IC/96/181

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

REPRESENTATIONS AND STRUCTURE OF GROUPSOF DIFFEOMORPHISMS OF NON-ARCHIMEDEAN BANACH

MANIFOLDS - II

Sergey V. Ludkovsky1

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

In this article new classes of groups of diffeomorhisms (GD) G(t, M) and Diff(t, M)of manifolds M with infinite atlases modelled on Banach spaces (BS) over non-Archimedeanfields K D Q p are defined and studied. Few classes of compact subgroups of GD areconstructed. Two different approaches are given for the construction of representations.Irreducible representations are considered unitary in Hilbert spaces over C and isometricalin non-Archimedean Banach spaces over locally compact fields L D Qs .

MIRAMARE - TRIESTE

September 1996

Permanent address: Theoretical Department, Institute of General Physics, Str. Vav-ilov 38, Moscow, 117942, Russian Federation.

1

1 Introduction.

GD plays a very important role in quantum mechanics on manifolds [13]. For solutionsof many problems it is necessary to know a structure of GD, quasi-invariant measureson it and its representations. At the same time a lot of articles appear by applicationsof non-Archimedean analysis to quantum mechanics, superstring theory and so on. Verylittle is known about representations of GD in the classical case and even less in thenon-Archimedean case. This shows that GD in non-Archimedean case appears to beimportant.

In the previous article [15] groups of diffeomorphisms (GD) G(t, M) and Diff(t, M)of manifolds with finite atlases At(M) := {(Ui, φi) : i £ Λ} modelled on BS X over non-Archimedean fields K, card(Λ) = n < Ho were studied. Cases with infinite atlases At(M),card(Λ) > Ho are considered here. For such M in §2 further new classes of GD G(t, M)and Dif f{t, M) are defined such that in general cases they are not metrizable. In additiona theorem is given about an analog of the van der Put base for infinite-dimensional Xover K, dimKM > H0- Classes of GD, which in particular for dimKX < Ho correspond tofunctions of the Mahler type are also discussed.

In §3 many results from [15] are generalized on GD G{t, M), but proofs differ strongly.In §4 new classes of compact subgroups are constructed such that they preserve analyticvector fields on M. This is the development of [23], where locally compact (LC) M overR and classical groups of analytical diffeomorhisms were considered.In §5 new series of irreducible representations unitary T : G(t, M) —> U(H) in a Hilbertspace H over C and isometrical f : G(t, M) -> IS(H) in BS H over L D Q s , where U(H)and IS(H) are the unitary group and the group of isometries.Results of §5 and their proofs differ strongly from [11], where classical GD of locallycompact manifolds M over R were considered. There were frequently used connectedcomponents. On the contrary, specific features of totally disconnected M are used here.All theorems are obtained for the first time and the main are 3.1-3.5, 3.7, 4.3, 5.17 and

5.23.

2 Notations and preliminary results.

2.1. Definitions. 1. Let M be an analytic manifold with ultrametric dM modelled on BSX over a non-Archimedean field K D Q p . Henceforth, atlas At(M) = {(Ui, φi) : i £ Λ}is composed from disjoint clopen Ui in M charts (Ui,φi), that is, Ui (~\ Uj = 0 for eachi 7 j £ Λ, φi(Ui) are clopen in the balls B(X,xi,ri), xi £ X, 0 < ri < oo, Λ C N, Λ isan ordinal, dM(x,y) = \\4>i(x) - (f>j(y)\\x for x and y £ Ui (see [15]).

2. For M and BS Y over K we denote by C*(t,M -> Y) for 0 < t < oo orfor t = (an, r) locally-K-convex space, that is, the strict inductive limit str — ind —lim[C*(t, (UE -> Y), πEF, Σ], where E £ Σ, Σ is the family of all finite subsets of Λ directedby the inclusion E < F if E C F, UE := \JjeE Uj, πEF : C*{t, UE -> Y) --> C*{t, UF -> Y)and ΠE : C*(t, UE —> Y) ^ C*(t, M —> Y) are uniformly continuous embeddings (isomet-rical for 0 < t < oo), * = 0 for spaces of the type C0(t, UE —> Y), * = 0 or simply is omit-ted for spaces of the type C(t, UE -^ Y) and * = c for spaces of the type Cc(t, UE -^ Y)(with 0 < t < oo in the last case). Here Cc(t,UE -> Y) := {f £ C0(t,UE ->

Y) : supiJ\a(m,fo(f)j

l)\J(t,m)pt =: \\f\\Cyt,uE < °o and limi+\m\+Ord(m)^oo\a(m, f i

(J)-1)\J(t,m)pi = 0 for t ^ oo and Cc(oo, f/E -> Y) := flieN Cc(/, f/E -> y ) .Here C*((an,r),UE —> Y) denotes the space of locally analytic functions f : UE —Y ,that is, for each x G Uj and j G E restrictions f o (f)jl\{B(X,(f)j(x),fj) n ^(t/?)] GC*((an,r),5(X,0 i (x),f i ) n φj(Uj) -> Y) =: C*(j;x) is analytic (see §2 in [15]), wherefj = m i n ( r , r j ) , \\f\\an,r,E •= s u p x e U . t j e E \\f o ^ H c a , * ) -

Evidently, C*(t, M —Y ) is the space of functions of class C*(t) with supports supp(f) :=cl{x G M : f(x) ^ 0} C U E ( f ) , E(f) G Σ and W (0 ^ W C O»(i, M -> Y)) is open if andonly if TVE1(W) D C*{t, UE -»• Y) is open for each E eE (see [15] and [22]). The spaceC*((an, r), M —Y ) consists of locally analytic functions with the radius of convergencenot less than r for decompositions in local coordinates with centres in each x G M.Particularly, (7*(t, M - - Y) for locally compact M are spaces of functions with compactsupport in classes of smoothness C* (t).

3. Let us denote by C*(la, M —Y ) the following inductive limit ind —(an, r),M -»• Y), ΠrR, Γ K \ (0))}, where n ^ : ^ ( ( a n , R), M -> Y) --> ^ ( ( a n , r), M -»• Y)are uniformly continuous embeddings for each 0 < r < R, since ||/||c,,((an,.R),M->Y) ^\\ f \\c4(an,r),M^Y), ΓK := {|x|K : x E K}, r and ReTK (see also [14]).

4. Let M and N be two manifolds modelled on BS X and Y as in 2.1.1, let also θ :M ^^ N be an embedding. That is, θ : M —>• (M) is a homeomorhism, θ(M) is closedin N, there exists a tangent mapping Tθ : TM —> TN, Txθ : TxM - - (x)N for eachx E M (see [3] and [12]), Tθ is locally analytic. This means that Txθ is analytic by xin non-void clopen subsets Ui Pi ^~1(V^) in M. Here T M and T N are analytic manifoldswith At(TM) = {{Ut X I , ^ X / X ) : J G Λ}, At(N) = {(Vj,ψj) : j G Q}, IX : X - . X isthe identity mapping, IXz = z.By (7*(t,M -> N) for 0 < t < oo or for t = la (see 2.1.2,3) we denote the follow-ing space of mappings f : M —N A" such that (fi — θi) G C*{t,M —Y ) for eachz G Q and fi = ψi ° f, θi = ψi ° θ. A uniformity in C*{t,M —N ) is inducedfrom C^it^M —Y ), that is, entourages of the diagonal are subsets S(W) := {(f,g) :

f and g G C**(t, M -^ N), (fi — gi) G W for each i G 11}, where W are neighbourhoods of0 G C*(t, M -^ Y) (see also [22], [6]).Let Hom(M) be the group of homeomorphisms of M. We introduce notations G(t, M) :=<7o(t,M -»• M) n Ham{M),_Diff{t,M) = C(t,M -> M) n Hom(M), Diff{t,M) :=

t, M -»• M) n Hom(M), GC(t, M) := (7c(t, M -> M) n Hom(M).

2.2. Theorem. Let M, N, X, Y and 0 < t < oo or t = la be such as in 2.1, letalso K be a spherically complete field, * = 0 or * = 0, or * = c; C p D K D Q p and K isa LC field for t = la. Then(1) <7*(t, M -»• Y), (7*(t, M -»• ), Di//(t , M), G(t, M), GC(t, M) are complete uniformspaces;(2) i fK is LC, card(Q) < Ho, X andY are BS of separable type overK, then the spaceslisted in (1) are separable;(3) Diff(t,M), G(t,M) andGC(t,M) are the topological groups.

Proof. (1). Since K is spherically complete ΓK is countable. Therefore, in 2.1.2,3 in-ductive limits reduce to limits of countable sequences, since A c N . Hence C**(t, M —Y )

are complete and non-metrizable (due to theorem about strict inductive limits of se-quences in Exer. 12.111 [22] and 2.16 [15]) for t ^ la. If (fγ : γ G α) is a Cauchy net inC*(la, M —> Y), consequently, there exist δ E α, E E Σ and r0 > 0 such that supp(fγ CU E and fγ G (7»((an,ro),M -> Y) for each γ > δ, since ΠRr : G\((an,R),UE -> K)°-> C*((an,r), U E —> K) are compact operators in view of 2.19 [15], where α is a limitordinal. From the completeness of (7*((an, r0), M —Y ) it follows that (fγ) convergesin C*(la,M —> Y), since Y, C*((an,r),M —> K) and c0(N,K) are complete, henceC*(la,_M -> Y) is complete. From definitions 2.2.3,4 it follows that G(t, M), Diff(t, M)and GC(t,M) are closed in C*(t,M —> M) for * = 0, * = 0 and * = c respectively,whence they are also complete.

(2). From additional assumptions in (2) it follows that M and N are separable,hence due to 2.16 [15] the spaces C*(t, UE —> N) are separable for each E E Σ. The space(7*(t, M —> Y) is isomorphic with the quotient space Z/P, where Z = 0 J € A C*(£> Uj —> Y),P is closed and K-linear in Z. From the separability of Z and Λ C N [6] it follows thatC*(t, M —> Y) is separable.

(3). From the formulas 2.16(i,ii) [15] it follows that GC(t,M) is the topologicalgroup for M with the finite atlas (see also [10, 16, 17]). For f and g e C*(t,M —>M) n Hom(M) for 0 < t < 00 or t = (an, r) there are E(f) and E(g) E Σ for whichsupp(f) := cl[x E M : f(x) ^ x] C UE(f), supp(g) C UE(g). In view of 2.16 [15](considering f(supp(f)) and g(supp(g)) C M homeomorphic with supp(f) and supp(g)correspondingly) we get g~l o f E C*(t,M -^ M) n Hom(M). If (fγ : γ e α) and(gγ : γ G α) are two convergent nets in G(t, M) (or Diff(t, M) or GC(t, M)) to f and grespectively, so for each neighbourhood W 3 id there exist E E Σ and β E α such that#~ : o / 7 G W n (7*(t, f/E -^ M) n Hom(M) for 0 < t < 00 or t = (an, r), where a is alimit ordinal. Therefore, for such t the mapping (f, g) —g t/"1 o f is continuous in G(t, M)(or Diff(t, M) or GC(t, M) respectively).For t = la let r = min(r(f), r(g)), where f and g G (7*(/a, M -^ M) n Hom(M), that is,there exist r(f) and r(g) G Γ K \(0) such that f G C*((an, r(f)), M -> M)nHom(M) andanalogously for g. Then 0 < r G Γ K \ (0) and g~l o f G ^ ( ( a n , r), M -^ M) n Hom(M).If (fγ : γ G α) converges to f and (gγ) to g, where α is a limit ordinal, hence for eachneighbourhood W 3 id in C*(la,M —> M) n Hom(M) there exist β G α and E G Σsuch that (supp(g~l o fγ)) U (supp(gγ)) U (supp(fγ)) C U E for each γ > β and r(gγ) > r,r(fγ) ^ r , since Γ K \ (0) is discrete and atlas At(M) consists of disjoint clopen charts.Therefore, (gγ ° fγ : γ G α) converges to g~l o f in C*(la, M —> M) n Hom(M), conse-quently, the last space is the topological group.

