remarks on the wonderful compactification of semisimple

Proc. Indian Acad. Sci. (Math. Sci.), Vol. 109, No. 3, August 1999, pp. 241-256. �9 Printed in India

Remarks on the wonderful compactification of semisimple algebraic groups

S SENTHAMARAI KANNAN

SPIC Mathematical Institute, 92 G N Chetty Road, T Nagar, Chennai 600017, India Email: kannan@ smi.ernet.in

MS received 13 November 1998

Abstract. We prove that if G is a semisimple algebraic group of adjoint type over the field of complex numbers, H is the subgroup of all fixed points of an involution a of G that is induced by an involution ~ of the simply connected covering G of G, then the wonderful compactification G/H of the homogeneous space G_[H is isomorphic to the G.I.T quotient -G~S(L)//H of the wonderful compactification G of G for a suitable choice of a line bundle L on G. We also prove a functorial property of the wonderful compactifications of semisimple algebraic groups of adjoint type.

Keywords. Involution; wonderful compactification; symmetric varieties; special dominant weight; line bundle.

1. Introduction

In [D-P], what are known as "wonderful compactifications" of symmetric varieties were constructed, and studied by De Concini and Procesi. More precisely, if G is a semisimple algebraic group of adjoint type over the field of complex numbers, H is the subgroup of all fixed points of an involution cr of G that is induced by an involution # of the simply connected covering G of G, then, they have constructed a complete embedding G/H of the homogeneous space G/H, with boundary being a union of normal crossing divisors. In particular, one gets such a compactification G for the group G (G being considered as (a x G)IA(G)).

Now, we consider the subgroup H of G as above and view G as a H variety (H acting on the right). We also have H linearised ample line bundles L, and one can therefore take the G.I.T quotients [cf [GIT]] ~ss (L) / / H of G (it is known that the connected component of the identity element in H=G" is reductive and hence the G.I.T qoutient -GSS(L)//H makes sense). This way also one obtains a compactification of G/H. A natural question is to get an explicit relationship between these compactifications and the "wonderful compactifications".

The aim of this paper is to prove the following result:

(a) There is a G-linearised ample line bundle L on G such that -GSS(L)//H is isomorphic to G/H. Continuing this line of investigation, we also obtain

(b) A natural functorial property of the wonderful compactifications G of G. [cf Theorem 4.7 for precise statement.]

The layout of this paper is as follows:

241

242 S Senthamarai Kannan

Section 2 consists of notations and basic theorems. In w 3, we prove result (a). In w 4, we prove result (b).

2. Notations and basic theorems

Through out w 2 and 3, we fix the following notations. Let G be a semisimple algebraic group of adjoint type over the field of complex numbers. Let 7r : G - - ~ G be a simply connected covering of G. Let a : G----*G be an automorphism of order two that is induced by an involution 6" of G , let T be a a stable maximal torus of G such that the dimension of the subtorus T1 = {t E T : or(t)= t -1) is maximal and let H = G ~, denote the invariants of ~r in G. Also, let H ' denote the invariants of # in G and let H denote the normaliser of 1t' in G. We note t ha t / t is actually the pullback 7r -1 (H). The group of characters of H' (resp. o f / / ) is denoted by X ( H ~) (resp. X(/-/)).

We have

Lemma 2.1. [cf[D-P] (p-4)]. One can choose the set ~+ of positive roots (with respect to T as above) in such a way that:

i f c~ E r +, then, either tr(~) = t~ or a(a) E ~ - .

Let ~+ be the set of positive roots as in Lemma 2.1, B be the corresponding Borel subgroup of G containing T. Let ~0 = {a E �9 : a ( a ) = a}, ~1 = ~ - if0.

Let F C r + denote the set of simple roots, let F0 = F M if0 and let Vl -- F N if1. We label these sets by:

F0 : {31, f ie , - . . , 3k} and Fl : {oq, O~2,.. . , a j ) .

Let (., .) denote the positive definite bilinear form on E = (Root lattice) | • induced by the Killing form of the Lie algebra of G and define (c~,3)= 2 ( a , 3 ) / ( 3 , 3)- Now, consider the fundamental weights. Since, they form a dual basis of the simple co-roots, we also divide them:

{r162 and {wl , . . . , ~ j } ,

where ~i is dual to 3i and wi is dual to r Using the Killing form, one can see that cr induces a permutation 6, of order two in the

indices { 1 , 2 , . . . , j} such that a(wi) = -w~(i). For a proof, one can see [[D-P] (pp 5--6)]. We recall from [D-P], the number I denotes the rank (G/H) and we have

l = the number of 6, orbits in { 1 , 2 , . . . , j}

= dim(T1).

Lemma 2.2. [cf (p-5) [D-P]]. For every ai E F1 we have that cr(c~i) is o f the form - (t~a(i) + ~ s ni,s3s) where hi, , 's are nonnegative integers.

DEFINITION 2.3

A dominant weight )~ is special if it is of the form J ~']~r=l nrWr with n~ = n~(~). A special weight is regular if n~ ~ 0 for all r = 1 , 2 , . . . , j.

One can check that A is special if and only if a (A)= -A. We have

Semisimple algebraic groups 243

Lemma 2.4. [cf [D-P] (p-6)~ Let A be a dominant weight and VA the corresponding irreducible representation of G with highest weight A. Then if VB~ denotes the subspace of f l invariant vectors in V~, one has dim V~ < 1, further, if V~ ~ O, then, )~ is special.

In [D-P], the Lemma (2.4) is stated in terms of the group H'. Since any/2/invariant vector in VA is also a HI invariant vector in Vx and since we will work with the group H, we have stated it in terms of f/.

We denote the weight lattice of G with respect to T by P. We denote the set of all dominant weights by P>0- We denote the lattice spanned by {c~i - cr(c~i) : i = 1 , 2 , . . . , 1} by P1- Let Psp denote the sublattice of special weights and Pspi denote the lattice spanned by the dominant weights A such that V~' ~ O. We denote the usual dominant ordering on the weight lattice P by _< and we order Psp by A < / / # if ~ - )~ is a nonnegative integral linear combination of the elements ai - a(c~i), i = 1 , 2 , . . . , I.

