Algebraic Weyl system and application

-In order to define an I-sheaf due to [BZ 76] on the finite-dimensional p-adic symplectic space, we define an algebraic Weyl system, and its properties are investigated. In particular, we prove some necessary and sufficient conditions for Weyl system to be irreducible. As application, we give another proof of the Ston-Von Neumann Theorem of the p-adic Heisenberg group. From the Schrodinger representation associated to a selfdual lattice, we construct a Weyl system depending on a selfdual lattice and a p-adic valued function. 1. Introducton. First we begin with the notations: Let No, Z, Q, R, C and T be the set of non-negative integers, the ring of integers, the rational number field, the real number field, the complex number field and the set of complex numbers of modulus 1, respectively. The field of p-adic numbers Qp are constructed as follows: For a fixed prime number p, the p-adic valuation ) . |p on Q is defined in the following way. At the first, we define it for natural numbers. Every natural number n can be represented as the product of prime numbers n = 2v23v3...pvp .... Then we define |n|p = we set |0|p = 0 and I nip = We extend the definition of p-adic valuation ] to all rational numbers by setting for m # 0, = |n|p/|m|p. The completion ofQ with respect to the metric dp(x, y) = y|p is a locally compact field Qp. The p-adic valuation satisfies the strong triangle inequality (1.1) |x + y|p ~ max(|x|p, |y|p). Any x E Qp can be expressed as x = p~ ~~° o with v E Z and a~ E Z satisfying 0 ~ aj ~ P 2014 1) O. ,To define the Fourier transform, an additive character ~p(03BBx) = exp(203C0i{03BBx}p) for every fixed A E Qp on Qp is used. Here = pv03A3-v-1j=0 ajpj is the fractional part of x. ’ In the Hilbert space L2(Qp) of C-valued square integrable functions on Qp, we introduce the standard inner product and the norm (1.2) (03C8,03C6)=~Qp03C8(x)03C6(x)dx, ~03C8~2=(03C8,03C8), where dx is the Haar measure on Qp such that the volume of the ring of p-adic integers Zp is 1. . 1991 Nlathematics Sub ject Classification. Primary 11Sxx; Secondary 11S80, 11585, 11599

ABSTRACT. -In order to define an I-sheaf due to [BZ 76] on the finite-dimensional p-adic symplectic space, we define an algebraic Weyl system, and its properties are investigated. In particular, we prove some necessary and sufficient conditions for Weyl system to be irreducible. As application, we give another proof of the Ston-Von Neumann Theorem of the p-adic Heisenberg group. From the Schrodinger representation associated to a selfdual lattice, we construct a Weyl system depending on a selfdual lattice and a p-adic valued function.
1. Introducton. First we begin with the notations: Let No, Z, Q, R, C and T be the set of non-negative integers, the ring of integers, the rational number field, the real number field, the complex number field and the set of complex numbers of modulus 1, respectively. The field of p-adic numbers Qp are constructed as follows: For a fixed prime number p, the p-adic valuation ) . |p on Q is defined in the following way. At the first, we define it for natural numbers. Every natural number n can be represented as the product of prime numbers n = 2v23v3...pvp .... Then we define |n|p = we set |0|p = 0 and Inip = We extend the definition of p-adic valuation ] to all rational numbers by setting for m # 0, = |n|p/|m|p. The completion ofQ with respect to the metric dp(x, y) = y|p is a locally compact field Qp. The p-adic valuation satisfies the strong triangle inequality (1.1) |x + y|p ~ max(|x|p, |y|p).
