infinitely divisible measures onp-adic groups

Journal of Theoretical Probability, Vol. 4, No. 2, 1991

Infinitely Divisible Measures on p-adic Groups

Riddhi Shah 1

Received December 18, 1989," revised June 25, 1990

We show that any infinitely divisible measure # on a p-adic algebraic group (p a prime) has a translate x#, by an element x centralizing the support of/~, such that x# can be embedded in a continuous real one-parameter semigroup {v,},> 0, as x#=vl .

KEY WORDS: p-adic algebraic group; probability measures; infinite divisibility; embedding; one-parameter semigroup.

1. INTRODUCTION

A probability measure # on a locally compact group is said to be infinitely divisible if for any natural number n there exists a probability measure 2 such that 2 n, the n-fold convolution power of 2, equals p. In the literature considerable attention has been paid to the question of whether infinite divisibility of a measure # together with certain structural aspects of the group implies that kt is embeddable in a continuous real one-parameter semigroup {/~,}~>o, as # = # 1 . The reader is referred to Chapter 3 of Heyer's book (5) and the recent papers of S.G. Dani and M. McCrudden (33'(4) for various results in this respect, especially on real Lie groups.

The aim of this paper is to analyze the question in the case of p-adic algebraic groups, namely, algebraic subgroups of GLn(Qp), n ~ N, where Qp is the field of p-adic numbers (for a prime p). To begin with it may be noted that even in the simplest case of Qp itself one does not expect embeddability of all infinitely divisible measures. Indeed, if p is the point measure supported on a nonzero element of Qp, then/~ is infinitely divisible but not

1School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India.

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392 Shah

embeddable. Observe however that such a measure has a translate, namely the point measure at 0, which is embeddable. The following theorem, which may be viewed as the main result of this paper, shows that a similar assertion holds for any p-adic algebraic group G and any infinitely divisible probability measure on G. We recall here that an element in GLn(Qp) is said to be unipotent if all its eigenvalues are equal to 1 and an element in an algebraic subgroup G of GL,,(Qp) is said to be unipotent if it is so in GLn(Qp).

Main Theorem. Let G be a p-adic algebraic group. Then #~MI(G) is infinitely divisible if and only if there exists a unipotent element x in Z(p), the centralizer in G of the support of p, such that xp is embeddable in a continuous, real, one-parameter, convolution semigroup.

The only if part is also upheld for infinitely divisible measures on any closed (not necessarily algebraic) subgroups of GLn(Qp), n ~ N (cf. Theorem 4). It might also be worthwhile to note that under certain conditions one can conclude the measure itself to be embeddable (cf. Proposi- tion 5). A knowledgeable reader may notice that these results for p-adic algebraic groups present a more complete picture than what is currently available for real algebraic groups. An interesting property of the p-adic groups (cf. Proposition 4) plays a crucial role in this regard.

The paper is organized as follows. In Section 2 we introduce the notion of N-root compactness, where N is any set of natural numbers, and prove the property for the group of p-adic numbers, for certain sets. We also note certain properties analogous to those appearing in Ref. 3. Section 3 is devoted to prove a general result on finding a "translate" of a given homomorphism into a topological semigroup from the semigroup of positive rational numbers, which is "locally tight" (cf. Theorem 1). The main theorem is deduced in Section4 after proving some results about factors and roots of measures in p-adic groups and using them together with a corollary of Theorem 1. In the final section we discuss some generalizations of the ideas to products of p-adic groups corresponding to different primes.

2. PRELIMINARIES

Let G be a locally compact (Hausdorff) topological group and let MI(G) denote the space of all probability measures on G, equipped with the usual weak topology. For 2, v~MI(G) we denote by )~v the convolution product of 2 and v and by 2 n, where n ~ N (any natural number), the n-fold convolution product of 2 with itself. A measure # �9 M~(G) is said to

Infinitely Divisible Measures 393

be infinitely divisible if for all n e N, there exists an nth root It,~ such that =

Let N be a subset of N. A locally compact group G is said to be N-root compact if the following condition holds: for any compact subset C of G there exists a compact subset Co of G such that if n e N, any sequence {xl ..... x,,} satisfying the conditions x. = e and

CxiCxjr~Cxi+jr foralli+j<~n

is contained in Co. Clearly a group is N-root compact if and only if it is a strongly root compact group (cf. Ref. 5, Definition 3.1.10). For any subset N of N we put

R(N, I t )= {2meMl(G)12~=itfor some n e N, m<,n}

Lemma 1. Let N be a subset of N and let G be an N-root compact group. Then for any # e MI(G), R(N, It) is relatively compact,

The proof is similar to that of ((i) ~ (ii)) in Theorem 3.1.4 of Ref. 5.

