FINITE APPROXIMATION OF WEYL SYSTEMS

FINITE APPROXIMATION OF WEYL SYSTEMS

T. DIGERNES, E. HUSSTAD and V. S. VARADARAJAN

Abstract

The functional analytic notion of approximation of Weyl systems, as introduced by Digernes and Varadarajan, is considered. It is shown that the Weyl system on any second countable lo- cally compact abelian group can be approximated by suitably chosen finite Weyl systems (Weyl systems on finite abelian groups).

1. Introduction

There has in recent years been considerable interest in quantum theories that are analogous to the conventional one, but differ from it in some of their main features. We mention, without aiming at completeness, the following works: Finkelstein [5], and Chan, Finkelstein [2] on q-deformed quantum theories; Vladimirov [14], and Vladimirov, Volovich, Zelenov [15] on p-adic quantum mechanics; Varadarajan [12] on quantum kinematics over general locally compact abelian groups treated from the point of view of deforma- tion and approximation. Quantum kinematics over finite abelian groups go back to Weyl [18], and Schwinger [9].

In this paper, we develop the point of view in [12] further, as we discuss approximations of quantum kinematics on locally compact abelian groups in more detail.

Our motivation for studying quantum models based on very general abe- lian groups does not arise solely, or even mainly, from any desire of gen- erality. Rather it stems from the work of Schwinger [9] on the classification of finite quantum systems, and its variations treated in Husstad [4], strongly influenced by Digernes and Varadarajan, and [12]. In [9] and [4], as well as in [18], unitary representations of a finite abelian group G and its dual G^ sa- tisfying Weyl commutation rules (ˆWeyl system) were studied, and it was shown that the conventional Weyl system associated to R^e may be approxi- mated (as in Section 2) arbitrarily well by Weyl systems on finite abelian groups. The approximation scheme of Schwinger gave remarkable numerical results on the level of generators. Motivated by this, the validity of this ap-

Received December 23, 1996.

proximation process was proved theoretically by Digernes, Varadarajan, Varadhan [3]. Consequently one could take the point of view that quantum theory associated with a finite abelian group is of much interest, and that the calculations overR^eare idealizations of the finite situation.

Our main Theorem 6.1 states that the Weyl system on any locally compact second countable abelian group is a limit of Weyl systems on finite abelian groups. This motivates dynamical considerations over other groups thanR^e, as they in this sense also give idealizations of finite quantum systems.

Moreover, this scheme makes it possible to obtain numerical results for more unconventional quantum dynamical systems.

It turns out (cfr. the comments in [12]) that the Weyl system on Z=pⁿZ converges to that associated to the p-adic field Q_p, as nÿ! 1. From this point of view one can for instance study ``harmonic oscillators'', and ``cou- lomb'' problems over local fields and rings. A path-integral formulation for vector spaces over division algebras over non-archimedean local fields has been established in Varadarajan [13].

The paper is organized as follows: In Section 2, Weyl systems and limits of such are defined. ``Continuity'' results for duality and direct sum are pre- sented, and the structure of finite Weyl systems is discussed.

In Section 3 we approximate any second countable locally compact abe- lian groupGby elementary groupsHN=Kn'R^eT^anFⁿ_NZ^bN, HN open compactly generated subgroup, while K_n is a compact subgroup for which G=K_n is the dual of a compactly generated group. The group Fⁿ_N is finite abelian. This follows from the more general results of van Kampen [11], and Pontrjagin [7]. LetN<n. Here,HN Hn which by [7] in particular induces an injection ⁿ_N :Z^bN ÿ!Z^bn, whereas the natural map G=KN ÿG=Kn in- duces an injectioncⁿ_N :Z^aN ÿ!Z^an. In the resulting mixed inductive/projec- tive limit description ofG, induced maps (for which ⁿ_N, andⁿ_N are two of the matrix coefficients) can be taken to be semi-aligned, but not diagonal in general.

Section 4 is used to define finite abelian groupsGn(and maps), candidates for approximating the Weyl system onG. We do a two-step approximation in the sense that we first take the diagonal H_n=K_n, and then construct G_n based on H_n=K_n and the matrix coefficients ⁿ_i and bⁿ_i, for all i<n. We choose to treat circle parts essentially as integer parts through Fourier transforms. In effect, the embedded finite translation on l²…Z^an†, and the embedded finite multiplication by character on L²…T^an†, are intertwined by thenon-finiteFourier transform fⁿ:l²…Z^an† !L²…T^an†.

The space of Schwartz-Bruhat [1] functionss…G†, functions which live on the elementary group Hn=Kn for some n, is introduced in Section 5 to deal with the analysis. The key point is thats…G†is invariant under the standard

Weyl system. This moves our calculations toH_n=K_n, and immediately shows that the Weyl system on G can be approximated by Weyl systems on ele- mentary groups. In this section, we get around problems with semi-align- ment: Given simple tensors in s…N† for which we control the support of their images in the discrete space l²…Z^aNF^N_NZ^bN†, we find that we also control the support of their images in l²…Z^anFⁿ_nZ^bn†for n>N (Lemma 5.3). Their support is governed by the mapsⁿ_N andcⁿ_N.

