Multiplicative Jordan Decompositionfe.math.kobe-u.ac.jp/FE/FE_pdf_with_bookmark/FE31-34-en...Multiplicative Jordan Decomposition $¥mathrm{a}¥mathrm{J}¥mathrm{d}:=$ additive Jordan

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Funkcialaj Ekvacioj, 31 (1988) 1-23

Multiplicative Jordan Decomposition in $¥mathrm{A}¥mathrm{u}¥mathrm{t}_{c}C[[x_{1},¥ldots, ¥chi_{n}]]$

and the Exponential Map

By

Kazuo UENO(Saga University, Japan)

Introduction

Let $M$ be the maximal ideal of $R:=C[[x_{1},¥ldots, x_{n}]]$ ; we consider $M$ as asub-C-algebra of $R$ . We have Aut $C$

$ C[[X_{1},¥ldots, X_{n}]]¥cong$ Aut $c(M):¥alpha¥leftrightarrow a|_{M}$ , and

Aut $(M)=$ $¥{¥alpha_{¥varphi} : ¥mathrm{x}_{i}¥mapsto¥varphi_{i}|¥varphi_{i}¥in M, ¥det (¥partial¥varphi_{i}/¥partial x_{i})¥in R¥backslash M(i=1,¥ldots, n)¥}$.

In this article we will study the structure of $AUT:=$ Aut$c(M)$ by means of theJordan decompositions and the exponential map.

Additive Jordan decomposition in $DER:=$ Der $c(M)=¥sum_{i=1}^{n}M¥cdot¥partial/¥partial x_{i}$ isknown ([6], [2]); we will study the multiplicative version in the case of $AUT$

(§1). The relationship between these Jordan decompositions and the exponentialmap from DER into $AUT$ is studied in §2?§3. A characterization of the image$¥exp$ DER in AUTis investigated in§4 by the use of the theory developed in §1?§3.

As shown in [8] and [7], $¥exp$ : $DER¥rightarrow AUT$ is not surjective (Theorem 3.3, (ii));therefore we construct the multiplicative Jordan decomposition in $AUT$ inde-pendently of the additive version in DER.

In§1 the base field $K$ is assumed to be algebraically closed. After preliminaryconsiderations (Notation 1.1-Definition 1.5), we recall the method of additiveJordan decomposition in DER given by [2] and then establish the multiplicativeJordan decomposition in $AUT$ (Proposition 1.6-Theorem 1.20). Lemma 1.9,(ii) plays an important role in characterization of unipotency in $AUT$ (Theorem1.13, $(¥mathrm{i}¥mathrm{i}))$ . Formal analytic diagonalization in DER and $AUT$ is considered(Corollary 1.21-Remark 1.24) in the light of Theorem 1.20. In the proofsof some of the multiplicative statements of §1, only “Multiplicative proof”is written when the proof can be performed just as that of the preceding additivestatement with obvious alteration to multiplicative version.

In §2 we assume merely $K$ to be of characteristic zero. We showthe exponential bijection from nilpotent derivations into unipotent automorphisms(Theorem 2.5) which is a generalization of the bijection from nilpotent matricesinto unipotent matrices: see Theorem 2.7. A logarithm from unipotent auto-morphisms into nilpotent derivations is defined by the use of Lemma 2.4, (ii)

Kazuo UENO

proved in [7; pp. 322-323]; we give another proof of this. For the exponentialbijection, see also [11; Part $¥mathrm{I}$ , §6].

§3 concerns the case $K=C$, where $gl^{¥infty}$ defined in §1 is provided with theentry-wise convergence topology. We study the relations between the exponentialmap from $g¥ell^{1}$ (resp. DER, $gl^{¥infty}$ ) into $GL^{1}$ (resp. $AUT$, $GL^{¥infty}$ ) and the correspond-ing Jordan decompositions (Theore $¥mathrm{m}$ $3.4$ -Corollary 3.7).

In §4 we consider characterization of $¥exp$ DER in $AUT$ for $K=C$. It is aninteresting subject and investigated from several points of view (Theorem 4.1-Corollary 4.11). In the one variable case, we obtain Theorem 4.10 which is aclear characterization of $¥exp$ $DER_{l}$ . Examples 4. 12, (iv) and 4. 13 show that $¥exp$ :$DER¥rightarrow AUT$ is not surjective (Theorem 3.3, $(¥mathrm{i}¥mathrm{i})$ ). For tlie relationship betweenthe map $¥exp$ : $DER¥rightarrow AUT$ and the theory of differential equations, see [1;Zweites Kapitel].

Part of the results in the present article has been announced in [10].

§1. Additive and Multiplicative Jordan decompositions over the maximal ideal

Notation 1.1. (i) Let $K$ be an algebraicaly closed field (of characteristicarbitrary), $R:=K[[x_{1},¥ldots, x,,]]$ , $M$ the $¥max$ imal ideal of $R$ , which we consideras a sub-algebra of $R$ , and $FM:=$ the $¥mathrm{M}$-adic filtration in $M$ :

$FM=¥{M^{j}|j¥in N^{¥times}¥}$ .

( $N^{¥times}:$ $=$ the set of all positive integers).For $i¥in N^{¥mathrm{x}}$ we set

$FM/M^{i+1}:=¥{M^{j}/M^{i+1}|j¥in N^{¥times}¥}$

(induced filtration in $M/M^{i+1}$ ),

$gl^{i}:=$ { $¥lambda:M/M^{i+l_{}}M/M^{i+l}|$ $K$-linear, $¥lambda(M^{j}/M^{i+1})¥subset M^{j}/M^{i+1}(j¥in N^{¥mathrm{x}})$}

and

$GL^{i}:=$ { $¥lambda¥in gl^{i}|$ invertible}.

(ii) { $gl^{j}¥rightarrow gl^{k}|$ natural projection for $j¥geqq k$ } is an inverse system. Takingthe inverse limit, we define

$¥mathrm{g}1^{¥infty}:=¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim(gl^{i})$ and $¥mathrm{G}¥mathrm{L}^{¥infty}:=¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim(GL^{i})$ .

Note that

$ gl^{¥infty}¥cong$ { $¥lambda:MM|K$-linear, $¥lambda(M^{j})¥subset M^{j}(j¥in N^{¥chi})$ },

$ GL^{¥infty}¥cong$ { $¥lambda¥in gl^{¥infty}|$ invertible}.

Multiplicative Jordan Decomposition

We denote by $P_{i}$ the natural projection: $gl^{¥infty}¥rightarrow gl^{i}$ (or $GL^{¥infty}GL^{i}$ ). $P_{i}$ arealgebra maps.

(iii) For $i=1$ , we have $M/M^{2}¥cong K¥overline{x}_{1}¥oplus¥cdots¥oplus K¥overline{x}_{¥mathrm{n}}¥cong K^{n}$ and $FM/M^{2}=¥{M/M^{2}$ ,0} reduces to a trivial filtration. We have the anti-isomorphism:

Mat: $g¥ell^{1_{¥rightarrow¥sim}}gl_{n}(K)$

$¥lambda-(¥lambda_{i,j})$ ,

where $¥lambda:¥overline{x}_{i}¥mapsto¥sum_{i^{=1}}^{n}¥lambda_{i,j}¥overline{x}_{j}$ . We have Mat(λμ)=Mat(μ)Mat(λ) $(¥lambda, ¥mu¥in gl^{1})$ .

Restricting $Maf$ to $GL^{1}$ , we have the anti-isomorphism:

Mat: $GL^{1_{¥rightarrow¥sim}}GL_{n}(K)$ .

We denote by $Tam$ the inverse map of Mat and by $Tdg(a)eg1$ the map $¥overline{x}_{i}¥mapsto a_{i}¥overline{x}_{i}$

$(i=1,¥ldots, n)$ , where $a:=(a_{1},¥ldots, a_{n})$.

(iv) $g¥ell^{1}$ , $g¥ell^{i}$ and $gl^{¥infty}$ have the $¥mathrm{K}$ -Lie algebra structures with Lie bracket$[¥lambda, ¥mu]:=¥lambda¥mu-¥mu¥lambda$, and $GL^{1}$ , $GL^{i}$ and $GL^{¥infty}$ have the natural group structures.

(v) We denote by DER the $¥mathrm{K}$ -Lie algebra of all $¥mathrm{K}$ -derivations of $M$ andby $AUT$ the group of all $¥mathrm{X}$-algebra automorphisms of $M$ . We have:

$DER=$ $¥{¥delta_{¥varphi} : =¥sum_{i=1}^{n}¥varphi_{i}¥partial_{i}|¥varphi_{i}¥in M¥}=¥sum_{i=1}^{n}M¥cdot¥partial_{i}$

$AUT$ $=¥{a_{¥varphi} : x_{i}¥mapsto¥varphi_{i}(i=1,¥ldots, n)|¥varphi_{i}¥in M, ¥det (¥partial¥varphi)¥in R^{¥times}¥}$ ,

where $¥partial_{i}:=¥partial/¥partial¥chi_{i}$, $¥partial¥varphi:=(¥partial_{j}¥varphi_{i})_{i,j}$ and $R^{¥times}=R¥backslash M$ . Finally we have:

$Aut_{K}(R)¥cong AUT$, $¥alpha-’¥alpha|_{M}$ .

Proposition 1.2. (i) We have the $K$-Lie algebra inclusions

$gl^{1}¥subset lD$ $DER¥subset g¥ell^{¥infty}$ ,

where $c_{D}(Tam(¥lambda_{i,j})):=¥sum_{i=1}^{n}$ $(¥sum_{j=1}^{n}¥lambda_{i,j}x_{j})¥partial_{i}$ ; we denote by $¥delta_{¥lambda}$ the right-handside of this.

(ii) We have the group inclusions:

$GL^{1}¥subset¥iota AAUT¥subset GL^{¥infty}$ ,

where $‘ A(Tam(¥gamma_{i,j}))(x_{¥mathrm{i}}):=¥sum_{j=1}^{n}¥gamma_{i,j}¥mathrm{x}_{j}(i=1,¥ldots, n)$ . The automorphism definedby the last formula is denoted by $¥alpha_{¥gamma}$ .

(iii) Let $¥gamma¥in GL^{l}(resp. ¥alpha¥in AUT)$. We set

$gl^{1}¥rightarrow gl^{1}$

$¥gamma_{*}:$ $¥lambda|¥rightarrow¥gamma^{-1}¥lambda¥gamma$

4 Kazuo UENO

(resp. $DER¥rightarrow DER$

$¥alpha_{*}:$ $¥delta|¥rightarrow¥alpha^{-1}¥delta¥alpha)$ .

Then we have the commutative diagram:

$gl^{1}¥subset lD$ DER$¥gamma_{*1}$ $¥downarrow lA(¥gamma)_{*}$

$gl^{1},¥subset D$ DER.

(iv) Under the notation of (iii), $¥gamma_{*}$ (resp. $¥alpha_{*}$ )defines also an automorphismof $GL^{1}$ (resp. $AUT$). The statement similar to (iii) holds for $c_{A}$ : $GL^{l}¥subset AUT$

and $P_{i}$ : $gl^{¥infty}¥rightarrow gl^{i}$ (or $GL^{¥infty}¥rightarrow GL^{i}$).(v) $P_{1}c_{D}$ : $gl^{l}DER$ $¥subset gl^{¥infty}->gl^{1}$ equals the identity map.(vi) $P_{1}c_{A}$ : $GL^{l}AUT¥subset GL^{¥infty}GL^{1}$ equals the identity map.

De inition 1.3. (i) Let $l$ be a $¥mathrm{K}$ -Lie algebra (not necessarily finitedimensional). We say that additive Jordan decomposition holds in $l$ if andonly if $l$ has the following properties:

(a) Semisimplicity and nilpotency are defined in $l$ .

(b) Denoting by $sl$ (resp. $lll$ ) the set of all semisimple (resp. nilpotent)elements of $l$ , we have $sl¥cap n¥ell=¥{0¥}$ .

(c) Each $¥lambda¥in l$ has a unique additive decomposition

(1) $¥lambda=^{s}¥lambda+^{n}¥lambda$

such that $s¥lambda¥in^{s}t^{n},¥lambda¥in^{n}l$ and $[^{S}¥lambda^{n},¥lambda]=0$ .

We call (1) the additive Jordan decomposition of $¥lambda$ , $ s¥lambda$ the semisimple part of$¥lambda$ and $ n¥lambda$ the nilpotent part of $¥lambda$ .

(ii) Let $G$ be a group (not necessarily algebraic). We say that multi-plicative Jordan decomposition holds in $G$ if and only if $G$ has the followingproperties:

(a) Semisimplicity and unipotency are defined in $G$ .

(b) Denoting by $sG$ (resp. $uG$) the set of all semisimple (resp. unipotent)elements of $G$ , we have $sG¥cap uG=¥{1¥}$ .

