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APPEARED IN BULLETIN OF THE

arXiv:math/9304215v1 [math.RT] 1 Apr 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 2, April 1993, Pages 215-252AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONSMarko Tadi´cIntroductionThe principal ideas of harmonic analysis on a locally compact group G whichis not necessarily compact or commutative were developed in the 1940s and early1950s. In this theory, the role of the classical fundamental harmonics is playedby the irreducible unitary representations of G. The set of all equivalence classesof such representations is denoted by bG and is called the dual object of G or theunitary dual of G.Since the 1940s, an intensive study of the foundations of harmonic analysis oncomplex and real reductive groups has been in progress (for a definition of reductivegroups, the reader may consult the appendix at the end of §2).

The motivation forthis development came from mathematical physics, differential equations, differen-tial geometry, number theory, etc. Through the 1960s, progress in the direction ofthe Plancherel formula for real reductive groups was great, due mainly to Harish-Chandra’s monumental work, while at the same time, the unitary duals of only afew groups had been parametrized.With Mautner’s work [Ma], a study of harmonic analysis on reductive groupsover other locally compact nondiscrete fields was started.

We shall first describesuch fields. In the sequel, a locally compact nondiscrete field will be called a localfield.If we have a nondiscrete absolute value on the field Q of rational numbers, thenit is equivalent either to the standard absolute value (and the completion is thefield R of real numbers) or to a p-adic absolute value for some prime number p.For r ∈Q× write r = pαa/b where α, a, and b are integers and neither a nor b aredivisible by p. Then the p-adic absolute value of r is|r|p = p−α.A completion of Q with respect to the p-adic absolute value is denoted by Qp.

It iscalled a field of p-adic numbers. Each finite-dimensional extension F of Qp has anatural topology of a vector space over Qp.

With this topology, F becomes a localfield. The topology of F can also be introduced with an absolute value which isdenoted by | |F (in §5 we shall fix a natural absolute value).

The fields of real andcomplex numbers, together with the finite extensions of p-adic numbers, exhaustall local fields of characteristic zero up to isomorphisms [We].Let F be a finite field. Denote by F((X)) the field of formal power series overF.

Elements of this field are series of the form f = P∞n=k anXn, an ∈F, for somec⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2MARKO TADI´Cinteger k. Fix q > 1. Very often q is taken to be the cardinal number of the finitefield F. One defines an absolute value of f by the formula|f|F((X)) = q−min{n;an̸=0}when f ̸= 0.

In this way F((X)) becomes a local field. Fields of formal power seriesover finite fields exhaust all local fields of positive characteristic up to isomorphisms[We].The fields R and C are called archimedean fields.

For any x, y ∈R× or x, y ∈C×,there is always a positive integer n such that |y| < |nx|. The above property doesnot hold for any other local field.

This is the reason that local fields which are notisomorphic to R or C are called nonarchimedean local fields.After Mautner in the 1960s, a series of people started to consider reductive groupsover nonarchimedean local fields. Let us recall that p-adic fields were introducedhistorically to enable one to consider a single equation over a p-adic field insteadof an infinite series of congruences mod pk.

Arithmetical problems also providedmotivation to consider representations of reductive groups over such fields. Thestrongest motivation comes from the Langlands program.

A unifying element inthis program is the representation theory of reductive groups. A nice introductionto the Langlands program is [Gb3].Let G be a reductive group over a local field.

Harish-Chandra created a strat-egy for obtaining the unitary dual bG through the nonunitary dual eG, where eG isthe set of all functional equivalence classes of topologically completely irreduciblecontinuous representations of G.Functional equivalence means that the matrixcoefficients of one representation may be approximated by matrix coefficients ofanother on compact sets, and vice versa. A complete definition of eG is in §2.

Toobtain bG, one needs to classify eG (the problem of the nonunitary dual) and toidentify bG ⊆eG (the unitarizability problem). In [L2] Langlands showed how toparametrize eG by irreducible representations with certain good asymptotic prop-erties (tempered representations) of reductive subgroups, when the field F is R.The tempered representations were classified for F = R by Knapp and Zuckerman[KnZu] on the basis of Harish-Chandra’s work, thus providing a complete pictureof eG.

Despite the Langlands classification of eG in the archimedean case, there wereno big breakthroughs in the classification of unitary duals for quite a long time.Borel-Wallach and Silberger proved that Langlands parametrization of eG in termsof tempered representations of reductive subgroups was valid for reductive groupsover all local fields [BlWh, Si1].In this paper, we shall be concerned with the unitarizability problem for reductivegroups over local fields. One usually breaks the unitarizability problem into twoparts.The first part is constructing elements of bG, and the second is showingthat the constructed representations exhaust bG (completeness argument).Thecompleteness argument is usually realized by showing that the classes of eG\ bG arenot unitarizable.

We may call such an approach to the completeness argumentindirect.Suppose that the field is archimedean. Then one can linearize the problems for bG(and eG); one can “differentiate” the representations and come to the infinitesimaltheory where the main object is the Lie algebra g of G. Also, for a maximal compactsubgroup K of G, the theory of compact Lie groups gives an explicit description

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS3of bK. Thus, one may try to understand (π, H) ∈eG by studying the restriction ofπ to K. These two points explain why, in the archimedean case, some problemsconcerning representations, and especially the unitarizability problem, were oftenapproached by studying the internal structure of representations.

Let us recall thatthe internal approach was very successful in the compact Lie group case (restrictionto a maximal torus). In the nonarchimedean case, there is no possibility of such aninternal approach to the unitarizability problem.

One of the reasons that there hasbeen much less study of the unitarizability problem is that the nonunitary dualsare not yet completely parametrized there.Despite the fact that unitary duals of a very restricted number of groups havebeen classified, it is interesting to note that in 1950 Gelfand and Naimark publisheda book [GfN2] in which they constructed what they assumed to be the dual ob-jects of the complex classical simple Lie groups. Their lists were very simple, andthe representations were also simple (although infinite dimensional).

Gelfand andNaimark were using functional analytic methods as tools in their analysis. In 1967Stein constructed, in a fairly simple manner, representations in GL(2n, C)ˆ whichwere not contained in the lists of Gelfand and Naimark [St].

For some other classicalgroups, it was even easier to see the incompleteness of the lists from [GfN2].The representations of Gelfand and Naimark of GL(n, C), complemented byStein, were not generally expected to exhaust the whole of GL(n, C)ˆ.The main aim of this paper is to present the ideas which lead first to the solutionof the unitarizability problem for GL(n) over nonarchimedean local fields [Td3] andto the recognition that the same result holds over archimedean local fields [Td2],a result which was proved by Vogan [Vo3] using an internal approach. Let us saythat the approach that we are going to present may be characterized as external.At no point do we go into the internal structure of representations.Let us present the answer.

We fix a general local field F. Let Du be the setof all functional equivalence classes of irreducible square integrable modulo centerrepresentations of all GL(n, F), n ≥1 (for definition see §3).Let ||F be themodulus of F (see §5). For each representation δ ∈Du of GL(n, F), and for eachm ≥1, consider the representation of GL(mn, F) parabolically induced (§1) by| det |(m−1)/2Fδ ⊗| det |(m−3)/2Fδ ⊗· · · ⊗| det |−(m−1)/2Fδfrom a suitable standard parabolic subgroup (i.e., from one containing the upper tri-angular matrices, §2).

This representation has a unique irreducible quotient whichwill be denoted by u(δ, m). For 0 < α < 1/2 let π(u(δ, m), α) be the representationof GL(2mn, F) parabolically induced by| det |αF u(δ, m) ⊗| det |−αF u(δ, m).Denote by B all possible u(δ, m) and π(u(δ, m), α).

Then the answer isTheorem. (i) Let τ1, .

. .

, τn ∈B. Then the representation π parabolically inducedbyτ1 ⊗· · · ⊗τnof suitable GL(m, F) is irreducible and unitary.

(ii) Suppose that ρ is obtained from σ1, . .

. , σn ∈B in the same manner as πwas obtained from τ1, .

. .

, τm in (i). Then π ∼= ρ if and only if n = m and thesequences (τ1, .

. .

, τn) and (σ1, . .

. , σn) coincide after a renumeration.

4MARKO TADI´C(iii) Each irreducible unitary representation of GL(m, F), for any m, can beobtained as in (i).A new, and at the same time very old, point of view that led to the papers [Td1–Td3] was that the unitarizability problem has a reasonably simple answer and thatthe unitary representations appear in simple and natural ways. In the completenessargument, instead of an indirect strategy, a direct argument was used.In thisway, a detailed study of nonunitary representations was avoided.

In all arguments,essentially only Hilbert space representations were necessary.Surprisingly, the statement of the theorem, which was first discovered in thenonarchimedean case in [Td3], says that in the case F = C the unitary dual ofGL(n, C) should consist of the representations of Gelfand, Naimark, and Stein.Not only the statement of the nonarchimedean case of the theorem, but also themethods of the proof in [Td3] made sense in the archimedean case. Thus, after acomplete proof had been written in the nonarchimedean case, we wrote in [Td2]the proof of the archimedean case of the theorem.

We have used there a theoremof Kirillov from [Ki1], for which he never published the complete proof (see §9).There are now many solutions of the unitarizability problems, especially forparticular reductive groups in the archimedean case. In general, they are based onideas different than the one that we present in this paper.

Two of them take adistinguished place (they solve completely the problem for a series of groups havingno bounds on their semisimple split ranks). The first is Vogan classification in [Vo3]of the unitary duals of GL(n) over R, C, and H. He proved a theorem equivalentto the statement of our theorem for archimedean F. The other one is Barbasch’sclassification of the unitary duals of the complex classical Lie groups in [Bb].Since we consider both the archimedean and nonarchimedean cases, it is naturalto recall Harish-Chandra’s Lefschetz principle: “Whatever is true for real reduc-tive groups is also true for p-adic groups” [Ha2].