In Section 6, the general approximation result is proved. First, pointwise convergence of characters is verified. The important point is that inZ-direc- tions, the approximation is exact from some n, and by semi-alignment we control the coordinate in these directions. In T-directions we have no such control, but the result follows as the approximation in T-directions is uni- form. The strong convergence of projections follows directly from the sup- port control of Section 5. The remaining statements essentially follow from pointwise convergence of characters, and the support control Lemma 5.3.

For the sake of completeness, we conclude with a proof for the conventional caseR^e. Finally, some applications to local fields and rings are mentioned.

2. Limits of Weyl Systems

Let G be a second countable locally compact abelian group, with G^ as its Pontrjagin dual. The Weyl representations V and U of G and G, respec-^ tively, are, forx2Gand2G, given by^

…V…x†f†…y† ˆf…yÿx†;

…U…†f†…y† ˆ hy; if…y†; f 2L²…G†; y2G:

This pair of strongly continuous unitary representations satisfies the Weyl relations;

U…†V…x† ˆ hx; iV…x†U…† x2G; 2G:^ …1†

The pair…V;U†is called thestandard Weyl systemonG. The standard Weyl system is irreducible; the resulting projective unitary representation ofGG^ has no non-trivial invariant subspaces inL²…G†.

Definition 2.1 (Limit of Weyl systems). LetfGng¹_nˆ1,Gbe second coun- table, locally compact abelian groups with associated standard Weyl systems f…V_n;U_n†gand…V;U†. Then we say that the sequencefG_ngconverges toGin the sense of Weyl systems (or that…V;U†onGis the limit of…V_n;U_n†onG_n) if the following conditions are satisfied:

i) There is a Hilbert space H and isometries In:L²…Gn† ÿ!H,

I:L²…G† ÿ!H, such that P_nÿ!Pstrongly. Here, P_n and Pare the ortho- gonal projections onI_n…L²…G_n††andI…L²…G††, respectively.

ii) Setting

U⁰…!† ˆ IU…!†I^ÿ1 on I…L²…G††

identity on I…L²…G††^? (

!2G;^

and defining V⁰, U_n⁰ and V_n⁰ similarly, there are, for each x2G, 2G, se-^ quencesfxⁿgandfⁿgsuch thatxⁿ2Gn,ⁿ2G^n, and

U_n^0…n† ÿ!U⁰…†; V_n⁰…xⁿ† ÿ!V⁰…x†

strongly.

If the conditions in Definition 2.1 are satisfied, we easily see that hxⁿ; ⁿi ÿ! hx; i, so pointwise convergence of characters is necessary for Weyl convergence.

Assume that the standard Weyl system on G is a limit of the standard Weyl systems on Gn. The Stone-von Neumann-Mackey Theorem says that up to multiplicity and unitary equivalence, the Weyl relations forG have a unique solution. Thus, in the natural sense we can approximate any Weyl system on G (ˆ any other solution of (1)) by Weyl systems onG_n. In this paper, we exclusively work with standard Weyl systems.

2.1. Duality. The standard Weyl system …V;^ U†^ on G^ (identify Gand its bidual) is connected to the standard Weyl system …V;U†on Gthrough the Fourier transform;V^ ˆfUf^ÿ1 and U^ ˆfVf^ÿ1:The Fourier transform f:L²…G† ÿ!L²…G†^ is for suitablef given by…ff†…† ˆR

x2Gf…x†hÿx; idx, 2G:^

Proposition2.2. If G is a limit…Definition2:1†for Gn, thenG is a limit for^ G^_nin the sense of Weyl systems.

Proof. Define ^In:L²…G^n† ÿ!H and ^I :L²…G† ÿ!^ H by ^InˆInf^ÿ1_n and

^IˆIf^ÿ1: By construction, P^_n ˆP_n and P^ ˆP. Let x2G. Then we easily see that ^I_nU^_n…xⁿ†^I_n^ÿ1ˆI_nV_n…xⁿ†I_n^ÿ1 on P_n…H† and ^IU…x†^ ^I^ÿ1ˆIV…x†I^ÿ1 on P…H†. Similar formulas forV^ prove the proposition.

2.2. Direct Sum. If G is decomposable, say the direct sum of two sub- groups, GˆG₁G₂;then the standard Weyl system ofG can be identified with the sum of the standard Weyl systems …VGj;UGj† on Gj, jˆ1; 2. This means, for xˆ …x1;x2† 2G, L²…G† 'L²…G1† L²…G2† and VG…x† ' V_G1…x₁† V_G2…x₂†:Similar relations hold forU. This extends to finite index sets.

Proposition 2.3. If…V^j;U^j†on G^j is the limit of …V_n^j;U_n^j†on G^j_n for j in a finite set, then…V;U†on Gˆ _jG^jis the limit of…V_n;U_n†on G_nˆ _jG^j_n.

Proof. The result follows from strong continuity of tensor products for uniformly bounded sequences of operators. Let us give some details in the case of two summands: DefineInˆI_n¹I_n², and IˆI¹I², both acting on H:ˆH¹H². Then PnˆP¹_nP_n² and PˆP¹P². Likewise, for xˆ …x1;x2† 2G, put xⁿˆ …xⁿ₁;xⁿ₂†: Applied to a simple tensor ˆ 1 22H, V_n⁰…xⁿ† ˆV_n⁰…xⁿ₁† ₁V_n⁰…xⁿ₂† ₂ and V⁰…x† ˆV⁰…x₁† ₁V⁰…x₂† ₂: As P^j_n and V_n⁰…xⁿ_j† are uniformly bounded in norm, we get the expected con- vergence. The arguments are the same forU when we takeⁿˆ …ⁿ₁; ₂ⁿ†for ˆ …1; 2† 2G^ ˆG^1G^2.

