数理解析研究所講究録1086 巻 1999 年 71-87

71

On the algebraic geometry of Kac-Moody groupsbyPeter Slodowy
Mathematisches Seminar
Universit\"at Hamburg
D-20146 Hamburg$/\mathrm{G}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{y}$These notes are a slightly elaborated version of a talk given at the RIMSSymposium on ”Topological Field Theory and Related Topies”, Kyoto, December
1996. Their aim is to give a survey of the main results obtained by Claus Mokler in his dissertation at Hamburg University ([8], October 1996) pertaining to anatural semigroup completion of Kac-Moody groups.

1.

))

’)
$\mathrm{A}\mathrm{b}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}$Kac-Moody groups

Starting point for the construction of Kac-Moody Lie algebras and associated-matrix $A=((a_{ij}))\in M_{l}(\mathbb{Z})$groups is a generalized Cartan matrix, i.e. ansatisfying$l\cross l$$a_{ii}$$a_{ij}$$a_{ij}$$=$$\leq$$=$

2$i\neq j$

$0$

$0$

$\Rightarrow\sim$$a_{ji}=0$

We shall assume, in addition, that is symmetrizable (cf. [2]). In fact, one might
to be symmetric for simplicity. Also, the generalized Cartan matrices
takearising in singularity theory and providing the original motivation for our researchin Kac-Moody groups (cf. [11], [13]) are symmetric, e.g. the matrix of typeencoded by the Coxeter-Dynkin diagram
$A$

$A$

$T_{pqr}$

72

Whereas the Kac-Moody algebra
the Lie algebra,$\mathrm{g}=\mathrm{g}(A)$is essentially generated by copies of$l$$\mathrm{s}\mathrm{l}_{2}(\mathbb{C})$$\langle e_{i}, h_{i}, f_{i}\rangle,$$=\dot{i}=1,$,$l$$\ldots,$subject to relations derived from , the corresponding Kac-Moody group $G=$
$G(A)$ is essentially generated bycopies of the Lie group. Here, therelations are either imposed abstractly (Tits,$[16]$ ) or by the ”integration”
of from the integrable representations of (Moody-Teo, Marcuson, Garland,and, in the most thorough way, Kac-Peterson [10], [3], [4] .
$A$$l$$SL_{2}(\mathbb{C})$$\mathrm{c}\mathrm{f}.[15],$$G$$\mathrm{g}$$)$The most important result about
as an abstract group is the existence of a$BN$”twin”-pair or ”twin” Tits system $(B^{+}, B^{-}, N, S)$ in providing us, among
others, with
$G$

$G$

$\bullet$positive and negative Borel subgroups$\bullet$a maximal torus $T=B^{+}\cap B^{-}=N\cap B^{+}=N\cap B^{-}$ ,$\bullet$a Weyl group $W=N/T$ with generating set$\bullet$$B^{+}$

and$B^{-}$$S$,,

Bruhat decompositions$G= \bigcup_{w\in W}B^{+}wB^{+}=\bigcup_{w\in W}B^{-}wB^{-}$,

and a Birkhoff-decomposition$G= \bigcup_{w\in W}B-wB^{+}$

Similarly, as in the case of the Lie algebra where one usually adjoins additional derivations to a”minimal” Kac-Moody algebra, the precise structure ofdepends on slightly finer data than . These data are given by an integral
realizationofwhich fixes the size of the maximal torus
and its
position inside .
$\mathrm{g}$

$G$

$A$

$(H, \square , \Pi^{\vee})$

$A$

$T$

$G$

Here, $H$ is the lattice of algebraic one-parameter subgroups
dual, the lattice of algebraic characters$\mathbb{C}^{*}arrow T$$P=H^{*}=\mathrm{H}\mathrm{o}\mathrm{m}_{\mathbb{Z}}(H, \mathbb{Z})$$\square =\{\alpha_{1}, \ldots, \alpha_{e}\}\subset P,$$\Pi^{\vee}=\{h_{1}, ..

., h_{l}\}\subset H$

into with, and$Tarrow \mathbb{C}^{*}$

$T$

73

are free subsets of simple roots in$P$, resp. of simple coroots in $H$ , related by$\alpha_{i}(h_{j})=a_{ij}$More explicitly,
Let
into

:

$\kappa_{i}$

$G$

$\Pi$

and

$SL_{2}(\mathbb{C})arrow G,$.

are given in our context as follows:$\Pi^{\vee}$$i=1,$ . .