One problem with the Lefschetzprinciple is that we usually obtain a result for archimedean F by one kind of con-siderations and for nonarchimedean F by very different methods, and after thatwe compare the results. An interesting problem is to explain the phenomenon ofthe Lefschetz principle, which is certainly related to our depth of understanding ofthese two theories.

An important task of this paper is to present a unified pointof view on the theorem and its proof. We shall discuss both the theorem and theproof, making no distinction concerning the nature of the field.

This is possible bythe use of the external point of view. In our approach we shall very often be closeto the point of view of the representation theory of general locally compact groups,and this will be a unifying point.Professor P. J. Sally suggested that I write a paper where the ideas would beextracted from the technical machinery as much as possible.I am thankful tohim for his advice and encouragement and for his generous help during the writingof various drafts of this paper, which he has read.I am also thankful for thehospitality and support of the University of Chicago during the summer of 1986when a sketch of this paper was prepared.

I would like to express my thanks toa number of mathematicians for helpful discussions or encouragement during thewriting of this paper and the preceding ones on the same topic. Let me mentionjust a few of them: M. Duflo, I. M. Gelfand, H. Kraljevi´c, D. Miliˇci´c, F. Rodier,P.

J. Sally, and M. F. Vigneras. H. Kraljevi´c gave very helpful comments on thefirst draft of this paper.

Discussions with D. Miliˇci´c and his suggestions were of

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS5great help in the preparation of the paper. The referee gave a number of valuableand helpful suggestions to the preceding draft of this paper.

Among others, thework of J. Bernstein and A. V. Zelevinsky greatly influenced the development ofsome ideas presented in this paper. The final step in preparation for publishingthis paper was done in G¨ottingen.

I am thankful for hospitality and support to theSonderforschungsbereich 170 and S. J. Patterson.We hope that this paper will illustrate a certain internal symmetry in the externalapproach to the unitarizability problem. We hope that some of our ideas will behelpful in dealing with the unitary duals of other nonarchimedean classical groups(and also nonunitary duals).

The papers [SlTd] and [Td8] indicate that this hopeis not without basis.Finally, we introduce some general notation which we shall use throughout thepaper. For a topological space X, C(X) will denote the space of all continuous(complex-valued) functions on X.

The subspace of all compactly supported con-tinuous functions will be denoted by Cc(X). If we have a measure µ on X, thenL1(X, µ) will denote the space of all classes of µ-integrable functions on X andL2(X, µ) will denote the Hilbert space of all classes of square integrable functionson X with respect to µ.

For a smooth manifold X the space of smooth functionson X will be denoted by C∞(X) and C∞(X) ∩Cc(X) will be denoted by C∞c (X).If X is a totally disconnected locally compact topological space, then C∞c (X) willdenote the space of all compactly supported locally constant functions on X. Thefields of real and complex numbers are denoted by R and C respectively. The ringof integers is denoted by Z, nonnegative integers are denoted by Z+, and positiveones are denoted by N.1.

Concept of harmonic analysison general locally compact groupsIn this section we shall outline some of the ideas of harmonic analysis on locallycompact groups.Let G be a locally compact group. We shall always suppose in this paper thatthe groups are separable.

A representation of G is a pair (π, V ) where V is a com-plex vector space which is not zero dimensional and π is a homomorphism of Ginto the group of all linear isomorphisms of V . By a continuous representation ofG we shall mean a representation (π, H) of G where H is a Hilbert space and themap (g, v) 7→π(g)v, G × H →H is continuous (we shall always assume that H isseparable).

A closed subspace H′ of H will be called a subrepresentation of a con-tinuous representation (π, H) of G, if H′ is invariant for all operators π(g), g ∈G.A continuous representation (π, H) of G is called irreducible if there does not exist anontrivial subrepresentation of (π, H) (i.e., different from {0} and H). A continuousrepresentation (π, H) of G will be called a unitary representation if all the operatorsπ(g), g ∈G, are unitary.

Two unitary representations (πi, Hi), i = 1, 2, of G arecalled unitarily equivalent if there exists a Hilbert space isomorphism ϕ : H1 →H2such that π2(g)ϕ = ϕπ1(g) for all g ∈G. The set of all unitarily equivalence classesof irreducible unitary representations of G will be denoted by bG and called theunitary dual of G. For a family (πi, Hi), i ∈I, of unitary representations of G,there is a natural unitary representation of G on the direct sum of Hilbert spacesLi∈I Hi.

This representation will be denoted by (Li∈I πi, Li∈I Hi). It is called adirect sum of representations (πi, Hi), i ∈I.

6MARKO TADI´CA continuous representation (resp. unitary representation) on a one-dimensionalspace is called a character (resp.

unitary character) of G.The main problem of harmonic analysis on the group G is to understand someinteresting unitary representations of G (such representations are usually given onfunction spaces). One way to study this problem is to break it into two parts (atleast for type I groups which will be described in the sequel of this section and whichwill be the only groups considered in this paper).

The first part is to understandirreducible unitary representations, i.e., bG, and the second part is to understandother unitary representations in terms of irreducible ones. This strategy is in thespirit of Fourier’s classical idea of fundamental harmonics.Let us explain what we mean by understanding general unitary representationsin terms of irreducible ones.

If G is a compact group, then a fundamental factis that each unitary representation of G can be decomposed into a direct sum ofirreducible unitary representations [Di, Theorem 15.1.3.]. Understanding a unitaryrepresentation π in terms of irreducible unitary representations means, in the com-pact case, to know how to decompose π into a direct sum of irreducible unitaryrepresentations.

In the noncompact case each unitary representation decomposesinto a direct integral of irreducible unitary representations, and understanding hereagain means to know how to decompose a given unitary representation into a directintegral of irreducible representations. We are not going to define here the notionof direct integral of representations because the definition is quite technical.

Theinterested reader may consult [Di, §8]. Let me only mention that the direct integralgeneralizes the notion of direct sum and that direct integrals are determined bymeasures on bG.

To consider measures on bG, one needs some σ-ring of sets. Thisσ-ring arises in a standard way from the natural topological structure on bG.

Nowwe shall define this topology.If (π, H) is a continuous representation of G and v, w ∈H, then the functiong 7→(π(g)v, w) on G is called a matrix coefficient of (π, H). We denote by Φ(π)the linear span of all matrix coefficients of (π, H).

The closure operator on subsetsof bG is defined as follows. Let π ∈bG and X ⊆bG.

Then π ∈Cl(X) if and only ifeach element of Φ(π) can be approximated uniformly on each compact subset of Gby elements from Sσ∈X Φ(σ). This topology on bG will be called the topology of theunitary dual of G.If G is a commutative group, then each irreducible unitary representation of Gis given on a one-dimensional space (this follows easily from the spectral theorem).Thus each π ∈bG is a function.

Pointwise multiplication of element of bG definesa group structure on bG. Together with the above topology, bG is again a locallycompact commutative group which is called the dual group of G. The role of thetopology of the unitary dual is crucial in harmonic analysis on locally compactcommutative groups.

This topology is a basis on which one builds the fundamentalfacts of the harmonic analysis on commutative groups. One of these fundamentalfacts is that G and (Gˆ)ˆ are canonically isomorphic (Pontryagin duality).In the case of noncommutative groups, the role of this topology on bG is less crucialthan in the commutative case, but it is still an important and natural object toconsider.

For example, G is a type I group if and only if bG is a T0-space, i.e., forany two different points in bG, at least one of them has a neighborhood which doesnot contain the other one [Di, 9.5.2. and Theorem 9.1]. It is important to noticethat, in general, bG is not topologically homogeneous and there exist significant

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS7connections between properties of irreducible representations and their position inbG with respect to the topology.Before we proceed further, we shall say a few words about some measures whichare natural to consider on locally compact groups. For a locally compact group G,there exists a positive measure invariant for right translations.

Such a measure willbe called a right Haar measure on G and it will be denoted by µG. Thus,ZGf(gx) dµG(g) =ZGf(g) dµG(g)for any f ∈Cc(G) and x ∈G.

Any two right Haar measures are proportional. Thebehavior of a right Haar measure for left translations is described by the modularfunction.

There exists a function ∆G on G such thatZGf(xg) dµG(g) = ∆G(x)−1ZGf(g) dµG(g)for any f ∈Cc(G) and x ∈G. The group G is called unimodular if ∆G ≡1,i.e., if µG is also invariant for left translations.

For more information about Haarmeasures and for proofs of the above facts one may consult [Bu2].Suppose that G is unimodular. The space Cc(G) becomes an algebra for theconvolution which is defined by the formula(f1 ∗f2)(x) =ZGf1(xg−1)f2(g) dµG(g),f1, f2 ∈Cc(G).

In a natural way one can extend the convolution to L1(G, µG).Then L1(G, µG) becomes a Banach algebra.For f ∈Cc(G) and a continuousrepresentation π of G setπ(f) =ZGf(g)π(g) dµG(g).Now π becomes a representation of the convolution algebra Cc(G), and this repre-sentation is called the integrated form of the representation π of G. If π is unitary,then the last formula also defines a representation of the algebra L1(G, µG). More-over, it is a ∗-representation if we define f ∗(g) = f(g−1).We have already mentioned that the basic problem of harmonic analysis onG is to classify bG and then to decompose interesting representations in terms ofbG.

Among the interesting representations, there is one that should be the firstto be understood, namely, the regular representation on L2(G, µG). By the ab-stract Plancherel theorem, for a unimodular group G there exists a unique positivemeasure ν on bG such thatZG|f(g)|2 dµG(g) =ZbGTrace(π(f)π(f)∗) dν(π)for all f ∈L1(G, µG) ∩L2(G, µG) [Di, Theorem 18.8.2].

The measure ν is calledthe Plancherel measure of G (it determines an explicit decomposition of L2(G, µG)into a direct integral of elements of bG).