Let X = G/[-I = G/H denote the wonderful compactification of the symmetric variety G/[-I = G/H. Let A E Pspi. Let La denote the line bundle on X associated to A and let {Di : i E { 1 , 2 , . . . , l}} be the G stable divisors of X, l = rank(G/H).

Then, we have

Theorem 2.5. [cf [D-P] (pp 30--31)1 1. H~ La) r 0 if and only if )~ = u + ~I=1 ti(ai - cr(ai) ) for some dominant weight u and ti E 7/>__0.

2. Further, if H~ Lx) ~ O, and if Vv is the irreducible G module of highest weight u, then H ~ (X, L~) = ~ V ~ , where the sum is taken over all dominant weights u of the form t /= /~ - ~'-~I=1 ti( O~i - cr(oq)), t ie 7/>_0.

Set Ai = wi + w~i) for iE { 1 , 2 , . . . , j}. Since the number l is the number of ~ orbits in the set {1 ,2 , . . . , j}, we can index the Ai's by the set {1 ,2 , . . . ,l}.

Now, consider the group G x G with the involution

r : G • G ,G x G defined by r ig , h ) = (h,g). (2.5.1)

It is clear that the invariants of -r in G x G is the diagonal A ( G ) = { ( g , g ) : gE G}. The map

F: G ,(G x G) /A(G) defined by F ( g ) = the coset (g ,e)A(G)

gives an isomorphism of G onto (G x G)/A(G) . (2.5.2) In this situation, we take T x T as a "r stable maximal toms of G x G, where T is

a a stable maximal toms of G as in the first paragraph of w 2, B x B- to be the Borel subgroup of G • G, where B is a Borel subgroup of G containing T with r as in Lernma 2. I. Observe that if u is any dominant weight (resp. regular dominant) with respect to B, then the pair ( u , - u ) is a special dominant weight (resp. regular special dominant) with respect to the involution T of G x G and the Borel subgroup B • B- of G x G.

3. Wonderful compactification as a quotient

In this section, we prove that there is an ample line bundle L on G such that the G.I.T. quotient (-G)SS(L)//H is isomorphic to the complete symmetric variety G/H by the following three steps.

Step (1) (PROPOSITION 3.1)

If )~, #E Pspi are two dominant weights, character if and only if A - iz E P1.

then, [-t acts on Vt~ ' and V if' via the same


Step (2) (Lemma 3.2). (i) There exists a regular dominant weight # E P which can be written as

l k

# : Z ai(wi + ~v~(i)) + Z b~fls i = 1 s = l

with each ai, bs necessarily positive integers having the following properties: (ii) Set #l := Y'~I=I ai(wi + ZZT#(i)). Then for every positive integer n and for every

special weight u, t, < n# if and only if v <_ n#l. (iii) We also have/zt E Pspi N X(T) and the action of[-1 on VS~' is trivial. (iv) If t/E Pspi and the action of [-I on lift' is trivial, then v < n# if and only if

v <H n#l.

Step (3) (Theorem 3.3). Let # be a regular dominant weight as in step (2) and let L := L(~,_,). Then -GS~(L)//[-I is isomorphic to G/H.

We now prove step (1) (Proposition 3.1). We have

P1 C Pspi C Psp. (3.0.1)

Here, the first inclusion is because the divisor Oi of X =- G/H is locally defined by ai - a(ai) and the second inclusion is by Lemma 2.4.

For any character X of ['I,^E x := (G • C ) / / / denotes /2/ orbit equivalence classes of (~ x C for the action o f h E H, (g,z) E G x C, defined by h(g,z)=(gh-J,x(h)z) .

PROPOSITION 3.1

Let A,/t E Pspi be two dominant weights. Then [-I acts on V~' and V~' via the same character if and only if A - # E Pl.

Proof. 3.1.1. We know that we have a natural inclusion VspiC Vic(X). Now, we prove that there is a natural exact sequence of abelian groups

(0) ,el ,espi--+x(~r). To do this, we consider the homomorphism eH: X(/-/) ,Pic(G/fI) defined by e/~(X):----

E x. By Proposition [[3.2] (i) (pp 81-82) [KSS]], the sequence

^ ^ 71"* ^

x(&) ~-~ x(~)-~ Pic(a/H)--%" Pic(a) is exact. Since G is simply connected semisimple algebraic group, we have X(G)= (0) and Pic((~)= (0) so the above exact sequence becomes

X([-I)-7-* Pic(G/[-I). (3.1.2)

Since (~/~/= G/H C X, the isomorphism in (3.1.2) gives a homomorphism

Pic (X) ---~X (/-/) (3.1.3)

whose kernel is the lattice of the divisors supported on X - G/H. This lattice is P1 by Corollary 8.2.3 of [D-P] [cf p-29 [D-P]].


Thus, from (3.0.1), (3.1.1) and (3.1.3), we have the required exact sequence

(0) 'Pl ----~espi ~ X ( / - / ) . (3.1.4)

We now describe the map ~b : Pspi---~X(/-/). Let A E P_>0 fl espi. Then V~ r is one dimensional. We note that by Lemma 1.6 of [D-P], the lattice espi is stable under the Weyl involution i. We claim that/: /acts on V~/' via the character ~p(i(A)), where i is the Weyl involution. For this claim, without loss of generality, we may assume that A is a regular special dominant weight. By Theorem 2.5, we have V~ C H~ Li(a)). By construction of X in [D-P], the evaluation map

ev : X x V~ 'Li(A)

is surjective and G equivariant. Therefore, the map ev : [/-/] x V~---~Li(~)I[[-I] is a nonzero H' invariant linear form on V~. By reductivity of H', the restriction of this linear form to V~' is nonzero and therefore we get a / - / equivariant isomorphism between Vff' and ~(~)1[/-/]. Since for any two elements A, # E Pspi n P>0, V~' is isomorphic to Vff a s / / modules if and only if v~) is isomorphic to V/~) as k modules, by (3.1.4), the proigosition follows. []

Thus, we have proved step (1).

We now prove step (2) (Lemma 3.2). We now recall from w 2, the usual dominant ordering on the weight lattice P and denote

it by <. On the other hand, we order Psp by setting A <n # if # - A is a nonnegative integral linear combination of the elements ai - a(ai), i = 1 ,2 , . . . , / .No t i ce that the set of fundamental weights { w l , . . . , wy} and that of simple roots {3l , . . . , /3k} are mutually orthogonal and each linearly independent, together they form a basis of P | Q.

