Classical theorem of probability on gelfand pairs khinchin theorems and cramer theorem

Classical theorems of probability on Gelfand pairs - Khinchin theorems and Cramer theorem

P. Graczyk and C. R. E. Raja

Abstract

We prove the Khinchin's Theorems for following Gelfand pairs 7^G;K8 sat- isfying a condition 7*8: 7a8 ^G is connected; 7b8 ^G is almost connected and Ad 7^G=M8 is almost algebraic for some compact normal subgroup ^M; 7c8 ^G admits a compact open normal subgroup; 7d8 7^G;K8 is symmetric and^Gis 2- root compact; 7e8^Gis a Zariski-connected ^p-adic algebraic group; 7f8 compact extension of unipotent algebraic groups; 7g8 compact extension of connected nilpotent groups. In fact, condition 7*8 turns out to be necessary and suHcient for^K-biinvariant measures on aforementioned Gelfand pairs to be Hungarian.

We also prove that Cramer's theorem does not hold for a class of Gaussians on Compact Gelfand pairs.

KEY WORDS Locally compact groups, Lie groups, algebraic groups, Gelfand pairs, proba- bility measures and factorization theorem, Khinchin's central limit theorem, limit theorem, Cramer theorem and anti-indecomposable measures, in;nite divisibility and embedding.

1 Introduction

A classical theorem of Khinchin known as Khinchin factorization theorem which we would call Khinchin's 5rst theorem says that any probability measure on ^R can be written as a countable product of indecomposable measures 9possibly in5nite: and a probability measure without indecomposable factors. Khinchin's factorization the- orem was extended to all commutative Hausdor@ metrizable groups by Ruzsa and Szekely 9see DRSE:. In DRSE Khinchin's factorization for measures on abelian Hausdor@

groups is achieved by proving that the semigroup of probability measures on such groups form a 5rst countable Hungarian semigroup. The notion of Hungarian semi- groups was introduced by Ruzsa and Szekely and it was studied in DRSE. It is shown in DRSE, that any element in a 5rst countable Hungarian semigroup is a countable

1991 Mathematics Subjects Classi;cation:

primary; 22A20, 43A05, 60B15 and 60F17 secondary; 22E30, 22E35 and 43A85.

1

product of indecomposable elements 0possibly in2nite3 and an anti-indecomposable element. It is shown in 9R; that semigroup of ^K-biinvariant probability measures on real or ^p-adic reductive symmetric spaces is a Hungarian semigroup and hence the factorization theorem holds for such semigroups.

Another classical theorem of Khinchin which we would call Khinchin's second theorem says that any antiindecomposable measure on ^Ris in2nitely divisible. This result was extended to many other groups by various authors. In 9RS;, Khinchin's second theorem is also proved for anti-indecomposable measures on 2rst countable abelian HausdorG groups by showing that semigroup of measures on such groups form a normable Hungarian semigroup.

At this point we would like to note that Delphic semigroup is another approach to prove the Khinchin's Theorems for abelian semigroup. It has been proved in 9G3;

that the semigroup of measures on noncompact symmetric spaces form a Delphic semigroup but it can easily be seen that measures on compact symmetric spaces do not form a Delphic semigroup 0see 9G3; for de2nition of Delphic semigroup3.

The study of probability questions on Gelfand pairs has been initiated by Letac in 9Le; and by Heyer in 9He1; and 9He2; where the author proves the Khinchin type factorization result for some class of Gelfand pairs.

In this article we attempt to prove Khinchin's Theorems for measures on Gelfand pairs. In section 2 we introduce the concept of Gelfand pair and we also prove some preliminary results which are needed in the succeeding sections to prove Khinchin's Theorems. In the section 3 we prove results on factor compactness which are needed in proving Khinchin's Theorems. In section 4, we prove Khinchin's Theorems for connected Gelfand pairs. In sections 5 and 6, we prove Khinchin's Theorems for certain Gelfand pairs which include discrete groups and doubly transitive groups and

p-adic algebraic groups.

One of the axioms of Hungarian semigroup is that the set of factors of an element is compact modulo the group of units. Some applications of this type of factor compactness in analysis and arithmetic of probability measures are limit theorems and embedding of in2nitely divisible measures: see 9S; and 9Te; for more details on limit theorems on general locally compact groups. In section 7 we obtain limit theorems for measures on certain Gelfand pairs and we also obtain the embeddability of in2nitely divisible meausres on certain Gelfand pairs: the embedding problem for general groups are studied by various authors 0see 9Mc;3.

