Algebraic Weyl system and application

A NNALES MATHÉMATIQUES B ^LAISE P ^ASCAL

Y ^OUNGHO J ^ANG

Algebraic Weyl system and application

Annales mathématiques Blaise Pascal, tome 4, n

^o

2 (1997), p. 27-40

<http://www.numdam.org/item?id=AMBP_1997__4_2_27_0>

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ALGEBRAIC WEYL SYSTEM AND APPLICATION

YOUNGHO JANG

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}

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

The n-dimentional

p-adic

space

Qnp

has the standard norm ⁼

max1~j~n|xj|p,

^{x =}

(x1,

^{x2, ’ ’ ’}

, xn)

^E

Qnp.

The Fourier transform is defined with respect ^to the character

~p((k,x))=03A0nj=1 ~p(kjxj),

^where

(k, x)

⁼

03A3nj=1 kjxj

^.

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 supersymmet-

ric 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 and Volovich

[VV 89] proposed

^a formalism of the

p-adic

quantum ^mechanics with C-valued functions

(cf.

^{see also}

[FO 88], [RTVW 89]

^and

[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 also}

~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

sym-

plectic

space

(V, B),

^{we define an}

algebraic Weyl system (H, W )

^on

(V, B),

^{and its} prop- erties 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.

2. The

general properties

^of

Weyl systems.

^{We review some} of well-known results of

Weyl

systems

(H, W )

^on the finite dimensional

p-adic symplectic

space

(V, B).

For

proofs

^{and more}

details,

^see

[Zel 91]

^and

[VVZ 94] (cf.

for the co-dimensional _case, see

[Zel 92]

^and

[Zel 94]).

For

simplicity

^{we assume that}

p 7~

^2.

By

the definition of ^a

p-adic symplectic

space is the

pair (V, B),

^{where V is a} finite demensional

Qp-vector

^{space and B is a nonde-}

generate antisymmetric Qp-bilinear

^{form on V. Then}

dimQp

^{V = 2n is even}

(cf.

^{for the}

oo-dimensional _case, see

[Zel ^94,

^p.

423]).

Given

0 ~

^{ei E}

V,

^{there must exist a x E V} for which

jS(ei,2-) ~ 0,

^{since B is}

nondegenerate.

^{We choose a a E}

Qp

^{so that =}

be1+ax,

^and

B(ei

⁼

aB(el, x)

⁼

1. Then the

hyperbolic plane hy = span{e1, en+1}

^{has matrix}

-(0 -1 1 0)

^with

^respect

to the basis

{e1, en+1}.

^Since

hy

^is

nondegenerate (i.e.,

^the

pair (hy, hy)

^is

^p-adic

symplectic space),

^{we have V}

= hy

^®

h,

^where

h~

^{is also}

nondegenerate. Hence,

^we repeat ^the

preceding

construction in to obtain an

orthogonal decomposition

^{of V of}

the form

(2.1)

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

formula

(2.2) Pjx

⁼

B(ej, x)en+j

^{+ x E}

V,

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:

(2.3) B(x, y)

^E

Zp ~x,y

^E

Xo

^{, dx E}

V,Xo, ~y

^E

Xo

^{such that}

B(x, y)

^E

QpBZp.

Zelenov

[Zel 91]

^{defined a}

Weyl

system ^on

(V, B)

^{as a}

pair (H, W )

^{of a}

complex

^Hilbert

space H and a continuous _map W : x ’-)-

W(x)

^{from V to the}

family

^of

unitary

operators

on H

satisfying

^{the condition}

(the Weyl relation)

(2.4) W(x)W(y) =

⁺

y)~

Two

Weyl

systems

(H, W )

^and

(H‘, W’)

^on

(V, B)

^are

unitary equivalent

^{if there exists}

an

intertwining unitary operator

^U : H -~ H’. I.e. U is an

unitary

operator ^{such that}

(2.5) W’(x), x

^{E V.}

A

Weyl

system

(H, W )

^{is said to} be irreducible if there exists no non-trivial

subspace

^of

H invariant under the

W(x), x

^{E V. . We} say that

(H, W )

^{can be}

represented

^{as a direct}

sum ~j~I(Hj,

W ) of Weyl systems (Hj, W )

^{if H can} be written ^{as a} direct sum

~j~IHj

of

subspaces HJ

^{which are} invariant under the action of

operators W(x).