We have

Lemma 3.2. (i) There exists a regular dominant weight lz E P which can be written as

l k

# : E ai(wi + w~(i)) + E bsfls i=1 s=l

with each ai, bs necessarily positive integers having the following properties: (ii) Set #I = ~I=1 ai(wi + wo(i)). Then for every positive integer n and for every special weight v, v < nl z if and only if v < nlz 1. (iii) We also have #l E Pspi NX(T) and the action of H on V n' is trivial. (iv) [f v E Pspi and the action of [-I on Vff' is trivial, then v < n# if and only if v <_n n#l.

Proof. (i) Set Po equal to the sum of all positive roots which are in the linear span of {El,.--,~k}- Clearly (P0, fls) = 2 while (Po, ai)<_ O. Then, we choose the ai's large enough so that ai > - min{(po, ai), (po, as(i))}. The rest of (1) is immediate.

(ii) This is a computation, v < n#l implies v < n# is trivial. We now prove the other side for the case n = 1, since the arguement is similar to all positive integers n. Set #o = # - #1. Assume v is special and v < #. Write

j k v = i z - E x i a i - E y s ~ s , x i , y s E 7/>_0.

i=1 s=l


Applying a, we have

j k --lJ ~- - -"1 "~- ~ -- Z XiO'(Oli) - E ysfs"

i=1 s=l

Thus, we have

j k 2#0 = Z xi(Oli -~- O'(O~i) ) "~- 2 Z y s e s .

i=1 s=l

Hence, we have

2#0 = z.... x i , q i - oLo(i)+ - xini,s + 2 ~..~y~__.~+:~ ~o i=1 s=l \ i=1 / s=l

From this it is clear that b~ < y~ for every s. Thus, we have u < #1. (iii) Since Pspi f') X ( T ) is a sublattice of P~ of f'mite index, we can choose ai's so that

#1 C Pspi fq X ( T ) . Since H ' is a subgroup of H of finite index, we can choose a i ' s so that [-I/H' act trivially on V~'.

(iv) follows from (3) and Proposition 3.1. [ ]

We now prove step (3) (Theorem 3.3). l Let G, T, B, H, H ' and/-/be as in w 2. Let # = Y ~ I ar(wr + zv~(r)) + ~ k I bsl3~ be the

regular dominant weight (with respect to T and B) as in Lemma 3.2(1). With this #, let #1 :=Y~/r=l a r ( W r + Z~7#(r)) be the regular special dominant weight (with respect to our

fixed involution cr). Let X = G / / - / = G/H, Y = G, and let M = L m be the very ample line bundle on X defined by #1. Let L = L(#_u) be the very ample line bundle on Y defined by regular special dominant weight (#, - # ) with respect to the twisted involution ~- of G • G [cf (2.5.2)] and the Borel subgroup B • B- of G x G. Let H~ n) denote the space of global sections of the line bundle M ~. We note that Y is naturally a G x space and so the vector space H~ L ~) is a G • G module and in particular a e x / : / module. Let H~ n denote the/-/ invariant global sections of L n on Y, where the action o f /4 on H~ L ~) is through e x f/.

We note that the pull backs of the line bundles L n (resp. M n) are trivial on G (resp. c//4).

Then, we have

Theorem 3.3. The G.LT. quotient ss ^ Y~(L)//H is isomorphic to the polarised variety (X,M). More precisely, the gradefl k algebra @nez>oH~ n) is isomorphic to the graded k algebra @neZ>_oH~ L~) n.

Proof. We will obtain, for every n E Z>~ an isomorphism

(bn : H~ Mn)---~H~ Ln) ~,

of G modules. We will show that the following diagram commutes

H~ m) | H~ ") ,H~ "+")

l l L,.) k | L")k---+ H~ L"+")


Here the horizontal arrows are the natural surjective maps, the vertical map on the left is q~,~ | ~b, and the vertical map on the right is @re+n- This will clearly prove the theorem.

To define (~n, we will first get for every n, injective ma~s ~p~ and 0,1 from H~ Ln) ~ and H~ ~) respectively into the C algebra R :---- C[G] of/-/invariant regular functions on G, where the action o f / - / o n C[G] is through the right regular action o f H on G and the homomorphism 7rl/-/:/-/---~H. We will then show that the images of %bn and 0~ are equal. We can then define (~ as ~b~-~0n. That the diagram above is commutative will follow once we show that the following diagram and its analogue for On are commutative:

t-,,~ k | 1 7 6 Ln) ~ --~ ~(r ,L , .+n) ~

l l R | - - ~ R

(3.3.1)

Here, the top horizontal arrow is the natural surjective map, the bottom horizontal arrow is the multiplication in R, the left vertical map is ~bm | ~bn and the right vertical map is ~bm+~.

We will now obtain %bn. Let D be a divisor in Y defining L. Then H~ ~) is the space of rational functions on Y that have poles of order atmost n on D. Since/z lies in the root lattice, the pull back of the line bundle L := L(#, - # ) is trivial on G and so D can be chosen so that support of D lies in the complement of G in Y. (For a proof: let D1, D2, . . . ,Dt be the irreducible divisors of Y which do not meet G. Write D = ~imiDi+ ~-~i hiD'i, with prime divisors D'i's meeting G. As L is trivial on G, D n G is a principal divisor and hence there exists a rational function f on G such that ~..ini(D'i fq G) = D n G = divo(f) . Therefore, we have d i v r ( f ) = ~,ieiDi + ~_,iniD'~ for some integers el, e2 , . . . , et. Thus, D - d i r t ( f ) = ~,i(mi - ei)Di (where d ivr ( f ) denotes the principal divisor defined by the rational function f in Y) is linearly equivalent to D, and the support D - d ivr ( f ) lies in the complemenf of G in Y).

Since G is dense in Y, we have therefore identified H~ L n) as a subspace of the ring k[G] of regular functions on G. By taking H invariants, we obtain ~bn. It should be clear from our definition of ~n that the diagram 3.3.1 commutes.