$l$
$,$

$,$

denote the basic homomorphisms of$SL_{2}(\mathbb{C})$, and let$h_{i}:\mathbb{C}^{*}$$arrow$$G$:$arrow$$G$$u_{i}$$\mathbb{C}$

be given byThen$h_{i}(\mathbb{C}^{*})\subset T$$h_{i}(s)$$:=$$\kappa_{i}()$,$s\in \mathbb{C}^{*}$$u_{i}(c)$$:=$$\kappa_{i}()$,$c\in \mathbb{C}$, i.e. $h_{i}\in H$ , and there is a character,,$\alpha_{i}\in P$

such that$tu_{i}(c)t^{-}=u_{i}(1\alpha i(t)_{C)}$for all $t\in T,$$c\in$C.

By its natural action on and , the Weyl group $W=N/T$ is identified withgenerated by the reflections $S=\{s_{1}, \ldots , s_{l}\}$the subgroup of
$P$

$T$

$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathbb{Z}}(P)$$s_{i}(\omega)=\omega-\omega(h_{i})\alpha i,$

Also,$s_{i}$$U_{i}$.

is given by the class of$\kappa_{i}()$in $N/T$ .

andmore explicit:

We can also make the groupsLet$\omega\in P$

denote the subgroup$B^{+}$$u_{i}(\mathbb{C})$$B^{-}$

and, for any real root $\gamma=w(\alpha_{i})(w\in W)$ , put$U_{\gamma}:=wU_{i}w^{-1}$real $=W(\Pi)$ of all real roots divides naturally into positive and
The setnegative roots,$\sum$

74$\Sigma$where$\sum$

real,-$=- \sum^{\mathrm{r}\mathrm{e}\mathrm{a}1,+}$real$=\Sigma^{\mathrm{r}\mathrm{e}\mathrm{a}1,+}\cup\Sigma$

real,-, and if we put$U^{\pm}=\langle U_{\gamma}|\gamma\in\Sigma^{\mathrm{r}}\mathrm{e}\mathrm{a}1,\pm\rangle$

(

$b,$

$\langle a,$$\ldots\rangle$

denoting the group generated by$B^{+}=T\ltimes U^{+}$

,

$a,$

$b,$

$\ldots$) we have$B^{-}=T\triangleright;U^{-}$

Finally, the anti-involution$SL_{2}(\mathbb{C})g$

can be lifted to all of$\bullet$$\bullet$$*(t)=t$ ,, i.e. there is an$SL_{2}{}^{t}g(\mathbb{C})$$\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}-\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}_{0}\mathrm{n}*:Garrow G$

such thatfor all $t\in T$$*(\kappa_{i}(g))=\kappa_{i}(^{t}g)$

In particular, one

2.

$G$

$-arrow$$g\in SL_{2}(\mathbb{C})$$\mathrm{h}\mathrm{a}\mathrm{s}*(U^{+})=U^{-},$”)), for all$\mathrm{A}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{c}$.

$*(U^{-})=U^{+}$.

Kac-Moody groups,
is a Cartan matrix of”finite type” (i.e. all components are of type, or ) then , as described in the last section, is a reductive algebraic groupis then a Hopf algebra,
over C. The algebraof regular functions on, inand the groupcan be completely recovered from the Hopf algebra
particular
.
If

$A$

$A_{n},$ $B_{n},$$G_{2}$$F_{n}$$\ldots$$G$$G$$\mathbb{C}[G]$$G$$\mathbb{C}[G]$$G= \mathrm{S}_{\mathrm{P}^{\mathrm{e}\mathrm{C}}}\max \mathbb{C}[G]=\mathrm{H}_{\mathrm{o}\mathrm{m}_{\mathbb{C}\mathrm{a}\mathrm{l}}}-\mathrm{g}(\mathbb{C}[G], \mathbb{C})$is a proper generalized Cartan matrix, then the associated algebra isIfshould be infinite-dimensional. A
of infinite dimension over C. Thus, also
proposal for an algebra of ”strongly regular” functions on was made by Kac
and Peterson in 1983 ([3]). As in the finite-dimensional case, this algebra is$A$$\mathrm{g}$

$G$

$G$

75

generated by the matrix coefficients of a suitable representation. Let us therefore
recall some basic facts about the irreducible highest weight representations of .
$G$

is of ”simply-connected
To simplify the presentation, we shall assume thatis a direct summand of $H$
type”, i.e. that the coroot lattice$G$$Q^{\mathrm{v}}=\mathbb{Z}.\Pi^{\mathrm{v}}$$H=Q\vee\oplus D$.

Then the set $P^{+}=\{\omega\in P|\omega(h_{i})\geq 0, i=1, \ldots , l\}$ of dominant weights can be
written as a direct sum$P^{+}=P^{0}\oplus\oplus i=1l$

N.