(c) Each $¥gamma¥in G$ has a unique multiplicative decomposition

(2) $¥gamma=^{s}¥gamma^{u}.¥gamma$

such that $s¥gamma¥in^{s}G$ , $u¥gamma¥in^{u}G$ and $ s¥gamma$ $¥cdot u¥gamma=¥gamma¥gamma u.s$ .

We call (2) the}nultiplicative Jordan decomposition of $¥gamma$ , $ s¥gamma$ the semisimplepart of $¥gamma$ and $ u¥gamma$ the unipotent part of $¥gamma$ .

Abbreviations 1.4. We use the following short for $¥mathrm{m}¥mathrm{s}$


$¥mathrm{a}¥mathrm{J}¥mathrm{d}:=$ additive Jordan decomposition.$¥mathrm{m}¥mathrm{J}¥mathrm{d}:=$ multiplicative Jordan decomposition.Jds: $=$ additive and multiplicative Jordan decompositions.$¥mathrm{s}¥mathrm{s}¥mathrm{p}:=$ semisimple.$¥mathrm{u}¥mathrm{p}¥mathrm{t}:=$ nilpotent.$¥mathrm{u}¥mathrm{p}¥mathrm{t}:=$ unipotent.to $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ $¥lambda$ in $l:=$ to decompose additively $¥lambda$ into $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ part and $¥mathrm{n}¥mathrm{p}¥mathrm{t}$ part asan element of $¥ell$ .

to $¥mathrm{m}¥mathrm{J}¥mathrm{d}$

$¥gamma$ in $G:=$ to decompose multiplicatively $¥gamma$ into $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ part and $¥mathrm{u}¥mathrm{p}¥mathrm{t}$ partas an element of $G$ .

Definition 1.5. Since $M/M^{i+1}$ is a finite dimensional $¥mathrm{K}$ -linear space, semi-simplicity and nilpotency (resp. unipotency) in $g¥ell^{i}$ (resp. $GL^{i}$ ) are defined inthe usual manner $(i ¥in N^{¥mathrm{x}})$ .

(i) $¥lambda¥in gl^{¥infty}$ is called $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ (resp. $¥mathrm{n}¥mathrm{p}¥mathrm{t}$) if and only if $P_{i}(¥lambda)¥in gl^{i}$ is $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ (resp.$¥mathrm{n}¥mathrm{p}¥mathrm{t})$ for all $i¥in N^{¥mathrm{x}}$ .

(ii) $¥gamma¥in GL^{¥infty}$ is called $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ (resp. $¥mathrm{u}¥mathrm{p}¥mathrm{t}$) if and only if $ P_{l}¥langle¥lambda$) $¥in GL^{i}$ is $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ (resp.$¥mathrm{u}¥mathrm{p}¥mathrm{t})$ for all $i¥in N^{¥times}$ .

Since $M/M^{i+1}$ are finite dimensional, nilpotency defined above is equivalent tothat of [2; Definition 1.4.].

Proposition 1.6. (i) $aJd$ holds in $gl^{¥infty}$ .

(ii) $mJd$ holds in $GL^{¥infty}$ .

Proof. It follows immediately from Definition 1.5 that $sgl^{¥infty}¥cap i1gl^{¥infty}=¥{0¥}$

and $sGL^{¥infty}¥cap uGL^{¥infty}=¥{1¥}$ .

(i) See [2; pp. 152-153]. Note that, for $¥lambda¥in gl^{¥infty_{S}},¥lambda$ (resp. ” $¥lambda$) is given by$¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim$ $(^{s}P_{i}(¥lambda))$ (resp. $¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim$ $(^{n}P_{i}(¥lambda))$ .

(ii) Let $¥gamma¥in GL^{¥infty}$ . Since $GL^{¥infty}¥subset gl^{¥infty}$ , we can $¥mathrm{a}¥mathrm{J}¥mathrm{d}$

$¥gamma$ in $gl^{¥infty}$ as $¥gamma=^{s}¥gamma+^{n}¥gamma$ .We have $s¥gamma¥in GL^{¥infty}$ . Since $[^{s}¥gamma^{ n},¥gamma]=0$ , $(^{s}¥gamma)^{-1}¥cdot{}^{¥prime ¥mathrm{t}}¥gamma$ is $¥mathrm{n}¥mathrm{p}¥mathrm{t}$ . Putting $u¥gamma=1+(^{S}¥gamma)^{-1}$ .$ n¥gamma$ , we have the $¥mathrm{m}¥mathrm{J}¥mathrm{d}$ of $¥gamma$ in $GL^{¥infty}$ : $¥gamma=^{s}¥gamma^{u}.¥gamma=^{u}¥gamma^{s}.¥gamma$ .

Proposition 1.7. (i) Let $¥lambda¥in g¥ell^{¥infty}$ , $f¥in M$ and $a¥in K$ . Then:

$s¥lambda(f)=af$ if and only $if¥lim_{i¥rightarrow¥infty}$$(¥lambda-a)^{i}(f)=0$ ,

where the limit is taken in the $M$-adic topology. Further:

$¥lambda(f)=af$ if and only if $s¥lambda(f)=af$ and $n¥lambda(f)=0$ .

In particular, if $¥lambda¥in^{n}gl^{¥infty}$ , then $¥lambda(f)=a.f$ implies $a=0$ or $f=0$.

(ii) Let $¥lambda¥in^{s}¥backslash ql^{¥infty}$ and $a¥in K$ . Put $M(¥lambda, a):=¥{f¥in M|¥lambda(f)=af¥}$ . Then:

$M=M-$adic $clo$ are of $¥bigoplus_{a¥in K}M(¥lambda, a)$ .

Kazuo UENO

For the proof, see [2; p. 154].

Definition 1.8. (i) Let $¥delta¥in DER$ . Since $DER¥subset gl^{¥infty}$ , we can $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ $¥delta$ in $gl^{¥infty}$ as$¥delta=^{s}¥delta+^{n}¥delta$ . We say that $¥delta$ is $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ (resp. $¥mathrm{n}¥mathrm{p}¥mathrm{t}$) in DER if and only if $¥Pi¥delta=0$ (resp.$s¥delta=0)$ . We have $sDER=DER$ $¥cap Sgl^{¥infty}$ , $nDER=DER$ $¥cap ngl^{¥infty}$ and $sDER¥cap n$DER$=¥{0¥}$ .

(ii) Let $a$ $eAUT$ Since $AUT¥subset GL^{¥infty}$ , we can $¥mathrm{m}¥mathrm{J}¥mathrm{d}$$¥alpha$ in $GL^{¥infty}$ as $¥alpha=^{s}¥alpha^{u}.¥alpha$ .

We say that $¥alpha$ is $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ (resp. $¥mathrm{u}¥mathrm{p}¥mathrm{t}$) in $AUT$ if and only if $u¥alpha=1$ (resp. $s¥alpha=1$ ). Wehave $sAUT=AUT¥cap SGL^{¥infty}$ , $uAUT=AUT¥cap uGL^{¥infty}$ and $sAUT¥cap uAUT=¥{l¥}$ .

We will show in Proposition 1.11 that in fact $ s¥delta$ and $ n¥delta$ (resp. $ s¥alpha$ and $ u¥alpha$)belong to DER (resp. $AUT$).

Lemma 1.9. (i) Let $¥delta¥in DER$ , $a$ , $b¥in K$ and $f$, $g¥in M$ . Then, for $k¥in N$ :

$(¥delta-(a+b))^{k}(fg)=¥sum_{i=0}^{k}$ $¥left(¥begin{array}{l}k¥¥i¥end{array}¥right)(¥delta-a)^{i}(f)(¥delta-b)^{k-i}(g)$ .

(ii) Let $¥alpha¥in AUT$, $a$ , $b¥in K$ and $f$, $g¥in M$ . Then, for $k¥in N$:

$(¥alpha-ab)^{k}(fg)=¥sum_{SR(k)}$$¥left(¥begin{array}{l}k¥¥i+j¥end{array}¥right)$ $(_{i}^{i+j})a^{i}(¥alpha-a)^{k-i}(f)b^{i}(¥alpha-b)^{k-j}(g)$ ,

where the summation ranges over $SR(k):=¥{(i, j)|0¥leqq i, j, i+j¥leqq k¥}$ .

Proof. Note that, since the binomial coefficients are rational integers, thelemma is characteristic-free.

(i) By induction on $k$ .

(ii) For $k=0$ , both sides are equal to $fg$ . For $k=1$ , since $¥alpha$ preservesmultiplication, we have:

(1) $(¥alpha-ab)(fg)=(¥alpha-a)(f)(¥alpha-b)(g)+(¥alpha-a)(f)$ . $bg+af¥cdot(¥alpha-b)(g)$ .

Suppose the formula for $k$ $(¥geqq 1)$ . Using (1), we compute:

$(¥alpha-ab)^{k+1}(fg)=(¥alpha-ab)^{k}(¥alpha-ab)(fg)$

$=(¥alpha-ab)^{k}((¥alpha-a)(f)(¥alpha-b)(g))+(¥alpha-ab)^{k}((¥alpha-a)(.f)¥cdot bg)$

$+(¥alpha-ab)^{k}(af¥cdot(¥alpha-b)(g))$ .

Applying the formula for $k$ to the last three terms, we obtain the formula for$k+1$ by manipulation of binomial coefficients.

Remark 1.10. Lemma 1.9, (i) is a generalized Leibniz formula for derivation,while Lemma 1.9, (ii) is an automorphism version for Lemma 1.9, (i).

Proposition 1.11. (i) Let $¥delta¥in DER$ . Put the $aJd$ of $¥delta$ in $gl^{¥infty}$ as $¥delta=^{s}¥delta+^{n}¥delta$ .Then $ s¥delta$ and $ n¥delta$ belong to DER.

Put the $aJd$ of $¥delta$ in $gl^{¥infty}$ as $¥delta=^{s}¥delta+^{n}¥delta$ .


(ii) Let $¥alpha¥in AUT$. Put the $mJd$ of $¥alpha$ in $GL^{¥infty}$ as $¥alpha=^{s}¥alpha^{u}.¥alpha$ . Then $ s¥alpha$ and$ u¥alpha$ belong to $AUT$

Proof, (i) See [2; p. 156].(ii) Let $¥alpha¥in AUT$ and its $¥mathrm{m}¥mathrm{J}¥mathrm{d}$ in $GL^{¥infty}$ be $¥alpha=^{s}¥alpha^{u}.¥alpha$ . It is sufficient to show

$s¥alpha¥in AUT$, which implies also $u¥alpha=(^{S}a)^{-1}¥cdot¥alpha¥in AUT$

Assume first $f¥in M(^{s}a, a)$ and $g¥in M(^{s}¥alpha, b)$ for some $a$ , $b¥in K$ (Proposition

1.7, $(¥mathrm{i}¥mathrm{i}))$ . By Lemma 1.9, (ii), we have

(1) $(¥alpha-ab)^{k}(fg)$

$=¥sum_{SR(k)}$$¥left(¥begin{array}{l}k¥¥i+j¥end{array}¥right)$ $(_{i}^{i+j})a^{i}(¥alpha-a)^{k-i}(f)b^{j}(¥alpha-b)^{k-j}(g)$ $(k ¥in N)$ .

Fix a large $p¥in N$. By Proposition 1.7, (i) there exists $q¥in N$ such that $(¥alpha-a)^{r}(f)$

and $(¥alpha-b)^{r}(g)¥in M^{p+1}(r¥geqq q)$ . Hence, by (1), $(¥alpha-ab)^{2q}(fg)¥in M^{p+1}$ . Since$¥alpha$ -ab respects $FM$, we have $(¥alpha-ab)^{r}(fg)¥in M^{p+1}(r¥geqq 2q)$ . This implies $¥lim_{i¥rightarrow¥infty}(¥alpha$

$-ab)^{i}(fg)=0$ , which is equivalent to $s¥alpha(fg)=ab¥cdot fg$ by Proposition 1.7, (i).Thus we see that $s¥alpha(f)=af$ and $s¥alpha(g)=bg$ imply $S¥alpha(fg)=ab$ $¥cdot fg=^{¥mathrm{s}}¥alpha(f)^{s}.¥alpha(g)$ .

Now we assume $f¥in M$ and $g¥in M(^{s}¥alpha, b)$ . By Proposition 1.7, (ii) we can$¥mathrm{w}¥mathrm{r}¥mathrm{t}¥mathrm{n}$ $f=¥sum_{a¥in K}f_{a}$ with $f_{a}¥in M(^{s}¥alpha, a)$ (summation in the $¥mathrm{M}$-adic topology). Since$ s¥alpha$ respects $FM$ , the above argument shows:

(2) $s¥alpha(fg)=s¥alpha(¥sum_{a¥in K}f_{a}¥cdot g)=¥sum_{a¥in K}s¥alpha(f_{a}¥cdot g)$

$=¥sum_{a¥in K}s¥alpha(f_{a})^{S}.¥alpha(g)=S¥alpha(f)^{s}.¥alpha(g)$ .