2.3. Definition. The groups Diff(t,M) and G(t,M), GC(t,M) from 2.2 we callthe groups of diffeomorhisms for 1 < t < 00 or t = la and groups of homeomorhismsfor 0 < t < 1 together with the usual group Hom(M). Evidently, for card(Λ) < Ho andX = K n , n E N, the group GC(t, M) coincides with Diff(t, M).

2.4. Notes. In [21] regular functions f were introduced, that is, f G C(0, B(Kn, 0,1) -K) fully determined by their restrictions on B(Qpn,0,1), where K D Q p . Their spaceis denoted C(reg). Mahler proved that (1) if K is not a totally ramified extensionof Q p , then C(reg) = C((an, 1), B(Kn, 0,1) -> K); (2) if K is a totally ramified ex-

4

tension of the field Q p , then C(reg) C C(oo, B(Kn, 0, 1) —> K) and is complete rel-

ative to the following norm: | |/ | | := sup{|a(m, f)P[m/p]/m!| : m G Non} < oo, where

P[m/p] = n«= 1pK«)/p] | P | = p-i/2, f ( x ) = E m e N S « ( ^ , / ) f c ) , a(m,f) G Qp and ex-

ists limi^i^oo a(m,f) P[m/p]/m! = 0. For locally compact K D Q p we can consider

analogously to 2.1 spaces C*(reg,M —N ) and C*(reg,M —> M) n Hom(M) with the

help of the substitution of J(t,m) on J(reg,m) = J((an,1),m) in the case (1) and

J(reg,m) = p~^m^p^2\l/m\\p in the case (2). Therefore, results of theorem 2.2 can be

transferred onto the case t = reg, since compositions x o x can be decomposed into

E , < i Q with s G Non, s(i) < m(i) + l(i), i = l,...,ra; < , G Q p.2.5. Definitions. 1. Let K be a field such that Q p c K c C P ) 5 c K " and S is a

perfect subset (that is, dense in itself and closed in K n , hence does not have isolated points1.7.10 [6]), we take some fixed 0 < Κ < 1. Let us remind that an approximation of theunity on S is a sequence of mappings {σn : S —S | n G N o } such that (1) σ0 = const, (2)σm o σn = σn o σm = σn for each m > n, (3) from |x — y| < κn it follows that σn(x) = σn(y)and |σn(x) — x| < κn, xn := σn(x) for x E S. For functions f G C(n — 1, S - - K) existsthe antiderivition P n f ( x ) := E™=0 E"=o / ( j ) ( ^ ) ( ^ + i " xm)j+1/(j + 1)! for each xeS(see 80.2 [27]).

2. For x G Z p there exists the standard orthonormal van der Put's basis (en : n G N o )in C(0,Z p -^ K) such that en(x) = 1 for nAx, en(x) = 0 for others x (see 62.2 [27]),where nAx if and only if n G (xo,xi,X3,...), x = Y^L-oo aipi G Q p , Xj = J2iZ-ooaiP\ai G {0,1, ...,p - 1}, [minai^o i] > -oo.

3. Let e^+V) := PiPi-i-Pie^) for l G N and x i G Z p , (%(x) := n i e a C ( ? ) ( ^ )for x G B(c0(α, Q p ), 0,1), |m| G N o , m(i) G N o for each i G α, x = (xi : i G α), where αis an ordinal, e°n(xi) := 1.

2.6. Theorem. Let K D Q p , X = c0(α,Qp) and Y = c 0(β,K) be BS. Then thefollowing set (e™(x)qj : |m| G (0,1,...,l + 1), j G β, w(i) and m(i) G N o for eachi i e a )is the orthogonal base in C0(l,B(X,0,1) —Y ), where (qj : j £ β) is the standard basis inY.

Proof. The functions e%{x) belong to C0(oo, B(X, 0,1) -»• K), since ewm((ii) )

C(oo, Z p -> K) due to theorem 81.3 [27] and |m| G N o . Indeed, Pnewm((ii))(xi) = Plewm((ii)forn > l > m(i). Let N(n, Z p -»• K) := {f G C(n, Z p -»• K) | dnf(x)/dxn = 0 for each x GZ p )} . In view of theorem 68.1 [27] it follows that (ewm((ii))(xi) : m(i) G (0,...,l + 1),w(i) G N o ) is the orthonormal base in C(l, Z p - - K) for l = 0 or l = 1, whence 2.6 is truefor these l. Moreover, (e™(x)qj : |m| < 1, j E β, m(i) and w(i) G N o for each i G α)is the orthogonal base in N(1, B(X, 0,1) -> K), where N(n, B(X, 0,1) ->• Y) := (f GC0(n,B(X,0,1) -»• Y) : dnf(x)/(dxi(-1\..dxi(-^) = 0 for each x G B(X,0,1), i(j) G α,j = 1,...,n).

Let us suppose that this theorem is proved for l = 0,1,...,n and (e™(x)qj : |m| < n,m(i) and w(i) G N o , i E α, j E β) is the orthogonal basis in N(n, B(X, 0,1) -^ Y).From 80.A [27] and definition 2.5 it follows that em(i)(z) are locally polynomial func-tions (for partition of Z p into clopen subsets) with degrees (m(i) — 1) by z G Z p .For f G N(n + 1,Zp ^ K) and g G C(0, Z p -+ K) we have | |/ + P r a + 1 P r a . . . P ^ | | c ( r a + 1 )

> | | (/ + P n + 1 . . .P1g)n+ 1 | | c ( 0 ) = |M|c(o) = \\Pn+iPn...Pig\\c(n+i), since due to 80.3 and 81.3[27] the mapping Pn+1...P1 from C(0, Z p —> K) into C(n + 1,Z p> K) is an isometry.In view of lemma 13.4 [27] we have | |/ + Pn+i---Pig\\c(n+i) > | |/||c(n+i), consequently,

N(n + 1, Z p -> K) is orthogonal to Pn+1...P1C(0, Z p -> K). Let f G C(n + 1, Z p -> K),

then f = f - Pn+lf + P n + i / ' and f'{x) = ET=o Z?=i <l, f)esl(x), where vs(l,f) G K,

x G Z p , g := f - P N + 1 F G iV(ra + 1,Zp -> K). From this it follows that g(x) =

J2T=oJ2?=iVs(l,g)el

s(x), Pn+1f(x) = J27=oZ?=iVs(l,f)Pn+iel

s(x), since from the proofs

of 81.1-3 [27] and the induction hypothesis it follows that the serieses for g(x) and

Pn+1f (x) converge. Therefore, (el

s(x) : l G (1,...,n + 2), s G N o ) is the orthogonal

basis in C(n + 1, Z p —> K). Hence from 2.16 [15] it follows the statement of theorem 2.6.

3 Structures of groups of diffeomorphisms and their

representations.

3.1. Theorem. Let G = G(t, M) be GD for 0 < t < oo or t = la is given by definitions

2.1,3. Then (1) there exists a clopen subgroup W in G such that each element g G W

belongs to a corresponding one-parameter subgroup for 1 < t < oo or t = la; (2) G is not

a Banach-Lie group; (3) G is simple and perfect.

Proof. (1). Let at first 1 < t < oo then from the definition of topology in G the

following set W := [f G G : supp(f) C UE(f), E(f) G E, pT

0UE(f)(f, id) < p~2 is the clopen

subgroup, since pl>uElfog){f ° g, id) < max(ρ0τ,UE(f)(f,id),ρ0τ,UE(g)(g,id)), where ρτ0,UE(f,g)

are ultrametrics in G(t, UE) (see 2.5 [15] and 2.1) inducing pseudoultrametrics in G, τ = t

for t 7 oo and 1 < τ < oo for t = oo. From theorem 3.1 [15] it follows the first statement.

For t = la let W := (f G G : supp(f) C U E ( f ) , U E ( f ) G E, p^ (

)

/ ) (/ ,«d) < p " 2 ,

/ G C0((an,r),M^ M), 0 < r < oo, r G Γ K ), where ρ(0an,r,UE)(f,g) := sup t€E H^"1 o f -id)itj\\an,r,E- Therefore, W is open in G and f|[Uj C\(f)jl(B(X,(f)j(x),fj))] belongs to someone-parameter subgroup < gz : z G B(K,0,1) > in G. Moreover, from the proof of 3.1[15] it follows, that one-parameter subgroups can be chosen consistent for f, since f arelocally analytic, At(M) is disjoint and for it exists a ramification atlas At'r(M) (see 2.1.1-3) with charts [Uj C\(j)j1(B(X,(j)j(x),fj))] =: Uj,x, since φj(Uj) are clopen and bounded in

BS X, ΓK is discrete in R. Indeed, for some suitable choice of Wj C Uj, x G Wj, At'r(M)

may be done disjoint: Uj,x (~\ Uj,y = 0 for each x ^ y G Wj, \Jx^w- Uj,x = Uj for each j. In

view of theorem 2.2 subsets W are closed in G (see also III. 1.14 [8]).

(2) each subgroup G(t,UE) for 1 < t < oo or G((an,r),Uj) for φj(Uj) C B(X, φj(x), r)

are closed in G and are not the Banach-Lie groups due to 3.1(2) [15], consequently, G is

not the Banach-Lie group [2].

(3) Considering the family of subgroups in G as in (2) and the minimal subgroup generated

by it, which is dense in G, from theorem 3.3 [15] we get that G is simple and perfect.

3.2. Theorem.Let G = G(t, M) be a group for 0 < t < oo, At'(M) and exp be as

in 3.4 [15]. Then (1) G is the analytic manifold for 1 < t < 00 and for it is defined the

following mapping Ev : Tη G> G such that EV(V) = exrpv(x) °Vr] from some neighbourhood

V-q = Vid°V of the zero section in TηG C TG on some neighbourhood Wη = Wi(iorq 3 η G G

and E by V belongs to C(oo) for 1 < t < oo, there are At(TG) and At(G) such that E

is locally analytic, Ev : Vv η> Wη is the local isomorphism; (2) in G for separable M for

0 < t < 00 there is the family [Gn : n G N] of compact subgroups such that Gn C Gn+1

for each n and UneN Gn =: Ga(t, M) is dense in G.