One more classical theorem of Khinchin which we would call Khinchin's third the- orem says that in2nitesimal limits are in2nitely divisible. This result was extended by Ruzsa and Szekely to abelian metrizable groups such that the set of characters separates points of the groups by showing that the semigroup of probability measures on such groups form a stable normable Hungarian semigroup 0see 9RS;3. In the sec- tion 8 we prove the normability which in turn proves second and third theorems of Khinchin for Gelfand pairs.

In section 9 we discuss Gaussian measures on compact Gelfand pairs and prove 2

that Gaussian measures are not in the class of anti-indecomposable measures. This in particular implies that Gaussian measures on certain compact Gelfand pairs do not satisfy Cram7er theorem: Cram7er theorem says that Gaussian measures on reals have only Gaussian factors and Cram7er theorem was generalized to abelian groups by various authors =see >Fe@A and to symmetric spaces of non-compact type by Graczyk

=see >G2@A. While proving this we obtain a class of measures which have indecompos- able factors. In the last section we make some remarks on central limit theorems of Lindeberg-Feller type for probabilities on Gelfand pairs.

2 Preliminaries

Let^Gbe a locally compact second countable group and ^K be a compact subgroup of

G. Then we say that the pair =^G;KA is a Gelfand pair if the convolution semigroup

P

K=^GA of all^K-biinvariant probability measures on ^G is a commutative semigroup;

see >BJR@,>F@, >GV@ and >MV@for more on harmonic analysis on Gelfand pairs. For any probability measure ^#, ^S=^#A denotes the support of^# and for any compact subgroup

M of ^G,^!M denotes the normalized Haar measure on^M.

Examples (1*

For any locally compact abelian group^Gand any compact subgroup

K of ^G, =^G;KA is Gelfand.

(2*

Semigroup of probability measures on a real reductive group ^G that are ^K- biinvariant for a maximal compact subgroup ^K of ^Gis commutative and hence the pair =^G;KA is a Gelfand pair =see >R@A.

(3*

The semigroup of probability measures on the Euclidean motion group ^G that are ^SO=ⁿA-biinvariant is commutative and hence =^G;SO=ⁿAA is a Gelfand pair.

Proposition 2.1

^{Let G} be a locally compact second countable group and ^K be a compact subgroup of ^G. Then the following are equivalent:

1. =^G;KA is a Gelfand pair;

2. for any ^x;y2G, ^KxKyK =^KyKxK; 3. for any ^x;y2G, ^xy2KyKxK;

4. the algebra ^L1K=^GA of ^K-biinvariant integrable functions on^G is a commutative algebra.

Proof

One may prove that =1A implies=2A by considering the^K-biinvariant measures

!

K ,

x

!

K and ^!K,y!K, for ^x;y2Gand that =2A implies =3A is obvious.

3

We now prove (3* implies (4*. We 1rst prove that (3* implies G is unimodular.

Letm be a left invariant Haar measure on G. Let U be a compact neighbourhood of e such that KUK = U. Then for g ²G,

m(Ug* =^R &^U(xg*dx

=^R &^E(gkx*dx (k depends on x*

=^R &^U(x*dx (ydx = dx; y ²G and KUK = U*

=m(U*

This proves that G is unimodular. The rest of the proof of (3* implies (4* is quite similar to Theorem 1.12 of CBJRG.

The implication (4* implies (1* follows from the existence of approximate identity sequence in L^1K(G* (see Lemma 1.6.8 of CGVG or Theorem 2.2.28 of of CBHG*.

Thus, the above result says that our de1nition of Gelfand pair agrees with the classical notion of Gelfand pair. We now prove that Gelfand pair property preserves quotients.

Proposition 2.2 Let (G;K* be a Gelfand pair and H be a normal subgroup of G. Let M = KH=K. Then (G=H;M* is also a Gelfand pair.

Proof Let x;y ² G. Since (G;K* is a Gelfand pair, by Proposition 2.1, xy ² KyKxK, that is there exist k¹,k² andk³ such thatxy = k¹yk²xk³. This implies that xHyH ²k¹HyHk²HxHk³H. Again by Proposition 2.1, (G=H;M* is also a Gelfand pair.

In this article we attempt to prove all three theorems of Khinchin for Gelfand pairs. This is achieved by applying the Hungarian semigroup theory: see CRG and CRSG for more details on Khinchin's theorems and Hungarian semigroups.

