Pierre CLARE

1 Introduction

The relation between the theory of unitary representations of topological groups and C∗-algebras is a classical topic(see[5])that has been developed over the years.An example of particular relevance here is the work of Rief f el who gave in[16]a description of induced representations in terms of Hilbert modules.

In an ef f ort to obtain elements of a C∗-algebraic formulation of the representation theory of semisimple Lie groups,Rief f el’s construction was adapted in[3]to describe parabolic induction,which resulted in the construction of Hilbert modules related to the principal series induced from a given association class of parabolic subgroups.After the results of generic irreducibility obtained in[3]in the spirit of F.Bruhat’s theory,the next step in the direction of an accurate description of the C∗-algebras related to a Lie group consists in developping the theory of intertwining operators.These objects were originally introduced by Knapp and Stein in[10–11](see also[19]and recently[4])and play a central role in the theory,as they not only allow to decide when principal series representations are reducible but also relate to the densities in the Plancherel formula and the construction of complementary series among other things.

The central piece in the classical study of intertwining operators is the normalization process,allowing to go from integral transformations analogous to Radon transforms and called standard intertwining integrals to unitary intertwiners that can often be expressed in terms of classical geometric transforms(see for instance[2,13,15]for recent examples).

Standard intertwining integrals were studied in the context of Hilbert modules in[3]and a result of normalization was obtained in this framework for SL(2)in[1],using Fourier transforms twisted by the non-trivial Weyl element.The point of the present paper is to extend this result to the case of maximally degenerate principal series of special linear groups:We will show howa Fourier transform defined on an appropriate space extends to a unitary intertwining operator between Hilbert modules and normalizes the standard intertwining integral in a sense that will be made precise.

The paper is organised as follows:Section 2 is devoted to the description of degenerate principal series of SL(n+1)in terms of Hilbert modules.More precisely,universal C∗-algebraic principal series are introduced in the discussion concluded by Definition 2.1 and Proposition 2.2 characterises a useful submodule of functions.Section 3 starts with the study of the Radon transform defining the standard intertwining integral.Then we introduce a Fourier transform adapted to the situation and prove in Theorem 3.1 that it extends to a unitary operator between Hilbert modules that normalizes the standard integral at the level of C∗-algebraic principal series.Finally,we explain in Section 4 how the results extend to the case of non-archimedean local fields.

The results presented here are part of an ongoing joint research project with members of the MAPMO(University of Orléans):A.Alvarez,P.Julg and V.Laf f orgue.This article greatly benefited from their help and comments.

2 C∗-Algebraic Universal Principal Series

2.1 Structure and notations

LetFbe the field R or C of real or complex numbers.Ifxis a matrix with coefficients inF,the matrix obtained by conjugating all the entries ofxwill be denoted byx.The real matrixwill be called the real part ofxand denoted by Re(x).

Troughout the article,GFwill denote the group SL(n+1,F)of matrices of sizen+1 with determinant 1,forn≥1.Letθbe the Cartan involution ofGFdefined byThe subgroupPFof upper block-triangular matrices of type(n,1)is a maximal parabolic subgroup ofGF.It admits a Langlands decompositionPF=LFNFwithθ-stable Levi component

and unipotent radical

whereIndenotes the identity matrix of sizen,so thatPFidentifies with the semi-direct product GLThe opposite parabolic subgroupis the image ofPFunderθ.It decomposes as

The groupis endowed with the Haar measure d×adefined by

where dadenotes the restiction of the Lebesgue measure of,and|·|is the usual absolute value in the real case and is defined byin the complex case.

2.2 The homogeneous spaces G/N and

The central result of this paper will rely on some analysis on the quotient spaceG/Ndescribed in this section.To avoid confusion,group actions will be denoted using dots(e.g.a.b),whereas matrix multiplications will be written without any symbol(e.g.ab).

Notation 2.1The vector space of matrices withplines,qcolumns and coefficients inFis denoted by Mp,q(F)(simply Mp(F)ifp=q)and we letXF=Mn+1,n(F).