$\Lambda_{i}$where$P^{0}=\{\omega\in P|\omega(h_{i})=0, i=1, \ldots, l\}\cong D^{*}$

and where$\Lambda_{i},$$i=1,$$l$$\ldots,$, are fundamental dominant weights$i,j=1,$$\Lambda_{i}(h_{j})=\delta_{i}i,$

uniquely determined modulo$P^{0}$$l$$\ldots,$,.

onto the set of isoAs in the finite-dimensional case there is a bijection ofmorphism classes of irreducible highest w.eight representations of$P^{+}$$L$$G$$\Lambda\in P^{+}rightarrow L(\Lambda)$

having a unique (up to scalars) highest weight vector
determined bywill be one-dimensional.), the moduleof weight A. (If$v_{\Lambda}\in$$L(\Lambda)$$\Lambda\in P^{0}$$L(\Lambda)\backslash \{0\}$$L(\Lambda)$

Any such module carries a nondegenerate contravariant form (essentially unique),i.e. a symmetric bilinear form$\langle$such thatanti-involution on$\langle v, gw\rangle=\langle g^{*}v, w\rangle$$G$,$\rangle$:$L(\Lambda)\mathrm{x}L(\Lambda)arrow \mathbb{C}$for all$v,$$w\in L(\Lambda),$ $g\in.

Let us call the function$c_{v,w}$:$Garrow \mathbb{C}$G$ ,and $g^{*}=*(g)$ the76given byfor some $w\in L(\Lambda)$ a matrix coefficient of. Kac and Peterson now define$c_{v,w}(g)=\langle v, gw\rangle$

respresentation$L(\Lambda))$

$v,$

$G$

(in the$:=$$\mathbb{C}[G]$and they prove the following”Peter-Weyl”-Theorem: The map$\bigoplus_{\Lambda\in F+}L(\Lambda)\otimes L(\Lambda)arrow \mathbb{C}[G]$induced by $v\otimes w\mapsto c_{v,w}$ is an isomorphism of $G\cross G$ -modules.
Here, the action of $G\cross G$ on
is given by $((g, h)f)(X)=f(g^{*}xh)$ . Alternatively, one might use the usual action of $G\cross G$ on functions on and let act
on the first factor
by the contragredient action$\mathbb{C}[G]$

$G$

$G$

$L(\Lambda)$$(g, v)-(g*)^{-1}v$ .

It turned out thatis not a Hopf algebra. There is neither a co-multiplication
nor an antipode (basically due to the infinite-dimenionality of theand theinequivalence between highest weight and lowest weight representations). Even
worse, Kac and Peterson exhibited elements in Specmax
not contained in(which injects into Specmax) (cf. [3] Remark 2.2). Thus they formulated
the following problem:$\mathbb{C}[G]$$L(\Lambda)$$\mathbb{C}[G]$$G$$\mathbb{C}[G]$$(1\mathrm{o}\mathrm{c}.

Determine Specmax
algebra$\mathbb{C}[G]$\mathrm{c}\mathrm{i}\mathrm{t}., 4\mathrm{H}\mathrm{b}))$(possibly with respect to a topological structure on the$\mathbb{C}[G])!$

Inspired by the deformation theory of certain singularities (cf. [13]) we conjectured$\overline{G}:=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}\mathbb{C}[G]=G.\overline{T}.G$

where$\overline{T}$is the closure of

$T$

in

$\overline{G}$realized as the torus embedding$T=\mathrm{s}_{\mathrm{P}}\mathrm{e}\mathrm{C}\mathrm{m}\mathrm{a}\mathrm{x}\mathbb{C}[P]\subset$

Specmax$\mathbb{C}[P\cap I]=\overline{T}$77, the Tits cone attached to . This embedding, or rather a domainof discontinuity for the action of $W$ , had been studied before by Looijenga
had turned out to be the base space of a semiuniversal
and the quotient
deformation for certain isolated singularities (cf. [6], [7]). Moreover, in [12], [13]as target spaces for an adjoint quotient of .
andwe realized

for

$G$

$I\subset P\otimes_{\mathbb{Z}}\mathbb{R}$$\mathcal{T}\subset\overline{\mathcal{T}}$$\overline{T}/W$$G$$\overline{T}/W$$\overline{\tau}/W$

During a stay at MSRI (1984), D. Peterson announced a proof of the above conbeing considered
jecture including a number of structural properties ofas the continuous spectrum with respect to some topology). In connection with
his infinite-dimensional algebraic-geometric approach to the flag manifolds of
Kac-Moody groups, M. Kashiwara also studied the abstract maximal spectrum), cf. [5]. Finally, C. Mokler ([8]) made a(without topology onofquite thorough study of in the context of some infinite-dimensional algebraic
geometry based on suitably topologized coordinate rings. In particular, he gavea detailed proof of our conjecture. This is what we want to report upon.
$\overline{G}([9],$$\overline{G}$$\mathbb{C}[G]$$\mathbb{C}[G]$$\overline{G}$3. A topology on the algebra of strongly$\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\backslash$functions