2.3.Finite Weyl systems. Letnˆp^r₁¹p^r₂² p^r_k^k be the prime expansion ofn (pj are different primes while rj are non-negative integers). Then, Zn'Z_p^r1

1 Z_p^r2

2 Z_p^rk

k, and Zn is indecomposable if and only if n is a prime power.

So, recalling the direct sum construction in the previous paragraph, the standard Weyl system …V;U† on Z_n is indecomposable in this geometrical sense, precisely when nˆp^r is a prime power. By their structure theory, namely as direct sums of finite cyclic groups, we can build the Weyl system on any finite abelian group from these geometrically indecomposable finite Weyl systems. Schwinger [9] started constructing this theory of finite degree of freedom. Finite quantum systems were also studied byStov|¨cek and Tolar [10], and later in [4].

3. Structure of Second Countable Locally Compact Abelian Groups

Recall structure theorems on l.c.a. groups: Pontrjagin [7], Section 39, Theo- rem 51 proves that any compactly generated group is of the form R^eCZ^b, where C is a compact abelian group (``compactly generated'' means ``generated by a compact neighbourhood of the identity''). In the same reference, Section 39, Proposition A, he proves that for any l.c.a. group G, and any compact set KG, there is a compactly generated open sub- group H such that KHG. Moreover, the structure theorem of van Kampen [11], Theorem 2, says that any l.c.a. groupGis of the formR^eG¹, whereG¹ contains a compact open subgroupK. For any other such decom- position, the exponenteis the same. Following Reiter [8], we say thatGis a G¹-group ifeˆ0 in this decomposition. In particular, a compactly generated G¹-group is of the formCZ^b, whereCis compact.

IfGis second countable and l.c.a., so isG^ and any subgroup and quotient

of G. Likewise, the property of being a G¹-group is preserved under these operations.

We have not found Proposition 3.2 in any standard source in topological group theory. That proposition follows from the next lemma, which is probably also stated somewhere.

Lemma 3.1. Let G¹ be a G¹-group, H G¹ a compactly generated open subgroup, and let K⁰ be a compactly generated open subgroup of Gc¹. If K:ˆ …K⁰†^?H …the annihilator is taken inGc¹†, then H=K'T^aFZ^b;an elementary group…Fis a finite abelian group†.

Proof. Use the structure theorem of Pontrjagin [7], Section 39, Theorem 51, for compactly generated groups on both H and K⁰. By duality, G¹=K'T^aD, where D is a discrete abelian group. Moreover, as K is compact, H=K'C=KZ^b, where C=K is a compact group. AsH G¹ is open, H=KG¹=K is open, and there is an open injection C=KZ^bÿ!T^aD. In particular, the compact open subgroupC=K f0g must map to a compact open subgroup. As T^a has no open subgroups but itself, the image ofC=K f0gis of the formT^aF, whereF Dis discrete, but also compact. ThusF is finite.

Proposition 3.2. Let G be a second countable locally compact abelian group. Then G'R^eG¹;where the following is true for the group G¹: There exists an increasing sequence of open subgroupsfHng¹_nˆ1such that[HnˆG¹, and a decreasing sequence of compact subgroupsfKng, H1 KnKn‡1, such that \K_nˆ f0g. Moreover, H_n=K_n'T^anF_nZ^bn;where F_n is a finite abe- lian group…a_nand b_n are non-negative integers†.

Proof. The first part follows from [11].

Any separable l.c.a. group can be written as a countable union of compact sets (take an open neighbourhood of 0 with compact closure, and translate this closure with elements from a countable dense subset ofG¹). Thus, by [7], Section 39, Proposition A, and Theorem 51, there is fH_n⁰⁰g such that [H_n⁰⁰ˆG¹, and H_n⁰⁰ is open and compactly generated. Let H_n⁰ ˆH₁⁰⁰‡ H₂⁰⁰‡ ‡H_n⁰⁰, this subgroup is also open and compactly generated. Then H_n⁰ H_n‡1⁰ . Likewise, construct fK_m⁰g such that K_m⁰ is open and compactly generated, and K_m⁰ %Gc¹. Thus, Km:ˆ …K_m⁰†^? is a compact subgroup of G¹ such thatKm& f0g. As K1 is compact andfH_n⁰g covers G¹, there is an in- teger N such that for n>N, K1H_n⁰. Let Hn:ˆH_N‡n⁰ . Then, by the pre- vious lemma, we are done.

The proof of Lemma 3.1 implies thatHn=Km'T^amF^m_n Z^bn; so we get elementary groups also ifn6ˆm. Here,Fⁿ_nˆF_n.