The groupsGFandLFact onXFfrom the left and the right respectively byg.x=gxandx.a=xaforginGF,ain GL(n,F)andxinXF.

Notation 2.2The set of elements of rankninXFis denoted by

This space is dense inXFand inherits the actions ofGFand GL(n,F).We denote by dxtheGF-invariant measure obtained by restricting the Lebesgue measure ofThe right action of GL(n,F)onXFtransforms the measure according to

Proposition 2.1The homogeneous space GF/NFidentifies withas a topologicaland measured space.The identification is(GF,LF)-equivariant.

ProofThe stabiliser subgroup inisNF,so the mapb:Gidentifieswithb(G)=G.x0in aGF-equivariant way.SinceLFnormalizesNFand

for anya,it is alsoL-equivariant.

To identify the image ofb,we observe that it is the map that extracts the left block of size(n+1,n)of a matrix in SL(n+1,F),that is,

foru∈XF,sob(g)has the maximal rank for allg.Surjectivity follows from the fact that any linearly independant family may be completed into a basis with determinant 1.Finally,bis compatible with the topologies induced onGFandby the standard ones on Mn+1(F)andXF,andGF/NFadmits essentially oneGF-invariant measure,which corresponds viabto the restriction of the Lebesgue measure toXF.

From now on,we will use freely the identi fication established in the above result and identify any element ofLFwith the corresponding element of GL(n,F).

An analogous description can be obtained for the “opposite”quotientidenti fied withthrough the mapwhere the matrixhas stabilisercarries actions ofGFandLFrespectively given byg.y=yg−1andy.a=a−1yforginGFandainLF.

2.3 The modules andand

From now on,we will systematically omit the subscriptF.The set of smooth functions with compact support on a spaceTwill be denoted by,and the Schwartz space of functions onT,all of whose derivatives are rapidly decreasing,byS(T).

In order to study intertwining operators in a C∗-algebraic context,we will work in the framework of universal principal series,introduced in[3].The basic objects will be Hilbert modules de fined as completions ofandrespectively.

Notation 2.3Iffis a function of the variablexandmis a group element such thatmxmakes sense,the right translation offbymis denoted bySimilarly,mf(x)=f(mx).

The actions ofGandLcan be defined for functions on the larger spaceX.Namely,for a functionfdefined onX,and elementsgofGandaofL,we let

which integrates to

for anyϕintegrable over GL(n,F).

Finally we consider the pairing

for suitable functionsfandhde fined onX,wheredenotes the usual inner product for square-integrable functions.Iffandhare smooth and compactly supported onG/N⊂X,thenis smooth and compactly supported onLand the above formulas de fine ona structure of the pre-Hilbert module overThe scalar factor involved in(2.3)shows thatfis actually considered as a half-density overG/N,and(2.4)then appears as a natural pairing.

The above formulas are special cases of the ones involved in the general construction of parabolic induction modules described in[3].Therefore,they can be used to define a Hilbert module structure as follows.

Definition 2.1The Hilbert module obtained fromby extending the actionto the reducedC∗-algebra ofGL(n,F),and then completing with respect to the norm induced by(2.4)is called the reducedC∗-algebraic universal principal series associated to the pair(G,P)and is denoted by E(G/N).

As explained in[3],this construction generalizes Rief f el’s construction of induction modules[16].It provides a global picture of principal series induced fromPin the sense that there exists aG-equivariant unitary specialisation map between the tensor product ofE(G/N)with the carrying space of any appropriate element ofand the corresponding principal series representation(see[3,Corollary 3.5]).

The next result allows us to consider Schwartz functions onXas a subspace ofE(G/N).It was suggested to me by Vincent Laf f orgue.

Proposition 2.2The inclusion map ι:extends continuouslyto S(X).