We will now obtain 0~. This construction is parallel to that of ~ . Let D be a divisor in X := G/H defining M. Then H~ ~) is the space of rational functions on X that have poles of order atmost n on D. By the hypothesis of #1 as in Lemma 3.2, we see that the pull back of M is trivial on G / t / = G/H, and so D can be chosen so that the support of D lies in the complement of G/H in X. Since G/H is dense in X, we have therefore identified H~ M n) as a subspace of the C-algebra C[G/H~of regular functions on the affine variety G/H. Now C[G/H] is just the C-algebra C[G] = R of H invariant regular functions on G. Since /:/ on through the homomorphism 7rl/~/:H----,H, we have R = C[G] B. We have therefore obtained On. It should be clear from the definition of 0n that the analogue of (3.3.1) for 0~ is a commutative diagram.

It only remains to show that the images of ~bn and 0~ are equal. For this, we first write R as a multiplicity free direct sum of irreducible G modules. We will then show that the images of ~bn and @~ are equal.

It is known that C[G]-~ @v(V~ | Vv) as G x G modules, where the sum runs over all dominant weights of G (with respect to the maximal torus and Borel subgroups that we have fixed as in Lemma 2.1) that are divisible by the order of the finite group of weight lattice modulo the root lattice of G.


We now take/~/ invariants on both sides of the above equation. The left hand side becomes R. By [D-P], if V* | (V~) H ~ 0, then it is isomorphic to V~* and v is special. This is an immediate consequence of the fact that the action of H on vr | Vv induced by the right action o f / : / o n Y is the action of /? /on the right factor V~ of V~, | V~.

Since " /~ dim V~ < 1 for any v, we have

(V* | V~)/~ # 0 implies (V~ | V~)t~-~ V~ and v E Pspi. (3.3.2)

Thus R -=- ~3~V~ = ~ V ~ | (V~) ~/, where the sum runs over certain special dominant weights, and so R is a multiplicity free direct sum of certain irreducible G modules.

Now, we show that the images of 0n and ~bn are equal and this will complete the proof of the theorem. By Theorem 2.5 applying to the involution a and the symmetric variety X, we have H~ M n) =- @~_<xnm V~*, where the sum runs over all special dominant weights v <H n/z1. Therefore, by the above decomposition of R = @~V~ | (V~) H,

the image of 0n is just the/-/invariants (~) V~* | (V~) t:/ . (3.3.3) specia l ~'<_Hn#l

On the other hand, by Theorem 2.5, applying to the involution ~- of G x G, and to the symmetric variety Y = G, we also have

H~ Mn)-- - ( ~ V ~ | V~, (3.3.4) /,I

where the sum runs over all dominant weights ~ < n/z. By Lemma 3.2 (4), we have

v<ntz if and only if g<H n/~l for any dominant weight vE Pspi. (3.3.5)

Thus, from (3.3.3), (3.3.4) and (3.3.5) the image of ~bn and the image of 0n are equal. Hence the theorem. []

4. Functoriality of the wonderful compactifications

In this section, we characterise when a morphism of semisimple algebraic groups of adjoint type ~b:H ~G can be extended to a H x H equivariant morphism of the wonderful compactifications ~b ̂ : H--*G. [cf. Theorem 4.7]

We first recall some facts from [D-P]. We fix some notations which will be used in the paragraph below. Let G be a semisimple algebraic group of adjoint type, T a maximal torus of G, B a Borel subgroup of G containing T, B- the opposite Borel subgroup of G determined by T and B, U and U- be the unipotent radicals of B and B- respectively. Also, let Fn = {ill, f12, �9 �9 �9 fit} denote the set of simple roots of G with respect to B and T.

By construction in [D-P], the wonderful compactification of G is the scheme theoretic closure of G • G orbit of the line through the identity element 1 ~ of End(Vx) in the projective space P(End(Vx)) where A is an arbitrary regular dominant weight with respect to T and B. In the case of the involution -r((g, h)) = (h, g) (as in 2.6.1), we have: (T x T ) I = {(t,s) E T x T : r ( ( t , s ) ) = ( t - l , s - 1 ) } = { ( t , t - l ) : t E T} and

0) -- �89 0) - 0))1 = �89 0) - (0, 1 = [ ( o , - A ) - = ( o ,


for every i E { 1 , 2 , . . . , l}. Therefore

(t, t -1 )-2[(3"~ = t-j31 (t-1)/3i • t-2Bi (4.0.1)

for every t E T and i E { 1 , 2 , . . . , l}. Let ~1 be the set of roots that are not fixed by "r. We then have

�9 1 = { u C ~ - ( B • ~ - (u )~u}={( -~ ,O) : ~E ~+(B)}

U {(0 ,~) : ~ ~+(B)}

and therefore the unipotent subgroup of G x G generated by the root groups {Ga,. : uE ~1} is U - • U.

Here, we fix A a regular dominant weight of G with respect to B. Le tp E (End(Va))* be the linear function on End(V~) defined by p(v) = the coefficient of the highest weight vector v(~,_a) in the expression of v (in terms of T x T weight vectors) for all vE End(V• and let V= {x E G : p ( x ) ~ 0} be the open set of G defined by the nonvanishing ofp. Since p is a lowest weight vector of the G • G module (End(V~))*, the open set V of G is B- x B stable. Consider the action of T • T on the affine space A t (l denoting the rank of G) defined by

(t, s) . (XI, X2, . . . , Xl) = ((ts-1)-fllXl, (ts-1)-/32X2,.. . , (is -1 )-13'Xl).

This action will induce an action of (T x T)I on the affine space A t, namely

(t, t - l ) �9 (xl, x2 , . . . , Xl) = (t-V~lxl, t -2Nx2 , . . . , t-Z~'xt), (4.0.2)

for ( t , t - l ) E ( T x T)I and (X1,X2, . . . ,Xt) E /5~ I. By Lemma 2.2 of [[p. 11] [D-P]] and (4.0.1), we have a (T x T)I equivariant isomorphism

f : closure of (T x T)I orbit of the line through the identity

element 1 ~ of End (V~) in V--~ A t. (4.0.3)

Now, consider the action of B- • B on U- • U • A l as follows:

For (u-{ t, sul) E B - x B with u{ E U-, ul E U, t, s E T

and (u-, u ,x)E U - x U • A t

we define (u{ t, sul)(u- , u, x) = (u~ u , uu-{ 1 , (t, s)x).