8MARKO TADI´CWhile the basic ideas of harmonic analysis on general locally compact groupswere laid down in the 1940s, one of the first breakthroughs in classifying unitaryduals was the work of Kirillov for nilpotent Lie groups at the beginning of the 1960s(see [Ki2, §§13 and 15]).Let us first recall that a Lie group is a group supplied with a structure of a (real)analytic manifold such that the group operations (i.e., multiplication and inversion)are analytic mappings. We can define the Lie algebra g of a Lie group G as thetangent space of G at the identity, supplied with a bracket operation [ , ] which canbe defined in the following way.

If X, Y ∈g are tangent vectors to curves x(t), y(t)for t = 0 respectively, then [X, Y ] is the tangent vector to the curvet 7→x(τ)y(τ)x(τ)−1y(τ)−1where τ = sgn (t)|t|1/2, at t = 0 [Ki2, 6.3.]. An element g ∈G acts on G by innerautomorphism.

The differential of this action is denoted by Ad(g). In this way Gacts on g, and this action is called the adjoint action of G on g.Let G be a connected simply connected nilpotent Lie group, and let g∗be thespace of linear forms on the Lie algebra g of G. There is a natural action of G on g∗.It is called the coadjoint action of G. Using the theory of induced representations(which we shall discuss a bit later in this section), Kirillov established a canonicalone-to-one mapping from the set of all coadjoint orbits onto the unitary dual of GG\g∗→bGwhich gives a simple description of the unitary dual.

With a natural topology on theleft-hand side, this is a homeomorphism. Kirillov theory also gives the charactersof irreducible unitary representations and the Plancherel formula for nilpotent Liegroups.In the second part of this section, we define some notions that we shall need inthe sequel.If G is a compact group, then each π ∈bG is given on a finite-dimensional space[Di, 15.1.4.].

As in the theory of finite group representations, the functionΘπ : g 7→Trace π(g),which is called the character of the representation π, completely determines theclass of π in bG. We have a right Haar measure on G. Thus the character functiondetermines a distribution on a compact Lie group G. It is easy to see that thisdistribution isf 7→Trace π(f), f ∈C∞c (G),which will be denoted by Θπ again.

If π is an infinite-dimensional representation,obviously the trace as a function is not well defined. Nevertheless, it may happenthat the above distribution is well defined.

For an arbitrary Lie group G, one cantake the above distribution for the definition of the character of π ∈bG, if π(f) istrace-class for f ∈C∞c (G). If π(f) has a trace for f ∈C∞c (G), then π(f) must be acompact operator for any f ∈L1(G, µG).

If all π ∈bG have characters in the abovesense, then π(f) is a compact operator for any π ∈bG and f ∈L1(G, µG).A locally compact group is called a CCR-group if π(f) is a compact operator forany π ∈bG and f ∈L1(G, µG). All CCR-groups are of type I [Di, Proposition 4.3.4.

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS9and Theorem 9.1]. A great number of very important groups are CCR-groups.

Themost important classes of CCR-groups are commutative groups, compact groups,nilpotent Lie groups, and reductive groups over local fields (reductive groups will bedefined in the appendix at the end of the following section). In particular, classicalgroups over local fields are CCR-groups.

One can characterize CCR-groups in termsof the topology of the unitary duals. A group G is a CCR-group if and only if bG isa T1-space, i.e., if the points are closed subsets of bG [Di, 9.5.3.

].For some important classes of groups one can show that they are CCR-groups byshowing that they have so-called “large” compact subgroups. Now we shall explainthe last notion.Let G be a locally compact unimodular group, and let K be a compact subgroup.For δ ∈bK and a continuous representation (π, Hπ) of G, let Hπ(δ) be the subspaceof Hπ spanned by all subrepresentations of π|K which are isomorphic to δ (hereπ|K denotes the representation of K obtained by restriction from G).

IfdimCHπ(δ) < ∞for all δ ∈bK, then π is called a representation of G with finite K-multiplicities.One calls K a large compact subgroup of G if for each δ ∈bK the functionπ 7→dimC Hπ(δ)is a bounded function on bG. If G has a large compact subgroup, then G is a CCR-group [Di, Theorem 15.5.2].Some very important classes of groups have largecompact subgroups, for example, reductive groups over local fields.For a continuous representation (π, H) of G, a vector v ∈H is called K-finite ifthe span of all π(k)v, k ∈K, is finite dimensional.In the rest of this section we shall discuss some parts of the theory of the inducedrepresentations for locally compact groups.

The notion of induced representationsfor locally compact groups generalizes the well-known notion of induced represen-tations for finite groups that was introduced and studied by Frobenius and Schur.Induction is one of the simplest and most important procedures for obtaining newrepresentations of locally compact groups.The most important case of induction for the purpose of this paper is parabolicinduction. To define this notion, it is enough to consider a closed subgroup P ofa unimodular group G and assume that there exists a compact subgroup K of Gsuch that PK = G. Let (σ, M) be a continuous representation of P. The space ofall (classes of) measurable functions f : G →H which satisfyf(pg) = ∆P (p)1/2σ(p)f(g),p ∈P, g ∈G,and||f||2 =ZK||f(k)||2 dµK(k) < ∞will be denoted by IndGP (σ).It is a Hilbert space.The group G acts by righttranslations on IndGP (σ).

This action we denote by R. Thus(Rgf)(x) = f(xg).

10MARKO TADI´CWith this action, IndGP (σ) is a continuous representation of G. It is unitary if σ isunitary.In our considerations G will be a reductive group over a local field, while P willbe a parabolic subgroup of G (these terms will be defined in the following section).One will take a Levi decomposition P = MN of P and a continuous representationσ of M. Since P/N ∼= M, we shall consider σ as a representation of P. Then weshall say that IndGP (σ) is a parabolically induced representation of G by σ .We shall also talk at some points in this paper about induced representationswhich are of more general type. Let G be a locally compact group which doesnot need to be unimodular, and let C be a closed subgroup of G. Suppose that aunitary representation (σ, H) of C is given (we may assume that σ is a continuousrepresentation only).

We denote by IndGC(σ) the space of all measurable functionsf : G →H which satisfyf(cg) = [∆C(c)∆G(g)−1]1/2σ(c)f(g),g ∈G, c ∈C.One square integrability condition is required also. This condition is more technicalthan in the case of parabolic induction [Ki2, 13.2].

Again G acts by right transla-tions on IndGC(σ). This is a unitary representation of G. We say that IndGC(σ) isunitarily induced by σ.Mackey obtained a simple criterion for testing if a given unitary representationof G is unitarily induced from C (Imprimitivity Theorem, [Ki2]).

There is an im-portant specialization of this theory.If G contains a nontrivial normal abeliansubgroup N, then Mackey theory implies a description of bG by irreducible unitaryrepresentations of smaller groups (G and N need to satisfy certain general topo-logical conditions). Let χ ∈bN and denote by Gχ the stabilizer of χ in G (G actson bN because G acts on N by automorphisms).

Note that bN consists of unitarycharacters. Take an irreducible unitary representation σ of Gχ such that σ|N is amultiple of χ.

Then IndGGχ(σ) is an irreducible unitary representation of G. Oneobtains all irreducible unitary representations of G by the above construction. Twosuch representations constructed from χ1, σ1 and χ2, σ2 are equivalent if and onlyif χ1, σ1 and χ2, σ2 are conjugate.

This specialization of Mackey theory is usuallycalled small Mackey theory. In the special case when G is a semidirect product ofa closed normal abelian subgroup N and a closed subgroup M, the small Mackeytheory describes bG more simply.

Here bG is parametrized by unitary characters ofN and irreducible unitary representations of their stabilizers in M, divided by anatural equivalence.For our purpose, the case of semidirect products is the most interesting one. It isimportant to observe that one obtains here automatically the irreducibility of someinduced representations.

For a more detailed exposition of small Mackey theory,one may consult [Ki2, 13.3].2. The nonunitary dual as a tool for the unitary dualIn the study of the representation theory of the general linear groups, and moregenerally, of the classical groups, the terminology of the theory of reductive groupsis very useful and natural.

We shall very briefly recall some of the terminology ofthis theory in the first part of the appendix at the end of this section. The readermay also skip over the general definitions and follow only the case of GL(n) wherewe shall not need these general definitions.

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS11We begin this section with a few definitions.We shall first define parabolicsubgroups in GL(n, F). Take an ordered partitionα = (n1, n2, .

. .

, nk)of n. Consider block-matricesA =A11.....A1k............................Ak1.....Akkwhere Aij is an ni by nj matrix. DenotePα = {A ∈GL(n, F); Aij = 0 for i > j},Mα = {A ∈GL(n, F); Aij = 0 for i ̸= j}.Let Nα be the set of all A ∈Pα such that all Aii are identity matrices.

Now Pα iscalled a standard parabolic subgroup of GL(n, F), Mα is called a Levi factor of Pα,and Nα is called the unipotent radical of Pα. The subgroup P(1,1,...,1) of all uppertriangular matrices in GL(n, F) is called the standard minimal parabolic subgroup.Take any g ∈GL(n, F) and any ordered partition α = (n1, n2, .

. .

, nk) of n.Then gPαg−1 is called a parabolic subgroup of GL(n, F). Set P ′ = gPαg−1, M ′ =gMαg−1, and N ′ = gNαg−1.

Then P ′ = M ′N ′ is called a Levi decomposition of P ′.The group M ′ is called a Levi factor of P ′, and N ′ is called the unipotent radicalof P ′. Similarly, a minimal parabolic subgroup is defined to be any conjugate ofthe standard minimal parabolic subgroup.We shall now introduce the nonunitary dual.

We denote by G a reductive groupover a local field F. There exists a maximal compact subgroup K of G such thatPminK = G for some minimal parabolic subgroup Pmin of G. We fix such a maximalcompact subgroup. The Iwasawa decomposition PminK = G holds for any maximalcompact subgroup K of G if F is an archimedean field.

If it is not, this may not betrue for all maximal compact subgroups. The group K is a large compact subgroupof G in the sense of the previous section.

In the case of GL(n, F) and F = R (resp.F = C), one may take for K the group of orthogonal matrices (resp. the groupof unitary matrices).