Finally assume $f$, $g¥in M$ . Writing $g=¥sum_{b¥in K}g_{b}$ with $g_{b}¥in M(^{s}¥alpha, b)$ , we have bya computation similar to (2):

$s¥alpha(fg)=s¥alpha(f)^{S}.¥alpha(g)$ .

Theorem 1.12. (i) $aJd$ holds in DER.(ii) $mJd$ holds in $AUT$

Proof. (i) Let $¥delta¥in DER$ and $¥delta=^{s}¥delta+^{n}¥delta$ the $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ of $¥delta$ in $g¥ell^{¥infty}$ . By Propo-sition 1.11 and Definition 1.8, we have $s¥delta¥in^{s}$DER and $ n¥delta$ c-$ ^{n}$ DER. Conversely,assume $¥delta=¥delta_{1}+¥delta_{2}$ where $¥delta_{1}¥in^{s}$DER and $¥delta_{2}¥in^{n}$DER. Since $¥delta_{1}¥in^{s}g¥ell^{¥infty}$ and $¥delta_{2}¥in$

$ngl^{¥infty}$ , the uniqueness of the $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ of $¥delta$ in $gl^{¥infty}$ gives $¥delta_{1}=^{s}¥delta$ and $¥delta_{2}=^{n}¥delta$ .

(ii) Multiplicative proof (see Introduction).

Theorem 1.13. (i) Let $¥delta¥in DER$ . Then: $¥delta¥in^{n}$DER if and only if $ P_{1}(¥delta)¥in$

$ng¥ell^{1}$ (cf. [6; p. 131, p. 135]).(ii) Let $¥alpha¥in AUT$ Then: $¥alpha¥in^{u}AUT$ if and only if $P_{1}(¥alpha)¥in^{u}GL^{1}$ .

Kazuo UENO

Proof, (i) Assume first $¥delta¥in^{n}$DER. By Definition 1.5, (i) with $i=1$ , wehave $P_{1}(¥delta)¥in^{n}g¥ell^{1}$ . Conversely, assume $P_{1}(¥delta)¥in^{n}gl^{1}$ . There exists $N_{1}¥in N^{¥mathrm{x}}$

such that $¥delta^{N_{1}}(M)¥subseteqq M^{2}$ . We show $ P_{¥iota}¥langle¥delta$) $¥in^{n}g¥ell^{i}(i¥in N^{¥times})$ by induction.Suppose $P_{i}(¥delta)¥in {}^{¥prime ¥mathrm{t}}g¥ell^{i}(i¥geqq 1)$ , i.e., $¥delta^{N_{¥mathrm{i}}}(M)¥subseteqq M^{i+1}$ for some $N_{i}¥in N^{¥mathrm{x}}$ . Consider

the commutative diagram:

$P_{i+1}(¥delta):M/M^{i+2}¥rightarrow M/M^{i+2}$

$ P_{i}^{i+1}¥downarrow$ $¥downarrow P_{i}^{i+1}$

$P_{i}(¥delta):M/M^{i+1}¥rightarrow M/M^{i+1}$ ,

with $P_{i}^{i+1}$ the natural projection. Since $0=P_{i}(¥delta)^{N_{i}}¥cdot P_{i}^{i+1}=P_{i}^{i+1}¥cdot P_{i+1}(¥delta)^{N_{¥mathrm{i}}}$ ,we have

(1) $P_{i+1}(¥delta)^{N_{¥mathrm{i}}}(M/M^{i+2})¥subset ¥mathrm{K}¥mathrm{e}¥mathrm{r}P_{i}^{i+1}=M^{i+1}/M^{i+2}$ .

By Le $¥mathrm{m}¥mathrm{m}¥mathrm{a}$ $1.9$ , (i) with $a=b=0$, we have

(2) $¥delta^{k}(fg)=¥sum_{j=0}^{k}$ $¥left(¥begin{array}{l}k¥¥j¥end{array}¥right)$ $¥delta^{j}(f)¥delta^{k-j}(g)$ $(k ¥in N, f, g¥in M)$ .

Let $h=¥sum_{F}f_{I}g_{I}¥in M^{i+1}$ with $f_{J}¥in M^{i}$ and $g_{I}¥in M$ ( $¥sum_{F}$ : finite summation). By (2)with $k=N_{1}+N_{i}$, we have

$¥delta^{N_{1}+N_{i}}(h)=¥sum_{F}¥sum_{j=0}^{N_{1}+N_{¥mathrm{i}}}$ $¥left(¥begin{array}{l}k¥¥j¥end{array}¥right)$ $¥delta^{j}(f_{I})¥delta^{N_{1}+N_{¥mathrm{i}}-j}(g_{I})$ .

Since $¥delta^{N_{1}}(M)¥subset M^{2}$ , $¥delta^{N_{i}}(M)¥subset M^{i+1}$ and $¥delta$ respects $FM$, we have $¥delta^{N_{1}+N_{i}}(h)¥in M^{i+2}$ ,which implies

(3) $P_{i+1}(¥delta)^{N_{1}+N_{i}}(M^{i+1}/M^{i+2})=0$ .

It follows from (1) and (3) that

$P_{i+1}(¥delta)^{N_{1}+2N_{i}}=0$.

(ii) Assume first $¥alpha¥in^{u}AUT$ By Definition 1.5, (ii) with $i=1$ , we have$P_{1}(¥alpha)¥in^{u}GL^{1}$ . Conversely, assume $P_{1}(¥alpha)¥in^{u}GL^{1}$ , i.e., $(¥alpha-1)^{N_{1}}(M)¥subseteqq M^{2}$ forso me $N_{1}¥in N$. We show $P_{i}(¥alpha)¥in^{u}GL^{i}(i¥in N^{¥times})$ by induction.

Suppose $P_{i}(¥alpha)¥in^{u}GL^{i}(i¥geqq 1)$ , i.e., $(¥alpha-1)^{N_{¥mathrm{i}}}(M)¥subset M^{i+1}$ for some $N_{i}¥in N$.Considering the same commutative diagram as in the proof of (i) with $¥alpha$ replacing$¥delta$ , we have $0=(P_{i}(¥alpha)-1)^{N_{¥mathrm{i}}}¥cdot P_{i}^{i+1}=P_{i}^{i+1}¥cdot(P_{¥mathrm{i}+1}(¥alpha)-1)^{N_{i}}$ , hence

(4) $(P_{i+1}(¥alpha)-1)^{N_{¥mathrm{i}}}(M/M^{i+2})¥subset ¥mathrm{K}¥mathrm{e}¥mathrm{r}P_{i}^{i+1}=M^{i+1}/M^{i+2}$ .

By Lemma1.9, (ii) with $a=b=1$ , we have for $k¥in N$ and $f$, $g¥in M$ :


(5) $(¥alpha-1)^{k}(fg)=¥sum_{SR(k)}$$¥left(¥begin{array}{l}k¥¥l+m¥end{array}¥right)¥left(¥begin{array}{l}l+m¥¥l¥end{array}¥right)$ $(¥alpha-1)^{k-l}(f)(¥alpha-1)^{k-m}(g)$ .

Let $h=¥sum_{F}f,g_{I}¥in M^{¥mathrm{i}+1}$ as in the proof of (i). By (5) with $k=N_{1}+N_{i}$ , we have

$(¥alpha-1)^{N_{1}+N_{i}}(h)$

$=¥sum_{F}¥sum_{SR(k)}(_{l+m}^{N_{1}+N_{i}})$ $¥left(¥begin{array}{l}l+m¥¥l¥end{array}¥right)(¥alpha-1)^{N_{1}+N_{¥mathrm{i}}-l}(f_{I})¥cdot(¥alpha-1)^{N_{1}+N_{i}-m}(g_{I})$ .

Since $(¥alpha-1)^{N_{1}}(M)¥subset M^{2}$, $(¥alpha-1)^{N_{i}}(M)¥subset M^{i+1}$ and $¥alpha-1$ respects $FM$, we have$(¥alpha-1)^{N_{1}+N_{i}}(h)¥in M^{i+2}$ , which implies

(6) $(P_{i+1}(¥alpha)-1)^{N_{1}+N_{i}}(M^{i+1}/M^{i+2})=0$ .

It follows from (4) and (6) that

$(P_{i+1}(¥alpha)-1)^{N_{1}+2N_{i}}=0$ .

Lemma 1.14. Let $¥lambda¥in gl^{¥infty}$ . Then we have $¥lambda¥in^{s}g¥ell^{¥infty}$ if and only if $¥lambda$ hasthe following property: if $N$ is a closed sub-K-linear space of $M$ such that $¥lambda(N)¥subset$

$N$ , we have a direct sum decomposition $M=N¥oplus N^{¥prime}$ , $¥iota vhereN^{¥prime}$ satisfies the sameconditions as $N$ .

For the proof, see [2; p. 151].

Theorem 1.15. Put

$ddg:=¥{¥delta¥in DER |¥delta=¥delta_{a} : =¥sum_{i=1}^{n}a_{i}x_{i}¥partial_{i}, a_{i}¥in K(i=1,¥ldots, n)¥}$

and

$ADG:=$ $¥{¥alpha¥in AUT|¥alpha=¥alpha_{b} : ¥chi_{i}¥mapsto b_{i}¥mathrm{x}_{i}, b_{i}¥in K^{¥times}(i=1,¥ldots, n)¥}$ .

Then:(i) $sDER=AUT_{*}ddg$ . (cf. [6; p. 131])(ii) $sAUT=AUT_{*}ADG$ .

Proof. (i) See [2; p. 157].(ii) Assume first $¥alpha¥in^{s}AUT$ By definition, $P_{1}(¥alpha)¥in^{s}GL^{1}$ . Since $M^{2}$ is a

closed sub-K-linear space of $M$ and $¥alpha(M^{2})¥subset M^{2}$ , we have a direct sum decom-position $M=M^{2}¥oplus W$ as stated in Lemma 1.14. Consider the commutativediagram:

$P_{1}$ $(¥alpha):M/M^{2_{¥rightarrow¥sim}}M/M^{2}$

$/|$ $|/$ $(¥dim_{K}W=¥dim_{K}M/M^{2}=n.)$

$¥alpha|_{W}$ : $W$ $¥simeq$ $W$.

$(¥dim_{K}W=¥dim_{K}M/M^{2}=n.)$

10 Kazuo UENO

As $P_{1}(¥alpha)$ is $¥mathrm{s}¥mathrm{s}¥mathrm{p}$ , so is $¥alpha|_{W}$ . We have $W=¥oplus_{i=1}^{n}K¥varphi_{i}$ and $¥alpha(¥varphi_{i})=b_{i}¥varphi_{i}$ for some $¥varphi_{i}¥in W$

and $b_{i}¥in K^{¥times}(i=1,¥ldots, n)$. Since $M=M^{2}¥oplus W=M^{2}+¥sum_{i=1}^{n}Rc¥rho_{i}$ , we have by theNakayama Lemma: $M=¥sum_{i=1}^{n}R¥varphi_{i}$ . Hence $¥det$ $(¥partial¥varphi)¥in R^{¥mathrm{x}}$ , and by Notation 1.1,(v), $¥alpha_{¥varphi}¥in AUT$ Putting $¥beta=¥alpha_{¥varphi}$ , we have:

$¥beta_{*}(¥alpha)(x_{i})=b_{i}¥mathrm{x}_{i}$ $(i=1,¥ldots, n)$ .

Conversely, let $oceAUT$ and assume $¥beta_{*}(¥alpha)=¥alpha_{b}$ : $x_{i^{}}b_{i}x_{i}$ for some $¥beta¥in AUT$

and $b_{i}¥in K^{¥times}(i=1,¥ldots, n)$ . Since $¥alpha_{b}¥in AUT$, we have $¥alpha_{b}(x^{j})=b^{j}x^{j}$ for $ j=(j_{1},¥ldots,j_{n})¥in$

$N^{:l}¥backslash ¥{0¥}$ . By passing to the quotient $M/M^{i+1}=¥oplus_{|j|=1}^{i}K¥overline{x}^{j}$, we have $P_{i}(¥beta)_{*}P_{i}$

$(¥alpha)(¥overline{x}^{j})=b^{j}¥overline{x}^{j}(i¥in N^{¥times})$ , which implies $P_{i}(¥alpha)¥in^{s}GL^{i}$ .

