Quasi-invariant measures on non-archimedean groups and semigroups of loops and paths, their representations. I

Loop and path groups G and semigroups S as families of mappings of one non-Archimedean Banach manifold M into another N with marked points over the same locally compact field K of characteristic char(K) = 0 are considered. Quasi-invariant measures on them are constructed. Then measures are used to investigate irreducible representations of such groups.

Loop and path groups are very important in differential geometry, algebraic topology and theoretical physics [2,5,19,25]. Moreover, quasi-invariant measures are helpful for an investigation of the group itself. In the case of real manifolds Gaussian quasi-invariant measures on loop groups and semigroups were constructed and then applied for the investigation of unitary representations in [17]. , In [12,14,16] quasi-invariant measures on non-Archimedean Banach spaces X and diffeomorphism groups were investigated. During the recent time non-Archimedean functional analysis and quantum mechanics develop intensively [21,27]. One of the reason for this is in the divergence of some important integrals and series in the real or complex cases and their convergence in the non-Archimedean case. Therefore, it is important to consider non-Archimedean loop semigroups and groups, that are new objects. There are many principal differences between classical functional analysis (over the fields R or C) and non-Archimedean (21., 22,24,28]. Then the names of loop groups and semigroups in the non-Archimedean case are used here in analogy with the case of manifolds over the real field R, but their meaning is quite different, because non-Archimedean manifolds M are totally disconnected with the small inductive dimension ind(M) = 0 (see §6.2 and Ch. 7 in [8]) and real manifolds are locally connected with ind(M) > l.
In the real case loop groups G are locally connected for dimRM dimnN, but in the non-Archimedean case they are zero-dimensional with ind(G) = 0, where 1 dimRN oo is the dimension of the tangent Banach space T~N over R for 1V. Shortly the non-Archimedean loop semigroups were considered in [15].
In this article loop groups and semi groups are considered. The loop semigroups are quotients of families of mappings f from one non-Archimedean manifold M into another N with limx~s0 03A603C5 f(x) = 0 for a v ~ t by the corresponding equivalence relations, where so and 0 are marked points in M and N respectively, M = M B ~s~}, ~v f are continuous extensions of the partial difference quotients ~~ f . Besides locally compact manifolds also non-locally compact Banach manifolds M and N are considered. This work presents results for manifolds At and N modelled on Banach spa.ces X and Y over locally compact fields K such that Qp C K C Cp, where Qp is the field of p-adic numbers, Cp is the field of complex numbers with the corresponding non-Archimedean norm, that is, K are finite algebraic extensions of Qp.
More interesting are groups constructed with the help of A. Grothendieck procedure of an Abelian group from an Abelian monoid. This produces the non-Archimedean loop group. Also semigroups and groups of paths are considered, but it is only formal terminology. Both in the real and non-Archimedean cases compositions of pathes are defined not for all elements, but satisfying the additional condition. Since the non-Archimedean field K is not directed (apart from R) this condition is another in the non-Archimedean case than in the real case. On the other hand, semigroups with units (that is, monoids) and groups of loops have indeed the algebraic structure of monoids and groups respectively. Quasi-invariant measures on these semigroups and groups are constructed in §3 of Part I and §2 of Part II. Then such measures are used for the investigation of irreducible unitary representations of loop groups in §3 of Part II.
To construct real-valued and also Qq-valued (for q ~ p) quasi-invariant measures specific anti derivations and isomorphisms of non-Archimedean Banach spaces are considered. Apart from the real-valued measures the notion of quasi-invariance for Qq-valued measures is quite different and is based on the results from [24]. For this a Banach space L(p,) of integrable functions defined for a tight measure ~ on an algebra Bco~X ) of clopen subsets of a Hausdorff space X with ind(X) = 8 is used. To construct measures we start from measures equivalent to Haar measures on K. The real-valued nonnegative Haar measure v on K as the additive group is characterised by the equation v(x + A) = v(A) for each x E K and A E where denotes the Borel u-field of X (see Chapter VII in [4]). Each bounded nonnegative Borel measure on a clopen compact subset of K may contain only countable number of atoms, but for it each atom may be only a singleton.