LetS be a commutative HausdorU semigroup with identity e. Let^! be a relation de1ned on S for x;y ²S, by

x^!y^,x = ry and y = sx

for some r;s² S. Any two elements x and y of S are said to be associates if x ^!y.

An element u of S is calledunit of S if it is invertible in S. Let S be the quotient semigroup corresponding to the relation ^!and 4:S ^!S be the canonical quotient map (this notation is followed throughout the article*. We say that the semigroupS is Hungarian if it satis1es the following properties:

(H-1* the set of associate pairs is a closed subset of S^$S;

(H-2* ifx and y are associates, then x = uy for some unit u in S;

(H-3* the set of divisors (factors* of any element inS is compact.

4

For any two subsets^Aand ^B of^S and any ^s;t2S, let us write^{At !sB} if for any

a 2A, there exists a ^b2B such that ^a =^sb and ^b=^ta. A Hungarian semigroup^S is calleduniformly Hungarian if for any ^s;t2S and subsets^A and ^B of ^S such that

A

s

!

t

B there exist units ^u and ^v in ^S such that ^{Au !vB}. The notion of uniformly Hungarian semigroup was introduced by A. Zempl=eni in >Z? to study the heredity of Hun and Hungarian semigroups.

A sequence A^xnB in a topological space^X is said to berelatively compactorbounded if it is contained in a compact subset of ^X.

We Drst prove following elementary results that are needed in proving the main results. First of such results characterizes all units in the semigroup of probability measures on Gelfand pairs.

Proposition 2.3 Let ^G be a locally compact group and ^K be a compact subgroup of

G. Suppose ^. and ^/ are ^K-biinvariant probability measures on ^G such that ^/. =

./=^!K. Then^.=^x!K for some ^x in^NA^KB, the normalizer of ^K. Suppose A^G;KB is a Gelfand pair and ^S is the semigroup of^K-biinvariant probability measures on^G. Then ^. is a unit in ^S if and only if ^.=^x!K for some ^{x 2N}A^KB.

Proof Let^. and ^/ be ^K-biinvariant probability measures on ^Gsuch that

."/ =^!K =^/".:

This implies that

SA^.B^SA^/B^#SA^/B^SA^.B^$K

and hence for any ^{g 2S}A^/B, ^.g is a left ^K-invariant probability measure supported on ^K. Thus, ^.g = ^!K. Since^SA^.B^$Kg?1, we have ^!Kx=^!Kg?1 for all ^{x2 S}A^.B.

Thus,

.=^!Kx for any ^x2SA^.B. Similarly we may prove that

.=^x!K for all ^{x 2S}A^.B. This implies that

x!

K x

?1 =^!K

and hence ^{x2 N}A^KB. Second part of the proposition follows from the fact that any measure of the form ^x!K, for ^x2NA^KB is ^K-biinvariant and^x?1!K is the inverse of

x!

K.

The following lemma is very useful and used often in the sequel without even referring to it.

5

Lemma 2.1

Let ^G be a locally compact group and ^K be a compact subgroup of ^G. Suppose ^G;K! is a Gelfand pair. Then any compact subgroup ^M containing ^K is normalized by ^{N K}!, the normalizer of ^K and ^G;M! is also a Gelfand pair.

Proof

Let ^{x 2 N K}!. Then ^x!K and ^!Kx?1 are ^K-biinvariant probability mea- sures. Since ^G;K! is a Gelfand pair, this implies that

!

xMx

?1 =^x!K!M!Kx?1 =^!Mx!Kx?1 =^!M!K =^!M:

Thus, ^xMx?1 = ^M. The second part of the theorem follows from the fact that

M-biinvariant probability measures are also ^K-biinvariant.

The next lemma determines when the semigroup of probability measures on a Gelfand pair satis@es H-2!.

Lemma 2.2

^LetGbe a locally compact group and^K be a compact subgroup of ^G. Let

S be the semigroup of all ^K-biinvariant probability measures on ^G. Suppose ^G;K! is a Gelfand pair. For any subgroup ^H of ^G, ^{N H}! denotes the normalizer of ^H in

G. Then the following are equivalent:

1. AH-2D holds for ^S;

2. for every compact subgroup^M of ^Gcontaining^K and^{x 2G}such that^xKx?1!

M, we have ^{x2N K}!^M;

3. for every compact subgroup ^M of ^G containing ^K, ^{N M}! =^{N K}!^M.