We denote

by Vo

⁼

{2:

^{E V : :}

1 ~

^{the open}

compact subgroup

^{of V. Let}

(H, W)

be a

Weyl system

^on

(V, B).

^A vector 03C60 E H is called a vacuum vector of

(H, W )

^{if the}

condition ⁼ _c~o is satisfied for all x E

Vo.

^{The set of vacuum} vectors of

(H, W)

forms the vacuum

subspace Hoof

^H.

Every Weyl system (H, W )

^on

(V, B) ^is,

^{in a certain} sense, determined

by

its restriction

(II,

^on

(Vo,

^Since

Vo

^{is a} compact ^abelian group, all irreducible

unitary representations ^{of Vo}

^are one-dimensional. Let

Vo

⁼

Hom(Vo, T)

^{be the} group of characters of Then we have

(2.6) V/V0

^{3 * E}

Vo

^,

a*(x) ;; ~p((03B1, x)),

^{x E}

vo,

where a is a

representative

^{of the} coset & E

VIVo. By

^the

theory

^of

unitary representations

of compact groups, the

representation

space H can be

expressed

^{as an}

orthogonal

^sum

(2.7)

^{H =}

~~V/V0H,

where

Ha

^{is the maximal}

subspace

^{on which}

Vo

^{acts as a}

multiple

^{of a*.}

Let us choose an element a from each coset & E

V /Vo

and denote the

family

^{of such}

elements

by Jo.

^Let

(2.8)

^{~ _ = E H : a} E

Jo}.

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:

Theorem 2.1

(cf. [Zel 91])

^.

Weyl systems

^{has the}

following properties:

(i)

^For any

Weyl system,

^{there exists} the vacuum vector.

(ii)

^A

Weyl system (H, W)

^is irreducible if and

only

^{if its vacuum}

subspace Ho is

^one- dimensional.

Otherwise,

^{if we choose some} orthonormal basis in

Ho (.H, W )

^can

be

represented

^{as a direct sum}

~i~I(Hi, W)

of irreducible

Weyl

systems

(Ht, W),

^where

subspace Hi

^{is the} span of the vectors ⁼

W(03B1)03C6i0 :

^o:

J0}i~I.

(iii)

^If

(H, W )

^is irreducible

Weyl

system with the vacuum vector then the system of coherent states E of

(H, W )

forms the orthonomal basis in H.

(iv)

All irreducible

Weyl

systems ^are

unitary equivalent.

Example

^2.2

(cf. ~VV 89]

^and

[Zel 89,

^{in case of} p ⁼

2]).

^For 2-dimensional _case, an

irreducible

Weyl

system is constructed as a

pair (L2(~r), W )

^on

(~~, ^B),

^{where the}

^unitary

operator W(z)

^{is defined}

by

W (z)~(x)

^{= +}

q)~

^{z =}

(q~p)

^E

~p,

^{4~ E}

the

symplectic

^{form B :}

: Q2p

^x

Q2p ~ Qp

^is

^{given by} B(z, z’) = qp’ - q’p (z’

⁼

(q’, p’)

^E

Q2p).

The vacuum vector has the form ⁼ where

f2(x)

^{is 1 if}

0 x

^{1 and 0 if} 2: > 1.

Example

^2.3

(cf. [VVZ ^94,

^p.

243]).

We denote the tensor

product

^{of n}

irreducible

Weyl

systems ^of

Example

^2.2

by (L2((~~ ),

^on

(~p~‘, ^B).

^Hence

W(n)(z)

⁼

W(zj),

^{z -}

(z1,..., zn)

^E

Q2np

and the vacuum vector has the form

n ^.

= =

(xl, ~ ..

^E

Q~.

j=1

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).

