数理解析研究所講究録924 巻 1995 年 208-253

208

July 17, 1995

Split -forms of irreducible prehomogeneous vector spaces$\mathrm{Z}$AKIHIKO GYOJAIntroduction.
Letbe a connected reductive group over$G$

representation, andvector space if$G$$V:=\mathrm{C}^{n}$$\mathrm{C},$$\rho$:$Garrow GL_{n}(\mathrm{C})$a rational. Such a triple $(G, \rho, V)$ is called a prehomogeneous

has a Zariski dense orbit in$V$. If $(G, \rho, V)$ is an irreducible, $(G, \rho, V)$is said to be irreducible. Now assume that $(G, \rho, V)$ is an irreducible prehomogeneous

vector space such that there exist a non-trivial rational character
and an irreducible polynomial function$f\in \mathrm{C}[V]$

on

$V$

$\phi\in \mathrm{H}\mathrm{o}\mathrm{m}(G, \mathrm{C}^{\mathrm{x}})$such that $f(gv)=\phi(g)f(v)$for all $g\in G$ and $v\in V$ . Put$\mathrm{A}\mathrm{u}\mathrm{t}(V, f):=$and$\mathrm{A}\mathrm{u}\mathrm{t}^{0}(V, f)${$(g,$ $\phi_{g})\in GL(V)\cross \mathrm{C}^{\cross}|f(gv)=\phi_{g}f(v)$be the identity component ofby the first projection coincides with$\rho(G)$$\mathrm{A}\mathrm{u}\mathrm{t}(V, f)$

for all $v\in V$ },. If the image of$\mathrm{A}\mathrm{u}\mathrm{t}^{0}(V, f)$, then $(G, \rho, V)$ is said to be saturated.

The purpose of this note is to classify and to describe the split -forms of the$\mathrm{Z}$saturated, irreducible prehomogeneous vector spaces. (See [G] for “split Z-form”.)
For this purpose, we need to describe a Chevalley system explicitly for each complex
simple Lie algebra. Such a description is given in \S 1, which would be useful in adifferent context, and so we have included some information which is not used in thepresent note. (For example, all information concerning$E_{8}$

is not necessary here.)

Notation. For a ring $A(\ni 1),$ $M_{n}(A)$ denotes the totality of
The group of units in$A$

is denoted by$A^{\cross}$. An element of$A^{\cross}$$n\cross$n-matrices.

is identified with the

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the

$n\cross n$-matrix whose $(i,j)$ -component is 1 and the other components are . We
$0$sometimes write$E_{i}$

for$E_{ii}$diagonal components are. We denote by diag$t_{1},$$\cdots,t_{n}$. For a set$(t_{1}, \cdots, t_{n})$$X$the diagonal matrix whose, its cardinality is denoted by $\# X$ .

\S 1. Chevalley system.
Let$\sum_{r\in R}\mathfrak{g}(r)$$\mathfrak{g}$be a simple Lie algebra over$\mathrm{C},$$\mathfrak{h}$a Cartan subalgebra,

the root space decomposition, $0\neq X(r)\in \mathfrak{g}(r)$ , and$\mathfrak{g}=\mathfrak{h}\oplus$$H(r)(\in \mathfrak{h})$thecoroot vector which corresponds to a root . A system $(X(r))_{r\in R}$ is called a Chevalley$r$system, if$[X(r),X(-r)]=H(r)$and, for$r,$ $s,$$(r\in R)$$r+s\in R$ ,[X $(r),X(_{S)}$ ] $=\pm px(r+s)$ ,

where

$p$

is the smallest positive integer such that $s+(p+1)r\not\in R$ .

The purpose of this section is to describe explicitly a Chevalley system foreach complex simple Lie algebra.
1.1. Type $A_{n-1}$ .

We may assume that$\mathfrak{g}=\{X\in Mn(\mathrm{C})|\mathrm{t}\mathrm{r}(X)=0\}$and$\mathfrak{h}=\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1}, \ldots,t_{n})|t_{i}\in \mathrm{C}, \sum t_{i}=0\}$

Then$R=\{\epsilon_{i}-\epsilon_{j}|i\neq j\}$

,

.

210

where
$\epsilon_{i}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t1, \ldots,t_{n}))=t_{i}$.

The coroots are given by$H(\epsilon_{i}-\epsilon_{j})=Ei-Ej$.

A Chevalley system is given by$X(\epsilon_{\dot{\iota}}-\epsilon_{j})=Eij$.

We may take as a root basis$\alpha_{i}=\epsilon_{i}-\epsilon_{+1}\dot{*}$$(1\leq i\leq n-1)$.

Then the Dynkin diagram is given by$\alpha_{1}^{\mathrm{O}-\mathrm{O}}\alpha 2$

1.2. Type$B_{n}$–. $..-\circ\alpha_{n-1}$.

Let us define an element

$J$

of

$M_{2n+1}(\mathrm{C})$by$J= \sum(E_{i,n}+i+E_{n}+:,i)+2nE_{2}n+1,2n+1$$i=1$

We may assume that$\mathfrak{g}=\{X\in M2n+1(\mathrm{c})|XJ+J^{t}x=0\}$and$\mathfrak{h}=\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t1, \ldots,tn’-t1, \ldots, -tn’ 0)\}$

.

.

211

Then
$R=\{\pm\epsilon_{i}\pm\epsilon_{j} (i\neq j), \pm\epsilon_{i}\}$

,

where
$\epsilon_{i}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1}, \ldots,tn’-t1, \ldots, -tn’ \mathrm{Q}))=t_{i}$.

The coroots are given by$H(\epsilon_{i}-\epsilon_{j})=(Ei-E_{j})-(E_{n+}i-E)n+j$$(i\neq j)$$H(\epsilon_{i}+\epsilon_{j})=(E_{i}+Ej)-(En+i+E_{n+}j)$$(i<j)$$H(-\epsilon_{i}-\epsilon j)=(-E_{i}-E_{j})-(-E_{n+i}-E_{n+j})$$(i<j)$$H(\epsilon_{i})=2(E_{i}-En+i)$$H(-\epsilon_{i})=-2(Ei-E_{n+i})$.

A Chevalley system is given by$X(\epsilon_{i}-\epsilon_{j})=Eij-E_{n}+j,n+i$$(i\neq j)$$X(\epsilon_{i}+\epsilon_{j})=Ei,n+j-E_{j,n+}i$$(i<j)$$x(-\epsilon_{\dot{0}^{-\epsilon}}j)=E_{n}+j,i-E_{n}+i,j$

$(i$

.

$<j)$

$x(\epsilon_{i})=E_{i,2n}+1-2E2n+1,n+i$$X(-\epsilon_{i})=2E_{2+i}n1,-E_{n+i,2n}+1$

We may take as a root basis of.

$R$$(1\leq i<n)$ ,$\alpha_{i}=\epsilon_{i}-\epsilon_{i}+1$$\alpha_{n}=\epsilon_{n}$Then the Dynkin diagram is given by$\alpha_{1}^{\mathrm{O}-_{\alpha_{2}}}\circ$$–...-_{\alpha_{n-1}^{\mathrm{O}}}\Rightarrow\alpha_{n}^{\mathrm{O}}$

.

.

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1.3. Type$C_{n}$.

Let$J= \sum_{i=1}^{n}(E_{i,i}n+-E_{n}+i,i)$.

We may assume that$\mathfrak{g}=\{X\in M_{2n}(\mathrm{C})|XJ+J^{t}X=0\}$and$\mathfrak{h}=\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t1, \ldots,t_{n}, -t_{1}, \ldots, -t_{n})\}$.

Then$R=\{\pm\epsilon_{i}\pm\epsilon_{j} (i\neq j), \pm 2\epsilon_{i}\}$

,

where
$\epsilon_{i}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1,\ldots,n}t, -t_{1}, \ldots, -t)n)=t_{i}$.

The coroots are given by$H(\epsilon_{i}-\epsilon_{j})=(E_{i}-E_{j})-(En+i-E_{n}+j)$$(i\neq j)$$H(\epsilon_{i}+\epsilon_{j})=(E_{i}+Ej)-(En+i+E_{n+}j)$$(i<j)$$H(-\epsilon_{i}-\epsilon j)=-(E_{i}+E_{j})+(E_{n+i}+E_{n+j})$$(i<j)$$H(2\epsilon_{i})=Ei-E_{n}+i$$H(-2\epsilon_{i})=-E_{i}+E_{n+}i$.

A Chevalley system is given by$X(\epsilon_{i}-\epsilon_{j})=Ei,j-E_{n}+j,n+i$$(i\neq j)$$X(\epsilon_{i}+\epsilon_{j})=E_{i},j+n+Ej,n+i$$(i<j)$$X(-\epsilon_{i}-\epsilon j)=E_{n+}j,i+En+i,j$$(i<j)$$X(2\epsilon_{i})=E_{1n}.)+i$$X(-2\epsilon i)=En+i,i$

.

213

We may take as a root basis$\alpha_{i}=\epsilon_{i}-\epsilon_{i+}1$$(1\leq i<n)$ ,$\alpha_{n}=2\epsilon_{n}$.

Then the Dynkin diagram is given by$\alpha_{1\alpha_{2}}^{\mathrm{O}-\circ}$

1.4. Type$D_{n}$$–...-_{\alpha_{n-1}^{\mathrm{O}}}\Leftarrow\alpha_{n}^{\mathrm{O}}$

.

.

Let
$J= \sum(E_{i,n+}i+E_{n+i}i,)n$

.

$i=1$

We may assume that$\mathfrak{g}=\{X\in M2n(\mathrm{C})|XJ+JtX=0\}$and$\mathfrak{h}=${diag( $t1,$$\ldots$,$t_{n},$$-t_{1},$$\ldots,$$-t_{n}$)}.

Then$R=\{\pm\epsilon_{i}\pm\epsilon_{j} (i\neq j)\}$

,

where

.

$\epsilon_{i}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t1, \ldots,tn-t_{n}, \ldots, -t_{n}))=ti$

The coroots are given by$H(\epsilon_{i}-\epsilon_{j})=(Ei-E_{j})-(E_{n+}i-En+j)$$(i\neq j)$$H(\epsilon_{i}+\epsilon_{j})=(E_{i}+Ej)-(En+i+E_{n+}j)$$(i<j)$$H(-\epsilon_{i}-\epsilon j)=-(E_{i}+Ej)+(E+i+nE_{n+}j)$

$(i$

.

$<j)$

.

214

A Chevalley system is given by$X(\epsilon_{i}-\epsilon_{j})=Eij-E_{n}+j,n+i$$(i\neq j)$$X(\epsilon_{i}+\epsilon_{j})=Ei,n+j-E_{j,n+}i$$(i<j)$$X(-\epsilon_{i}-\epsilon_{j})=En+j,i-E_{n}+i,j$$(i<j)$ .

We may take as a root basis.

.

$(1\leq i$$\alpha_{i}=\epsilon_{i}-\epsilon_{1+1}$$<n)$ ,$\alpha_{n}=\epsilon_{n-}1+\epsilon_{n}$.

Then the Dynkin diagram is given by$\alpha_{n,0}$$–...-_{\alpha_{n-2^{-}}^{\mathrm{O}}}|\alpha_{n-}01$$\alpha_{1}^{\mathrm{O}-\circ}\alpha 2$

Up to now, we have worked with the vector representation of the simple Lie algebraof$\mathrm{t}$)$\mathrm{y}\mathrm{p}\mathrm{e}D_{n}$, but we also need to work with the half-spin representation. In theremainder of this paragraph, we freely use the notations of [$\mathrm{S}\mathrm{K}$,pp.110-114], where abrief account of the theory of the spin representation is given.
The representation space$\Lambda(E)=\Lambda(\mathrm{C}^{n})$Grassmann algebra of the vector space$e_{i_{1}}$A$e_{i_{2}}\wedge\cdots$A$e_{i_{k}}$of the spin representation is the$E=\oplus_{i=1}^{n}\mathrm{C}ei$. We write$e_{i_{1}}ei_{2}\cdots ei_{k}$for. Let us consider two kinds of linear operators which are defined asfollows:$e_{i}(e_{i_{1}i_{2}\cdots i}eek)=e_{i}ei1e_{i_{2}\cdots i_{k}}e$.

$(-1)^{p-1}e_{i_{1}}\ldots e^{\wedge}i\cdots e_{i_{k}}\mathrm{p}$’if $i=i_{p}$ for some ,$p$$f_{i}(e_{i}e_{i_{2}}\ldots e_{i_{k}})1=\{$$0$,otherwise.