Proof

Suppose^S satis@es H-2!. Let^M be a compact subgroup of ^Gcontaining^K. Suppose ^x2G is such that

K and ^{xKx?1! M: i}!

Consider

+ =^!M and ^, =^{!K-x!M: ii}! Then ⁺ and ^, are in ^S. Let

.

1 =^!K-x!K and ^.2 =^!K-x?1!K: Then ^{.1;.2 2S} and by ⁱ! we get that

, =^!K-x!M =^!K-x!K!M =^.1+ and

+=^!M =^!K!M =^!K-x?1!K-x!M =^.2,:

6

Thus, and ! are associates. Then = u! for some unit u in S. By Proposition 2.3, = u! = g!^K! = g!

for some g ²N8K9. Thus, by substituting 8ii9, we have

!^K)^x!^M =g!^M

and hence KxM = gM for some g ² N8K9. This implies that x ²gM ^! N8K9M.

This proves that 819 implies 829.

Suppose 829 holds. LetM be a compact subgroup of G containing K. Then xKx^{?1 !}xMx^?1 =M

for all x ² N8M9 and hence by assumption x ² N8K9M for all x ² N8M9. This implies that N8M9^! N8K9M. Since K ^!M, N8K9 normalizes M, that is N8K9^! N8M9. Thus, N8M9 = N8K9M. This proves that 829 implies 839.

Suppose for every compact subgroup M of G containing K, we have N8M9 = N8K9M. We now prove that S satisDes 8H-29. Let !, , -¹ and -² be inS. Suppose

! = -¹ and = -²!:

Then ! = -¹-²!

and hence by Theorem 1.2.7 of HHeI,S8-¹9S8-²9^!fg ²G^jg! = ! = !g^g=M, say.

Since ! is K-biinvariant, K ^!M. Replacing -ⁱ by!^{M %}-ⁱ, for i=1,2, if necessary we may assume that -^i%!^M =-ⁱ, for i = 1;2. Then we have,

-¹-²=!^M =-²-¹:

By Proposition 2.3, -ⁱ = xⁱ!^M for some x^{i 2} N8M9 for i = 1;2. This implies that x^{i 2}N8K9M = MN8K9 for i = 1;2. This impliesthat -ⁱ =gⁱ!^M for someg^{i 2}N8K9 for i = 1;2. Thus, ! = g¹ for g^{1 2}N8K9. This proves that 839 implies 819.

We say that a pair 8G;K9 consisting of a locally compact group G and a compact subgroup K of G^{satis$es condition **,}if 829 or 839 of Lemma 2.2 is satisDed. Thus, a Gelfand pair 8G;K9 satisDes condition 8*9 if and only if the semigroup of K-biinvariant probability measures onG satisDes 8H-29. We will see that this condition plays a vital role in proving Khinchin's Theorems.

It is easy to see that when K is a maximal compact subgroup, condition 8*9 is satisDed. It is also easy to see that 8G;K9 satisDes condition 8*9 when G is a connected Lie group and K is a maximal torus which may be seen as follows: suppose M is a compact group containingK, then for x²N8M9, xKx^?1 =mKm^?1for somem² M and hence N8M9 = N8K9M. Also if there exists a compact group L contained in K

7

such that &G;L' satis)es condition &*', then &G;K' also satis)es condition &*' which follows from the equation that

N&M' = N&L'M = N&L'KM = N&K'M for any compact subgroup M containing K.

We now prove that the Gelfand pair &G;K' satis)es the condition &*' when G=K is a compact Riemannian symmetric space. We )rst observe the following:

Proposition 2.4

^Let G=K be an irreducible Riemannian symmetric space. Then K is a maximal proper compact connected subgroup of G.

Proof

Let H be a compact connected subgroup of G containing K properly. Let

G;^K and ^H be the Lie algebras of G, K and H respectively, with ^G =^K#P. Since G=K is irreducible, Ad &K' acts irreduciblyon the subspace^P where Ad is the adjoint representation of G on its Lie algebra ^G. Since H contains K, the subspace ^H%P is a Ad &K'-invariant subspace of ^P and hence ^H%P = &0' or ^P. This implies that H = K or H = G. This proves the proposition.

Lemma 2.3

^Let G=K be a compact Riemannian symmetric space. Then the Gelfand pair &G;K' satis9es the condition :*<.

Proof

Let &G;K' be a compact Riemannian symmetric pair. Let ~G be the simply connected covering of G and p: ~G^!G be the covering map of G. Let ~K = p^?1&K'.