ProofAccording to[9,Chap.VII§3],we can use the KAK decomposition to write any element of GL(n,F)as a productk1ak2,wherek1andk2are elements in a maximal compact subgroup of GL(n,F)andais a diagonal matrix with positive entriesa1,···,an.Then forfandhinS(X),we prove that there exists a constantCsuch that:

By polarization,we may assume thatf=h.Moreover,sinceKis compact,it is enough to treat the casek1=k2=1.IdentifyingXto(Fn+1)nviax=(x1,···,xn),the assumption thatfis rapidly decreasing implies the existence of positive bounded integrable functionsfionFn+1such that|f(x)|≤f1(x1)···fn(xn).Observe thataacts onxby multiplying eachxibyai,so that it is enough to consider the case of onefi.One has,forfpositive bounded and integrable onFn+1anda>0,

Since

the same computation yields

so that

which implies(2.5).As a consequence,we will prove thatbelongs to the reduced C∗-algebra of GL(n,F).

Following[14,§4.4],we consider the generalized Harish-Chandra Schwartz space associated to a real reductive groupLconsisting of functionsFonLsuch that for anyp,there exists aconstantCpsuch that the inequality

is satisfied for allk1ak2in the KAK decomposition ofL,whereρdenotes the half sum of positive roots for(l:a).We will use the fact,whose proofis recalled in[14],that forplarge enough,there existsCpsuch that(2.6)defines a norm that yields a Banach algebra which is a subalgebra ofIn the case at hand,denotingdF=dimR(F),one has

sinceand min≤1 for alli.Therefore,(2.5)implies thatsatisfies(2.6)for anyp,and hence defines an element of

Next we prove that functions inS(X)can be approximated by elements offor the(GL(n,F))-valued norm used to build the Hilbert moduleE(G/N).Letμdenote the Lebesgue measure ofX.Sincethere exists a sequenceof open subsets ofXcontainingsatisfyingand such thatbe the family of closed balls centered at the origin inXwith radiusmfor the usual norm and consider a sequenceof compactly supported non-negative smooth functions onXsuch that

Letfbe inS(X).For allm,the pointwise productfm=f.χmis compactly supported away from singular matrices,and hence de fines a function in.We will prove that the-valued norm off−fmconverges to 0,so thatfcan be viewed as the limit inE(G/N)of the sequence{fm}m≥1.To do so,we denoteBy the same argument used above,(2.5)implies thatϕmbelongs to the generalized Harish-Chandra Schwartz space of GL(n,F).Furthermore,(2.5)and(2.7)imply that

for some constantCthat can be chosen independently ofmby the construction offm.It follows that for anyε>0,there exists a compactZin GL(n,F)such that

By the definition offm,the sequence{ϕm}m≥1converges to 0 uniformly on any compact subset of GL(n,F),such asZ.Therefore,

formlarge enough,so thatϕmconverges to 0 in the generalized Harish-Chandra Schwartz space of GL(n,F),and hence in,which concludes the proof.

Remark 2.1The advantage ofS(X)compared withas a submodule ofE(G/N)is the fact that,unlike compactness of the support,the properties of asymptotic decay of elements inS(X)will be preserved by the Fourier transform used in the normalization process of Section 3.3,thus allowing to properly define an operator between Hilbert modules.

Remark 2.2Starting from the pre-Hilbert module structure on,it is possible to consider various completions,for instance,with respect toor the full C∗-algebra C∗(L).However,the estimate(2.5)on which the proof of Proposition 2.2 relies does not allow to exhibitS(X)as a submodule for these norms.

The universal principal seriesassociated to the paircan be described in the same way.More precisely,all the arguments of the above discussion hold,working onand using the following formulas for the actions and the inner product:

forfandhfunctions on,ginG,ainLandϕinCc(L).As in Proposition 2.2,the inclusion map:extends to

3 Intertwining Operators

The purpose of this section is to study and normalize standard intertwining integrals at the level of the Hilbert modules introduced above.

3.1 Standard intertwining integrals

The study of intertwiners between principal series induced fromPandrespectively relies on the construction of operators transformingN-invariant functions into-invariant ones(see[8,10–11]).Working formally,that is outside of convergence considerations,one is naturally led to consider integrals of the form

It was observed(see[12])that such operators can be interpreted as Radon transforms and understood in the context of double fibrations as in[7].The present situation is described bythe diagram

Figure 1

wherepNanddenote the natural projections.Lemma 6.2 in[3]implies that(3.1)defines a mapwhich does not extend to an operator between Hilbert modules,the latter fact being directly related to reducibility phenomena occurring in the principal series(see[1,Appendix]).