If F is nonarchimedean, one may take K in GL(n, F) to beGL(n, OF ), where OF is the ring of integers in F, that is, OF = {x ∈F; |x|F ≤1}.In a general linear group over any local field, all maximal compact subgroups areconjugate.We shall always assume in the sequel that continuous representations of G thatwe consider have finite K-multiplicities.For a continuous representation π of G, we have denoted by Φ(π) the linear spanof all matrix coefficients of π. This is a subspace of C(G).

Denote by Cl Φ(π) theclosure of Φ(π) with respect to the open-compact topology on C(G). Then twocontinuous representations π1 and π2 of G are said to be functional equivalent ifCl Φ(π1) = Cl Φ(π2).

12MARKO TADI´CWe denote by eG the set of all functional equivalence classes of continuous irreduciblerepresentations of G which have finite K-multiplicities. These representations areprecisely the topologically completely irreducible representations of G (for a def-inition of the last notion, one may consult [Wr, 4.2.2.]).

The set eG is called thenonunitary dual of G. We could use Naimark equivalence to define eG instead ofthe functional equivalence. We would get the same object.

Two continuous repre-sentations (π1, H1) and (π2, H2) are called Naimark equivalent if there exist densesubspaces V1 ⊆H1 and V2 ⊆H2 which are invariant for integrated forms and aclosed one-to-one linear operator ϕ from V1 onto V2 such that(π2(f)ϕ)(x) = (ϕπ1(f))(x)for any f ∈Cc(G), x ∈V1.For an irreducible continuous representation π of G which has finite K-multiplicities,the operator π(f) is of trace-class if f ∈C∞c (G). The linear formf 7→Trace(π(f))is denoted by Θπ and called the character of the representation π.

Two irreduciblecontinuous representations with finite K-multiplicities are functionally equivalentif and only if they have the same characters. Furthermore, characters of represen-tations from eG are linearly independent.The natural mapping bG →eG is one-to-one (G is a CCR-group).

Therefore, weshall identify bG with a subset of eG. A class π ∈eG will be called unitarizable orunitary if π ∈bG ⊆eG.

One supplies eG with a topology in the same way as we havesupplied bG with the topology of the unitary dual.The reader may consult the second part of the appendix at the end of this sectionfor standard realizations of the set eG. Those realizations depend on whether thefield is archimedean or not.The idea of Harish-Chandra was to break the problem of describing bG into twoparts: the problem of the nonunitary dual and the unitarizability problem.

Theproblem of determining the nonunitary dual appeared to be much more manageablethan the unitarizability problem.Besides the above general strategy, there could be other strategies for getting bG.It would be interesting to obtain a classification of the unitary duals dealing withnonunitary representations as little as possible or perhaps not at all. A strategyof such classification for GL(n) over archimedean fields dealing with only unitaryrepresentations can be based on a paper from 1962 by Kirillov [Ki1].

We shallreturn to [Ki1] in §9. Now we shall outline the strategy.Let Pn be the subgroup of GL(n, F) of all matrices with the bottom row equalto (0, 0, .

. .

, 0, 1). An interesting property of Pn follows from the small Mackeytheory: bPn is in a bijection withGL(n −1, F)ˆ∪GL(n −2, F)ˆ∪· · · ∪GL(2, F)ˆ∪GL(1, F)ˆ∪GL(0, F)ˆ(GL(0, F) denotes the trivial group).

Gelfand and Naimark showed already in 1950that certain irreducible unitary representations of SL(n, C), which they expectedto exhaust SL(n, C)ˆ, remain irreducible as representations of P ′n where P ′n de-notes the subgroup of SL(n, C) of all matrices with the bottom row of the form

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS13(0, . .

. , 0, x), x ∈C×.

They obtained this result from the explicit formulas for thoserepresentations. Kirillov’s idea was to first find a general proof of the irreducibilityof π|Pn for π ∈GL(n, F)ˆ when F = R or C. Then one has an inductive procedurefor classification.

After classifying GL(m, F)ˆ for m ≤n −1, one also has a classifi-cation of bPn by the above remark about bPn. Thus, the second part of the strategy isto find all possible extensions of representations from bPn to unitary representationsof GL(n, F).

This strategy was used by I. J. Vahutinskii in his study of irreducibleunitary representations of GL(3, R).At the end of this section we shall say a few words about the characteristicsof some approaches to the unitary duals of certain groups in two relatively simplecases.We shall first consider the case of a compact Lie group G. We shall assume thatG is connected. One starts from an irreducible unitary representation (π, H) ofG, and studying the internal structure of H, one comes to exact parameters whichclassify bG .

Let us give a rough idea of how this approach goes. A closed connectedcommutative subgroup of G is called a torus in G. Each torus is isomorphic to someTn where T is the group of all complex numbers of norm one.

We fix a maximaltorus T in G. Then each element of G is conjugate to an element of T , i.e., eachconjugacy class of G intersects T . Suppose that π is a continuous representationof G on a finite-dimensional space H.One can choose an inner product on Hinvariant for the action of G. Thus the representation π|T decomposes into a directsum of some unitary characters χ1, .

. .

, χm. These unitary characters are called theweights of π with respect to T .

Take now (π, H) ∈bG. We have already mentionedthat the character Θπ of the representation π determines the class of π.

Since thecharacter Θπ, as a function on G, is obviously constant on conjugacy classes, π isalready determined by π|T , i.e., π is determined by its weights. Further analysis ofthe structure of the representation π on H requires the study of the representationof the Lie algebra (the formula (L.A.) in the appendix at the end of this section,defines the action of the Lie algebra).

It gives that there is a particular weightamong all weights of π which already characterizes π (the highest weight). In thisway one obtains a parametrization of bG by a certain subset of characters of T (thedominant weights).

For an exposition of this nice theory of Weyl and Cartan, onemay consult, among many nice expositions, the seventh paragraph of [Bu1].We shall present now a simple strategy for solving the unitarizability problem forG = SL(2, R). One may take for K the group SO(2) of all two-by-two orthogonalmatrices of determinant one.

Note that SO(2) ∼= T . The unitary dual of K is givenby the charactersσn :cos(ϕ)−sin(ϕ)sin(ϕ)cos(ϕ)7→einϕ,when n runs over Z.

If (π, H) ∈eG, then it is not hard to show that multiplicitiesof the representation π|K are one, i.e., H(σn) are either zero- or one-dimensionalsubspaces. This can be seen in a similar way as one shows that the space of K-invariant vectors in an irreducible representation is one dimensional if (G, K) is aGelfand pair [GfGrPi, Chapter III, §3, no.

4]. Thus, there is a basis {vn; n ∈S} ofthe Hilbert space H, parametrized by a subset S of Z.

Suppose that π is a unitaryrepresentation with a G-invariant inner product ( ,) on H. The formula (L.A.)from the appendix defines the representation of the Lie algebra g of G on K-finitevectors.Differentiating the relation (π(g)v, π(g)w) = (v, w), g ∈G, along one-

14MARKO TADI´Cparameter subgroups, one gets that the representation on K-finite vectors satisfies(π(X)v, w) = −(v, π(X)w)for all X ∈g. Put ||vn|| = cn.

Since vn, n ∈S, are orthogonal, the inner productis completely determined by numbers cn, n ∈S. One can solve the unitarizabilityproblem in the following way.

Take (π, H) ∈eG and check if there exist positivenumbers cn, n ∈S, such that the inner productXλivi,Xµivi=Xc2i λi ¯µisatisfies (π(X)v, w) + (v, π(X)w) = 0 for all K-finite vectors v and w in H and allX ∈g. All π for which there exist such numbers form the unitary dual.

Clearly,to be able to solve the above problem, we need to know explicitly the internalstructure of irreducible representations of G.In the above two examples, one solves the problem by study of the internalstructure of representations.AppendixAlgebraic groups. We shall recall very briefly in the first part of this appen-dix some definitions from the theory of algebraic groups.

For precise definitionscontaining all details, one should consult [Bl].A linear algebraic group G is a Zariski closed subgroup of some general lineargroup with entries from an algebraically closed field. A linear algebraic group iscalled unipotent if it is conjugate to a subgroup of the upper triangular unipotentmatrices.

A linear algebraic group G is called reductive if G does not contain anormal Zariski closed unipotent subgroup of positive dimension. A linear algebraicgroup G is called semisimple if it is reductive and if it has a finite center.

If anoncommutative linear algebraic group G does not contain a proper normal Zariskiclosed subgroup of positive dimension, then G is called a simple algebraic group. Ifthere is a group isomorphism of G onto some GL(1)n which is also an isomorphismof algebraic varieties, then G is called a torus.A Zariski closed subgroup P of G is called a parabolic subgroup if the homoge-neous space G/P is a projective variety.By a reductive group, we shall mean the group G(F) of all F-rational points ofa reductive group G which is defined as an algebraic variety over a local field F.By a parabolic subgroup of a reductive group G(F), we shall mean the group of allrational points of a parabolic subgroup of G, which is defined over F. A minimalparabolic subgroup of G(F) is a parabolic subgroup which does not contain anyother parabolic subgroup.

All minimal parabolic subgroups in G(F) are conjugate.If we fix a minimal parabolic subgroup Pmin of G(F), then the parabolic subgroupscontaining Pmin are called standard parabolic subgroups. Each parabolic subgroupin G(F) is conjugate to a standard parabolic subgroup.Let P be a parabolicsubgroup of G(F).

Among Zariski connected normal unipotent subgroups of Pthere is the maximal one. It is called the unipotent radical of P. Denote it by N.There exists a reductive subgroup M of P such that P = MN and M ∩N = {1}.Note that P is then a semidirect product of N and M. One says that P = MN isa Levi decomposition of P. Also, one says that M is a Levi factor of P.

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS15Since a reductive group G(F) is a closed subgroup of GL(n, F), G(F) is in anatural way a locally compact group. If F is an archimedean field (i.e., F = R orC), then G(F) is a Lie group in a natural way.