215

Here

$e_{i_{1}}\ldots e_{i_{\mathrm{p}}\cdots e}^{\wedge}|k$means$e_{i_{1}}\ldots e_{i_{\mathrm{p}1}}-e_{i\mathrm{p}}+1\ldots e_{i_{k}}$$e_{i}f_{j}$$e_{i}e_{j}$$f_{j}f_{i}$Then$\tilde{\mathfrak{g}}$. Let$(1\leq i,j\leq n)$$\tilde{\mathfrak{g}}$,$(1\leq i<j\leq n)$,$(1\leq i<j\leq n)$.

is a Lie algebra and an isomorphism between
diag $(t_{1}, \ldots, t_{n}-t_{1}, \ldots, -t_{n})$be the linear span of$\mathfrak{g}$and$\tilde{\mathfrak{g}}$is given as follows:$\frac{1}{2}\sum t_{i}(e_{i}fin-f_{i}ei)$$rightarrow$

.

$i=1$

$X(\epsilon_{i}-\epsilon_{j})=E_{ij}-E_{n}+j,n+i$$rightarrow$$X(\epsilon_{i}+\epsilon_{j})=E_{i,n+j}-Ej)n+i$$rightarrow$$X(-\epsilon_{i}-\epsilon j)=En+j,i-En+i,j$$rightarrow$

Thus a Chevalley system of$\tilde{\mathfrak{g}}$$(i\neq j)$$e_{i}f_{j}$,$(i<j)$ ,$e_{i}e_{j}$$(i<j)$ .

$f_{j}f_{i}$is given by,$X(\epsilon_{i}-\epsilon_{j})=eif_{j}$$(i\neq j)$$X(\epsilon_{i}+\epsilon_{j})=e_{i}ej$$(i<j)$ ,$X(-\epsilon_{i}-\epsilon_{j})=f_{j}f_{i}$$(i<j)$ .

As is easily seen$\Lambda^{odd}=\Lambda^{odd}(E)=\sum_{k=odd}\Lambda^{k}(E)$and$\Lambda^{even}=\Lambda^{eve}n(E)=k\sum_{=even}\Lambda^{k}(E)$are -stable subspaces of$\tilde{\mathfrak{g}}$$\Lambda(E)$. These -modules$\tilde{\mathfrak{g}}$$\Lambda^{odd}$

and$\Lambda^{even}$

are known to beirreducible and are called the odd half-spin representation and the even half-spin
representation, respectively.

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We define an involutory automorphism
erated by$\{e_{1}, \ldots, e_{n}, f_{1}, \ldots, fn\})$by$\iota(e_{i})=f_{i}$$\iota$of the Clifford algebra $C(Q)$ (gen-and $\iota(f_{i})=e_{i}(1\leq i\leq n)$ . Then$\iota$induces an automorphism of $Spin_{2n}$ , which we shall denote by the same letter . See$\iota$[$\mathrm{S}\mathrm{K}$, pp.110-114] for the Clifford algebras and the spin groups.
1.5. Type$G_{2}$.

We may assume that$(_{f}^{a}0dceb$$x_{0}x_{C}x_{21}-b23111d$$\mathfrak{g}$is the totality of the matrixes$x_{3}x_{2}x_{0}-C2a12e_{2}2$$x_{0}x_{2}\mathrm{a}x_{33}-a2fb13$$-x-x-x_{1}-2a_{12}0ef131$$-x-x_{2}-x-d2f0b22213$$-x-X_{3}-x_{31}-e2_{C}d0_{33}2)$with $x_{11}+x_{22}+x_{33}=0$ , and$\mathfrak{h}=\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(0,t1, t2,t3, -t1, -t2, -t_{3})|t_{1}+t_{2}+t_{3}=0\}$.

Then$R=\{\epsilon_{i}-\epsilon_{j} (i\neq j), \pm\epsilon_{i}\}$

,

where

.

$\epsilon_{i}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(0, t1, t2, t3, -t1, -t_{2}, -t\mathrm{s}))=t_{i}$The coroots are given by$H(\epsilon_{i}-\epsilon_{j})=(E_{1+i}-E_{1+}j)-(E4+i-E4+j)$$(i\neq j)$,. $-E1+j-E1+k)-(2E_{4i}+-E4+j-E_{4}+k)$ ,$H(\epsilon_{i})=(2E_{1+}|$$H(-\epsilon_{i})=-(2E_{1}+i-E1+j-E1+k)+(2E_{4i}+-E4+j-E_{4}+k)$ ,217

where $\{i,j, k\}=\{1,2,3\}$ . A Chevalley system is given by,$X(\epsilon_{i}-\epsilon_{j})=E1+i,1+j-E4+j,4+i$$X(\epsilon_{i})=E_{1+i},1+2E_{1,4+}i+E_{4+k,1+j}-E_{4}+i,1+k$,$X(-\epsilon_{i})=E4+i,1+2E1,1+i+E1+j,4+k-E1+k,4+j$ ,where $(i,j, k)$ is an arbitrary even permutation of (1, 2, 3). In fact,$[X(\epsilon_{i}-\epsilon_{j}), X(\epsilon_{j}-\epsilon_{k})]=X(\epsilon_{i}-\epsilon_{k})$$[X(\epsilon_{i}-\epsilon_{j}), X(\epsilon_{j})]=X(\epsilon_{i})$$[X(\epsilon_{i}-\epsilon_{j}), X(-\epsilon_{i})]=-X(-\epsilon_{j})$$[X(\epsilon_{i}), X(-\epsilon j)]=3X(\epsilon_{i}-\epsilon_{j})$$[X(\epsilon_{i}), X(\epsilon_{j})]=2X(-\epsilon_{k})$,,$[X(-\epsilon_{i}), X(-\epsilon j)]=-2X(\epsilon_{k})$.

In the last two commutation relations, $\{i,j, k\}=\{1,2,3\}$ . Let
algebra ($=\mathrm{t}\mathrm{h}\mathrm{e}$algebra of Cayley numbers) over$u_{1}=e_{0}$,$u_{3}=e_{1}+\sqrt{-1}e_{6}$$\mathrm{C}$be the octonion[F,l.l]. Define a basis of$\mathrm{C}$$\not\subset$by$u_{2}=e_{7}$,$u_{6}=-e_{1}+\sqrt{-1}e_{6}$$u_{4}=e_{2}+\sqrt{-1}e_{5}$,,$u_{7}=-e_{2}+\sqrt{-1}e_{5}$$u_{5}=e_{4}+\sqrt{-1}e_{3}$,,$u_{8}=-e_{4}+\sqrt{-1}e_{3}$.

Here we use the notations of [F,1.5]. With respect to this basis, the Lie algebra of theinfinitesimal automorphisms of$C$

is identified with the Lie algebra

We may take as a root basis$\alpha_{1}=\epsilon_{1}-\epsilon_{2}$,$\alpha_{2}=-\epsilon_{1}$.

$\mathfrak{g}$defined above.

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Then the Dynkin diagram is given by$\alpha_{1}\ni\alpha_{2}$1.6. Type$F_{4}$.

.

In this paragraph, we use the notations of [F]. Define a basis of(1.6.1)$f_{1}=e_{0}+\sqrt{-1}e7$,$f_{50+}=-e\sqrt{-1}e_{7}$,$f_{2}=e_{6}+\sqrt{-1}e_{1}$,$f_{6}=-e_{6}+\sqrt{-1}e_{1}$,$f_{3}=e_{5}+\sqrt{-1}e_{2}$,$f_{7}=-e_{5}+\sqrt{-1}e_{2}$,$f_{4}=e_{3}+\sqrt{-1}e_{4}$,$f_{8}=-e_{3}+\sqrt{-1}e_{4}$

.

$C$

by

The multiplication table is given by$f_{1}2f_{1}f_{1}$(1.6.2)$2f_{2}f_{2}$$2f_{3}f_{3}$$2f_{4}f_{4}$$0f_{5}--0f_{6}$$0f_{7}$$f_{8}0$$f_{2}$$0$$0$$-2f_{8}$$2f_{7}$$-2f_{2}$$2f_{1}$$0$$0$$f_{3}$$0$$-2f_{8}$$0$$-2f_{6}$$-2f_{3}$$0$$2f_{1}$$0$$f_{4}$$0$$-2f_{7}$$2f_{6}$$0$$-2f_{4}$

$0$

$0$

$2f_{1}$$f_{5}$

$0$

$0$

$0$

$0$

$-2f_{5}$$-2f_{6}$$-2f_{7}$$-2f_{8}$$f_{6}2f_{6}$$-2f_{5}$

$0$

$0$

$0$

$0$

$2f_{4}$$-2f_{3}$$f_{7}2f_{7}$$0$$-2f_{5}$

$0$

$0$

$-2f_{4}$$0$$2f_{2}$$f_{8}2f_{8}$

$0$

$0$

$-2f_{5}$$0$$2f_{3}$$-2f_{2}$$0$e.g., $f_{1}f_{3}=2f_{3},$ $f_{3}f_{1}=0$ . Let us identi$f\mathrm{y}$a linear endomorphism ofcorresponding matrix with respect to the basis$\{f_{i}\}$, e.g.,$E_{ij}f_{j}=f_{i}$$\mathrm{C}$with the. Let us describe

219

the automorphisms$\lambda$$X=(_{z}^{X}x_{0}31x_{41}Zz_{1}\mathrm{o}_{21}11423$

and$\lambda^{2}$$-z_{4}x_{0}xx_{42}z_{2}Z_{2}\mathrm{o}_{3}321212$

of$\mathfrak{D}_{4}$$–\mathcal{Z}xxx_{0}z_{3}\mathrm{o}_{13}z_{4}43231323$[F,2.2.4] in the matrix form. For$-z-z_{1}-xxx_{0}\mathrm{o}_{24}\mathcal{Z}_{34}3424144$$-X_{14}-x-x_{13}-y1-y_{3}-y41\mathrm{o}\mathrm{o}2121$$-X_{2}-x_{23}-X_{24}-y4-y3y0021221$$-x_{34}-x_{3}-x-y43yy310032321$$-x_{4}-x_{4}-X_{41}y42y41y\mathrm{o}04323$),

we have(1.6.3)$\lambda(X)=(_{-X}^{-Z}-x3-x_{13}-z0z_{1}\mathrm{o}24342124$$-y_{3}43-yXx_{12}x_{42}y400_{1}312$$-y_{4}x_{2}x_{4s_{1}}x_{0}yy210_{1}4233$$-x_{41}--y43xx_{42}x0z\mathrm{o}_{1}32312$$-X-\mathcal{Z}xxx_{41}y4\mathrm{o}_{2}\mathrm{o}_{21}241333$$-y_{3}32-xx_{0}x_{0}y_{3}1y2121444$$-x_{2}-X_{4}-x-z_{13}zz_{0}\mathrm{o}_{3}2141432)$$-y4xXx_{0}y_{4}y20_{3}21413132$$-x_{13}-x_{23}-x_{24}-Zz_{0}z0341241$$-x_{2}-X_{32}-x-z-zz_{0}\mathrm{o}_{34}1412341$

and(1.6.4)$\lambda^{2}(X)=(_{-y}^{-y3}-y211-y_{4}1y43y300422$$-x-y_{21}2-z_{3}xxx_{0}\mathrm{o}_{3}2414431$$-x_{1}-x_{4}-xxy4z\mathrm{o}_{43}\mathrm{o}2133122)$$-z-z_{4}z_{2}z_{1}zz_{1}\mathrm{o}_{34}\mathrm{o}_{3}12423$$-x_{21}-x-xzy_{X_{2}}\mathrm{o}_{1}\mathrm{o}_{23}344134$$-x_{12}-x_{32}-x_{34}-z_{2}x_{0}y_{3}10144$

220

Let

$=$

$= \frac{1}{2}$ .

,Then, for
$X=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1},t_{2}, t3, t4, -t1, -t_{2,3}-t, -t_{4})$,we have.

(1.6.5)$\lambda^{j}(X)=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(tt_{2’ 3}t, t_{4}, -(j)(j)(j)-t^{(}(j)(j)t_{1’ 2}, -1’ j)(it-3’ 4)t^{(j)})$As in [F,4.5.9], 3 denotes the exceptional simple Jordan algebra. We may assume
that{infinitesimal automorphisms of 3}.