Let be a complex vector space. Then we may view the symmetric algebra $S(V^{*})$
weas the coordinate ring of the variety . Ifof its dual space
have
.
$V$

$V$

$V^{*}$$\dim_{\mathbb{C}}V<\infty$$\mathrm{H}\mathrm{o}\mathrm{m}_{k-}\mathrm{a}(s(V^{*}), \mathbb{C})=\mathrm{H}\mathrm{o}\mathrm{m}(V*, \mathbb{C})=V**=V$

However, if $\dim_{\mathbb{C}}V=\infty$ we have $V\subset V^{**},$ $V\neq V^{**},$ , and Specmax $S(V^{*})$ isstrictly larger than
the algebra $S(V^{*})$ :$V$. To remedy this defect we put the following topology on

A basis of neighborhoods of$\mathrm{O}\in S^{*}(V^{*})$

is given by the ”cofinite” ideals{ $J(V’)|V’\subset V$ a finite-dimensional subspace},$J(V’)=\{f\in S(V^{*})|f|V’\equiv 0\}$.

Now, the continuous maximal spectrum

SpecmoS(V*)$=\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{C}\circ}\mathrm{n}\iota-k-\mathrm{a}(s(V*), \mathbb{C})$is easily identified with (i.e. Hilbert’s Nullstellensatz gives $V’=S(V^{*})/J(V’)$
for all finite-dimensional $V’\subset V$ ).
$V$

78

To put a topology on
constructed as follows:$\mathbb{C}[G]$

we embed

We fix contravariant forms,
them to a form, also denoted by$\langle$$\rangle$$\langle$$G$, and finally$\overline{G}$, into a larger space $M$

on all modules,A,, on the direct sum$L(\Lambda)$$\in P^{+}$, and extend$\rangle$$L:= \bigoplus_{+\Lambda\in P}L(\Lambda)$

by requiring
andto be orthogonal for A
subalgebra of End $(L)$ satisfying$L(\Lambda)$$\bullet$$\bullet$$\varphi(L(\Lambda))\subset L(\Lambda)$

the adjoint

We let$\varphi^{*}$

offor all$\varphi$denote the$\Lambda\in P^{+}$$\neq\Lambda’$. Let $M$ denote the,with respect to$\langle$,$\rangle$exists.

-algebra generated by all matrix coefficients:$w\in L,$, and consider the ”cofinite” topology on
given by the neighborhood basis of$\mathbb{C}[M]$$Marrow \mathbb{C},$$\mathbb{C}[M]$$L(\Lambda’)$$\mathbb{C}$$c_{v,w}$$c_{v,w}(\varphi)=\langle v, \varphi w\rangle$

$v,$

$0$

{ $J(M’)|M’\subset M$ a subspace of finite dimension},$J(M’)$ being the vanishing ideal of $M’$ .

Then we have$\bullet \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{m}^{\mathrm{o}}\mathbb{C}[M]=M$$\bullet$is a”weak” algebraic monoid (i.e. right and left multiplication on $M$ by
given elements of $M$ are ”morphisms” of $M$ ; note that there is no comultiplication on.
$M$$\mathbb{C}[M])$By the definition of the contravariant forms on theand we have a naturalembedding. Moreover,is the image ofunder the restriction$M$fromto . We now put the quotient topology with respect toonand we obtain$L(\Lambda)$$G^{\mathrm{c}}arrow M$$G$$\mathbb{C}[G]$$\mathbb{C}[G]$$L$$\mathbb{C}[M]$$\mathbb{C}[M]arrow \mathbb{C}[G]$79$\mathrm{S}_{\mathrm{P}^{\mathrm{e}\mathrm{c}\mathrm{m}^{\mathrm{O}}}}\mathbb{C}[G]=\overline{G}=\mathrm{Z}\mathrm{a}\mathrm{r}\mathrm{i}_{\mathrm{S}\mathrm{k}\mathrm{i}}$$\bullet$$\overline{G}$$\bullet$-closure of$G$

in $M$ ,is a”weak” algebraic monoid (in the sense above).

4. The Tits cone and the closure of the maximal torus
for allbe the ”real” character group,
the union of all $W-$
a fundamental Weyl chamber, and$i=1,$
translates of . Then is a convex solid cone, called the Tits cone. The interiorof is a domain of discontinuity of W. (For details, cf. [2]).