Proof. (1) Let us take V := (V G C0(t,M -> TM) : supp(V) C UE(V), E(V) G Σ;( ) < 1/p, 7r oVv = i] for each ?? G G) for 1 < t < 00. Here

r = t for t ^ oo and l < r < o o f o r t = o o , 7 r : TG —> G is the natural projection,V , = V" n TηG. If V e TG, then there exists v E TM such that % = v o η for each ?? G .Let us consider exp on At'(M) analogously to the proof of theorem 3.5(iv) [15]. There-fore, Wid :=[g EG : supp(g) C UE(*\ E{g) E Σ, ρτ

0,UE(g)(g,id) < 1/p) for 1 < t < oo,Wη = Wid o η (see notations in the proof 3.1), hence Ev : Vv η> Wη is the uniform iso-morphism. Indeed, if V E V C (7o(t,M -> TM), then | | F | [ / E | | C o ( T ) f / ^ T M ) < 1/p andVη = wor , v is a vector field on M. From B(X, y, r) C < -(£/"j) and f/j C UE it follows thatn oVv = r] and | |p 2 o v o r](z) - p 2 ovor](z')\\x < \\<f>j(vU)) ~ <f>j(v(z'))\\x/p < r/p. Hereφj(η(z)) E B(X,y,r), <f>j(ri(z')) = y, π : TM -> M and p 2 : T M -^ X are the naturalprojections. Therefore, Ev : TηG -^ G are continuous and EV(V) = expv(x)V o η(x). FromB(X, 0, r) + B(X, y, r) = B(X, y, r) it follows that exp(v o ??([/j)) = ^(f/j) for each j E Q.Let us consider in Co(t, M -^ TM) the following family of subsets [PE

:= (SE\US3FCE,F^E

E E Σ], SE := (V E C0(t,M -»• TM), supp(V) C UE), where Σ is the family of all fi-nite subsets of the ordinal Λ, At(M) = {(Ui,φ i) : i E Λ}. Since SE are clopen inC0(t, M -> TM) by 2.1 and the family (F E Σ : F C E, F ^ E) is finite, so PE are clopenin G0(t, M -> TM) and P B n P F = 0 for each E ^ F E Σ. From the proof of 3.5(iv)[15] it follows that each PE has some disjoint analytic atlas for 1 < t < 00, which may bedone with the help of || * \\CO(T,UE^TM) with τ = t for t < 00 and 1 < τ < 00 for t = 00.Then E induces a disjoint analytic atlas with clopen charts on G, since charts (Vj,ψj) ofan atlas of G0(t, M —> TM) may be chosen such that -E"!^ be analytic, since this is truefor exp on At'(M) and [e:zp7?(*)i' o r/(*)] o r/" 1^) is an isometry on M for V E PE and

(2) Let us take Gn := (f E G : supp(f) C UEn,f|U(En) E C0,a(t,UE -+ M),n), where UE := \JjeE Uj, En = (1,..., n) C Λ, n E N, At(M) has countable Λ C N, sinceM is separable. In view of theorem 3.7 [15] and 2.2 each subgroup Gn is compact in G.Since Unewif £ G : supp(f) C U E n ) is dense in G, so Ga(t, M) := Ura G n is dense in G. Iff, g E Gait, M), then there exists n with supp(f)Usupp(g) C U E n and # - 1 o / G Ga(t, M),hence Ga(t, M) is the subgroup in G.

3.3. Theorem. Let G = G(t, M) be defined as in 2.1, M be separable, 0 < t < 00;T : G ^ U(H) be unitary and T : G —> IS(H) be isometrical representations which let bestrongly continuous (see [15]), where H is a Hilbert space over C, H is BS over a locallycompact field L D Q s , K D Q p , p ^ s. Then T and T have decompositions into directsums of irreducible representations.

Proof. Let us take the sequence of compact subgroups from 3.2(2). Using the proofsof theorems 5.2 and 5.3 [15] we get decompositions of T and T into irreducible stronglycontinuous representations.

3.4. Theorem.Let G = Diff((an,r), M), M C B(X,0,r), X = c0(J,K), J be anordinal, Q p C K C C p , K be a complete field (may be not LC, see 2.15.1 [15]). Thenthe following subgroup W := B(G, id, 1/p) given with respect to the ultrametric ρ on G isisomorphic with the projective limit of some sequence (in the topology of box type [22], [6],[19] inherited from their product) of groups B(G((an,r), Mn), id, 1/p) =: Wn of manifoldsMn C B(Xn,0,r), Xn = c 0 ( J n , K n ) are BS over LC fields Q p C K n C K such thatUneN^n is dense in K.

Proof. In view of 2.15.1 [15] each element g E G has the decomposition g(x) =YsiesjeNgi(j; x)ei, where for each b > 0 the set {(i,j) : \\gl(j', *)\\rj > b} is finite. Since(i) gi(j;x) = hi(j; x, ...,x), hi(j;x1,...,xj) : M j> K are polylinear and multihomoge-

neous, so they are completely defined by their values hi(j; es(1), ..., es(j)) E K, {es : s G J}is the standard basis in X over K, s(n) G J. The norms | |/^(j;*)| | are equivalent (ii)\\hz(j; * ) ! ! ' := sups{n)eJn=lt tj |hi(j;es(1),...,es(j))| [24]. In K C C p there exists a sequence

of LC subfields K n C K n + 1 that are finite algebraic extensions of Q p , such that LLGN K n

is dense in K [19].Let us take BS Xn = c0(J, K n ) and Mn := M n θn(Xn), consequently, X = pr —lim{Xn,πln,N) and M = pr — lim{Mn,πln,N), where the projective limits are takenin the box topology, θn : c0(J, K n ) °-> c0(J, K) are natural embeddings induced byK n °-> c 0 (J, K), πln : Xn —> Xl are continuous operators linear over Kl for each n > l, forM n are taken their natural restrictions.From this and formulas (i, ii) it follows that (iii) C((an, r), M —> X) = pr—lim{C((an, r), Mn

Xra); vrj1; N}, where vrf are continuous operators (linear over Kl) induced by vrf1, vrf :C((an,r),M n> Xn) —> C((an,r),M l> Xl) for each n > l. From (iii) it follows thatW = pr - lim{Wra;7rP,N) in the box topology, since (E~l(W) - E~l(id)) is the clopenneighbourhood of 0 in C((an, r),M -> X), where Wn := W n E9n(C((an, r),Mn^ Xn))and #ra : C((an, r), M - - Xra) ^ C((an, r), M - - X) are the natural embeddings. From||#n - ^||c((«n,r),Mn^xn) < 1/p for each gn G Wn (omitting here £", see 3.2) it follows thatgn : Mn -^ Xn are isometries, whence gn E Hom(Mn) and Wn are the clopen subgroupsin Diff((an,r),M).

3.5. Theorem.Let G = G(t,M) be the same as in 2.1, 1 < t < oo; also let a LCfield K be a finite algebraic extension of a LC field L, L D Q p . Then there are BS Yover L, an analytic manifold N modelled on Y and a clopen subgroup W in G such thatW can be embedded as a closed subgroup into G(t, N).

Proof. In view of 2.15 [15] and 2.1 each element in BS C0(t, UE -> X) has the fol-lowing decomposition in the local coordinates x & φj(Uj): f(x) := 52{a(m, fl)Qm(xn)ei\i E J,m E N o n , n E I, xn E K n } , where J is an ordinal, X = c0(J, K), a(m, fi) E K.The field K/L is finite with its elements denoted {&i,...,&&}, consequently, each y E Khas the decomposition y = Yli=ihyh where yi E L, bi + L = &», 1/p < |BI|K < 1 (seech.III in [24]). Therefore, X as L-linear BS is isomorhic with Y = c0(J x {1, ...,k},L)with the isomorphism denoted ξ : X —> Y. Its orthonormal base relative to L is {ej ® bi :j E J,i E (1,...,k)}. From this it follows that (i) f(x) = E(« L (m, / ( i ' i ) Q m (x r a )e i ® bi|j E I,i E (l,...,fc), xra G Ln, n G I x (1,...,k), m G Non), where aL(m,f(j,i)) G Lare the finite linear combinations of a(rh, fj) with \m\ < \m\, f(j)(x) = J2i=i f(j,i)(x)bi,P G K, f(j,i) G L. Indeed, Ql(y) = E i , ^ , ^ , - , ^ ) ^ , s = (s1,...,sk), |s| < l G N,si G N o , c{% G L. From the definition of Φvf (see §2 [15]) and formula (i) it fol-lows that C0(t, UE —> X) =: AE can be embedded as the closed L-linear subspace intoCo(t,U'E E> Y) =: FE, where <f>'j(U'j) are considered as clopen subsets in BS Y over Linstead of φj(Uj) C B(X,0,rj) in X, 0; : Uj -> ^(C/j) C B(y,0,^), O < r < prj,^. = ξ o φ j . Then At(M) generates the family {(f/j,^) : j G Λ} =: At(N) andA := UjeA U'j. Taking the strict inductive limits (by E) of AE and FE we get the embed-ding θ of the space C0(t, M -^ X), considered as L-linear, to C0(t, N ^ Y) as the closedsubspace (see Exer. 12.111 [22]). Let W be Wid from the proof of 3.2(1) [ with p " 2 insteadof 1/p], whence we get the statement of theorem 3.5, since \\9OTI2OE~1 (g)\\Co^TtUE^Y) ^ 1/pfor each g G W, where τ = t for t ^ oo and 1 < τ < oo for t = oo, Π2 : TM —> X is thenatural projection.

3.6. Definition. Let G(ar,k) denotes the following subset of Gar for M with fi-nite At(M): Ga(r,k) := {f G G

)pz(i)] =: ρta,k(f, id) < r} ,where z(i) = k for i = 1,..., k and z(i) = 0 for i > k, 0 < t < ooor t = (an,R) (see 3.6 and 3.7 [15]).

3.7. Theorem.Let Ga and G^'^ be given by 3.6. Then the group Ga is filtered.The group Gar is pro-p-group and G(ar,k) are clopen normal subgroups such that Gar is theprojective limit of the inverse sequence {Ga/Ga(r,k) =: Hr,k| k E N} .

Proof. From UreN Gr

a = Ga and Gr

a C G ' for each r < r' it follows that Ga isfiltered (see §III.2.1 in [4]). From the formula 2.16(ii) and the proof of 3.7 [15] it followsthat G^'^ are subgroups of Gr

a. Indeed, (Gr

a, ρt0) are compact and Ga(r,k) are compact andclosed under multiplication, consequently, G^'^ are subgroups by proposition III. 1.17[8]. Then Gr

a C W for r < 1/p, where W = B(G(1, M),id,p~2) for 0 < t < oo andW = B(G(an, R),id,p~2) for t = (an, R).In view of the proof of theorem 3.1 [15] each element of g E Gr

a lies on a one-parametersubgroup, {gq : q E B(K,0,1)} C Gar, g1 = g, since Gr

a is complete and on each iterationstep (ZiA\(j;x)ei) E C0,a(t,M -> TM), glq(j;x) E Gr

a. Let X and Y be infinitelydifferentiable vector fields for 0 < t < oo or analytic for t = (an, R), Y,i Xiei and Y,i YieiE C0,a(oo,M -»• TM) (or in C0,a((an, R), M -»• TM) respectively) we have | | ( e a X e 6 y -e 6 y e a X )x | | = \\J2n+m>hn>itnsm(XnYm-YmXn)x/(n\m\)\\ < \\X\\. Using the mappingE of theorem 3.5(iv) we get that (i) ρta,k(f o g o f~l o g~l, id) < r for each f E Ga(r,k) andg E Gar, since pik(fio9lo fflog~l, id) < r for each fl E G^DG(r, M),9IE Gr

aDG(r, M)for 0 < t < oo and τ = oo, where liml fl = f, limlgl = g in Ga(r,k) and Gar respectively.Therefore, G^'^ is the normal subgroup in Gr

a for each fceN and r < 1/p.Now let r > 1/p. For each submanifold M h C M, dimKMh = h E N there existsfc E N such that p~fc < min j,w r j,w, where every φj(Uj,h) is the disjoint finite union ofB(Kh,xj,w,rj,w), Uj,h := Uj nMh'. Therefore, for each g E Gar\ Ga(r,k) with gi(x) = xi inlocal coordinates of every Uj for each i > h (in the analytic case At(M) = {(U1, φ1)} thereexist g1 e Gra\ Ga(r,k) and f1 E Ga(r,k) such that g = gio f1 and g1 (Uj) = Uj for each j . Sinceρa,k(g1 °f,f°gi)<r for each / G G^ we get gG^g~l = Ga(r,k) due to the formula (i).The family of such g is dense in Gr

a, whence G(ar,k) is the normal subgroup in Gar. FromUk^G{a'k) = {^} we get that Gr

a = pr - l imj^fcjTr^N}, where TT£, : Hr>k ->• ^fc',TTfc : Ga - - Hr,k are the quotient mappings for each k > k' E N, Hr,k := Gr

a/G(ar,k) arethe quotient groups. Evidently, using decompositions of g E Gr

a in local coordinates (see§2 [15]) we get that the indices [Gr

a : Ga(r, k)] = pm with m E N, since K considered asBS over Q p is isomorhic to Qpn, n E N and for the additive group B(K n ,0,r) we have[B(Kl, 0, r) : B(Kl, 0, rp~k)] = plnk.

