M h . M a t h . 115, 191-213 (1993)

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P r i n t e d in A u s t r i a

Semistable Measures and Limit Theorems on Real and p-adic Groups

By

Riddhi Shah, Bombay

(Received 4 May 1992; in final form 1 September 1992)

Abstract. For any locally compact group G, we show that any locally tight homomorphism from a real directed semigroup into MI(G) (semigroup of probability measures on G) has a 'shift' which extends to a continuous one-parameter semigroup. If G is a p-adic algebraic group then the above holds even i f f is not locally tight. These results are applied to give sufficient conditions for embeddability of some translate of limits of sequences of the form {v kn} and # E M 1 (G) such that z(#) = #k, for some k > 1 and zeAu t G (cf. Theorems 2.1, 2.4, 3.7).

Introduction

Let G be a locally compact (Hausdorff, second countable) group and let MI(G) denote the topological semigroup consisting of all probabili ty measures on G with weak topology and the convolut ion as the semigroup operation. Let Aut G be the group of all bicontinu- ous au tomorphisms of G. A cont inuous one-parameter semigroup {#t}t>~0 in MI(G) is said to be (z, c)-semistable, for some z~Aut G and for some c e ]0, 1 [ w ] 1, o0 [, if r(#t) = Pct for all t i> 0. Given a # e M 1 (G) and a z e A u t G such that z(#) = #k for some k~N\{1} (where N is the set of all natural numbers), we are interested to know if # can be embedded in a (~, k)-semistable cont inuous one-parameter semigroup {#t}t>~o as #1 = #. We prove this to be so in the case when G is a connected aperiodic (real) Lie group (cf. Corollary 2.2). This general- ises a result of NOBEL who has proved it for every aperiodic strongly root compact group G (cf. [19], Remark 4). We also show that if G is a connected real Lie group or a p-adic algebraic group (for a prime p) with -c, # and k as above (z is an algebraic morph i sm in the p-adic case) then there exists an element x in G, such that x# can be embedded in a (z,k)-semistable cont inuous one-parameter semigroup {#t}t~>o as

1991 Mathematics Subject Classification: 60B15, 22E35, 60B10.

192 R. SHAH

#~ = xp (cf. Theorems 2.1, 2.4). Semistable one-parameter semigroups of measures are characterised, in terms of their supports and the contraction groups, by HAZOD and SIEBERT in the real case (cf. [12]) and, by DANI and SI-IAH in the p-adic case (cf. [8]).

In Sect. 1, we develop a criterion for shift embeddability for dense real directed semigroups. Namely, given a locally tight homomorphism f from such a semigroup M to M ~ (G) we find a shift of the map from M which has a continuous extension to R+ (see Theorem 1.4). For a connected Lie group, by a result of DANI and MCCRUDDEN (cf. [6], Theorem 2.1), any abstract homomorphism from a real directed semigroup is locally tight and hence the preceding conclusion holds without the local tightness assumption. We show that in the case of p-adic algebraic groups also, the conclusion holds without the local tightness assumption (see Theorem 1.5); this in- cludes a p-adic analogue of the result of DANI and MCCRUDDEN.

In Sect. 3, we discuss the embeddability of the limit measures on real and p-adic groups. We prove in particular that if G is a connected locally compact group and #eMI(G) is such that supp# is not contained in any proper closed subgroup of G and # = lira,_, ~o v*," for some sequence {v,} in MI(G) and for some unbounded sequence {k,} in N, then there exists x ~ G such that x# is embeddable in a continu- ous one-parameter semigroup { #t } t/_- o as # 1 = x# (cfi Theorem 3.7 for a more general result, including a p-adic analogue). NOBEL showed that such a # is embeddable if G is an aperiodic strongly root compact group (cf. [19], Theorem 1). This can be deduced from our result (see Theorem 3.6).

1. Criterion for Shift Embeddability for Submonogeneous Semigroups

Here and throughout all the sections Q +, Q*, R + and R* denote respectively the additive semigroups of positive rational, nonnegative rational, positive real and nonnegative real numbers equipped with the usual topology. We denote by Qp, P a prime number, the field of p-adic numbers equipped with the p-adic topology. For x E Qp, Ix Iv denotes the p-adic absolute value of x. For any x = (Xl, . . . ,xr)~

u Qp( r~N) , [[x[[p=max{Ixi[pll <~i<<.r} denotes the norm of x. We refer the reader to [4] for details.

A semigroup M is called submonogeneous if it is generated by {mi[ i ~ N} such that, for each i, there exists r i E N such that m~_ 1 --- mi;

Semistable Measures and Limit Theorems on Real and p-adic Groups 193

equivalently, it is a union of an increasing sequence of infinite cyclic semigroups.

A real directed semigroup M is a subsemigroup of R* such that, given r, s e M there exist t e M and m, n e N such that r = m t and s = nt. Any real directed semigroup is submonogeneous and conversely (cf. [17], pp. 314). For any real directed topological semigroup M and a topological semigroup S, a semigroup homomorphism f : M--* S is said to be locally tight if for each rneM, the set {f(m')]m' <<. m} is relatively compact in S.

In this section we develop a criterion for shift embeddability for submonogeneous semigroups. Namely, given a dense submono- geneous semigroup M of Q*, and given a locally tight homomor- phism f : M ~ M I ( G ) there exist a continuous homomorphism q~:R+--*MI(G) and a homomorphism e: m ~ G such that ~b(m)= e(m)f(m) = f(m)e(m) for all m e m (see Theorem 1.4). K. H. HOVMANN had derived the criterion for shift embeddability for the case when M is the whole semigroup Q*. We first find a shifting map ~b: Q+ Mt(G) thereby reducing it to Hofmann's criterion (cf. Corollary 1.3).

We show that the above assertion also holds i f f is an abstract (not necessarily locally tight) homomorphism in the case when G is a p-adic algebraic group (see Theorem 1.5).

We first observe the following equivalent conditions for a dense submonogeneous subsemigroup of Q*.

Proposition 1.1. Let M be a subgroup of Q* . Then M is sub- monogeneous and dense in Q* if and only if it satisfies the following conditions:

(a) I f a, b e M are such that a > b then a - b e M . (b) There exists an integer a e M and a sequence {ak} in M such that

I i m k ~ a k = 0 in Q+ and for all n e N , (a - ak)/n! e M for all large k.

Proof. Let M be submonogeneous and dense in Q*. Then M is generated by {a /mieMTmi+l=mir i for some r ieN, i e N and limi_.~ m~ = oo}. Clearly M satisfies condition (a). Since {ml} is rela- tively compact in Qp for each prime p, by a diagonalisation process, one can find a subsequence {mj} of {m~} which is convergent in Qp for all primes p. Now suppose for each prime p, limj_, oo mj ~ 0 in Qp. Then since limi_~oo mj-- 0o in the usual sense, there exists a subsequence {rag} such that lim k_~ oo (inkling + 1 ) = 0 in Q + and limk_~ ~ (mk/mk + 1 ) = 1 in each Qp. Let for each keN, a k = amk/mk+ 1 which is in M. Also

194 R. SHAH

limk-* co ak = 0 in Q + and lim k_, o0 (a - ak) = 0 in each Qp. Hence for all neN, (a - ak)/n! e M for all large k.