3.1.Semi-alignment for G¹-groups. LetG¹be a second countableG¹-group with fH_ng and fK_mg from Proposition 3.2. Let _m:G¹ÿ!G¹=K_m and lm:G¹=Kmÿ!G¹=Kl be the natural maps (m>l). We have l ˆmlm, and the kernel oflm is the compact groupKl=Km. From the proof of Lem- ma 3.1, there are subgroups Zn;CnG¹, where Zn'Z^bn, Cn is compact, such that H_nˆC_n‡Z_n (direct sum). Likewise, there are subgroups T_m;D_mG=K_m with T_m'T^am, D_m is discrete, such that G=K_mˆT_m‡D_m (direct sum). For m>l and k>n we then have the following commuting diagram, which describes the structure ofG¹in terms of elementary groups;

the top row givesG¹ as an inductive limit, while the right-most column de- scribesG¹ as a projective limit:

C_n‡Z_n C_k‡Z_k G¹

??

ymj_Hn ??ymj_Hk ??ym

T_m‡F_n^m‡Z_n T_m‡F_k^m‡Z_k T_m‡D_m

??

y ??y ??y T_l‡F_n^l‡Z_n T_l‡F_k^l‡Z_k T_l‡D_l: …2†

All sums are direct. By the subgroupZnG¹=Km we mean the isomorphic image ofZn under m. The subgroups F_n^l 'F^l_n are finite. This setup follows from the comment following Proposition 3.2. However, in this diagram we select a basis for allZ^bn-parts from that in the top row, and a basis on allT^an- parts from that in the right-most column. We make no particular choice for the finite parts. Thus, in this basis the inclusion Hn=KmHk=Km is re- presented by a 33-matrix

i_nk^m ˆ id

^m_nk ^m_nk ^k_n 0

B@

1 CA;

…3†

for some morphisms ^m_nk:F_n^mÿ!F_k^m, ^m_nk:Znÿ!F_k^m, and ^k_n :Znÿ!Zk. The empty places represent 0-maps. The 22-matrix in the lower right corner is triangular becauseZ_khas no finite subgroups.

Lemma 3.3. In the preceeding matrix representation, ^m_nk, ^k_n are both in- jective.

Proof. The map^m_nkis injective as the finite part is only mapped into the finite part.

Let 06ˆz2Z_n. Let‰zŠbe the image ofzinZ_n=Ker…^m_nk† 'Im…^m_nk†. As this quotient is a finite abelian group, there is some integer k such that

‰kzŠ ˆk‰zŠ ˆ0. Consequently, kz2Ker…^m_nk†: Thus, by injectivity of i^m_nk, 06ˆi_nk^m…0;0;kz† ˆ …0;0; ^k_n…kz††, so^k_n…z† 6ˆ0. Thus,^k_n is injective.

Let us recall some generalities on dual groups. For l.c.a. groups fG_ig (i runs over a finite set), setGˆ _iG_i. ThenGb ˆd_iG_i' _iGb_i. Duality between GandGb :ˆ _iGb_i is set up with

h…x_i†;…_j†i ˆY

i

hx_i; _ii_Gi …4†

where for xi2Gi and i2Gbi, hxi; ii_Gi is some duality betweenGi and Gbi. Let :Gÿ!X be a morphism between l.c.a. groups Gˆ iGi and Xˆ jXj. Thus, ˆ …ij†, where ij:Gjÿ!Xi is a morphism. Then the dual mapb:Xb ÿ!Gb (under (4)) has matrix representation bˆ …c_ji†in the natural dual basis on both Xb, and G. Here,b c_ij :X_iÿ!G_j is the dual map under h;i_Gj, and h;i_Xi. Thus, the rule is the same as the usual one for the adjoint in matrix algebras.

So,lmj_Hn=Km ˆ:jⁿ_lm:Hn=Kmÿ!Hn=Kl has matrix representation

jⁿ_lmˆ

^m_l ⁰ⁿ_lm ⁰ⁿ_lm

id 0

B@

1 CA:

…5†

The reason is that the dual ofj_lmⁿ is an open injection; of the same type asi^m_nk. Using (4):

Corollary 3.4. In this matrix representation for j_lmⁿ , ^m_l :T_mÿ!T_l and ⁰ⁿ_lm:F_n^mÿ!F_n^l are both…non-zero† surjective. The morphism⁰ⁿ_lm:F_n^mÿ!T_l could be0.

The reason why^k_n, and^m_l do not depend onmandn, respectively, is that all diagrams in (2) are commutative. In fact,^k_n represents the inclusion ofZ_n in Hn intoZk inHk(top row). Similarly,^m_l is the matrix coefficient in lm

mappingTm ontoTl (right most column).

Later, we need some additional technical properties on the description of G¹.

Lemma3.5.Let i:Z^bÿ!Z^b0 be an injection. Then there is a complemented submodule B such that i…Z^b† BZ^b0, B'Z^b.

Proof. LetBbe thosez2Z^b0 for whichnz2i…Z^b†for somen2Z:This is the torsion closure of the image ofi. It is easy to see thatBis a submodule.

Let non-zero n_ju_j2i…Z^b†forn_j2Z, u_j 2B, wherej runs over a finite set. If fnjujgis dependent overZ, thenfujgis also dependent overZ. Conversely, as

i…Z^b† B, the rank ofB over Z equals the rank of i…Z^b†over Z. As i is in- jective, this rank isb. Moreover, it follows from the definition ofBthat the quotientZ^b0=Bis torsion free. This implies thatBis a direct summand (with bgenerators).