$\mathfrak{g}=$Let us identi$f\mathrm{y}$an element$\delta$

where$\delta_{i}=\lambda^{i-1}(\delta)$($\overline{x_{2}3}\xi_{1}$of$\mathfrak{D}_{4}$with the element)$\frac{\delta x0}{\delta_{1^{X}1}}33$$\xi=(^{0}\overline{\delta_{33}x_{2}\delta_{2^{X}}}$$x_{3}\overline{x_{1}}\xi_{2}\overline{x_{2}x_{3}1}$$\delta$of$\mathfrak{g}$defined by$\overline{\delta_{2^{X}2}\delta_{11}X}10$’. We may assume that$\mathfrak{h}=$where we identi$\delta${diag$f\mathrm{y}\mathfrak{h}(\subset \mathfrak{D}_{4})$$(t_{1},$$t_{2},$$t_{3},$$t4,$ $-t_{1},$ $-t2,$ $-t3,$ $-t_{4})$},with a subalgebra of via the above defined identification.
$\mathfrak{g}$Then$R= \{\pm\epsilon_{i}\pm\epsilon_{j}(\dot{i}\neq j), \pm\epsilon_{i}, \frac{1}{2}(\pm\epsilon 1\pm\epsilon_{2}\pm\epsilon \mathrm{s}\pm\epsilon 4)\}$

,

221

where
$\epsilon_{i}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t1,t_{2},t_{3},t4, -t_{1}, -t_{2}, -t_{3}, -t_{4}))=t_{i}$.

The coroots are given by$H(s_{i}\epsilon_{i}+s_{j}\epsilon j)=si(Ei-E_{4+}i)+s_{j(}Ej-E_{4}+j)$$(i\neq j)$$H(s_{i}\epsilon_{i})=s_{i}(2Ei-2E_{4+i})$(1.6.6)$H( \frac{1}{2}(_{S_{1}}\epsilon 1+s_{2}\epsilon 2+s_{3}\epsilon_{3}+s4\epsilon_{4}))=\sum Si(E_{i}4-E4+i)$

,

$i=1$

where $s_{i}=\pm 1$ . For$a\in \mathbb{C}$, let$(a)_{1}=,$ $(a)_{2}=$ , $(a)_{3}=$ .
For$X\in \mathfrak{M}^{(3\rangle}$, define a linear endomorphism$\tilde{X}$of 3 by$\tilde{X}(\mathrm{Y})=\frac{1}{2}(X\mathrm{Y}+Y^{*}X^{*})$

where(1.6.7)$X^{*}$,

is the transposed conjugate of $X$ [F,4.1]. A Chevalley system is given by$X(\epsilon_{i}-\epsilon j)=Eij-E4+j,4+i$$(i\neq j)$$X(\epsilon_{i}+\epsilon j)=Ei,4+j-E_{j,+}4i$$(i<j)$$X(-\epsilon_{i}-\epsilon_{j})=E_{4}+j,i-E4+i,j$$(i<j)$$X(\epsilon_{i})=(f_{i})_{1}\sim$$X(-\epsilon_{i})=(f_{4}+i)_{1}\sim$$X(\epsilon_{i}\circ\lambda)=(fi)_{2}^{\sim}$$X(-\epsilon\circ\lambda)=(f4+i)_{2}^{\sim}$$X(\epsilon_{i}0\lambda^{2})=(f_{i})3\sim$$X(-\epsilon\circ\lambda^{2})=(f_{4+i})_{3}^{\sim}$

.

222

Note that$= \frac{1}{2}$and$= \frac{1}{2}$ .
Let us give explicitly the commutation relations. Let
form$X(\pm\epsilon_{i}\pm\epsilon_{j})(i\neq j)$(1.6.4). Here

(1.6.8)$\delta_{j}f_{i}$. Thenare of the$\delta_{1},$$\delta_{2},\delta_{3}$$f\mathrm{o}\mathrm{r}\mathrm{m}\pm f_{k}$are of the$\delta$be an element of , of the$\mathfrak{g}$$f\mathrm{o}\mathrm{r}\mathrm{m}\pm X(\pm\epsilon_{i}\pm\epsilon_{j})$by (1.6.3) and. By [F,4.9.4],$[\delta, (f_{i})_{j}^{\sim}]=(\delta_{j}f_{i})^{\sim}j=\pm(fk)j\mathrm{o}\sim \mathrm{r}0$.

The signature appeared in (1.6.8) can be easily determined by using (1.6.3), (1.6.4)
and (1.6.7). A direct calculation shows that$[(f_{i})_{1}\sim, (fj)_{1}\sim]=2(E_{i}j^{\prime-}Eji’)$(1.6.9)$[(f_{i})_{2}^{\sim}, (f_{j})_{2}^{\sim}]=2\lambda^{2}(Eij’-E_{j}il)$$[(f_{i})_{3}^{\sim}, (fj)^{\sim}3]=2\lambda(E_{i}j’-Eji^{\prime)}$where$i+4$ ,$(i\leq 4)$$i-4$ ,$(i>4)$ ,$i’=\{$

and(1.6.10)$[(a)_{i}^{\sim}, (b)_{j}^{\sim}]=(- \frac{1}{2}\overline{ab})_{k}\sim$

,

,

223

for each even permutation $(i,j, k)$ of (1, 2, 3).

is of the$\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}-\frac{1}{2}\overline{f_{i}fj}$$f\mathrm{o}\mathrm{r}\mathrm{m}\pm f_{k}$

of ,
$0$

(1.6.8), (1.6.9) and (1.6.10) together with the results of (1.4), give the commutation
relation among the,Chevalley system given above. We may take as a root basis$\alpha_{1}=\epsilon_{2}-\epsilon_{3},$$\alpha_{234}=\epsilon-\epsilon,$$\alpha_{4}=\frac{1}{2}(\epsilon_{1}-\epsilon 2-\epsilon 3-\epsilon_{4})$$\alpha_{3}=\epsilon_{4}$,.

Then the Dynkin diagram is given by$\alpha_{1}\alpha 0-0_{2}\Rightarrow\alpha_{3}^{\mathrm{O}-\circ}\alpha 4^{\cdot}$

1.7. Type$E_{6}$.

In this paragraph, we use the notations of [F]. We may assume that$\mathfrak{g}=\mathrm{G}_{6}=${linear endomorphisms of which (infinitesimally)$\mathrm{J}$preserves $\det(X, \mathrm{Y}, z)\}$[F,8.1]. The Lie algebraLet$\mathfrak{h}_{4}$$S_{4}$

of infinitesimal automorphisms ofbe the Cartan subalgebra of$ff_{4}$$\mathrm{J}$is contained in$\mathfrak{g}$.

which is given in (1.6). We may assume that$\mathfrak{h}=\mathfrak{h}_{4}+\{|t_{5}+t_{6}+t7=0\}$.

Let$h(t_{1}, \ldots, t_{7})=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1}, t_{2}, t_{3}, t4, -t1, -t_{2}, -t3, -t_{4})+$

224

and
$\epsilon_{i}(h(t_{1}, \ldots,t_{7}))=t_{i}$Let us define endomorphisms $\alpha_{ij}(1\leq i,j\leq 3)$ of$(1\leq i\leq 3)$$\alpha_{ii}=0$$\alpha_{23}=1$$\alpha_{32}=\kappa$[F,2.2]. Note that every$\alpha_{ij}$,$\alpha_{31}=\lambda$,$\alpha_{13}=\mathcal{K}\lambda$

by$\alpha_{12}=\lambda^{2}$,,$\alpha_{21}=\kappa\lambda^{2}$. Let$(1\leq i\leq 3)$$A_{ii}=0$$A_{23}=1$$\mathfrak{h}_{4}$$\mathfrak{D}_{4}$,,preserves.

,,$A_{31}= \frac{1}{2}$ ,$A_{12}= \frac{1}{2}$ ,$A_{32}=$ ,$A_{13}= \frac{1}{2}$ ,225$A_{21}= \frac{1}{2}$ ,and$=A_{ij}$ .
Then.

$\alpha_{ij}h(t_{1}, t_{2}, t3, t4,0,0,0)=h(t_{1}, t_{2}^{\dot{8}}, t_{3}^{i},t_{4}, \mathrm{o}, 0,0)ijjjij$

We identi$f\mathrm{y}$an element$\delta$of$\mathfrak{D}_{4}$with a linear endomorphism of M3 [F,4.1] as follows:$\delta(\sum x_{ijij}3E(3))-\sum(\delta_{ijj}x_{i})E^{(3})3ij$’$i,j=1$

where$\delta_{ij}=\alpha_{ij}(\delta)$$i,j=1$[F,4.9]. Let$(1\leq i,j\leq 3)$$R_{ij}=\{\pm\epsilon_{k}\circ\alpha ij|1\leq k\leq 4\}$

.

Then

$R= \bigcup_{i\neq j}(R_{ij}+\frac{1}{2}(\epsilon 4+i-\epsilon_{4j}+))\cup\{\pm\epsilon i\pm\epsilon j|1\leq i<j\leq 4\}$$= \{\pm\epsilon_{8}\pm\frac{1}{2}(\epsilon 6-\epsilon_{7})$.

$(1 \leq i$ $\leq 4)$$\frac{1}{2}\sum s_{i}\epsilon_{i}\pm 4\frac{1}{2}(\epsilon_{5}-\epsilon_{7})$$i=1$$\frac{1}{2}\sum s_{i}\epsilon_{i}\pm\frac{1}{2}(\epsilon 5-\epsilon 6)4$$i=1$$\pm\epsilon_{i}\pm\epsilon_{j}$,$( \prod s_{i}=4-1)$

,

$i=1$

$( \prod s_{i}=41)$

,

$i=1$

$(1 \leq i<j\leq 4)\}$

,

226

where $s_{i}=\pm 1$ . Define an order by$\sum s_{1}\epsilon_{i}>07.$,$|.=1$

if $s_{\sigma(1)}=\cdots=s_{\sigma(k-1)}=0$ and $s_{\sigma(k)}>0$ for some 1 $\leq k\leq 7$ , where a$(1 \leq i\leq 4)$$\pm\frac{1}{2}\sum_{i=1}4Si\epsilon i+\frac{1}{2}(\epsilon 5-\epsilon 7)$$( \prod_{i=1}^{4}s_{i}=-1)$$\pm\frac{1}{2}\sum s_{i}\epsilon_{i}+4\frac{1}{2}(\epsilon 5-\epsilon 6)$$( \prod^{4}=1)$$.$,$(1 \leq i<j\leq 4)$$\epsilon_{i}\pm\epsilon_{j}$$\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}.\mathrm{s}$

,

$i=1$

$i=1$

and simp.le,$\pm\epsilon_{i}+\frac{1}{2}(\epsilon_{6}-\epsilon_{7})$

,

. are

$r_{1}=- \epsilon_{1}+\frac{1}{2}(\epsilon_{6}-\epsilon 7)=-\epsilon_{1}\mathrm{O}\alpha_{2}3+\frac{1}{2}(\epsilon 6-\epsilon_{7})$$r_{2}=\epsilon_{3}-\epsilon_{4}$$r_{3}=\epsilon_{1}-\epsilon_{2}$$r_{4}=\epsilon_{2}-\epsilon_{3}$$r_{5}=\epsilon_{3}+\epsilon_{4}$.

$r_{6}= \frac{1}{2}(-\epsilon_{1}-\epsilon_{2}-\epsilon_{34}-\epsilon)+\frac{1}{2}(\epsilon_{5}-\epsilon_{6})=\epsilon_{1}\circ\alpha_{12}+\frac{1}{2}(\epsilon 5-\epsilon 6)$

Let $h_{1}=h(1,0, \mathrm{o},0,0,0),$ $h_{2}=h(0,1,0,0,0, \mathrm{o})$ etc. The coroots are given by$H( \sum c_{i}\epsilon_{i})=\sum c_{i}h_{i}+2\sum c747ih_{i}$

$i=1$

$i=1$

$i=5$

,

$=$

227

where

$\sum_{i=1}^{7}c_{i}\epsilon_{i}\in R$. Especially$H(r_{1})=-h_{1}+(h_{67}-h)$ ,$H(r_{2})=h_{3}-h_{4}$,$H(r_{3})=h1-h2$ ,$H(r_{4})=h2-h3$ ,$H(r_{5})=h3+h4$ ,$H(r_{6})= \frac{1}{2}(-h_{1}-h_{2}-h_{3}-h_{4})+(h_{5}-h_{6})$.