3.8. Note. Let F p k n be a finite field of pkn elements [29] and M = B(c 0(N,K,0,1)

then Gar/Ga(r,k) =: Hr,k for 0 < r < 1/p is isomorhic with the following group of polynomial

bijections / : (Fpkn)kr,k ^ ( F p k n ) k such that / = E t i 1%, x £ ( F p k n ) k , x = E t i x%, xi E

F p k n , u(i, j) := u(i, j)+B(K, 0,p~k) (see 2.3 [15]), f(x) = J20<\m\+Ord(m)<k a{m, fi)Qm{x)ei

\a(m, fl)-5mfr\pJ(t, m) max(pi,p|m|+Ord(m) <T/xr,mE Nko, where Qm(x) := n t i Qm^(

Qm(i)(xi) = Pm(i)(xi)/Pm(i)(u(i,m(i)), Pm(i)(xi) = (xi - u(i, 0 ) ) . . . ^ - u(i,m(i) - 1))) are

polynomials on F p k n with values in F p k n , for them u(i,j) are chosen pairvice distinct, | * p

on F p k n is induced by | * p on K, n := dimQpK, 0<t<ooort= (an, R), a E F p k n . In

the latter case we have also decompositions f(x) = J2o<\m\+ord(m)<k, i<i<kb(m> fi)xmei,

\b(m, fi) - δm,ei| max(p i,p|m |+O r d(m)) <pkxr, where b E F p k n .

3.9. Corollary.(1) Let M = B(c0(α,K,0,1), α C N, 0 < r < 1/p, then Gar(t,M)for 0 < t < oo and Gar((an, 1), M) are the isomorphic topological pro-p-groups.

9

(2) If M has finite atlas At(M) (see §2 [15]) then there exists 0 < r < oo such thatGar(t, M) and Gar(1, M) are isomorphic for each 0 < t < oo.

Proof. The first statement follows from 3.8, the second from 3.8 and the proof of 3.7.Indeed, for M with the finite atlas At(M) = {(Uj,φj) : j E Λ} there exists 0 < r < oosuch that g(Uj) = Uj for each g E Gar and j E Λ, since Gr

a C G(1, M) for r < 1/p andM can be embedded into C0(Α,K) = X with disjoint clopen φj(Uj). For such r we haveGra/Ga(r,k) = r W ^ f c ) ^ Hr,k = (Hr,k)j for each j , since φj(Uj) is clopen in B(X,xj,rj)and fj,j are fully defined by a(m, fj,ji) in bijection with coefficients of decompositionsairnJlj) o f / i n (Hr,k)j.3.10. Corollary. The group Gr

a has the Haar measure with values in Q s for each s ^ p.Proof. The group Gr

a is s-free, hence has the Haar measure JJL : Bf(Gr

a) —> Q s [24].

4 Subgroups of a diffeomorphisms group preserving

vector fields.

In this section we consider the modification of results in [23] for manifolds over R intoanalogous results in the non-Archimedean case. To avoid repeating we give here generaldifferences in definitions and proofs, because using theorems of non-Archimedean analysisfrom [22], [27] and [24] may be lightly reproduced simple details.

4.1. Definitions. Let MN be an analytic manifold modelled on BS X = K N , MN

is a clopen subset in B(X,0,r), where r > 0, and K be a field such that Q p C K C C p

(see §2 [15]). For a covector field A := {Aα(x) : α = 1, ...,N} on MN a differen-tial 1-form (i) A = Aα(x)dxα is called a potential structure, where summation is bya = 1,...,N. It is called analytic if A E C((an,r),MN -> K N ) . It is called non-degenerate, if (ii) det(Fα,β) = 0 for each x, where (iii) F = dA = Fα,βdxαΛdxβ/2. Letg E Diff((an,r),MN) and xα = gα(x), x = (x1,...,xN), xi E K, x E MN. If (iv)Aa = A^dg^/dxa or (v) Fafi = F^v{dgtl/dxa){dgv/dx'3). The groups of such g are de-noted by GA or GF respectively and are called potential or symplectic group respectively.Corresponding to them Lie algebras of vector fields are denoted by LA and LF (see theproofs of theorems 3.3 and 3.5 [15], [23], [2].

4.2. Note. It is evident that there are accomplished analogs of proposals 1 and 2[23], since GA C GF and LA C LF. It is necessary to note that an analog of theorem 4.3given below is not true in G{la,MN) and, moreover, in G(t,MN) for 0 < t < oo, sincethe space of locally constant f : MN —> K is infinite-dimensional over K, d^f = 0, alsobecause of theorem 3.2(1) and 4.1 (iv).

4.3. Theorem. Let MN with N = 2n, n E N and GA, LA be defined as in 4.1such that (i) A = cα,νx

νdxα, here cα,ν = —cv,a = const, det(cα,ν ^ 0; or (ii) A = Aαdxα,Aα = T.T=iCk

a,Vl,...,v'kxvl-xVk, here c*^,...,^ E K, c1α,ν = -c\a for each α,ν = 1,...,Nα,

det(c1

α,ν) ^ 0. Then (i) dimKLA = N(N + 1)/2, GA = Sp(2n,K) := {g E GL(2n,K)|gteg = e} is the symplectic group; (ii) dimKLA < N(N + 1)/2, where gt denotes thetransposed matrix.

Proof. (i) Let us consider at first ca,^ = ta^, where ta,a+i = 1, ea+i,a = —1 fora = 1,..., N — 1, ea>p = 0 for others (α,β). Therefore, are true analogs of formulas (10-13) [23] with a$lt_>Vk E K. The matrices B g - ; in lemma 1§2 [23] have integer elements,consequently, are true analogs of formulas (15,16) [23] for the field K, since an analyticvector field ξ is in LA if and only if (iii) ^d^Aa^ A^8^ =: LξAα = 0.

10

In general the form A can be reduced to (iv) A = —\eatVxvdxa/2 by some operator

j E GL(N,K) in B(X,0,r), where λ G K, j(B(X,0,r)) = B(X,0,r) (since {cα,ν} is theskew-symmetric matrix, j α , ν and λ may be expressed through {cα,ν

: α,ν = 1,..., N} withthe help of the arithmetical operations, Q is dense in Q p , K C C p , see also §IX.13 [9]).(ii) For a potential structure A given by 4.3(ii) the analogs of equations (24 — 26) [23]are satisfied.

4.4. Note. Theorem 2 [23] may also be modified, but should be rather lengthy.

5 Irreducible representations of a group of diffeo-

morhisms that are induced by their action on a

manifold.

5.1. Definition. A measure \i : Bf(M) —> [0, oo) with 0 < /i(M) < oo is calledquasi-invariant relative to a subgroup G C G(t,M) for 0 < t < oo, if /j,g is equivalentto fj, for each g E G and its quasi-invariance factor (density) dfj,(g,x) := /ig(dx)//i(dx) iscontinuous by (g,x) E G x M, where M is an analytic manifold from 2.1 with Λ C N,H9{A) := nig'1 A) for each A G Bf(M), Bf(M) is the Borel σ-algebra on M. ByAf(M,/i) we denote the completion of Bf(M) by \i.The measure JJL is called σ-finite, if there exists a countable family UjeN Mj = ^ Mj GAf(M,n), such that //(M,-) < oo for each j .

5.2. Theorem.Let M be an analytic manifold with Λ C N from 2.1 and X be aseparable BS over a LC field K, 1 < t < oo. Then on M there exists a σ-finite measurefi that is, quasi-invariant relative to GC(t,M).

Proof. In view of theorems 5.12 and 5.16 [24] BS X is isomorphic with c 0 (α,K),where α C N. Let T : X —> X be a compact operator such that (i) Tei = siei, 0 si G K,lim^oo \sip~c'Xl\cp = 0, lim^oo |p c x V s i | c p = 0, where {ei : i G N} is the standard or-thonormal base in X, c and C / G Q , 0 < C ' < C < I are fixed. Therefore, for each r > 0 andi? > 0 there exists a finite subset A C X such that T(B(X, 0, r)) C co(A) + B(X, 0, R),where co(A) = {z = Y,i^i%i : xi E A, λi E B(K,0,1)} (see theorem 1.2 and consequence2.5 [28].L e t S ( j , n ) : = p j B ( K , 0 , 1 ) \ p

j + 1 B ( K , 0 , 1 ) f o r j e Z a n d j < h ( n ) , S ( h ( n ) , n ) =ph(n )B(K,0,1), where h(n) = -(ordpsn), that is, | s n | K = ph(n) . Also let mn be a familyof non-negative measures on K such that mn(dx) = fn(x)w(dx), where w is the Haarmeasure on K with w(B(K,0,1) = 1, fn(x) = a(j,n) = const > 0 for x E S(j,n).For example, let a(j,n) = 2h(-nKj-h(-n^ x (1 - 2"/l(ra))(l - l/p)p-h^ for j < h(n),a(h(n),n) = (1 - 2"2/l(ra))p-/l(ra). Then mn(K) = 1, mn(K \ S(h(n), n)) = 2~2h^\qm := J2n=m^~Hn) < J2n=m^~2[c'n], consequently, (ii) lim^oo qm = 0, where [b] is theintegral part of b E R.

Let Pjx = (x(1),...,x(j)) be projectors from X onto K j, j E N, x = Y,ieax("i)ei- Alsolet νj(dz) := <8>i=iTOi(< ( )) be measures equal to products of measures on K j, z E K j,z = (z(1), ...,z(j)), z(i) E K. Then there exists a cylinder measure \i that is, generatedby a consistent family of measures yj(A) := νj(E) on σ-algebras P~l(Bf(K^)) 3 A forA = P~\E), E E Bf(Kj) [5], [1], that is, un(p-](E)) = νj(E) for each n> j andprojectors Pn,j : K n - - K j, P n , j o Pn = Pj, Pj,j are the identity operators. From theformulas (i,ii) and compactness of the operator T it follows that for each e > 0 there

11

exists a compact L := {x G X : |x(i)| < pk(i) for i = 1, ...,m; |x(i)| < |SI| for i > M } such

that 0 < fj,(X \ L) < e, where k(i) G Z, lim^*, k(i) = -oo. If # G G(1,K -> K),

then mg(dx)/m(dx) = I ^ G r V ) ) ! ^ ^ ) ) / / ^ ) ] =: ρm(x - g-\x),x)\g'(x)\K, where

TO : Bf(K) —> [0, oo) is a σ-additive measure such that m(dx) = f(x)v(dx), v is the Haar

measure on K, f(x) is a continuous real positive function on K, mg(E) := m(g~1(E)) for

£" G Bf(K), mg(dx) = f(g~1(x))v(dx), ρm(z,x) := mg(dx)/m(dx) for g" 1^) := x — z

for each i e K .