We define a Hilbert

subspace

^of

L2 (V ) by

(2.9)

⁼

{f

^E

L2(V) : f(x

⁺

y) = y)/2)f(x) ^dy

^E

V0},

and an

operator W(x ) by

(2.10) W(x)f(y)

⁼

~p(B(x,y)/2)f(y

^-

x) (x

^E

va f

^E

L~p2(V0)).

Example

^2.4

(cf. [Zel 91]

^and

(VVZ 94,

p.

243]).

^The

^pair

^{is an} irreducible

Weyl

system ^on

(V, B),

^{and the vacuum vector} has the form

03C60(x)

⁼

Let

Sp(2n, Qp)

^{be the group of}

automorphisms

^{of the} space

(V, B)

^that preserve the

symplectic

form B. We fix a basis of V and _{express any g E}

Sp(2n, Qp) by

^{a matrix}

(9jk)

^{E Let}

(2.I1) ~g~’

^"

max1~j,k~2n|gjk|p.

Then G ⁼

{g

^E

Sp(2n, ~P) : =1}

^{forms a maximal} compact

subgroup

^of

Sp(2n, Qp).

^.

Theorem 2.5

(cf. [Zel 91J).

^Let

(H, W )

^{be an} irreducible

Weyl

system ^on

(V, B).

^Then

a

family

^of operators

{!7(~) : ~

^E

_{G} satisfying}

(2.12) U(g)W(x)

^{= z G V}

forms an

unitary

representation ^{of G in}

H,

^and any ^vacuum vector of

(H, W )

^{is an}

eigen-

vector of the

U(g).

3.

Algebraic Weyl

system. In this

section,

^{in order to} define an /-sheaf

(V, .~’)

on

(V, B),

^{we define an}

algebraic Weyl system,

^and prove ^some necessary and sufficient conditions for

Weyl

systems

(H, W)

^on

(V, B)

^{to be} irreducible.

Let

(H, W )

^{be a}

Weyl

system ^on

(V, B)

^{with a vacuum} vector ~po E

Ho

^and

S(V)

^the

space of Schwartz-Bruhat functions on V. Then the convolution

product

(3.1) _fg(x)

⁼

y)/2)f(y)g(x

^-

y)dy

makes

S(V)

^an associative

C-algebra

without the unit element. For each

f

^E

S(V),

^we

define ^a linear

endomorphism Ew( f)

^{of H}

by (3.2) EW(f)03C6

=~vf(x)W(x)03C6

dx,

03C6 ~ H.

Since

f

^is

locally

^{constant on V} with compact support, ^this

integral

^is well-defined. It is easy to ^see that

EW(fg)

⁼

Ew( f)Ew(g). Indeed,

^for

f

^, g E

S(V)

^and 03C6 E

H, using

^the

Weyl

^relation

(2.4)

^and

(3.2),

^we

get

EW(fg)03C6

⁼

/ v / v y)W(x)03C6 dxdy

= ~ ~

⁺

^dtdy

=

9’(t) f /

^dt

=

EW(f)EW(g)03C6.

Hence H is a

S(V)-module.

Definition

3.1.

a) Weyl system (J7,~)

^{is said to be}

algebraic

^if there exists an open

compact subgroup K

^{of V such that} for

all 03C6

^{H and x 6} K. For some open compact

subgroup K

^of

V,

^let

^F~

⁼

{~

^{6 H : V~}

_~}

be the

subspace

of K-invariant vectors in H. Then it is clear that

W)

^{is an}

algebraic Weyl system

called the

algebraic part

^{of W. In}

particular, (Ho

⁼

W~)

^{is the}

algebraic

part ^{of W.}

b) Weyl

system

(H, W)

^{is said to} be admis.sible. if it is

algebraic

and if for each _open compact

subgroup K

^of

V, ^HK

^{is finite} dimensional.

Algebraic Weyl

system

gives

^{an I-sheaf on}

(V, B)

^{as follows:}

Proposition

^3.2.

Weyl

system

(H, W)

^is

algebraic

^{if and}

only

^if

S(V)H

^{= H.}

Proof.

^Let

(H, W)

^{be an}

algebraic.