Proposition 1.16. (i) $c_{D}$ : $gl^{l_{}}DER$ is compatible with $aJd$ : let $¥lambda¥in gl^{1}$

and $¥lambda=^{s}¥lambda+^{n}¥lambda$ be its $aJd$ in $gl^{1}$ . Then $c_{D}(^{S}¥lambda)=^{s}c_{D}(¥lambda)$ and $‘ D(^{i¥mathrm{t}}¥lambda)=^{n}c_{D}(¥lambda)$ .

(ii) $c_{A}$ : $GL^{l}AUT$ is compatible with $mJd$ : let $¥gamma¥in GL^{1}$ and $¥gamma=^{s}¥gamma^{¥iota¥iota}.¥gamma$

be its $mJd$ in $GL^{1}$ . Then $c_{A}(^{S}¥gamma)=^{s}c_{A}(¥gamma)$ and $c_{A}(^{¥mathcal{U}}¥gamma)=^{u}c_{A}(¥gamma)$ .

Proof, (i) Since $s¥lambda¥in^{s}gl^{1}$ , we have $¥mu_{*}(^{s}¥lambda)=Tdg(a)$ for some $¥mu¥in GL^{1}$

and $a¥in K^{1¥mathrm{i}}$ . By Proposition 1.2, (iii), we have $c_{A}(¥mu)_{*}¥ell_{D}(^{s}¥lambda)=¥delta_{a}$, hence byTheorem 1.15, (i) $‘ D(^{s}¥lambda)¥in^{s}$DER. As to the $¥mathrm{n}¥mathrm{p}¥mathrm{t}$ part, by Theorem 1.13 (i), andProposition 1.2, (v), we have $¥ell_{D}(^{n}¥lambda)¥in^{I¥mathrm{t}}$DER. Since $[c_{D}(^{s}¥lambda), c_{D}(^{n}¥lambda)]=c_{D}[^{S}¥lambda^{n},¥lambda]$

$=0$ , the uniqueness of $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ gives $c_{D}(^{S}¥lambda)=^{s}c_{D}(¥lambda)$ and $‘ D(^{n}¥lambda)=^{n}¥ell_{D}(¥lambda)$ .

(ii) Multiplicative proof.

Proposition 1.17. (i) $P_{1}$ : $DER¥rightarrow gl^{1}$ is compatible with $aJd$ .(ii) $P_{1}$ : $AUT¥rightarrow GL^{1}$ is compatible with $mJd$ .

Proof, (i) Let $¥delta¥in DER$ and $¥delta=^{s}¥delta+^{n}¥delta$ be its $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ in DER. By Theorem1.15, (i), we have $¥beta_{*}(^{s}¥delta)=¥delta_{a}$ for some $¥beta¥in AUT$ and $a¥in K^{n}$ ; Hence $P_{1}(¥beta)_{*}P_{1}(^{s}¥delta)=$

$Tdg(a)$ with $P_{1}(¥beta)¥in GL^{1}$ , which implies $P_{1}(^{s}¥delta)¥in^{s}gl^{1}$ . Since $P_{1}(^{n}¥delta)¥in^{n}gl^{1}$

(Theorem 1.13, $(¥mathrm{i})$ ) and $[P_{1}(^{s}¥delta), P_{1}(^{n}¥delta)]=P_{1}[^{s}¥delta^{ ll},¥delta]=0$ , the uniqueness of$¥mathrm{a}¥mathrm{J}¥mathrm{d}$ proves our assertion.


By the definition, the inclusion $DER¥subset gl^{¥infty}$ (resp. $AUT¥subset GL^{¥infty}$) is compatiblewith $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ (resp. $¥mathrm{m}¥mathrm{J}¥mathrm{d}$ ). Collecting and combining Theorem 1.13-Proposition 1.17,we obtain the following Theorem 1.18-Theorem 1.20.

Theorem 1.18. (i) The diagram

$gl^{1}¥leftarrow¥subset P_{1}lD$

$DER¥subset gl^{¥infty}$

is compatible with $aJd$ .

Multiplicative Jordan Decomposition 11

(ii) The diagram

$GL^{1^{l}}¥leftarrow¥subset P_{1}A$

$DER¥subset GL^{¥infty}$

is compatible with $mJd$ .

Theorem 1.19. (i) For $¥beta¥in AUT$, $¥beta_{*}:$ $DER¥rightarrow¥sim DER$ is compatible with $aJd$ .

(ii) For $¥beta¥in AUT$, $¥beta_{*}:$ $AUT¥rightarrow¥sim AUT$ is compatible with $mJd$ .

Theorem 1.20. Let $¥delta¥in DER$ (resp. $¥alpha¥in AUT$). We call the eigenvalues of$P_{1}(¥delta)$ (resp. $P_{1}(¥alpha)$) the eigenvalues of $¥delta$ (resp. of $¥alpha$).

(i) Let $¥{a_{1},¥ldots, a_{n}¥}$ be the eigenvalues of $¥delta$ . Then for some $eeAUT$ and$N¥in^{n}gt^{1}$ :

$¥epsilon_{*}(^{s}¥delta)=s(¥epsilon_{*}¥delta)=¥delta_{a}=¥sum_{i=1}^{n}a_{i}¥chi_{i}¥partial_{i}$ ,

$P_{1}(¥epsilon_{*}(^{n}¥delta))=nP_{1}(¥epsilon_{*}¥delta)=N$

and Mat(N) $¥in^{n}gl_{n}(K)$ is an upper-triangular nilpotent Jordan matrix i.e.,direct sum of matrices of the type

( $0..1.$ . .$1$ ).

Further, putting $¥epsilon_{*}(^{n}¥delta)(x_{i})=¥sum_{|j|¥geqq 1}¥varphi_{i,j}x^{j}$ , we have

$(a_{i}-¥sum_{k=1}^{n}a_{k}j_{k})¥varphi_{i,j}=0$ $(i=1,¥ldots, n, j¥in N^{n}¥backslash ¥{0¥})$ .

(cf. [6; pp. 135-138], [2; pp. 157-158])(ii) Let $¥{b_{1},¥ldots, b_{n}¥}$ be the eigenvalues of $¥alpha$ . Then, for some $¥epsilon¥in AUT$ and

$U¥in^{u}GL^{1}$ :

$¥epsilon_{*},$$(^{s}¥alpha)=^{s}(¥epsilon_{*}¥alpha)=¥alpha_{b}$ : $¥mathrm{x}_{i}|¥rightarrow b_{i}x_{i}$ $(i=1,¥ldots, n)$ ,

$P_{1}(¥epsilon_{*}(^{u}¥alpha))=uP_{1}(¥epsilon_{*}¥alpha)=U$

and Mat(U) $¥in^{n}GL_{n}(K)$ is an upper-triangular unipotent matrix expressed asdirect sum of matrices of the type

$¥left(¥begin{array}{lll}1¥ddots c & & ¥¥ & ¥ddots¥ddots & ¥¥ & & 1c¥end{array}¥right)$

with $c=b_{i}^{-1}$ for some $i$ . Further, putting $¥epsilon_{*}(^{u}¥alpha)(x_{i})=¥sum_{|j|¥geqq 1}¥psi_{i,j}x^{j}$ , we have

$(b_{i}-b^{j})¥psi_{i,j}=0$ $(i=1,¥ldots, n, j¥in N^{n}¥backslash ¥{0¥})$ .

12 Kazuo UENO

Proof, (i) By Theorem 1.15, (i), we have $¥beta_{*}(^{s}¥delta)=¥delta_{a}$ for some $¥beta¥in AUT$;

besides, linear algebra gives $¥gamma¥in GL^{1}$ such that $¥gamma_{*}P_{1}(¥beta_{*}¥delta)=Tdg(a)+N$ with$[Tdg(a), N]=0$ and $N$ satisfying the above stated conditions (Jordan canonicalform). Hence, $Tdg(a)=^{s}(¥gamma_{*}P_{1}(¥beta_{*}¥delta))=¥gamma_{*}P_{1}(¥beta_{*}(^{s}¥delta))=¥gamma_{*}Tdg(a)$ and $N=^{n}(¥gamma_{*}P_{1}$

$(¥beta_{*}¥gamma))=¥gamma_{*}P_{1}(¥beta_{*}(^{n}¥delta))$ . Putting $¥epsilon:=¥beta¥cdot c_{A}(¥gamma)¥in AUT$ and using Proposition1.2, (iii) and (vi), we compute: $¥epsilon_{*}(^{s}¥delta)=c_{A}(¥gamma)_{*}¥beta_{*}(^{s}¥delta)=c_{A}(¥gamma)_{*}¥delta_{a}=_{A}‘(¥gamma)_{*}¥circ c_{D}$

$(Tdg(a))=c_{D^{¥circ}}¥gamma_{*}Tdg(a)=_{D}‘(Tdg(a))=¥delta_{a}$ and $P_{1}(¥epsilon_{*}(^{n}¥delta))=(P_{1^{¥circ}}c_{A}(¥gamma))_{*}P_{1}(¥beta_{*}(^{n}¥delta))=$

$¥gamma,|.{}_{¥backslash :}P_{1}(¥beta_{*}(^{n}¥delta))=N$ . The last assertion follows from $0=$ $[¥epsilon_{*}(^{s}¥delta), ¥epsilon_{*}(^{n}¥delta)]=$ $[¥delta_{a}$ ,$¥epsilon_{*}(^{n}¥delta)]$ .

(ii) Multiplicative proof (without using Proposition 1.2, $(¥mathrm{i}¥mathrm{i}¥mathrm{i})$ ).

We will state relations between the theory of $¥mathrm{a}¥mathrm{J}¥mathrm{d}$ (resp. $¥mathrm{m}¥mathrm{J}¥mathrm{d}$ ) and formalanalytic diagonalization in DER (resp. $AUT$).

Corollary 1.21. Notation as in Theorem 1.20. It follows immediately

from Theorem 1.18-Theorem 1.20 that:(i) If $P_{1}(¥delta)¥in^{s}g¥ell^{1}$ and $a_{i}-_{k=l}^{n}a_{k}j_{k}¥neq ¥mathit{0}$ ( $ i=1,¥ldots$ , $n$ , $j¥in N^{¥mathfrak{l}l}$ with $|j|¥geqq 2$),

then $¥delta¥in^{s}$DER.(ii) If $P_{1}(¥alpha)¥in^{s}GL^{1}$ and $b_{i}-b^{j}¥neq 0$ ( $i=1,¥ldots,$ $n$ , $j¥in N^{il}$ with $|j|¥geqq 2$), then

$¥alpha¥in^{s}AUT$

Proposition 1.22. Under the assumptions of Corollary 1.21, we have:(i) $¥{¥beta¥in AUT|¥beta_{*}(¥delta)=¥delta_{a}¥}$

$=$ $¥{¥epsilon Tam(¥gamma_{i,j})|(a_{i}-a_{j})¥gamma_{i,j}=0(i, j=1,¥ldots, n)¥}$ .

(ii) $¥{¥beta¥in AUT|¥beta_{*}(¥alpha)=¥alpha_{b}¥}$

$=$ $¥{¥epsilon Tam(¥gamma_{i.j})|(b_{i}-b_{j})¥gamma_{i,j}=0(i, j=1,¥ldots, n)¥}$ .

$Proo/$. (i) Let $¥beta¥in AUT$ be such that $¥beta_{*}(¥delta)=¥delta_{a}=¥epsilon_{*}(¥delta)$ . Since $¥epsilon^{-1}¥beta¥cdot¥delta_{a}=$

$¥delta_{a}¥cdot¥epsilon^{-1}¥beta$ , we have $a_{i}¥theta_{i}=¥delta_{a}(¥theta_{i})$ $(¥theta_{i} : =¥epsilon^{-1}¥beta(x_{i}), i=1,¥ldots, n)$ . The assumption ofCorollary1.21, (i) gives $¥epsilon^{-1}¥beta¥in GL^{1}$ . Putting $ Tam(¥gamma_{i,j}):=¥epsilon^{-1}¥beta$ , we have $a_{i}¥sum_{j=1}^{n}$

$¥gamma_{i,j}x_{j}=¥sum_{j=1}^{n}¥gamma_{i,j}a_{j}x_{j}$ , hence $(a_{¥mathrm{i}}-a_{j})¥gamma_{i,j}=0(¥mathrm{i}, j=1,¥ldots, n)$. The converse is clear.(ii) Multiplicative proof.

Corollary 1.23. Under the assumptions of Corollary 1.21,(i) if, moreover, $P_{1}(¥delta)=Tdg(a)$ , then $¥beta_{*},(¥delta)=¥delta_{a}$ for some $¥beta¥in AUT$ with

$P_{1}(¥beta)=1$ .

(ii) if, moreover, $P_{1}(¥alpha)=Tdg(b)$, then $¥beta_{*}(¥alpha)=¥alpha_{b}$ for some $¥beta¥in AUT$ with$P_{1}(¥beta)=1$ .