In the notations of Figure 1,the standard intertwining integral can be written as

In order to study this integral at the level of Hilbert modules,it will be convenient to work inXandrather thanG/NandG/,so Figure 1 will be replaced by

Figure 2

First,we observe that two elementsxandyinXandXcome from the sameginG,if and only ifx=gx0andy=y0g−1,with the notations used in the proof of Proposition 2.1.Denoting byδthe Dirac distribution on Mn(F),it follows that the integral operatorIis given by the distributional kernelkIdefined onX×by

in the sense that

for any suitable functionf,that is

The following properties ofkIwill be useful in the normalization process studied in the next section.Formulas(2.3)andimply thatkIis invariant under the diagonalL-action:

forain GLn(F).Moreover,the homogeneity property of the Dirac distribution implies that

Our main purpose will be to construct aG-equivariant unitary operator between Hilbert modules that dif f ers fromIonly by a convolution overL.In order to state a more precise definition,let us introduce one more notation.

Definition 3.1If T is a function on L,the operator CTof convolution by T acts ona function f on G/N by

In the present situation,implies that

Definition 3.2A bounded operator U:E(G/N)→E(G/N)is said to normalize the standard intertwining integral I defined by(3.2)if

(i)it is unitary and G-equivariant;

(ii)there exists a function γon L such thatfor fin a dense subspaceof E(G/N).

Remark 3.1Definition 3.2 above should be compared with Definition 3.1 in[1],where normalization is defined for standard intertwining integrals associated to Weyl elements.

3.2 Fourier transform on X

The use of the Fourier transform on matrix spaces to study principal series representations goes back to the work of Stein[18].Let us briefly collect here the main properties of the integral operator that will give rise to the C∗-algebraic intertwiner studied in the next section.

The transform defined by

mapsS(X)toS(X).It satisfies the relation

3.3 Normalization

We can now establish the main result of the article,relating the Fourier transform of the previous paragraph to the standard intertwining operator at the level of the C∗-algebraic universal principal series.

Theorem 3.1The transform F between S(X)andextends to a unitary operator ofHilbert modules

that normalizes I in the sense of Definition3.2.The corresponding normalizing distribution isdefined byfor a in L.

ProofTheG-equivariance is a direct consequence of theG-invariance of the measure.To establish thatFisCc(L)-linear,it is enough to show that it isL-equivariant.

In view of Proposition 2.2 and Remark 2.1,we work at the level of Schwartz functions onX.ForfinS(X),one has,according to(2.3),

The fact thatFpreserves the-valued inner products also follows from a straightforward computation relying on its equivariance properties,namely

the last line relying on the Plancherel equality for square integrable functions.

Let nowγbe a function onL.According to(3.5),the compositionI◦Cγacts on a functionfby

Therefore,the kernelk◦defining the composition can be written as

Comparing this last expression with the kernel of the Fourier transform of the previous paragraph

it is easily seen that the result follows from choosing

Pursuing the comparison with the case of SL(2)suggested in Remark 3.1,we observe that this result is consistent with the one obtained in the casen=1 in[1,Theorem 3.2],where the compositionwas proved to be equal to a convolution operator by a distribution analogous toγ1.

4 The Non-archimedean Case

The purpose of this last paragraph is to describe how the construction and the main result presented above extend to the non-archimedean case.

Let F be a non-archimedean local field,that is,a finite extension of Qpor Fp((t)).We fixanon-trivial continuous character χ:F →U(1)and a Haar measure dx on F so that the Fourier transform Fχdefined by

is an isometry ofL2(F).

A normis fixed onFthat coincides with the modular function related to the Haar measure(see[20]).It is specified by the relationwhereπis a uniformizer andqis the cardinality of the residual field.