With this action of B- • B on U- • U • A t and for the canonical action of B- • B on V, [by Proposition [2.3] [p. 12] [D-P]] the morphism ~b : U- • U • At-----~V defined by

~p((u-, u ,x ) )= u - f - l ( x ) u -1 [ f as in (4.0.3)] is a

B- • B equivariant isomorphism. (4.0.4)

Let {D'~ : i E {1 ,2 , . . . ,l}} be the set of all G x G divisors o f G [cf Theorem 3.1 [p. 14] [D-P]]. By Theorem 3.1 [p. 14] of [D-P], we also have

-1 (O', n V ) = U - x U • A1-1 w h e r e A I - ' = { ( x l , x2 , . . . , x t ) E A l: X i = 0} .

(4.0.5)

We also have properties (4.0.3), (4.0.4) and (4.0.5) for the wonderful compactification H of H. In the case of H, we denote the open subset of H similar to the open set V of G by


VH, (T x T)I by (TH X TH)1, A t by A IH, the H • H stable divisors of H by D1, D2 , . . . , Dr, and so on.

We use the following basic result from the paper [C] in our proof.

Theorem 4.1. [ cf [ p. 17] [C]]. A connected semisimple algebraic group G has a finite set of connected closed normal subgroups G1, G2 , . . . , Gk such that:

(a) each Gi is simple. (b) [Gi, Gj] --- 1 for i ~ j.

k ,G is an isogeny. (c) the multiplication map # : IIi=lGi

Now onwards, J will denote the set {1 ,2 , . . . ,k} where k is as in Theorem 4.1.

Note 4.2. If G is a semisimple algebraic group of adjoint type, then the multiplication map # : IIiejGi---*G is an isomorphism and any closed normal subgroup of G is isomorphic tO 1-IiEiG i (via #-1) for some appropriate subset I of J.

Proof of Note 4.2. By Theorem 4.1, # : 1-[i~jGi-----~G is an isogeny. Since G is of adjoint type, each Gi is of adjoint type, and so is the product IIi~ jGi. The kernel K = ker # being a finite normal subgroup of IIiEjGi, w e must have K ----- 1. Hence # is an isomorphism.

Let N be any closed normal subgroup of G. Then, the connected component of the identity N O is isomorphic to lI idGi for some I C J. Therefore, G/N ~ is isomorphic to ((Hi~jGi)/(Hi~lGi)--~HictGi). So, N / N ~ is isomorphic to a finite normal subgroup of 1-[iq~tG i. N o w , Hir i being semisimple of adjoint type, N / N ~ = 1.

In the paragraph below, we will make some elementary observations which will be used later.

Remark 4.3. It is known that there exist a maximal toms TH of H, a Borel subgroup BH of H containing TH and a maximal toms To of G containing TH. Let ~t~ (resp. ~o) denote the set of all roots of H (resp. G) with respect to TH (resp. To). The set ePo is the disjoint union of the following two sets: ~ o : = {/3 E ~G: i* (/3) = 0} and ~ := {/3 E ~o : i* (/3) ~ 0}. It is easy to see that the one dimensional root groups {G~,a :/3 EcI '~ generate a reductive subgroup of G, say Go, and let <b~ denote the set of all positive roots with respect to a Borel subgroup of Go containing the maximal torus (To r Go) ~ (= the connected component of identity element in To fq Go) of Go. By identifying X(TH) with Z t'~ via the basis FB,, there is a canonical total order " > " on X(Tu) induced by the lexicographic order on 7fl H.

We fix Tu, BH, To as in remark above and hence we fix Go and ~+ also as above. For o0 /3 E X(To), we denote i* (/3) > 0 to represent i* (/3) is positive in the canonical total order " > " on X(TH).

With these notations, we have

Lemma 4.4. The set of roots ~ := {/3 E ~ : i*(/3) > 0} t2 ~+0 is the set ofpositive roots of G with respect to a Borel subgroup Bo of G containing To and B~I.

Proof of Lemma. For any root a E ~+, we denote the one dimensional root space corrresponding to a by ~ and let X~ denote a nonzero element of ~ . Let L := @ ~ e . ~ , . Since L is a Lie(To) submodule of Lie(G), Lie(To) is abelian, dim(Lie(To))+ dimension of a maximal nilpotent subalgebra (namely nilpotent radical of a Borel subalgebra) is


equal to the dimension of a Borel subgroup of G and the sum L @ Lie(T6) is direct, to prove the claim, it is sufficient to prove that L is a maximal nilpotent subalgebra of Lie(G).

We will now prove some observations which will be used to prove that L is a maximal nilpotent suhalgebra of Lie(G).

(4.4.1) I f v = )-'~aer m~a, with every m~ is a nonnegative integer and at least one m~ is positive, then we must have v # 0.

Proof of observation 4.4.1. Let v = )-~aeo+ mac~ be such that each ma is a nonnegative integer and atleast one m,~ is positive. I f eve~ry c~ such that m,~ is positive belongs to 0 + o, then the sum ) - ~ ' ~ + m ~ a is nonzero, since O ~ is precisely the set o f positive roots o f Go with respect to a GBorel subgroup of Go containing T6 M Go. Otherwise, the sum ~-~er m,~a is strictly positive in the canonical total ordering " > " and hence the sum ~-~0~ m~a is nonzero. [ ]

(4.4.2) I f a and/3 are two elements of �9 + such that cz + 3 E O6, then we must have a + fl E O~. The proof of this follows immediately from (4.4.1).

(4.4.3) We have �9 + U - 0 + = OG, where - O ~ :---- {a E O 6 : - - a E 0+} .

Proof of the observation 4.4.3. I f c~ E OG, then either we have c~ E �9 ~ or we have a E �9 1. In the first case, either we have a E O ~ C O~ or we have -c~ E O~0 C O~ (since 0 + 0 is precisely the set of all positive roots of Go with respect to a Borel subgroup of Go containing TM Go). In the second case, we have i*(c~) # 0. Since " > " is a total order on X(Tn), we have either i*(a) > 0 or i * ( - c 0 = - i*(tx) > 0. Therefore, either we have a E �9 + 6 or we have -c~ E �9 +. [ ]

From the observations (4.4.1), (4.4.2) and (4.4.3), it is easy to see that L is a nilpotent Lie algebra of Lie(G) and O~ is the set o f positive roots with respect to the Borel subalgebra L @ Lie(T6) of Lie(G).