^I.e. any 03C6 ~ H is fixed

by

^some open compact

subgroup,

say

K,

of V. Let

03BEK

⁼

volume(K)-1

characteristic function of K. Then

= p.

Conversely,

^{let =} H. We can construct an open

compact subgroup

K of V such that

W(x)03C6=03C6

^{for all} y ~ H and a: ~ K ^as follows: Let _{p 6} H. Then _03C6

can be written as y ⁼

~;, E~(/.)~,. Then, using

^the

linearity of W, (3.2)

^{and the}

Weyl

relation

(2.4),

^{we have}

= =

03A3W(x)EW(fi)03C6i

!

=

W(x)~Vfi(y)W(y)03C6i ^dy

⁼

~Vfi(y)W(x)W(y)03C6i ^dy

=

~Vfi(y)~p(B(x,y)/2)W(x+y)03C6i ^dy

=

E/

^dt.

Since

/,

^is Schwarz-Bruhat

function,

there exists a

positive integer

^/

sufliciently large

^such

that

supp fi

^c

p-lX0

^{and for} any t 6

supp fi, /,( - j:)

⁼

|x|p

^{~ p-l, where}

Xo

^{is a}

Zp-span

^{of the}

symplectic

^{basis of V}

(see §2).

^{Let K =}

plX0.

^{Then K is an} open

compact

subgroup

^of V and for all t and _T,

B(.r, t)

⁶

Zp.

^{Thus we have} = ~ for all _{w e} H and .r 6

A’, t

Let K be an open

compact subgroup

^{of V. Then ~ n}

Xo

^{is also an} open compact

subgroup

of V. Let 0 ⁼

{~

ⁿ

Xo :

^{~ is an} open

compact subgroup

^of

V}

^{and Y 6}

0,

and let

03BEY

⁼

volume(Y)-1

characteristic function of Y. Then

03BEY

^{is an}

idempotent

ⁱⁿ

S(V)

^and

S(Y)

^{= is an} associative

C-algebra

^{with the unit element}

_~y.

Let

/(x)

^and

r(a-)

^{be a} translations of

S(V) given by

(3.3)

⁼

!/)/2)/(~ - ~),

⁼

!/)/2)/(t/

⁺

r).

The

following

^is

easily proved.

Proposition

^{3.3. .} We have the

following properties:

(i)

^{= for all 3-} 6 V and

/

^e

S(V).

(ii) f *(l(x)9) = (r(-x)f )*9~ (r(x)f )*9 =

^{for all}

f

^~ 9 E

s(v).

(iii)

⁼

(l(x)f)g, r(x)(fg)

⁼

f(r(x)g)

^{for all}

f,

^, 9 E

S(V).

(iv) l(x)03BEY

^{= =}

03BEY

^for all ^x E Y..

By (iv)

^of

Proposition 3.3, S(Y)

^{is the space} of elements

f

^of

S(V)

^{such that}

l(x) f

⁼

r(x) f

⁼

f

^{for all x E Y.}

Clearly,

^the

image EW(03BEY)H

^{coincides with the}

subspace ^HY

of Y-invariant vectors in H. It

follows, inparticular,

^{that for} any exact sequence of

S(V)-

modules 0 ^-t

Hi

^-t

H2

^-+

H3

^-t 0 the sequence 0 ~

HY1 ~ HY2 ~ Hi

^{~ 0 is also}

exact.

Let

Hw(Y)

⁼

{03C6

^{E H : =} 03C6

Vy

^E

y},

Then the kernel of

EW(03BEY)

^{is the}

space

H(Y) spanned by

^{all vectors of} the form

Hw(Y).

^For

it is clear that ⁼ ₀ and that Y act

trivially

^on

H/H(Y)

^since

.H’~H(Y)

^{^-_’}

Im(EW(03BEY))

⁼

HY,

^{so that}

_EW(03BEY)

^{is the}

_identity

_map on

H/H(Y).

Theorem 3.4. Let

(H, W )

^{be a}

Weyl

system ^on

(V, B).

^Then

(H, W )

^is irreducible if and

only if

^for any open

compact subgroup

^{Y E 0 of V either}

^HY

⁼ 0 or

(HY, W )

^is

irreducible.