Now suppose for some prime q, limj_~ 00 mj = 0 in Qq. Then a/q" e M for all n e N . Hence there exists a subsequence {nk} such that limk-, oo (1/q "k) = 1 in each Qp such that p va q. In this case let ak = a/q "~ for all k e N . Then it is easy to see that {ak} satisfies condi t ion (b).

Now we prove the converse. Let M satisfy condit ions (a) and (b) of the proposi t ion. Let b be the smallest integer in M. T h e n if m/ne M, gcd(m, n) = 1 then using condi t ion (a), it is easy to see that b/n e M and b divides m. Hence we have that {b/nln is such that bin e M} generates M. N o w given b/r, b / seM let u = lcm(r, s). Then u = r's' for some r', s' which divide r, s respectively and gcd(r', s ' )= 1. Hence we have that b/r',b/s'eM and b / r '+b / s '=b t /ueM for some t e n such that gcd(t, u ) = 1. Hence condi t ion (a) implies that b/ueM. Hence b/r, b/s are contained in the infinite cyclic semigroup generated by b/u. This implies that M is an increasing union of cyclic semigroup and hence M is submonogeneous . F r o m condi t ion (b), for m/neQ*, (am--ak) / an e M for all large k and it converges to m/n in Q * . Hence M is dense in Q* .

Theorem 1.2. Let M be a dense submonogeneous subsemigroup of Q* and S be any metrizable topological semigroup. Let f : M-~ S be a locally tight semigroup homomorphism. Then there exist a locally tight homomorphism q~:Q+ ~ S and a homomorphism a: M-- .S such that

~b(Q+) c f (m), a(m) is contained in a compact subgroup o f f (m) and O(m) = tT(m)f(m), VmeM.

Proof. The proof is somewhat similar to that of Theorem 1 in [-21]. Let

K = (-] f(]O,x[c~M) and K ' = f ( ] O , l [ n M ) . x ~ R *

It is clear that K' is compact , K is a compact abelian subgroup o f f ( M )

and the identity e of K is the identity of f(M). Without loss of generality we may assume that 1 e M and that

{ 1/m i e M[ m i + 1 : miri for some r i e N, i e N and limi_~ ~o ml -- ~ }. Also from the previous proposi t ion we have a sequence {ak} in M such that limk_,~a k = 0 and for each nEN, (1 -- ak)/n!~M for all large k. Since 1/m i, (1 l ak)/mieM for all large k, condi t ion (a) of Proposi t ion 1.1 implies that ak/m i e M for all large k.

Semistable Measures and Limit Theorems on Real and p-adic Groups 195

Now we construct homomorphisms ~b and a. First we construct inductively for all n e N , the subsequences {a~ ")} of {ak} such that {f((1 - a~"))/n!)} converges for all n e N and {f(a~")/n)} converges if n = m i for some i. Since limk_~ oo ak = 0 and K' is compact there exists a subsequence {a~ 11} of {ak} such that {/(a~l))} and { / ( 1 - - a(kl))} converge in K'. Having found for an n e N , the subsequences

(lh a k ; , . . . , {a~ ")} with the desired properties, we proceed to define {a~,+ 1)} as follows. Consider the sequence {a~"l/(n + 1)!}, which is eventually in M. Again since K' is compact there exists a subsequence {a(k,+ i)} of {a(k ")} such that {f((1 --a(k "+ 1))/(n + 1)!2~ converges in K' a n d i f n + 1 = mi for some i, then we can choose {a~ )} in such a way that {f(a~ "+ 1)/(n + 1))} converges. We thus get {a~ ")} for all h e N , with the desired properties.

We define 4~(1/n!) to be the limit of {f((1 - a~"))/n!)} in K'. It is obvious that 4~(1/(n+ 1)!) "+1 =~b(1/n!) for all heN . Hence as {(1/n!):neN} generates Q * , by Lemma 3.1.30 of [13], q~ can be extended homomorphical ly to Q * . Let qS(0)= e, the identity of K.

Obviously ~b(Q + ) c f (M) . Since K' is compact, ~b is locally tight. Now we define a : M ~ S . Since limk_.~ak=O, for each i,

{f(a~ml)/mi)} converges to (say) x ieK, and since K is a group, x/-1 exists and is in K. Let o-(1/m,) = x~- 1 for each i. Now using the relation mi+ l = r~mi for some teeN, we get

x ~ l = lira f { a i .... } = lim f = = x~. k oo \ m i + l J k oo + k m

Hence we have a(1/mi+l) r'= e(1/mi) for each i. And since {1/mi} generates M, by Lemma 3.1.30 of [13], a can be extended homomor-

phically to M. And a(M) c K, a compact subgroup of f (M) . We next show that ~b(m) = f(m)a(m) for each m e M. It is enough to

show that dp(1/mi) = a(1/mi)f(1/mi) for each i.

k f k a f ( l ' ] a ( l ' ] = f ( 1 i a (mi)\ /a(mO\ / 1 \ ; ) lw) I ; ) =

( ; ) - - a k lim f i = l i m f 1 (,,0", k-*o~ a~m') X - l =

196 R. SHaH

This completes the proof of the theorem.

Remark. I came to know from W. HAZOD that he had verified that HOFMANN'S proof for shift embeddability for the rational semigroup Q* also goes through for any dense submonogeneous subsemigroup of Q* and that he has used the general version in [9] and [10].

Corollary 1.3. Let G be a locally compact second countable group and f : M ~ M I ( G ) be a locally tight homomorphism for some dense submonogeneous subsemigroup of Q*. Then there exist a locally tight homomorphism q~:Q+ ~ MI(G) and a homomorphism ~: M ~ G

such that q~(Q+)cf(M), a(M) is contained in a compact subgroup of G and O(m)=e(m) f (m)=f(m)a(m) ,VmeM. (/-/ere the elements of G are identified with respective Dirac measures.)

Proof. The proof is similar to that of Corollary 1 in [21]. By the previous theorem there exist a locally tight homomorphism q~:Q+~MI(G) and a homomorphism a : M ~ M I ( G ) such that

q~(Q +) c f (M) and q~(m) = a(m)f(m). Let K be the group defined as in the proof of the previous theorem. Then a(M) c K. The identity of K has to be q~(0) = co H, the Haar measure of some compact subgroup/-/ of G (cf. [13], Theorem 1.2.10) and the elements of K are of the form XCOH(= 6x'COn, 6x being the Dirac measure at x), x eN(H), the nor- maliser of/-/ in G (cf. [13], Theorem 1.2.13).

