О пределах алгебраических подгрупп Текст научной статьи по специальности «Математика»

On limits of algebraic subgroups

E.B. Vinberg

We shall consider complex algebraic varieties and groups. Both analytic and Zariski topology will be used. Unless stated otherwise, the analytic topology is meant.

For elements gi,...,gp of an algebraic group G , we denote by (gx,. ..,gp) the Zariski closure of the subgroup generated by gu .. ,,gp . If (gi,.. .,gp) = G , we say that G is Zariski generated by gx,..., gp . Any algebraic group is Zariski generated by finitely many elements. In particular, any connected reductive group is Zariski generated by two elements [Vi]. Closeness of algebraic subgroups can be evaluated by the closeness of their Zariski generating sets.

As usually, we denote the tangent Lie algebras of Lie groups G,H,... by the corresponding gothic letters 0, fj,...

We set Gp = G x ■■■ x G .

'-----V-----'

p

Let H be an algebraic subgroup of an algebraic group G , and gi,g2, ■ ■ ■ G G . Suppose that there exists a limit

[ = lim Ad(0„)f)

in the relevant Grassmanian, and set

L = \\mgnHg~l = {limgnhng~l : hx, h2, ■ ■ • G H}.

(Here hi,h2,... are supposed to be-chosen in such a way that lim gnhng~l should exist.) Obviously, L is a subgroup of G .

Theorem 1. L is an algebraic subgroup with tangent algebra [ .

Theorem 2. If H is reductive, then any reductive algebraic subgroup S C L is conjugate to a

subgroup of H .

Page and Richardson [PR] proved the following stability property of semisimple subalgebras of Lie algebras:

Let s be a semisimple subalgebra of a Lie algebra i) , and [)' a sufficiently small deformation of f) .

Then there exists a subalgebra s' C t)1 which is isomorphic to s and close to s (as a subspace).

A simple proof of this was given by Neretin [Ne, Lemma in Section 1.4].

Since close semisimple subalgebras of a Lie algebra are conjugate, the above stability-proper ty implies the following theorem:

(*) Let s be a semisimple subalgebra of a Lie algebra Q . Then any subalgebra of g , containing a subspace of dimension dims sufficiently closc to 5 , contains a subalgebra conjugate to s and close to 5 .

Making use of this theorem, we prove the following version of it for algebraic groups.

Вестник ТГУ, т.2. вьіп.4. 1997

Theorem 3. Let S be a connected semis,mple subgroup of an algebraic group G , and s = („ ' « )

a Zanskigen eratmgset of S. For any neighbourhood U of e ,n G there easts a neighbourhood y

G reductive subgroup HcG satisfying the condition wnvte conta „„

a subgroup gSg~ with geU . contains

It seems that the assumption o„ reduct,v,ty of H in Theorems 2 and 3 is superfluous, but I cannot avoid it.

This „„,k was supported by the grant 95-01-00783a of the Russian Foundation of Fundamental Re search and the grant RM t-206 of the Civilian Research and Development Foundation. '

1. A property of the exponential mapping. Let G c GL„(C) be an algebraic linear group. Proposition 1. There exist, a positive number c = cc ( » sue/, that the exponential mapping

exp : 0 -c G

maps diffeomorphically the (open, set U(„, c) of elements of g , „hose eigenvalues Л satisfy the condition |/„A| < c , onto the (open) set U{G, c) of element, of G , „hose eigenvalues A satisfy the

condition | arg A| < c . ~

Proof. The assertion is known for GLn(C) , with c = ,r [MN], It follows that for any positive c < *■

the exponential mapping maps diffeomorphically U(5,c) onto an open subset of U(G, c) . We are'to prove that, for some c ,

* Є b (gln (6), c) fc exp * ЄС7 implies * Є g. ^

Let * = *5 + *„ be the additive Jordan decomposition of * . Then exp* = exp* • exp* is the multiplicative Jordan decomposition of exp* . If * Є U(3ln(C),c) , then 6,6, Є U(B!n(C),c), and if exp, Є , 11ЄП exp6, exp*„ є G . So it suffices to prove (1) for semisimple and nilpotent elements.

If £ is nilpotent and exp* £ G , then exp/* Є G for any t Є C and hence * Є 0 .

Let. now * be semismiple. If the element exp* belongs to the connected component of (7 containing t .e unit, then ,t belongs to a maximal torus I of G . In a basis consisting of weight vectors, I is udincd b) equations of the form

П Ai'J = 1 (г = 1. . . ., m; ntJ E Z)

J

ІП U,R diag°naI entrlRS Values) ’s- tangent algebra t of T is defined by the equations

X>;A=0 (i=l,...,m).

Obviously, * is diagonal in the same basis. Let ------------ A„ be its eigenvalues. Then

Ц(еА>)"-> = i (i=

whence

(3)

If * £ U(Qln(C), c) with

2tt

(4)

then (3) implies (2), i.e., * £ 0 .