Proo f. Suppose

^{that for} any open compact

subgroup

^{Y E d of}

V, ^HY ~

^{0 and}

W )

is reducible. Then there exists a non-trivial

S(Y)-submodule

^{a of}

^HY

that is invariant with respect ^{to the action} of the operators

W(x), x

^{e V. . Let b be a}

S(V)-submodule

^of

H

generated by

^{a. Since a} C

b,

^{b is} non-trivial.

Every

~o E b is

represented

^{in the} form

s t

03C6 ⁼

03A3 ^EW(fi)ai

⁺

^{(s, fi}

^E

^S(V),

^ai,

bj

^{E a,} n J ^{j ~}

Z).

^.

i=I j=I

Then, using (i)

^of

Proposition 3.3,

^we have for x E

V,

s t s t

= L W(x)EW(fi)ai

⁺ + ^°

’=1 j=I ^~~1 j=I

_

Since i E

S(V )

^{and E} a,

W(x)03C6

^{E 6. Hence}

_{(H, W )}

is reducible.

Conversely,

suppose that b C H is a non-trivial

S(V)-submodule

^of

(H, W ).

^For any open

compact subgroup

^{Y E 0 of}

V,

^the sequence 0 ^~

HY

^-->

(H/b)Y

^{~ 0 is exact. Hence for}

Y E 0 with 0,

6Y

is a non-trivial

S(Y)-submodule

^of

Let

(H, W)

^{be an}

algebraic Weyl

system ^on

(V, B)

^{and H* =}

HomC(H, C)

^{the dual}

space of H. We define a

Weyl

system

(H.*, W*)

^on

(V, B) by (3.4) _03C6, W*(x)03C6*

^_

W(-x)03C6, 03C6*,

for x E E

H, c~*

^{E H* and}

(, )

^{is the natural}

_pairing

between H and H*. This

Weyl

system

(H*, W *)

^{is not}

algebraic,

^{so we take its}

algebraic

part. ^More

precisely,

^let

H*al

⁼

UYEo(H*)Y

^{and =}

W*(x)IH* , x

^{E V. Then}

(H*al, W*al)

^{is an}

algebraic Weyl

system ^on

(V, B). Clearly, EW*al ^(f)03C6*

^{= for}

f

^E

S(V)

^where

=

f (‘x).

Proposition

^{3.5. Let}

(H, W )

^{be an} admissible

Weyl system

^on

(V, B).

^{Then we have:}

(i) (H*al, W*al)

^is also admissible.

(ii) (H, W)

^is irreducible if and

only

^if

(H~j,

^is.

Proof. (i)

We show that for all Y e

0,

^{is finite} dimensional. Let

p*

^E

(H*al)Y.

Then we have for cp E

H,

~SP, ~*~ _

^{= = .}

This

^{show that}

^(H:1)Y

⁼

^(HY)*.

^I.e.

^(H:1)Y

^{is finite}

dimensional,

^since

(H,W)

^{is an}

admissible.

(ii)

^{If a is a} non-trivial

S(V )-submodule

^of

H,

^{then =}

~y~*

^E

Ha~ :

^:

(a, c~*~

⁼

0}

^{is a}

non-trivial

S(V )-submodule

^{of Hence}

(H*al, W*al)

^is reducible. The converse follows from that H ^-- is an

isomorphism, i.e.,

⁼

W. !

Proposition

^3.6

(Schur’s Lemma).

^If

^{(H, W )}

^{is an} irreducible

Weyl

system ^on

(V, B),

then D ⁼

HomS(V)(H, H)

^{is a division}

ring.

^{In other}

words,

^{D =}

C, i.e.,

^{if o : H ~ H is}

a

S(V)-module homomorphism,

^{then o- is a scalar}

multiple

^{of the}

identity morphism.

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 - 03BB1 . Id)-103C6, (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 Stone-Von Neumann Theorem.