Lemma 2. Let N be a subset of N and let H be a closed central subgroup of G; let ~I: G--, G/H be the natural projection and let #~M1(G). If H is N-root compact and ~I(R(N, #)) is relatively compact, then R(N, it) is relatively compact.

The proof is similar to that of Proposition 8 in Ref. 9. We denote by Qp, where p is a prime number, the field of p-adic

numbers equipped with the p-adic topology. For xeQp, IXlp denotes the p-adic absolute value of x. Through the rest of this section we note some general results about groups of matrices with p-adic entries. We refer the reader to Refs. 2 and 11 for generalities on p-adic groups.

Lemma 3. For n e N let Am= {meNlpn~m}. Then under addition Qp is Nn-root compact for all n E N.

Proof Let n e N and m e N,. Let C be a compact subset of Qp and let 0 > 0 be such that ]Xfp <~ 0 for all x ~ C. Let C' and Co be the subgroups of all elements of absolute value at most 0 and pn- 10, respectively. Then C' and Co are compact subgroups. Let {xl,..., xm} be a sequence in Qp with xm = 0 such that

(C+xi)+(C+xj)c~(C+xi+j)r foralli+j<~m (2.1)

By (2.1) we have

x~+xj-xi+jeC' foralli+j<.m

394 Shah

Adding the elements xl + x l - x 2 , x l + x 2 - x 3 , . . . , Xl"~-Xm_ 1, all of which belong to C', one gets that rnx~ ~ C' or equivalently ImXllp <<, O. Since pn~m, this implies that ]Xllp<~pn-lO. Since x l + x l - x 2 , x~+x2-x3 , . . . , xj + Xm-1~ C', by the non-Archemedian property of p-adic addition, the condition further implies that [xil p ~< pn - N 0 for all i = 2,..., m. Thus x~ ,..., Xm are all contained in Co and therefore Qp is Nn-root compact Vn e N. []

It may be remarked here that Qp is not strongly root compact. Clearly C = {x E Op]lXlp <~ ] } is a compact subset of Qp and, for any n e N, Eq. (2.1) holds for r n = p ~ and xi=i/rn for l<<.i<~m-1 and Xm--0, but there is no compact subset of Qp containing {i/p n [ i, n ~ N }.

Let W be a finite dimensional vector space over Qp and GL(W) the group of nonsingular linear transformations, both equipped with their respective p-adic topologies. Let d e t a denote the determinant of a ~ GL(W).

A sequence {xn} in a locally compact topological space X is said to be divergent if it has no limit point in X, and it is said to be bounded if it is relatively compact.

The following two results are analogues of Propositions 1.4 and 1.5 of Ref. 3 for vector spaces over Qp.

Proposition 1. Let {an} be a divergent sequence in GL(W). Suppose that there exists an a > 0 such that Idet a,[p ~< a for all n. Then there exists a subsequence {a '} of {~n} and a proper subspace W 1 of W such that the following holds: If {wn} is a bounded sequence in W, then any limit point of the sequence {•'Wm} is contained in W 1.

Proof. By fixing a basis {el ..... ea} for W, GL(W) may be realized as GLd(Qp) (the group of dx d nonsingular matrices with entries in Qp). By Cartan's decomposition, any element x e GLa(Qp) can be expressed as, x = k a k ' , where k, k ' e GLa(Zp), which is compact, and a = diag(cq,..., c~a) (cf. Ref. 8, Theorem 2.6.11). The proposition can now be deduced by the same argument of Proposition 1.4 in Ref. 3, using such a decomposition instead of the polar decomposition used there. []

The following proposition can be deduced from Proposition 1 along the same lines as the deduction of Proposition 1.5 from Proposition 1.4 in Ref. 3.

Proposition 2. Let {~} be a divergent sequence in GL(W). Suppose that there exists a b > 0 such that Idet % L p ) b for all n. Then there exists a proper subspace W~ of W such that the following holds: If 0 e M~(W) is such that {%(0)} is relatively compact in M I ( W ) , then the support of 0 is contained in WI.


3. A C R I T E R I O N F O R SHIFT E M B E D D A B I L I T Y

Let Q be the (topological) group of rational numbers and let Q* (resp. Q+) be the subsemigroup of positive (resp. nonnegative) rationals. In this section we prove the following general result on extension of semigroup homomorphisms, up to a shift.