Therefore, the Haar measure v certainly has not any atom. The Qq-valued Haar measure w on K is characterised by w(z + A) = w(A) for each z E K and each A E Bco(K) (see Chapter 8 in [21]). In view of Monna-Springer Theorem 8.4 [21] a non-zero Qq-valued invariant measure w on Bco(K) exists for each q ~ p, but does not exist for q = p.
Pseudo-diSerentiability of measures with values in R and Qq also is considered, because in the non-Archimedean case there is not any non-trivial differentiable function Qq for q ~ p. This notion of pseudo-differentiability of real-valued measures is based on Vladimirov operator on the corresponding space of functions f K --~ R [26,27].
Semigroups and groups of loops and paths are investigated in §2 and Part II respectively. Here real-valued and also Qq-valued measures are considered (for q ~ p). Unitary representations of loop groups are given in Part II.
The loop groups are neither Banach-Lie nor locally compact and have a structure of a non-Archimedean Banach manifold (see Theorem 11.2.3). A.
Weil theorem states that, if there exists a non-trivial non-negative quasiinvariant measure on a topological group G relative to left shifts Lg for all g E G, then G is locally compact, where Lgh = gh for each g and h ~ G (see also Corollaries 111.12.4,5 [9]). Therefore, the loop group and the loop semigroup has not any non-zero Haar measure. In Part I manifolds with disjoint atlases modelled on Banach spaces are considered. This is sufficient for many purposes. Moreover, in §II.3.4 it is shown, that arbitrary atlases of the corresponding class of smoothness of the same manifolds preserve loop groups and semigroups up to algebraic topological isomorphisms. In Part II loop and path groups for manifolds modelled on locally K-convex spaces also are discussed.
The notation is summarized in §II.6.

Loop semigroups.
To avoid misunderstandings we first give our definitions and notations. They are quite necessary, but a reader wishing to get main results quickly can begin to read from §2.6 and then to find appearing notions and notations in § §2.1-5.
2.1. Notation. Let K be a local field, that is, a finite algebraic extension of the p-adic field Qp for the corresponding prime number p [28]. For b E R, 6 (4) h; () := (F(z + ~h) -E Y~b e partial difference quotients of order b for 0 b I, ac + (h E U, ~h ~ o, :~-F, where Y~p is a Banach space obtained from Y by extension of a scalar field from K to Ap. By induction using Formulas (1 -4) we define partial difference quotients of orders n + 1 and n + b: where o t ~ R, sign(y) = -1 for y 0, sign(y) = 0 for y = 0 and Bign(y) = 1 for y > ~.
Then the locally K-convex space C(oo, U ~ Y) := ~~n=1 C(n, U ~ Y) is supplied with the ultrauniformity given by the family of ultranorms il * ~C(n,U~Y). 2.4.2. Let X, Y and M be the same as in §2.4.1 for a local field K. When X or Y are infinite-dimensional over K, then the Banach space C(t, M ~ Y) is in general of non-separable type over K for 0 t E R. For constructions of quasi-invariant measures it is necessary to have spaces of separable type. Therefore, subspaces of type Co are defined below. Their construction is analogous to that of Co from I°~ by imposing additional conditions (see for comparison [21]). Proof. This follows immediately from the definition, since where Co(t, M -~ K). In view of Formulas (1 2014 5) the space Co(t, M --~ Y) is of separable type over K, when card(a x ~ x A,~) ~ Evidently, for compact M the spaces Co (t, M -~ Y) and Co(t, ~I --~ Y) are isomorphic. This induces the continuous injective homomorphism for (1) lim diam{x EM: : A C M C X. On the other hand, from limn~~(fn V gn) = f V g it follows that limn~~ fn = f and limn~~ g" = g. Using Formulas (1 -4) we get '~o o f >x,~= f >x,~ and wo >x,~= e is the unit element in ~V)s ince f >x,~ ° g >x,~= fog >x,~ for each f and g E Co (f , (M, so) "(~:  (2) if At(M) is finite, then there are Ãt(TG) and Ãt(G) for which # is locally analytic. Moreover, G is not locally compact for each o t.