Heisenberg

group N

=1V (V, B)

^{of a}

p-adic smplectic

space

(V, B)

^{is the set of}

pairs (t, x)

^e

Qp

^{x V with the}

multiplication

^law

(~~ x)

^~

(t’? x’)

^_

(t

^{+ t’ +}

~’)l ~f

^{x +}

~’),

and it satisfies ^an exact sequence 0 ^--~

Qp

^--‘-~

N(V, B) ^~

^{V -}

0,

^{where c and r~ are}

given by

⁼

(t, 0)

^{and =} x,

respectively.

The _group

Im(~) == {(t, o) : t

^E

Qp}

^{is the center and the} commutator

subgroup

^{of 1V.}

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 r~

on

Im()

^{and w induces the}

identity

map ^on l.

Explicitly,

^{such w is}

given by

(4.1)

⁼

(t

^E

Qp,

^{x E}

~).

We denote

by (F~,T~)

^{= the}

unitary representation

^{of N induced}

by

^the

character ~ of L.

Hence,

the Hilbert _space is the

completion

^{of the} space of all functions

~ : TV 2014~ C such that

(4.2)

⁼

(n

⁶

N, h L),

(4.3)

^{p 6 for the Haar measure dfi on}

N/L,

and the

unitary operator T03C9(n0)

^on

H03C9 (n0 ~ N)

^{is the left}

multiplication

^of

n-10:

(4.4)

^’

We have the Stone-Von Neumann Theorem

(in

^{case of}

real,

^see

[Car 66,

p.

368]):

Theorem 4.1

(The

Stone-Von Neumann

Theorem).

(a) (.H~.T~)

^{is an} irreducible

unitary representation

^ofN.

(b)

^{For any Hilbert space}

H

every

unitary representation (H, T)

^{of N}

satisfying (4.5) T(t, 0) = ~p(t) . ^IdH

^for

(t, 0)

^E

Im()

is a

multiple of T(.ù.

Remark 4.2. It can be seen from

[Per ^81,

^pp.

371-372]

that the Stone-Von Neumann Theorem is connected with Weil

[Wei 64]

^{as follows: If V == ~}

^{(B ~}

^{is a}

decomposition

^of

(V, J5)

^{into a sum of two}

Lagrangian subspaces,

^{then we can define a} map

H03C9

3 p

~ 03C6|l’ ^L2(l’).

This is an

intertwining unitary operator

^{for the}

unitary representation

^{and the}

irreducible

unitary representation (Ir2(~),~)

^of

^N,

^{where $ is}

given by (~)/)(~)

^{= +}

(u,~) - (u,~)/2)/(~ ~ u~)

for x ⁼

(u, u* )

^{V and}

f

^e

L2(l’)

^{if l’ ~ l*}

(Pontrjagin

^{dual of}

l)

^and

( , )

is the canonical bilinear form on V.

Let us consider also the

subgroup

^{A =}

{(t,0) :

^{t 6}

Zp}

^{of the center}

Im()

^{of N. Thus}

all

unitary representations

^{of N}

satisfying (4.5)

^{are trivial on A. Hence we may consider}

them ^as

representations

^{of =}

/0394.

^{We can}

identify

^{N with T x V via}

(t, x)

^~

(03B1,x).

Then the

multiplication

^{law of N is}

given by

(4.6) (c~ 2~). (/?, ?/)

^{= +}

~/).

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).

Another

proof

of the Stone-Von Neumann Theorem. Let

(hr~,,

^{be the}

unitary representation

^{of the}

Heisenberg

^{group N.} Each element of the

Heisenberg

^{group N is}

written

uniquely

^as

(t, x) - (0,~) . ~ (t, o). Hence,

^if ~ E

H~,, (4.2) implies

⁼

x) ~ (t, o))

⁼

x). Thus ~

^is determined

by

its restriction to V. Hence the

mapping

H03C9 03C6

03C6/V

~

L2(V)

is an

intertwining unitary

operator. ^The

unitary representation T(n)

⁼

RTW(n)R"1

^acts

on

L2(V ) by

^the

following

^{formula: For n =}

(t, x) ^EN,

= =

=

B(x~ y)I2~ y - x)

=

x) ~ (-~ - B(x, y)I2~ ~))

=

y)/2)03C6|V(y

^-

x).