As in the theorem, without loss of generality we may assume that I ~ M and {1/mi~ Mlmi + 1 = mir i for some r i~N and limi~o~m i = oe } generates M. Let x i be such that a(1/mi) = xico H. One can construct a homomorphism e: M ~ G such that o-(m)= e(m)co n = cope(m) for all m e M (see [21], Corollary 1 or [13], Theorem 3.6.2). Hence we get that ~b(m) = e(m)f(m) = f(m)e(m) for all meM.

Theorem 1.4. Let G be a locally compact second countable group. Let M be any dense real directed semigroup and let f : M--* MI(G) be a locally tight homomorphism. Then there exist a continuous one-par- ameter semigroup qS: R+ --, M I ( G) and a homomorphism c~: M -~ G such

that 4~(R+) c f(M), the image of c~ is contained in a compact subgroup of G and ~)(m) = e(m)f(m) = f(m)c~(m), Vm~ M (as earlier, the elements of G are identified with respective Dirac measures).

Since any real directed semigroup is submonogeneous and it is contained in rQ* for any r e M, it is isomorphic to a submonogeneous

Semistable Measures and Limit Theorems on Real and p-adic Groups 197

subsemigroup of Q* (cf. [17], pp. 314). Hence it is easy to prove this theorem using Corollary 1.3 and Theorem 3.5.1 of [131.

By a p-adic algebraic group, p a prime, we mean an algebraic subgroup of GL,(Qp) for some h e n (cf. [1], [14] or [15] for generali- ties on algebraic groups). For #eMI(G), let G(#) denote the closed subgroup generated by the support of # in G and let G(#) denote the Zariski closure of G(#) in G. Let Z(#) denote the centralizer of G(#) in G. Obviously Z(#) is an algebraic group.

For a #~MI(G), F(#)= { 2 ~ m l ( G ) 1 2 , v = v , 2 = # for some v~ MI(G)} is said to be the factor set of#. I fH is any closed subgroup of G and tl: G ~ G/H is the natural projection, then for E ~ MI(G), E/H denotes the subset {tl(2)]2eE}(=q(E)) of MI(G/H).

A sequence {x,} in a topological space is said to be divergent if it does not have any convergent subsequence.

By a result of DANI and MCCRUDDEN for a connected real Lie group G, any abstract homomorphism f from a real direct semigroup to MI(G) is locally tight (cf. [6], Theorem 2.1). The following theorem can be considered as a p-adic analogue of this result. To prove this theorem we follow some techniques used in the real case by DANI and MCCRUDDEN (cf. [6]) upto a certain extent and then use some results in [21] and some properties of p-adic groups.

Theorem 1.5. Let G be any p-adic algebraic 9roup and Ma(G) be the semigroup of all probability measures on G. Let M be any dense real directed semigroup. Let f : M --* MI(G) be an abstract homomorphism of the semigroups. Then there exist a continuous one-parameter semigroup qS:R+--*MI(G) and a homomorphism c~:m~G such that ~b(m)= c~(m) f (m) = f (m)c~(m), V m ~ M.

Proof. As earlier, without loss of generality we may assume that M is a dense submonogeneous subsemigroup of Q* with I ~ M and {1/mi6Mlmi+ x = mir i for some ri6N , i6N and limi_~m i = oo} is the set of generators of M. Let us denote f(m) by #,n for each meM.

Step 1. We first claim that there exists an s ~ M such that for all m ~ M, #m is supported on Z(Z(#s)), namely, the centraliser of Z(#s) in G.

Since {Z(#,,)},,~M consists of p-adic algebraic groups, it is easy to prove this along the same lines as the proof of Lemma 2.3 in [6].

Now since Z(Z(#~)) is a p-adic algebraic group, without loss of generality we may assume G to be Z(Z(#~)). This implies that Z(#~) = Z the center of G.

198 R. SHAH

Step 2. We show that {#,, [m ~< s}/N is relatively compact where N r is a central subgroup isomorphic to Qp for some r/> 0.

From Theorem 2 of [21] we know that F(#~)/Z is relatively compact (as Z(#~) = Z). Let Z ~ be the Zariski connected component of Z. Since Z/Z ~ is finite, we get that F(#~)/Z ~ is relatively compact. Hence {#,,[ m <~ s}/Z ~ is relatively compact. As Z ~ is a Zariski connec- ted abelian group, it is isomorphic to N x N1, where N ~ Qp for some r i> 0 (cf. [1], Theorems 4.7, 10.6) and N 1 is a compactly generated group (cf. [3], Theorem 13.4). Hence by Proposition 8 of [16], {#m[m ~< s}/N is relatively compact.

Step 3. If r = 0 or if {mi} is a relatively compact sequence in Qp then we show that f is locally tight and the theorem in this case follows by Theorem 1.4.

If r = 0, that is, if N is trivial then {#,,Ira <~ s} is relatively compact and hence f is locally tight. Now let r :A 0 and let {ml} be a relatively compact sequence in Qp" For any n e N let N, = {m~N[p"Xm} and for #eMI(G), let R ( N n , # ) = { P [ 2 m - - # for some meN,,m>~a}. Then R(N,, #~) is relatively compact for each h e n (cf. [21], Lemmas 2, 3). Then since {mi} is a relatively compact sequence in Qp we have that {#m[m ~< s} is contained in R(N,,#~) for some hEN. This implies that {#,. [m ~< s} is relatively compact and hence f is locally tight. Now one can apply Theorem 1.4 and get the maps ~ and q5 with the desired properties.

Step 4. Let r 4:0 and suppose that {mi} is not a relatively compact sequence in Qp. Let rI:G~G/N be the natural projection. Then t/({#m I m ~< s}) is relatively compact, that is, r/(f) is locally tight. Take 2, = f(1/p") for all n ~> 0. If {2,} is relatively compact then it will be clear from Step 5 of this proof that f is locally tight and we can proceed as in Step 3.

Now suppose that {2n} is not relatively compact. Then by a well-known result (see [20], Ch. III, Theorem 2.2) there exists a sequence {x,} in G such that {x,} is not relatively compact but {2,x,} is relatively compact. Since r/(f) is locally tight, we have that {t/(2,)} and hence {t/(xn) } is relatively compact. Hence we get x, = ynzn for all h e n where {y,} is relatively compact in G and {zn}, ~hich is in N, is not relatively compact. Since N is central and {2,x,} is relatively compact we get that {2,z,} is relatively compact. We can assume that

Semistable Measures and Limit Theorems on Real and p-adic Groups 199

z o = e, the identity in G. Now

(~n+iZn+ 1) p P p p = "~n+ IZn+ i ~--" V~nZn+ 1"

So we have {2,z,} and {2,z,P+ 1} are relatively compact . This implies - 1 p that {z, z,+ 1} is relatively compact . Hence there exists a compact

subset that C of N such that z,-1 p z, + 1 e C for all n. Let L be the sub- group generated by C in N. Now

-- i pn ZO Zn ZO 1 p - p p2 - p n - 1 pn Z1Z 1 Z 2 " " Z n - 1 Z n =

z g l p - i. = z l ( z l r-

p" Since L is a group and z o = e we get that z, e L for all n. Now since

u N = Qp, L is compact and we get for all n, z, = l,/p" for some l, eL.