It is well known that, traditionally, we use ordinary real numbers in theoretical and mathematical physics, since lengths of segments and angles etc. from Archimedean axiom should be measured precisely. However in quantum gravity and in string theory it was proved that a measurement of distances smaller than the Planck length (it is the smallest distance that can be measured, approximately 10-33cm) is impossible. Vladimirov and Volovich [VV 84] proposed to consider the superanalysis and corresponding supersymmetric field theories not only over the field R but also over the field Qp and other locally compact fields. The interest in physics of n.a. quantum models is based on that the structure of space-time for very small distances less than Planck length might conveniently be described by n.a. numbers. There are different mathematical ways to describe this violation of the Archimedean axiom. One of them is given by p-adic analysis. Vladimirov [VVZ 94]). This formalism is based on a triple (L2(Qp), W (z), U(t)). Here, L2(Qp) is the Hilbert space of C-valued square integrable functions on Qp, W (z) is the Weyl representation of the commutation relations, z is a point in the classical phase space and U(t) is the time evolution operator where the time t is a p-adic number. The proposed formalism was extended, by Zelenov [Zel 91,92,93,94], to the case of many-and infinite-dimensional quantum mechanics, in which notion of Weyl system (H, W ) on the p-adic symplectic space (V, B) was used, and the representation theory of the p-adic Heisenberg group was investigated (cf. see alsõ Nleu 91]).
We recall that the definition of I-sheaf on the I-space by Bernshtein and Zelevinskii [BZ 76, pp. 6-9] which is an introduction to the representation theory of p-adic groups: A topological space X is said to be an I-space if it is Hausdorff, locally compact, and zero-dimensional. Denote by C~(X) and S(X) the space of all locally constant C-valued functions on X and the space of Schwartz-Bruhat functions on X, respectively. We say that an l-sheaf is defined on X if with each x E X there is associated a C-vector space Fx and there is defined a family .~' of cross-sections (that is, mappings cp defined on X such that E .~x for each x E X) such that the following conditions hold: (1) F is invariant under addition and multiplication by functions in C~(X).
(2) If y~ is a cross-section that coincides with some cross-section in ~' in a neighbourhood of each point, then c~ E ~'.
(3) If cp E E X, and = 0, then ~p = 0 in some neighbourhood of x.
(4) For any x E X and ~ E there exists a c~ such that = ~.
The I-sheaf on X is denoted by (X, .~'). The spaces are called stalks, and the elements of F cross-sections of the sheaf. We call the set supp 03C6 = {x E X : 03C6(x) ~ 0} the support of the cross-section 03C6 E J'. Condition (3) guarantees that supp 03C6 is closed.
A cross-section 03C6 E F is called finite if supp 03C6 is compact. We denote the space of finite cross-sections of (X,F) by Fc. It is clear that Fc is a S(X )-module, and that = Fc.
It turns out that this property can be taken as the basis for the definition of an I-sheaf. Proposition 1.1 (cf. [BZ 76, Proposition 1.14]). Let M be a S(X)-module such that S(X)M = M. Then there exists one and up to isomorphism only one I-sheaf (X, ~') such that M is isomorphic as an S(X)-module to the space of finite cross-sections Fc.
Proposition 1.1 means that defining an I-sheaf on X is equivalent to defining an S(X)module M such that S(X)M = M.
In this paper, in order to define an I-sheaf on the finite-dimentional p-adic symplectic space (V, B), we define an algebraic Weyl system (H, W ) on (V, B), and its properties are investigated, and we give an application. This paper is organized as follows: In §2, we summerize the general properties of Weyl systems {H, W ) on (V, B). In §3, we introduce the concept of an algebraic Weyl system (H, W ) on (V, B), and prove some necessary and sufficient conditions for Weyl system (H, W ) to be irreducible. In §4, as application, we prove the Stone-Von Neumann theorem. In §5, we construct a Weyl system depending on a selfdual lattice and a Qp-valued function.
When it is irreducible, our construction coincides with the one of Zelenove [Zel 94].
The author thanks the referee for pointing out an error in the proof of Lemma 4.1 in an earlier version of the paper.
Finally, thanks are due to Professor Yasuo Morita for invaluable advice. where each hy is a hyperbolic plane. Thus there is a basis {e~ : : 1 j 2n) for V for which the matrix of the form is ( -E'~ , where E~ is the n x n unit matrix and 0 is the n x n null matrix. Also the map V --> hy, j = 1,... , , n, which is defined by the is the orthogonal projection map on hy. Let Xo be a Zp-span of the symplectic basis : 1 j 2n~. Then Xo is an open compact Zp-submodule of V and has the following properties: If al and a2 belong to the same coset a E then, using (2.4), the definition of the vacuum vector and (2.8), we obtain 03C603B11 = W(03B11)03C60 = W (a2 + (03B11 -03B12))03C60 = Therefore, a change of the element a in â induces only a scalar multiplication of We call E the system of coherent states of (H, W ).