Let M be a compact subgroup of G containing K such that xKx^{?1 '} M for some x ² G. We now claim that x² N&K'M. Let ~G¹; ~G²;⁾⁾⁾; ~Gm be )nite set of simple Lie subgroups of G¹ such that

G = ~G^1* ~G^2*)))* ~Gm:

Now for each i, 1⁺i⁺ m, there exists a compact subgroup ~Ki of ~G such that ~Gi= ~Ki

is a irreducible Riemannian symmetric space and

~K = ~K^1* ~K^2*)))* ~Km: Now let ~M = p^?1&M'. Then by Proposition 2.4, we have

~M⁰ = ~K^1*)))* ~Kr^* ~Gr^+1*)))* ~Gr

for some r, 0 ⁺r ⁺m where ~M⁰ is the connected component of identity inM. Now let y = &x¹;x²;⁾⁾⁾;xm' be inp^?1&x'. Since ~K is connected, we have that y ~Ky^?1 and

~K are contained in ~M⁰. This impliesthatxi~Kix^?1_i ^' ~Ki, for 1⁺i⁺r and hence since

~K is a connected Lie group, we have xi ² N& ~Ki', for 1 ⁺ i ⁺ r. This implies that y ² N& ~K' ~M and hence p^?1&x'^' N& ~K' ~M. Thus, x² p&N& ~K' ~M' = p&N& ~K''M ^' N&K'M. This proves condition &*' for any compact Riemannian symmetric space.

8

Remark

The following gives an example of a Gelfand pair which does not satisfy the condition 6*8. Let^Q2 be the additive group of 2-adic integers and^j!jbe the 2-adic norm on ^Q2. Let ^L=^{fx2 Q2 jjxj}= 1^g. Let ^K be the subgroup of automorphisms generated by the automorphism ^{x 7! ?x}. Let ^G be the semidirect product of ^K and ^Q2. Let ^{x0 2 Q2} be such that ^{x0 62 L} but 2^{x0 2 L}. Then the semigroup of all ^K-biinvariant probability measures on ^G is isomorphic to the semigroup of all symmetric probability measures on ^Q2. Thus, 6^G;K8 is a Gelfand pair. Let ^M be the compact subgroup of ^G generated by ^L and^K. Then since 2^{x0 2L}, it is easy to see that ^x0 normalizes^M. Since ^N6^K8 =^K and ^{x0 62M}, we get that ^N6^K8^M =^M is a proper subgroup of ^N6^M8. Thus, 6^G;K8 is a Gelfand pair which does not satisfy the condition 6*8.

We now present various types of Hungarian semigroups which are useful in proving the heredity of Hungarian semigroups and limit theorems. For any subset ^C of a semigroup ^P, let^TC be the set of all factors of elements of ^C. Let ^*:^{S !S} denote the canonical quotient map. A Hungarian semigroup ^S is called stable if for every compact set ^C of ^S , ^TC is compact. A Hungarian semigroup ^S is called division compact if for any two compact subsets ^C and ^L of ^S, the set ^C=L = ^{fs 2 S j} there exists a ^{l 2 L;sl 2 Cg} is compact. It is shown in KRSM that the semigroup of all compact-regular probability measures on an abelian HausdorN topological group

G is a stable division compact Hungarian semigroup 6see Chapter 3, Theorem 1.1 of KRSM8.

A Hungarian semigroup ^S is called strongly stable if for any compact set ^C of

S, there is a compact set ^L of ^S such that ^*6^TC8 = ^*6^L8. It should be noted that strongly stable Hungarian semigroups are stable. A. ZemplReni introduced the notion of strongly stable Hungarian semigroups in KZeM. It is shown in KZeM that for a lo- cally compact Srst countable abelian group ^G, the semigroup ^P6^P6^!!!6^G8^!!!88 is a strongly stable division compact uniformly Hungarian semigroup with Prohorov prop- erty. In KRM, it is proved that ^P6^P6^!!!6^S8^!!!88 is a strongly stable division compact uniformly Hungarian semigroup with Prohorov property when ^S is the semigroup of

K-biinvariant probability measures on a real reductive symmetric space. Here we prove a similar result for certain Gelfand pairs.