Similarly we have for all m, neN , z2 az, .+.e~, p" t that is, (1,, +, - l,,)/p m e L. Hence if p" = max { 1[ x [] v [x e L} then [[ l,, +, - l,, [[ p ~< pa-m for all re, h e N . Hence {/,} is a convergent sequence in L. Let lim,_. ~o I, = l, which is in L. Let b, = I/p" for all n >f 0. {b,} is a sequence in N. Then for any m ~> 0,

z m b m 1 __ I m - - l _ lim Im- l" + " e L. pm ,-~ ~ pm

Hence {zmb~, ~ } is relatively compact . This implies that {2,b,} is relatively compact .

N o w define ~ ' : M ~ G and c~' :M-,MI(G) as c((m)=ml and (~'(m) = f (m) , ' (m)= f(m)(ml) for each m e M . Then the image of e' is central in G and {qS'(1/p")},~N is relatively compact .

Step 5. We prove that ~b' is locally tight and hence the proof in this case follows from Theorem 1.4.

To prove that ~b' is locally tight we only use the fact that {qS'(1/p")},~ N is relatively compact . Since (p'(m)N = f ( m ) N in G/N for each m e M and t l(f) is locally tight, we have that t/(~b') is also locally tight.

Let I = {(i,j)] 1 / j e m and i ~<j}, then I is countable. 21d = (~'(i/j) for each ( i , j )el . Then {2ij} is a sequence in MI(G) and there exists a sequence {hi,j} in N such that {21jhij} is relatively compact in Ma(G). h,i,,j = hi,j for each h e n such that (1/nj)eM. Since {2~,p.} is same as {q~'(1/p")} which is relatively compact . Hence we can assume that h~,p, = 0 = h(1, 1) for each n e N . Now since N is central, arguing as in

200 R. SHAH

the proof of Proposi t ion 8 of [16], we get that there exists a compact subset K of N such that

h,,j + hk, j - h i + k , j E K for i + k ~<j.

r Since N = Qp for some r :~ 0, we may assume K to be a compact subgroup. Le t j = m d for any fixed generator m d of M. Let mn= ap b for some fixed aEN, pXa and b ~> 0. Now using the above equat ion we get that hid + hid - hz,j , h i , j q- h2, j - h3 , j , . . . , h i , j q- h a- 1,j - - had~K. Since K is a group and had = hi,pb = O, by adding the above equations we get that ah~d~K. Since pXa, we get that h i j~K . Again using the above equat ion we get had + hi_ 1,fi K and hence hi_ 1,fi K. Succes- sively we get hi, i lK for all i < j . Since this is true for any generator m a =j, we get that hid6K for all (i,j)sI. Hence {hi,j} is relatively compact . This implies that {2id } is relatively compact and 4)' is locally tight. Hence by Theorem 1.4 there exists a cont inuous h o m o m o r - phism qS:R+ ~ M I ( G ) and a h o m o m o r p h i s m a": M ~ G such that

~b(R+) c q~'(M) and q~(m) = ~"(m)dp'(m) = dy(m)a"(m) for all m~M. Since a'(M) is central, we can define a h o m o m o r p h i s m a: M - . G as a(m)= a'(m)a"(m) for~each m~M. Thus we have ~b and ~ with the desired properties.

Remark. Using the same technique as in Theorem 4 of [21], one can easily show that the above theorem also holds for any closed (not necessarily algebraic) subgroup G of GL,(Q,) for any n e N.

2. Semistable Measures on Real and p-adic Groups

For a probabil i ty measure # on G let s u p p # denote the suppor t of #, let G(#) be the closed subgroup generated by s u p p # in G. Let the invariance group of# in G be defined as I(#) = {x E G Ix# = #x = #} and let the invariance group of# in Aut G be defined as J ( # ) = {z~Aut {r

= #}.

Theorem 2.1. Let G be a connected locally compact 9roup and let #EMX(G). Let z ~ A u t G be such that z(#)= #k, for some k~N\{1} . Then there exists an element x contained in a compact subgroup of G such that x# can be embedded in a continuous one-parameter semigroup {#t},~> 0 as #1 = X# and {#t}t~> 0 is (% k)-semistable, that is, z(Ut) = #kt for all teR+.

Semistable Measures and Limit Theorems on Real and p-adic Groups 201

Moreover, if G is a Lie group then there exists an n 6 N such that gcd(k, n) = 1 and #" can be embedded in a continuous (z, k)-semi- stable one-parameter semigroup {/~t}t~>o as #1 = I~ and #o = con, the Haar measure of the compact group H = I(#).

Proof. Let M={a/kb laeN , b>~O}. Clearly M is a dense sub- monogeneous subsemigroup of Q * with generators {1/kblbeN}. Now define f : M ~ MI(G) as f (a /k b) = "c-b(#a) for all a e N and b i> 0. Clearly f is a homomorph i sm. Since G is connected, there exists a compact normal subgroup L of G such that G/L is a connected Lie group. Let rc:G-,G/L be the natural projection and let jT: M--*MI(G/L) be defined by f(m) = zc(f(m)) for all m~M. Then j~is a h o m o m o r p h i s m and by Theorem 2.1 of [6], f is locally tight. Since L is compact , this implies that f itself is locally tight. Hence by The- orem 1.4, there exist a cont inuous h o m o m o r p h i s m ~b: R+ ~ M I(G)

and a h o m o m o r p h i s m ~: M ~ G such that ~b(R +) c f (M), the image of is contained in a compact subgroup of G and ~b(m) = e(m)f(m) =

f(m)c~(m),Vm~m. Moreover, since z(2)= 2 k for all 2 e f ( m ) , we have ~(~b(t)) = (o(t) k = 4)(kt) for all t e R + . Let e(1) = x (say), then q~(1) = x#. Also qS(0) = c%r , the Haar measure of some compact group H of G, is

the identity of f (M). This proves the first assertion in the theorem.

Now suppose that G is a Lie group. Let K = Ny~R*+ f ( ] 0 , y [ n M). Then K is a compact abelian group whose elements are of the form &a,oo~, aEN(H), the normaliser of H. Hence K is a Lie group with only finitely many connected components . Let K ~ be the connected componen t of the identity in K. Then K ~ ~ T ~ for some l ~> 0, where T = { y e C [1 y l = 1 } (where C is the field of complex numbers). Since ~(f(m))=f(km), for all m e m , we have z(K)= K. Thus z is an au tomorph i sm of K. But "c(2) = 2 k for all 2 ~ K, k > 1, can not be a au tomorph i sm of K ~ unless K ~ is a trivial group. Hence K is a finite group. Let n be the order of K. Then gcd(n, k) = 1. F r o m Corollary 1.3 and Theorem 3.5.1 of [-13] it is clear that e(m).coHeK for all meM. Hence (x#)" = x"#"= x"coH#" = #". Hence by reparametrising ~b we get that #" can also be cont inuously embedded in a (~, k)-semistable one parameter semigroup as Pl = #" and # o - c~ It follows from Proposi t ion 1.2 of [10] that H = I(#).