Hence the Dynkin diagram is given by$r_{1}-r_{3}-r_{4}-r_{5}-r_{6}$$1$$r_{2}$

Let$(a)_{ij}=aE_{i}^{(3)}j$$\mathfrak{M}_{3}^{r}=${$T\in \mathfrak{M}$$(1\leq\dot{i},j\leq 3, a\in \mathbb{C})$,with real diagonal elements},$\chi=x_{11}+x_{22}+x_{3}3$$\tilde{T}(X)=\frac{1}{2}(TX+XT^{*})$

Every element of$\mathfrak{g}$$(X\in \mathrm{J}, T\in \mathfrak{M}_{3})$can be uniquely expressed as$\delta+\tilde{T}$

,

.

228

where

$\delta\in \mathfrak{D}_{4},$$T\in$

M3 and $\chi(T)=0$ [F,8.1.1]. A Chevalley system is given by$X(\epsilon_{i}-\epsilon_{j})=E_{i},j-E4+j,4+i$$(i\neq j)$$X(\epsilon_{i}+\epsilon j)=Ei,4+j-Ej,4+i$$(i$$X(-\epsilon i-\epsilon j)=E_{4}+j,i-E4+i,j$$(i<j)$.

$<j)$$X( \epsilon_{i}0\alpha_{k1}+\frac{1}{2}(\epsilon_{4+k}-\epsilon_{4+l}))=(f_{\dot{8}})kl\sim$$X(- \epsilon_{i}\circ\alpha_{kl}+\frac{1}{2}(\epsilon_{4+k}-\epsilon_{4+l}))=(f_{4+i})_{kl}^{\sim}$

1.8. Type$E_{7}$$(1\leq i\leq 4,1\leq k, l\leq 3, k\neq l)$.

.

In this paragraph, we use the notations of [H]. Let

$X=$

{

$(x,$ $y)|x,$ $y$

are alternating$8\cross 8$

matrices}.

Define linear endomorphisms of $X$ by$(x, y)arrow p(px+x^{t}p, -^{t}py-yp)$ ,(1.8.1)

where

$p$

is an

$8\cross 8$

matrix with trace , and$0$$((x_{ij}), (yij)) arrow\theta((\sum 8\theta^{ijmn}ymn), (- \sum 8\theta_{ijn}n^{X}m)m)$(1.8.2)$m,n=1$

where$\theta$,$m,n=1$denotes a tensor, antisymmetric in its indices, and upper, lower indices satisfy

the relation$\theta_{i_{1},\ldots,i_{4}}=\frac{1}{4!}\sum_{j_{1},\ldots,j_{4}}I_{i1,\ldots,4}\theta 1,\ldots,8j_{1},\ldots,ji,j_{1},\ldots,j_{4}4$

.

229

Here

$I_{k_{1},,k_{8}}^{1,...\cdot.’.8}$$\{1, \ldots, 8\}$denotes the signature of the permutation

, and

$0$

otherwise. Then, we may assume that$\mathfrak{g}=\mathrm{G}_{7}$is the linear span ofthese linear endomorphisms, whose Lie algebra structure is given by$\mathrm{r}_{P,p’}]=pp’-p^{l}p$$\lceil p,$$\theta]=\theta’,$, where $pp’$ denotes the matrix multiplication,

where $( \theta’)^{ij}kl=\sum(\theta^{mjk}\iota_{p}im+\theta^{imkl}pjm+\theta^{ijml}pkm+\theta^{ijkm}plm)$,$m$$[\theta, \theta’]=p,$where$p \dot{f}j=\frac{2}{3}\sum(\theta^{lmni}(\theta l)lmnj-\frac{1}{8}(\sum\theta^{lm}nr(\theta’)lmnr)\delta_{i}j)$$l,m,n$.

$r$

Hereafter, we identify $p\in Lie(SL_{8}(\mathrm{C}))$ with the element of$\mathfrak{g}$defined by (1.8.1). We

may assume that$\mathfrak{h}=\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1}, \ldots, t8)|\sum t_{i}=0\}8$

.

$i=1$

Let
$\epsilon_{i}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{i}, \ldots, t8))=t_{i}$.

Then$R=\{\epsilon_{i}-\epsilon_{j}$$(1 \leq i,j\leq 8, i\neq j)$$\epsilon_{i}+\epsilon_{j}+\epsilon_{k}+\epsilon_{l}$,$(1 \leq i<j<k<l\leq 8)\}$ .

The coroots are given by$H(\epsilon_{i}-\epsilon_{j})=E_{i}-Ej$,.

$H( \epsilon_{i}+\epsilon_{j}+\epsilon k+\epsilon\iota)=(Ei+E_{j}+E_{k}+E\iota)-\frac{1}{2}\sum_{=i1}8E_{m}$Let$\theta(ijkl)$

be the tensor, with$(ijkl)- \mathrm{C}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{C}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}=1$, all others zero (but to preserve

the anti-symmetry of ), e.g., $\theta(ijkl)ijk\iota=1$ . A Chevalley system is given by$\theta$$X(\epsilon_{i}-\epsilon_{j})=E_{ij}$$(i\neq j)$$X( \epsilon_{i}+\epsilon_{j}+\epsilon_{k}+\epsilon_{l})=\frac{1}{2}\theta(ijkl)$$(i\leq i<j<k<l\leq 8)$ .

230

In fact$[X(\epsilon, -\epsilon_{j}), X(\epsilon_{j}-\epsilon_{k})]=X(\epsilon_{i}-\epsilon_{k})$$[X(\epsilon_{i}-\epsilon_{j}), X(\epsilon_{j}+\epsilon_{k}+\epsilon_{l}+\epsilon_{m})]=X(\epsilon_{i}+\epsilon_{k}+\epsilon_{l}+\epsilon_{m})$$[X(\epsilon_{i}+\epsilon_{j}+\epsilon_{k}+\epsilon_{l}), X(\epsilon_{i}+\epsilon_{m}+\epsilon_{n}+\epsilon_{r})]=X(\epsilon_{i}-\epsilon_{s})$,where different letters indicates different numbers. (Note that if 7 indices ( klmnr)$ij$

are given, then the remaining index, say , is uniquely determined.) The other com$s$

mutators are all zero. By these commutation relations, we can show that there exists
a unique involutory automorphism of$\iota$such that$\mathrm{C}_{7}$$\iota(p)=-^{t}p$$(p\in sl_{8}(\mathrm{C}))$,and$\iota(\theta(ijkl))=\theta(mnrs)$

where (mnrs) is chosen so that

$I_{1}!_{k}jlmnrs2345678=1$

,

. Let

$\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}$$\alpha_{8}=\epsilon_{5}+\epsilon 6+\epsilon 7+\epsilon_{8}$.

Then we may take as a root basisor$\{\alpha_{i}|\dot{i}\neq 1\}$$\{\alpha_{i}|i\neq 7\}$.

In fact, the extended Dynkin diagram is given by$\alpha_{1^{-\alpha_{2}}}-\alpha_{3}-\alpha 4-\alpha_{5}-\alpha_{6}-\alpha_{7}$$1$$\alpha_{8}$The involutory automorphism induces the unique non-trivial automorphism$\iota$of the extended Dynkin diagram.

231

1.9. TypeIn this$E_{8}$.
, we use the notations of [VE]. Let us consider three kinds$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{r}\mathrm{a}_{\mathrm{P}}\mathrm{h}$of tensors$X=(x_{j}^{i})_{1}\leq i,j\leq 9$with $\sum x_{i}^{i}9=0$ ,$i=1$$X_{*}=(x_{ijk})1\leq i,j,k\leq 9$,$X^{*}=(x^{ijk})_{1\leq}i,j,k\leq 9$.

Here all the tensors are assumed to be antisymmetric in the covariant indices and in
the contravariant indices. We may assume that$\{X^{*}\}$

is the vector space $\{X\}\oplus\{X_{*}\}\oplus$$\mathfrak{g}$, which is equipped with a Lie algebra structure by[X,$\mathrm{Y}$] $=Z$ ,.

$z_{jj\cdot j}^{i}=x.y^{i}-y.xi$] $=Z_{*}$ ,[X,$Y_{*}$[X,$Y^{*}$zijkl] $=z*$ ,$z^{ijk}=- \frac{1}{2}I^{i}..j.k_{X}.\cdot y\ldots$$[X*, Y_{*}]=Z$ ,$z_{j}^{i}= \frac{1}{2}(xy_{j}..-\frac{1}{9}x\ldots y\ldots I^{i}i\cdot\cdot j)$$[X^{*}, \mathrm{Y}^{*}]=Z_{*}$$[X_{*}, Y_{*}]=^{z^{*}}$$= \frac{1}{2}I_{ij}\ldots Xk\cdot.y\ldots$,,$z_{ijk}= \frac{1}{36}I_{ij}\ldots\ldots x\ldots y\ldots$$z^{ijk}= \frac{1}{36}I^{ijk}\ldots\ldots x\ldots y\ldots$

Here we used the notations of the first two sections of [VE]. We may assume thatis the set of the diagonal$X’ \mathrm{s}$. Let$\epsilon_{i}(\sum t_{i}Ei)=t_{i}9$

.

$j=1$

The root system is given by$R=\{\epsilon_{i}-\epsilon_{j} (1 \leq i,j\leq 9, i\neq j), \pm(\epsilon_{i}+\epsilon_{j}+\epsilon_{k}) (1 \leq i<j<k\leq 9)\}$.

$\mathfrak{h}$232

The coroot are given by$H(\epsilon_{i}-\epsilon_{j})=Ei-Ej$,$H( \pm(\epsilon i+\epsilon_{j}+\epsilon k))=\pm\{(Ei+Ej+E_{k})-\frac{1}{3}\sum E_{m}9\}$

.

$m=1$

Let $X_{*}(ijk)$ (resp. $X^{*}(ijk)$ ) be the tensor of type$=1$

$X_{*}$(resp.

, all others zero (but to preserve the anti-symmetry of$X^{*}$$X_{*}$), with $(ijk)$ -coefficient(resp.

$X^{*})$

system is given by$X(\epsilon_{i}-\epsilon_{j})=Eij$$X(\epsilon_{i}+\epsilon j+\epsilon_{k})=x_{*}(ijk)$,.

$X(-\epsilon i-\epsilon j-\epsilon k)=^{x*}(ijk)$

Let.

$\alpha_{i}=\epsilon_{i}-\epsilon_{i+}1$$(1\leq i$$\alpha_{9}=-\epsilon 1-\epsilon_{2}-\epsilon_{3}$$\leq 8)$,.

Then we may take as a root basis$\{\alpha_{i}|i\neq 8\}$.

In fact, the extended Dynkin diagram is given by$\alpha_{1}-\alpha_{2}-\alpha 3-\alpha_{4}-\alpha_{5}-\alpha 6-\alpha 7-\alpha_{8}$$1$$\alpha_{9}$). A Chevalley

233\S 2. Split Z-forms.
The purpose of this section is to classi$f\mathrm{y}$and describe the split$\mathrm{Z}$-forms of

saturated, irreducible, prehomogeneous vector spaces $(G, \rho, V)$ over C. Here we use
the definitions and the results of [G].

According to [G], first, we should choose highest weight vectors
$V$

and

$V^{}$

and$v_{0}$$v_{0}^{}$

ofso that
$V_{\max}(\mathrm{Z})\cap \mathrm{C}v_{0}=V_{\min}(\mathrm{Z})\cap \mathrm{C}v_{0}=\mathrm{Z}v_{0}$where, by definition,
We shall describe$V_{\min}(\mathrm{Z})=u_{\mathrm{z}\cdot v_{0}}$$V_{\min}(\mathrm{Z})$classify the graded$\mathcal{U}_{\mathrm{Z}}$

and

and

$V_{\max}(\mathrm{Z})$-modules$V(\mathrm{Z})$$V_{\max}(\mathrm{Z})$,is the dual lattice of$\mathcal{U}\mathrm{z}\cdot v^{\mathrm{v}}0[\mathrm{G}]$.

explicitly for each case. Our next task is towhich are -lattices of$\mathrm{Z}$$V_{\min}(\mathrm{Z})\subset V(\mathrm{Z})\subset Vmax(\mathrm{z})$

$V$

and

.