Proof. (A.) Let at first M be with a finite atlas At(M). . Let I § E TqG for each q, e G, V e Co(g, G -+ TG), suppose also that oV~ = q be the natural projection such that £ TG -+ G, then V is a vector field on G of class Indeed, let © = (g E Co(/, (M, so) -+ (N, y0)) : I /p) . Then -(N, yo)) and Co(( , (M, so) -+ (TN, yo x o)) have disjoint atlases with clopen charts, since there are neighbourhoods P 3 tvo in Co(f, M -N) and fi 3 (wo © 0) in Co(/, M -+ TN) homeomorphic to clopen subsets in Co(/, M -+ Y) and Co (/, M -+ Y x Y) , where P and # are such that they may be embedded into A := #; Co (/ , U; -+ Y) and B := %; U; -Y x Y) respectively. Moreover, there are the natural embeddings A P Co(£, M -Y) and B P Co(£, M -Y x Y).
The disjoint and analytic atlases At(C0(03BE, M ~ N)) and M T N) induce disjoint clopen atlases in G and TG with the help of the corresponding equivalence relations, since the metrics in these quotient spaces satisfy Inequality 2.7.(8). These atlases are countable, since G and TG are separable.
Let us suppose, that G has a compact clopen neighbourhood W. of e.
3.1. . Definition. Let G denote the Hausdorff totally disconnected group. A function f : K --~ Y is called pseudo-differentiable of order b, if there exists the following integral: (1) where g(z,y,b) :=| zy |-1-b for Y = C and g(z,y,b):= q(-1-b) ordp(x-y) for Y = Aq with the corresponding Haar Y-valued measure v and b ~ C (see also §2.1). We introduce the following notation for such integral by B(K, fl, ~) instead of the entire K.
3.2. Remarks. 1. For a Hausdorff topological space X' with a small inductive dimension ind(X') = 0 [8] the Borel u-field is denoted B f (X'). Henceforth, measures ~ are given on a measurable space (X', .E), where E is a u-algebra such that E ~ Bf(X') and has values in R or in the local field Kq ~ Qq. The completion of B f (X ') relative to is denoted by Af(X',p,). The total variation of p, with values in R is denoted by | |(A) for A E A f (X', ~). If ~c is non-negative and ~(X') ~. ~, then it is called a probability measure.
We recall that a mapping : E ~ Kq is called a measure, if the following conditions are accomplished: (1) ~c is additive and p(0) = 0, Tight measures (that is, measures defined on an algebra E such that E D Bco(X')) compose a Banach space ~I (X') with a norm Let also G' and G" be dense subsemigroups in G such that G" C G' and a topology T on M is compatible with G', that is, p ~ h is the homomorphism of (M, F) into itself for each h E G', where := p,(h o A) for each A E B f (G). Let T be the topology of convergence for each E E B f (G). If P E (M, F) and h ~ P for each h E G' then is called quasi-invariant on G relative to G'. We shall consider p with the continuous quasi-invariance such that x* is in the class of smoothness C(oo). Therefore, there is the linear mapping (differential)   In view of Theorems 6.13 and ~. 16 [21] the Banach space H is isomorphic with c0(03C90, ff ). The Borel u-algebras of c0(03C90, K) relative to the norm and weak topologies coincide. Suppose that there exists a sequence of finitedimensional over K distributions 03BDLn on which means by the definition, that Ln := spK(e1, ..., en) is a sequence of finite-dimensional over K subspaces such that Ln C Lm for each n m, Un Ln is dense in co, 03BDLn is a family of measures on all with values in one chosen field among either R or Kq satisfying the following condition (13) ~ Lm) = 03BDLn(A) for each A E and each n m, where 03C0n : co ~ Ln are projections such that = x = xjej E co, a:n = xjej and xj ~ K.
When the sequence of finite-dimensional over K distributions (14) = ~nj=103BDj(dxj) generates a measure v on co we write  for each A E B f(G), where o r 1 for real ~i or r = q for ~i with values in Kq. This ~ is the desired measure, which is quasi-invariant and pseudo-differentiable of order b relative to the submonoid ?" = G' (see § §3.3, 3.4).