Theorem 1. Let S be a metrizable topological semigroup and f : Q* --, S be a semigroup homomorphism. Let {M.} be a sequence of semigroups in Q* such that the following conditions are satisfied:

1. M, cM~+1 fn and ~ M ~ = Q * .

2. I fnEN, anda, b e M . a r e s u c h t h a t a > b t h e n a - b e M . .

3. VneN, f(]0, 1[ c~M,,) is a compact subset of S.

4. There exists a sequence {ak} in M~ such that l i m k ~ ak=0 and for all heN, (1 --ak)/n!eM~ for all large k.

Then there exist homomorphisms r Q + --, S and a: Q --* S such that r 1 [ c~ Q) is compact, the images of r and a are contained in f (Q*) , and r = a(q) f(q), Vq e Q*.

Proof For all n e N let

K~=f(]O,I[c~M,,) KI,= ('] f ( ] O , x [ n M n ) and K'= U K" x~R~ heN

We first show that K' is a subgroup of the semigroup S. To begin with, it is clear that each K;, is a nonempty compact Abelian subsemigroup of S. Let n ~ N be arbitrary. From the general theory of semigroups it follows that E(K'~), the set of idempotents in K',, is nonempty. Let e(~)eE(K',). Then there exists a sequence {c~ n)} in M, such that

lim c(k ") = 0 and lim f (c(k ~ ) = e ~)

Let te ( ]0 , I [c~M,) be arbitrary. There exists n, e N such that c~")<t Vk > n,. Then t - c~ ~) e M, and

f ( t ) = f ( t - c~)) f(c~ ~)) - (c ~' (3.1) - f k )f(t--c(k ~) Vk>n,

Since K n is compact, the sequence {f( t -c(n)) t k ~ has a convergent subsequence converging to, say, x(n)eKn. Then by Eq. (3.1), f(t)=x(n)e(n)= e(n)x(n)ee(")Knc~K,e (n). Since this holds for each t~(]0, l[c~Mn), K~c e(n)K, ~ K , e ~) and hence K, c e(")Se ~). This implies that K', is a closed subsemigroup of e(")Se ~). Hence e (~) is the identity element in K'~. As this

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holds for any element of E(K',) and the identity of a semigroup is uniquely determined, we get that E(K',)= {e(')}. As K', is a compact semigroup with the unique idempotent e (') as its identity, it must be a subgroup with e (') as the identity. We now apply this for all n. Since M , c M , +a u we see that e (') has to be same for all n, say, e (') = e Vn, and that K' is a subgroup with e as the identity. We also note here for future use that e is the identity in f(Q* ).

To prove the theorem, consider the sequence {ak} in M 1 as in condi- tion4. Recall that a k ~ 0 and VneN, { (1 -ak ) /n ! } is eventually in M1. For every n ~ N we fix a Jn e N such that ( l /n!)e Mj . Then by condition 4 in the hypothesis, for all n, {a~/n!} is eventually in Mj . We now obtain a homomorphism a: Q ~ S by defining the images of (l/n!) as follows.

First we construct inductively subsequences {a~ ")} of {ak} such that each {a~ ")} is a subsequence of {a~"-1)} and the sequence {f(a~.")/n!)} converges to an element in K'. Since {ak} c M1, ak ~ 0 and KI is compact,

J n "

there exists a subsequence {a~ 1)} of {ak} such that {f(a~l))} converges to an element in K'~. Having found for n s N subsequences {a(kt)},..., {a~ n)} with the desired property, we proceed to define ~,,(~+ ( ~ k 1)} as follows. Con- sider the sequence {a~nl/(n + 1)! }, which is eventually in Mjo+~. Since Ki,+~ is compact, there exists a subsequence {a~ "+~)} of {a~ ')} such that {f(a(k"+~l/(n+ 1)!)} converges and the limit is evidently an element in

Jn + 1 "

We thus get sequences {a~ ")} for all n ~ N, with the desired properties. Now let x , e K ' be the limit of {f(a~')/n!)}. We define a(1/n!) to be x~ 1

Jn

the inverse of x, in the group K'. It is obvious that

VneN

Hence, as {(I/n!): n E N} generates Q * , by Lemma 3.1.30 of Ref. 5, a can be extended homomorphically to Q* . Then K' is a subgroup containing a(Q*); hence cr can be extended homomorphically to Q.

