Groupe de travail d’analyse ultramétrique

Groupe de travail

d’analyse ultramétrique

Y ASUO M ORITA

A non-archimedean analogue of the discrete series

Groupe de travail d’analyse ultramétrique, tome 9, n

^o

3 (1981-1982), exp. n

^o

J13, p. J1-J4

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A NON-ARCHIMEDEAN ANALOGUE OF THE DISCRETE SERIES

By

Yasuo MORITA

(Y.

^Go

CHRISTQL,

^Po

ROBBA)

9e année, 1981/82,

^fasc.

3,

^n°

J13,

^{4 p.}

Journée d’Analyse p-adique [1982. Marseille-Luminy]

septembre ¹⁹⁸²

[ Tohoku University]

The purpose of this papers is ^to define a non-archimedean

analogue

^of the discrète séries of We will refer to what is

conjectured,

^{what can be}

proved,

^and

what are the difficultien in

studying

^our

représentations.

^For

proofs

^and

details, we quote MORITA-MURASE [ 5 ]

^&#x3E;

1 o Cl.assical case.

Let (3 and R be

complex

^{number field} and the real number

field, respective- ly.

For any field

F 00FF

^let

P (F) =

^{F u be the one} dimensional

projective

space

over F . Then

P _{( C) -} P _(R)

is the

disjoint

union of the upper half

plane

H == (2:

^e

C ; Im(z)

&#x3E;

0}

and the lower half

plane H = {z ~ C ; Im(z) 0) .

^For

any

integer

s ~

-- 2 ~ put

f E

V+

and z ⁼ x +

iy (x ,

y E

R) .

^Then

V+

becomes a Hilbert _space with

s ^- s

the and is a uni

tary operator

^on

V+ . s

^{Hence defines a uni-}

representation

^{oi the}

locally compact

group ^on

if’ .

^can also define

1

by modifying

^{the norm}

suitably. Further,

^{if we use H} instead of

H ,

^then

-1 "’ - +

Ble obtain another

unitary representatinn

any

integer s ~ -

^{1 . It is well}

THEOREM C. - The irreducible

representations

^oF

SL2(R) ,

^{and no}

^{two of}

tnem for various s are

equivalent.

20 Definition of rr

p-adic

^cases.

s ^{- --}

Now we are

going

^{to construct}

p-adic analogues

^{of the} _s ^ide

replace R by

- ^a

finite extension L of the

p-adic

^number

field Qp ,

^and

replace C by

^{a non-}

archimedean field

(k, ! I. 1) containing

L such that k is

complete

^with

respect

(*) Yasuo MORITA, Institut,

^Tohoku

University, SENDAI ⁹⁰⁰

J13-02

(

^and

algebraically

^{closed. Hence we} consider continuous

représentations

^of

G =

SL2(L)

^{on linear}

topological

^k-vector spaces.

Put D ⁼

pl _{(le) -} P _{(L) .}

Since L is

locally compact, P _(L)

is

compact. Hence,

for any

positive integer

m ~

P _(L)

is covered

by (z e P _{(k) ;} ) z ( &#x3E; )p" ) )

^and

a finite number of

mutually disjoint

open balls

Put

Then

{D m}~m=1

^{is an}

increasing

sequence of subsets of D and D ⁼

~m D .

^Let

be the space of k-valued functions on

Dm

nf the fnrm

m

..

vhere c

m and

c

^{are elements of}

^{k ,}

^end we assume that this limit converges

(uniformly)

^on

D .

It is known that becomes a Banach space with the supre-

inum

norm |f|

⁼

supDm| ^{) f(z) j .}

We say that ^a k-valued function f on D is

analytic if,

^and

nnly if,

^{the res-}

trictinn nf f to each

D

_m

^belnngs

^to

;(D ) .

_m

^(This

is the définition of

analytic

functions in the

theory

^nf

rigid analytic

spaces

(cf.

^MORITA

[4]).)

Let V be the space nf all k-valued functions ⁿⁿ D. Hence V is the

projective

limit of the

o(D )

^with

^respect

^{to the} restriction map

o(D )

2014&#x3E;

0(D ) .

^{Therefore V has a}

natural

Fréchet topology.

^In

particular,

^{V is a}

complete

^linear

topological

space.

consider this space ^{V as the}

analogue

^nf

V±

3 for any ^{s .}

s s

Let s be a

négative integer.

^For any

put

Then n _s defines ^a continuous

représentation

^{of the}

locally compact

group G ^on the

Fréchet

space V. This is ^our

analogue

^{of the classical} discrète séries.

