor-

mation.) Hence there are at most 2 split$\mathbb{Z}$-forms. More precisely, there is only one

split -form for the prehomogeneous vector space of type (1), (3), (5), (6), (7), (9),$\mathbb{Z}$(10), (11), (12), (13), (15D), (20), (21), (23), (24), (27), (28), (29). There are 2 split$\mathbb{Z}$-forms for the type (2), (4), (8), (14), (15B). Here the number refers to that of [6,\S 7]. The type (15),if$n$$\mathrm{i}.\mathrm{e},$is referred to as (15B) (resp.

$(SO_{n}\cross GL_{m}, \mathrm{C}^{n}\otimes \mathrm{C}^{m})$$(15\mathrm{D})$)is odd (resp. even). See [3] for the detail.
2.6.

Example. If $G=GL_{n}$ andis the totality of$V$$n\cross n$symmetric

matrices. Then

$V_{\min}(\mathbb{Z})=\{(xij)\in V|xij\in \mathbb{Z}\}$,$V_{\max}(\mathbb{Z})=\{(xij)\in V|x_{ii}\in \mathrm{z}, 2x_{ii}\in^{\mathrm{z}}(i\neq j)\}$2.7. Geometric meaning of split$\mathbb{Z}$$\mathbb{Z}$-form. Let.

$(T,\mathcal{U}\mathrm{z}, V(\mathbb{Z}))$be a split-form. Then, we get(1) a Chevalley-Demazure group schemesuch that$\mathbb{Z}M$$G_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathrm{c}=G_{\mathbb{C}}$and which contains$G_{\mathrm{Z}}$($=\mathrm{a}$split reductive group scheme)$T_{\mathbb{Z}}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{Z}M$

as a maximal torus, whereis the group ring of $M$ (cf. (2.1) for $M$ ),(2) a vector bundle

algebra of$V^{\vee}(\mathbb{Z})$$\otimes_{\mathrm{Z}}\mathrm{C}$, where$S(V^{}(\mathbb{Z}))$is the symmetric, and(3) the action

after$V_{\mathrm{Z}}=\mathrm{S}_{\mathrm{P}^{\mathrm{e}\mathrm{C}}}S(V^{\vee}(\mathbb{Z}))$

which becomes the original action $G\mathrm{x}Varrow V$$G_{\mathbb{Z}}\cross V_{\mathrm{Z}}arrow V_{\mathrm{Z}}$.

Let us add some explanation about (3). By (2.1, (3)), from a split

get a$\mathcal{U}_{\mathrm{Z}}$$T_{\mathrm{Z}}$$T_{\mathrm{Z}}$-action on$V_{\mathbb{Z}}$. Roughly speaking,$G_{\mathbb{Z}}$( . semisimple part), and hence we get a$=$-action and the$\mathcal{U}_{\mathrm{Z}}$-action on$V(\mathbb{Z})$.

consists of$G_{\mathbb{Z}}$$T_{\mathrm{Z}}$-action on($V_{\mathrm{Z}}$$\mathbb{Z}$-form we can$=\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l}$torus) andcombining the above

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2.8. General -forms. Now, let$\mathbb{Z}$$\mathbb{Z}$$(G_{\mathrm{Z}}’, V_{\mathbb{Z}^{/}})$-group scheme and a vector bundle overnumber of$a\mathbb{Z}$$\mathbb{Z}$is 1) such that$\mathrm{s}_{\mathrm{p}\mathrm{e}\mathrm{c}}\mathbb{Z}$be an arbitrary pair of a reductive(i.e., a$(G_{\mathbb{Z}}’, V_{\mathrm{Z}}’)\otimes \mathrm{C}=(G, V)$$\mathbb{Z}$-lattice, since the class. Such a pair-form of $(G, V)$ , and can be obtained from a split$(G_{\mathbb{Z}}, V\mathrm{z})$$(G_{\mathbb{Z}}’, V_{\mathrm{Z}}^{l})$is called

by twisting it using

non-abelian \’etale cohomology.
2.9. Remark. (1) In [1], we obtained (2.8) assuming the irreducibility ofIn order to obtain (2.8) assuming only that $(G, V)$ is multiplicity free, it is$(G, V)$ .