In view of the Prokhorov theorem §IX.4.2 [1] the cylinder measure on an algebra U

:= UjeN Uj with fi(A) = yj(A) for A G Uj has the countably additive extension on

Bf(X) and, consequently, on Af(X,p) such that fi(X) = 1 and (iii) /j,(z + A) = p(A) for

z(i) G ph(i)B(K, 0,1) for each i G N, p(z + A) is equivalent to // for z(i) = 0 for a finite

family of i G N. Therefore, //(z + A) is equivalent to // for z G X0 := sp{ei : i G N} + Xs,

where Xs is the set of z fulfilling (iii), hence exists the density pM(z,x) := /iz(dx)//i(dx).

Then the following measure (iv) p,(S) := Y^J^A ^(fijiS ^ Uj))//i((f)j(Uj)) is σ-finite, since

A C N, 0 < //([/,) < oo for each j, M = [JjeAUj> ,φj(Uj) are clopen in B(X,0,rj),

rj > 0, where S G Bf(M), S C UE, E e Σ. Further, fi on Af(M,p.) will be writ-

ten simply as [i. Let us consider the family J of elements g G GC(t, M) such that

supp(g) C Uj, 5f|t/j> G C(t, Uj,n -^ Ui,n), g : Uj,n -^ g(Uj,n) C Ui,n and g are homeomor-

phisms of Uj,n onto g(Uj,n) for some n G N, where Uj,n = (f)jl(B(Kn,Q,rj)). Therefore,

/j,g(dx)//j,(dx) = \det(g'(g~l(x)))\-KPn(x — g~l(x),x), where the difference (x — g~l(x)) G X

is taken in local coordinates inherited from X, g'(x) = (dgl(x)/dxm : l,m = 1, ...,n) and

x m are the local coordinates as in 3.4 [15].

In view of the Kaplansky theorem the minimal group gr(J) generated by the fam-

ily J is dense in GC(t,M) (see also [15]). If g G gr(J), so g = g1 o .... o gn for

some g1,...,gn G J and there exists /ig(dx)//i(dx) due to the chain rule of differen-

tiation and (x — g^l(x)) G X0, since (v) /ig(dx)//i(dx) = p(g~l o ... o g^ldx)/'/i(dx)

= \Hg(dx)/Hg^og^dx)}... [ngi(dx)/n(dx)} = Y^^det^g-1 (x))|Kρ(g1 o ... o ^_i(x)-

fl'trl(fl'io-o^-i(a;)),fl'io-ofl'i-i(a;))), since ^(5-r1o...ofl(1-

1(fl(1o...ofl(i_1(a;))) = ^(^"1)W)>

where ^ = id.

Let (/ G GC[t, M), then there exists a sequence gi £ J such that limre_>oo(<jfno<jfn_1o...o<jf1) =

<jr (is convergent in the topology of the group GC(t, M)). From the construction of measure

[i, the decomposition of g by {Qm^i} m local coordinates and formulas (i—v) it follows that

there exists a sequence {gn : n G N} such that there is m G N with /ign(dx)//i(dx) = 1

for each n > m and x G M, since for isometries gn : M ^ M with ρc,Eτ(gn)(gn,id) <

p- f c min i e E ( f l n )(r i ; 1) and p 1" f c < sup i e N(|si |K,P~ c7l si |K) we have | d e t ( ^ ) | K = 1 and

P^y-9nl{y),v) = l , s i n c e ^i+\m\+ord(m)^oc sup i e E ( f l ) I a ^ ^ ^ f / ^ l J ^ m y = 0, where

supp(gn) C UE(gn), ρc,E(g)(g,id) := s u p i e A j e E ( f l ) ^(m,^!^)]J(τ,m)p i ; UE := UjeB^>

£" G Σ (see 2.1), τ = t for t ^ oo and 1 < τ < oo for t = oo. In view of the theorems

of Fubini, Fatou and the construction of /j, there exists pg(dx)/p(dx) =: d^(g,x) and

redefining it on some //-negligible set we get that the density d^ is continuous by (g,x)

for g G {GC(t, M) : supp(f) C UE} fl W for some E G Σ and an open neighbourhood

W 9 id in GC(t,M), where x G M. Therefore, for each g0 e GC(t,M), x0 G M and

e > 0 there exist neighbourhoods V 3 g and S 3 x0 such that \dn(g,x) — dn(go,xo)\ < e

for each (g,x) 6 ^ x S . If x ^ supp(g), so pJg(dx)/pJ(dx) = 1. From the definition of the

topology in GC(t, M) and pJgig2(dx)/pJ(dx) = [pJgig2(dx)/pJgi(dx)][pJgi(dx)/p(dx)] it follows

that dfj,(g,x) is continuous (see also [18], [20].

In the case of X = Kn, n G N a measure // may be chosen on Bf(M) such that // is in-

12

duced by ν on X and ν|φj(Uj) were restrictions of the Haar measure from Bf(B(X, 0, rj))on Bf(φj(Uj)), whence d^iz^y) = 1 for such JJL and each z, y E B(X,0,rj).

5.3. Definitions. 1. Thereafter M is an analytic manifold modelled on a BS Yof separable type over a LC field K D Q p , that is, dimKX < Ho; At(M) with Λ = N,Mi = M for each i E N (see 2.1). Let us denote X = Tlien Mi a n d t a k e ^ G Af(M>v)?E = YlieN Ei. A subset E C X is called R-unital (product subset), if it satisfies twoconditions:(UP1) E i e N HEi) - 1| < oo and fi(Ei) > 0 for each i E N,(UP2) Ei fl Ej = 0 for each i =£ j , where \i is the measure from theorem 5.2. We call asubset E K-unital, if it satisfies (UP2) and (UP3) Ei = \Jj€A{i) Ei,j, Λ(i) C N, EitjnEitj> =0 for each j =£ j ' , for each Ei,j there exists l with Ei,j C Ul, φl(Ei,j) = B(Y, yi,j, ri,j) is theball in Y with radius ri,j > 0, yi,j E Y, supE :(UP3) a n d i e N infi,j ri,j =: t(E) > 0 and there

are Q(i) C N, card(Q(z)) < Ho such that EE C f/n(i), where {^j : (UP3) and i G N} isa family of all such partitions of E. Evidently, Ei are clopen in M. We endow hereafterE with the Tychonoff topology of the product [6], if it will not be specified in anothermanner.2. A R-( or K)-unital E is called R-cofinal or strongly R-cofinal (or K-cofinal) tocorresponding E' C X, if

(SCF) there exitsts t i e N such that ^(EiAE^) = 0 for each i > n; or(KCF) there exists n G N such that Ei = E\ for each i > n correspondingly, whereEAS := (E \ S) U (S \ E). These equivalence relations are denoted ERE', E=E' andEKE'. The equivalence class for E given by (CF) or (KCF) is denoted Σ(E) or ΣK(E)respectively.3. For a R-unital E and £" G Σ ( E ) let vE{E') := FlieN M-^OJ w n e r e FI converges dueto (UP1). In view of the Kolmogorov theorem [5] the additive measure ΝE has the σ-additive extension from the minimal algebra generated by the family {n~l(Bf (E'i))\ Ei GΣ(E)} := A(E) onto the minimal σ-algebra M(E), generated by A(E), where ΠI : E' -^ Eiare the natural projections.4. Let G = GC(t, M), where 1 < t < oo and T.^ (Soo) be the infinite symmetric groupof all (finite respectively) permutations of N, Σ n be the symmetric group of {1,...,n},where t i G N . For g G G, σ G Soo and x = (xi G M : i G N) G X there exist the actions<jr:z := (gxi : i G N), xσ := (xσ(i) : i G N) of G and SQO on X. Evidently, they commute(gx)σ = g(xσ).

5.5. Lemma. If E is R- (or K)-unital, then (gE)RE (or (gE)KE), Ea=E (or(Eσ)KE respectively) for each g G G and σ G T,^.

Proof. (K). For each g G G there exist k and n G N such that |gji(x) — gji(y)|Y =x i — V%W for each i > n and (i) gj(4>jl(B(Y,y,r)) D B(Y,gj(y), |det( g'(y))\-Kr') for each

j G N, infyeM \det(g'(y))|K > 0, since this is true for each gn G J (see the proof of 5.2and [27]), where (xi : i G N) are local coordinates in Uj C M, ^ := φj o g, 0 < r < rj,y G φj(Uj), r' = p~kr, k G N o is not dependent from Uj. For each g E G and locallycompact M we have supp(g) is also compact, hence each gEi is compact and is containeda finite union of Uj.

(R). Let Sg := supp(g), then EiAgEi C S g and EiAgEi C (Ei n Sg) U ( ^ ^ n Sg) and< 2fi(Sg) < oo.

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5.6. Theorem. Let E be a R-unital subset in X. Then for each g E G and x =

(xi) E E'RE the following equality holds: (νE)g(dx)/νE(dx) = pE{g~l\x) := IlieN PM(g~l] x

where pu(g~l\ y) := l^g(dy)/fi(dy) for each y E M.

Proof. Let {gn o ... o g1 : n E N} be a sequence from the proof of 5.2, then there are

constants 0 < c1 < c2 < oo such that (i) c1 < ρM(gn ° ... ° g1; y) < c2 for some suitable

clopen subset S in M contained in a finite union of charts and each y E gn o ... o g1S,

n E N. From (i) it follows that (ii) c1 < ΡM(G;Y) < c2 for each y E Sg := supp(g) and

ΡM(G; y) = 1 for y ^ gS, since there exists L E Σ such that UL D gn° ... ° g1S for each n.

Further we may proceed analogously to the proof of 1.5 [11].

5.7. Definition. A locally finite subset in M is called a configuration. The set of all

infinite configurations in M is denoted ΓM. Let X be the set of all ordered configurations

in M, that is, X consists of elements x E X with Xi ^ xj for each i / j e N and

a sequence {xi : i E N} does not have accumulation points in M. This means that

^ / E O O = F M and the quotient mapping exists: 9^ : X -^ ΓM. Here X is considered with

the metric given by the formula (2.19) [11].

Evidently, lemmas 1.1, 1.2, 1.6 and formula (1.15) [11] can be lightly transferred onto the

case considered here. We get the quotient mapping TTQ : X —> Q« := X/T.^ [6], here X

is considered in the box topology, that is, its base consists of open subsets of the form

V = WjLi Vj, Vj are open in M.

A subset B E QM is called E-measurable if ?r^1(-B) E M(E), the family of such B is

denoted MQ(£ I). It can be supplied with a measure VE,Q{B) = ΝE(L) for each 7TQ(£>) =

Uo-eSoo Lσ (disjoint union), where M(E) 3 L is called the fundamental subset for B.

Evidently, is true the analog of lemma 1.7 [11] for the considered here case, hence TTQ is

measurable and Eoo acts measurably on X.

5.8. Proposition. For each R-unital E = ]T^i Ei there exists a R-unital subset

E(0) E Σ(E) satisfying the following conditions:

(UP4) Ei(0) are clopen in M;

(UP5) for each submanifold Mn C M, dimKMn = n E N and i E N there exists g E G

such that (pj(gEl ' n Uj,n) are balls in the subspace Kn C Y for each j E N, where

Uj,n = U3C\Mn.

Proof. The measure \i is Radon and Borel regular, hence for each i e N there exists

a clopen subset Ei C M such that n(EiAE{) < 2"\ Let E\ := E[ \ U^E'j, so {E”i}

is the family of disjoint clopen subsets in M and EiAE"i C (EiAE^) U (U5=i(- i n Eft),

consequently, J2Zi v(^AE\) < E ~ i ^ ( ^ A ^ ' ) + E i ^ M ^ n ^ . ) , E i ^ M ^ n ^ . ) <

E^li M(U~ i + i Ei) n^-) < E ^ i viE'^Ej), whence Ei n{EiAE"i) < 2 and the conditions

(UP1,2) are satisfied, so E"RE, where E" = Y\iE\. That is, E(0) = E" satisfies (UP3).