If the element exp* belongs to another connected component of G , say, G\ , then it belongs to a maximal toric subvariety S of Gi (see [Vi]), which is still diagonal in some basis and is defined by equations of the form

2. A criterion for algebraicity of a complex Lie group. Let G be a complex Lie group, and Go its connected component containing the unit.

It is known that, for an algebraic group G ,

(A) if g = gsgu is the (multiplicative) Jordan decomposition of an element g £ G , then gs £ G, gn £

Let us call a semisimple linear operator compact, if its eigenvalues are of modulus 1, and positive, if

where gc (resp. gp ) is a compact (resp. positive) semisimple linear operator and gcgp = gpgc . Let us call (7) the polar decomposition of g .

It is easy to see that, for an algebraic group G ,

(B) if g = gcgP is the polar decomposition of a semisimple element g £ G , then gc £ G, gP £ Go ■

It is also easy to see that, if p : G —► H is a homomorphism of algebraic linear groups and g — grgp is the polar decomposition of an element g £ G , then ip(g) = <p(gc)ip(gp) is the polar decomposition of the element <^(<7) £ H . The analogous property of the Jordan decomposition is well-known.

j

where ’s are some roots of 1 , not all of them being equal to 1 . Obviously, * is diagonal in the same basis. Let Ai, . . ., A„ be its eigenvalues. Then

J

Let pk be a primitive q -th root of 1 , where q > 1 . If * £ U(gln(C),c) with

27T

(6)

then (5) cannot be satisfied, which is a contradiction.

Thus, if c satisfies the inequalities (4) or (6) for all connected components of G , the implication (1) holds. □

G 0 ■

they are positive. Any semisimple linear operator g can be uniquely represented in the form

9 - 9c9P,

(")

Proposition 2. A complex linear Lie group G is algebraic if and only if (A) and (B) hold.

Proof. Let (A) and (B) hold, and let G be the Zariski closure of G .

It is known (see, e.g., [VO]) that the group (Go, Go) is algebraic. Obviously, it is normal in G and hcnce in G . Passing to the quotient G/(Go,Go) we may assume that Go is abelian.

Let Go be the Zariski closure of Go . We have Go = T x U , where T is an algebraic torus and U an abelian unipotent group. In view of (A) the subgroup Go C G'o is the direct product of its projections to T and U . Since any connected Lie subgroup of U is algebraic, we have Go = T' x U , where T' is a connected Lie subgroup of T .

Let Tc (resp. Tp ) be the real Lie subgroup of T consisting of the elements whose eigenvalues are of modulus 1 (resp. positive). Then T = Tc x Tp and, for the tangent algebras, we have t„ = itc , so T is the complcx hull of the compact torus Tc . In view of (B) T' is the complex hull of a compact subtorus Tc C Tc . Since the complex hull of a compact torus is an algebraic torus, we have T' = T , so G0=TxU = G0 .

Passing to the quotient G/Go , we may assume that Go = {e} , i.e. G is discrete. In this case it follows from (A) and (B) that G is periodic. By a theorem of I.Schur (see, e.g., [CR]). any periodic

subgroup of GLn(C) is conjugate to a subgroup of Un . If, in addition, it is discrete, it is finite. So

under our assumption G is finite and hence algebraic. □

3. Proof of Theorem 1. For any r) £ ( there exist rji, t/v. ■ ■ • £ h such that

lim Ad(gn)r]n = rj

and hence

li m (exp rj7t )f7n 1 = exp ij.

It follows tlini

exp ( C L. (8)

Let c = c/y be chosen as in Proposition 1. Take any h £ U(L,c) . Let hi.h2

litr; gnhn9nl = h-

We may assume that hn £ U(H,c) for any n . Then hn = exp rjn and hence

gnhng~l - exp Ad(gn)rjn for some T]n £ U(f), c) . In view of (9) we must have

lim Ad(gn)rjn = 77 £ [, exp rj= h.

Thus,

U(L, c) C exp I. (10)

It follows from (8) and (10) that L is a (complex) Lie group with tangent algebra I . To prove that

it is algebraic, we are to check that (A) and (B) hold for L .

£ II be such that

(9)

Let h G L be defined by (9). Then

lim</n(/tn)j<7n — hs> hm <7n(/i„)u<7n — /iu,

so hs, hu £ L . Moreover,

limgn{hnfug~l = h[e L

for any i G C , whence hu € Lq .

The property (B) is checked in the same manner. □

4. Some invariant theory. Let us recall that, for an action of an algebraic group on an algebraic

variety, any orbit is Zariski open in its Zariski closure (see, e.g., [VO]). It follows that the Zariski closure of an orbit coincides with its closure in the analytic topology. In particular, an orbit is Zariski closed if

and only if it is closed in the analytic topology.