enough to replace the “highest weight vector”$v_{0}$

appearing in the definition of the“\’epinglage of a representation of a reductive group scheme” [1, (3.6)] by the “maximal

weight vectors$\{v_{i}\}_{i}$”(see (2.3) for$\{v_{i}\}_{i}$).

Problem C. Prove (2.8) without assuming the multiplicity freeness.
The essential difficulty is how to define “\’epinglage of a representation ofa reductive group scheme”. (Even without assuming the multiplicity freeness, we
can prove that, \’etale locally with respect to the base scheme, a representation of areductive group scheme over$\mathbb{Z}$can be obtained similarly as in (2.7). An “\’epinglage’’is a device which is used to patch together these local data to obtain a globally split
object.)(3) Although Problem$\mathrm{C}$is unsettled, “the multiplicity freeness” does notseem to be very harnfful for our application in the theory of prehomogeneous vector
spaces. In fact, if $(G, V)$ is a prehomogeneous vector space, and $V=\oplus_{i=1}^{N}V_{i}$ isan irreducible decomposition of$V$, consider $(\tilde{G}, V):=(G\cross GL_{1}^{N}, \oplus_{i=1}^{N}Vi)$ , where

the i-th factor of $GL_{1}^{N}=\{(x_{1}, \cdots, x_{N})\}$ acts ontrivially on the remaining$V_{j}’ \mathrm{s}$. Then$(\tilde{G}, V)$$V_{i}$

as a scalar multiplication andis a multiplicity free, and as is easily seen,

the relative invariant polynomials on the prehomogeneous vector spaces $(G, V)$ and$(\tilde{G}, V)$$G$

are the same. Since the relative invariants are of our main interest, replacingwith the larger group$\tilde{G}$, we can escape the difficulty.

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2.10. Polynomial with$\mathbb{Z}$-coefficients. Assume that a -form of $(G, V)$$\mathbb{Z}$is given (cf. (2.8)). Then we can consider the lattice
An element ofalgebraof polynomial functions onfunctions onfunction on

gives a linear function on$V^{\vee}(\mathbb{Z})(\subset V^{})$$\mathbb{Z}[V_{\mathrm{Z}}]:=S_{\mathrm{Z}}(V^{}(\mathbb{Z}))$

with

$V$

$V^{}$$\mathbb{Z}$$V(\mathbb{Z})$generated by$V$$V^{\vee}(\mathbb{Z})$$V$

and its dual lattice$V^{\vee}(\mathbb{Z})$.

, and hence the symmetric(over Z) can be regarded as a ring

. We shall consider an element of$\mathbb{Z}[V_{\mathbb{Z}}]$

as a polynomial-coefficients. In the same way, we can consider a polynomialwith Z-coefficients.

3. Application to the theory of prehomogeneous vector spaces–

Leading coefficients of $b(s)$ .
From now on, we assume that $(G, V)$ (cf.

\S 2) is a prehomogeneous vector

space. Concerning the prehomogeneous vector spaces, we use the notations of [2,(1.4) freely.
$]$

3.1. Take$\phi\in \mathrm{H}\mathrm{o}\mathrm{m}(G, \mathrm{C}^{\mathrm{x}})$invariant whose character is$\phi$. Let(resp.

not consider a -form of $(G, V),$$\mathbb{Z}$$f$$f\in \mathrm{C}[V]$$\phi^{-1}$and(resp.