Fortunately, it will turn out that our second task is almost nothing. In fact, ourcalculation will show that such a$V(\mathrm{Z})$coincides with

$V_{\min}(\mathrm{Z})$or$V_{\max}(\mathrm{Z})$.

In course of our calculation, we need to fix a Chevalley system, a basis of aroot system etc. In such a case, we always use those given in the first section. If a

non-degenerate bilinear form (, is defined on$\rangle$

the vector space
where the le$\mathrm{Z}$$f\mathrm{t}$

$V$

$V$

, we identi$f\mathrm{y}$the vector space

via the isomorphism $I:V^{\vee}-V\simeq$ defined byhand side is the natural pairing. (Note that

$I$

$V^{}$

with$\langle v^{}, v\rangle=(I(v^{}), v)$,does not preserve the-structure.) For the sake of a convenience for later calculations, we will give anon-degenerate bilinear form such that $\rho(G)=\rho^{\vee}(G)$ , if we identify$V^{}$

with

In $(2.1)-(2.15)$ , we shall treat reduced prehomogeneous vector spaces.

2.1. Type (1).

$V$

.

234

The representation space

matrices$M_{m}(\mathrm{C})$$V$

can be identified with the totality of $m\cross m$. We may assume that $G=GL_{m}\cross GL_{m}$ . The action of$G$

is givenby$(X\in M_{m}(\mathrm{C}),g=(g1,g_{2})\in G)$$\rho(g)x=g_{1}x^{t}g_{2}$Then a highest weight vector is given by $v_{0}=E_{11}$ . By applying

Then the action of

$G$

on

$V^{}$$V^{}$

of$V$$\rho^{\vee}(G)$is identified withhighest weight vector of$V^{}$$V$by X,$V_{\min}^{\vee}(\mathrm{Z})=M_{m}(\mathrm{Z})$$\mathrm{Y}\rangle$$\langle$$=\mathrm{t}\mathrm{r}(^{t}X\mathrm{Y})$$v_{0}$

for $X,$$(Y\in Mm(\mathrm{c}),g=(g1,g_{2})\in G)$, we have$Y\in M_{m}(\mathrm{C})$.

.

via the above identification $V=V^{}$ . A$\rho(G)$is given by$v_{0mm}^{\mathrm{v}_{=E}}$

Hence

to

is given by$\rho^{\vee}(g)Y=^{t_{g\mathrm{Y}g_{2}^{-1}}}1-1$

Note thatwith$\mathcal{U}_{\mathrm{Z}}$.

$V_{\min}(\mathrm{Z})=M_{m}(\mathrm{z})$We identify the dual space

.

.

, and

$V_{\max}(\mathrm{Z})=Mm(\mathrm{Z})$Hence there is only one split -form. A -basis of$\mathrm{Z}$$\mathrm{Z}$$E_{ij}$.

$V_{\min}(\mathrm{Z})=V_{\max}(\mathrm{Z})$$(1\leq i,j\leq m)$

and its dual is$E_{ij}^{}=E_{i}j$$(1\leq i,j\leq m)$.

is given by

235

2.2. Type (2).

The representation space can be identified with the totality of$n\cross n$symmetric

matrices $V=\{X\in M_{n}(\mathrm{C})|t_{X}=X\}$ . We may assume that $G=GL_{n}$ . The action
is given by$\rho(g)x_{=}gx$tg$(X\in V,g\in G)$.

A highest weight vector is given by $v_{0}=E_{11}$ . By applying$V_{\min}(\mathrm{Z})=\{x\in Mn(\mathrm{Z})|t_{X=X\}}$

We identi$f\mathrm{y}$the dual space$V^{}$

of

$V$

with

$V$

$\mathrm{Y}\rangle$$v_{0}$, we have.

$=\mathrm{t}\mathrm{r}X\mathrm{Y}$. The action of

$G$

on

is given by$V^{}$$\rho^{\vee}(g)Y=t11g^{-}\mathrm{Y}g^{-}$

Note that$\rho^{\vee}(G)$is identified withhighest weight vector of$\mathrm{Y}\}$by {X,

to$\mathcal{U}_{\mathrm{Z}}$$\rho(G)$is given by$V^{}$$(Y\in V^{},g\in G)$.

via the above identification $V=V^{}$ . A$v_{0}^{\vee}=E_{nn}$. Since$V_{\min}^{}=\{\mathrm{Y}\in M_{n}(\mathrm{Z})|t_{\mathrm{Y}=}$,$V_{\max}( \mathrm{Z})=\sum_{=i1}\mathrm{Z}Ei+\sum_{<ij}\mathrm{z}\cdot\frac{1}{2}(Eij+E_{ji})n$We can show that

$V_{\max}(\mathrm{Z})/V_{\min}(\mathrm{Z})$is a simple graded$\mathcal{U}_{\mathrm{Z}}$.

-module. (It is enough toconsider the action of the Weyl group.) Hence there are exactly two split -forms. A$\mathrm{Z}$$\mathrm{Z}$-basis of$V_{\min}(\mathrm{Z})$is given by$E_{i}$$(1 \leq i\leq n)$,$E_{ij}+E_{ji}$$(1 \leq i<j\leq n)$.

Its dual basis is given by$E_{i}^{}=E_{i}$which is a basis of,$(1 \leq i\leq)$$V_{\max}(\mathrm{Z})$.

$(E_{ij}+Eji)^{\mathrm{v}_{=}} \frac{1}{2}(E_{i}j+E_{ji})$$(1 \leq i<j\leq n)$

,

236

2.3. Type (3).

The representation space can be identified with the totality of $2m\cross 2m$
skew-symmetric matrices $V=\{X\in M_{2m}(\mathrm{C})|t_{X}+X=0\}$ . We may assume that
$G=GL_{2m}$. The action of$G$

is given by$\rho(g)X=gX^{t}g$$(X\in V,g\in G)$.

A highest weight vector is given by $v_{0}=E_{12}-E_{21}$ . By applying$V_{\min}(\mathrm{Z})=\{X\in M_{2m}(\mathrm{Z})|t_{X}+X=0\}$

We idention$V^{}$$f\mathrm{y}$the dual space$V^{}$

of

$V$

with

$V$

by X,
$\langle$$Y\rangle$$=- \frac{1}{2}$$\mathcal{U}_{\mathrm{Z}}$to$v_{0}$, we have.

tr $XY$ . The action of$G$

is given by$\rho^{\vee}(g)Y=t_{g^{-1}Yg^{-1}}$

Note that
vector of$\rho^{\vee}(G)$$v^{}$$\mathrm{Y}+t_{Y=0\}}$is identified with$\rho(G)$$(\mathrm{Y}\in V^{\vee},g\in G)$.

via our identification. A highest weightis given by $v_{0}^{\vee}=E_{2m}-1,2m-E2m,2m-1$ . Since$V_{\min}^{\vee}(\mathrm{Z})=\{\mathrm{Y}\in M_{2m}(\mathrm{Z})|$,$V_{\max}(\mathrm{Z})=\{X\in M_{2m}(\mathrm{Z})|X+t_{X=0\}}$Hence there is only one split -form. A -basis of$\mathrm{Z}$$E_{ij}-E_{ji}$$\mathrm{Z}$.

$V_{\min}(\mathrm{Z})=V_{\max}(\mathrm{Z})$$(1\leq i<j\leq 2m)$.

Its dual basis is$(E_{ij}-Eji)\vee=E_{i}j-E_{ji}$$(1 \leq i<j\leq 2m)$.

is given by237

2.4. Type (4).

The representation space can be identified with the third symmetric product$S^{3}(\mathrm{C}^{2})$Thenof a two dimensional vector space. We may assume that $G=GL_{2}=GL(\mathrm{C}^{2})$ .

$G$

acts naturally on

vector is given by$v_{0}=e_{1}^{3}$$S^{3}(\mathrm{C}^{2})$. Let$e_{1}=\mathrm{t}1,0$. By applying

to$\mathcal{U}\mathrm{Z}$$v_{0}$) and$e_{2}=\mathrm{t}0,1$, we have$V_{\min}(\mathrm{Z})=\mathrm{Z}\cdot e_{1}^{3}+\mathrm{Z}\cdot 3e_{1}^{2}e_{2}+\mathrm{Z}\cdot 3e_{1}e_{2}^{2}+\mathrm{Z}\cdot e_{2}^{3}$

We identi$f\mathrm{y}$the dual space$V^{}$

of

with

$V$

$V$

). A highest weight.

itself by$\langle e_{1}e_{2’ 12}ee-b\rangle as_{-a}b3=\{$$(a\neq b)$$0$

If we denote the actions of

$G$

on

$V$

and

$V^{}$

by$\rho$

and$\rho^{}$.

, respectively, then$\rho(^{t}g^{-1})$. In particular, $\rho^{\vee}(G)=\rho(G)$ . A highest weight vector of$v_{0}^{\vee}=e_{2}^{3}$. We have$V^{}$$\rho^{\vee}(g)=$is given by$V_{\min}^{}(\mathrm{Z})=^{\mathrm{z}\cdot e_{1}}3+\mathrm{Z}\cdot 3e_{12}^{2}e+\mathrm{Z}\cdot 3e_{12}e^{2}+\mathrm{Z}\cdot e_{2}^{3}$and$V_{\max}(\mathrm{Z})=\mathrm{Z}\cdot e_{1}^{3}+\mathrm{Z}\cdot e_{1}^{2}e_{2}+\mathrm{Z}\cdot e_{12}e^{2}+\mathrm{Z}\cdot e_{2}^{3}$

We can show that

$V_{\max}(\mathrm{Z})/V_{\min}(\mathrm{Z})$is a simple gradedexactly two split -forms. A -basis of$\mathrm{Z}$$\mathrm{Z}$$V_{\min}(\mathrm{Z})$$\mathcal{U}_{\mathrm{Z}}$.

-module. Hence there are

ais given by$e_{1}^{3},3e_{1}^{2}e_{2},3e_{1}e_{2}^{2},$$e_{2}^{3}$.

Its dual basis is given by$(e_{1}^{3})^{\vee}=e_{1}^{3},$which is a basis of$(3e_{1}^{2}e_{2})^{\vee}=e_{1}^{2}e_{2},$$V_{\max}(\mathrm{Z})$.

$(3e_{1}e_{2})^{\vee}2=e_{1}e_{2}^{2},$$(e_{2}^{3})^{\vee}=e_{2}^{3}$

,

238

2.5. Types (5),(6) $,(7),(9),(10)$ and (11).
Let $(l, m, n)=(3,6,1),$ $(3,7,1),$ $(3,8,1),$ $(2,6,2),$ $(2,5,3)$ or (2, 5, 4) for the

prehomogeneous vector space of type (5),(6) $,(7),(9),(10)$ or (11), respectively. Thenthe representation space can be identified withthe l-th Grassmann product ofwhich acts naturally on
standard bases of$v_{0}=$($e_{1}$A). We may assume that$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\wedge^{l}(\mathrm{C}^{m})$is$G=GL(\mathrm{C}^{m})\cross GL(\mathrm{C}^{n})$,. Let $\{e_{i}|1\leq i\leq m\}$ and $\{f_{j}|1\leq j\leq n\}$ be theand$\mathrm{C}^{m}$$e_{2}\wedge\cdots\wedge e_{l}$$V$$\mathrm{C}^{m}$$V=\wedge^{l}(\mathrm{c}m)\otimes \mathrm{C}^{n},$$\mathrm{C}^{n}$, respectively. A highest weight vector is given by. By applying$\otimes f_{1}$$\mathcal{U}_{\mathrm{Z}}\mathrm{t}\mathrm{o}_{-}v_{0}$(, we have$ei_{1}\wedge\cdots$A)$e_{i_{1}}$$\otimes fj$.