Lemma 3.6. Let Z^b1 ÿ!¹ Z^b2 Z^bn ÿ!ⁿ Z^bn‡1 ÿ!^n‡1 . . . where fng consists of injections. Then we can choose the basis on eachZ^bn such that for any pair m>n, for^m_n :ˆmÿ1 n,^m_n…Z^bn† Z^bn f0g^bmÿbn;maps into the first bn

factors.

Proof. Choose a basis onZ^b1. Then use Lemma 3.5 to find complemented ₁…Z^b1† B₁Z^b2. Now it is clear that we can choose a basis as wanted on Z^b2. Again from Lemma 3.5 we find complemented ₂…Z^b2† B₂Z^b3. In particular, ₂…B₁† B₂. Use Lemma 3.5 once more to find complemented 2…B1† B₂¹B2. Now we can obviously choose a basis onZ^b3 where Z^b2 is mapped into the first b2 coordinates, while Z^b1 is mapped into the first b1

coordinates ofZ^b3. The proof is completed by continuing this construction.

Define the elementary groups Enm:ˆT^amF^m_n Z^bn, Fⁿ_nˆFn and E_n:ˆE_nn:ThenH_n=K_m'E_nm.

Let us return to Diagram (2). Using this, we can assume that the induced injection (same notation) i_nk^m :E_nm ÿ!E_km has a matrix representation like (3), and the induced surjectionjⁿ_lm:Enmÿ!Enl has a matrix representation like (5).

As^k_n:Z^bnÿ!Z^bkis given as in Lemma 3.6, we apply that lemma tofZ^bng.

We use this change of basis on Z^bn in any E_nl. Thus, the matrix representa- tion ofi^m_nkis of the same type as before. As a dual condition (use (4) between Z^an and T^an), we make the kernel of^m_l contain T^amÿal f0g^al. The annihi- lator of the image of a morphism is the kernel of the dual map. The ob- servation that relates allZ^bn (inHn=Kl) toHn is crucial at this point. Order- ing problems would otherwise occur for the two-dimensional array (those …k;l† 2Z² for whichk;l>0), and we could not get the analogue of Lemma 3.6.

Assume thatn>Nare positive integers. RenameiNn :ˆi_Nnⁿ andjNn :ˆj^N_Nn. The next result summarizes our discussion:

Proposition 3.7. …semi-alignment†: Let G¹ be a G¹-group with structure given by Proposition 3.2. Then HN=Kn'ENn ˆT^anFⁿ_NZ^bN, where for n>N the induced injection i_Nn :E_Nnÿ!E_n, and the induced surjection j_Nn:E_Nn ÿ!E_Ncan be assumed to have matrix representations

i_Nnˆ id

ⁿ_N ⁿ_N ⁿ_N 0

B@

1

CA and j_Nn ˆ

ⁿ_N ⁰ⁿ_N ⁰ⁿ_N

id 0

B@

1 CA:

The maps _N :F_Nÿ!F_n and ⁿ_N :Z^bN ÿ!Z^bn are injective, and _Nn…Z^bN† Z^bN f0g^bnÿbN. Furthermore,⁰ⁿ_N:Fⁿ_N ÿ!F_N andⁿ_N :T^an ÿ!T^aN are surjec- tive. Also, the kernel of ⁿ_N contains T^anÿaN f0g^aN. The morphisms ⁿ_N:Z^bnÿ!Fnand⁰ⁿ_N :Fⁿ_N ÿ!T^aN could be0.

Asⁿ_N and⁰ⁿ_N in general are non-zero, the finite parts may intertwine in a non-trivial way. This causes technical problems in the rest of this paper.

3.2.The dual system. This paragraph contains definitions.

From standard topological group theory, Z^anFⁿ_NT^bN 'HNd=Kn' K_n^?=H_N^?: The sequence fK_n^?g %Gc¹ consists of open subgroups while the elements of fH_N^?g & f0g are compact subgroups. So, the (non-unique) se- quences fH_Ng and fK_ng single out a special decomposition for the dual groupGc¹.

Fix a dualityh;ibetweenG¹ andcG¹. Letx2HN and2K_n^?. Then hx‡K_n; ‡H_N^?i_Nnˆ hx; i

…6†

(well) defines the dualityh;i_Nn betweenH_N=K_nandH_Nd=K_nˆK_n^?=H_N^?. Define the standard duality h;i_s between ENnˆT^anFⁿ_NZ^bN and EdNnˆZ^anFⁿ_NT^bN (dual basis) as follows: Let xˆ …t;f;z† ˆ ……t_i†;…f_j†;…z_k†† 2E_Nn, andˆ …u;g;s† ˆ ……u_i†;…g_j†;…s_k†† 2Ed_Nn. Then

hx; i_sˆ ht;ui_Tanhf;gi_Fⁿ

Nhz;si_Zbn ˆY

i

t^u_iⁱY

j

e^2ifjgjnj Y

k

s^z_k^k; …7†

Fⁿ_Nˆ jZnj from the structure theorem of finite abelian groups.