A locally compact group is said to be aperiodic if it does not contain any nontrivial compact subgroup.

202 R. SHAH

Corollary 2.2. Let G, z, #, k be as in Theorem 2.1. Suppose that G is aperiodic. Then # itself can be embedded in a (z, k)-semistable continu- ous one-parameter semigroup {#t}t>~o as #1 = #. Moreover, G(#) is a nilpotent Lie group.

This follows immediately from the above theorem, Proposi- tion 4.2 of [11] and Corollary 2.4 of [22].

A measure #E MI(G) is said to be B-stable if for every k e N there exists ZkeAUt G such that z(#) = #k.

Corollary 2.3. Let G be a connected Lie group and let #e MI(G) be B-stable. Then # is embeddable i f J (# ) is compact.

Proof. Let for each k e N , ZkeAUt G be such that Zk(#) = #k. Fix a keN\{1}. Let M, f : M ~ M I ( G ) and K be defined as in the proof of Theorem 2.1 for z = z k. Then f is a locally tight homomorphism such that f ( 1 ) = # and zk( f (m))= f (km) for all m eM.

Now if J ( # ) is compact then, as in the proof of Lemma 4 of [19], one can easily show that Zk(f(m))= f (km) for all k e n and m e M . Since K is a finite group and Zk(2 ) = 2 k for all keN. Hence K is trivial and f extends continuously to R*. And Zk(f(t)) = f (k t ) for all k e N and t e R * .

Remark. If G is a Lie group without any compact central sub- group of positive dimension and # e M 1 (G) is such that G(#) = G then J (# ) is relatively compact; more generally see Theorem 1.8 of [5]. Moreover, if # is B-stable then by Corollary 2.3, # is embeddable.

Theorem 2.4. Let G be a p-adic algebraic group for some prime p and #eMI(G) . Let z be an automorphism of G as a p-adic algebraic group. Suppose that z(#) = #k for some k e N\{ 1 }. Then there exists an element x e G(#) such that x# can be embedded in a continuous one- parameter semigroup {#t}t~>o such that #1 = x# and {#t}t~>0 is (z, k)- semistable, that is, z(#t) = #kt for all t e R + .

Proof. Since z(#) = #k we have that z(G(#)) c G(#) and z(G(#)) is a subgroup of finite index in G(#). Hence since z is an automorphism of G as a p-adic algebraic group, we have that -c(G(#)) = t~(#). Hence without loss of generality we may assume that G = G(#).

Now let M = {a/kb[a e N, b >f 0}. Clearly M is a dense submono- geneous subsemigroup of Q* with generators {1/kb[beN}. Now

Semistable Measures and Limit Theorems on Real and p-adic Groups 203

define f : M ~ M: (G) as f ( a / k b) = -c -b(#,) for all a G N and b ~> 0. Clearly f is a homomorph i sm, f (1) = # and ~(f(m)) = f ( km) for all m e M .

Now suppose that pXk. Then since G = (~(#), Proposi t ion 3 of [21] implies that f ( ]0, 1 [ c~ M) is relatively compact and hence f is locally tight. Hence by Theorem 1.4 there exist a cont inuous h o m o m o r p h i s m ~b: R+ ~ M : ( G ) and a h o m o m o r p h i s m ~: M ~ G such that ~b(R+)c

f ( m ) and c~(m) = e(m)f (m) = f(m)o~(m), VmeM. Since z(2) = 2 k for all

2 e f ( m ) , we have that z(dp(t)) = r k = dp(kt) for all t~R+ . Now let plk. Then by Theorem 1.5, there exist a cont inuous

h o m o m o r p h i s m ~b: R + ---, M ' (G) and a h o m o m o r p h i s m e : M ~ M 1 (G) such that r o~(m)f(m)= f(m)e(m). To conclude the proof we only have to show that z(4)(t)) = c~(kt) for all t e R + . As in Step 5 of Theorem 1.5 there also exist a locally tight h o m o m o r p h i s m ~b': M M'(G) and a h o m o m o r p h i s m e': m --. G such that r = e'(m)f(m),

qS(R+) ~ r and e'(M) is central in G, as G = G(#) = G(f(m)) for each m. In fact i f Z is the center of G, then e'(M) : N, where N is as in

r the proof of Theorem 1.5, a closed central subgroup of G and N = Qp, for some r e N . Obviously z(N) = N. Hence it is enough to show that z(O'(m)) = r for all m ~ m .

z(r = :( f (m) )z(c( (m) ) =

= f (km)a ' (km)c((km)-I z(a'(m)) =

= d/(km) c((km)- 1 z(a'(m)).

Since r is locally tight so is -c(r Hence the above equality implies that A = {~'(km)-:z(~'(m))[rnem, m ~ 1} is relatively compact in N. Moreover since p divides k, we have that if a e A then alp ~ A. Hence A = {e}, where e is the identity. This implies that z(~'(m)) = ~'(km) and hence, since z is a morphism, "c(~b'(m))= (~'(km) for all m ~ M . Let 0~(1) = x (say). Then ~b(1) = xf(1) = x#. Now, since G is totally discon- nected, as in the p roof of Theorem 4 of [21], we get supp q~(t) c G(#) for all t ~ R + . Since supp r x# = x supp# , this implies that xeG( ).

3. Limit Theorems for Measures on Real and p-adic Groups

In this section we study the possible limit laws of any sequence {vk, ~ of probabili ty measures on connected locally compact groups and p-adic algebraic groups, where {k,} is a sequence in N. NOBEL has

204 R. SHAH

made a detailed study of the limit laws on any simply connected ni lpotent Lie groups (cf. [19-1). We give certain general condit ions under which the limiting measure is root compact , infinitely divisible or more generally embeddable.

A measure # e M 1 (G) is said to be infinitely divisible if for each n ~ N there exists a measure 2 e M I ( G ) such that 2" = #; such a 2 is called an n th-root of p. A measure # e M I(G) is said to be root compact if the root set of #, namely R(#) = {2mI2eMI(G), 2" = # for some n >/m} is rela- tively compact .

T h r o u g h o u t this section for a locally compact group G, let {v,} be a sequence in MI(G) and {k,} be a sequence in N such that

lim k. Vn =f l

for some #eMI(G) . Let A = {vml m <~ k,, kn, neN}.

Theorem 3.1. Let G be a connected locally compact group. Let {v,}, {k,},# and A be as above. Let N be a connected nilpotent normal subgroup of G. I f there is no proper closed subgroup containing N and the support of #, then A is relatively compact. In particular, # is root compact.

To prove this theorem we use techniques similar to those develop- ed by MCCRLrDDEN in [16]. Before proving it let us state some useful results.

L e m m a 3.2. Let G be any locally compact group and {v,}, {k,}, # and A be as above. Let C be a closed subgroup of G and q: G ~ G/C be the natural projection. Let C be either a compact subgroup or strongly root compact central subgroup of G. Then A is relatively compact if and only if q(A) is relatively compact.