Proof, (i) Since $Tdg(a)=P_{1}(¥delta_{a})=P_{1}(¥epsilon)_{*}P_{1}(¥delta)=P_{1}(¥epsilon)_{*}Tdg(a)$, $P_{1}(¥epsilon)^{-1}$ and$Tdg(a)$ commute. Hence $¥beta:=¥epsilon¥cdot P_{1}(¥epsilon)^{-1}$ satisfies the condition in Proposition1.22, (i) and $P_{1}(¥beta)=1$ .



Remark 1.24. $P_{1}(¥delta)¥not¥in^{s}gl^{1}$ implies $¥delta¥not¥in^{s}$DER. On the other hand, ifthe eigenvalues of $P_{1}(¥delta)$ satisfies the condition in Corollary 1.21, (i), then $¥delta$ canbe formal analytically linearized; see the proof of Corollary 1.21, (i). Similarremark applies to the multiplicative case Corollary 1.21, (ii), which is relatedto the formal analytic solvability of Schroder equation.

Example 1.25. We give a formal analytically solvable case of the Schroderequation in one variable. We denote by $AUT_{l}$ , $GL_{1}^{1}$ , etc. the group $AUT$, $GL^{1}$ ,

etc. in the one variable case. Let $¥alpha¥in AUT_{1}$ with $¥alpha(x)=bx+$ terms of higher order$(b ¥in K^{¥mathrm{x}})$ . If $b^{p}¥neq 1(p ¥in N^{¥times})$ , then $¥alpha¥in^{s}AUT_{1}$ (note $GL_{1}^{1}=^{s}GL_{1}^{1}$ and use Corollary1.21, $(¥mathrm{i}¥mathrm{i}))$ .

Example 1.26 (cf. [6; p. 131, p. 138]). Let $¥delta¥in DER$ , $f¥in M$ and $a¥in K$ .

Then, by Proposition 1.7, (i):

$¥delta(f)=af$ if and only if $s¥delta(f)=af$ and $n¥delta(f)=0$ .

In particular, if $¥delta¥in^{n}$ DER, then $¥delta(f)=af$ implies $a=0$ or $f=0$ .

Examples 1.27. Here we take $K=C$.

(i) Let $¥delta¥in DER$ and $¥{a_{1},¥ldots, a_{n}¥}$ be its eigenvalues. If there exists $¥theta¥in[0$ ,$2¥pi[¥mathrm{s}¥mathrm{u}¥mathrm{c}¥mathrm{h}$ that $¥mathrm{R}¥mathrm{e}$ $(¥exp (¥sqrt{-1}¥theta)a_{i})<0(i=1,¥ldots, n)$ , then $¥delta¥in AUT_{*}Der_{c}(C[¥mathrm{x}_{1},¥ldots, ¥mathrm{x}_{1},]$

$¥cap M)$ .

$Proo/$. Since $¥sum_{k=1}^{n}¥mathrm{R}¥mathrm{e}$ $(¥exp (¥sqrt{-1}¥theta)a_{k})¥cdot j_{k}¥rightarrow-¥infty(|j|¥rightarrow+¥infty)$, $¥{j$ $¥in N^{n}¥backslash ¥{0¥}|a_{i}$

$=¥sum_{k=1}^{n}a_{k}j_{k}¥}$ is a finite set $(i=1,¥ldots, n)$ ; hence, using the notation of Theorem1.20, (i), we have $¥epsilon_{*}(^{s}¥delta)=¥delta_{a}$ and $¥epsilon_{*}(^{n}¥delta)(¥mathrm{x}_{i})¥in C[x_{1},¥ldots, x_{n}]¥cap M$ .

(ii) Let $¥alpha¥in AUT$ and $¥{b_{1},¥ldots, b_{n}¥}$ be its eigenvalues. If either $0<|b_{i}|<1$

$(i=1,¥ldots, n)$ or $|b_{i}|>1(i=1,¥ldots, n)$ , then $¥alpha¥in AUT_{*}Aut_{C}(C[x_{1},¥ldots, x_{n}]¥cap M)$ .

$Proo/$. Since $|b^{j}|¥rightarrow 0$ $(|j|¥rightarrow+¥infty)$ (resp. $|b^{j}|¥rightarrow+¥infty$ $(|j|¥rightarrow+¥infty)$ ), $¥{j$ $¥in N^{n}¥backslash ¥{0¥}|$

$b_{i}=b^{j}¥}$ is a finite set $(i=1,¥ldots, n)$ ; hence, using the notation of Theorem 1.20,(ii), we have $¥epsilon_{*}(^{s}¥alpha)=¥alpha_{b}$ and $¥epsilon_{*}(^{u}¥alpha)(¥mathrm{x}_{i})¥in C[X_{1},¥ldots, X_{n}]¥cap M$.

§2. Correspondence between nilpotency and unipotency

In this section, $K$ denotes a field of characteristic zero (not necessarily alge-braically closed). Note that nilpotency and unipotency defined in Definition 1.5have meanings over an arbitrary field $K$ .

We use the powerseries

$¥exp X:=¥sum_{k=0}^{¥infty}(1/k¥dagger)X^{k}¥in Q[[X]]$

and

14 Kazuo UENO

$¥log(1+X):=¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)X^{k}¥in Q[[X]]$ .

Lemma 2.1. We have(i) $¥exp(¥log(1+X))=1+X$ in $Q[[X]]$ ,

(ii) $¥log(¥exp X)=X$ in $Q[[X]]$ ,

(iii) $¥exp(X+¥mathrm{Y})=¥exp X$ . $¥exp Y$ in $Q[[X, ¥mathrm{Y}]]$ ,

(iv) $¥log((1+X)(1+Y))=¥log(1+X)+¥log(1+¥mathrm{Y})$ in $Q[[X, ¥mathrm{Y}]]$ .

Lemma 2.2. We have the bijection: $ng¥ell^{1}¥frac{¥rightarrow}{¥log}¥exp uGL^{1}$ (a theorem in linear

algebra).

Proposition 2.3. We have the bijection: $ng¥ell_{¥frac{¥rightarrow}{¥log}}^{¥infty^{¥exp}u}GL^{¥infty}$.

Proof. It follows from Lemma 2.2 that $ilgl^{i¥underline{¥rightarrow}u}¥exp GL^{i}$ is a bijection$¥log$

$(i ¥in N^{¥times})$ . Taking the inverse limit, we obtain the desired result.

Lemma 2.4. Let $A$ be an associative $K$-algebra.

(i) Let $¥Delta$ be a $K$-derivation of A. If $¥exp¥Delta$ is well-defined as a $K$-linearendomorphism of $A$ , then it is a $K$-algebra endomorphism of A. Moreover, ifalso $¥exp$ $(-¥Delta)$ is well-defined, then $¥exp¥Delta$ is a $K$-algebra automorphism of $A$

and $(¥exp¥Delta)^{-1}=¥exp$ $(-¥Delta)$ .

(ii) Let $¥Phi$ be a $K$-algebra endomorphism of A. If $¥log¥Phi$ is well-definedas a $K$-linear endomorphism of $A$ , then it is a $K$-derivation of A. (cf. [7; pp. 322-323])

Proof. (i) The Leibniz rule of derivation gives $(¥exp¥Delta)(fg)=(¥exp¥Delta)(f)$ .

$(¥exp¥Delta)(g)(f, g¥in A)$ , hence $¥exp¥Delta$ preserves multiplication. By Lemma 2.1,(iii), we have $ 1=¥exp¥Delta$ . $¥exp$ $(-¥Delta)=¥exp$ $(-¥Delta)$ . $¥exp¥Delta$ in $Q[[X]]$ . If both $¥exp¥Delta$

and $¥exp$ $(-¥Delta)$ have meanings as $¥mathrm{K}$ -linear endomorphisms, then $¥exp¥Delta$ is invertibleas a morphism and $(¥exp¥Delta)^{-1}=¥exp$ $(-¥Delta)$ .

(ii) Put $¥Psi:=¥Phi-1$ , a $¥mathrm{K}$ -linear endomorphism of $A$ . Let $a$ , $b¥in A$ . Wewill show:

$(¥log¥Phi)(ab)=(¥log¥Phi)(a)$ . $b+a$ . $(¥log¥Phi)(b)$ .

Define a $¥mathrm{K}$ -linear space $S$ :

$K^{(N¥times N)}¥underline{¥nearrow¥sim}S$

$(s_{i,j})-¥sum s_{i,j}¥Psi^{i}(a)¥Psi^{j}(b)$ ,

where $(i, j)$ runs over $N¥times N;¥Psi^{i}(a)¥Psi^{j}(b)$ corresponds to a lattice point $(i, j)¥in$

$N¥times N$ and $S$ is isomorphic to $K^{(N¥times N)}$ . Define two $¥mathrm{K}$-linear endomorphisms of $S$ :


$e_{1}$ : $¥Psi^{i}(a)¥Psi^{j}(b)|¥rightarrow¥Psi^{i+1}(a)¥Psi^{j}(b)$ ,

$e_{2}$ : $¥Psi^{i}(a)¥Psi^{j}(b)|¥rightarrow¥Psi^{i}(a)¥Psi^{j+1}(b)$ , $(i, j)¥in N¥times N$

Note that $e_{1}e_{2}=e_{2}e_{1}$ . Since $¥Phi=¥Psi+1$ preserves multiplication, we have in $A$ :

(1) $¥Psi(¥Psi^{i}(a)¥Psi^{j}(b))=¥Psi^{i+1}(a)¥Psi^{j+1}(b)+¥Psi^{i+1}(a)¥Psi^{j}(b)+¥Psi^{i}(a)¥Psi^{j+1}(b)$

$(i, j¥in N)$ .

By means of (1), we define the action of $¥Psi$ on $S$ denoted by $¥Psi_{S}$ ; we have $¥Psi_{S}=$

$e_{1}e_{2}+e_{1}+e_{2}$ , hence $(¥Psi^{k})_{S}=(¥Psi_{S})^{k}=(e_{1}e_{2}+e_{1}+e_{2})^{k}(k¥in N)$. We compute:

$¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)(¥Psi_{S})^{k}=¥log((1+e_{1})(1+e_{2}))$

$=¥log(1+e_{1})+¥log(1+e_{2})=¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)((e_{1})^{k}+(e_{2})^{k})$ ,

by Lemma 2.1, (iv), which is a well-defined $¥mathrm{K}$ -linear endomorphism of $S$ ; hencewe have:

(2) $¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)(¥Psi_{S})^{k}$(ab)

$=¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)((e_{1})^{k}+(e_{2})^{k})$ (ab)

$=¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)(¥Psi^{k}(a)¥cdot b+a¥cdot¥Psi^{k}(b))$ .

The assumption that $¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)¥Psi^{k}$ is a well-defined $¥mathrm{K}$ -linear endo-morphism of $A$ implies that (2) can be considered as a sum in $A$ . Definitively,we have:

$(¥log¥Phi)$ (ab)= $¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)¥Psi^{k}$(ab)

$=¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)(¥Psi^{k})_{S}(ab)$

$=(¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)¥Psi^{k}(a))¥cdot b+a¥cdot(¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)¥Psi^{k}(b))$

$=(¥log¥Phi)(a)$ . $b+a$ . $(¥log¥Phi)(b)$ .

Theorem 2.5. We have the bijection: $n$DER $¥frac{¥rightarrow}{¥log}¥exp$ uAUT

Proof. By Proposition 2.3, it suffices to show $¥exp$ $(^{n}DER)¥subset^{u}AUT$ and $¥log$

$(^{u}AUT)¥subset ^{n}$DER; these inclusions follow from Theorem 1.13, Lemma 2.2 andLemma 2.4.

16 Kazuo UENO

Corollary 2.6. Let $¥alpha¥in^{u}AUT$. If $¥alpha^{¥mathrm{z}}=1$ for some $z¥in Z¥backslash ¥{0¥}$ , then $¥alpha=1$ .

Proof. Since $1=¥alpha^{z}=¥exp(z¥delta)$ with $¥delta=¥log¥alpha¥in^{n}$ DER, we have $z¥delta=0$ byTheorem 2.5; hence $¥delta=0$ and $¥alpha=1$ .