3’ Conj ecture.

Let U be the _{space of} rational functions of ^z

(with

coefficients in

k ) which

have ^nt)

poles

^{in D . Then U is a dense}

subspace

^{of V.} Further the

subspace

^U

of V is G-invariant because

(bz

⁺

d) belongs

^{to U} for any g ^E G .

Since k is

algebraically closed,

any élément f of U can be

expressed

^{as a}

partial

fractional séries of the form

Then va can prove that U is a closed G-invariant

subspece

^{or U . Let V} be

s s

the closure in V of the

subspace

U . It is obvious that V is a closed a-

s s

invariant

subspace

of V. Hence we obtain

représentations

^{nf G nn V and}

V/V.

s s

Now we have the

following conjecture (cf.

^Theorem

C) :

CONJECTURE V.

(i)

^{V and}

V/V

^are

(topologically)

irreducible G-modules.

s ^"’ s ^{2014 - ’* - ’-} ’"-~---2014- 2014M~20142014

(ii)

No twn of them for varions s are

G-equivalent.

4. The result and remarks.

Thnugh

^{we can} not prove the

conjecture

nnwy ^{we can} prove the

corresponding

^asser-

tion for the dense

subspace

^{U of V.}

Let g ⁼

(X

⁼

tr(X)

⁼

O} ,

^and

put

Then this séries converges ⁱⁿ

M~(L)

^{tn an} élément of G if the

eigenvalues

db X of X

satisfies X ) Ip1/( -1)1 .

^It follows that

is well-defined for any X e g ^{and and}

d03C0s

^{is a}

représentation

^{of the}

Lie

algebra g

^on the space V . It ^is obvious that any closed G-invariant sub-

space of

^{U is}

Let 0 be the

integer ring

^nt

L,

^and

put

^{K =}

8L2(0) .

^{Then K is a maximal}

compact subgroup

of G. Since K is an open

subgroup

^of

G ~

^{a closed} K-invariant

subspace

^{of U is}

( g ~ K)-invariant.

Now our main result can be stated as :

THEOREM U.

(i)

^{U and}

U/U

_s ^are

algebraically irreducible (g, ^(i.

^e.

^they

have no nontrivial

(g , K)-invariant k-subspaces).

(ii)

^No two of them for varinus s are

isomorphic

^as

Obviously

^{this theorem}

implies topological

irreduciblities of U and

U/Us ,

^and

the

non-équivalence

^{of them.}

Remark.

(i).

^The difficulties in

proving

^the

conjecture

^{lie in the tact that s lemma}

J13-04

does not hold in our case. Fnr

example,

^{we can} prove ^that the

only intertwining

^ope-

rators t)f V are the scalar

operators, though

^{V has the closed}

G-invariant

sub- space

V 5 .

^{We prove our theorem}

by corstructing

^{the space} from any ^non-zero élément

(cf.

MORITA-MURASE

[5J, 3-3).

(il).

^{It is} remarkable that U is infinité dimensional and irreducible as a K-

s

nodule

though

^{K is a}

compact

group. This

phenomenon

^{causes a}

difficulty

ⁱⁿ

defining

the admissible

représentations.

[1] Automorphic forms, representations

^and

L-functions, part

^{1-2. -}

Providence,

American mathematical

Society, ¹⁹⁷⁹ (Proceedings

^of

Symposia

in pure Mathema-

tics, 33).

[2]

^{GEL’ FAND}

^{( I.} M.),

^GRAEV

^{(M. I.)}

and VILENKIN

(N. Ya) . -

Generalized

functions,

vol., 5 :

Integral geometry

^and

representation theory. -

^New

York, London,

^Aca-

demic

Press, ¹⁹⁶⁶

[3]

^GERRITZEN

^(L.)

and VAN DER PUT

(M.). - Schottky

groups and Mumford ^{curves. -}

Berlin, Heidelberg,

^New

York, Springer-Verlag, ¹⁹⁸⁰ (Lecture

Notes in Mathema-

tics, 817).

[4]

^MORITA

^{(Y.). - Analytic}

^{functions on an} open subset of

P1 (k) ,

^J.

^für

^reine

und angew.

Math.,

^t.

311-312, 1979,

p.

361-383.

[5]

^MORITA

^{(Y. )}

and MURASE

(A.). - Analytic representations

^of

SL over

^a

^p-adic

number

field,

J. of Fac. of

Sc.,

^{Univ. of}

Tokyo,

^Section

1A,

^t.

28, 1982,

p.

891-905.