$f^{}\in \mathrm{C}[V\vee]$) be a relative). (See [2, (1.4, (10), and (11)].) If we do$f^{}$are determined only up to$\mathrm{C}^{\cross}$. Hence theleading coefficient of $b(s)$ does not have a much significance. Now consider a Z-form$(G_{\mathrm{Z}}, V\mathrm{z})$of $(G, V)$ , and assume that(1) some constant multiples of

$f$

and

$f^{}$

are polynomial functions with Z-coefficients.

(This condition is automatically satisfied if $(G,$ $V)$ is irreducible.) Then first assume

that

$f$

and

$f^{}$

are of -coefficients, and next single out the common factor of the$\mathbb{Z}$coefficients. In this way, we can normalizecoefficient$f^{}$$b_{0}$

of $b(s)$ has a meaning up$\mathrm{t}\mathrm{o}\pm 1$

$f$

and

$f^{}$

up to$\pm 1$. Now multiplying, we may assume $b_{0}>0$ , and then the leading coefficient. Then the leading$\mathrm{s}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\pm 1$$b_{0}$

to

$f$

and

of the b–function is

uniquely determined without any ambiguity. In $(3.2)-(3.4)$ below, we assume that ,$f$$f^{},$$b(s)$

and$b_{0}$

are normalized in this way.

3.2. A strange formula. If $(G, V)$ is a (not necessarily reduced nor regular)

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irreducible prehomogeneous vector space and if the -form$\mathbb{Z}$$(G_{\mathrm{Z}}, V\mathrm{z})$is split, then (3.1,(1) is satisfied and$)$$b_{0}= \prod_{j\geq 1}(jj)e(j)$

,

where
$b^{\exp}(t)= \prod_{\geq j1}(tj-1)e(j)$See [2, (1.4, (24))] for$b^{\exp}$.

and $e(j)$ .

3.3. Example. Let $(G, V)$ be a reduced irreducible regular prehomogeneous

vector space of type (11) [6, \S 7, Table I]. Then a conjecture of I.Ozeki says$b(s)=(s+1)^{8} \{(S+\frac{2}{3})(_{S}+\frac{4}{3})\}4\{(s+\frac{3}{4})(s+\frac{5}{4})\}4\{(_{S}+\frac{5}{6})(S+\frac{7}{6})\}4$$\cross\{(_{S+\frac{7}{10}})(s+\frac{9}{10})(s+\frac{11}{10})(S+\frac{13}{10})\}^{2}$

.

Hence
$b^{\exp}(t)=\phi 18\emptyset_{3}^{4}\emptyset 4\phi_{6}44\phi^{2}10$

’

where
$b_{0}$$\phi_{j}$and$\prod_{j\geq 1}(j^{j})^{e}(j)=2^{562410}35$,

is the j-th cyclotomic polynomial (e.g., $\phi_{3}=t^{2}+t+1$ ). On the ther hand$\dot{\mathrm{o}}$is calculated by J.Murakami (1984.8.20) using a computer based on the method(3.7) below: $b_{0}=256324510$ .
3.4. Remark. I expect that (3.2) holds without assuming the irreducibility.

See [4, Remarks 7-9].
3.5. Even if we admit a degeneration of the geometric fibres ofleading coefficient$\prod_{j\geq 1}(t^{j}-1)^{e(j)}$primes at which$b_{0}$

of $b(s)$ seems to be divisible by$\prod_{j\geq 1}(j^{j})^{e(j)}$$G_{\mathbb{Z}}$, the, where $b^{\exp}(t)=$, and moreover the quotient seems to be a product of (powers of)$G_{\mathbb{Z}}$degenerates. In other words,$b_{0}/ \prod_{j\geq 1}(j^{j})^{e(j)}$seems to control

the bad reduction of a prehomogeneous vector space $(G, V)$ together with .
$f$

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3.6. Example. Let $f=x_{1}^{2}+\cdots+x_{n}^{2}$ and $f^{}=y_{1}^{2}+\cdots+y_{n}^{2}$ . Then$b(s)=4(_{S}+1)(s+ \frac{n}{2}),$$b0=4$ ,$(t-1)^{2}$if$(t^{2}-1)$

if

$n$

is even,$b^{\exp}(t)=\{$

if

$n$

4 if

$n$

1

is odd,is even,$n$$\prod_{j\geq 1}(j^{j})e(j)=\{$

3.7. The leading coefficient$b_{0}$

is odd.