Theorem 2.7. We have the commutative diagram:

$ngl^{1}¥subset^{n}lDDER¥subset ngl^{¥infty}$

$¥exp¥downarrow|$ $¥downarrow¥uparrow$ $¥downarrow¥uparrow¥log$

$liGL^{1}lA¥subset uAUT¥subset 1^{l}GL^{¥infty}$

Proof. Commutativity of the right-hand half of the diagra $¥mathrm{m}$ follows fromProposition 2.3 and Theorem 2.5. As for the left-hand half, it suffices to show$¥exp¥cdot c_{D}=c_{A}¥cdot¥exp$, which implies also $c_{D}$ . $¥log=¥log¥cdot c_{A}$ . For $¥lambda¥in g¥ell^{1}$ , we have$C_{D}(¥lambda)^{k}(X_{i})=(Mat(¥lambda)^{k}(¥mathrm{x}))_{i}(k¥in N)$ , where $x=(x_{1},¥ldots, ¥mathrm{x}_{n})$ and $(¥cdots)_{i}$ denotes the $¥mathrm{i}$ -thentry. We compute:

$¥exp¥cdot c_{D}(¥lambda)(x_{i})=¥sum_{k=0}^{¥infty}(1/k^{1})(Mat(¥lambda)^{k}(¥mathrm{x}))_{i}=$ $(Mat(exp ¥lambda)(¥mathrm{x}))_{i}$

$=¥sum_{j=1}^{n}$ Mat$(exp ¥lambda)_{i,j}¥chi_{j}=¥ell_{A}¥cdot¥exp$ $(¥lambda)(x_{i})$ $(i=1,¥ldots, n)$ .

Examples 2.8. By restricting the bijection of Theorem 2.5 to subsets, weobtain:

(i) $¥{¥delta|P_{1}(¥delta)=0¥}^{¥underline{¥rightarrow}}¥exp$$¥{¥alpha|P_{1}(¥alpha)=1¥}$ .

$¥log$

(ii) In the one variable case, Theore $¥mathrm{m}$ $2.5$ reduces to (i). Properties of this

bijection: $1l$ DER $1¥underline{¥rightarrow¥exp}¥iota¥iota AUT_{1}$ (cf. Example 1.25) as well as formal analytic$¥log$

structure of $uAUT_{1}$ have been studied by many authors; see [3], [7], etc.(iii) { $¥delta|¥mathrm{P}_{1}(¥delta)=¥mathrm{T}¥mathrm{a}¥mathrm{m}(¥mathrm{u}¥mathrm{p}¥mathrm{p}¥mathrm{e}¥mathrm{r}$ -triangular nilpotent matrix)}

$¥underline{¥rightarrow¥exp}$ { $¥alpha|¥mathrm{P}_{1}(¥alpha)=¥mathrm{T}¥mathrm{a}¥mathrm{m}(¥mathrm{u}¥mathrm{p}¥mathrm{p}¥mathrm{e}¥mathrm{r}$ -triangular unipotent matrix)}.$¥log$

(iv) We have $¥sum_{¥acute{i}^{l}=1}¥chi_{i}M¥cdot¥partial_{i}¥subset^{n}$DER; moreover,

$¥exp$ $(¥sum_{i=1}^{n}¥chi_{i}M¥cdot¥partial_{i})=$ $¥{¥alpha|¥alpha(x_{i})¥in¥chi_{i}(1+M)(i=1,¥ldots, n)¥}¥subset¥iota¥iota AUT$.

Proof. For $¥delta¥in¥sum_{i=1}^{n}x_{i}M¥cdot¥partial_{i}$ , we have $¥delta^{k}(x_{i})¥in¥chi_{i}M^{k}(k¥in N)$ by induction;hence $¥exp¥delta(x_{i})¥in¥chi_{i}(1+M)(i=1,¥ldots, n)$ . Conversely, let $¥alpha¥in^{u}AUT$ with $¥alpha(x_{i})¥in$

$X_{i}(1+M)(i=1,¥ldots, n)$ . We have $(¥alpha-1)^{k}(x_{i})¥in x_{i}M^{k}(k¥in N)$ by induction; hence$¥log¥alpha(x_{i})¥in x_{i}M(i=1,¥ldots, n)$ .

(v) $¥sum_{i=1}^{n}x_{i}M¥cdot¥partial_{i}$ is a sub-K-Lie algebra of DER and $¥exp$ $(¥sum_{i=1}^{n}x_{i}M¥cdot¥hat{a}_{i})$

a subgroup of $AUT$ In the one variable case, they coincide with $nDER_{l}$ and$uAUT_{1}$ respectively; if $n¥geqq 2$ , then they do not.


§3. Exponential map and Jordan decompositions

In this section we assume $K=C$. We give $gl^{i}$ the usual topology inducedfrom the inclusion $gl^{i}¥subset gl(M/M^{i+l})$ and $¥mathrm{g}1^{¥infty}=¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim(gt^{i})$ the Frechettopology defined through the inverse limit.

Theorem 3.1. (i) If $¥lambda¥in g_{¥acute{¥prime}}^{n¥infty}$ , then $¥exp¥lambda$ is well-defined and belongs to$GL^{¥infty}$ .

(ii) If $¥delta¥in DER$ , then $¥exp¥delta¥in AUT$ (see Lemma 2.4, $(¥mathrm{i})$ ).(iii) We have the commutative diagram:

(1) $gl^{1}¥subset lD$ $DER¥subset gl^{¥infty}$

$¥exp¥downarrow$ $¥downarrow¥exp$ $¥downarrow¥exp$

$GL^{1}lA¥subset AUT¥subset GL^{¥infty}$ .

(See the proof of Theorem 2.7.)(iv) The diagram of Theorem 2.7 with $K=C$ is the restriction of (i)

Remark 3.2. $¥exp:gl^{¥infty}¥rightarrow GL^{¥infty}$ is a rather generic notion; for example, wehave the commutative diagram:

$R$ $¥subset gl^{¥infty}$

$¥exp¥downarrow$ $¥downarrow¥exp$

$R^{¥times}¥subset GL^{¥infty}$

with $R=C[[x_{1},¥ldots, ¥chi_{n}]]$ acting on $M$ by multiplication.

Theorem 3.3. (i) $¥exp:gl^{l_{}}GL^{1}$ and $¥exp:gl^{¥infty}¥rightarrow GL^{¥infty}$ are surjective.(ii) $¥exp$ : $DER¥rightarrow AUT$ is not surjective.

Proof, (i) The first assertion is well-known ([9; pp. 313-315]). Thesecond assertion follows from the first one by means of an inverse limitargument.

(ii) (cf. [8; pp. 453-454], [7; p. 330]). In §4 we will give explicit examplesof $¥alpha¥in AUT¥backslash ¥exp$ DER (Examples 4.12, (iv), Example 4.13) verifying thestatement.

Theorem 3.4. $¥exp:gl^{1}$ (resp. DER, $gl^{¥infty}$) $¥rightarrow GL^{1}$ (resp. $AUT$, $GL^{¥infty}$ ) iscompatible with $Jds$ : if $¥lambda=^{s}¥lambda+^{n}¥lambda$ is an $aJd$ , then $ s¥exp¥lambda=¥exp$ $(^{s}¥lambda)$ and $u¥exp¥lambda=$

$¥exp(^{n}¥lambda)$ .

Proof. By Theorem 1.18 and Theorem 3.1, (iii), it suffices to study the case

18 Kazuo UENO

$¥exp:g¥ell^{¥infty}¥rightarrow GL^{¥infty}$ . Since $[^{S}¥lambda^{n},¥lambda]=0$ , we have $¥exp¥lambda=¥exp$ $(^{¥mathrm{s}}¥lambda)$ . $¥exp$ $(^{n}¥lambda)=$

$¥exp$ $(^{¥mathfrak{l}¥mathrm{t}}¥lambda)¥cdot¥exp$ $(^{s}¥lambda)$ . By the definition, $¥exp$ $(^{s}¥lambda)$ equals $¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim(¥exp^{s}P_{l}¥langle¥lambda))$ and $¥exp$

$(^{n}¥lambda)$ equals $¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim(¥exp^{1l}P_{i}(¥lambda))$ ; see the proof of Proposition 1.6, (i). Since$¥exp P_{i}(¥lambda)=¥exp^{s}P_{l}¥langle¥lambda$) $¥cdot¥exp^{n}P_{i}(¥lambda)=¥exp^{n}P_{i}(¥lambda)¥cdot¥exp^{s}P_{i}(¥lambda)$, we have $s¥exp P,¥langle¥lambda)=$

$¥exp^{s}P_{i}(¥lambda)$ and $u¥exp P_{i}(¥lambda)=¥exp^{n}P_{i}(¥lambda)$ . Combining together these equalities, wecompute:

$s¥exp(¥lambda)=¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . liln$ ^{s}P_{i}(¥exp (¥lambda))=¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim^{s}¥exp P_{i}(¥lambda)$

$=¥mathrm{i}¥mathrm{n}¥mathrm{v}$ . $¥lim¥exp^{s}P_{i}(¥lambda)=¥exp$ $(^{s_{¥lambda}^{¥cap}})$ ;

similarly, we have $ u¥exp$ $(¥lambda)=¥exp$ $(^{ll}¥lambda)$ .

Corollary 3.5. Let $(l, G)$ denote $(gl^{1}, GL^{1})$ , (DER, $AUT$) or $(gl^{¥infty}, GL^{¥infty})$ .

By Theorem 3.4, we have $¥exp$ $(^{s}¥ell)¥subset^{S}$G. Moreover, $¥exp:^{s}¥ell¥rightarrow^{S}G$ is(i) not injective,(ii) surjective in the cases of the first two pairs.

Proof, (i) is clear. Linear algebra shows $¥exp$ $(^{s}gl^{1})=^{s}GL^{1}$ . By Theorem1.15, we have $¥exp$ $(^{s}DER)=^{s}AUT$

Remark 3.6. The author has not known whether $¥exp$ $(^{S}gl^{¥infty})=^{s}GL^{¥infty}$ ornot; se misimplicity in this case has a different character.

Corollary 3.7. Notation as in $Corol/ary3.5$ . Then:

$¥{¥lambda¥in l|¥exp¥lambda¥in^{s}G¥}=s¥ell$ .

Proof, Let $¥lambda¥in l$ with $¥exp¥lambda¥in^{s}G$ . By Theorem 3.4, we have $1=^{u}¥exp¥lambda=$

$¥exp$ $(^{¥mathrm{J}1}¥lambda)$ ; hence, by Theorem 2.7, $l¥uparrow¥lambda=0$ .

Example 3.5. Let $¥delta¥in DER$ . Set $¥exp(C¥delta):=¥{¥exp(t¥delta)|t¥in C¥}¥subset AUT$, theone-parameter subgroup generated by $¥delta$ in $AUT$ With $¥{a_{1},¥ldots, a_{n}¥}$ being theeigenvalues of $P_{1}(¥delta)$ , we have:

(i) If $n¥delta¥neq 0$ , then $¥exp(C¥delta)¥cong C$ . (cf. Corollary 2.6)(ii) If $ n¥delta=0=^{S}¥delta$ , then $¥exp(C¥delta)=¥{1¥}$ .

(iii) If $ n¥delta=0¥neq^{s}¥delta$ , then $¥exp(C¥delta)¥cong C/¥bigcap_{a_{i}¥neq 0}(2¥pi¥overline{-1}/a_{i})Z$ ; hence we have$¥exp(C¥delta)¥cong C$ or $C/Z$ according as $¥bigcap_{a_{¥mathrm{i}}}¥neq 0(2¥pi¥sqrt{-1}/a_{i})Z=0$ or not.

$Proo/$. We have $¥exp(C¥delta)¥cong C/¥mathrm{K}¥mathrm{e}¥mathrm{r}_{¥delta}$ with $¥mathrm{K}¥mathrm{e}¥mathrm{r}_{¥delta}:=¥{t¥in C|¥exp(t¥delta)=1¥}$ .

(i) Suppose $t¥in ¥mathrm{K}¥mathrm{e}¥mathrm{r}_{¥delta}$ . Since $t^{s}.¥delta=^{s}(t¥delta)$ and $t^{n}.¥delta=^{n}(t¥delta)$ , we have byTheorem 3.4 that $1=^{u}¥exp(t¥delta)=¥exp(t^{n}.¥delta)$ ; hence, by Theorem 2.5, $ 0=t^{n}.¥delta$ .$n¥delta¥neq 0$ gives $t=0$ .

(ii) Clear.(iii) Since $¥delta=^{s}¥delta¥in^{s}$DER, we have by Theore $¥mathrm{m}$ $1.20$ , (i) that $¥beta_{*_{¥backslash }}(¥delta)=¥delta_{a}$ for some


$¥beta¥in AUT$; hence $¥exp(C¥delta)¥cong¥exp(C¥delta_{a})$ . The assertion follows from $¥mathrm{K}¥mathrm{e}¥mathrm{r}_{¥delta_{a}}=$

$¥bigcap_{a_{i}}¥neq 0(2¥pi¥sqrt{-1}/a_{i})Z$.

Remark 3.9. In Example 3.8, (iii), $¥delta=^{s}¥delta¥neq 0$ implies some $a_{i}¥neq 0$ ; hence,we have always $¥exp(C¥delta)¥cong C/Z$ in the one variable case, while not necessarily soif $n¥geqq 2$ .