$V_{\max}( \mathrm{Z})=\sum_{\leq j}1\leq i_{11}<\cdots<i\leq n^{\mathrm{t}}\leq m\mathrm{Z}\cdot$

We identify the dual space$\langle$($e_{i_{1}}\wedge\cdots$A$e_{i_{\mathrm{t}}}$$V^{}$)

of$\otimes f_{j},$

$V$

with

$(e_{i_{1}’}\wedge\cdots\wedge e_{i_{\mathrm{t}}^{\prime)}}\otimes f_{j’}\rangle=\delta_{i_{1}i_{1}^{\prime\delta_{iijj^{l}}}}\ldots\iota_{\iota}l\delta$

where $\dot{i}_{1}<\cdots<i_{l},$ $i_{1}’<\cdots<i_{l}’$ andaction of

$G$

on

$V$

and

$V^{}$

by$\rho$

by

$V$

and

$p^{}$,

is the Kronecker’s delta. Denote the$\delta$, respectively. Then$\rho^{\vee}(g1,g_{2})=\rho(t_{g_{1}^{-}}1, tg2)-1$

for $(g_{1},g_{2})\in G$ . In particular, $\rho^{\vee}(G)=\rho(G)$ . A highest weight vector ofis given byand$v_{0}^{\vee}=$($e_{m-\iota+1}\wedge\cdots$$V_{\max}(\mathrm{Z})=V_{\min}(\mathrm{Z})$$V_{\min}(\mathrm{Z})=V_{\max}(\mathrm{Z})$(A$e_{m}$)$\otimes f_{n}$. Then we have

$V_{\min}^{\vee}(\mathrm{Z})=V_{\max}(\mathrm{Z})$. Hence there is exactly one split -form. A -basis of$\mathrm{Z}$$\mathrm{Z}$is given by$e_{i_{1}}\wedge\cdots$

A$V^{}$$e_{i_{\iota}}$)$\otimes f_{j}$$(1\leq i_{1}<\cdots<i_{l}\leq m, 1\leq j\leq n)$

Its dual basis if given by$((e_{i_{1}}\wedge\cdots \mathrm{A} e_{i_{1}})\otimes f_{j})^{\vee}=$($e_{i_{1}}\wedge\cdots$A$e_{i_{\mathrm{t}}}$)$\otimes f_{j}$

.

.

239

2.6. Type (8).

The representation space can be identified withmay assume that${}^{t}(1,0,0),$$G=GL(\mathrm{C}^{3})\cross GL(\mathrm{C}^{2})$, which acts naturally on$e2=t(0,1,0),$ $e_{3}=$ tO, $0,1$ ), $F_{1}={}^{t}(1,0)$ andvector of$V$

is given by$v_{0}=e_{1}^{2}\otimes f_{1}$$V=S^{2}(\mathrm{C}^{3})\otimes \mathrm{C}^{2}$. By applying$\mathcal{U}_{\mathbb{Z}}$$f_{2}=\mathrm{t}0,1$

to

$V$

of

$V^{}$

$V$

with

$V$

, we have$v_{0}$

Denote the ctions of

$G$

on

$\rho(^{t-1-1}g_{1},{}^{t}g_{2})((g_{1},g_{2})\in G)$$V^{}$

is given by,

if $(a_{1}, a_{2}, a_{3}, a)=(b_{1}, b_{2}, b3, b)$$e_{1}^{b}ee^{b_{3}}1b232\otimes f_{b}\rangle=\{$$0$$\dot{\mathrm{a}}$.

by$\frac{a_{1}!a_{2}!a_{3}!}{2!}$$\langle e_{1}^{a_{1}}e_{23}^{a_{2}a_{3}}e\otimes f_{a},$. Let $e_{1}=$). A highest weight$V_{\min}( \mathbb{Z})=(\sum \mathbb{Z}\cdot e_{i}1\leq i\leq 3+21\leq<\sum_{ij\leq 3}\mathbb{Z}\cdot 2e_{i}ej)\otimes(\mathbb{Z}f_{1}+\mathbb{Z}f2)$

We identify the dual space

. We

$V$

and

$V^{}$,

byotherwise.
$p$

and

$\rho^{}$, respectively. Then $p^{\vee}(g1,g2)=$. In particular, $p^{\vee}(G)=\rho(G)$ . A highest weight vector of$v_{0}^{\vee}=e_{3}^{2}\otimes f_{2}$. We haveand$V_{\min}^{}(\mathbb{Z})=V_{\min}(\mathbb{Z})$.
$V_{\max}( \mathbb{Z})=\sum_{\leq 1\leq k2}\mathbb{Z}\cdot e_{i}1\leq i\leq j\leq 3e_{j}\otimes f_{k}$We can show that

$V_{\max}(\mathbb{Z})/V_{\min}(\mathbb{Z})$is a simple gradedexactly two split -forms. A -basis of$\mathbb{Z}$$\mathbb{Z}$$V_{\min}(\mathbb{Z})$$\mathcal{U}_{\mathbb{Z}}$-module. Hence there are

is given by,$e_{i}^{2}\otimes f_{k}$$(1 \leq i\leq 3,1\leq k\leq 2)$$2e_{i}e_{j}\otimes fk$$(1 \leq i<j\leq 3,1\leq k\leq 2)$.

Its dual basis is given by,$(e_{1}^{2}\otimes f_{k})^{\vee}=e^{2}i\otimes f_{k}$$(1 \leq i\leq 3,1\leq k\leq 2)$$(2e_{i}e_{j}\otimes f_{k})\vee f=e_{i}e_{j}\otimes k$$(1 \leq i<j\leq 3,1\leq k\leq 2)$

which is a basis of$V_{\max}(\mathbb{Z})$

.

,

240

2.7. Type (12).
The representation space can be identified withmay assume that$\{f_{j}|1\leq j\leq 2\}$

$G=GL(\mathrm{C}^{3})\cross GL(\mathrm{C}^{3})\cross GL(\mathrm{C}^{2})$be the standard bases ofvector is given by$\mathrm{C}^{3}$and$\mathrm{C}^{2}$$V=\mathrm{C}^{3}\otimes \mathrm{C}^{3}\otimes \mathrm{C}^{2}$. We. Let $\{e_{i}|1\leq i\leq 3\}$ and, respectively. A highest weight. We have$v_{0}=e_{1}\otimes e_{1}\otimes f_{1}$$\sum$$V_{\min}(\mathrm{Z})=$$\mathrm{Z}\cdot e_{i}\otimes e_{j}\otimes f_{k}$.

$1\leq i,j\leq 3$$1\leq k\leq k$

We identify$V^{}$

with

$V$

by

$\langle e_{i}\otimes e_{j}\otimes f_{k},$The action$G$$\rho^{}$

of

$G$

on

$V^{}$.

$e_{i’}\otimes e_{j’}\otimes f_{k’})=\delta_{ii}’\delta_{j}j’\delta_{k}k’$

is given by$\rho^{\vee}(g_{1},g2,g3)=p(^{t_{g_{1}}-1t_{g_{2}^{-}}1-1},, t)g_{3}$. In particular, $p^{\vee}(G)=\rho(G)$ . A highest weight vector of$e_{1}\otimes e_{1}\otimes f_{2}$. Then we have

$V_{\min}^{\vee}(\mathrm{Z})=V_{\min}(\mathrm{Z})$is exactly one split -form. A -basis of$\mathrm{Z}$$\mathrm{Z}$$e_{i}\otimes e_{j}\otimes f_{k}$and$V^{}$

for $(g_{1},g2,g2)\in$

is given by$V_{\max}(\mathrm{C})=V_{\min}(\mathrm{Z})$$V_{\min}(\mathrm{Z})=V_{\max}(\mathrm{Z})$$v_{0}^{\vee}=$. Hence thereis given by$(1 \leq i,j\leq 3,1\leq k\leq 2)$.

Its dual basis is given by$(e_{i}\otimes e_{j}\otimes f_{k})^{}=ei\otimes e_{j}\otimes f_{k}$.

2.8. Type (13).

The representation space can be identified withassume that

$G=Sp_{2n}(\mathrm{C})\cross GL_{2m}(\mathrm{C})$$V=\mathrm{C}^{2n}\otimes \mathrm{C}^{2m}$. We may. Here we realize the symplectic groupas in (1.3), i.e.,$Sp_{2n}(\mathrm{c})=\{g\in GL_{2}n(\mathrm{c})|gJ^{t}g=J\}$.

$Sp_{2n}(\mathrm{C})$

241

Then

acts naturally on$G$standard bases ofby$v_{0}=e_{1}\otimes f_{1}$$\mathrm{C}^{2n}$

$V$

and

. Let $\{e_{i}|1\leq i\leq 2n\}$ and $\{f_{j}|1\leq j\leq 2m\}$ be the$\mathrm{C}^{2m}$, respectively. A highest weight vector ofis given$V$. We have.
$V_{\min}( \mathrm{Z})=11\leq i\leq\sum_{j\leq m}\leq_{2}2n\mathrm{Z}\cdot e_{i}\otimes f_{j}$We identify$V^{}$

with$V$

by the skew-symmetric bilinear form defined by$\langle e_{i}\otimes f_{j},$$e_{n+k}\otimes f_{l}\rangle=\delta_{ik}\delta_{jl}$,$\langle e_{i}\otimes fj, ek\otimes fl\rangle=\langle e_{n}+i\otimes f_{j}, e_{n+}k\otimes f\iota\rangle=0$

for $1\leq i,$ $k\leq n$ and $1\leq j,$ $l\leq 2m$ . The action$\rho(g_{1},$of$V^{}$$t_{g_{2}^{-1})}$

of

$G$

on

is given by$V^{}$$\rho^{\vee}(g1,g_{2})=$

for $(g_{1},g_{2})\in G$ . In particular, $\rho^{\vee}(G)=\rho(G)$ . A highest weight vector

is given by$V_{\min}(\mathrm{Z})$$\rho^{}$,$v_{0}=e_{1}\otimes f_{2m}$. Then we have

$V_{\min}^{\vee}(\mathrm{Z})=V_{\min}(\mathbb{Z})$. Hence there is exactly one split -form. A -basis of$\mathrm{Z}$$\mathrm{Z}$and$V_{\max}(\mathrm{Z})=$$V_{\min}(\mathbb{Z})=V_{\max}(\mathrm{Z})$isgiven by$(1\leq i\leq 2n, 1\leq j\leq 2m)$$e_{i}\otimes f_{j}$.

Its dual basis is given by$(e_{i}\otimes f_{j})^{}=e_{i’}\otimes fj$

,

where
$i+n$

$(1\leq\dot{\iota}\leq n)$$i-n$$(n+1\leq i\leq 2n)$$i’=\{$.

2.9. Type (14).

Let$\{e_{i}\}_{1\leq i}\leq 6$

be the standard basis of$\mathrm{C}^{6}$. The representation space can be

identified with
$V= \{ \sum x_{ijk}e_{i}\wedge e_{j}\wedge e_{k}|x_{i14}+x_{i25}+x_{i36}=0 (1\leq i\leq 6)\}$$1\leq i<j<k\leq 6$

,

242

where we regard$Sp_{6}(\mathrm{C})$, where$(x_{ijk})$

as an alternating tensor. We may assume thatis realized as in (1.3). Then

$Sp_{6}(\mathrm{C})$weight vector is given by$v_{0}=e_{1}\wedge e_{2}\wedge e_{3}$$6’=3,\overline{\mathrm{a}}\mathrm{n}\mathrm{d}ijk=e_{i}\wedge e_{j}\wedge e_{k}$$G$acts naturally on$V$$G=\mathrm{C}^{\cross}\cross$. A highest. Let $1’=4,2’=5,3’=6,4’=1,5’=2$ ,. Then a -basis of$\mathrm{Z}$$V_{\min}(\mathrm{Z})$123, $1’23,12’3,123’,$ $12’3’,$ $1’23’,$

is given by$1’2^{\prime_{3}},1’2^{\prime_{3’}}$,$122’-133’,$ $211’-233^{l},$ $311’-322^{l}$ ,(2.9.1)$1^{\prime_{22}/}-1’33l,$$2^{\prime_{11^{l}-}}2\prime 33’,$

Let us define a skew-symmetric bilinear form

$3’11^{l}-3/_{22’}$$\mathrm{o}\mathrm{n}\wedge^{3}(\mathrm{c}^{6})$.

by$(ijk,$ $lmn\rangle=\mathrm{S}\mathrm{g}\mathrm{n}$ ,where sgn is the signature on the symmetric group

sgn

$=0$

Note that $X(r)$ acts$\mathrm{o}\mathrm{n}\wedge^{3}(\mathrm{C}^{6})$$S_{6}$

which is extended by

if $\{ijklmn\}\neq$ {123456}.