Let$\overline{C}=\{\omega\in V|\omega(h_{i})\geq 0$$V=P\otimes_{\mathbb{Z}}\mathbb{R}$$\ldots,$$I=W.\overline{C}$

$l\}$

$I$

$\overline{C}$$I$$I^{\mathrm{o}}$Example: Let$A$

be the ”hyperbolic” matrixof signaturedefines a symmetric bilinear form on
Now, the matrix
may be be identified with the$(+, +, -)$ , and with respect to some convention
interior of the positive light cone. The Weyl group $W$ is isomorphic to.
acting as a group of hyperbolic motions on the unit
$V\cong \mathbb{R}^{3}$$A$$I^{\mathrm{O}}$$PGL_{2}(\mathbb{Z})$$\mathrm{d}\mathrm{i}_{\mathrm{S}\mathrm{c}\cong}\mathrm{P}(I^{\mathrm{O}})\subset \mathbb{P}(V)$The boundary of is of particular interest for us. A subset $I’\subset I$ is called a)(resp. a(rational) boundary component of if there is asuch that$I$$\bullet$$\bullet$$\omega(\gamma)\geq 0$

for all$\omega(\gamma)=0$

for$\gamma\in H$$\gamma\in V^{*}=H\otimes_{\mathbb{Z}}\mathbb{R}$$I$$\omega\in I$$\omega\in I$

implies$\omega\in I’$.

It is possible to classify all boundary components of in terms of a special subsetof them:or if all connected components ofis called pure if either
A subset $0$$0$ (in an obvious sense) are of infinite type.
$I$$\Theta=\emptyset$$\subset\Pi$

To any pure subset $0$$I(\Theta)=${$\subset\Pi$

we may associate the following subset$\omega\in I|\omega(h_{\mathrm{i}})=0$

for all$i$such that$\alpha_{i}\in\Theta$$I(\Theta)$

}.

of :
$I$

80

We now have the following result, essentially due to Looijenga ([6]):Theorem:i) Let$\Theta\subset\Pi$be pure. Then

$I(\Theta)$

is a rational boundary component of .
$I$ii) Letbe a boundary component. Then there is a unique pure $0$
and a $w\in W$ such that $I’=w.I(\Theta)$ . In particular, all boundary components of are rational.
$I’\subset I$$\subset\Pi$$I$

Example: We take up the previous example. There are 3 pure subsets of,$\emptyset$,$\Theta=\{\alpha_{1}, \alpha_{2}\}$$\Pi=\{\alpha_{1}, \alpha_{2}, \alpha_{3}\}$$\square$:.

The corresponding boundary components are$I$,

all rational half-lines on
the positive light cone,$\{0\}$.

To determine theof in we first have to describe the restriction of
to . Since all weights of a module, are contained in $I\cap P$ ,and since
we obtain$\mathrm{c}1_{\mathrm{o}\mathrm{S}}\mathrm{u}\mathrm{r}\mathrm{e}\overline{\tau}$$T$$\mathbb{C}[G]$$T$$\overline{G}$$L(\Lambda),$$\Lambda\in P^{+}$$\overline{C}\cap P=P^{+}$$\mathbb{C}[G]|_{T^{=}}\mathbb{C}[P\mathrm{n}I]$,

the semigroup algebra of $P\cap I$ . It is easily seen that the induced topology on
is discrete, thus$\mathbb{C}[P\cap I]$$\overline{T}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{C}\mathrm{m}\mathbb{C}0[P\cap I]=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{m}\mathbb{C}[P\cap I]$.

is not finitely generated its maximal spectrum can be deterThroughmined similarly as in the usual ”finite type” theory of torus embeddings (cf. e.g.
[1] , i.e. one has$\mathbb{C}[P\cap I]$$)$$\overline{T}=\bigcup_{I’}T/\mathrm{A}\mathrm{n}\mathrm{n}(I’)=\cup\ominus\bigcup_{w\in W}T/w\mathrm{A}\mathrm{n}\mathrm{n}(I(\Theta))w^{-1}$where{ $t\in T|\omega(t)=1$ for all
through all rational boundary components of$\mathrm{A}\mathrm{n}\mathrm{n}(I’)=$(resp. $0$ ) runs
(resp. all pure subsets of ).

$\omega\in I’$$I$} and where

,

$I’$

$\square$81

As a subset of $M$ , the completion
realization:
Let $0$$\subset\Pi$$\overline{T}$has a quite natural representation theoretic

be a pure subset. We define the pojection operator $e(\Theta)\in M$ by$v$$e(\Theta)v=\{$
$0$

if
if

and
and

$v\in L(\Lambda)_{\mu}$$v\in L(\Lambda)_{\mu}$$\mu\in I(\Theta)$$\mu\not\in I(\Theta)$.

ofis realized as the -orbit T.
Then the boundary stratum
as a boundary point ofunder left multiplication by . To realize
forsuch thatchoose a one-parameter subgroup,. Since for all
exactly whenandall
we have$T$$T/\mathrm{A}\mathrm{n}\mathrm{n}(I(\Theta))$$\overline{T}$$T$$e(\Theta)$$e(\Theta)$$e(\Theta)$$\omega(\gamma)\geq 0$$\gamma\in H=\mathrm{H}\mathrm{o}\mathrm{m}(\mathbb{C}^{*}, T)$$\omega\in I$$s\in \mathbb{C}^{*},$$\omega\in I(\Theta)$$\omega(\gamma)=0$$\gamma(s)v=Sv\omega(\gamma)$$v\in L(\Lambda)_{\omega}$,

we clearly obtain (in $M$ )$\lim_{sarrow 0}\gamma(S)=e(\Theta)$.