In the previous section we found an isomorphism H_N=K_n 'E_Nn giving Proposition 3.7. Let

x‡K_n 2H_N=K_nÿ!x_Nn2E_Nn; in particular x_n:ˆx_nn …8†

denote this isomorphism. There is an isomorphism betweenK_n^?=H_N^?andEdNn, ‡H_N^?2K_n^?=H_N^?ÿ!_Nn2Ed_Nn; in particular _n:ˆ_nn;

…9†

such that forx2x‡K_n, and2‡H_N^?,

hx; i ˆ hx‡Kn; ‡H_N^?i_Nnˆ hxNn; Nni_s: …10†

This is because there is only one dual pairing modulo automorphisms (for any isomorphism K_n^?=H_N^?'Ed_Nn, (12) defines some dual pairing, compose this isomorphism with the appropriate automorphism).

AssumeN<n. Thenic_Nn (under the standard dual pairing) is the surjection induced from the natural map K_n^?=H_n^?ÿ!K_n^?=H_N^?, and jc_Nn is the injection induced from the inclusion K_N^?=H_N^?K_n^?=H_N^?: Let x2HN, 2K_N^?. Then hx‡KN; ‡H_N^?i_NN ˆ hx‡Kn; ‡H_N^?i_Nnˆ hx‡Kn; ‡H_n^?i_nn from (8).

Thus, the dual of the inclusion HN=KnHn=Kn is the natural map K_n^?=H_N^? ÿK_n^?=H_n^?, and the other way around for the original natural map.

As the standard dual pairing in particular is of the form (4), we have in the dual basis,

ic_Nn ˆ id

cⁿ_N cⁿ_N bⁿ_N 0

BB

@

1 CC

A and jc_Nnˆ cⁿ_N c⁰ⁿ_N c⁰ⁿ_N

id 0

BB

@

1 CC A:

The meaning of cⁿ_N etc. is clear from the definition of the standard dual pairing.

The dual map of ⁿ_N, cⁿ_N, is an injection Z^aNcⁿ_N ÿ!Z^an: The form of the standard dual pairing shows that cⁿ_N…Z^aN† Z^aN f0g^anÿaN. The maps fⁿ_Ng andfcⁿ_Ng are used in the definitions of the next section.

3.3. The inclusion L²…HN=Kn† ÿ!L²…G¹†. Later, we apply the structure theory on the level of functions. Let us still work with G¹. As H_N=K_n is needed to describe the relation betweenH_n=K_n andH_N=K_N, we incorporate H_N=K_n in the analysis of this situation. Define for any positive integersN,n the linear map

L²…HN=Kn† ÿ!^SnN L²…G¹† by …S_Nⁿ†…x† ˆ …x‡Kn† if x2HN

0 otherwise

: The norm ofS_Nⁿis finite as HN is an open subgroup andKn is a compact subgroup (explained in Paragraph 3.3.1).

LetG¹ have some fixed Haar measure. Then there is a unique Haar mea- sure onH_N=K_n such thatSⁿ_N is an isometry. It turns out that we do not need to know more about these measures for the main approximation result, Section 6. Nevertheless, the next paragraph gives an explicit description of these measures, and for convenience we will use these measures in the rest of this paper. To simplify the notation we setS_n:ˆSⁿ_n.

3.3.1. Measures. If B is a subgroup of the abelian group A, we say that …A;B;A=B†is a Weil triple ifA,B, and A=Bhave Haar measures satisfying Weil formula, symbolically writtend_A=Bd_Bˆd_A. IfK is a compact group, by normalized measure, we mean the Haar measure on K such that total measure ofK is one.

We make the following choice: HN has restricted measure as an open

subgroup ofG¹,K_n has normalized measure, and…H_N;K_n;H_N=K_n†is a Weil triple. This defines the correct measure onH_N=K_n:

Z

G¹j…S_Nⁿf†…x†j²dxˆ Z

HN

j…S_Nⁿf†…x†j²dx

ˆ Z

HN=Kn

jf…x‡K_n†j²d…x‡K_n† meas_Kn…K_n†

ˆ Z

HN=Kn

jf…x‡Kn†j²d…x‡Kn†:

We omit the proof of the next lemma as this description is not strictly necessary for Theorem 6.1. By counting measure, we mean counting measure with point weight one.

Lemma 3.8. Let G¹ be a G¹-group with structure f…H_N;K_n†g, H_N=K_n' ENnˆT^anFⁿ_NZ^bN, andFⁿ_nˆ:Fn.

i) There is a Haar measure on G¹such that the map S1is isometric when E1

has the product measure whereZ^b1 has counting measure, and bothT^a1, andF₁ have normalized measures.

ii) If G¹ has the measure of i†, for any positive integer n, S_n is isometric when En has the product measure whereZ^bn has counting measure, andT^an has normalized measure. The measure onFnhas the same point weight as the point weight onFⁿ₁whenFⁿ₁ has normalized measure.

Part ii) of this lemma is illustrated in the examples of Section 7.