,

as

(1)

$iarrow j$,$j’arrow-i’$ ,(2)$i’.arrow j$,$j’arrow i$,$karrow 0$$(k\neq i,j’)$$karrow 0$$(k\neq i’,j^{/})$

or

(3)

$iarrow j’$,$jarrow i’$where $i,j\in\{1,2,3\},$ $1\leq k\leq 6,$,$karrow 0$$(k\neq i,j)$$-i=-e_{i}\mathrm{a}\mathrm{n}\mathrm{d}-i’=-e_{i’}$$\langle ijk, i^{/}j’k’\rangle=1$

and,. Note also that$\langle ijk’, i^{l\prime}jk\rangle=-1$

243

for $i,j,$ $k\in\{1,2,3\}$ . By using these facts, we can show that our bilinear form is$Sp_{6}(\mathrm{C})$

of

$G$

-invariant. We identify

on$V^{}$

is given by

$V$

and

$V^{}$

by this bilinear form. Hence the action$\rho^{\vee}(g1,g2)=\rho(g_{1}^{-1},g_{2})$for$(g_{1},g_{2})\in G=\mathrm{C}^{\cross}\cross Sp_{6}(\mathrm{C})$particular, $\rho^{\vee}(G)=\rho(G)$ . A highest weight vector of
we have$V_{\min}^{}(\mathbb{Z})=V_{\min}(\mathrm{Z})$is given by$v_{0}^{\vee}=123$. In. Then. The dual basis of (2.9.1) is$(1’23)^{}=-12^{\prime_{3’}},$$(123)^{\vee}=1’2^{;_{3}}/,$$V^{}$$\rho^{}$$(12’3)^{\vee}=-1’23’,$$(1’23’)^{}=-12’3,$$(12’3’)\mathrm{v}=-1^{\prime_{23}},$$(123’)^{}=-1’2’3$ ,$(1’2^{\prime_{3}})^{\mathrm{v}}=-123’,$$(1^{l}2^{l}3/)^{\mathrm{v}}=123$(2.9.2)Hence$(122’-133^{l})^{}= \frac{1}{2}(1’2^{\prime_{2}}-1’3’3)$

etc.

$(1’22’-1 \prime 33’)^{\vee}=\frac{1}{2}(12’2-13\prime 3)$etc.

$V_{\max}(\mathbb{Z})$is the free -module generated by (2.9.2). We can show that$\mathrm{Z}$is a simple graded$\mathcal{U}_{\mathrm{Z}}$$V_{\max}(\mathrm{Z})/V_{\min}(\mathrm{Z})$-module. Hence, there are exactly two split Z-forms.

2.10. Type (15B).

The representation space can be identified with

$V=\mathrm{C}^{2k+1}\otimes \mathrm{C}^{m}$. We mayassume that $G=so_{2k+1}(\mathrm{C})\cross GL_{m}(\mathrm{C})$ . Here we realize the special orthogonal group$so_{2k+1}(\mathrm{C})$as in (1.2), i.e.,$SO_{2k+1}(\mathrm{c})=\{g\in GL2k+1(\mathrm{c})|gJ^{t}g=J\}$

Then

$G$

acts naturally on

standard bases ofby$v_{0}=e_{1}\otimes f_{1}$$\mathrm{C}^{2k+1}$$V$.

. Let $\{e_{i}|1\leq i\leq 2k+1\}$ and $\{f_{j}|1\leq j\leq m\}$ be theand$\mathrm{C}^{m}$, respectively. A highest weight vector of. We have.
$V_{\min}( \mathrm{Z})=\sum_{i1\leq,1\leq j\leq\leq 2km}\mathbb{Z}\cdot e_{i}\otimes f_{j}+\sum_{\leq 1\leq jm}\mathrm{z}\cdot 2e2k+1^{\otimes f_{j}}$$V$is given244

Let us identify

with

$V$

$M_{2k+1,m}(\mathrm{c})$by$\sum_{p,q}a_{pq}e_{p^{\otimes}q}farrow(a_{pq})$

The induced$G$-action on$V^{}$

with$V$

is given by$M_{2k+1,m}(\mathrm{c})$$(g_{1},g_{2})\in G=SO_{2k+1}\cross GL_{m}$$varrow g_{1}v^{t}g2$

We identify.

.

by the symmetric bilinear form defined by$(v_{1}, v_{2}\rangle=\mathrm{t}\mathrm{r}(tv_{1}J^{-}1v_{2})$Then

1$\langle e\otimes pfq’ re\otimes f_{S}\rangle=\{$$\frac{1}{2}$$0$.

$(p=r’\neq 2k+1, q=s)$$(p=r’=2k+1, q=s)$otherwise,

where$i’=\{$

The action$\rho^{}$

of

$G$

on

$V^{}$$i+k$$(1\leq i\leq k)$$\dot{i}-k$$(k+1\leq i\leq 2k)$

$2k+1$$(i=2k+1)$ .

is given by$\rho^{\vee}(g1,g_{2})=\rho(g_{1},g_{2})t-1$

particular, $\rho^{\vee}(G)=\rho(G)$ . A highest weight vector ofThen we have

$V_{\min}^{\vee}(\mathrm{Z})=V_{\min}(\mathrm{Z})$. A -basis of$\mathbb{Z}$$V^{}$$V_{\min}(\mathrm{Z})$is given by

is given by$e_{i}\otimes f_{j}$$(1 \leq i\leq 2k, 1\leq j\leq m)$$2e_{2k+1}\otimes f_{j}$$(1\leq j\leq m)$.

for $(g_{1},g_{2})\in G$ . In,

$v_{0}^{\vee}=e_{1}\otimes f_{m}$

.

245

Its dual basis is given by$(e_{i}\otimes f_{j})^{}=e_{i’}\otimes fj$$(1 \leq i\leq 2k, 1\leq j\leq m)$$(2e_{2k+1}\otimes fj)^{}=e_{2k1}+\otimes f_{j}$$(1\leq j\leq m)$

,

.

Hence

.
$V_{\max}( \mathrm{Z})=1\leq i1\leq j\sum_{2\leq k+1}\mathbb{Z}\cdot e_{i}\otimes f_{j}$We can show that

$V_{\max}(\mathrm{Z})/V_{\min}(\mathbb{Z})$is a simple graded$\mathcal{U}_{\mathrm{Z}}$-module. Hence, there areexactly two split Z-forms.

2.11. Type (15D).
With a trivial modification of (2.10), we have1$(p=r’, q=s)$$\langle e_{p}\otimes fq’ re\otimes fS)=\{$$0$$v_{0}=e_{1^{\otimes f_{1}}}$, otherwise,,$v_{01^{\otimes f_{m}}}^{\mathrm{v}_{=e}}$,,$V_{\min}( \mathrm{Z})=V_{\max}(\mathrm{z})=1\leq i\leq_{m}1\leq j\sum_{\leq^{2}}\mathrm{z}\cdot e_{i}k\otimes f_{j}$and$(e_{i}\otimes f_{j})^{}=e_{i’}\otimes fj$

In particular, there is exactly one split Z-form.

.

246

2.12. Types (20), (21), (23) and (24).
Let $(m, n)=(2,5),$ $(3,5),$ $(1,6),$ $(1,7)$ , if we are considering a prehomoge-

neous vector space of type (20), (21), (23), (24), respectively. Then the representation
space can be identified$\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\wedge^{even}(\mathrm{C}^{n})\otimes \mathrm{C}^{m}$. Here and below in this paragraph, we

use the notations of (1.4). We may assume that $G=Spin_{2n}\cross GL_{m}$ , which acts
naturally on$\mathrm{C}^{n}$and$l=2[ \frac{n}{2}]$$\mathrm{C}^{m}$$V$. Let $\{e_{i}|1\leq i\leq n\}$ and $\{u_{j}|1\leq j\leq m\}$ be the standard bases of, respectively. A highest weight vector is given by$e_{1}e_{2}\ldots e_{l}\otimes u_{1}$, where. We have$\sum$$\sum$$V_{\min}(\mathrm{Z})=$$\mathrm{Z}\cdot e_{i_{1}}e_{i_{2}}\ldots e_{i_{k}}\otimes u_{j}$.

$0\leq k\leq^{\iota}1\leq i_{1}<\cdots<i_{k}\leq^{\iota}$$k:\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}$We identify$V^{}$

with$V$$1\leq j\leq m$by$\langle e_{a:}\ldots e_{a_{k^{\otimes,ee}}}uib_{1}\cdots b\iota^{\otimes}uj\rangle$

1$(\{a_{1}, \ldots, a_{k}\}=\{b1, \ldots, bl\}, i=j)$$0$(otherwise),

$=\{$

where

$1\leq a_{1}<\cdots<a_{k}\leq n$$1\leq b_{1}<\cdots<b_{l}\leq n$$1\leq i.,j\leq m$

Then the action$p^{}$$G=Spin_{2n}\cross GL_{m}$

of

$G$

on

$V^{}$,,.

is given by$\rho^{\vee}(g1,g_{2})=\rho(\iota(g_{1}), t-g_{2}1)$for$(g_{1},g_{2})\in$. Here is the involutory automorphism of $Spin_{2n}$ given in (1.4).
$\iota$In particular, $p^{\vee}(G)=p(G)$ . A highest weight vector is given by$1\otimes u_{m}$. Then we

247

have
$\mathrm{Z}$$V_{\min}^{\vee}(\mathrm{Z})=V_{\max}(\mathrm{Z})$-form. A -basis of$\mathbb{Z}$and$V_{\min}(\mathrm{Z})$$V_{\max}(\mathrm{Z})=V_{\min}(\mathrm{Z})$. Hence there is exactly one split

is given by( $1\leq i_{1}<\cdots<i_{k}\leq n,$$e_{i_{k}}\ldots e_{i_{k}}\otimes u_{j}$$k$: even, $1\leq j\leq m$ ),and its dual basis is given by$(e_{i_{1}}\ldots e_{i_{k}}\otimes u_{j})\mathrm{v}=ei1\ldots eik\otimes u_{j}$.

2.13. Types (27) and (28).
Let $n=1,2$ if we are considering a prehomogeneous vector space of type (27)or (28), respectively. The representation space can be identified with$\mathrm{J}\otimes \mathrm{C}^{n}$. Hereand below in this section, we use the notations of (1.6) and (1.7). We may assume
that $G=G(E_{6})\cross GL_{n}$ , where$G(E_{6})=$See [F,8.1]. Then{linear automorphism of which preserves $det(X,$$\mathfrak{J}$$G$

acts naturally onhighest weight vector of$V$is given by

$V$

. Let

$\{u_{i}\}$

$Y,$

$z)$

be the standard basis of$v_{0}=E_{11}^{(3)}\otimes u_{1}$$(a)_{i}$. We identify$V^{}$

with$V$$\mathrm{C}^{n}$. A. We have$V_{\min}( \mathrm{Z})=\otimes\sum_{n1\leq k\leq}\mathbb{Z}\cdot u_{k}$See (1.6) for}.

.

by the symmetric bilinear form defined by$(X\otimes u_{j}, Y\otimes u_{k})=\delta_{jkx}.(X\mathrm{o}Y)$(X, $Y\in fl$ ),

248

where

$\chi$

is the trace function of$\mathrm{J}$(see (1.7)), and$X \mathrm{o}\mathrm{Y}=\frac{1}{2}(X\mathrm{Y}+YX)$. A direct

calculation shows$\langle(_{X}\overline{x_{2}3}\xi_{1}$$\overline{x_{2}X_{1}\xi \mathrm{s}}1,$$x_{3}\overline{x_{1}}\xi_{2}$$\overline{y_{2}y_{1}})\eta_{3}\rangle=\sum_{i=1}^{3}\{\xi_{i}\eta_{i}+2(xi, yi)\}$$(_{y_{2}}^{\eta}\overline{y_{3}}1$$\eta_{2}y_{3}\overline{y_{1}}$

,

where
$(x, y\in \mathrm{C})$$(x, y)= \frac{1}{2}(x\overline{y}+y\overline{x})=\frac{1}{2}(\overline{x}y+\overline{y}x)$Let$\rho^{}$

be the dual of . Since$\rho$$\chi(X\mathrm{o}\mathrm{Y})$is$S_{4}$-invariant$[\mathrm{F}$.

, 4.5.13 ,$]$$p^{\vee t_{g}-1}(g_{1},g_{2})=\rho(g1,2)$

for $(g_{1},g_{2})\in G(F_{4})\cross GL_{2}$ . Here $G(F_{4})$ is the subgroup of $G(E_{6})$ which corresponds
to the Lie subalgebra$S_{4}(.\subset \mathrm{G}_{6})$of the infinitesimal automorphisms of the Jordan

algebra 3. A direct calculation shows thatX$(i\neq j, a\in (!), X\in \mathrm{J},$ $Y\in \mathrm{J})$$(a_{ij}^{\sim_{X\mathrm{o}}}\mathrm{Y})+\chi(X\mathrm{o}(-\overline{a})_{j}^{\sim}iY)=0$

Hence we can define an involutory automorphism of$\mathrm{C}_{6}$$\iota$$\iota((a)_{ij}\sim)=(-\overline{a})_{ji}^{\sim}$.

by$(i\neq j, a\in C)$

and$\iota|S_{4}\equiv \mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{y}$.