of $b(s)$ can be calculated by the method

used in the proof of Proposition 2.7 of [7]. Let us explain it. In our notation,$b\mathrm{o}f(v)^{-1}=f^{\vee 1}((\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\log f)(v))=f(f(v)-.(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f)(v))=f(v)-d.f\mathrm{v}_{(}(\mathrm{g}\Gamma \mathrm{a}\mathrm{d}f)(v))$

,

i.e.,

(1)

$b_{0}=f(v)^{-}d+1f\vee((\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f)(v))$.

Take some , which is suitable for the calculation, and then evaluate the right hand$v$side of (1).
4. Second application –Hessian of $\log f$ .

Take a -form of a prehomogeneous vector space $(G, V)$ . Then we can con$\mathbb{Z}$

sider

put

$V(\mathbb{Z})$and its dual lattice$x_{i}:=v_{i}^{}$

and regard$V^{\vee}(\mathbb{Z})$. Let$\{x_{1}, \cdots, x_{n}\}$$\{v_{1}^{\vee}, \cdots, v_{n}^{}\}$

be a free -basis of$\mathbb{Z}$as a linear coordinate system of$V$$V^{\vee}(\mathbb{Z})$,. Then we

can consider(1)$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(\log f)=\det(\frac{\partial\log f}{\partial x_{i}\partial x_{j}})_{1\leq i,j\leq n}$

If we took an arbitrary linear coordinate system$\{x_{1}, \cdots, x_{n}\}$(1) depends on the choice of the coordinate and hence

defined over$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{S}(\log f)$$\mathrm{C}$, thenhas a meaning

only up to a constant multiple. However, under the normalization as above, for two

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coordinate systems
and especially$\{x_{1}, \cdots, x_{n}\}$$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(\log f)$and$\{x_{1}’, \cdots, x_{n}’\}$, the Jacobian$\det(\partial x_{i}/\partial x_{j}’)\mathrm{i}_{\mathrm{S}}\pm 1$,

is independent of the choice of the coordinate. Therefore

it is interesting to know its explicit form. This calculation is complicated, but can besomewhat simplified by using(1)$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(\log f)=(1-d)^{-}1f(x)-n\mathrm{H}\mathrm{e}\mathrm{S}\mathrm{S}(f)$,(cf. the proof of Proposition 10 of [6, pp.62-64]). Indeed, the right hand side is easierto calculate, although it is still difficult. Note that this quantity and some otherrelated quantity appear in [2, Theorem$\mathrm{C}$].

References[1] A.Gyoja, Representations of reductive groups schemes, Tsukuba J. Math. 15(1991), 315-346.
[2] A.Gyoja, Prehomogeneous vector spaces over finite fields, in the same volume.
[3] A.Gyoja, Split$\mathbb{Z}$-form of irreducible prehomogeneous vector spaces, in the same

volume.
[4] A.Gyoja, Construction of invariants, Tsukuba J. Math. 15 (1991), 335-346.
[5] S\’eminaire de g\’eom\’etrie alg\’ebrique, Sch\’emas en groupes, Lecture Notes in Math.

151, 152, 153 (1970), Springer.

[6] M.Sato, T.Kimura, A classification of irreducible prehomogeneous vector spaces

and their relative invariants, Nagoya Math. J. 65 (1977), 1-155.
[7] T.Shintani, On Dirichlet series whose coefficients are class numbers of integral

binary cubic forms, J. Math. Soc. Japan 24 (1974), 132-188.
[8] R.Steinberg, Lectures on Chevalley groups, Yale University.