Let

W(x)

⁼

Xp(-t)T(t, x).

^{It does not}

depend

^{on t} and satisfies the

Weyl

^relation

(2.4).

Thus

(L2 (V ), W )

^{is a}

Weyl

system ^on

(V, B).

^In

particular,

^if

03C6|V

^{E then}

(L2(V), W )

⁼

(vo ), W )

^{is an} irreducible

Weyl

system ^on

(V, B) (see Example 2.4).

To

complete proof,

^{we must show that}

(L2(V), W )

^{is an} irreducible. Let H ⁼

L2 (V )

^and

Y E 0. Then

HY

⁼

5’(V) ~ 0,

^{since =} ~p ⁼

W(x)cp

^{for all x Let}

fo

^{be a}

vaccum vector of

(S(Y), W).

^{Then we} have for x E

Vo

^and y E

~)

^,

/o(2/)

⁼

W(x)f0(y)

⁼

x).

Since

f o E S(Y),

^for any

point y

^{E V there exists an}

integer

^{1 such that}

f o(y

^-

x)

⁼

f0(y), |x|p pj.

^Thus

supp f0

^C

^Br

⁼

{x

^{E V :}

|x|p

^{r = and}

f0(y)

⁼

constant, y

^E

Br.

^Therefore

fo(Y)

^{= , c E} C. It follows from

(ii)

^{of Theorem}

2.1 and Theorem 3.4 that

(L2(V ), W)

^{is an} irreducible

Weyl

system ^on

(v: B). I

5. A

Weyl

system

depending

^{on a selfdual}

Zp-lattice

^{and a}

Qp-valued

^func-

tion. Let £ be a lattice in

(V, B).

^The dual lattice ,~* is defined

by

(5.1)

^{,~* =}

{x

^{E V :}

B(x, y)

^E

Zp

^{for all y E}

,~}.

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.}

Theorem 5.I

(cf. [LV 80,

p.

143] ).

^{There exists a canonical}

isomorphism

^between

H03C9

^and

Hr intertwining

^the

representations T03C9

^and

Tr

^:

’

(5°2) (e2,1%’)16’~)

^~

£ ’ (°’ ’~ ))°

x~/(~l)

For _{any p e}

Hr,

^{z e V and} y e

£,

^{we have}

+

v)

^"

k’( (° ’ ~~’ ) ° ( ~B( ~~" v )/2> Y) ) " ~ ~ ~ (

=

xp(B(z , y) /2)T ~~ (0, y)p(0, z).

Let

H(£, a)

be the Hilbert _space obtained

by completing

^the space of continuous func- tions

f :

^{V - C}

satisfying

^the

following

^{two conditions}

(5.3) f(z

⁺

y)

⁼

~p(B(x, y) /2

^-

a(y)) f(z) (z

^E

V,

y E

£),

where a is a

Qp-valued

^{function on V}

satisfying (a(z

⁺

y)

^-

u(z) -

^e

Zp ; (5.4) f

^e

L2 (V/£)

for the Haar measure di on

V/.

We define a

unitary

operators

W£,a(z) ^by

5.5) W,03C3(x)f(y) ^’ Xp(B(Z> l/)/2

- 03C3(x))f (y-

x),

V E

v> f

^E

H(,03C3).

Example

^{5.2. The}

pair (H(£, a), W£,a)

^{is a}

Weyl

system ^over

(V, B). Indeed,

^the

unitarity

^{of the}

W£,a(z)

^is obvious. It is sufficient to check the

Weyl

relation. We have

w£,uZ)w£,u(Y),f(Z)

= W,03C3(x)~p(B(y,z)/2- 03C3(y))f(z

^-

V)

=

a(z))xp(B(y, z

^-

^z)/?

^-

a(y))f(z -

^{z -}

y)

~ + l/>

~)/2

^~

~(~°) ~

^~

(~

⁺

Y))

=

~p(B(x,y)/2)~p(B(x

^{+ y,}

z)/2 - u(z

⁺

,y)) f(z - (z

⁺

y))

=

~p(B(x,y)/2)W,03C3(x

^{+ °}