4. Setup

Let GˆR^eG¹; where G¹ is a second countable G¹-group. Recall Defini- tion 2.1. The purpose of this section is to define finite approximands for …U;V† on G. Because G is a finite direct sum, we use the construction in Proposition 2.3. We first introduce some notation which will be explained below. Here,nis an odd positive integer throughout this section, and for any such odd positive integer we definen throughnˆ2n‡1:

Finite abelian group GnˆZ^e_nG¹_n; Isometry InˆR^e_nI_n¹; Group element yⁿˆ …rⁿ;xⁿ† 2G_n;

Dual group element ⁿˆ …dⁿ; ⁿ† 2Gb_nˆZ^e_nGb¹_n:

For an odd positive integer j, Z_jˆ fÿj;ÿj‡1;. . .;ÿ1;0;1;. . .;jg, con- sidered as a finite cyclic group. The e-th power of Z_j is denoted by Z^e_j. Moreover, self-duality is set up withhk;li ˆe^2ikl=j fork;l2Zj.

4.1.Schwinger embedding. The real partR^e is handled by the groups and scalings_n of Schwinger [9]. We follow Schwinger as we use hx;yi ˆe^ixy for x;y2R. Then we get the same scalings n as he used. The finite group has already been taken asZ^e_n.

4.1.1.The maps R^e_n. LetR^e_n:l²…Z^e_n† ÿ!L²…R^e†be theetimes tensor map of the operator Rn:l²…Zn† ÿ!L²…R†, where for k2Zn, I_fkgÿ!

…n†^ÿ1=2I_{‰…kÿ1=2†n;…k‡1=2†n†}ˆ …n†^ÿ1=2I_In^k; where nˆ 

p2=n

: Characteristic function for the Borel setEis denoted byI_E.

4.1.2.Group element rⁿand dual group element dⁿ. Letj j₁ denote the sup- normjrj₁ ˆmax_iˆ1;...;jjr_ij for rˆ …r_i†. For rˆ …r_i† 2R^e, we approximate by rⁿ2Z^e_n in the following way: If jrj₁ …n‡1=2†_n, then define rⁿˆ …rⁿ₁;. . .;rⁿ_e† 2Z^e_n, where rⁿ_i is the unique integer such that ri2 …r ⁿ_i ÿ1=2†n;…rⁿ_i ‡1=2†n

. Otherwise,rⁿ is by definition 0. We identify R^eand its dual group, and the approximation of d2R^eis given by the same procedure as that forr.

We turn to the G¹-group part of the set up. Here, the idea is to use the structure theory of the previous section to get hold of finite abelian groups G¹_n. This set up will relate to the standard duality (7).

4.2. The groups G¹_n. Let F_kmn, k;m;n odd positive integers, be given by F_kmnˆZ^a_mⁿF_nZ^b_kⁿ:The numbersb_n, a_n and the groupsF_n come from the elementary group structure of G¹, Proposition 3.2; H_n=K_n' EnˆT^anFnZ^bn for some choice of fHng and fKng. For j odd, let F_j^b consist of thosek2Z^b for whichjkj₁j, thej-cube inZ^bcentered in origo.

Recall that we in Section 3 found ⁿ_i :Z^bi ÿ!Z^bn for i<n. Let kn be the smallest odd integer such thatⁿ_i…F_n^bi† F_k^b_nⁿ for all i<n:Equivalently, k_n is the smallest odd integer such that jkj₁ n implies jⁿ_i…k†j₁ k_n for any i<n. Likewise, asbⁿ_i :Z^ai ÿ!Z^anfori<n, letmnbe the smallest odd integer such that bⁿ_i…F_n^ai† F_m^an_n for all i<n: For n odd define G¹_n:ˆFknmnnˆZ^a_mⁿ_nFnZ^b_kⁿ_n:

4.3.The embeddings I_n¹. We start by constructingIkmn:l²…Fkmn† ÿ!L²…G¹†:

These maps are defined through l²…Fkmn† 'l²…Z^a_mⁿ† l²…Fn† l²…Z^b_kⁿ†

ÿÿÿÿÿÿÿ!^TmanidZbnk L²…T^an† l²…F_n† l²…Z^bn† 'L²…H_n=K_n† ÿ!^Sn L²…G¹†:

Here we identify I_kmn with S_nT_m^anidZ^b_kⁿ

, S_n is the lift of Paragraph 3.3. Finally, letI_n¹:ˆI_knmnn:

The measures onZ^bn, T^an, andFn are those of Lemma 3.8. The maps Z^b_kⁿ

and T_m^an are defined below in order to be isometric when Z^b_kⁿ has the usual counting measure, whileZ^a_mⁿ has normalized measure.

4.3.1.The maps Z^b_kⁿ. The embeddingsZ_k^bn:l²…Z^b_kⁿ† ÿ!l²…Z^bn†are defined in the obvious way by sendingI_fig toI_figin l²…Z^bn†fori2Z^b_kⁿ:

4.3.2. The maps T_m^an. Let T_m^anˆfⁿZ_m^an…fmn†^ÿ1; where fmn:l²…Z^a_mⁿ† ÿ!

l²…Z^a_mⁿ†and fⁿ:l²…Z^an† ÿ!L²…T^an†are Fourier transforms. This is a varia- tion of the approach in Proposition 2.2.