In order to achieve Khinchin's second and third Theorems, that is any anti- indecomposable measure or any inSnitesimal limit is inSnitely divisible, Ruzsa and Szekely introduced the concept of normable Hungarian semigroups. For any ^s in a Hungarian semigroup ^S, deSne ^H6^s8 as the maximal idempotent factor of ^s in ^S 6see 22.11 of KRSM for the existence of ^H6^s88. A normable Hungarian semigroupis a Hungarian semigroup ^S satisfying the condition that for every ^{s 2 S} that is not an associate of an idempotent, there exists a map X^s:^Ts!K0^;18 such that

X^s6^ab8 = X^s6^a8 + X^s6^b8 6^p8 for any ^{a;b 2 Ts} with ^{ab 2 Ts}, X^s6^s8 ³0 and X is continuous at ^H6^s8 where ^Ts is the set of factors of ^s. Any map satisfying condition 6^p8 is called a partial homo-

9

morphism. It is proved +RS., that the semigroup of probability measures on a locally compact abelian group is a normable Hungarian semigroup. Combining the Kendall homomorphism of +G. with the results in +R., we get that the semigroup of probability measures on a reductive symmetric space is a normable Hungarian semigroup. In the next sections we prove that the semigroup of ^K-biinvariant probability measures on Gelfand pairs is also normable.

Another application of normable stable Hungarian semigroup is the inCnite divis- ibility of an inCnitesimal limit, that is Khinchin's third Theorem. We will answer this question aHrmatively in the section 8.

3 Factor compactness

The following lemma is an important tool in proving the factor compactness which is useful in establishing the strong stability and limit theorems: see +DM., +DR. and +M.

for results on factor compactness for measures on general locally compact groups.

Lemma 3.1

Let ^N be a connected nilpotent Lie group and^A be a group acting on ^N by automorphisms such that the induced action on the Lie algebra of ^N is semisimple.

Let ^X be a subset of ^N such that for any sequence M^xnN in ^X, the sequence M^xn%M^x?1n NN

is relatively compact for every ^{% 2 A}. Then for each ^{x 2 X}, there exists ^ax and ^bx such that

x=^bxax;

fb

x g

x2X is relatively compact and ^%M^axN = ^ax for all ^{x 2 X} and ^{% 2 A}. In other words, ^X is relatively compact in ^{N =NA} where ^NA denotes the group of all ^A-=xed points in ^N.

Proof

Let ^LM^NN be the Lie algebra of ^N. We Crst consider the case when ^N is abelian. There is no loss of generality in assuming that ^N is a vector group. Let ^U be the subspace of ^LM^NN consisting of all ^v2LM^NN such that

d%M^vN =^v

for all ^{% 2A}. Then there exists a^A-invariant subspace ^W of ^LM^NN such that

LM^NN =^U#W:

Now for each ^{x 2 X}, there are ^ax and ^bx in the exponential image of ^U and ^W respectively, such that

x=^bxax: 10

Suppose ^fbxgx2X is not relatively compact, then there exists a sequence 4^xn5 in

X such that

b

n =^{bxn !1:}

Let ^Yn2W be such that

exp4^Yn5 =^bn: Since exp is a di:eomorphism, we have

Y

n

!1

and 4^d'4^Yn5^?Yn5 is relatively compact for all ^'2A and hence

d'4 ^Yn

jjY

n jj

5^{? Yn}

jjY

n jj

!0

for all^'2Awhere^jj'jjis the Euclidean norm on ^L4^N5. By passing to a subsequence, if necessary we may assume that

Y

n

jjY

n jj

!Y

and hence ^Y is a nonzero vector in ^W such that

d'4^Y5 =^Y

for all^'2A. This is a contradiction. This proves that^fbxgx2X is relatively compact.

We now consider the general case. The rest of the proof is based on induction on dimension of ^L4^N5. Suppose dimension of ^L4^N5 is one, the result follows from the abelian case. Now let ^Z be the center of ^N and ^L4^Z5 be the Lie algebra of ^Z. Since the action of ^A on ^L4^N5 is semisimple, there exists a ^A-invariant subspace ^W of ^L4^N5 such that

L4^N5 =^L4^Z5^(W:

Since ^N is nilpotent,^Z4^N5 is of positive dimension, now applying induction hypoth- esis to ^{N =Z}4^N5 yields that for each ^x2X, there are ^ax, ^bx and ^zx such that

x=^bxaxzx

where^axZ is Gxed by all elements of^A,^zx is in^Z for all^xand ^fbxgx2X is a relatively compact subset of ^N. Let exp be the exponential map of ^L4^N5 into ^N. Since ^N is a connected nilpotent Lie group, by Theorem 3.6.1 of KVM, exp is an onto map. Since

L4^N5 =^L4^Z5 +^W, for each ^x2X, there exists a ^{vx 2L}4^Z5 and ^{wx 2W} such that exp4^vx+^wx5 =^ax

and hence, since ^vx belongs to the center of the Lie algebra, by Corollary 2.13.3 of KVM, we have

exp4^wx5exp4^vx5 =^ax 11

for all ^{x 2 X}. Thus, for each ^{x 2 X}, replacing ^ax by ^axexp4^?vx5, we may assume that

a

x

2exp4^W5 for all ^{x 2X}.