If F is a nonarchimedean field, thenG(F) is a totally disconnected group.In the sequel, a reductive group G(F) will usually be denoted simply by G.The most important examples of reductive groups are the classical groups suchas general linear groups GL(n, F), special linear groups, symplectic groups andorthogonal groups.The groups GL(n, F) form the simplest series of reductivegroups and one of the first series to be considered.Even if one is interested in harmonic analysis on some particular class of classicalgroups, it is usually necessary to study a broader class of groups because someimportant constructions involve subgroups which may not belong to the consideredclass. Such subgroups are reductive.

This is the reason why it is convenient touse the terminology of reductive groups even if we consider some specific class ofgroups.Realizations of eG. The set eG has the following realizations, depending on whetherthe field is archimedean or not.

We shall not use these realizations in the sequel, sothe reader can also skip over these definitions. We note that the following notionsare very important in the theory.

It is also interesting to note how different eG looksin the following archimedean and nonarchimedean realizations.Suppose that F is an archimedean field. First, we shall give a definition of a(g, K)-module (g is the Lie algebra of G).

A representation π of a Lie algebra g isa real-linear map from g into the space of all linear operators on a complex vectorspace V such thatπ([X, Y ]) = π(X)π(Y ) −π(Y )π(X)for any X, Y ∈g. If we have a Lie group G and a continuous representation π of Gon a finite-dimensional space V , then the following limit exists(L.A.)π(X)v =ddt[π(x(t)v)]t=0,v ∈V,and it defines a representation of the Lie algebra g of G on V (in the above formulaX ∈g and X is the tangent vector to the curve x(t) at t = 0).

We call this Liealgebra representation the differential of π. Suppose now that G is a reductivegroup over F and k the Lie algebra of K.Let (π, V ) be a pair where V is acomplex vector space and π = (πg, πK) is again a pair consisting of a Lie algebrarepresentation πg of g on V and of a representation πK of K on V , such that thefollowing three conditions are satisfied.

(a) For each v ∈V the vector space W spanned by all πK(k)v, k ∈K, is finitedimensional and the representation of K on W is continuous. (b) The differential of the representation πK of K equals the restriction of theLie algebra representation πg to k.(c) For any k ∈K, X ∈g, and v ∈Vπg(Ad(k)X)v = πK(k)πg(X)πK(k−1)v.

16MARKO TADI´CThen (π, V ) is called a (g, K)-module. An irreducible (g, K)-module is a (g, K)-module which has no nontrivial subspaces invariant both for actions of K and g.Two (g, K)-modules (π′, V ′) and (π′′, V ′′) are equivalent if there is a one-to-onelinear mapping ϕ from V ′ onto V ′′ such that ϕπ′K(k) = π′′K(k)ϕ and ϕπ′g(X) =π′′g(X)ϕ for any k ∈K and X ∈g.

Now eG is in a natural bijection with the setof all equivalence classes of irreducible (g, K)-modules. If (π, H) is an irreduciblecontinuous representation of G (with finite K-multiplicities, which we always as-sume), then one takes for V the space of all K-finite vectors in H. The formula(L.A.) defines an action of g on V , and V becomes an irreducible (g, K)-module inthis way.Suppose now that F is nonarchimedean.

A representation (π, V ) of G is calledsmooth if for each v ∈V there is an open subgroup Kv of G such that π(k)v = vfor any k ∈Kv. Again we say that smooth representation (π, V ) is irreducible ifthere is no nontrivial vector subspace invariant for the action of G. Two smoothrepresentations (π1, V1) and (π2, V2) of G are equivalent if there exists a one-to-one linear map ϕ from V1 onto V2 such that π2(g)ϕ = ϕπ1(g) for any g ∈G.As before, there is a natural one-to-one correspondence from eG onto the set ofall equivalence classes of irreducible smooth representations of G. If (π, H) is anirreducible continuous representation of G, one takes again for V the space of allK-finite vectors v in H. Now the restriction of the action of G on H to V definesan irreducible smooth representation of G on V .3.

Some simple constructions of unitary representationsOne would like to have rather simple and natural constructions of unitary repre-sentations which produce the whole of bG. For nilpotent Lie groups such a systematicprocedure consists of unitary induction by one-dimensional unitary representations.For the groups we consider, the situation is not so simple, but it is not too badeither.

For example, one obtains the whole of SL(n, C)ˆ by parabolic induction withone-dimensional, in general nonunitary, representations (see §9).We have introduced the topology of bG in §1. In the construction of new uni-tary representations, the hardest problem is to find new connected components ofbG.

Since bG is not topologically homogeneous, there may exist special connectedcomponents, those consisting of only one point. These representations are usuallycalled isolated representations.

To avoid the influence of the commutative harmonicanalysis coming from Gab = G/Gder where Gder denotes the derived group of G,we shall define the notion of representations isolated modulo center. Let Z(G) bethe center of G. For (π, H) ∈eG there exists a character ωπ ∈Z(G)˜ such thatπ(z) = ωπ(z) idH for all z ∈Z(G).

The character ωπ is called the central characterof π. For a character ω ∈Z(G)˜ seteGω = {π ∈eG; ωπ = ω}andbGω = eGω ∩bG.The representation π ∈bG (resp.

π ∈eG) will be called isolated modulo center (resp.isolated modulo center in the nonunitary dual) if {π} is an open subset of bGωπ(resp. an open subset of eGωπ).

In the sequel, by isolated representation we shallmean isolated modulo center. According to what we have said about the topologyof bG, we may say roughly that matrix coefficients of isolated representations arenot similar to other matrix coefficients of elements of bG.

The following example

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS17indicates that.If G is compact, then bG is discrete, and matrix coefficients ofdifferent representations are L2-orthogonal. Kazhdan proved in [Ka] that the trivialrepresentation is isolated when G is a simple group of split rank n ≥2.

The splitrank is the highest n such that G possesses a Zariski closed subgroup defined over Fwhich is isomorphic over F to GL(1, F)n. As opposed to the trivial representation,other isolated representations are usually not easily constructible. In fact, isolatedrepresentations of bG or eG are very distinguished representations in known examples.Certainly, each isolated representation in the nonunitary dual, which is unitary, isalso isolated in the unitary dual.The first isolated representations that one usually meets in the representationtheory of reductive groups are square integrable.

An irreducible unitary represen-tation (π, H) of G is called square integrable modulo center if for any v, w ∈H, thefunctiong 7→|(π(g)v, w)|is a square integrable function on G/Z(G) with respect to a Haar measure. Weshall use the term square integrable instead of square integrable modulo center.Actually, the unitarity of some (π, H) ∈eG may be obtained from the above squareintegrability condition (note that the unitarity of the central character of π mustbe assumed to be able to formulate the above square integrability condition).

Thesquare integrable representations are crucial for Plancherel measure and importantfor parametrizing the nonunitary dual. In known examples, they are very oftenisolated (in the unitary dual).The known examples show that the construction of a connected component, orat least a big part of it, reduces to the construction of isolated representationsof reductive subgroups of G attached to parabolic subgroups and some standardsimple and well-known constructions.

Now we shall recall these standard simpleconstructions. The first and the oldest one is:(a) Unitary parabolic induction.

Let P = MN be a Levi decomposition of a para-bolic subgroup P of G. For a continuous representation σ of M, we have consideredσ also as a representation of P using the projection P = MN →P/N ∼= M. ThenIndGP (σ) was called a parabolically induced representation of G. If σ is a unitaryrepresentation, then this process will be called unitary parabolic induction. In fact,we always take σ ∈cM.

Then IndGP (σ) is a unitary representation which is a directsum of finitely many irreducible representations. It is usually irreducible.

In theconstruction (a), we shall always assume that P is a proper subgroup of G. Ingeneral, if ( ,) is an M-invariant hermitian form on the representation space Uof σ, then(f1, f2) 7→ZK(f1(k), f2(k)) dµK(k)is a G-invariant hermitian form on IndGP (σ), and it is positive definite if the formon U was positive definite.Unitary parabolic induction has been used since the first classification of theunitary duals of reductive groups [Bg], [GfN1]. Gelfand and Naimark started to usesystematically unitary parabolic induction for the classical simple complex groups,while Harish-Chandra started a systematic study of this induction.The following construction was used also in the first classifications of unitaryduals of reductive groups [Bg, GfN1].

18MARKO TADI´C(b) Complementary series. It happens that some representations induced by nonuni-tary ones become unitary after a new inner product is introduced on the representa-tion space.

The idea is the following. One realizes on the same space a “continuous”family (πα, Hα), α ∈X, of irreducible induced representations which possess G-invariant nontrivial hermitian forms.

Let X be connected. Suppose that some παis unitary.

The fact that a continuous family of nondegenerate hermitian forms ona finite-dimensional space parametrized by X, being positive definite at one pointof X, must be positive definite everywhere enables one to conclude that all con-structed representations are in bG (here one reduces arguments to finite-dimensionalspaces by considering spaces L H(δ), where δ runs over fixed finite subsets of bK).Positivity at one point is obtained in general from (a). The delicate point is theconstruction of a continuous family of G-invariant hermitian forms, and it is basedon the theory of intertwining operators for induced representations.For the above construction some authors use the term deformation [Vo4].

Wehave chosen rather a more traditional name.Let us recall that a topological space X is quasi-compact if each open coveringof X contains a finite open subcovering. Note that in the above definition of aquasi-compact topological space, the Hausdorffproperty is not required (this is thedifference between quasi-compactness and compactness).

A topological space is lo-cally quasi-compact if each point has a fundamental set of neighborhoods which arequasi-compact. The fundamental fact about the topology of bG (actually, about thedual of any C∗-algebra) is local quasi-compactness.

This fact essentially, togetherwith some understanding of the topology of the unitary dual, implies that bG cannotbe complete without(c) Irreducible subquotients of ends of complementary series. This fact was firstobserved and proved by Miliˇci´c.