Let now a reductive algebraic group G act on an affine algebraic variety X . The algebra of polynomials on X is denoted by C[Ar] , and the subalgebra of G -invariant polynomials by C[X]G . The categorical quotient of X with respect to the action of G , i.e. the spectrum of C[X]G , is denoted by X//G , and the factorization morphism X —► X//G defined by the embedding C[X]G C C[X] is denoted by ttq . A standard fact of invariant theory is that each fiber of no contains exactly one Zariski closed orbit. (For details see, e.g., [VP].)

Consider the action of a reductive group G on Gp by simultaneous conjugations. Denote by 7rG the factorization morphism

■KG : Gp - Gp//G.

It is known [Ri] that the orbit of a p -tuple g = (g^,.. .. gp) G Gp is closed if and only if the subgroup

(g) — (ffi i ■ ■ ■ i Up) Zariski generated by g is reductive.

For any reductive subgroup H C G the embedding Hp C Gp gives rise to a morphism

Hp//II - Gpl/G.

The main result of [Vi] is that this morphism is finite. In particular, its image 7rc(//p) is closed in

Gp!/G .

5. Proof of Theorem 2. First, reduce the proof to the case when G is reductive. Let G be a maximal reductive subgroup of G containing H , and V the unipot.ent, radical of G . We have

G = UG (a semidirect product).

Let

gn - un9n (un G U, gn G G).

Passing to a subsequence, we may assume that there exists a limit

h = lim Ad(<jFn)f)

and thereby a limit

Li = lim gnH g~l.

Let L be the projection of L to G . Obviously, L C L\ .

now 3 be The projecdon^T^TTTT^----------

'" ^>‘^are conjugate in as andlhemo “^“d S 'Active subgroup*

more m G . We have

so if the theorem holds for reductive groups S7 i, , • *

« conjugate to a subgroup of H in G . ’ * ‘° * °f H in G and> hence, 5

Suppose now that G is reductive. Let s = {si, , > g „ b 7 . , .

exist p —tuples hn G HP such that generating set of S . There

s = lim^h^-1.

Applying itg gives

^(s) = lim 5rG(h„).

Since 7rG(Hp) is closed in Gpl/G tho j-

. ^ III* (see the preceding section), we get

*g(s) 6 TTG(HP),

I.e. there exists a p-tuple h G Hp such that

7TG(s) =: 7TG(h).

The fiber of 7Tff containing h contains , j r, , .

this orbit (lying in the same !C -orbit. Replacing h with a representative of

•l-nbgroup «C1 Zarist, herald ^ r; r ‘‘h'

Since any fiber of IO co„tams on|y om c|Mai r‘ “i‘'K ‘ lhe C'“b'1 of !> » closed.

“ th'! “groups Zariski generated by » and h ° °*S “h' G_“b"s of s “J 1* coincide.

».■ C°"1UmU- ThUS’ 5 <• “".>-S«e to a subgroup of

6. -<ariski dense subgroups Thp fnli ■ i-

1 ",S “Ill,ar-" r'“11 15 «**•< f< •!,. P™f of Theorem .,

Proposition 3. An, Zmsl„ demr c°nt„„ „ finitely ,enmUd Zar„t, „ J„/ °f ‘ <■'?*«„ ,ro,r G

Proof. Let us first prove that r •

l^1 ^ SUbgrOUP' 161

Then the connected component G10 of G doP t u * maximal dimension.

Mows tha, Gl0 ,s,„„rmaiSnbg“ of G p fT'radd'"St° ^ '>™i« of F , It

*<- = H • This means tha, a„ col!bh H'"8 °, ™d“'° ^ ^ «.n*

impossible, unless G = je) , ’ ° “bgroup of r is finite, which is obviously

Let now r be countably generated', and let r, r T h « .

closure rs = G, has the maximal dimension As ab ‘ T “bSr°“P Wh“e Za'isti

pns that genetated - T -r the proof to the case C„ = {e) .

0 = M ■ is hn.te. In th,s case we are to prove that

We may assume G r GL (C\ a . jan integer m (depending on ^ (**’ **- «»» ««*

subgroup of index < m , or, equivalently admits a hoi” °f G£n(C) co"talns an »bel™ normal

°'*r <m. y’ adm,'S 4 homomorphism with an abelian kernel to a group of

Under our assumption we have Г = (J~ l Г, , where

riCTjC...

are finite groups. Passing to a subsequence, we may assume that each Г,- admits a homomorphism pi with an abelian kernel to one and the same group Д of order < m . Again passing to a subsequence, we may assume that any 7 G Г has one and the same image in Д under all p,’s for sufficiently large i . Denote this image by p{7) . In such a way we obtain a homomorphism p : Г —> Д , whose kernel N is obviously abelian. Since Г is Zariski dense in G , the Zariski closure of N is an abelian normal subgroup of G . Hence N is finite, which is impossible, unless G = {e} . □

7. Proof of Theorem 3. Suppose the conclusion of the theorem is false. Then for some neighbourhood U of e in G there exist reductive subgroups Hi, H2, ■ ■ С G and p-tuples hb h2,... (h„ G H?) such that

s = lim hn, (11)

but for any n and g G U

Hn}gSg-1. (12)

We are going to show that one may assume all Hn’s to be conjugate to one and the same connected semisimple subgroup, and then to apply the theorem of Page and Richardson cited in the introduction.