Proof. It is obvious that if A is relatively compact then t/(A) is relatively compact . N o w let t/(A) be relatively compact . Then if C is compact , the proof follows from a well-known criterion for relative compactness of a set of probabil i ty measures (cf. [20], Ch. II, The- orem 6.7). If C is a strongly root compact central subgroup then it can be proved along the same lines as Proposi t ion 8 of [16] (see [13], 3.1.10 for the definition of strongly root compact groups).

Semistable Measures and Limit Theorems on Real and p-adic Groups 205

Proposition 3.3. Let G be an almost algebraic subgroup of GLn(R ) (resp. an algebraic subgroup of GL,(Qp)) for some n ~ N. Suppose that the center of G is finite. Let {x,} be a divergent sequence in G. Then there exist a subsequence {Ym} of {x,} or {xn -1} and a proper algebraic subgroup B of G such that for every relatively compact sequence {zm} in G, any limit point of the sequence { ymZmy2 2 } is contained in B.

In the real case this can be easily derived from Theorem 3.2 in [-7-1 and in the p-adic case, it can be derived from Theorem 2 in [21].

Proposition 3.4. Let G be a connected Lie group (resp. a p-adic algebraic group) and Z be the center of G. Let C be a normal closed (resp. algebraic) subgroup of G such that C is a vector group and Z c~ C is discrete. Let tl: G ~ G/C be the natural projection and suppose that {xn} is a sequence in G such that

(c) {xn} has no convergent subsequence in G and (d) {t/(xn)[n ~ N} is relatively compact in G/C. Then there exist a subsequence {Ym} of {X~} and a proper closed

(resp. algebraic) subgroup B containing the centralizer of C in G, such that for every relatively compact sequence {z,n } in G, any limit point of the sequence { YmZmY21} is contained in B.

The proof is same as that of Proposition 9 of [16] and it is easy to see that the subgroup B chosen as in that proof contains the central- izer of C in G.

Proof of Theorem 3.1. Since G is a connected locally compact group there exists a compact normal subgroup K of G such that G/K is a Lie group (cf. [18], pp. 118). Let t l : G ~ G / K be the natural projection. Then t/(N) is a connected nilpotent normal subgroup of G and the closed subgroup generated by the support of t/(/~) and t/(N) is the whole of G/K. Hence, by Lemma 3.2, without loss of generality we may assume G to be a Lie group.

Now we will prove the theorem by induction on the dimension of the Lie group G. If dim G = 1 then G is abelian. Then G is strongly root compact (cf. [13], 3.1.12) and hence A is relatively compact (cf. [19], Lemma 1). Assume that the theorem is true for all Lie groups G with dim G < n. Now let dim G = n. We may assume that the center Z of G is zero dimensional; otherwise Z would contain a closed connected subgroup C of positive dimension and as, such a C is strongly root

206 R. SrIaH

compact, the assertion follows from the induction hypothesis and Lemma 3.2.

Now let G be semisimple. Then since G/Z has trivial center and Z is compactly generated, by Lemma 3.2, we can assume that G is a semisimple group with trivial center. Hence G can be realised as an almost algebraic subgroup of GL,(R) for some heN. Also since G is semisimple, N = {e}, where e is the identity of G. Let us suppose that A is not relatively compact, that is, there exists a sequence {b~[b~ <~ k~} such that {v~'} is a divergent sequence in A. Hence by a well-known result (cf. [20], Ch. III, Theorem 2.2), one can choose a divergent sequence {x~} in G such that {v~'x~},

bz 1 ez) cz - {xzvl }, {Xl vl t and {vz x~ 1} are relatively compact, where - - 1 kl ) kz - 1 } c~ = k 1 - b I for each 1. This implies that /x~ vz xz~ and {xffz xl

are relatively compact. F rom Proposition 3.3, there exist a subsequ- ence {y,,} of {x~} or {x~ -1} and a proper closed subgroup B of G, such that for every relatively compact sequence {zm} in G, any limit point of the sequence {y,,,z,,y21 } is contained in B. Since N is trivial, supp # generates whole of G. Hence #(G\B)r O. Then there exists a compact set K such that K n B = ~ and #(K) = 6 > 0. Then there exists an open relatively compact set U such that KUc~B= ;Z5. Then there exists N e N such that v~'(Kff)>61 > 0 for all n i> N (cf. [-20], Ch. II, Theorem 6.1). Since {Ym} is a sub-

- - 1 - - 1 k m �9 " sequence of {x~} or {xt }, {y,~ v,, y,,} is relatively compact; (here k m k! {v m } is a subsequence of {v I }). Hence there exists a compact set

K 1 such that y~lvk"y, .(K1)>l--61/2 and for all l, that is, k m z ~T - 1 \ Vm (ym~ly ,~)>1--61/2 for all I. This implies that y,K~y~,~c~

Kt? ~ IZ/. Hence there exists a relatively compact sequence {z,,) in K~ such that {YmZmY~ 1 } is contained in a compact set KtT, hence {y,,z,y~ :~ } has a convergent subsequence whose limit is not contained in B. This is a contradiction. Hence A is relatively compact.

Now suppose that G is not semisimple. Without loss of generality we may assume N to be the maximal connected nilpotent normal subgroup of G. Hence N is nontrivial. Let C be the connected component of the identity in the center of N. (C is nontrivial since N is nontrivial, connected and nilpotent). Since C is abelian C = R m x K, where m is a nonnegative integer and K is a compact subgroup ([18], Theorem 26). If K is nontrivial then by Lemma 3.2, it is enough to show that r/(A) is relatively compact, where '1: G ~ G/K is the natural projection, which follows from the induction hypothesis as

Semistable Measures and Limit Theorems on Real and p-adic Groups 207

dim G/H < n. If K is trivial then C = R m for some m > 0, that is, C is a normal vector group. Let 11: G ~ G/C be the natural projection. Sup- pose that A is not relatively compact. Then arguing as before we conclude that there exist a sequence {b,[b I <~ kz} in N and a sequence {x~} in G such that {@} and {xl} are divergent sequences and {v~'x,} and {x~-l@xi} are relatively compact. Since dim G/C< n, by the induction hypothesis t/(A) is relatively compact. This implies in particular, that {t/(vTz)} and hence {t/(xt) } are relatively compact. Hence by Proposition 3.4 there exist a subsequence { Ym} of {x,} and a proper closed subgroup B containing the centralizer of C in G, such that for every relatively compact sequence {zm} in G, any limit point of the sequence {y,,z,,y~, 1} is contained in B. Since C is central in N, B contains N. This implies that #(G\B) ~ O. As in the semisimple case, this leads to a contradiction. Hence A is relatively compact. The proof is complete by induction.

Theorem 3.5. Let G be a Zariski connected p-adic algebraic group r cbn and {v,}, {kn}, # and A be as above. For each meN, set A,~= iv,

AIb.pZ= k , , l<m, ceN}. Let N be the maximal Zariski connected nilpotent normal subgroup of G. I f there is no proper algebraic subgroup containing N and the support of#, then A m is relatively compact for each meN.