Hence the lemma. [ ]

We have

PROPOSITION 4.5

Let H be a semisimple subgroup of adjoint type of a semisimple algebraic group of adjoint type G. Then, there is a pair of triples (Tn, Bn, BH) and (TG, BG, B~) in H and G respectively such that the following hold:

(a) TH C_ TG, BH C B G and B H C_ BG, (b) the intersection of the two monoids,

7/>0 span of/*(FAG) f-I 7/> 0 span of i*(FB~ = -Fn~) is zero.

(Here, by a triple (T,B,B-), we mean a maximal toms T, a Borel subgroup B, opposite Borel subgroup B- determined by T and B and i* denotes the restriction map X(TG) X(Tn), where X(TG) and X(TH) denoting the character groups Of TG and TH respectively


and Fna and ~ denoting the sets of simple roots of G with respect to Ba and B G respectively.)

Proof of Proposition. Let TH, B~t be as in Remark 4.3. Let B6 be the Borel subgroup of G corresponding to our ~ as in Lemma 4.4.

We now prove that Lie(BH) is contained in Lie(Bt). For, let c~ be a positive root of G with respect to BH and TH. Write X~ E Lie(BH)C Lie(G) as a linear combination ~ a c ~ caXa, with c~c C for each t i c ~c. For any/3E ~c such that c a is nonzero, we must have i* (fl) = c~ and hence/3 E ~b+.

Therefore, BH is contained in BG, The proof of B~ containing B~ is similar to the proof of Bt~ C BG. Thus, we have proved (a).

The proof of (b) follows from the fact that a nonzero element of X(TH) cannot be both positive and negative in the lexicographic ordering.

Hence the proposition. []

We will now make a remark which will be used in Theorem 4.7.

Remark 4.6. If H = H' x K, where H', K, H are semisimple algebraic groups of adjoint type, then, there is a canonical isomorphism of the wonderful compactification H onto the product of the wonderful compactifications H ~ • K of H t and K.

Proof. There is a natural Segre map H~x K----+H'x K = H induced by End(V;~)• End(V,)----~End(Va+,) = End(Vx) | End(V,), where A, # are regular dominant weights for H t and K respectively. It is easy to see that the Segre map is a H • H equivariant isomorphism, and we identify H with H' x K through this isomorphism. We denote by the map H'---+H given by x ~ (x, e), where e is the identity element of K. []

Now, for a homomorphism q~ : H ,G of semisimple algebraic groups of adjoint type, we define a natural morphism ~ : ~b(H)----~H which will be used in the statement of Theorem 4.7.

To do this, let ~ : H ~G be a homomorphism of semisimple algebraic groups of adjoint type, let K be the kernel of the homomorphism and let ~(H) denote the image of H in G. Since the homomorphism ~ : H ,if(H) is surjective, by Note 4.2, H must be isomorphic to ~b(H) • K. Thus by Remark 4.4 applying to the situation/'t ' = ~b(H), we have a natural map dp(H)-----~H = O(H) x K given by x~--~(x,e), where e is the identity element of K. We denote this map by i.

We now prove the main theorem. Now, let ~ : H----~G be a homomorphism of semisimple algebraic groups of adjoint

type. Consider the natural map ~p(H)--~H = q)(H) • K sending x to (x, e), where e is the identity element of K, the kernel of the homomorphism ~b. We denote this natural map (b(H)---.H = cb(H) x K sending x to (x, e) by i and we use in the statement of the following theorem.

Theorem 4.7. For a morphism ep : H----*G of semisimple algebraic groups of adjoint type, the following statements are equivalent:

(1) ~b can be extended to a H x H equivariant morphism ~ : H---~-G with the property that the composition ~ o i : (b(H)---~G (where ~i is as above) is an isomorphism onto the scheme theoretic closure of ~p(H) in -G.


(2) There are two pairs (TH, Btt), (To, Bo) in H and G respectively such that ck(Tn) C_ TG, ck(Bt~)C Bo and ck*(FB~)C 7/>_0 span of FB,. (Here, by a tuple (T,B), we mean a maximal torus T and a Borel subgroup B containing T, Fan (resp. FBG) denotes the set of simple roots of H (resp. G) with respect to TH and BH (resp. To and Bo).)

Proof. We will first reduce to the case when ~ = i is an inclusion. For this, we first note that if ~b is a surjection, then both the statements (1) and (2) hold: statement (1) can be seen to be true from Remark 4.6, and statement (2) holds obviously. For a general ~b, we can break it up as i o 7r where 7r is a surjection and i is an inclusion. It is easy to see that statement (1) (respectively (2)) for i is equivalent to statement (1) (respectively (2)) for ~b (because both statements (1) and (2) are true for 7r as we have seen above). We are therefore justified in assuming from now on that H is a subgroup of G and ~b is the inclusion.

(1)==~(2): Let 0 : H-7-~ Y be a H x H equivariant isomorphism such that the restriction of ~ to

the group H is the inclusion ~ : H~-*G. We start by making some observations. Fix triples (Tn,Bn, B~) and (T6,Bo, BG)

satisfying Tn C_ To, BH C_ Bo and B~ C_ B~ (such triples exist by Proposition 4.5(a)). Let U and U- be the unipotent radicals of Bo and BG respectively. Let Un and U~ be unipotent radicals of BH and B~ respectively. Let Y be the scheme theoretic closure of H in G. Let A be a regular dominant weight with respect to the Borel subgroup Bo of G. Let V be the B~ x Bo stable open subset of G as in the beginning of this section. Set Vr := V fq Y. Since V is a B- x B stable open subset of G, B~ x BH is contained in B~ x Ba, Y is closed in G and the isomorphism ~b of (4.0.4) is B~ x BH equivariant (in fact B- x B equivariant) ~b-l(Y N V) is B~ x BH stable closed subset of U- x U • A t. Consider the map U~x UHx(TH x TH)I-----*U-• l~ t defined by (u-,u,(t,t-1))~-* (u-, u, (t-23~,.. . , t-23~)). Since Y N V is an open subset of Y, we have

dim(Y N V) = dim(Y) = dim(U~ x Un x (Tn x Tn)l).