We next define ~b: Q+ ~ S by setting

r V q e Q ] r

Since o ( Q ) ~ f ( Q * ) , which is an Abelian semigroup with e as the identity, and since a and f are homomorphisms, ~b is also a homomorphism. Now VneN,


where

x, = lira, f \-~-. ] E K}~ (3.2)

Hence

Vk (3.3)

Since {f((1-a~['))/n!)} is eventually in Kl, it has a convergent subsequence. By Eqs. (3.2) and (3.3), ~b(1/n!) is the limit of a subsequence { f ( ( 1 - a(fl))/n!)}. Since ~b and f a r e homomorphisms, this implies that for all (m/n)e (]0, 1[ n Q), O(m/n), which is the same as O(m(n- 1)!/n!), is the limit of a subsequence of { f ( m ( 1 - a(fl))/n)} and hence belongs to K1. This shows that, ~b(]0, I [ ~ Q ) is contained in K~ and in particular that it is compact. It is obvious that the image of ~b is contained in f ( Q * ). []

Corollary 1. In the above theorem, let S=M~(G) for some locally compact second countable group G and let f:Q*-+MI(G) be a homomorphism. Let M, be a sequence of semigroups in Q * satisfying conditions 1, 2, 3, and 4 of Theorem 1. Then there exist homomorphisms ~b: Q+ ~ MI(G) and ~: Q ~ G such that ~b(]0, 1[ r~ Q) is relatively compact, the image of ~b is contained in f ( Q + ) , and (~(q)=ct(q)f(q)= f(q) c~(q), fq e Q*. (Here the elements of G are identified with respective Dirac measures.)

Proof By Theorem 1 there exist homomorphisms ~b: Q+ ~ S and c r : Q ~ S such that ~b(]0, 1Ec~Q) is compact, the images of ~b and ~r are contained in f ( Q * ) , and ~b(q)= a(q)f(q), f q e Q * . Let K' be the group defined as in the proof of Theorem 1. Then the identity of K' has to be of the form ~oH, the Haar measure of some compact subgroup H of G (of. Ref. 5, Theorem 1.2.10), and the elements of K' are of the form xcott (=6x * con, 6x being the Dirac measure at x), xeN(H), the normalizer of H in G (cf. Ref. 5, Theorem 1.2.13).

For f n ~ N , let xn~N(H) be such that a(1/n!)=xncoH. Consider H,,=x,H, h e N . They are compact subsets of G and if, for all n e N , fn:Hn+~--*Hn are the maps defined by fn(h)=k "+L fheH~+~, then {(Hn, fn): n e N} is a projective system of compact spaces with continuous projections. Hence its projective limit is not empty. This means that there exists a sequence {yn} such that y , e H,, and

n + l Y,, + x = Y~ Vn e N

398 Shah

In turn, this implies that there exists a homomorphism ~: Q ~ G such that ~(1/n!) = y~, Vn ~ N. Moreover since y, ~ N(H) for all n ~ N, ~(Q) c N(H) and hence a(q)--c~(q)m/~ = mH~(q), Vq ~ Q. Thus we have

~(q)=~(q)f(q)=f(q) ~(q) Vq6Q~ []

4. M E A S U R E S O N p-ADIC G R O U P S

Throughout this section we let G be a p-adic algebraic group, where p is a prime; namely, G is an algebraic subgroup of GL~(Qp) for some n E N (cf. Refs. 1, 6, or 7 for generalities on algebraic groups). For/~ ~ MI(G), let G(#) denote the closed subgroup generated by supp/~, and G(/~) be its Zariski closure in G. Let N(p) and N(/~) be the normalizers of G(/~) and 5(#), respectively. Let Z(#) be the centralizer of supp #; clearly Z(#) is a Zariski-closed subgroup of G.

If H is a closed subgroup of G and ~l: G~G/H is the natural projection, then for E ~ MI(G), E/H denotes the subset {y/(2)[2 ~E} of M~(G/H).

For a measure I~M~(G) we denote by F(#) the set of two-sided factors of # in MI(G), i.e., F(~)={2~MI(G)I2v=v2=# for some v EMI(G)}. The following theorem is the p-adic analogue of the corresponding result of S. G. Dani and M. McCrudden for real algebraic groups (cf. Ref. 4, Theorem 1.1). The proof is a straightforward adaptation of a more recent proof of that theorem by the same authors, which is to appear in a forthcoming paper.

Theorem 2. Let # ~ M I(G). Then F(t~)/Z(#) is relatively compact.

Proof We realize G as an algebraic subgroup of GLd(Qp), for some d (cf. Ref. 1, p. 101, Proposition 1.10). Let Md(Qp ) be the vector space of dx d matrices with entries in Qp. Then we have G ~ GLd(Qp) c Md(Qp ). Let W be the (vector) subspaee spanned by G(/~) in Md(Qp). Then nwn-l~ W for all nEN(/~) and w~ W. Hence we get a homomorphism Q: N(I~)~ GL(W) defined by o(n)(w)= nwn 1, for all n ~N(#) and we W. It is clear that 0 is algebraic as a homomorphism of the algebraic groups and that k e r 0 = Z ( # ) . Hence 0(N(#)) is closed in GL(W) (cf. Ref. 10, Lemma 1.22). Let ~: N(I~)/Z(~)--* GL(W) be the homomorphism induced by 0. Then the image of ~ is closed.