Before we give a brief argument why the rep-resentations in (c) must be included in bG, we shall give a simple but suggestiveexample.Let P be a minimal parabolic subgroup in G = GL(2, F) (one may take for P theupper triangular matrices in G). We have denoted by ∆P the modular characterof P. RepresentationsIα = IndGP (∆αP ),−1/2 < α < 1/2are irreducible.

If α = 0, then I0 is unitary by (a) (∆0P is the trivial representation,so it is unitary). The family Iα, −1/2 < α < 1/2, is a “continuous” family ofirreducible representations with nondegenerate G-invariant hermitian forms.

Thus,they belong to bG by (b).We shall pay attention now to the representation at the end of these complemen-tary series I−1/2 = IndGP (∆−1/2P). From the definition of I−1/2, it follows that thetrivial representation of G is a subrepresentation of I−1/2 (the trivial representa-tion of some group G is a representation on a one-dimensional space V where eachelement acts as the identity on V ).

This is the unique proper nontrivial subrepre-sentation of I−1/2. Since I−1/2 is infinite dimensional, there is no inner product onI−1/2 for which I−1/2 is a unitary representation (in a unitary representation foreach subrepresentation there is another subrepresentation on the orthogonal com-plement).

Nevertheless, the representation on the quotient of I−1/2 by the trivialrepresentation is unitary (actually, it is square integrable, and this implies that it

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS19is unitary). So, though I−1/2 is not unitary, each irreducible subquotient of therepresentation at the end of the complementary series is unitary.

This is the casein general.The representation I−1/2 from the above considerations is a representation whichis not irreducible, but also, it is not very far from being irreducible. This is an exam-ple of a representation of finite length.

Before we proceed further with explanationof the construction (c), we shall recall the definition of a representation of finitelength. Suppose that we have a continuous representation (π, H) of a reductivegroup G, which has finite K-multiplicities.

Then we say that π is of finite length ifthere exist subrepresentations{0} = H0 ⊆H1 ⊆· · · ⊆Hn = Hof H, such that the quotient representations of G on Hi/Hi−1 are irreducible rep-resentations of G, for i = 1, 2, . .

., n. Parabolic induction carries the continuousrepresentations of M of finite length to the continuous representations of G offinite length.Now we shall give a brief argument for the unitarity of representations in (c).We shall omit technical details. Suppose that we have a complementary series ofrepresentations πα, α ∈X.

We may consider the following situation. Let Y be atopological space with a countable basis of open sets, and let X be a dense subsetof Y .

Assume that to each α ∈Y is attached a nontrivial continuous representationπα of G of finite length such that the functionsα 7→Θπα(ϕ),Y →Care continuous, for all ϕ from the space Cc,∗(G) of all continuous compactly sup-ported functions which span a finite-dimensional space after translations by el-ements of K (left and right).Suppose also that the πα are irreducible unitaryrepresentations for all α ∈X. Take any α ∈Y .

Let (αn) be a sequence in Xconverging to α. Miliˇci´c proved that in general a sequence (πn) in bG has no accu-mulation points if and only if limn Θπn(ϕ) = 0 for all ϕ [Mi, Corollary of Theorem6]. Since Θπα ̸≡0, (παn) has a convergent subsequence.

If S is the set of all limitsof subsequences of (παn) in bG, then Miliˇci´c’s description of the topology of bG saysthat there exist positive integers nσ, σ ∈S, such thatlimn Θπαn (ϕ) =Xσ∈SnσΘσ(ϕ)for all ϕ in Cc,∗(G) [Mi, Theorems 6 and 7]. Also, S is a discrete and closed subsetof bG.

ThusΘπα(ϕ) =Xσ∈SnσΘσ(ϕ)for all ϕ in Cc,∗(G).Since the character of πα is the sum of characters of itsirreducible subquotients, one can obtain easily that S is precisely the set of allirreducible subquotients of πα. Since S ⊆bG, each irreducible subquotient of πα isunitary.

Thus (c) provides unitary classes.For a direct proof without use of the topology, one may consult [Td5]. The proofis based on the fact that a group of unitary operators on finite-dimensional Hilbert

20MARKO TADI´Cspace is finite dimensional. One uses in the proof the fact that reductive groupshave large compact subgroups.While the constructions of (a) and (b) provide bigger continuous families of uni-tary representations, (c) provides smaller families, but they are often important inthe construction of unitary representations.

Representations obtained by construc-tions (a), (b), or (c) are never isolated. We shall describe now a simple constructionfound by Speh that may produce isolated representations.

This construction is par-ticularly useful when it is combined with some other constructions, for example,with (a), (b), and (c). Contrary to previous constructions, here one gets unitarity ofrepresentations of smaller groups from unitarity of representations of bigger groups.Before we describe the construction, we need a notion of hermitian contragradient.For a continuous representation (π, H) of G, ¯π will denote the complex conjugateof π.

It is the same representation, but the Hilbert space is the complex conju-gate of H. The contragradient representation of π will be denoted by eπ. It is therepresentation on the space of all continuous linear forms on H with the action[eπ(g)f](v) = f(π(g−1)v).

Set π+ = ¯eπ. Then π+ will be called hermitian contragra-dient of π.

A continuous irreducible representation π will be called hermitian if πand π+ are in the same class in eG. It is easy to see that all unitary representationsare hermitian.

In the classifications of eG it is usually easy to check whether π ishermitian or not.Let P = MN be a proper parabolic subgroup of G. It is very easy to prove thefollowing fact:(d) Unitary parabolic reduction. If we have hermitian σ ∈fM such that IndGP (σ) isirreducible and that its class in eG is unitarizable, then the class of σ is unitarizabletoo.We shall now give a rough argument explaining why (d) provides unitarizablerepresentations.

Let (σ, H) be an irreducible nonunitarizable hermitian represen-tation of M. Then there is a nondegenerate M-invariant hermitian form ψ on H.Now H decomposes H = H+⊕H−as a representation of K ∩P, where ψ is positivedefinite on H+ and negative definite on H−. Clearly, H+ ̸= {0} and H−̸= {0}.Note that IndGP (σ) and IndKK∩P (σ|K ∩P) are isomorphic as representations of KandIndKK∩P (σ|K ∩P) ∼= IndKK∩P (H+) ⊕IndKK∩P (H−).Since unitary induction carries unitary representations to unitary, we see that thehermitian form induced by ψ is indefinite.

This form is also G-invariant. Sincea G-invariant hermitian form on an irreducible representation is unique up to ascalar, we see that if IndGP (σ) is irreducible, then it is not unitarizable.Roughly speaking, the construction (d) enables one to construct sometimes froma component of cM1 a component of cM2, where Pi = MiNi are two parabolicsubgroups of G.There are also some simple constructions based on the geometry of a group orgroups.

For example, if σi ∈bGi, then σ1 ⊗σ2 ∈(G1 ×G2)ˆ (and conversely). Thereare also irreducible unitary representations which appear already in the classifica-tion of the nonunitary dual—the square integrable ones.It is interesting to ask which constructions must be added to (a)–(d) to gener-ate the whole unitary dual of the classical groups, starting with square integrable

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS21representations. We shall see that for the first class, the case of GL(n, F), theconstructions (a)–(d) are enough.Remarks.

(1) Fell introduced in [Fe] a notion of nonunitary dual space for arbi-trary locally compact group. It is a topological space consisting of so-called linearsystem representations.

We have studied eG as a topological space in [Td6] if G isa reductive group over a nonarchimedean field F. It was shown that eG coincideswith Fell’s nonunitary dual.The set bG is a closed subset of eG.A representa-tion π ∈eG is isolated modulo center if and only if there is a nontrivial matrixcoefficient which is compactly supported modulo center. Therefore, one may in-terpret Jacquet’s subrepresentation theorem [Cs, Theorem 5.1.2] in the followingway.

Each element of eG can be obtained as a subrepresentation of IndGP (σ), with σisolated modulo center representation of M for some parabolic subgroup P = MNof G. Each isolated modulo center representation of G in eG is essentially unitary(i.e., it becomes unitary after twisting by a suitable character of G); actually it isessentially square integrable. Certainly, all these facts about the topology of thenonunitary dual should hold over archimedean fields with the proofs along the samelines.

In the archimedean case, the representations with matrix coefficients com-pactly supported modulo center can exist only if G/Z(G) is compact. Constructionof representations of such groups is essentially solved by the case of compact Liegroups.

(2) Suppose that F is nonarchimedean. Let P = MN be a parabolic subgroupin G. Let σ ∈fM be isolated modulo center.

Denote by 0G the set of all g ∈Gsuch that the absolute value of µ(g) is one for any homomorphism µ : G →F ×which is also a morphism of algebraic varieties defined over F. Then a characterχ : G →C× is called unramified if χ is trivial on 0G, i.e., if χ(0G) = {1}. Let U(M)be the set of all unramified characters of M. Then U(M)σ = {χσ; χ ∈U(M)} is aconnected component of fM containing σ, and the set of all irreducible subquotientsof IndGP (τ), τ ∈U(M)σ, is a connected component of eG.

All connected componentsof eG are obtained in this way [Td6]. So, for eG the set of connected componentsreduces to the set of isolated representations modulo center in the nonunitary dualof the reductive parts of parabolic subgroups.

Note that a difficult problem in thenonarchimedean case is the construction of representations isolated modulo center(i.e., of supercuspidal representations).4. Completeness argumentIn the last section, we have outlined constructions (a)–(d) of unitary representa-tions of a reductive group G. It seems that those constructions provide a remarkablepart of the unitary duals of the classical groups.

This leads to one of the most in-teresting questions about unitary representations of reductive groups: How can oneconclude that a set X ⊆bG is a significant piece of bG, or even all of it? At thepresent time there is no satisfactory strategy for answering such density questions.Recall the simple answer for finite groups: One needs to check if the sum of squaresof degrees of representations in X is equal to the order of the group or not.Suppose that a set X ⊆bG is constructed and suppose also that one expectsthat it is the whole unitary dual.