Since $G(E_{6})(\supset\{\omega|\omega^{3}=1\})$ is simply connected,$G(E_{6})$$\iota$induces an automorphism of, which we shall denote by the same letter . Then we have$\iota$$\chi(gX\mathrm{o}\mathrm{Y})=\chi(X\mathrm{o}\iota(g)\mathrm{Y})$$(g\in G(E_{6}))$

.

249

Hence

$\rho^{\vee}(g1,g_{2})=\rho(\iota(g_{1}), t_{g_{2}}-1)$$\rho^{\vee}(G)=\rho(G)$have

for $(g_{1},g_{2})\in G=G(E_{6})\cross GL_{n}$ . In particular,. A highest weight vector of$V_{\min}^{\vee}(\mathrm{Z})=V_{\min}(\mathrm{Z})$. A -basis of$\mathrm{Z}$$E_{ii}^{(2)}\otimes u_{k}$$( \frac{1}{2}f_{j})_{i}\otimes u_{k}$$V^{}$

is given by$V_{\min}(\mathrm{Z})$is given by$v_{0}^{\vee}=E_{33}^{(2)}\otimes u_{n}$. Then we$(1 \leq i\leq 3,1\leq k\leq n)$$(1 \leq i\leq 3,1\leq j\leq 8,1\leq k\leq n)$.

Its dual basis is given by$(E_{i}^{(3)}i\otimes u_{k})^{}=E_{ii}^{(3)}\otimes u_{k}$,$(( \frac{1}{2}f_{j})_{i}\otimes u_{k})^{}=-(\frac{1}{2}f_{\sigma(}j))_{i}\otimes u_{k}$

where

Hence

$\sigma=$$V_{\max}(\mathrm{Z})=V_{\min}(\mathrm{Z})$.

. Hence there is exactly one split Z-form.

2.14. Type (29).

In this paragraph, we use the notations of (1.8). The representation spacecan be identified with $X$ . We may assume that $G=G(E_{7})\cross GL_{1}$ , where $G(E_{7})$ is the

subgroup of $GL(X)$ which corresponds to the Lie subalgebra C7 of$\mathfrak{g}1(x)$weight vector is given be $v_{0}=(0, E_{18}-E_{81})$ . Here and below, we choose
as a basis of$R$$\{\alpha_{2}, \ldots\alpha_{8}\}$. We have$V_{\min}( \mathrm{Z})=\sum_{\leq 1i<j\leq 8}(\mathrm{Z}\cdot(Eij-E_{j}i, \mathrm{o})+\mathrm{z}\cdot(0, E_{ij}-E_{ji}))$

We identify. A highest$V^{}$$\langle(x_{1}, x_{2}),$

with

$V$

.

by the symmetric bilinear form defined by$(y_{1},y_{2}))=- \frac{1}{2}(\mathrm{t}\mathrm{r}(x_{1y_{1}})+\mathrm{t}\mathrm{r}(x_{2}y2))$,$(x_{1}, x_{2}),$$(y_{1}, y_{2})\in X$

.

250

Since $G(E_{7})(\supset\{\pm 1\})$ is simply connected, the involutory automorphism$\iota$definedin (1.8) induces an involutory automorphism of $G(E_{7})$ , which we shall denote by thesame letter . Then the action$\iota$$(g_{1},g_{2})\in G=G(E_{7})\cross GL_{1}$of$V^{}$

is given by$p^{}$

of

$G$

on

. In particular,$v_{0}^{\vee}=(E_{18}-E_{81}, \mathrm{o})$is given by$V^{}$$\rho^{\vee}(g1,g_{2})=p(\iota(g_{1}),g_{2}^{-1})$$\rho^{\vee}(G)=\rho(G)$. We havefor. A highest weight vector$V_{\min}^{\vee}(\mathrm{Z})=V_{\min}(\mathrm{Z})$. A -basis of$\mathrm{Z}$is given by$V_{\min}(\mathrm{Z})$$(E_{ij}-E_{j}i,0),$ $(0, E_{ij}-Eji)$$(1 \leq i<j\leq 8)$

and its dual basis is given byHence$(E_{ij}-E_{j}i, 0)^{\mathrm{v}}=(E_{ij}-E_{j}i, 0)$, and$(\mathrm{o}, Eij-Eji)^{\vee}=(0, E_{ij}-Eji)$.

$V_{\max}(\mathrm{Z})=V_{\min}(\mathrm{Z})$. Hence there is exactly one split Z-form.

Non-regular prehomogeneous vector space with a relative

2.15.

invariant.
There is a unique non-regular irreducible reduced prehomogeneous vector

space which has a non-trivial relative invariant, which we refer to as the type($(\mathrm{N}\mathrm{R})$-regular) provisionally in this paper. The representation space can be identified$=\mathrm{n}\mathrm{o}\mathrm{n}$with$V=\mathrm{C}^{2n}\cross S^{2}(\mathrm{C}^{2})$$Sp_{2n}(\mathrm{C})$$\mathrm{C}^{2n}$(resp.

$\{f_{1}, f_{2}\})$

byis realized as in (1.3). (Note thatacts on$\mathrm{C}^{\cross}$. We may assume that$V$

as scalar multiplications,$S^{2}(\mathrm{C}^{2})$$v_{0}=e_{1}\otimes f_{1}^{2}$. A -basis of$\mathrm{Z}$$e_{i}\otimes f_{1}^{2}$,$\mathrm{C}^{2n}$$V_{\min}(\mathrm{Z})$$e_{i}\otimes 2f_{1}f_{2}$$SL_{2}(\mathrm{c})/\{\pm\}=SO_{3}(\mathrm{C}).$(resp.

$Sp_{2n}(\mathrm{C})$), and hence we get abe the standard basis of$G=\mathrm{C}^{\cross}\cross Sp_{2n}(\mathrm{C})\cross SL_{2}(\mathrm{c})$$G$(resp.

-action$\mathrm{C}^{2}$$\rho$

on$SL_{2}(\mathrm{C})$$V$) The first factor) acts naturally on. Let$\{e_{i}\}_{1\leq i\leq}2n$(resp.

). A highest weight vector is given

is given by,, where$e_{i}\otimes f_{2}^{2}$,$(1 \leq i\leq 2n)$

.

251

We identify$V^{}$

withby the skew-symmetric bilinear form on$V$$(e_{i}\otimes f_{p}f_{q},$ $e_{j}\otimes frfS\rangle=\langle e_{i}, e_{j}\rangle(f_{p}f_{q},$ $frf_{S}\rangle$$\langle e_{i}, e_{n+j}\rangle=-\langle e, e_{i}n+j\rangle=\delta ij$$\langle e_{i}, e_{j}\rangle=(e_{n+i},$,,$(f_{1}f_{2},$$f_{1}f2 \rangle=\frac{1}{2}$for the other cases,$\langle f_{p}f_{q}, f_{r}f_{S}\rangle=0$

bydefined by$e_{n+j}\rangle=0$$\langle f_{1}^{2}, f12\rangle=\langle f_{2}^{2}, f_{2}^{2}\rangle=1,$

for $1\leq i,j\leq 2n$ and $1\leq p,$$V$$q,$ $r,$ $s\leq

2$. Then the action$p^{}$

of$G$$\rho^{\vee}(g1,g2,g3)=\rho(g_{1}^{-1t},g_{2},g_{3}^{-1})\in G=\mathrm{C}^{\cross}\cross Sp_{2n}(\mathrm{C})\cross SL_{2}(\mathrm{c})$$\rho^{\vee}(G)=\rho(G)$have. A highest weight vector of$V_{\min}^{\vee}(\mathrm{Z})=V_{\min}(\mathrm{Z})$. A -basis of$\mathrm{Z}$$e_{i}\otimes f_{1}^{2}$

We can show that,

$V_{\max}(\mathrm{Z})$$e_{i}\otimes f_{1}f_{2}$$V_{\max}(\mathrm{Z})/V_{\min}(\mathrm{Z})$$V^{}$,

is given byon$V^{}$

is given. In particular$v_{0}^{\vee}=e_{i}\otimes f_{2}^{2}$. Then we

is given by$e_{i}\otimes f_{2}^{2}$,$(1 \leq i\leq 2n)$is a simple graded$\mathcal{U}_{\mathrm{Z}}$.

-module. Hence there areexactly two split Z-forms.
2.16. Let $(G_{i}, \rho_{i}, V_{i})(i=1,2)$ be two irreducible representations andtheir duals. We assume that a Borel subgroup of eachbe highest root vectors ofbilinear form
$V_{i}$$\langle, \rangle$$V_{8}$

andis given for each

$V_{i}^{}$$V_{i}$$G_{i}$

is given. Let$v_{i}$$V^{\mathrm{v}_{)}}$$(G_{i,\rho_{ii}^{\vee}},$and$v_{i}^{}$, respectively. Assume that a non-degenerate

and that$\rho_{i}(G_{i})=\rho_{i()}^{\vee}G_{i}$, if we identify$V_{i}^{}$

withvia this bilinear form.
Let us consider the irreducible representation $(G, \rho, V)=(G_{1}\cross G_{2,\rho_{1}}\otimes$$\rho_{2},$$V_{1}\otimes V_{2})$vectors ofand its dual

$V_{1}\otimes V_{2}$and$(G, \rho^{\vee}, V^{\vee})=(G_{1}\cross G_{2}, \rho_{1}^{\vee}\otimes\rho_{2’ 1}^{\vee}V\vee\otimes V_{2}^{})$$V_{1}^{\vee}\otimes V_{2}^{\mathrm{v}_{\mathrm{a}}}\mathrm{r}\mathrm{e}$given by$v_{1}\otimes v_{2}$and$V_{\min}( \mathrm{Z})=V1,\min(\mathrm{z})\otimes V_{2},\min(\mathrm{Z})$$v_{1}^{}\otimes v_{2}^{\vee}$,$V_{\max}( \mathrm{Z})=V1,\max(\mathrm{z})\otimes V2,\max(\mathrm{Z})$.

. Highest weight. Then we have252

A non-degenerate bilinear form on$V$is given by$\langle v_{1}’\otimes v_{2}’, v_{1}//\otimes v_{2}’’\rangle=\langle^{;/l}v_{1}, v_{1}\rangle\langle v_{2}’, v_{2}’’\rangle$for$v_{i}’,$$v_{i^{l}}’\in V_{i}(i=1,2)$. Then$V^{}$

can be identified with$V$

and $\rho^{\vee}(G)=\rho(G)$ .

Combining this fact with the calculations in $(2.1)-(2.15)$ , we have the following theorem.

2.17. Theorem. Let $(G, \rho, V)$ be an irre ducibl prehomogeneo us vector$e$spa ce. Then there are at most two split$V_{\max}(Z)$. The$ex\mathrm{a}ct$$\mathrm{Z}$-forms which are given by $V_{\max}(Z)$ an$d$number of split -forms of each $(G, \rho, V)$ is given in the following$Z$ta $ble$ . (The first row indica tes the type of $(G, \rho, V)$ and th second row indica tes the
$e$

$n$

(

umber of split Z-forms.)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

1

2

1

2

1

1

1

2

1

1

1

1

1

$(15B)$$(15D)$

(20)

(21)

(23)

(24)

(27)

(28)

(29)

$(NR)$

2

1

1

1

1

1

1

1

1

2

$14\rangle$$2$

References[F] H.Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometries, Mimeographed

Note (1951).
[G] A.Gyoja,$\mathrm{Z}$-forms of representations of$reduCt_{\dot{i}}ve$groups and prehomogeneous

vector spaces, in the same volume.
[H] Stephen J.Harris, Some irreducible representations of exceptional algebraic groups,

Amer. J. Math. 93 (1991), 75-106.

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

and their relative invariants, Nagoya Math. J. 65 (1977), 1-155.
[VE] E.B.Vinberg, A.S.Elashvili, Classification of trivectors of 9-dimensional space (inRussian), Trudy Sem. Vector. Tenzor. Anal. 18 (1978).