The investigation of Weyl systems on p-adic symplectic spaces is essentially based on the notions of vacuum vector and the system of coherent states as follows:  On the group V, we normalize the Haar measure d.r by the condition ~V0 dx = 1. In the Hilbert space L2 (V ) of C-valued square integrable functions on V, the standard inner product and the norm are given by (1.2)   (ii) f *(l(x)9) = (r(-x)f )*9~ (r(x)f )*9 = for all f ~ 9 E s(v).  Proof. Clearly, any non-zero cr E D is bijective, hence is invertible. Consequently, every non-zero element of D is a unit and thus D is a division ring. Also, if 0' is not a scalar multiple of the identity map Id, then 03C3 -03BB . Id is invertible for any A E C. Let 0 ~ 03C6 E H.
If a sequence (a -0 3 B B 1 . I d ) -1 0 3 C 6 , (a -a2 03C6,..., for a; ~ C distinct, of elements in H is linearly dependent, then there exists a sequence z~ , z2, ~ ~ ~ of elements in C such that not all the z=, say zi and z2, are equal to 0 and 03BB1 . Id)-lcp + z2(o -03BB2 . Id)-lcp = 0. It implies (z~ + z2 ) o -zl-~ ~ Id) p = 0, which contradicts the fact that (1 -A . Id is invertible for any A E C. Thus (c--A . Id)-103C6 (A E C) are linearly independent. But, indeed, by (iii) of Theorem 2.2.1 H is spanned by a E Jo, which is countable.
So H can not contain uncountablly many linearly independent vectors. Therefore 7 is a scalar multiple of the identity morphism. 4. Application. In this section, using Theorem 3.4 we give another proof of the Stone-Von Neumann Theorem of p-adic Heisenberg group.
The Thus a character ~ of Im() is given by the formula ~((t)) = ~p(03BBt) for some a E Qp. From now we assume A == 1. Let l be a Lagrangian subspace of V. Then L = Qp l is an abelian subgroup of lV, and there exists a unique character w of L such that w induces rõ n Im() and w induces the identity map on l. Explicitly, such w is given by (4.1) = (t E Qp, x E ~). We call ./V the p-adic Heisenberg group of (V, B). The center of TV consists of the elements is a subgroup of the commutative group L. Let w' be a character of L extending the character (a, 0) ~ a of the subgroup C(N). We can consider w' as a character of L satisfying (4.1). Remark 4.3. If (H, W ) is a Weyl system on (Y, .8), then the family of operators T(t,x) = p( t ) W ( x ) , (t, x) E N (resp. T(a,x) = 03B1W(x), (a, x) E N), forms an unitary representation of N (resp. N) on H. Conversely, if T(t,x), (t, x) E N (resp. T (a, x), (a, x) E N), is some unitary representation of N (resp. N) on the Hilbert space H satisfying the condition T(t, o) = IdH (resp. T{a, 0) = a ~ IdH), then the pair (H, W), W(x) = xp(--t)T (t, x) (resp. W(x) = T (1, x)), x E V, is a Weyl system on (V, B). If £ = £* , then £ is called selfdut. From now we consider only case where £ is a selfdual lattice. We consider a commutative subgroup F = ((t, z) : : t e Qp, °z e £) of N. Let. I°' be the image of the group F in N. The fact that the lattice £ is selfdual is equivalent to the fact that I°' is a maximal commutative subgroup of N. Let T be a character of I°' extending the character (a, 0) F-+ a of the subgroup T. By the Stone-Von Neumann Theorem, (Hr , Tr ) = Ind(N, F; T) is an irreducible unitary representation of N. (ii) For the Heisenberg group T x V of a p-adic symplectic space (V, B) of arbitrary dimension, £-representation corresponding Weyl £-system, which is analogues of Fock representations of commutation relations, was constructed by Zelenov [Zel 94].