The structure theory of the groupGFremains the same in this context.The definitions of functional spaces in the case of a totally discontinuous spaceTneed to be adapted as follows:Cc(T)still denotes a continuous function with compact support onT,while(andS(T))denotes a locally constant compactly supported function onT.Up to these modifications and replacing absolute values byeverywhere,the Hilbert module construction of Section 2.3 providing the reduced C∗-algebraic universal principal series can be carried out in the same way.

Finally,the proof of the analogue of Proposition 2.2 is straightforward and the normalization process can be achieved by defining the Fourier transformFbetweenS(X)andS()by

then extending it as an operator of Hilbert modulesas in Theorem 3.1.The normalising functionγnis then given by

for anyainL.

Concluding remarkThe point of view on principal series and intertwining operators presented here relates to other works on the subject.More precisely,the distributionγnin(4.2)allows to recover localγfactors introduced by Godement and Jacquet in[6]and the Hilbert module operator obtained from(4.1)by Theorem 3.1 is a way of considering simultaneously a whole family of the normalized intertwiners studied by Shahidi in[17].The connection with these results will be studied in detail in future works.

[1]Clare,P.,C∗-algebraic intertwiners for principal series:Case of SL(2),J.Noncommut.Geom.,2011,to appear.

[2]Clare,P.,On the degenerate principal series of complex symplectic groups,J.Funct.Anal.,262(9),2012,4160–4180.

[3]Clare,P.,Hilbert modules associated to parabolically induced representations of semisimple Lie groups,J.Operator Theory,69(2),2013,483–509.

[4]Clerc,J.L.,Intertwining operators for the generalized principal series on symmetricR-spaces,2012,arXiv:1209.0691.

[5]Dixmier,J.,Les C*-algèbres et leurs représentations,Gauthier-Villars and Cie,Editeur-Imprimeur,Paris,1964.

[6]Godement,R.and Jacquet,H.,Zeta Functions of Simple Algebras,Lecture Notes in Mathematics,260,Springer-Verlag,Belin,New York,1972.

[7]Helgason,S.,The Radon Transform,Second Edition,Progress in Mathematics,5,Birkhäuser,1999.

[8]Knapp,A.W.,Representation Theory of Semisimple Groups,Princeton Mathematical Ser.,36,Princeton University Press,Princeton,NT,1986.

[9]Knapp,A.W.,Lie Groups,Beyond an Introduction,Second Edition,Progress in Mathematics,140,Birkhäuser Boston,Inc.,Boston,MA,2002.

[10]Knapp,A.W.and Stein,E.M.,Intertwining operators for semisimple groups,Ann.of Math.,93,1971,489–578.

[11]Knapp,A.W.and Stein,E.M.,Intertwining operators for semisimple groups II,Invent.Math.,60(1),1980,9–84.

[12]Kobayashi,T.and Mano,G.,The Schrödinger model for the minimal representation of the indefinite orthogonal groupO(p,q),Mem.Amer.Math.Soc.,213(1000),2011,132 pages.

[13]Kobayashi,T,Ørsted,B.and Pevzner,M.,Geometric analysis on small unitary representations of GL(N,R),J.Funct.Anal.,260(6),2011,1682–1720.

[14]Laf f orgue,V.,K-Théorie bivariante pour les algèbres de banach et conjecture de baum-connes,Invent.Math.,149(1),2002,1–95.

[15]Ólafsson,G.and Pasquale,A.,The Cosλand Sinλtransforms as intertwining operators between generalized principal series representations of SL(n+1,K),Advances in Math.,229(1),2012,267–293.

[16]Rief f el,Marc A.,Induced representations of C*-algebras,Advances in Math.,13,1974,176–257.

[17]Shahidi,F.,Fourier transforms of intertwining operators and Plancherel measures for GL(n),Amer.J.Math.,106(1),1984,67–111.

[18]Stein,E.M.,Analysis in matrix spaces and some new representations of SL(N,C),Ann.of Math.,86,1967,461–490.

[19]Vogan,D.and Wallach,N.,Intertwining operators for real reductive groups,Advances in Math.,82(2),1990,203–243.

[20]Weil,A.,Basic Number Theory,Classics in Mathematics,Springer-Verlag,Berlin,1995.