Let {/~n} be a sequence in F(#). The supports of #n, n ~> 1, and that of # are contained in N(p) (cf. Ref. 3, Proposition 1.1). Hence by Proposi- tion 1.2 in Ref. 3, there exists a sequence {xn} in N(#) such that {xn#x# 1 } and {x#l#xn} are relatively compact and if ~/ :~(#)~N(/~)/Z(#) is the


natural projection, then {t/(/~,)} is relatively compact in MI(N(y)/Z(I~)) if and only if {q(x,)} is bounded in N(#)/Z(#). Suppose that {t/(x,,)} is not bounded; replacing it by a suitable subsequence, we may assume {~/(x,)} to be a divergent sequence. Let r;, = O(fl(x,)) for all n. Since the image of 0 is closed and {q(x,,)} is divergent, {cr,} is a divergent sequence in GL(W). Passing over to a subsequence one can assume that either I det o-~ I p ~> 1 for all n or I det o-~ 1[ P ~ 1 for all n.

First assume that Met G, Ip >> - 1 for all n. Let ~6M~(G) be defined by = # ( E -~) for all Borel sets E and F be the set of all probability measures

on G of the form

~) = /~]Cl/.~]Iyk2]~[2,.. ykr~ Ir

where r ~> 1 and kl,..., kr and ll,..., lr are nonnegative integers [by conven- tion 20=C5e, Dirac measure at the identity, for all 2eM~(G)] . Since suppTcG(y) , VTeF, F may be realized as a subset of M~(W) in an obvious way.

Since {x~yx, l } is relatively compact, so is {x, Txs ~ } for every 7 s F. Then {~r,(?)} is relatively compact for every 7 s F. Hence, by Proposition 2, there exists a proper subspace W 1 of W such that supp 7 c W1, V7 s F. Since

G(y )= U supp? y ~ F

this means that G(y )= W1. Since W1 is Zariski closed, we must, in turn, have ~ ( y ) c WI. But this is a contradiction since G(y) spans W.

Similarly if [deto-,-l],~>l for all n, then one can use the fact that {xs is relatively compact and arrive at a contradiction. Hence {~/(x,)} is hounded and hence {~/(~tn)} is relatively compact in MI(N(~)/Z(#)). Since this holds for any sequence {/G} in F(y) we get that F(y)/Z(IO is relatively compact. []

Proposition 3. Let/~ e M I(G) be such that G(#)= G. For n ~ N let

iV,= {m+N)~pn)~rn}

Then R(N,, y) is relatively compact, Vn + N.

Proof Since (~(y)=G, Z ( # ) = Z , the center of G. Hence, from Theorem 2, we know that F(#)/Z is relatively compact; i.e., fl(F(#)) is relatively compact, where fl: G -~ G/Z is the natural projection. Let Z ~ be the Zariski connected component of Z in G and let fl: G--* G/Z ~ be the natural projection. Since Z/Z ~ is finite, we get that ~/(F(/I)) is relatively

400 Shah

compact. In particular q(R(N,, #)) is relatively compact. The subgroup Z ~ is an Abelian Zariski connected algebraic group over Qp and hence Z ~ H x Qpk, for some k, for a suitable compactly generated subgroup H (cf. Ref. 1, p. 156, Theorem 4.7). Therefore, by Lemma 3 and the strong root compactness of compactly generated Abelian groups, it follows that Z ~ is Nn-root compact for all n eN. Hence by Lemma 2, R(N,, #) is relatively compact Vn e N.

Proposition 4. If # is infinitely divisible on G, then # is infinitely divisible on G(#).

Proof Let t/:.N(#)-->.N(#)/G(#) be the natural projection. Since N(#)/G(#) is a p-adic algebraic group, one can assume that N(#) /G(#)c GLd(Qp), for some d (cf. Ref. 1, p. 101, Proposition 1.10). It is well known that, for given p and d, there exists an N O such that the order of any finite subgroup of GLd(Qp) is less than or equal to No (cf. Ref. 11, p. LG 4.35, Theorem 1). In particular, there exists an no such that if x EGLd(Qp), x n= e, the identity, for some n then xn~ e.

Now let # be infinitely divisible on G. Let n ~ N be arbitrary. Let 2 be an (nno)th root of #. Then 2 is supported on a coset aG(#), where a ~ N(#) is such that ann~ G(#). Then t/(a) nn~ e and hence, by the choice of no as above, tl(a) ~~ e. Hence a"~ G(#). This implies that 2 n~ which is an nth root of p, is supported on G(#). Thus for all n there exists an nth root of # supported on G(#), which means that # is infinitely divisible on G(#).