(4.3.1) Therefore, the U~ x [In x (Tn x Tn)~ orbit of the point (1, 1), (1, 1 , . . . , 1)) in ~b-l(Y N V) is open and dense in ~p-1(Vr = Y N V). Since Uff (resp. Un) is closed in U- (resp. in U), the closure of Uff x Un x (Tn x Tn)I orbit of ((1, 1), (1, 1 , . . . , 1)) in U- x U • A t is U~ • Un x Z, where Z is the closure of the (Tn x TH)I orbit of (1, 1 , . . . , 1) in A t. Therefore, by (4.3.1), we have ~b-l(Vr = Y N V ) = U~ x [In x Z; note that Unx [In x Z is closed in U- • U x A t since Uff • [In is closed in U- x U and Z is closed in A t.

By hypothesis, Y is isomorphic to H and therefore smooth. So, Vr is smooth and so also Z (being a factor of Vr). Being a smooth toric (Tn x Tn)l variety, Z is isomorphic to the product of an affine space and a toms (cf Proposition 2.1 [p. 29] [Fu]). We will show below the following:

(a) Z is in fact an affine space. (b) The image of 1 x 1 • ~ t , under q~ o ~bn equals the image of 1 x 1 x Z under ~b (as

subsets of Y).

We then prove (1)==~(2) by using (a) and (b). We now prove (a). Consider the weights of (TH x TH)I module. Since k[Z] is a

k-algebra, this set of weights is a monoid. Since the restrictions of the co-ordinate functions xl,x2,. . . ,xt of A t (as in (4.0.2)) generate the k-algebra k[Z], the above


mentioned monoid is generated by the (Tn x Tn)l weights of the restrictions of the functions xl ,x2, . . . ,Xl and hence the monoid is generated by {~b*(2/3k) : 3~ ~ Fo~}. By Lemma 4.3(b), there is no nonzero weight X such that both X and -X belong to this monoid. Thus, the units of the k-algebra are constants and hence the toms part of Z is empty, which proves (a).

We now prove (b). Let z~, z2, �9 �9 �9 zt,, be In co-ordinate functions of the affine space Z which are (TH x TH)I

weight vectors (this exist since Z is a (Tn x Tn) toric variety that is an affine space). Let z0 denote the unique point of Z where all the zi's vanish. It is clear that z0 is a Tn x TH fixed point of Z. Since the divisors defined by the zi's are In distinct TH x TH stable irreducible divisors of 1 x 1 x Z, each intersection ~k(Di) M ~b((1, 1) x Z) is nonempty and these divisors/9i f3 ~b((1, 1) x Z) of (7((1, 1) • Z) are just the divisors defined by the zi's.

(4.5.1) Therefore, the TH x TH fixed point ~b((1, 1,z0)) must lie in the intersection nl" l~(Oi).

Now, consider the isomorphism ~n of (4.0.4) and the point (1, 1,0) of U~ x UH• A l. (in the case of H). Since ~n ((1, 1,0)) is a Tn • TH fixed point of nl~ IDi = (H x H ) / (BH x B~), we have

~PH((1, 1 ,0))= (W,W')(BH x B•)

W ~ for some w, E WH. Since the isotropy of ~bH((1, 1,0)) in U~ x Un is identity, we have wBnw -1 N U~ = (1) and w'B;l(W')-lM Un = (1). Therefore the set of roots {a E ~+(Bn) : w(a) E ff-(Bn)} is empty and hence w is the identity element of WH and wBnw -1 = BH.

t t - - 1 Similarly, w' is the identity element of WH an__d w B~l(W ) = B~. Hence the isolxopy of ~bH((1, 1,0)) in H x H is Bn • B;I. Since q~: H---oY is a H • H equivariant isomorphism

^ A - I

and ~b((1, 1) • Z) contains a T n x T n fixed point of fql"lcb(Di)~-~ MtiX=l Di = (H x H) / (Bn • B~) (by (4.5.1)), namely ~b((1, 1, z0)), by the above argument (since the isotropy of ~l 1, 1, z0) in Uff x UH is trivial), the isotropy of z~((1, 1, z0)) in H • H is Bn x B~I. Since Mi"__lDi = (H x H)/(BH x B~) is the unique closed H x H orbit of H, ~bn((1, 1,0)) is the unique point of H whose isotropy in H x H is Bn • B~. Since ~ : ~_Z_,y is a H x H equivariant isomorphism, ~b((1, 1,z0)) is the unique point of Y whose isotropy in H x H is BH x B~ and ~(~bn((1, 1,0))) = ~b((1, 1,Z0)). Therefore, ~(~bH((1, 1) • Atn)M ~b((1, 1) xZ)) is a smooth affine (TH • Tn)l toric variety containing the unique (Tn x TH) 1 fixed point ~((1, 1,z0)) of ~b((1, 1) x Z). Hence, [by Proposition 2.1 [p. 29 [Fu]], ~bl~n ((1, 1) • Aln):~bn((1, 1) X Ah,) ~ ,~b((1, 1) xZ) is a (TH x TH)I equivariant isomorphism.

We now complete the proof of (1)==V(2). Let xl, x2 , . . . , xl be the coordinate functions of the affine space A t as in (4.0.2) and let

Yl, Y2,.. . , Yln (in the case of H ) be the corresponding coordinate functions of the affine space At,. Then, we have

(t,t-1)xi= t2j3ixi for all i E {1,2 , . . . ,l}, t E T and

(t,t-1)yi = t2~'yi for all i E {1,2 , . . . ,ln}, t C TH.