Examples 3. 10. (i) Set $ddg(R):=$ $¥{¥delta_{a}¥in ddg| a_{i}¥in R (i=1,¥ldots, n)¥}$ and$ADG(>0):=$ $¥{¥alpha_{b}¥in ADG|b_{i}>0 (i =1,¥ldots, n)¥}$ . Then $¥exp:AUT_{*}ddg(R)¥rightarrow$

$AUT_{*}ADG(>¥mathit{0})$ is bijective.

$Proo/$. It suffices to show the injectivity. Suppose $¥alpha_{*}¥exp¥delta_{a}=¥exp¥delta_{b}$ $(¥alpha¥in$

$AUT$, $¥delta_{a}$ , $¥delta_{b}¥in ddg(R))$ . Putting $¥alpha(x_{i})=¥sum_{|j|¥geqq 1}¥varphi_{i,j}x^{i}$ $(i=1,¥ldots, n)$ , we have$(e^{b_{¥mathrm{i}}}-¥exp (¥sum_{k=1}^{n}a_{k}j_{k}))¥cdot¥varphi_{i,j}=0$ ; since $a_{i}$ , $b_{i}¥in R$ , the last equality is equivalent to$(b_{i}-¥sum_{k=1}^{n}a_{k}j_{k})¥cdot¥varphi_{i,j}=0$ , which implies $¥alpha_{*}¥delta_{a}=¥delta_{b}$ .

(ii) The inverse map of $¥exp:AUT_{*}ddg(R)AUT_{*}ADG(>¥mathit{0})$ is not ex-pressed as the powerseries of logarithm.

(iii) For $¥alpha_{b}¥in ADG$ , $¥log¥alpha_{b}:=¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)(¥alpha_{b}-1)^{k}$ is convergent in $gl^{¥infty}$

if and only if $¥{b_{i}¥}_{¥mathrm{i}=1,..,n}$ is contained in]0, 1]; if this condition is satisfied, thenwe have by Lemma 2.4, (ii) that $¥log a_{b}=¥delta_{a}¥in ddg(R)$ with $¥exp(a_{i})=b_{i}$ . Note that,when dealing with $P_{1}(¥alpha_{b})=Tdg(b)¥in GL^{1}$ , the convergence of $P_{1}(¥log¥alpha_{b})=$

$¥sum_{k=1}^{¥infty}((-1)^{k-1}/k)(Tdg(b)-1)^{k}¥in gl^{1}$ follows from the condition $|b_{i}-1|¥leqq 1$

$(i=1,¥ldots, n)$ ; in the case of $AUT$ and DER, their actions on all the monomials$¥{X^{i}||i|¥geqq 1¥}$ are considered.

(iv) Put DER(≦ 0): $=$ {$¥delta¥in DER$ $|$ all the eigenvalues of $¥delta¥subset]-¥infty$ , 0]} and$AUT(]0,1]):=$ {$¥alpha¥in AUT|$ all the eigenvalues of $¥alpha¥subset]0,1]$ }. DER(≧ 0) and$AUT([1, +¥infty[)$ are similarly defined. Then we have $¥exp$ (DER(≦ 0))=AUT$(]¥mathit{0}$ ,

1]) and $¥exp$ (DER(≧ 0))=AUT$([¥mathit{1}, +¥infty[)$ .

$Proo/$. Since $AUT([1, +¥infty[)=$ $¥{¥alpha¥in AUT|¥alpha^{-1}¥in AUT(]0,1])¥}$ , it suffices toshow the first case. Let $¥alpha¥in AUT(]0,1])$ and $ b_{i}¥in$ ] $0,1$ ] its eigenvalues. ByTheorem 1.20, (ii), we have $¥beta_{*}(^{s}¥alpha)=¥alpha_{b}$ for some $¥beta¥in AUT$; hence, by (iii), $¥log¥beta_{*}(^{s}¥alpha)$

$=¥delta_{a}¥in DER(¥leqq 0)$ with $¥exp(a_{i})=b_{i}$ . By Theorem 2.5 and Theore $¥mathrm{m}$ $1.19$ , (ii), wehave $¥epsilon:=¥log¥beta_{*}(^{u}¥alpha)¥in^{n}$DER. Since $¥delta_{a}$ . $¥epsilon=¥epsilon¥cdot¥delta_{a}$ , we have $¥delta_{a}=^{s}¥delta$ and $¥epsilon=^{n}¥delta$ with$¥delta:=¥delta_{a}+¥epsilon¥in DER$ ; hence $¥alpha=^{s}¥alpha^{u}.¥alpha=¥exp((¥beta^{-1})_{¥backslash *},¥delta)¥in¥exp$ (DER(≦ 0)).

§4. Characterization of $¥exp$ DER in $AUT$

We assume $K=C$ in this section.

Theorem 4.1. $¥exp$ $ DER=¥{¥alpha¥in AUT|^{¥mathrm{s}}¥alpha=¥exp¥epsilon$ , $ u¥alpha$ . $¥epsilon=¥epsilon^{u}¥cdot¥alpha$ for some $¥epsilon¥in$

DER}.

20 Kazuo UENO

Proof. If $¥alpha=¥exp¥delta¥in¥exp$ DER, then we have, by Theorem 3.4, $ s¥alpha=¥exp¥epsilon$ and$[¥log^{u}¥alpha, ¥epsilon]=0$ with $¥epsilon=^{s}¥delta$ . Note that $[¥log^{1l}¥alpha, ¥epsilon]=0$ is equivalent to $ u¥alpha$ . $¥epsilon=¥epsilon^{u}.¥alpha$ .

Conversely, putting $¥delta:=¥epsilon+¥log^{u}¥alpha$ , we have $¥alpha=¥exp¥delta¥in¥exp$ DER.

Lemma 4.2. For $¥beta¥in AUT$, we have $¥beta_{*}¥exp$ $ DER=¥exp$ DER.

Notation 4.3 for Lemma 4.4-Example 4.9. Let $¥alpha¥in AUT$ By Theorem 1.20,(ii), we have $¥beta_{*}(^{s}¥alpha)=¥alpha_{b}$ for some $¥beta¥in AUT$ with $b_{i}$ the eigenvalues of $¥alpha$ . ByLemma 4.2, we can assume $¥alpha=¥alpha_{b}¥cdot¥gamma=¥gamma¥cdot¥alpha_{b}$ with $¥gamma¥in^{tl}AUT$ without loss of generality.We fix $a_{¥mathrm{i}}¥in C$ such that $¥exp(a_{i})=b_{i}(i=1,¥ldots, n)$ . For $z=(z_{1}.., z_{n})¥in Z^{n}$ , we put$a_{i}(z):=a_{i}+z_{i}¥cdot 2¥pi¥sqrt{-1}$ and $¥sigma_{z}:=¥delta_{a(z)}¥in^{s}$DER with $a(z):=(a_{1}(z),¥ldots, a_{n}(z))$ (cf.Theorem 1.15). Finally, $C_{AUT}(¥alpha_{b})$ denotes the centralizer of $¥alpha_{b}$ in $AUT$

Lemma 4.4. $¥{¥delta¥in DER |¥exp¥delta=¥alpha_{b}¥}=$ $¥{¥beta_{*}¥sigma_{z}|¥beta¥in C_{AUT}(¥alpha_{b}), $z $¥in Z^{¥mathfrak{l}¥ddagger}¥}$ .

Proof. Assume $¥exp¥delta=¥alpha_{b}$ . By Corollary 3.7 and Theorem 1.15, (i), we have$(¥beta_{1})_{*^{¥prime}}¥delta=¥delta_{c}$ for some $¥beta_{1}¥in$ AUTwith $c_{i}$ the eigenvales of $¥delta$ ; hence $¥{¥exp(c_{¥mathrm{i}})¥}_{i=1,..,n}=$

$¥{b_{i}¥}_{i=1}$, ’

$¥mathfrak{l}1$

. Adjusting by a permutation of indices (which belongs to $AUT$) as$¥exp(c_{i})=b_{i}$ , we have $c_{i}=a_{i}(z)(z_{i}¥in Z, i=1,¥ldots, n)$ , which implies $¥delta=(¥beta_{2})_{*}¥sigma_{z}$ forsome $¥beta_{2}¥in AUT$ Since $¥alpha_{b}=¥exp¥delta=(¥beta_{2})_{i¥mathrm{r}_{¥backslash }}¥exp¥sigma_{¥mathrm{z}}=(¥beta_{2})_{*}¥alpha_{b}$ , we have $¥beta_{2}¥in C_{AUT}(¥alpha_{b})$ .

The converse is clear.

Proposition 4.5. $¥alpha_{b}$ . $¥gamma¥in¥exp$ DER if and only if $¥beta_{*}¥gamma¥cdot¥sigma_{¥mathrm{z}}=¥sigma_{z}¥cdot¥beta_{*}¥gamma$ for some$Z¥in Z^{n}$ and $¥beta¥in C_{AUT}(¥alpha_{b})$ . Moreover, if this is the case, then $¥alpha_{b}$ . $¥gamma=¥exp((¥beta^{-1})_{*}¥sigma_{Z}+$

$¥log¥gamma)$ (cf. [5; p. 219, Satz 4]).

Proof. Apply Theorem 4.1 and Lemma 4.4 with $s¥alpha=¥alpha_{b}$ and $ u¥alpha=¥gamma$ .

Lem ma 4.6. $C_{AUT}(¥alpha_{b})=¥{¥alpha|¥alpha(x_{i})=¥sum_{|j|¥geqq 1}¥varphi_{i,j}x^{j}$ with $(b_{i}-b^{j})¥varphi_{i,j}=0$ $(i=$

$ 1,¥ldots$ , $n$ , $|j|¥geqq 1)¥}$ (cf. Theorem 1.20, $(¥mathrm{i}¥mathrm{i})$).

Corollary 4.7. If there exists $Z¥in Z^{n}$ such that

$a_{k}(z)-¥sum_{i=1}^{n}a_{i}(z)j_{i}¥not¥in 2¥pi¥sqrt{-1}(Z¥backslash ¥{0¥})$

$(k=1,¥ldots, n, j¥subset-N^{n}¥backslash ¥{0¥})$ , then $¥alpha_{b}$ . $¥gamma=¥exp$ $(¥sigma_{Z}+¥log¥gamma)¥in¥exp$ DER (cf. [8; p. 455,Theorem 1], [5; p. 209] $)$ .

Proof. By virtue of the assumption, $b_{k}-b^{j}=0$ is equivalent to $a_{k}(z)-$

$¥sum_{i=1}^{1l}a_{i}(z)j_{i}=0$. Since $¥gamma¥in C_{AUT}(¥alpha_{b})$ , we have, by Leinina4.6, $(b_{k}-b^{j})¥varphi_{k,j}=0$ with$¥gamma(X_{i})=¥sum_{|j|¥geqq¥iota}¥varphi_{i,j}x^{j}$ ; hence $(a_{k}(z)-¥sum_{¥acute{i}=1}^{l}a_{i}(z)j_{i})¥varphi_{k,j}=0(k=1,¥ldots, n, .j ¥in N^{n}¥backslash ¥{0¥})$ .

This implies $¥sigma_{z}$ . $¥gamma=¥gamma¥cdot¥sigma_{z}$ . Thus the condition in Proposition 4.5 is satisfiedwith $¥beta=1$ .

Remark 4.8. Since $¥beta_{*}¥alpha$ and $¥alpha$ have the same eigenvalues $(¥beta¥in AUT)$ ,


Corollary 4.7 can be applied to $¥alpha¥in AUT$ in general.

The converse to Corollary 4.7 is false:

Example 4.9. Put $b:=(-1,1,¥ldots, 1)$ and $a:=(¥pi¥sqrt{-1},0,¥ldots, 0)$ . Then$¥alpha_{b}=¥exp$ $(¥pi¥sqrt{-1}x_{1}¥partial_{1})¥in¥exp$ DER, while the condition in Corollary 4.7 is notsatisfied with $k=1$ and $j=(3,0,¥ldots, 0)$ .

Theorem 4.10. In the one variable case, we have:

$¥exp$ DER $l=^{s}AUT_{l}¥cup uAUT_{l}$ .

(cf. Example 1.25.)