Proposition 5. Let G be a p-adic algebraic group. Let #sMI(G) be such that Z(#)c~ G(#) is compactly generated. Then # is root compact on G(p); namely, the set {2m~Ml(G)[supp,).cG(#) and 2~=# for some n e N, m <~ n } is relatively compact. If, further, # is infinitely divisible, then # can be embedded in a continuous, real, one-parameter, convolution semigroup.

Proof Clearly in proving the proposition, without loss of generality we may assume that G(#)= G; for the second assertion this follows from Proposition 4. The assumption also entails that Z(#) c~ G(#) = Z, the center of G, and by hypothesis it is compactly generated. Now the root compactness of # follows from Proposition 8 of Ref. 9 and Theorem 2 above. If, further, # is infinitely divisible, then, since G is totally disconnected, Theorem 3.5.4 of Ref. 5 implies that # can be embedded in a continuous, real, one-parameter, convolution semigroup. [3

For a locally compact group G, a homomorphism f : Q* ~Ml(G) is said to be locally tight if for any q s Q * , f ( ]0 , q [ n Q * ) is relatively compact.


Lemma 4. Let f : Q--+ GLd(Qp ) be an abstract homomorphism of the groups. Then f ( Q ) consists of unipotent elements. If f l Q * (the restriction of f to Q*+) is locally tight, then f is trivial.

Proof Consider the Zariski closure H o f f ( Q ) in GLa(Qp). It is an Abelian algebraic group. Since Q has no subgroups of finite index, H must be Zariski connected. Hence H = H s x H~, where H s (resp. Hu) is the group of all semisimple (resp. unipotent) elements of H (cf. Ref. 1, p. 156, Theorem 4.7). We note that H~. can be diagonalized over a finite extension k of Qp, i.e., Hs c (k*) ~ for some n. It can be easily shown that k* does not contain any nontrivial divisible element. Hence H consists only of unipo-

n tent elements. Hence H ~ Qp, for some n. Now, for any q e Q* and m e N, f(q/pm)=f(q)/pm, the latter being the unique pmth root of f(q) in H. If f ( q ) r for some q E Q * , then this implies that f ( ]0 , q [ c ~ Q * ) is unbounded, contradicting the local tightness o f f [ Q * . Hence f ] Q * is trivial. Hence since f is a homomorphism on Q, f is trivial. []

Theorem 3. Let G be a p-adic algebraic group and let # e MI(G) be infinitely divisible. Then there exists a continuous, one-parameter, convolution semigroup r and a homomorphism e: Q ~ Z ( # ) c ~ ~(#) such that ~b(1) = a(1)#.

Proof In view of Proposition 4, we may assume that (7;(#) = G. Then by Proposition 3, R(n, #), the set of all nth roots of/z, is relatively compact, VneN. Hence, there exists a semigroup homomorphism f : Q * - + M I ( G ) , such that f(1 ) =/~ (cf. Ref. 5, Theorem 3.1.32).

Let N,={meNlp"~m}

and

M,,= {a/beQ* [ a e N , beN,} = {x eQ* l lXlp < p"}

Then Mn are (additive) semigroups, MncM,+~ for all n and 0n M, = Q~_. Also condition 2 of Theorem 1 is satisfied for each M, . By Proposition 3, VneN, R(N~,p) is relatively compact and hence f ( ]0 , 1[ c~M,) is relatively compact. Let a~= 1/(1 + pk), VkeN. Then {ak} satisfies condition 4 of Theorem 1. Therefore, by Corollary 1, there exists homomorphisms ~b: Q+ --+ MI(G) and ~: Q--+ G such that

(b(q)=c~(q)f(q)=f(q) ~(q) V q e Q * (4.1)

and ~b(]0, 1[ c~ Q ) i s compact. Let

K = (~ q~(]O,x[mQ) x~>O

402 Shah

Being a divisible compact group, K is connected. Also all elements of K are of the form xo), , x E G, where co, is the Haar measure on a compact subgroup H of G; co H is the identity of K (cf. Ref. 5, Theorem 1.2.13). Since K is connected and G is totally disconnected, this implies that K is a trivial group, namely, K = {coil}. This implies that ~b is continuous and extends to a unique, real, one-parameter semigroup (cf. Ref. 5, Lemma 3.4.4, Theorem 3.4.6).