5. Unipotent subgroups
as well as those of generalof
To study the unipotent radicals
parabolic subgroups we have to take a closer look at the action of on
generated by
as a variety with the coordinate ring
. We consider, and equipped with the appropriate:
the functions”cofinite” topology. Then, for any fixed $v\in L(\Lambda)$ , the orbit map$B^{+},$ $B^{-}$$U^{+},$ $U^{-}$$G$$P^{+}$$\Lambda\in$$\mathbb{C}[L(\Lambda)]$$L(\Lambda)$$c_{w}$$L(\Lambda),$$L(\Lambda)arrow \mathbb{C},$$c_{w}(v)=\langle v, w\rangle$$M$$arrow$$L(\Lambda)$$m$$\mapsto$

$mv$

). We
is a morphism of varieties (with continuous comorphism
shall make use of the following results of Kac and Peterson ( $[10],[3]$ Lemma 4.3)$\mathbb{C}[L(\Lambda)]arrow \mathbb{C}[M]$$\bullet$

The Kostant cone.
closed in$\mathcal{V}(\Lambda)=(Gv\mathrm{o})\cup\{0\}$, with$v_{0}\in L(\Lambda)_{\Lambda}\backslash \mathrm{f}\mathrm{o}\}$, is Zariski$L(\Lambda)$$\bullet$If A is a regular dominant weight, A $\in P^{++}$ (i.e. $\Lambda(h_{i})>0$ for $i=$
runis generated by the matrix coefficients, thenis the, wherening through all elements in (in fact,$1,$$\ldots,$$l)$$c_{xv0,v0},$ $x$$\mathbb{C}[G]|_{U^{-}}$$\Sigma^{-}$$\mathrm{g}$$x \in \mathrm{g}^{-}=\bigoplus_{\alpha\in\Sigma^{-}}\mathrm{g}_{\alpha}$system of all negative roots, is sufficient).

82

Theorem ([8] Satz 5.6,1) : The groups$U^{+}$$)$

Proof: Because of the existence of theto consider
AssumeThis implies$c_{v_{0},v_{0}}(u)=1$
for all
for allund

are Zariski closed in $M$ .

$U^{-}$

it is sufficientchosen such that.

$\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}-\mathrm{i}\mathrm{n}\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}*:Garrow G$$U^{-}$$v_{0}\in L(\Lambda)_{\Lambda}\backslash \{\mathrm{o}\}(\Lambda\in P^{++})$$c_{v_{0},v_{0}}(\varphi)=1$$u\in U^{-}$$\langle v_{0}, v_{0}\rangle=1$, and$\varphi\in\overline{U^{-}}$

Let
Then
implies. Since $Marrow L(\Lambda),$ $m\mapsto mv_{0}$ ,is continuous andis closed in
we get. Thus, using
the Birkhoff decomposition of , we find $u\in U^{-},$ $n\in N$ such that$\varphi\in\overline{U^{-}}$$\langle v_{0}, \varphi v_{0}\rangle=1$$\varphi v_{0}\neq 0$$L(\Lambda)$$\mathcal{V}(\Lambda)$$\varphi v_{0}\in Gv_{0}\subset \mathcal{V}(\Lambda)$$G$$\varphi v0=u^{-}nv0$

Because of$(u^{-})^{*}\in U^{+}$.

we have$1=\langle v_{0}, \varphi v\mathrm{o}\rangle=\langle(u^{-})^{*}v0, nv_{0}\rangle=\langle v_{0},$ $nv_{0\rangle}$

and thus $n=1$ , oror $\varphi=u^{-}\in U^{-}$ , q.e.d.

$\varphi v_{0}=u^{-}v_{0}$Recall that any subset$\Psi\in\Pi$. This implies $c_{xv_{0},v}0(\varphi)=c_{xv_{0},v}0(u^{-})$ for all

gives rise to a Weyl subgroup$W_{\Psi}=\langle S_{\alpha_{i}}|\alpha_{i}\in\Psi\rangle$

and parabolic subgroups$P_{\Psi}^{+}=\langle B^{\pm}, W_{\Psi}\rangle$

with unipotent radicals$U_{\Psi}^{\pm}=\cap w\in W\Psi wU^{\pm}w^{-1}$

It is obvious that

$U_{\Psi}^{\pm}$

are Zariski closed in $M$ , as well.