We now claim that ^ax is :xed by all elements of ^A. Let ^{wx 2 W} be such that exp4^wx5 =^ax. Since ^axZ is :xed by elements of ^A, we have

(4exp^q4^wx+^L4^Z55 = exp^q4^wx+^L4^Z55

for all ^{(2 A} where exp^q denotes the exponential map of the Lie group ^{N =Z}. Since

N is a connected nilpotent Lie group, exp^q is a diAeomorphism of the Lie algebra

L4^N5^=L4^Z5 onto ^{N =Z} 4see Theorem 3.6.2 of EVG5. This implies that

(4^wx +^L4^Z55 =^wx+^L4^Z5 for all ^{( 2A} and hence

(4^wx5^{?wx 2L}4^Z5

for all ^{( 2A}. Since ^{wx 2W} which is an ^A-invariant subspace, we have

(4^wx5^{?wx 2W "L}4^Z5 = 405

for all ^{( 2A}. This implies that ⁽4^wx5 =^wx for all ^(2A and hence

(4^ax5 =^ax for all ^{( 2A} and all^x2X.

Now for any sequence 4^zxn5, and for each ^(2A,

x

n

(4^x?1n 5 =^bxnaxnzxn(4^zx?1n5⁽4^a?1xn5⁽4^b?1xn5 =^zxn(4^z?1xn5^bxn(4^b?1xn5^:

This implies that 4^fzxn(4^zx?1n55 is relatively compact. Now the result follows from the abelian case.

Lemma 3.2

^{Let U} be a unipotent algebraic group and ^K be a compact group of automorphisms on ^U. Let ^X be a subset of ^U such that for any sequence 4^xn5 in ^X, the sequence 4^xn(4^x?1n 55 is relatively compact. Then ^XUK is relatively compact in

U=U

K where ^UK is the group of all ^K-;xed points of ^U.

Proof

Since ^U is a unipotent algebraic group, exponential is a diAeormorphism of the Lie algebra of ^U onto ^U. Since ^K is compact, the induced action of^K on the Lie algebra of ^U is semisimple. Thus, one may prove the lemma by arguing as in Lemma 3.1.

The next result extends Lemma 3.1, to connected solvable groups with a faithful representation and when the group of automorphisms is a compact connected group.

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Lemma 3.3

Let ^G be a connected solvable Lie group with a faithful representation and ^K be a compact connected group of automorphisms of ^G. Suppose ^X is a subset of ^G such that for every sequence ^xn! in ^X, the sequence ^xn^{$ x}n!^?1! is relatively compact for all ^$2K. Then for each ^x2X, there exist ^bx!and ^ax!in ^G such that

x=^bx^ax^;

fbx^{j x2Xg} is relatively compact and ^{$ a}x! = ^ax for all ^$2K and all ^x2X.

Proof

Let^Gbe a connected solvable Lie group with a faithful representation. Then there exists a compact connected abelian subgroup ^T of ^G and a simply connected normal subgroup ^H of^G such that^G=^TH see <Ho>!. Let^N be the nilradical of^H and ^H be the Lie algebra of ^H. Let ^H1 be the Lie subalgebra of^H such that

H

1 =^{fv2Hj$ v}! =^v; for all ^{k 2Kg:}

Let ^H1 be the connected Lie subgroup of ^G corresponding to the subalgebra ^H1. Then ^H1 is a simply connected closed subgroup of ^H see Theorem 3.18.12 of <V>!

and ^{$ h}! = ^h for all ^{h 2 H1} and all ^{$ 2 K}. By Leptin's Theorem see <BJR>!,

H =^NH1. Now the lemma follows from Lemma 3.1.

We need the following lemmas which are quite useful in establishing the main result of this section.