Since f(1/n)"=f(1)=# for all n~N, Eq. (4.1) readily implies that c~(q)#c~(q)-l=# for all q s Q . Thus each c~(q) is an element of F(#), when viewed as a Dirac measure. Let c~: Q--,G/Z be the induced homomorphism. Then, since Z(p)=Z, by Theorem 2, c~(Q) is relatively compact. Since G and Z are both algebraic groups, G/Z is contained, as a closed subgroup, in GLa(Qp), for some d. Hence, by Lemma 4 ~ is trivial. This implies that the image of c~ is contained in Z, which in the case at hand is the same as Z(#)c~ ~(#). []

Theorem 4. Let H be a closed (not necessarily algebraic) subgroup of GLn(Qp) for some n~N. Let # ~ M I ( H ) be infinitely divisible. Then there exists a unipotent element x in the center of G(#) (the closed subgroup generated by supp #) such that x# is embeddable in a continuous, real, one-parameter, convolution subsemigroup of M I(H).

Proof Let G be the Zariski closure of H in GLn(Qp). Then, by Theorem 3 and Lemma 4 there exists a unipotent element x in Z(#) c~ G(#) [namely, e(1) as in Theorem 3] such that x# is embeddable in a continuous, real, one-parameter, convolution semigroup, say, {vt}t>o, in MI(G). We shall now show that supp v, is contained in G(#) for all t > 0 . Since v~ = x#, this implies in particular that x ~ G(#) and, as x e Z(#), it is contained in the center of G(#). Thus the proof of the theorem will be complete if we prove the preceding assertion.

By Proposition 1.1 of Ref. 3, for all q ~ Q * , supp Vq is contained in N(x#), the normalizer of G(x#) in G. By continuity, therefore, supp vt is contained in N(x#) for all t > 0 . Let t/: N(x#)~N(x#)/G(x#) be the canonical quotient homomorphism. Then {t/(v,)},>o is a continuous, real, one-parameter, convolution semigroup on N(x#)/G(x#) and t/(vl)= t/(x#) is a Dirac measure. Therefore all r/(vt) must be Dirac measures, and since N(x#)/G(x#) is a totally disconnected group, it follows that they are all supported on the identity subgroup. Thus supp vt c G(x#) for all t > 0.

Now let C be the closed subgroup generated by G(x#) and {x}i Then G(g) is a closed normal subgroup of C. Let O: C --,, C/G(#) be the canonical quotient homomorphism. Considering {0(v~)}t> o and arguing as above, we get that supp v~ c G(#) for all t > 0. This completes the proof. []


Now we prove the main theorem.

Proof of the Main Theorem. One way implication is clear from Theorem 4. Now let x eZ(#) be a unipotent element such that x# is infinitely divisible. By Proposition 4, x/~ is infinitely divisible on G(x/0, the algebraic subgroup generated by the support of x/~. Let U x be the algebraic subgroup generated by x in G. Since x is unipotent, U~ ~ Qp (cf. Ref. 7, p. 96, Lemma C). Moreover since Z(x/~), the centralizer of supp(x/0, is algebraic and x also belongs to Z(xl~), U~ ~ Z(xlt) and hence x is divisible in Z(x#). Now if n e N, x~ is an nth root of x in Z(x#) and #, is an nth root of x/~ supported on G(x#), then we see that x,~ 1#~ is an nth root of #. Hence p is infinitely divisible. This proves the converse.

Remark. We note that the converse implication as in the main theorem is not true for measures on arbitrary closed subgroups of GL,(Qp), n c N, e.g., if H is a compact subgroup containing a nontrivial unipotent element, say, x, then for p = 6x we see that x l~t is embeddable but ~t is not infinitely divisible; the latter part follows from the fact that a nontrivial unipotent element of GLn(Qp ) has a unique p~th root for any r e N and the set of all y t h roots is not relatively compact.

5. MEASURES ON PRODUCT GROUPS

The method used in the proof of Theorem 3 can also be applied to study embeddability of measures on products of p-adic groups corresponding to different primes. In this section we briefly discuss the general situa- tion.

Let G=I~=~Gi, for some l~N , where Gi=Gi(Qp,), for some algebraic group (~i over Qp,, for some primes p~, i = 1 ..... l. Let ~i: G --+ Gi be the natural projection for every i.

A sequence {/~n} in M1(G) is relatively compact if and only if {~ri(/~n)} is so for every i. In particular, p is root compact (resp. factor compact) if ~i(~) is root compact (resp. factor compact) for every i.

Proposition 6. For # ~ M I ( G ) , F(#)/Z(It) is relatively compact.

Proof. Since ni(Z(#))= Z(ni(#)) for every i,

l

G/z(~) ~ I~ G,/Z(~i(~)) i=1

By Theorem 2, F(~t(/~))/Z(~i(#)) is relatively compact for every i. Hence F(It)/Z(lt) is relatively compact.

404 Shah

Proposition 7. Let # E MI(G) be such that the smallest algebraic subgroup containing supp 7ci(#) is Gi, i.e., ~ (~ i (# ) )= G~ for every i. For n E N let

N,= {rnENqp'~m Vi= 1 ..... l}

Then R(N,, #) is relatively compact Vn E N.