Proof. We prove this by the induction on dim G. Let dim G = 0, then G = {e} and MI(G) is compact. Hence A is relatively compact. Now let the theorem be true for all G with dim G < k. Let dim G = k. If G is semisimple (not necessarily Zariski connected), then G has a finite center and N = {e}, the trivial group. Hence G(#)= G. Then using Proposition 3.3 and arguing as in the real case, we get that A is relatively compact. If G is reductive then the center Z is compactly generated (cf. [3], Theorem 13.4) and Z ~ = R(G), the radical of G (cf. [3], Proposition 2.2). Hence G/Z is an open subgroup ofa semisimple algebraic group (of. [1], Theorem 6.8); in fact it is a subgroup of finite index (see [2], Proposition 1.12, Theorem 6.2). Hence if q: G ~ G/Z is

kn the natural projection then t/(v, )~t/(#) and (~(t/(#)) contains G/Z. Hence t/(A) is relatively compact. Using Lemma 3.2, we get that A is relatively compact.

Now suppose that G is not reductive. Then N = U x T, U is the (nontrivial) unipotent radical of G and T is the maximal torus in N. Let C be the center of U. Then C is a normal algebraic vector group,

208 R. SHAH

G/C is an open subgroup of a Zariski connected algebraic group (cf. [1], Theorem 6.8); in fact G/C itself is algebraic (see [2], Proposit ion 1.12, Theorem 6.2(b)). Now dim G/C < k. If Z is compactly generated then C n Z = J25. Let 1"1: G --* G/C be the natural projection. Then t/(N) is a Zariski connected nilpotent normal subgroup of G/C. We also have limn_, oo t/(v, kn) = t/(#) and there is no proper algebraic subgroup of G/C containing t/(N) and supp t/(#). Hence using the induction hy- pothesis, we get that t/(Am) is relatively compact for each m ~ M. Now using Proposit ion 3.4 and arguing as in the real case, we get that A,, is relatively compact for each m ~ N. Suppose that Z is not compactly generated. Let m ~ N be arbitrary and let Nm = {n e N: pm)[n}. As in the proof of Proposition 3 of [21], Z ~ is N,,-root compact (cf. [21] for the definition). Let rl: G ~ G/Z ~ be the natural projection. Using induc- tion as earlier, we get that t/(Am) is relatively compact for each m e M . Using this fact and arguing as in Proposition 8 of [16], we get that A,, is relatively compact for each m e M . This completes the proof by induction.

F rom now on until the end of this section let {k,} be an un- bounded sequence in N. Without loss of generality we can assume that {k,} is a strictly increasing sequence.

Theorem 3.6. Let G be any locally compact group and let {v,}, {k,}, # and A be as above. I f A is relatively compact then there exists an element x contained in a compact subgroup of G such that x# is embeddabte in a continuous one-parameter semigroup {g(t)}t>.o as #1 = x # .

Moreover, for each t > O, there exist {nj} c N and x t in a compact subgroup of G such that

#(t) -- lim v~knjtl * Xt = Xt * lim vtknjtln~ �9 (3.1) j~0o j-ooo

Proof. The proof is somewhat similar to that of Theorem 1 in [19]. Since A is relatively compact without loss of generality we can assume that {v,} is convergent and let lim._,~ovn=v. Let B = {v"lneN}. Then B c A is relatively compact. Hence v is sup- ported on some compact subgroup C of G (cf. [19], Lemma 2). Let B' = (']k~N {v"ln >>. k}. Then B' is a group and the elements of B' are of the type XCOH(----O~BX) where co n, the Haar measure of some compact group H, is the identity of B'. Then it is easy to see that o n , # =

Semistable Measures and Limit Theorems on Real and p-adic Groups 209

P* con = # and for all n e N, there exists x, e C, co n �9 v" = x, �9 co n for all h e N .

Now {v~k"(1/k~)llne N}, which is contained in A, is relatively com- pact for all k e N . Hence by diagonalisation process there exists a subsequence {%} of {v,} such that for l e N there exists n( / )eN and

lim V [knJ(1/t!)l ,j = # ~ e A and # = #z~.v"(~ j - -* co

Clearly, for each I e N, co n* #~ = #~ * con = #~ and hence # = #~l ~ �9 v "a) �9 l!

con = # t *Xn(0 for some Xn(l)eC. Let y be some limit point of x,~t 1) in C. Then

y# = #y = lira #~. l--* oo

Hence y# is infinitely divisible in A which is relatively compact. Now one can construct a locally tight homomorphism f : Q* --. MI(G) such that f ( 1 ) = y # ( = # y ) and f(1/n!)=limr_.o~#~; ~/"r in A (cf. [13], Lemma 3.1.30). Hence by Theorem 3.5.1 of [13], there exist a continu- ous one-parameter semigroup {#(t)}t~o and a homomorphism c~: Q --, G such that #(q) = f(q)c~(q) = c~(q)f(q) for all q e Q * , {#(t)}~o c

f ( Q * ) and e(Q) is a compact subgroup of G. Let e (1)=z . Then #1 = yz#. Moreover, both y and z commute with all the elements of f ( Q * ) . Hence x = yz belongs to some compact subgroup of G and x# = #x is embeddable.

To prove equation (3.1), it is enough to show that, for each t > 0 there exists x t which commutes with #(t) and

[knjt] , (3.2) #(t) = lim v,j x t j ~ o o

for some sequence {n j} depending on t. We have

]j[kna/b] [kn/Ir!]lr!a/b * n = Vn Pr

for some Pr e B. Hence for each r e N there exists xr e C such that

lim - [knja/b] . I t ! a / b , . Vnj ~-- [~Ir JLr"

j - , ov

Going to a suitable subsequence, limi_.o~v~kj"#b]=f(a/b)*x(a/b), where x(a/b)eC. Hence equation (3.2) holds for t = a / b e Q + , x t = x(a/b)-lc~(a/b) for each j. As MI(G) is second countable one could choose the same sequence {n j} for all t e Q * .

210 R. SHah

Fix t in R * . Let {rm} c Q * and {tm} c R* be such that rm--*t , tin--*0 as m--,oo and t = r m + t , , , V m . Then for each m, v~k"tl= 11[k~r ,n l . . [ k n t m ]

, ~v, *Pro for some PmeB. Hence for all m, using equation (3.2) for rm, for a suitable subsequence we get that

lim V.jtk"~t]---- #(rm)*X,-ml *p~ = #(rm)*p~, j--, oo

t t k n t m ] , ~ a n d " ' where pm=l imj_ ,~v , t 'm p m = X r m * P m . As in the proof of Theorem 1 of [19], if p is a limit point of p,~ then p is contained in a compact subgroup of A. One can easily show that for all q e Q * , p , f ( q ) = f ( q ) , p = f ( q ) , x for some x contained in a com- pact subgroup of G. And hence limi_, ~o v[k"JO = #(t)* X(t) for some x(t) . j contained in a compact subgroup of G. So equation (3.2) holds for t with x t = x(t)- 1. From our choice of x t it is clear that #(t) , x t = x t , #(t) for all t > 0. This completes the proof of the theorem.