Lemma 3.4

Let ^G be a locally compact second countable group and ^K be a compact subgroup of ^G. Let ^/n! be a sequence of automorphisms of ^G. Suppose ^/n ^!K!! is a relatively compact sequence in ^{P G}!. Then ^/n ^k!!is a relatively compact sequence in ^G, for all ^{k 2K}.

Proof

Suppose ^/n ^k0!! is not relatively compact for some ^{k0 2 K}. Then by pass- ing to a subsequence, if necessary we may assume that ^/n ^k0!! has no convergent subsequence and ^/n ^!K! ^{! 1 2 P G}!. Since each of ^/n ^!K! is an idempotent, ¹ is an idempotent and hence ¹=^!M for some compact subgroup ^M of ^G see Theorem 1.2.10 of <He>!. Since ^G is second countable, ^{P G}! is metrizable. Let ^Ui! be a de- creasing sequence of compact neighbourhoods of ^M. Since ^/n ^!K! ^{! !}M, for each

i'1, there exists an ⁿi such that

!K ^/n^{?1i U}i!!⁶1^? 1 2ⁱ

see <P>! and we may assume that ⁿi ⁷ⁿi⁺¹ for all ^i'1. Let

B =⁾¹_m^=1*1_i⁼_m^/_n^{?1i U}i!^: Then ^B is a Borel subset of ^G and

!K ^GnB!^,X1

i⁼m

21ⁱ 13

for all m 1. This implies that !K/B0 = 1. Let b ²B. Then there exists a m 1 such that $nⁱ/b0 ² Ui for all i m. Since /Ui0 is a decreasing sequence, we have

$nⁱ/b0²Um for alli m and hence /$nⁱ/b00 is relatively compact for all b²B. Let H be the set of all k in K such that /$nⁱ/k00 is relatively compact. Then H is a co-null subgroup of K. By Proposition B.1 of CZiE, H = K. This implies that /$n/k⁰00 has a convergent subsequence. This is a contradiction. Thus, we prove the lemma.

We make the following observation which is essentially Lemma 2.1 of CDRE.

Lemma 3.5

^Let V be a %nite-dimensional algebra over real or p-adic %eld. Let /$n0 be a sequence of algebra automorphisms of V. Then there exists a subalgebra W of V such that

1. W = ^fw²V ^j/$n/w00 is bounded ^g and

2. if /$n/.00 is relatively compact for .^2P/V 0^{, then} . is supported on W^.

Proof

Since V is of Knite-dimension, there exists a vector subspace W of V such that /$n/w00 is bounded if and only if w²W. Now let .^2P/V 0 be such that /$n/.00 is relatively compact. We now claim that the support of. is contained in W, in other words, for each v ² S/.0, /$n/v00 is a bounded sequence. Suppose for some v ² V , the sequence /$n/v00 is not bounded. Then there exists a subsequence /$kⁿ0 of /$n0 such that

$kⁿ/v0^!1 and $kⁿ/.0^!1

for some 1 ^{2 P}/V 0. Then by Lemma 2.1 of CDRE, there exists a subspace W⁰ of V such that /$kⁿ/w00 converges for all w ²W⁰ and . is supported on W⁰. This implies in particular, that /$kⁿ/v00 converges. This is a contradiction. Thus, v ² W. This proves the lemma.

Proposition 3.1

Let G be an almost connected Lie group and G⁰ is a semisimple Lie group. Let K be a compact subgroup of G^{. Suppose} /G;K0 is a Gelfand pair and S is the semigroup of all K-biinvariant probability measures on G. Let /.n0 be a relatively compact sequence in S and /4n0 be a seqeunce such that for each n 1, 4n

is a factor of .n. Then there exists a sequence /xn0 from the center of G such that /xn4n0 is realtively compact.

Proof

Since /.n0 is relatively compact, by Theorem 1.2.21 of CHeE, there exists a sequence /gn0 in G such that /gn4n0 is relatively compact. Let Ad be the adjoint representation ofG and let H = Ad /G0. Then H is a connected algebraic semisimple group. Thus, the center of H is Knite and H ^'G=Z where Z is the center of G. By Proposition 2.2, we have /H;KZ=Z0 is also a Gelfand pair. Let M be a maximal compact subgroup of H containing KZ=Z. Then /H;M0 is a Gelfand pair. Let p:G ^! H be the cannonical quotient map. Then /p/.n0⁾!M0 is relatively compact and p/4n0⁾!M is a factor of p/.n0 for all n 1. Then there exists a sequence /hn0

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