$x\in \mathrm{g}$,83

6. The main result
For any $i\in\{1, \ldots, l\}$ we fix a highest weight vectorbythe principal open subset$v_{i}\in L(\Lambda_{i})_{\Lambda_{i}}\backslash \{0\}$and define$D_{i}\subset\overline{G}$$D_{i}=\{\varphi\in\overline{G}|c_{vv_{i}}(i,\varphi)\neq 0\}$We can almost cover by these sets. Let
is the ”orthogonal” union of the set, i.e.
type.

$\Pi_{\infty}\subset\Pi$$\overline{G}$

$\Pi$

$\Pi$

$\Pi_{\infty}$.

the maximal pure subset ofof finite
and a subset$\Pi\backslash \Pi_{\infty}$

Proposition A ( $[M]$ , Satz 5.16): We have$\bigcup_{i=1}^{l}\bigcup_{g,h\in G}gD_{i}h=\overline{G}\backslash \tau_{e(\Pi}.\infty)$

.

.
and
Proof: To simplify our presentation, we shall assumeThen $e(\square )=e(\Pi_{\infty})\in M$ is characterized by the property $e(\Pi)v=0$ , for all
for all $i\in$and assume$v\in L(\Lambda)$ , A. Consider$h\in G$ . Then
$P^{\mathrm{O}}=\{0\}$$\Pi=\Pi_{\infty}$$\varphi\not\in gD_{i}h$$\varphi\in\overline{G}$$\in P^{+}\backslash \{0\}$$\{1, \ldots, l\},$ $g,$$\langle gv_{i}, \varphi hv_{i}\rangle=0$, for all

$i,$

$g,$

$h$

.

$g\in G$ , we obtainis spanned by allof theis made up from tensor
reduction, we getfor allSince$L(\Lambda_{i})$$L(\Lambda),$ $\Lambda\in P^{+}\backslash \{0\}$. Since any
and subsequent$\varphi|_{L(\Lambda_{i})}=0$$gv_{i},$$\mathrm{p}\mathrm{r}.\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{s}$$L(\Lambda_{i})$$\Lambda\in P^{+}\backslash \{0\}$$\varphi|_{L(\Lambda)}=0$

or $\varphi=e(\Pi)$ .
(This proof can be easily adopted to the general case.)

As a next step, we shall determine the structure of the open setsthat recall the parabolic subgroups$P_{i}^{\pm}=P_{\Pi\backslash \{\}}^{\pm}\alpha_{i}$with unipotent radicals$U_{i}^{\pm}=U_{\Pi\backslash \{\}}^{\pm}\alpha_{\mathfrak{i}}$’$D_{i}\subset\overline{G}$

. For

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Levi subgroupand Weyl group. Thenis theKac-Moody group attached to the realization. Let
denote the algebra of strongly regular functions onand letdenote thealgebra of restricted functions fromto the subgroup . Then the function
restricts to the character
on , and representation theoretic arguments
quickly show (cf. [8], section 5.1.2):$G_{i}=P_{i}^{+}\cap P_{i}^{-}$$G_{i}$$W_{i}=W_{\Pi\backslash \{\alpha_{i}\}}$$(H, \Pi\backslash \{\alpha_{i}\}, \Pi\iota\{h_{i}\})$$G_{i}$$\mathbb{C}[G]_{i}$$G_{i}$$\mathbb{C}[G]$$G_{i}$$\Lambda_{i}$$c_{v_{i},v_{i}}$Lemma: The inclusion
ization ofwith respect to$\mathbb{C}[G]_{i}\subset \mathbb{C}[G_{i}]$$\mathbb{C}[G]_{i}$$\mathbb{C}[G_{i}]$to$\Lambda_{i}$induces an isomorphism from the local:$\mathbb{C}[G_{i}]$$(\mathbb{C}[G]_{i})_{\Lambda_{i}}arrow \mathbb{C}[\sim G_{i}]$.