4.4. Approximate group element in G¹. Given x2G¹, we associate to it x^kmn2F_kmn. As G¹ˆ [H_n, x2H_Wx for some smallest integer W_x. For nWx, surject (recall Equation 8) xÿ!x‡Knˆxnˆ …tn;fn;zn† 2En: Then letx^kmnˆ …t^mn;fn;z^kn†if nWx, and 0 otherwise, where the elements z^kn2Z^b_kⁿ andt^mn 2Z^a_mⁿ are defined below. Finally, let xⁿ:ˆx^knmnn;tⁿ:ˆt^mnn andzⁿ:ˆz^knn:

4.4.1. The integer part z^kn. If zn2F_k^bn; let zn ˆ:z^kn2Z^b_kⁿ. Otherwise, put z^kn to 0.

4.4.2. The circle part t^mn. Here we apply root functions. Parametrize the circle T by zˆe²ⁱ, 2 ‰ÿ1=2;1=2†. For our t_nˆ …t_n;j† 2T^an, let t^mnˆ …t^mn_j † 2Z^a_mⁿ, where t^mn_j is the unique element in Z_m such that _n;j2 ‰…t^mn_j ÿ1=2†=m;…t^mn_j ‡1=2†=m†fort_n;jˆe^2in;j:

4.5.Approximate character inGc¹.Recall the dual construction and defini- tions of Paragraph 3.2. Therefore, for 2Gc¹, ^kmn2Z^a_mⁿF_nZ^b_kⁿ (dual basis) is chosen by the same procedure as for the group element case. Let us fix some notation: For nW^ (W^ chosen analogous to Wx), _n ˆ …u_n;g_n;s_n† 2cE_n(recall Equation 9). Then^kmnˆ …u^mn;g_n;s^kn†ifnW^, and 0 otherwise. Here the circle parts^kn2Z^b_kⁿ and the integer partu^mn 2Z^a_mⁿ are defined by the procedures in the last paragraph (with reverse notation).

Finally, we setⁿ:ˆ^knmnn,uⁿ:ˆu^mnn andsⁿ :ˆs^knn:

5. The Space of Schwartz-Bruhat Functions

LetGbe a l.c.a. group. Recall from Section 3 the existence of pairs…H;K†, whereH=K is elementary,H is an open subgroup, andKH is a compact subgroup.

Bruhat [1] introduces the Schwartz-Bruhat space of functions onG,s…G†, as those complex valued functions which have support in some H, and are locally constant on some corresponding K. Thus is naturally defined on the elementary group H=K, here should look like an ordinary Schwartz function. This means: Let P be a polynomial function on H=K, and D a

translation invariant differential operator. Then2s…H=K†ifis smooth, and all the seminorms jjPDfjj₁ are finite. Alternatively, this can be for- mulated by the tensor product of Grothendieck for locally convex spaces (Schwartz spaces are nuclear). We useTHK to denote

2L²…G¹† ÿ!THK2L²…H=K†; …THK†…x‡K† ˆ…x†;

x2x‡K2H=K:

Notice that T_HK is the inverse of S_Nⁿ (Paragraph 3.3) for HˆH_N and KˆK_non the range ofS_Nⁿ. IfH H⁰andK⁰K, thenSchwartz-Bruhat on …H;K†impliesSchwartz-Bruhat on…H⁰;K⁰†as well. There are ``large'' enough pairs…H;K†for s…G†to be dense in L²…G†, and the Fourier trans- form leaves this space invariant; ifis Schwartz-Bruhat on…H;K†, then^ is Schwartz-Bruhat on…K^?;H^?†.

AsGby [11] is of the formR^eG¹, whereG¹ is aG¹-group, the definition ofs…G¹†is really what is new in this extension of Schwartz functions.

LetG¹be a second countableG¹-group. Use Proposition 3.2 to findfHNg, fKng. It suffices to defines…G¹†on the pairsf…Hn;Kn†g:

Lemma 5.1. Let G¹, fHng and fKng satisfy the conclusions of Proposition 3.2. If…H;K†is some other pair in the definition ofs…G¹†, then there is a non- negative integer n such that HnH and KnK.

Proof. First,H=K'T^aFZ^b, so, as the quotient is compactly gener- ated and K is compact,H itself is compactly generated (the pre-image of a compact set is compact asK is compact). IfCis compact and generatesH, then G¹ˆ [H_n covers C, andCH_N for some integer N. Thus, HH_N. Next, K^?Gc¹ is open while H^?K^?cG¹ is compact. Moreover, K^?=H^?'H=K, sod K^? is compactly generated, and the same reasoning as before givesK^?K_M^? andKMK. So the claim follows withnas the larger ofNandM.

We use 2s…n†to denote 2s…G¹†supported in Hn=Kn. Notice that s…n† s…n⁰†whenn<n⁰. Also, putⁿ:ˆT_HnKnfor2s…n†. LetVⁿand Uⁿ denote the standard Weyl system onL²…H_n=K_n†.

Lemma5.2.Assume x2G¹,2Gc¹ and2s…G¹†. Then we can find an n such that , V…x† and U…† are all contained in s…n†, …V…x††ⁿˆVⁿ…x‡K_n†ⁿ, and…U…††ⁿˆUⁿ…‡H_n^?†ⁿ.

Proof. Assume 2s…n⁰†. Since H_n%G¹, x is contained in some H_n⁰⁰, and by the group property of any H_n it follows that V…x†2s…maxfn⁰;n⁰⁰g†. Replacingxbyx‡k, fork2Kn, clearly makes no difference. That multiplication by character is locally constant is a special