Proof For a fixed n and i=l,.. . ,l let N(~)={mENlpT~m}. Then N, c N~, ~) Vi. Hence

~z~(R(N., it)) c R(Nn, rt~(#)) c R(N( n, rc~(#)) Vi

By Proposition3, R(N(,,i),ue(#)) is relatively compact Vi. Therefore u~(R(N., #)) is relatively compact Vi. Hence R(N., #) is relatively compact, Vn EN. []

Lemma 5. Let ItEMI(G) be infinitely divisible. Then # is infinitely divisible on 1-[I= 1 G(rci(#)).

It is easy to prove this lemma along the same lines as Proposition 4.

Theorem 5. Let # E M I(G) be infinitely divisible. Then there exists an element x , E Z(#) such that x~# can be embedded in a continuous, real, one-parameter, convolution semigroup.

Proof By Lemma 5 one can assume that G(rci(#) ) = Gi, Vi. Then by Proposition 7, R(n, It) is relatively compact Vn E N. Hence there exists a homomorphism f: Q* ~ M I ( G ) such that f ( 1 ) = # (cf. Ref. 5, Proposi- tion 3.1.32).

For n E N let N.= {meN[ p'~m Vi= 1,..., l} and

M,,= {a/bEQ* l aEN, b e N . }

Then M. are semigroups of Q * , M. c M . + 1 Vn, and U. M n = Q * . Also condition 2 of Theorem 1 is satisfied for each/14.. By Proposition 7, Vn E N, R(N., It) is relatively compact, and hence f ( ]0 , 1E ~ Mn) is compact. Let

i = l

Then {ak} satisfies condition 4 of the Theorem 1. Since G is totally disconnected, as in the proof of Theorem 3, there exists a real, one-parameter, convolution semigroup ~b: R+ ~MI(G) , and a homomorphism c~: Q--, G, such that

~b(q) = ~(q) f(q) = f(q) a(q) Vq E Q +


Let c~(1)= x , . Then ~b(1) = x , # . Let o~i:7~ioo~, k/i. As in the Theorem 3, we get e i ( Q ) / Z i , where Z i is the center of Gi, is b o u n d e d Yi. Hence by L e m m a 4 , c q ( Q ) c Z i , Yi. Hence x u e Z , the center of G. In the case to which we have reduced the proof, the asser t ion is equivalent to x~ e Z( / l ) fI ' / :

Remark. One can also get a charac te r i za t ion of infinitely divisible measures on p roduc t g roups as in the ma in theorem. However , we will not go into tha t as it gets ra ther technical.

A C K N O W L E D G M E N T S

M y hear ty thanks to S. G. Dan i for the va luable help and guidance.

I t hank M.S . R a g h u n a t h a n for suggest ing the p rob lem and T . N . V e n k a t a r a m a n a and Kir t i Joshi for helpful discussions. I would also like to t hank M. M c C r u d d e n and the referees for va luable comments and sugges- t ions on a p re l iminary version of the paper .

R E F E R E N C E S

1. Borel, A. (1969). Linear Algebraic Groups, W. A. Benjamin, New York. 2. Cassels, J. W. S. (1986). Local Fields. London Math. Soc. Student Texts 3, Cambridge

University Press, Cambridge-London-New York. 3. Dani, S. G., and McCrudden, M. (1988). Factors, roots and embeddabflity of measures

on Lie groups, Math. Z. 199, 369-385. 4. Dani, S. G., and McCrudden, M. (1988). On the factor sets of measures and local

tightness of convolution semigroups over Lie groups, J. Theor. Prob. 1, 357-370. 5. Heyer, H. (1977). Probability Measures on Locally Compact Groups, Springer-Verlag,

Berlin-Heidelberg. 6. Hochschild, G. P. (1981). Basic Theory of Algebraic Groups and Lie Algebras, Springer-

Verlag, Berlin-Heidelberg. 7. Humphreys, J. E. (1975). Linear Algebraic Groups, Springer-Verlag, Berlin-Heidelberg. 8. Macdonald, I. G. (1971). Spherical Functions on a Group of p-adic Type, Publication 2,

Ramanujan Institute, Madras, India. 9. McCrudden, M. (1981). Factors and roots of large measures on connected Lie groups,

Math. Z. 177, 315 322. 10. Prasad, G. (1979). Lattices in semisimpte groups over local fields, In G.C. Rota (ed.),

Studies in Algebras and Number Theory, Academic Press, New York-San Francisco- London.

ll. Serre, J.-P. (1964). Lie Algebras and Lie Groups, W. A. Benjamin, New York.

infinitely divisible measures onp-adic groups

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