Any reductive group H is a product of its connected component #0 and some finite group (see, e.g-, [Vi]). In view of this the Jordan theorem (see the preceding section) implies the existence of an

integer m (depending on G ) such that for any reductive subgroup Я С G the group H/H0 admits

a homomorphism with an abelian kernel to a group of order < m .

Passing to a subsequence, we may assume that for each n the group Hn/Hn0 admits a homomorphism with an abelian kernel to one and the same group Д of order < m . Denote by фп the composition of this homomorphism and the canonical homomorphism Hn >• Hn/Hn0 . Again, passing to a subsequence, we may assume that фп{h„) is one and the same p-tuple [8г,..., Sp) G Др .

Let F be a free group on p generators and ф : F —► Д the homomorphism taking the i-th generator to Si . Its kernel is a (normal) subgroup of finite index in F and hence finitely generated. Let Wi,...,w4 be some generators of it. These are some words in p letters.

The subgroup generated by the elements tui(s),. ..,wq(s) G S has finite index in the subgroup Г generated by si, . . ., sp and hence is Zariski dense in 5 . At the same time the subgroup generated by iui(h„), . . ., wq(hn) is contained in the kernel of фп . Replacing s with the 5-tuple (u^i(s), . . ., wq(s)) and hn with the q -tuple (ил(Ь„),. .., w,(hn)) , we reduce the proof to the case when the group

Hn/Hno is abelian for any n .

Assuming this and coming back to the former notation, consider the commutator subgroup Г' of Г . Since Г is Zariski dense in 5 , Г' is Zariski dense in S' = S . By Proposition 3 Г contains a finitely generated subgroup Гх , which is still Zariski dense in 5 . Let F be a free group on p generators and wu...,wqe F' some words such that iu^s), . .., iu,(s) generate Гх . Note that under our assumptions адЦЬп),..., w?(h„) G #n0 ■ Replacing s with the p-tuple (wi(s),...,№,(s)) and hn with the p-tuple (u>i(h„), . . ■, wq{h„)) , we reduce the proof to the case when the group Hn is connected for any

71 .

Repeating this trick, we reduce the proof to the case when Hn is connected and semisimple for any n . "

Since there are only finitely many conjugacy classes of connected semisimple subgroups in G , we may assume that each subgroup Hn is conjugate to one and the same connected semisimple subgroup H C G . Furthermore, we may assume that there exists a limit

[ lim f)n

and thereby a limit

L — In;; Iin

in the sense of this paper (see the introduction). By Theorem 1 I is an algebraic subgroup with tangent algebra [ .

It follows from (11) that L D S . Hence IDs, and, for sufficiently large n , \jn contains a subspace of dimension dim s arbitrarily close to s . By the theorem (*) stated in the introduction this implies that, for sufficiently large n, f,„ contains a subalgebra Ad(g)s (and, hence, Hn contains the subgroup g S g 1 ) with g £ U , which contradicts (12).

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lCRi PubUl962)iS’ L Reiner’ L'neaT representati0ns °! fin,ie gTOU?s and “‘’ociative algebras, New York: Interscience ^ T NOI1°’ Gn tke hgaTlthm,C functlons of matrices, I., J. Sci. Hiroshima Univ. Ser. A 14 (1950),

.V; Vu.A. Neretin, An estimate of the number of parameters defining an n-dimensional algebra, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 306-318 (Russian); English transl.: Math. USSR Izvestiya 30 (19S8), 283-291.

!lj StabU subal^ebTas °S Lie algebras and associative algebras, Trans. Aincr. Math. Soc. 127

Richardson- Conjugacy classes of n-tuples in Lie algebras and algebraic groups, Duke Math. J. 57 (1988),

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' " onq'MnierS'^v;-^TOV’ w'Bn’ttB‘ ike0rV' ln: It0g! nauki 1 tekhniki- S°vr. probl. mat. Fundam. napr. 55. 137-

^ Mosk™' VIMTI (1989) (Russian); English transl.: V.L. Popov, E.B. Vinberg, Invariant theory, in: Algebraic Geometry IV (Encyclopaedia Math. Sci. 55), 123-284, Springer-Verlag (1994).