If one wants to prove that, then a usual strategyhas been to prove that in eG\X all representations are nonunitarizable. One checksfor each representation in eG\X that it cannot be unitarizable considering various

22MARKO TADI´Cproperties of that representation. The simplest properties that one can considerare: if the representation is hermitian, if it has bounded matrix coefficients, etc.The construction (d) can be used also for getting nonunitarity (from fM to eG).

Theabove strategy we shall call the indirect strategy (of proving completeness of a givenset of unitary representations).The indirect strategy becomes less satisfactory for groups of larger size. At thesame time, the indirect strategy is not completely satisfactory from the point ofview of harmonic analysis: the stress is not on unitary representations, which are ofthe principal interest, but on nonunitary ones.

Actually, one needs a very detailedknowledge of the structure of representations in eG\ bG, and the set eG\ bG is usuallymuch, much larger than the set bG. The indirect strategy does not develop directlythe intuition about unitary representations.

In dealing with eG, it is very usefulto algebracize the situation (the algebraic description of eG was presented in theappendix at the end of §2).Later on we shall present a completeness argument for GL(n), dealing simulta-neously with all GL(n) and having only one argument rather than various cases.This will be an example of the direct strategy of proving completeness.In the following section we shall explain the sense in which the set of represen-tations of Gelfand, Naimark, and Stein is “big”.5. On the completeness argument: the example of GL(n, C)We shall first introduce some general notation for the general linear group andthen pass to the complex case.In the first part of this section, F denotes any local field.For x ∈F ×, there exists a number |x|F > 0 such that|x|FZFf(xg) da(g) =ZFf(g) da(g)for all f ∈Cc(G) (da(g) denotes an additive invariant measure on F).

Set |0|F = 0.Then | |F is called the modulus of F. Note that | |R is the usual absolute valueon R, | |C is the square of the usual absolute value on C (i.e., |z|C = z¯z), while|x|Qp = |x|p (see the Introduction). For g ∈GL(n, F) setν(g) = | det(g)|F .Clearly, ν : GL(n, F) →R× is a character.For n1, n2 ∈Z+, we have denoted by P(n1,n2) the parabolic subgroup of GL(n1 +n2, F) consisting of the elements g = (gij) for which gij = 0 when i > n1 and j ≤n1.Also we have denoted M(n1,n2) = {(gij) ∈P(n1,n2); gij = 0 for j > n1 and i ≤n1}.

ThenM(n1,n2) ∼= GL(n1, F) × GL(n2, F)is a Levi factor of P(n1,n2).For two continuous representations τi of GL(ni; F), i = 1, 2, we consider τ1 ⊗τ2as a representation of M(n1,n2) ∼= GL(n1, F) × GL(n2, F) in a natural way. Themapping mn 7→(τ1 ⊗τ2)(m), where m ∈M(n1,n2) and n ∈N(n1,n2), defines arepresentation of P(n1,n2).

This representation of P(n1,n2) was again denoted byτ1 ⊗τ2. Thusτ1 ⊗τ2 :g1∗0g27→τ1(g1) ⊗τ2(g2)

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS23for g1 ∈GL(n1, F) and g2 ∈GL(n2, F). Now the parabolically induced represen-tationIndGL(n1+n2,F )P(n1,n2)(τ1 ⊗τ2)will be denoted by τ1 × τ2.

It is a standard fact that (τ1 × τ2) × τ3 is isomorphic toτ1 × (τ2 × τ3) (i.e., there exists a continuous intertwining which is invertible). Thisis a consequence of a general theorem on induction in stagesIndH3H2(IndH2H1(σ)) ∼= IndH3H1(σ).Therefore, it makes sense to write τ1 × τ2 × τ3.Before we explain an important property of the operation ×, we need the notionof associate parabolic subgroups and associate representations.

Suppose that wehave two parabolic subgroups P1 and P2 of some reductive group G. If we have Levidecompositions P1 = M1N1, P2 = M2N2, and w ∈G such that M2 = wM1w−1,then we say that P1 and P2 are associate parabolic subgroups. Suppose that σ1 andσ2 are finite length continuous representations of M1 and M2 respectively, suchthat σ2(m2) ∼= σ1(w−1m2w) for all m2 ∈M2.

Then we say that σ1 and σ2 areassociate representations.For a continuous representation (π, H) of G of finitelength, consider a sequence of subrepresentations{0} = H0 ⊆H1 ⊆· · · ⊆Hk = Hof H where the quotient representations on Hi/Hi−1 are irreducible representationsof G, for i = 1, 2, . .

., k. Denote by R(G) a free Z-module which has for a basiseG. We shall denote by J.H.

(π) the formal sum of all classes in eG of the repre-sentations Hi/Hi−1, i = 1, 2, . .

., k. We shall consider J.H. (π) ∈R(G) and callit the Jordan-H¨older series of π.

The element J.H. (π) ∈R(G) does not dependon the filtration Hi, i = 0, 1, 2, .

. ., k as above (one can see that from the linearindependence of characters of representations in eG).

Suppose that P1 = M1N1and P2 = M2N2 are associate parabolic subgroups. Let σ1 and σ2 be associaterepresentations of M1 and M2 (as before, we consider σ1 and σ2 as representa-tions of P1 and P2 respectively).

A standard fact about parabolic induction fromassociate parabolic subgroups by associate representations is that representationsIndGP1(σ1) and IndGP2(σ2) have the same characters, which implies that these tworepresentations have the same Jordan-H¨older series. The formula for the characterof a parabolically induced representation from a minimal parabolic subgroup, whenF = R, is computed in [Wr, Theorem 5.5.3.1].

A similar formula holds withoutassumption on parabolic subgroup. In the nonarchimedean case we have a similarsituation.Suppose that τ1 and τ2 are continuous representations of finite length of GL(n, F)and GL(m, F) respectively.

The above fact about induction from associated para-bolic subgroups by associate representations implies that τ1 × τ2 and τ2 × τ1 havethe same Jordan-H¨older series.Set Irru = Sn≥0 GL(n, F)ˆ. To solve the unitarizability problem for the GL(n, F)-groups, one needs to determine Irru.In the rest of this section we shall assume F = C. Recall that | |C is the squareof the standard absolute value that we usually consider on C.Let χ0 : C× →C× be the character x 7→x|x|−1/2C.

Since each π ∈GL(n, C)ˆ hasa central character and GL(n, C) is a product of the center and of SL(n, C), the

24Marko Tadi´crestriction of representations from GL(n, C) to SL(n, C) gives a one-to-one mappingofGL(n, C)ˆχ0 ∪GL(n, C)ˆχ20 ∪· · · ∪GL(n, C)ˆχn0onto SL(n, C)ˆ. Therefore, in order to understand SL(n, C)ˆ it is enough to under-stand GL(n, C)ˆ (and conversely).

In the rest of this paper we shall deal only withGL-groups and interpret the Gelfand, Naimark, and Stein representations in termsof GL(n).The first obvious irreducible unitary representations of GL(n, C)ˆare one-dimensionalrepresentations χ ◦detn where χ is a unitary character of C× and where detn de-notes the determinant homomorphism of GL(n). Gelfand and Naimark obtainedalso the following series of irreducible unitary representations(ν−αχ) × (ναχ) = [ν−α(χ ◦det1 )] × [να(χ ◦det1 )],χ ∈(C×)ˆ, 0 < α < 12,the complementary series representations for GL(2, C).

These complementary seriesstart from representationsχ × χ = (χ ◦det1 ) × (χ ◦det1 ),χ ∈(C×)ˆ.The following unitary representations of GL(n, C) will be obtained by parabolicinduction using the above representations. Gelfand and Naimark showed in [GfN2]that the unitary representations obtained by parabolic induction using the rep-resentations above are irreducible.They assumed that in this way one gets allirreducible unitary representations of GL(n, C), up to unitary equivalence.We can interpret the above remarks in the following way.

SetB0 = {χ ◦detn , [ν−α(χ ◦det1 )] × [να(χ ◦det1 )]; χ ∈(C×)ˆ, n ∈N, 0 < α < 12}.Gelfand and Naimark showed that for τ1, . .

. , τk ∈B0, the representations τ1×· · ·×τk are in Irru.

Their assumption was that in this way one can get any representationfrom Irru.Stein showed that the Gelfand and Naimark complementary series representa-tions for GL(2, C) are just the first of the complementary series representationswhich exist for all GL(2n, C) [St]. He showed that[ν−α(χ ◦detn )] × [να(χ ◦detn )] ∈Irrualso for n ≥2, 0 < α < 1/2, χ ∈(C×)ˆ, and he showed that these representationswere not obtained by Gelfand and Naimark.

These complementary series start fromrepresentations(χ ◦detn ) × (χ ◦detn ),n ≥2, χ ∈(C×)ˆ,which were already well known to Gelfand and Naimark.Now it is natural to complete the Gelfand and Naimark list by the Stein com-plementary series representations. Therefore, putB = {χ ◦detn , [ν−α(χ ◦detn )] × [να(χ ◦detn )]; χ ∈(C×)ˆ, n ∈N, 0 < α < 12}(i.e., B is just B0 completed by the Stein complementary series representations).Now using arguments similar to those of Gelfand and Naimark, one may concludethat for τ1, .

. .

, τk ∈B the representations τ1 × · · · × τk are in Irru(see [Sh]). Letus denote by {G.N.S.} the set of all representations obtainable in this way.We can explain now in what sense {G.N.S.} is big in Irru.

It is easy to prove thefollowing fact (and we shall prove it later):

An external approach to unitary representations25(D) For any π ∈Irru, there exist τ1, τ2 ∈{G.N.S.} such that π × τ1 and τ2 havea composition factor in common.It is clear that (D) plays a role in the completeness argument.Regarding unitary parabolic induction for GL(n, C), the simplest situation wouldbe if it were always irreducible.

This is what Gelfand and Naimark expected tohold in 1950:(S1) Unitary parabolic induction for GL(n, C) is irreducible, i.e.,τ1, τ2 ∈Irruimpliesτ1 × τ2 ∈Irru .Let us suppose that (S1) holds. Because of (S1) and (D), for each π ∈Irru thereexist τ1, τ2 ∈{G.N.S.} such that π × τ1 = τ2.