Proposition : For any $i\in\{1, \ldots , l\}$ we have an isomorphism of infinitedimensional varieties$\mathrm{B}$$D_{i}=U_{i}^{-}\cross \mathrm{S}_{\mathrm{P}^{\mathrm{e}\mathrm{C}\mathrm{m}^{\mathrm{o}}}}\mathbb{C}[c_{i}]\cross U_{i}^{+}$

Proof: Let us first look at$D_{i}\cap G$. Then the Birkhoff-decomposition$G= \bigcup_{w\in W}U^{-}wTU^{+}$gives$D_{i} \cap G=\bigcup_{w\in W_{i}}U^{-}wTU^{+}=U_{i}^{-}.G_{i}.U_{i}+$(direct product)

Recall that theare closed in $M$ , therefore in
and in. By the Lemma,
the closure ofincan be identified with Specmo. This gives the claim.
$\overline{G}$$U_{i}^{\pm}$$G_{i}$$D_{i}$$D_{i}$$\mathbb{C}[G_{i}]$Applying downward induction to Propositions A andresult.
$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\Gamma \mathrm{e}\mathrm{m}$$\mathrm{B}$

we arrive at our main([8], Satz 5.18): We have$\overline{G}=${$ge(\Theta)h|\Theta\subset\Pi$

pure

$g,$

$h\in G$}$=G.\overline{T}.G$.

Remarks: Proposition for the case of the minimal parabolic
may alreadybe found in [3], Lemma 4.4. Its general version for arbitrary parabolics is due toKashiwara ([5], Proposition 5.3.5), who has also given a form of Proposition A ina somewhat different context ([5], Proposition 6.3.1).
$\mathrm{B}$$B^{+}$

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7. An application
In [8] one finds many more results on the structure of . Here, we want toconclude with an application to the adjoint quotient of studied in [12], [13],admits a ”parabolic” partition[14] (details are forthcoming). Recall that$\overline{G}$

$G$

$G$

$G=\cup G(\Theta)$$\ominus\subset^{\mathrm{n}}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}$parallel to a stratification of$\overline{T}/W$$\overline{T}/W=\bigcup_{\subset\ominus\Pi}(\overline{T}/W)(\Theta)$($(\overline{T}/W)(\Theta)$

the image of$T/\mathrm{A}\mathrm{n}\mathrm{n}(I(\Theta))$in$\overline{T}/W$).

The adjoint quotient defined in [12], [13] is a conjugation invariant map$x$:$Garrow\overline{T}/W$. With the help of a theory of
for any puretomapping”optimal one-parameter semisubgroups” in the partition and the map canwith the following
be extended to a conjugation invariant map :properties, basic in geometric invariant theory:$G(\Theta)$$0$$(\overline{T}/W)(\Theta)$$\subset\Pi$$\overline{G}$$\chi$$\overline{\chi}$$\bullet$$\bullet$Every fibre of$\overline{\chi}$$\overline{G}arrow\overline{T}/W$

contains a unique closed conjugacy class,are mapped to the same point in
two elements
if the closures of their conjugacy classes meet,$\psi\in\overline{G}$$\overline{T}/W$$\varphi,$$\overline{(Ad(G)\varphi)}\cap\overline{(Ad(G)\psi)}\neq\emptyset$

if and only.

these statements hold only for the
Remarks: 1) If one considers :mapping onto $T/W$ .
”classical” part
orbits in all fibres of are given as the2) The closed
pure, $t\in T$ .
orbits of the elements$\chi$$Garrow\overline{T}/W$$G(\emptyset)$$(=\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{a}1=\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}_{\mathrm{S}}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e})$$t.e(\Theta),$$\Theta\subset\Pi$$\overline{\chi}$

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References[1] W. FULTON:

Introduction to toric varieties; Ann. of Math. Studies 131, Princeton University Press, 1993
[2] V.G. KAC:

Infinite-dimensional Lie algebras; Cambridge University Press, 1990
[3] V.G. KAC, D. PETERSON:
Regular functions on certain infinite-dimensional groups; in ”Arithmetic andGeometry”, Progress in Math. 36, Birkh\"auser, Basel-Boston, 1983, 141-166[4] V.G. KAC, D. PETERSON:
Defining relations on certain infinite-dimensional groups; Ast\’erisque numero
hors s\’erie, 1985, 165-208[5] M. KASHIWARA:

The flag manifold of Kac-Moody Lie algebra; Amer. J. of Math. 111 (Supplement), 1989, 161-190[6] E.J. LOOIJENGA:
Invariant theory for generalized root systems; Inventiones math. 61 (1980),

1-32[7] E.J. LOOIJENGA:
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267-322[8] C. MOKLER:
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Letter to the author, May 1984

[10] D. PETERSON, V.G. KAC:
Infinite flag varieties and conjugacy theorems; Proc. Natl. Acad. Sei. USA
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Chevalley groups over$\mathbb{C}((t))$

and deformations of simply elliptic singularities;

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Singularit\"aten, Kac-Moody-Algebren, assoziierte Gruppen und Verallgemeinerungen; Habilitationsschrift, Universit\"at Bonn, 1984

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An adjoint quotient for certain groups attached to Kac-Moody algebras;in ”Infinite-dimensional groups with applications”, MSRI-Publ. Vol. 4,
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