Proof. By Theorem 2.5 and Corollary 3.5, (ii), we have $ sAUT_{l}¥cup uAUT_{l}¥subset$

$¥exp$ $DER_{l}$ . Conversely, assume $¥alpha=¥exp¥delta¥in¥exp$ $DER_{l}$ . By Theorem 3.4, wehave $ S¥alpha=¥exp^{s}¥delta$ and $ u¥alpha=¥exp^{n}¥delta$ . By Theorem 1.13, (i), Theorem 1.19, (i) andTheorem 1.20, (i), we have $¥beta_{*}^{s}¥delta=ax¥partial_{X}$ and $¥beta_{*}^{n}¥delta=¥varphi¥partial_{x}$ for some $¥beta¥in AUT$, $a¥in C$

and $¥varphi=¥sum_{j¥geqq 2}¥varphi_{j}x^{j}¥in M_{1}^{2}$ with $a(1-j)¥varphi_{j}=0(j¥geqq 2)$. If $a=0$, then we have $s¥alpha=1$ ,

i.e., $¥alpha¥in^{u}AUT$; if $a¥neq 0$ , then we have $u¥alpha=1$ , i.e., $¥alpha¥in^{s}AUT$

Corollary 4.11. Let $¥alpha¥in AUT_{1}$ with $b$ its eigenvalue; by Theorem 1.15,(ii), we have $¥beta_{*}^{s}¥alpha=¥alpha_{b}$ for some $¥beta¥in AUT_{1}$ . Put $ p:=¥inf¥{k ¥in N^{¥mathrm{x}}|b^{k}=1¥}=¥inf¥{k¥in$

$ N^{¥times}|(^{S}¥alpha)^{k}=1¥};p=+¥infty$ if $b^{k}¥neq 1(k ¥in N^{¥mathrm{x}})$ . Then the following are equivalent:(i) $¥alpha¥not¥in¥exp$ DER, ,

(ii) $ 2¥leqq p<+¥infty$ and $¥iota¥iota¥alpha¥neq 1$ ,

(iii) $ 2¥leqq p<+¥infty$ and $¥alpha^{p}¥neq 1$ .

(cf. [7; p. 330].)

Proof. By Theorem 4.10, (i) is equivalent to $ u¥alpha¥neq 1¥neq^{s}¥alpha$ ; this implies byExample 1.25 that $b^{k}=1$ for some integer $k¥geqq 2$ , hence (ii). Supposing (ii), we haveby Corollary2.6 that $¥alpha^{p}=(^{s}¥alpha)^{p}(^{u}¥alpha)^{p}=(^{u}¥alpha)^{p}¥neq 1$ . Finally, (iii) implies $ s¥alpha¥neq 1¥neq^{u}¥alpha$ .

Examples 4. 12 (one variable), (i) Let $¥alpha¥in AUT_{1}$ . If $¥alpha^{k}=1$ for some $k¥in N^{¥times}$ ,

then $¥alpha¥in^{s}AUT_{1}$ . In fact, putting $c(x):=¥sum_{i=0}^{k-1}b^{-k}¥alpha^{k}(x)¥in M_{1}¥backslash M_{1}^{2}$ with $b$ theeigenvalue of $¥alpha$ , we have $¥gamma¥in AUT_{1}$ : $¥mathrm{x}¥mapsto c(¥mathrm{x})$ , and $¥gamma_{*}¥alpha=¥alpha_{b}$ (cf. [4; p. 65]).

(ii) Let $¥alpha¥in AUT_{1}$ : $x¥mapsto¥zeta_{p}x(1-qx^{q})^{-1/q}$ with $p$ , $q¥in N^{¥times}$ , $p$ not dividing $q$

inN, and $¥zeta_{p}$ a primitive $¥mathrm{p}$-th root of unity. Then $¥alpha¥in^{s}AUT_{1}$ ; in fact, $¥alpha^{i}¥neq 1$

$(1¥leqq ¥mathrm{i}¥leqq p-1)$ and $¥alpha^{p}=1$ (cf. (iv) below).(iii) We have $¥exp(tx¥partial_{X})¥in^{s}AUT_{1}$ and $¥exp(tx^{p+1}¥partial_{X})¥in^{u}AUT_{1}(t¥in C, p¥in N^{¥times})$ .

Put $¥varphi(p;t):=¥exp(tx^{p+1}¥partial_{x})(x)(t¥in C, p¥in N)$ . Then $¥varphi(p;t)$ is the solution tothe equation:

$dy^{f}(t)/dt=y(t)^{p+1}$ with $y(0)=x$ .with $y(0)=x$ .

22 Kazuo UENO

We have $¥varphi(0;t)=e^{t}x$ and $¥varphi(p;t)=x(1-tpx^{p})^{-1/p}(p¥geqq 1)$ . (cf. [1; p. 40].)(iv) An example of $¥alpha¥in AUT_{1}¥backslash ¥exp$ $DER_{l}$ . Putting $¥alpha:x¥mapsto¥exp(2¥pi¥sqrt{-1/}p)$ .

$x(1-p¥mathrm{x}^{p})^{-1/p}$ , $¥beta:x¥mapsto¥exp(2¥pi¥sqrt{-1}/p)x$ and $¥gamma:x¥mapsto x(1-px^{p})^{-1/p}(p¥geqq 2)$, we have$¥alpha=¥beta¥gamma=¥gamma¥beta$ , $¥beta¥in^{s}AUT_{1}$ and $¥gamma¥in^{1l}AUT_{1}$ ; hence $¥beta=^{s}¥alpha$ and $¥gamma=^{u}¥alpha$ . Since $¥beta¥neq 1¥neq¥gamma$ ,

we have $¥alpha¥not¥in¥exp$ $DER_{l}$ by Theorem 4. 10 (cf. (ii) above).(v) With the notation of (iv), we have $¥beta=¥exp((2¥pi¥sqrt{-1/}p)x¥partial_{x})$ and $¥gamma=$

$¥exp(x^{p+1}¥partial_{X})$ . On the other hand, solving the equation:$dy(t)/dt$ $=cy+y^{p+1}$ with $y(0)=x$ $(c ¥in C^{¥mathrm{x}})$ ,

we obtain $y(t)=¥exp(t(cx+x^{p+1})¥partial_{X})(¥mathrm{x})=¥exp(ct)x(1-(¥exp(pct)-1)x^{p}/c)^{-1/p}$ .

Substituting $t:=1$ and $c:=2¥pi¥sqrt{-1}/p$ , we have $¥exp(((2¥pi¥sqrt{-1}/p)x+¥chi^{p+1})¥partial_{X})=¥beta$ .

Note that $¥lim_{c¥rightarrow 0}¥exp((cx+¥chi^{p+1})¥partial_{X})=¥gamma$ .

Example 4. 13 (several variables). Consider the equation:

$dy_{i}(t)/dt=y^{p}¥cdot y_{i}$ with $y_{i}(0)=¥chi_{i}$ $(i=1_{ },.., n)$ ,

where $p=(p_{1}, .., p_{n})¥in(N^{¥times})^{n}$ . The solution is given by

$y_{l}(t)=¥exp(tx^{p}x¥cdot¥partial)(x_{i})=x_{i}(1-t|p|x^{p})^{-1/|p|}$

$(i=1,¥ldots, n)$ with $x$ . $¥partial:=¥sum_{j=1}^{Il}x_{j}¥partial_{j}$ .

Putting $¥alpha:x_{i}¥mapsto¥exp(2¥pi¥sqrt{-1}/p_{i})X_{i}(1-|p|x^{p})^{-1/|p|}$ , $¥beta:¥chi_{i}¥mapsto¥exp(2¥pi¥sqrt{-1/}p_{i})x_{i}$

and $¥gamma:¥chi_{i}¥mapsto x_{i}(1-|p|x^{p})^{-1/|p|}$ $(i=1,¥ldots, n)$ , we have $¥alpha=¥beta¥gamma=¥gamma¥beta$ , $¥beta¥in^{s}AUT$ and$¥gamma¥in^{u}AUT$; hence $¥beta=^{s}¥alpha$ and $¥gamma=^{u}¥alpha$ .

Put $q:=$ G.C.D. $(p_{1},¥ldots, p_{l},)¥in N^{¥times}$ . We will show:

(1) $¥alpha¥in¥exp$ DER if and only if $q$ divides $n$ in $N$.

For $p_{1}=¥cdots=p_{n}:=n+1$ , we have $¥alpha¥not¥in¥exp$ DER by (1); hence $¥exp$ : $DER¥rightarrow AUT$ isnot surjective (Theorem 3.3, $(¥mathrm{i}¥mathrm{i})$ ) and $¥exp$ DER is not a subgroup of $AUT$ $(¥beta,$ $¥gamma¥in$

$¥exp$ DER but $¥alpha=¥beta¥gamma¥not¥in¥exp$ DER) for any number of variables.

Proof of (1). Since $¥log¥gamma=x^{p}x¥cdot¥partial$ , it follows from Theorem 4.1 that $¥alpha¥in$

$¥exp$ DER if and only if:

(2) $¥beta=¥exp¥epsilon$ and $[x^{p}¥mathrm{x}¥cdot¥partial, ¥epsilon]=0$ for some $¥epsilon¥in DER$ .

Assume (2) with $¥epsilon=¥sum_{i=1}^{n}¥varphi_{i}¥partial_{i}(¥varphi_{i}¥in M)$ . $[x^{p}x¥cdot¥partial, ¥epsilon]=0$ is equivalent to:

(3) $x_{i}¥sum_{j¥neq i}p_{j}¥varphi_{j}/x_{j}=(¥mathrm{x}¥cdot¥partial-p_{i}-1)¥varphi_{i}¥in M$ $(i=1,¥ldots, n)$ .

It follows that $¥varphi_{j}=x_{j}¥psi_{j}$ for some $¥psi_{j}¥in R(j=1,¥ldots, n)$ ; substituting these into (3),we have $x$ . $¥partial(¥psi_{i})=¥sum_{¥acute{j}=1}^{l}p_{j}¥psi_{j}(i=1,¥ldots, n)$ and

(4) $0=¥sum_{i=1}^{n}p_{j}c_{j}$


with $c_{j}:=¥psi_{j}(0)$ . Since $¥beta=¥exp¥epsilon$, we have $ Tdg(¥exp$ $(2¥pi¥sqrt{-1}/p_{1}),¥ldots$ , $¥exp(2¥pi¥sqrt{-1/}$

$p_{n}))=P_{1}(¥beta)=¥exp(P_{1}(¥sum_{i=1}^{ll}x_{i}¥psi_{i}¥partial_{i}))=Tdg(¥exp(c_{1}),¥ldots, ¥exp(c_{l},))$ ; hence

(5) $c_{i}=2¥pi¥sqrt{-1}((1/p_{i})+z_{i})$

for some $z_{i}¥in Z(i=1,¥ldots, n)$ . It follows fro $¥mathrm{m}$ $(4)$ that $q$ divides $n$ . Conversely,if $n+¥sum_{¥acute{j}=1}^{l}p_{j}z_{j}=0$ for some $z_{j}¥in Z$, we have (2) with $¥epsilon:=¥sum_{i=1}^{n}2¥pi¥sqrt{-1}((1/p_{i})+$

$z_{i})x_{i}¥partial_{i}$ ; in fact, we can reverse from (5) to (3).

References

[1] Grobner, W., Die Lie-Reihen und ihre Anwendungen (2nd edition), VEB DeutscherVerlag der Wissenschaften, Berlin, 1967.

[2] Gerard, R. et Levelt, A. H. M., Sur les connexions a singularites regulieres dans le casde plusieurs variables, Funkc. Ekv., 19 (1976), 149-173.

[3] Labelle, G., Sur l’inversion et l’iteration continue des series formelles, Europ. J.Combinatorics, 1 (1980), 113-138.

[4] Reznick, B., When is the iterate of a formal power series odd?, J. Austral. Math.Soc. (Series A), 28 (1979) 62-66.

[5] Reich, L. und Schwaiger, J., Uber einen Satz von Shi. Sternberg in der Theorie deranalytischen Iterationen, Monatsh. Math., 83 (1977), 207-221.

[6] Saito, K., Quasihomogene isolierte Singularitaten von Hyperflachen, Invent. Math.,14 (1971), 123-142.

[7] Scheinberg, S., Power series in one variable, J. Math. Anal. Appl., 31 (1970), 321-333.[8] Sternberg, S., Infinite Lie groups and the formal aspects of dynamical systems, J.

Math. Mech., 10 (1961), 451-474.[9] Sugiura, M., Jordan canonical forms and the theory of elementary divisors I, II (in

Japanese), Iwanami, Tokyo, 1976-1977.[10] Ueno, K., On multiplicative Jordan decomposition in $¥mathrm{A}¥mathrm{u}¥mathrm{t}{}_{C}C[[x_{1}, _{ }., x_{n}]]$ and the

exponential map (in Japanese), in RIMS Kokyuroku 533, RIMS, Kyoto University,Kyoto, 1984.

[11] Ueno, K., Umbral calculus and special functions, to appear in Advances in Math.

After having finished the manuscript, I found the following paper closelyrelated to this article:

Praagman, C., Iterations and logarithms of formal automorphisms, Aequa-tiones Math., 30 (1986), 151-160.

nuna adreso:College of Liberal ArtsSaga UniversitySaga 840Japan

(Ricevita la 27-an de januaro, 1986)(Reviziita 16-an de junio, 1986)(Reviziita la 26-an de majo, 1987)

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