Each φj(Ei(0 ) (~\ Uj =: £i j is clopen in BS Y over the LC field K, consequently, Eij are

disjoint unions of balls \JkB(Y,yitjtk,ritjtk)- Each B(Y,y,r) is the infinite disjoint union

of balls B(Y,yi,r/p), where y and 'y* G F, r > 0, B(Y,y,r) = (y - x) + B(Y,x,ξR) for

ξ|R = r and ξ G K for each x and y G Y, r > 0, R > 0. Let us consider the local

coordinates in Uj. Using disjoint clopen partitions {4>Jl(B(Y,yijtk,rijtk)} for E\ ' and

partitions of each Uj into a disjoint family (f)J1(B(Y, zj,l, Rj,l)) with the help of formula

5.5(i) we find suitable gi E G. Indeed, locally K-linear mappings on clopen subsets of K n

are in C(oo,Kn —> K) and Uj,n are clopen in Kn C Y, gi may be taken equal to id on

M\Ei(

14

5.9. Note . Hereafter, Π : Σ —> U(V(Π)) denotes a unitary representation on aHilbert space V(Π) over C, II' : Σ —> IS(c0(N,L)) denotes an isometrical represen-tation on a BS c 0(N,L), where L is a LC field, L D Q s [15], [25]. Evidently, E ^ isthe LC group, since it is discrete. Analogously to §2.1 [11] we define the Hilbert spaceH($2) as the completion of Ue'es(E) HR,, with the scalar product of the type (2.5) [11]< 0i,02 > = EaeSoo JEmnE(Va < fa(x)> n ^ ) " 1 ^ ^ " 1 ) ) >v(n) ^E(dx), where Hfe, :=L2(E'; M(E)\E';vE\E';V(Tl)) is a space of functions on E' with values in V(Π), £ :=(U;n,E); E'RE, E is R-unital. Let us denote by Ty(g)(f)(x) := pE{g~l\x)l/24){g~lx) arepresentation, since results of §2.2 [11] may be lightly transferred onto the case consid-ered here, where g E G, φ E H(Y/).For a K-unital E and E' E ΣK(E) and a LC manifold M let EfE, := BUC(E ->c0(N,L)) be a space of uniformly continuous bounded mappings f : E —> c0(N, L) withII/HE' = sup x e B , | | / | | C O (N,L)- We denote by H(J2) the completion of \JE>€HK(E) H^>Lbe-cause from E” D £" it follows that there exists the isometric embedding HR, ^^ HR«,where the norm on -ff(E) is induced by the family {|| * \\E> : E' E ΣK(E)}, that is, for4> E Ue'esK(E) H$, is defined by \\<f>\\B := sup_x e ij,e E K ( ij ) ) < 7 e E o o ^ ( ^ " ^ ^ ( N . L ) - There-fore, we define an isometrical representation Ty(g)f(x) := f(g~lx) for f e H(J2) and

fE

5.10. Proposition. The representations defined in 5.9 Ty : G - - U(H(Y/)) and: G ~^ IS(H(J2)) are strongly continuous.

Proof. (Ty). The space H(J2) is isomorphic with the completion H'(J2) of Ue'es(Ewith the scalar product < f1,f2 >H>'-= IF < f1(x),f2(x) >V(Π) νE(dx), where fi G

i), E( E E(E), F E M(E), Fσ for σ e E M are disjoint and supp(f1(x)f2(x)) C?cr- Here i/'^/ is a space of functions / = Qu<P, where <p £ -^IE' a n d (0 Qn0 :=(v)n(v))fa (Qn(<t>)(x<r) = n(a)-V(x); (n) R{a)<l>{x) := 0(xa); (m) n(a)0(x) :=

U(a)((f)(x)), | | / | | 2 = /E/ \\f(x)\\ym'ji'E(dx) < oo, since £"cr for a E SQO are pairvice disjointfor different pairs of σ. Therefore, as in 2.1 [11] we get < Ty(g)fi, f2 >=< v1,v2 >V(Π)

Let us fix J E Σ and take U(J) = Ujej Uj C M. As in the proof of theorem 5.6 we canfind a neighbourhood W 3 id in G and 0 < c1 < c2 < oo such that c1 < PM{9~1] y) < c2

for each y E UJ and PM{Q~1'-, y) = 1 for each y <£UJ for each g E W with supp(g) C U J .Hence for each e > 0 there exists W 3 id such that | < Ty(g)f\, f2 > — < f1, f2 > | < e,consequently, due to the Banach-Steinhaus theorem (11.6.1 [22], [26]) there exists a neigh-bourhood V 3 id such that | |(7y(g) — /)/i | | < t and T y is strongly continuous.

(Ty). The space H(J2) is isomorphic with the completion H'(J2) of Ue'esK(E) H'\E,,

where H'\E, are generated from BR, by the formulas (i—iii) with | |/ | | = sup^gg/ ||/(x) ||CO(N,L) <

oo for f E H'\E,, since E'o are pairvise disjoint for different pairs of σ E EQO. Suppose

/ E B(^ ' (E),0, l ) , soforeache > 0 there exist E' E ΣK(E) and/such that | | / - / | | < e,/ E B(H!E),0,1) and / is uniformly continuous on M. Each E[ is compact in M, henceE' is compact, since M is locally compact in this case. Therefore, for each e > 0 thereexist n E N and x E E' such that (iv) \f(y) — f(yi,...,yn,xn+i,xn+2,...)\ < t for eachy E E', where y = (yi E Ei : i E N) and U?=i Ei is compact in M. From the proof of 5.2and lemma 5.5 it follows that there exist J E Σ with J C (1, ...,n) and neighbourhoodWJ 3 id in G for which gE' = E' for each g E WJ with supp(g) C U J . Using the family

15

{WJ Pi [g E G : supp(g) C El] : J, i} we find for each e > 0 a neighbourhood W 3 id suchthat \\f(g~1x) — f(x)\\ < t and T y is strongly continuous.

5.11. Definition. Let {Ei : i = 1,...,r} be a family of disjoint clopen subsets inM, H1 := ®l=lL

2(Et;dii\Et;C), G1 := IILi G|Ei with G|Ei = {g E G : supp(g) C Ei}.Then we introduce the natural representations L1 : G 1> U(H1) such that (L1(g)f)(y) :=

(nLiPMtor1;^)172)/^-1*/) for / e #i> : = idiVi -i = h-,r), v e FlU-For r = oo and R-unital (or K-unital ) E = n*=i m X we define GE

:= U.iliG\E>the weak direct product, that is, the family g = (gi E G|Ei

: i E N such that gi ^ idfor a finite set {i}). Evidently, there is the natural embedding of GE into G, so GE maybe considered in the hereditary topology.Let E' E Σ(E) a n d H\E, : = L2(E',M(E)\E',uE\E',C), so we define (LE,(g)f)(x) : =

PE{9~1'-,X)1^2 f{g~lx). We denote by G((E')) a subgroup in G of elements g E G suchthat there are n G N, submanifold Mn °-> M with d im K M n = n < oo and σ E T.^ withgE'i>n = E!

a(l),n for each i E N, where E'i>n = E[ n Mn, g |(M \ Mn) = id|(M \ Mn).For a LC manifold M let us consider BS BUC(E —> L) of uniformly continuous f : E —>• Lmappings with E open in M and | |/ | | := supxeE\f(x)\jJ. Then we define (L2(g)f)(x):= /(^-1x) for x E Yll=iEt, f E H2 := ®l=1BUC(Et -+ L) and g G G1, that is,L2 : G ->• IS(H2), where L is a LC field, L D Q s . For r = oo let H 2 := BUC(E -> : L),where E = n t ~i Ei is K-unital in X. We define also LE : GE-> IS(H2) by the formula(LE(g)f)(x) = f(g-lx).

5.12. Lemma. The representations Li (i = 1,2) of the group G1 and LE>, LE> of the

group GEi defined in 5.11 are irreducible.

Proof. (L1) If f G H1, so for each e > 0 there exists a simple function fn = Z ^ = 1 bkχBk ^H1 such that | |/ — /n||i?i < e, where bk E C, Bk are clopen in 111=1 ^ = : E(r) and disjointfor different k, ΧB(X) = 1 for x E B and ΧB(X) = 0 for X <£ B. For each clopen subsets Aan B in Ui and Uj such that φi(A) and φj(B) are balls in Y, for each δ > 0 there existsg E GC(t, M) =: G with supp(g) C ^ U f/, such that ^ ^ ( G - 1 ^ ) - XB{X)\\I?(M,H,C) < δ,where 1 < t < oo is fixed. Indeed, 0 < /i(Ui)/i(Uj) < oo, from quasi-invariance of JJLrelative to G and the proof of theorem 5.2 it follows, that for each γ > 0 there existn E N and cylinder subsets A'n = P-\An), B'n = P~\Bn), An and Bn E Bf(Kn) suchthat n(AAAn) + /i(BABn) < γ, where An = 4)~l{A'n), Bn = 4)~l(B'n). Moreover, foreach w > 0 there exists g1 E gr(J) such that fi(BnAgiAn) + fi(giBnAAn) < w, since inC(oo, B(Kn,0,1) -^ K) are dense locally constant and locally K-linear functions (thatis, locally equal to C1 + C2(x — x0), where Ci G K, x E B(Kn,x0,r), 0 < r < 00 aresufficiently small), also using decompositions of g in local coordi nates. Therefore, thereexists g E G1 with \\f(gx) - bkXBk(x)\\Hl < γ for some k. Then spK{χBk(gx) : g E G1} isdense in H1, consequently, L1 is irreducible.(L2, LE). Let at first r < 00. From formulas (i, ii) 2.16 [15] it follows that f(gx) E H2 foreach g E G1 and f(x) E H2. In BS BUC(Ei —> K) are dense locally constant functionsdue to the formula (iv) 5.10, since in BUC(B(Kn,0, r) -^ K) are dense locally constantfunctions [27]. There exists g E J such that g(An,r,j) = Am,R,i for all An,r,j C Uj andAm,R,i C Ui of the form An,r,j := {x E Uj : Pn o(f)j(x) = (x1, ...,xn) G B(K n ,0,r)}with o o > r > 0 , o o > i ? > 0 . We assume Uj,n := Pn(φj(Uj)), where P n : Y —> K n

is the projector. Hence for every f G BUC(Ei —> K), supp(f) C Uj and dependentonly from (x1, ...,xn) G Uj,n and each e > 0 there are c E K, g1 and g2 E J such that

c(f(g1x) ~ f(g2x)) - ΧB1\B2(X)| < e for each x E Ei, where Bi = A r a W ) r , w j ( i ) , i = 1,2,

0 < r(2) < r(1)/p. Each set Uj,n is closed in B(K n ,0,1), hence is compact and B1 has

16

finite covering by sets of the type B2, B1 = \J£=l B2,k, B2,k n B2,l = 0 for each k ^ I.Therefore, J2T=i χB1\B2,k = mχB1 - Exs 2, f c = (m - 1)ΧB1, where m > p > 1. Thismeans that spK{f(gx) : g G G1} is dense in H 2 and L2 is irreducible. For r = oo eachm = 1 G|E i =: Rm can be embedded into GE, ®™=l BUC(Ei -> K) =: Sm has the naturalembedding into H2 such that {jm(Rm x eH°) is dense in GE and UTO Sm is dense in H 2.(LE>). From the irreducibility of LE>\G\E> and in view of (L1) analogously to the proof of3.3 [11] we get the irreducibility of LE>.