T h e o r e m 3.7. Let G be a connected locally compact (resp. a Zariski connected p-adic algebraic) group. Let {v,}, {k,} and # be as above. Let N be a connected (resp. the maximal Zariski connected) nilpotent normal subgroup of G. I f there is no proper closed (resp. algebraic) subgroup containing N and the support o f# then there exists an element x contained in a compact subgroup o fG (resp. x contained in G(#)) such that x#( = #x) is embeddable in a continuous one-parameter semigroup { # ( t ) } t ~ > o as #1 = x#.

Moreover, for each t > 0 there exist sequences {n~} c N and {xta} in G such that

#(t) = lira v[,~ "jt],xtJ. (3.3) j~oo

Proof. If G is a connected locally compact group then the asser- tion follows from Theorems 3.1 and 3.6.

Now suppose that G is a p-adic algebraic group and A be as above. Let A m be as in the hypothesis of Theorem 3.6. Then Am is relatively compact for each m e N . Consider the sequence {1/k,} in Qp. If {1/k,} is relatively compact in Qp then A = A m for some m e N , which is relatively compact and the assertion follows from Theorem 3.6.

Now suppose that {1/k,} is not relatively compact in Qp. Then there exists a subsequence of {k,}, denote it again by {k,}, such that lim,_, o0 k, = 0 in Qp, that is, for each m e N, P"[ k, for all large n. Let M = {a/pmlaeN, b >~0}. Then M is a dense submonogeneous sub-

Semistable Measures and Limit Theorems on Real and p-adic Groups 211

semigroup of Q * . Since A,, is relatively compact for each m e N , by a diagonalisation process we can find a subsequence of {v,}, denote it again by {v,}, such that for each m e N , lim,_~ o~ vk, "/p" = #,, (k,/p" is an integer for all large n). Obviously, P #,. + 1 = #,, for all m e N. Let f : M ~ M I ( G ) be a h o m o m o r p h i s m such that f (1/p")=#, , and f(1) = # (cf. [13], L e m m a 3.1.30). Then by Theorem 1.5, there exist a cont inuous one-parameter semigroup {#(t)}t~0 and a h o m o m o r - phism ~:M-*G such that #(q)=f(q)~(q)=~(q)f(q) for all qeM. #(1) = e(1)f(1) -- x# where a(1) -- x(say). Arguing as in the proof of Theorem 2.4, we get that x e G(#).

As in the proof of Theorem 3.6, it is easy to show that equa- t ion (3.3) follows for some fixed sequence {n j} of N all teM. Now fix t in R*. Since M is dense in R * , t = r m+ t,~ for each m, where { r , , } c M , { t , , } c R * ; r m - * t and t m ~ 0 as m---~oe. Clearly for any l i t "[kntl knrm [kntral ,v, = v, *v, for all large n. Also v~kjl*ytd converges for some sequence {Ytd} in G and a subsequence of {n j}, which we denote by {n j} again (cf. [20], Ch. III, Theorem 2.2). Again taking a suitable subsequence of {n j},

�9 k n T m , [ k n j t m ] , lim v~k7 ~tl* Ytd ---- !im v,j v, Ytd

= f(rm)* a(r~)* ~(rm)- 1.2m _--

= #(t)* lim a(rm)- ~*2m, m--+ GO

where 2 m = limj_.oov~k"*ml*yt,j for each m. Let 1imm_~ooa(rm)-12m = 2. Now if #(0) -- con, the Haar measure of some compact group H, then

r "~ knja 3 o) H = limj_~ o~ a(aj)f(aj) = limj_~ ~o ataflv,j for some {a j} c M, aj ~ 0 as j --* oo. Since a(q)f(q') = f(q')a(q) for all q, q' e M and lim~_~ ~ t m = 0, it is easy to show that co~,2 = 2 and 2 , 2 ' = a~ n for some 2 'eMI(G). This implies that 2 = ~o~ �9 x for some x e G. Hence equat ion (3.3) holds for xtd = ytd ,x-1 for eachj . This completes the proof.

One could also show that for each t > 0,

_, . . tk.,tl (3.4) #(t) = lira xt,j v.j j-~oo

! for some sequence {xtj} in G.

Remark. In the hypothesis of Theorems 3.6 and 3.7, if G is a connected real Lie group then there exists m e N such that #" is embeddable; # itself is embeddable if G is a connected ni lpotent or

212 R. SHAH

aperiodic Lie group. Let {#(t)}t>to and x be as constructed in the p roof of Theorem 3.6 such that #(1) = x#. If G is aperiodic then x = e and #1 = #. Clearly, x ,# ( t ) = # ( t ) , x for all t e R + . Let G(x) be the closed subgroup generated by x in G. Then G(x) is a compact abelian Lie group with finitely many connected components . For some me N, x " e G(x) ~ the connected componen t of identity in G(x). Then there exists a cont inuous one-parameter semigroup {at}t~ o c G(x) ~ such that a 1 = x - " . Let 7(0 = at*#(mt) for all t e R + . Then {7(t)}t>~o is a cont inuous one-parameter semigroup with 7(1) = #m. NOW let G be a connected ni lpotent Lie group. Then the center Z of G is connected (cf. [23], Corollary 3.6.4) and x, being in a compact subgroup of G, is contained in Z. Hence there exists a cont inuous one-parameter semigroup {at}t~ 0 c Z such that a 1 = x -1. Let 7(0 = at*#(t) for all t e R +. Then {7(0 }t~> 0 is a cont inuous one-parameter semigroup with 7(1) = #. Clearly, equat ion 3.1 holds for 7(0 in place of#(t) for all t > 0.

The following is an easy consequence of Theorem 3.7 and the above remark.

Corollary 3.8. Let G be a connected locally compact (resp. Zariski connected p-adic algebraic) group and #eMI(G) . Let {k.} be an unbounded sequence in N such that # has a (k.)th-root for each n e N. I f G(#) = G (resp. 5(#) = G) then there exists an element x contained in a compact subgroup of G (resp. x contained in G(#) ) such that x#( = #x) is embeddable; # itself is embeddable if G is aperiodic. I f G is a connected real Lie group with G(#) = G then there exists an m e N such that #m is embeddable. I f G is a connected nilpotent real Lie group then # itself is embeddable.

Acknowledgement. I thank S. G. DANI, W. HAZOD and M. MCCRUDDEN for helpful discussions and S. G. DAN/and C. S. ARAVINDA for help in preparing this manuscript. I thank Sonderfor- schungsbereich-170, "Geometric und Analysis", GSttingen and the International Centre for Theoretical Physics, Trieste, Professors M. DENKER and A. VERJOVSKV in particular, for hospitality while this work was being done. I also thank the referee for valuable comments and suggestions which led to Corollary 2.3 and improvement of Theorems 3.6 and 3.7.

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RIDDHI SHAH School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400 005 India