Since each xi is a (T x T)I weight vector, each xi is also a (Tn • TH)I weight vector and hence (~ o ~H)*(xi l~3((1, 1) x Z)) is also a (Tn x TH)l weight vector of the the co-

. 1--lrl H mi,r ordinate ring of A IH and therefore (~ o ~bn) (xil~b((1, 1) • Z)) is a monomial •r=l yr , with each mi, r E ~->_0. Hence, for every t E Tn, we have

Semisimple algebraic groups

t 2~' (~9 o ~tt)*(xil~b((1, 1) • Z))----- (t, t - l ) . ($ o CH)*(Xi[r l) • Z))

= (t,t-1) ymi,r ~_ t~f~rymri,r r=l

255

In = o • z))

r=l

and therefore 2~iJTn = 2(F_,,l~l mi,rc~r). Thus, we have

/n diIT. = ~_'ni,r,~, ~ Z>__O span of FB..

r=l

Therefore, we have ~b*(FsG) C_ Z>0 span of FB,, ( 2 ) ~ ( 1 ) : Let (Tn,Bn) and (To, Bo) be a pair of tuples in H and G respectively such that

TH C To, BH C Bo and i*(FBG) C_ ~7>0 span of FB,. We first recall some facts from [D-P]. By construction in [D-P], the wonderful

compactification G is the scheme theoretic closure of G • G orbit of the line through the identity element 1~ of End(V~) in P(End(V~)), where >, is a regular dominant weight of G with respect to To and Bo. For our convenience, we choose ), so that )~ is a character of TG.

We will prove the following:

(c) End(Va) is a direct sum of End(V,) and a H • H module W, where 3 is the restriction of )~ to Tn.

(d) The identity element lx of End(V~) is a sum of two vectors v and w with v a nonzero vector in End(V6) and w is a linear combination (Tn • Tn)l weight vectors of weight ( 6 - ~,il"=l miai, - ( 6 - ~il"=l miai) ), where ai's are simple roots of H with respect to Tn, BH above and mi's are nonnegative integers.

We will then prove ( 2 ) ~ ( 1 ) by using (c), (d) and Lemma 4.1 of [D-P] [cfpp [16-17] [D-P]].

We now prove (c): Since ~b*(FsG) c_ Z_>0 span of Fs , , we must have B~ C_ B~ and hence Bn x B~ C_

B6 x B~. Therefore, the smallest H x H submodule of End(V~) containing the highest weight vector vx | v_~ is the H x H irreducible submodule of End(Vx) of highest weight (6, -6 ) , namely End(V6). By, complete reducibility of the H x H module End(V• there is a H x H submodule W of End(V~) such that End(V~) = End(V,) @ W.

Thus (c) is proved. We now prove (d): Since the coefficient of the highest weight vector vx | v_~ in the expression of 1~, the

identity element of End(V~) (in terms of To x To weight vectors) is nonzero, 1~ must be a sum of a nonzero vector v in End(V~) and a vector w in W. Since 1~ is a A(H) (in fact A(G) invariant) invariant vector both v and w are also A(H) invariant vectors. Therefore, w is a linear combination of (Tn x Tn)l weight vectors of (Tn • Tn)l weights of the form (v, - v ) , where v is Tn weight. Since w (being an element of End(V• is a linear combination of TG X T6 weight vectors of the form ( X - X"tG n i f l i , - ( )~- tG , E i = I ?li/~i) ) ' E..~i= 1 where hi, n'i's are nonnegative integers and/3i's are simple roots of G with respect to BG, any such TH weight v as above is the restriction of TG weight of the form A - ~'~.i nifli,


where ni's are nonnegative integers. But, by hypothesis, the restrictions ~b*(3i)'s must lie in Z>0 span of FB,,. Therefore, any such Tt4 weight v as above must be of the form

- - IH - ~i=, miai. Thus, we have proved (d). We now complete the proof of ( 2 ) ~ ( 1 ) : By Lemma 4.1 [pp [16--17] [D-P]], the canonical rational map from the projective

space P(End(V~)) into the projective space P(End(V6)) (which exists by (c)) is defined on the scheme theoretic closure Y of the H x H orbit of the homogeneous point of flZ(End(V~)) defined by the vector la -- v + w and is an isomorphism onto the scheme theoretic closure of the H • H orbit of the homogeneous point defined by the vector v in P(End(V~)), which is actually the wonderful compactification H of H.

Thus, we have proved the theorem. []

Remark 4.8. (1) Let H denote a semisimple algebraic group of adjoint type over C that is not isomorphic to PGL2. Consider the adjoint representation of H; 4) : H~--~G :---- PGL(Lie(H)). For this q~, the statement (2) of Theorem 4.7 is not satisfied.

(2) Let H denote the adjoint group associated to the symplectic group Sp(2n) over C. For the natural inclusion ~b : H~-+G :----- PGL2n, the statement (2) of Theorem 4.7 is satisfied.

Acknowledgements

We thank Prof. C S Seshadri for suggesting this problem, his encouragement and his helpful discussions. We thank Prof. C Procesi, V Balaji, K N Raghavan, S P Inamdar, P A Vishwanath, and P Sankaran for useful discussions. We thank Prof. M Brion for reading the preliminary version of this paper carefully and giving helpful comments and suggestions and we also thank him for pointing out that a result similar to Theorem 3.3 has been obtained by Nicolas Ressayre, a student of his. We thank Prof. C De Concini for reading the preliminary version of this paper carefully and we also thank him for suggesting Proposition 3.1 and its proof which plays a crucial role in Theorem 3.3.

References

tB]

[C] [D-P]

[Fu]

[Grr]

[Har] [Hum]

[K]

[KSS]

Borel A, Linear representations of semi-simple algebraic groups, AMS, Proc. Symp. Pure Math. 29 (1974) 421--440 (American Mathematical Society, Providence) (1975) Carter RW, Finite Groups of Lie Type (New York: John Wiley, 1993) De Concini C and Procesi C, Complete symmetric varieties, 1--43, Invariant Theory, Lect. Notes Math. 996 (Springer-Verlag) Fulton W, Introduction to Toric Variety, Ann. Math. Stud. No. 131, Princeton University Press (1993) Mumford D, Fogarty J and Kirwan F, Geometric invariant theory, Third Edition (Springer- Verlag) (1994) Hartshome R, Algebraic Geometry (Springer-Verlag) (1977) Humphreys J E, Introduction to Lie Algebras and Representation Theory (Springer-Verlag) (1972) Knop F, The Luna-Vust theory of spherical embeddings, Proc. of the Hyderabad Conference on Algebraic Groups (1991) 225-249 Kraft H, Slodowy P and Springer T A, Algebraic Transformation Groups and Invariant Theory (Birkhauser, Verlag) (1989)

remarks on the wonderful compactification of semisimple

Documents