5.13. Lemma. Let subset E' = FL° i E[ in X be R-unital and satisfies (UP4, 5) in5.8 or E' be K-unital. Then for each σ E T.^ and a submanifold Mn C M there existsg E G such that gE'in = E',i)n for each i e N and g\E'in = id\E'it if σ(i) = i, whereE'i>n = El n Mn, dimKMn = nEN,g\(M\ Mn) = id|(M \MN).

Proof. Since σ E Soo, so there is m E N such that σ(i) = i for each i > m. Letus consider a clopen subset U Li E'i>n =: E0 which is clopen in M, so E0 n Mn =: E0,n isopen and compact in Mn. Each E'in is the finite disjoint union of subsets Li,n,j,k C Uj,U(j,k)ev(i,n) Li,n,j,k = E'i>n, φj(Li,n,j,k) = B(Kn,x,r), where V(i,n) C N 2 , since E'i>n fulfils(UP4, 5) or is K-unital. Evidently, B(Kn,x,r) = (x-y)+B(Kn,y,£R) with |ξ|KR = r, xand y E K n , 0 < r < oo, 0 < R < oo. For f E C(1, B(K, x, 1) ->• K) with f(x) ^ 0 thereis r dependent on \f(x)\ and ||/||c(i) for which f(B(K,x,r)) = B(K,f(x), \f'(x)\r) (see[27]). Therefore, we can take the partition (Li,n,j,k) with sufficiently small r = r(i, n,j, k)and find necessary g E G, since 1 < t < oo (see 2.1). The last condition g|(M \ Mn) =id|(M \ Mn) may be fulfilled using normal coordinates , since K n is complemented inc0(N, K), that is, c0(N, K) = (c0(N, K) 0 K n ) 0 K n ) .

5.14. Note. Let E, E', (f> | H^, and H^, be the same as in 5.9, so for f = QΠφ

(see the proof of 5.10) and x E E' we have (T^(g)f)(x) = pE(g-1\x)1/2U(a)4>(g-1xa) and

(Ty^(g)f)(x) = H(a)(j)(g~1xa) that induces representations TE>(g)(f)(x) = pE(g~1\x)1^2H(a)(j)(g~1xa)

and fE'{g)<j){x) = U(a)(p(g-lxa), that is, TE> : G((E')) -»• U(H'fE>) and fE, : G((S/)) -»•

IS(H'\E,) respectively (see 5.11), where σ E Soo is chosen fulfilling the condition gE' =E'a for dimKM < oo or as in 5.13 with some Mn.

5.15. Lemma. For R-unital subset E' fulfilling (UP4,5) (for K-unital E' in X)representations T'E (TE> respectively) of G((E')) are irreducible.

Proof. BS H'fE, and H'fE, are isomorphic with H\E.®V(U) and BUC(E -> L)0c o (N, L)respectively, where H\E, = L2(E', M ^ ) ! ^ ' , vE\E\ C). Then TE,(g) = LE,(g) 0 IV(Π) andfE,{g) = LE,(g) 0 I c 0 ( N , L ) for each g E G(E') := G((E')) D n ^ e N G ® , where G ( ^ ) areisomorphic with G|e', IV is the unit operator on BS V. Assume A is an intertwiningoperator, so A o (LE>(g) 0 IV) = (LE>(g) 0 IV) o A, g E G(E'), analogously for L. Inview of 5.12 these representations are irreducible, consequently, A = 1# 0 A1 with abounded operator A1 on V(Π). In view of 5.13 for each σ E T.^ and a submanifoldMn C M there is g E G((E')) with ^ n = E'a{i) na and let us take in 5.14 φ = ψ 0 v,where V G ^ E ' , t; G V(U) or V G BUC(E' -»• K),'w G c o(N,L). Hence V'® (^in(a)w) =•0' 0 (n(<7)Aii>) with ^'(x) = pE(g~l\x)ll2tl)(g~lx(j) or ^'(x) = vjj(g~1xa) respectively,consequently, A1Π(σ) = Π(σ)A1 for each σ E T,^. The representation Π is irreducible,hence A1 = λ x I, λ = const.

5.16. Lemma. Suppose G is a topological group, T : G —> IS(H) is a stronglycontinuous representation in the group of isometries of BS H over a LC field L D Q s .Suppose also {Gδ : δ E A} and {Hδ : δ G A} are families of subgroups and subspacesrespectively satisfying the following conditions: (a) Hδ are Gδ-invariant and topologically

17

irreducible components T|Gδ; (b) in H is dense U<seA Hδ; (c) for each δ and 8' E A thereexists a finite sequence δ1 = δ, δ2, ...,δr = 8' with Hδi n Hδi+1 ^ (0) for each 1 < i < r;(d) for some δ0 E A topologically irreducible representation T|Gδ0 : Gδ 0> IS(Hδ0) doesnot appear in H Q Hδ0. Then representation (T, H) of G is irreducible.

Proof. Due to (d) each G-invariant subspace in H either contains Gδ0-irreduciblesubspace Hδ0 or is contained in H 0 Hδ0. Therefore, there exists G-irreducible subspaceH', H' D Hδ0. For each δ E A in view of (c) there are δ1, ...,δs with Hδi fl Hδi+1 = (0)for each 0 < i < s, where δs+1 = δ. Since H' D Hδ0 D Hδ1 = (0), so due to (a) we haveH' D Hδ1 and by induction H' D Hδi for i = 2,..., s, hence H' D Hδ for each δ E A andfrom the condition (b) and strong continuity of T it follows that H' = H.

5.17. Theorem. The representations T^ and f^ of G = GC(t, M) with 1 < t < 00

described in 5.3 and 5.9 on BS H(52) and H(52), respectively, are irreducible.Proof. We can use lemmas 4.3 [11] and 5.16 above. If E is R-unital, so there is R-

unital subset E' in X fulfilling (UP4, 5) and E(0) RE such that E'=E^, since fi is a Radonand Borel regular measure (see [5], [7] and 5.2, 5.3, 5.8 above). Let us consider the familiesA = {E1 : E'=EM and E' is R- unital and satisfies (UP4,5) } for ( T y ( 0 ) , ff(E(0))); A =

{E1 : E'KE} for ( f ^ , ^ ( E ) ) with K-unital E. We take Gδ = G{{E')), Hδ = H'fE, or\E'

Hδ = H',E, respectively for δ = E' E A or in A. Using the transfinite induction andanalytical atlas on M we get a sequence of submanifolds Mn in M such that \Jn Mn isdense in M. Therefore, G((E)) is dense in {g E G : supp(g) C Uien Ei}. From lemma 5.15it follows that (a, b) are satisfied. Let E' and E" G A, H 1 = H',E,, H3 = H'\E» (or E' and

E" G A, Hx = H'™E,, H3 = H'^E»). Then there exists N EN such that E[ = E”i = Ei(

(or E[ = E" i = Ei) for each i > N, consequently, there are σ E ΣN and N1, 0 < N1 < N,with Fi := F/ n E”σ(i) 7 0 for N1 < i < N and {Ej U F " ^ ) =: Fj| 1 < j < N1} is some

disjoint family. If E™ = E', E^ = E"σ, E^ = (H?^ Fi) x E>N(0), E>N(0) := Ut>N Ei

(or E>N(0) := Ili>iv Ei), so E^ E A (or in A respectively). For H2 = H'^{2) (or H'^{2)) we

get Hi n i7j+i = (0) for i = 1 or 2, since νE(E( i F ( i + 1 ) ) = (0) in the first case, and

due to E(2)KE(i) for i = 1,3 and K-unitality of E in the second case. Whence (c) is

accomplished.

Let us fix δ0 = E' and take arbitrary δ = E" G A (or in A). There is N E N such that

E' = (Flili Ai) x E>N(0) and E" = (nili Bi) x F^i , where Ai n Aj = 0 and B n B, = 0 forz 7 j , Aj and Bj are clopen in P(0) := M\c/(Ui>iv E\ ) (or in P := M\ (Ui>w Ei). From

(UP3) it follows that P is clopen in M, since {Ej} is a locally finite family [6]. Suppose

j , Ci,0 = Aj\c/(Uili -Bj), C0,j = Bj\d(\$=l Ai) (omitting d in the second case

Nj=0 Ci,j a n d Bj = U=o Ci,j).Then B := FI Li Bj a r e disjoint unions of subsets of the form D = Y\f=1 Dj with Dj = Cij

for suitable ij. Let DB be the family of all such I ^ D C B and [D] := D x E>N(0), hence

-fffe- = 0 D G D B - [[D] (analogously for H due to theorems 5.13 and 5.16 [24]). Each Dj for

D E DB either is contained in some Ai or is disjoint with all Ai. Hence (i) E'f^i C H',E,

in the case Da C A for some a E T,N; or (zz) H'^D^ C H'(52) 0 -ff' / in the case when

some Dj is disjoint with every Ai or Dj U -Dj/ C A for some j 7 j ' (analogously for

# ) . Therefore, (zzz) ^ = F j 0 Hi H} C ^ 0 , H} C i?'(E) 0 ^ 0 , where H\ and F |

are the direct sums of the spaces -f mi in the cases (i) and (ii) respectively (analogously

for H). Let Gδ0,δ = UDGD^ GD, GD := Yli=iG\Di, hence in view of lemma 5.12 the

18

representation of the subgroup Gδ0,δ C Gδ0 := G((E')) on Hδ0 = H'\E, is disjoint withHδ C H'{ E ) 0 Hδ0 =: F. Therefore, due to (iii) in BS F is dense U<seA Hδ2 (analogouslyin the case of H). Hence (d) is also satisfied.

5.18. Lemma. Fora E SQO the mapping (Raf)(x) = f(xa) gives the isomorphism ofrepresentations [T^H'iY.)) with(T a<£,H'( a E ) and (f^, # ' ( £ ) ) wiih(T a<£,H'( a E ) )of the group G, where " E = ( a

Π /i, Ea~l) in the first case and a E = ( a Π, F a " 1 ) inthe second case, aσ := aver1, aΠ(σ) := U(a~1aa).

Proof. If a subset E is R-unital (or K-unital) in X, then the same is true forEa = Yl^Zi Ea(i). From formulas (i — iii) in 5.10 their equivalence follows.

5.19. Proposition.Suppose Tv^ =Tv^ orT^>=T-^> (that is, they are equivalent),

where E i = (Ri;/i,E), E2 = (Π2; //, F) and E, F are H-unital; E i = (Π1;E), (E2 =(Π2, F) and E, F are K-unital. Then ER(Fb) or EK(Fb) respectively for some b E Soo-The proof consists of few steps and is the modification of §5 [11].

Step 1. From the proof of theorem 5.17 it follows that E and F can be taken R-

unital satisfying (UP4, 5) (or K-unital), also for -ff'(Ei) and H'{J22) as the completions of

UE(I)=E ffSi) and \JFW=F Hypm subsets E(1) and F(1) can be chosen R-unital satisfying

(UP4,5) (see the proof 5.10). For -ff'(Ei) as the completion of UEWKE H\EW ^ u e to 5.3

and E(1)KE we have that E(1) is K-unital, analogously for F. Let A : H'(Ei) -»• ^ ' ( E 2 )be a nonzero intertwining operator of TV with TV , hence A is the isometry, since these

representations are irreducible. There is the projector PEΠ(1): -ff'(Ei) -^ EΠ(1)1, that in-

duced by the decomposition A' 0 B) 0 B = A' for BS A' and B over the LC field L D Q s

[24]. Then AF(1),E(1) := P|ΠF2

(1) o A o P|ΠE1(1) intertwines T^JG^ with T^JG^ 1 ), where

n G((F(1)) (analogously for f ) . Moreover, there are σ and τ 6 S M

and a submanifold M n in M with gEi,n(1) = EaL n and gFi,n(1) = F^,l n for each i. Let us

denote hul(E(1)) := C / ^ N ^ and E$ = F$ = Ei D F^\ E\% = Ei(1) \ hul(F(1)),Fj!l = Fj(1) \ hul(E(1)), whence (hul(E(1)))n and (hul(F(1)))n are invariant relative to g

and g(Ei,j(1))n = (Eσ(1)(i),τ(j))n, g(E^)n = ( £ $ ) > 0 0 ) n > g(FJ%)n = ( F $ ) > 0 0 ) n > where the index

n means the intesection with Mn and in the case of T the index n and the closure cl may

be omitted. Considering different g we can obtain Mn such that their union is dense in

M.

Then we consider the following disjoint unions Ei(2) := \J°°=1E£}, Fj(2) := U ^ i ^ -

The projectors Pn : Y -^ K n induce the projectors Pn : M -^ Mn for suitable Mn,

which may be produced with the help of normal coordinates in Uj, that is, Pn\Uj =

(pj1 ° Pn ° φj : Uj —> Uj,n. Therefore, {Pn : n} induces the consistent family of measures

/in = /i\P-\Bf(Mn)) and /i(p-\E^AE^)) = 0, analogously for F. In view of 5.2

/liE^AE^) +/i(F J

( 1 ) AFf ) ) = 0 for T (or Ei(1) = Ei(2) and Fj(1) = Fj(2) for f ) , conse-

quently, H'^l) = H'$2), H'^l) = H[% and analogously for H, where E

Step 2. Let Ei for i = 1,..., r and Fj for j = 1,..., s are clopen in M, E^ilEi = 0 andF i n Fl = 0 for each i ^ I. Assume d = YYl=l G|Ei, G2 = n,-=i G|Fj, H1 = (g)[=1 H(Ei),H2 = <8>s

j=iH(Fi)> w h e r e H(Ei) : = L2(Et;n\Et,C) (or H(E i ) := BUC(E i ^ L) inthe second case). Put Bi,j = Ei n Fj, Bi)OO = E i \ ( U ^ i ^ ) , Boo,,- = F j \ ( U L i ^ ) ,/oo = {1, 2,..., r, 00} and JQO = {1, 2,..., s, 00}. Suppose that (i) [i(Ei \\Jjejoo Bi,j) +

19

V>(F3 \ Uie/oo Bi,j) = 0 (or (ii) Ei = {jjeJoo Bi,j and Fj = \Ji€loo Bi,j respectively). ThenH{Ei) = ®^JooH{BhJ), define G1>2 = Ui€iooj€jooG(Bi>j) C d n G2, where B^ =0, G(Bi,j) = (e) for Bi,j = 0, G(B) = G|B. Each G(5i)OO acts trivially on H2, since-Bi,oo n Fj = 0 for each j . Each G(Bi,j) acts naturally on H(Bi,j) and trivially on H(Bi>j>)if (^)i') 7 (i,j). There are the natural isomorphisms Φj,i : H(i) —> H(j) ® #([«] \ [j])for (m) [I] D [j], where j = (ji> •••>>); « = ( « i , - , O , #(3) = <8£=i #(#*,*) , # « =

= {(i,ji) : 1 < i < r}, [I] = {(ij,j) : 1 < j < s}, //([I] \ [j]) :=] H (it;) [i] \ []] C {(oo,j) : 1 < j < s}, r < s. From 5.12 we get the

following.

5.20. Lemma.Suppose there are given the representations Li : G i> U(Hi) (orIS (Hi)) with an intertwining operator A : H 1> H 2 of the representation L1|G12 w * ^L2|G 12 and the condition (i) (or (ii) of step 2) is fulfilled. Then A is the direct sum ofλj,i o $jl o (Idj ® ψ) relative to some isometry, where ψ G H([i] \ [j]), λj,i e C (or Lrespectively), [i] and [j] satisfy (iii,iv) of step 2 (see also 3.2 [11]).

Step 3. Let G^ := (fl^ G{E$)) x (nj G(E{^)) x (n; G(JF,(

)

1

ci)) c G(1), N G N, sothere are the natural embeddings H N := (<8>i<N,j<N H(Ei,j)) ^ -ff|e(2) and analogouslyfor /f. In view of lemma 5.18 and the fact that L W N HN is dense in H|E(2) we get theanalogs of lemmas 5.4 and 5.5 [11] and the following.

5.21. Lemma.Suppose there exists a nonzero intertwining operator A of Ty^ with

Ty . Then the family AF(1),E(1) = PJj?m o AoP|ΠE1(1) strongly converges to A, where

F(1)KF, Ei(1) and Fj(1) are clopen in M and hul(E(1)) = hul(F(1)).

Proof. In the considered case E\% and F^l are clopen and Ei(3) = EI(1) U

Fj(3) = Fj(1) U E^l produce K-unital E(3) = [lieN Ei(3) D E(1). We take Mn = M, so from

the definition of G(1) in step 1 and lemma 5.13 we get E(3)KE(1) and [ L ^ N Fj(3) D F(1),

F(3)KF(1). We can choose hul(E(1)) = hul(F(1)), consequently, hul(E(3)) = hul(F(3)

and these sets are clopen in M. For E\]} := \Jj&i E\]], Ef] = E\]} U E\% there are

isomorphisms BUC(Ei -> L) = BUC(Ei,f -»• L)0 BUC(EI, -»• L), since Ei(2) is

open and compact in M due to (UP3) and £"{ C Ei(1) . In view of 5.20 AF(2),E(2)ξ = 0

deletes all components from ξ e H(Ei^). On the other hand, there exists the isometricalembedding of H!E into H'n(J2), L is locally compact, hence there exists P|ΠE (see step 1).

5.22. Corollary. Suppose Ty =Ty (or Ty =Ty ) . Then substituting R-unital

(or K-unital ) subsets E in J2i or F in Y.2 ont° R-cofinal (or K-cofinal) to them subsets

in X one may assume that hul(E) = hul(F).

Step 4. In view of step 3, 5.19 and 5.20 we may restrict our consideration of AF(1),E(1)

to hul(E(1)) = hul(F(1)), consequently, E& = 0, F^l = 0 and Ei(2) = \Jj€SE\% Fj(2) =

\ Ei(2))+ M^,(1) \ Ff>) = 0 for Ty- Ei(1) = Ei(2) and

for Ty. Then H'^ = H[E. ® V(Πi) and G(2) acts trivially on V(Πi) for i = 1 ory.

2 (analogously for H with c 0(N,L)). Then (i) H|E(2) = © ^ ©^. e N H(EI,J(2)I) and (ii)

H|F(2) is isomorphic with (©£L'i©j.eN H(Fj,i))<8>j>N' H(Fj )), since X is isomorphic

to ]Jj>N,(M)j, where (M)j = M for each j , analogously for H). Denote by P^> the

projectors onto ©^^ © ^ ^ N H(Fi,j(2)i). Each Fi(2) is compact in the second case, hence suchdirect sums are finite.It is enough to show that lim;v'->oo-FV' = 0 (strong convergence) if

20

(or E(2) ^ /ΣK(F(2)b)) for any b E Soo, but this means AF(1),E(1) = AF(2),E(2) = 0 and

A = 0 by step 3 (or lemma 5.21). This contradicts the condition imposed on A. Put

ci,j = M ^ S ) = M*l?) , di = M 4 2 ) ) = E ^ i C i j , ej = ^ 2 ) ) = XZtCij. From thesolution of the combinatorial type problem 5.8 [11] we get the final part of the proof of5.19 above for T ^ .

Step 5. Now let M be locally compact (the second case) and evidently Ty (g) ^ I

for each g ^ id in G(t, M) with supp(g) C hul(E(1)). In view of (UP3) hul(E(1)) is notcompact. If (i) E £ ΣK(Fb) for each b 6 E M then E ^ g(F) for each g E G(t,M),since for E = gF must be noncompact supp(g). The group G(t, M) is simple and perfect(see theorem 3.1(3) above) and Ty : G(t,M) —> IS(3(52)) is strongly continuous. The

group SQO acts naturally on G((E(1)) and G(2). In view of step 4, G(2) acts trivially on

c0(N,L). Then G^/Soo and G((JB(1)))/SOO are also simple and perfect. From (UP3) it

follows that hul(E(1)) is not compact, consequently, there are not any g E G and n e N ,

a E Soo such that gEi(1) = F^l for each i > n.

The existence of the intertwining operator A of 7V ' with TV / leads to the conclusion

that (ii) APN'A~l is strongly (continuously) convergent to the bounded invertible operator

while N' tends to oo, if P^> is strongly convergent to the bounded invertible operator.

From 5.13 and 5.16 [24] and formulas (i,ii) of step 4 it follows that the operator A can

be represented in the block form, where the blocks A(i),(j) : H(E\ •) —> H(Fj,i(1)) are

the isomorphisms. In view of the condition (i) of step 5 subsets E(j) := Yl^Li Ei,j(1)i a n d

F(i) '•= YlJLi (1)j are closed, but not open in E(1) and F(1) respectively for each sequences

j := (ji : i E N) and i = (ij : j E N) corresponding to Ei,ji =£ 0 and Fj,ij =£ , since

card{i : card(j : E\' = \JjEij, Ei,j ^ 0) > 1} = H0- Therefore, the characteristic

functions of E(j) and F(i) do not belong to BUC(E( -»• L) and BUC(F^ -> L)

respectively. Therefore, (ii) of step 5 is not satisfied, hence {-P/v} cann't be convergent

in view of (ii) in step 4 for invertible A with bounded A and A~l. There remains only

one possibility that PN> converges strongly to 0, that is, also impossible in view of step 4.

Thus 5.19 is proved for Ty also.

5.23. Theorem.Let ( T ^ , H''(£*)) [and (f^/, i?(E-))7 be the irreducible represen-

tations defined in 5.9 with J2i = (T1I;IJL,E) and 522 = (n^;//, F) [and J2[ = (Π1;E) and

J2'2 = (Π2; F)]. Then Ty =Ty [and Ty =Ty J if and only if there exists a E T.^ such

that IIi= a n 2 ; ER(Fa~l) [and EK(Fa~1) respectively].Proof. Necessity. In view of 5.19 and 5.8 we have ER(Fa~l) such that E and F

satisfy (UP4,5) [and EKFa'1] for some a E t^. From 5.14 and 5.17 it follows that

n : = «n2.Sufficiency. The space BUC(E^ -> L) is isomorphic with BUC(F^ -> L), since

hul(E(1)) = hul(F(1)), £ « and F(1) are K-unital and BUC(Ei -> L) is isomorphic

with BUC(Fi

(1) -»• L) (see also theorems 5.13, 5.16 [24]). If E(1) E ΣK(F(1)b) then the

isometrical isomorphism can be chosen such that it establishes a bijective corespondence

between characteristic functions of all respective clopen subsets of E(1) and F(1). Hence

there exists an intertwining operator A of Ty with Ty , since Diff(t, B(Kn, x, r)) andDiff(t, B(Kn, y, R)) are isomorphic topological groups for each x and y E K n , 0 < r < ooand 0 < R < oo. Also the case Ty^ is obvious using JJL and L2(E; /i, C).

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[28] W.H. Schikhov. "On p-adic compact operators". Report 8911 (Dept. Math. Cath.Univ., Nijmegen, The Netherlands, 1989).[29] A. Weil. "Basic number theory". (Springer-Verlag, Berlin, 1973).

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