Thus, to obtain Irru, it is enough toknow how representations from {G.N.S.} can be parabolically induced.We shall call π ∈bG primitive if there is no proper parabolic subgroup P = MNand σ ∈cM so that π ∼= IndGP (σ).

Certainly, each π ∈bG is unitarily equivalent tosome IndGP (σ) where σ ∈cM is primitive. We have mentioned that if P1 and P2 areassociate parabolic subgroups and σ1, σ2 associate representations, then IndGP1(σ1)and IndGP2(σ2) have the same Jordan-H¨older series.

The simplest situation wouldbe if the converse were true for σ1 and σ2 primitive (because of the induction instages, it is necessary to assume that σ1 and σ2 are primitive). For GL(n, C) thiswould mean (having (S1) in mind):(S2) Ifboth families τi ∈GL(ni, C)ˆ,i = 1, .

. .

, n, and σj ∈GL(mj, C)ˆ, j =1, . .

. , m, consist of primitive representations, and ifτ1 × · · · × τn = σ1 × · · · × σm,then m = n, and after a renumeration, the sequences (τ1, .

. .

, τn) and (σ1, . .

.. . .

, σm) are equal (all ni, mj are assumed to be ≥1).We shall assume that this holds.A very plausible hypothesis is:(S3)The Stein representations[ν−α(χ ◦detn )] × [να(χ ◦detn )];χ ∈(C×)ˆ, n ∈N, 0 < α < 12,are primitive.Certainly, if we assume (S3), then all elements of B are primitive. Now it isobvious that (D), together with (S1), (S2), and (S3), impliesIrru = {G.N.S.

}.It remains to find a way to prove (S1), (S2), and (S3). Note that until now only(classes of) unitary representations were necessary, and statements (S1), (S2), and(S3) are essentially analytic.

So, if one could prove (D), (S1), (S2), and (S3) dealingonly with unitary representations, one would have a classification of GL(n, C)ˆcompletely in terms of unitary representations.

26Marko Tadi´cAs we shall see, (D) is easy to prove using noncomplicated parts of the nonunitarytheory. The following strategy for proving (S2) and (S3) simultaneously can beused.The set Irru can be embedded in a suitable ring which is factorial andwhere multiplication corresponds to parabolic induction.

Then one can prove thatelements of B are prime or close to being prime. This would imply (S2) and (S3).This strategy would also include nonunitary theory (but again, not the complicatedparts).Finally, we leave the discussion about (S1) for §9.

Let us say that the first ideasfor proving (S1) are due to Gelfand, Naimark, and Kirillov.In the following sections we shall elaborate in more detail the above strategy andoutline such a strategy for GL(n) over general local field F.6. The nonunitary dual of GL(n, F)In this section, we state some basic facts about a parametrization of GL(n, F)∼.Besides Irru which was introduced in the previous section, we introduceIrr =[n≥0GL(n, F)∼.The set of all classes of square integrable representations in Irru of all GL(n, F), n ≥1, will be denoted by Du.

The set of all essentially square integrable representationswill be denoted by D. More precisely,D = {(χ ◦det) δ; χ ∈GL(1, F)∼, δ ∈Du}.For a set X, M(X) will denote the set of all finite multisets in X. These are allunordered n-tuples, n ∈Z+.

This is an additive semigroup for the operation(a1, . .

. , an) + (b1, .

. .

, bm) = (a1, . .

. , an, b1, .

. .

, bm).By WF we shall denote the Weil group of F if F is archimedean and the Weil-Deligne group in the nonarchimedean case.For the purposes of this paper, itwill not be essential to know exactly the definition of WF . We denote by I theset of all classes of irreducible finite-dimensional representations of WF .

By thelocal Langlands conjecture for GL(n) (which generalizes local class field theory),there should exist a natural one-to-one mapping of Irr onto classes of semisimplerepresentations of WF , i.e., onto M(I)Irr →M(I).Under such a mapping, D should correspond to I.Thus, there should exist aparametrization of Irr by M(D). Let us write one such parametrization.Let a = (δ1, .

. .

, δn) ∈M(D). We can writeδi = νe(δi)δui ,e(δi) ∈R, δui ∈Du.After a renumeration, we may assume e(δ1) ≥e(δ2) ≥· · · ≥e(δn).

The represen-tationλ(a) = δ1 × · · · × δn

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS27has a unique irreducible quotient (possibly λ(a) itself), whose class depends onlyon a [BlWh, Jc1]. Its multiplicity in λ(a) is 1.

We shall denote this class by L(a).Nowa 7→L(a),M(D) →Irris a one-to-one mapping onto Irr. It is a version of the Langlands parametrizationof the nonunitary duals of GL(n)-groups.

Certainly, for the existence of such aparametrization, it is crucial that the parabolic induction by square integrablerepresentations is irreducible for GL(n).The formula for the hermitian contragradient in the Langlands classificationbecomesL((δ1, . .

. , δn))+ = L((δ+1 , .

. .

, δ+n )).If we write δi = νe(δi)δui (e(δi) ∈R, δui ∈Du), thenδ+i = (νe(δi)δui )+ = ν−e(δi)δui .Also, it is easy to show thatναL((δ1, . .

. , δn)) = L((ναδ1, .

. .

, ναδn)),α ∈C.Let Rn be the free abelian group with basis GL(n, F)∼, i.e., Rn = R(GL(n, F)).For each finite-length continuous representation π of GL(n, F), we have denoted byJ.H. (π) its Jordan-H¨older series which is an element of Rn.

SetR =Mn≥0Rn.Now R is a graded additive group which is free over Irr. LetRn × Rm →Rn+mbe the Z-bilinear mapping defined on the basis Irr by(σ, τ) 7→J.H.

(σ × τ).This defines a multiplication on R, which will be denoted again by ×. We havementioned in the previous section that τ × σ and σ × τ have the same Jordan-H¨older sequences.

This just means the commutativity of R. Since (τ1 × τ2) × τ3 ∼=τ1 × (τ2 × τ3) (§5), R is also associative.Certainly, {L(a);a ∈M(D)} is abasis of R. It is a standard fact that {λ(a);a ∈M(D)} is a basis of R (i.e.,standard characters form a basis of the group of all virtual characters). This meansnothing else than the following fact which was first noticed by Zelevinsky in thenonarchimedean case [Ze, Corollary 7.5].6.1.Proposition.

The ring R is a Z-polynomial algebra over all essentially squareintegrable representations (i.e., over D).In particular, R is a factorial ring.It is a natural question to ask if it is possible to relate the operation of summingrepresentations of WF with some operation on representations from Irr in the cor-respondence a 7→L(a); i.e., is there a relation between L(a+b) and representationsL(a), L(b)? The answer is very nice:L(a+b) is always a subquotient of L(a)×L(b).One may find more information about the Langlands classification on the level ofgeneral reductive groups in [BlWh].

For the Langlands philosophy one may consult[Gb3].

28MARKO TADI´C7. Heuristic constructionIn this section we shall try to see what would be the part of the unitary dual forthe GL(n)-groups over arbitrary local field F, generated by classical constructions(a)–(d) of §3.

Our principle will be to expect a situation as simple as we couldassume, bearing our evidence in mind.So, let us start with Du. The representation δ × δ is irreducible, so δ × δ ∈Irruby the construction (a).

Then we have the complementary series which starts fromδ × δ:[ναδ] × [ν−αδ],0 < α < 1/2.This is an example of construction (b). We require α < 1/2 to ensure that theinduced representation is irreducible.

At the end of complementary series [ναδ] ×[ν−αδ], there will be unitary irreducible subquotients. To be able to identify atleast one, let us recall the relation mentioned at the end of §6; namely,L(a + b) is a subquotient of L(a) × L(b) for a, b ∈M(D) .We shall assume that this holds in the rest of this section.

Therefore, we havethat L((ν1/2δ, ν−1/2δ)) is unitary since it is at the end of the above complementaryseries (construction (c)).To proceed further, in order to be able to form a new complementary series, letus suppose that for general linear groups unitary parabolic induction is irreducible.Then one has L((ν1/2δ, ν−1/2δ)) × L((ν1/2δ, ν−1/2δ)) ∈Irru, and further, one hasa new complementary series[ναL((ν1/2δ, ν−1/2δ))] × [ν−αL((ν1/2δ, ν−1/2δ))], 0 < α < 1/2.At the end of this complementary series[ν1/2L((ν1/2δ, ν−1/2δ))] × [ν−1/2L((ν1/2δ, ν−1/2δ))]= L((νδ, δ)) × L((δ, ν−1δ))is the unitary subquotientL((νδ, δ, ν−1δ, δ))by the assumption that L(a + b) is a subquotient of L(a) × L(b). It is natural toask if the above representation is primitive.

But L((νδ, δ, ν−1δ, δ)) is a subquotientofL((νδ, δ, ν−1δ)) × L((δ)).Note that L((νδ, δ, ν−1δ)) ⊗L((δ)) is hermitian. If we take the simplest possibility,that is, L((νδ, δ, ν−1δ)) × L((δ)) irreducible, then the construction (d) implies thatL((νδ, δ, ν−1δ)) ⊗L((δ))is unitary.

Clearly then, L((νδ, δ, ν−1δ)) ∈Irru.At this point it is convenient to introduce some notation. Seta(γ, n) = (ν(n−1)/2γ, ν(n−3)/2γ, .

. .

, ν−(n−1)/2γ)

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS29for γ ∈D and n ∈Z+ andu(γ, n) = L(a(γ, n)).We shall writeνα(δ1, . .

. , δn) = (ναδ1, .

. .

, ναδn).Now we proceed further. We already have u(δ, 1), u(δ, 2), u(δ, 3) ∈Irru.

Oneconsiders a complementary series[ναu(δ, 3)] × [ν−αu(δ, 3)], 0 < α < 1/2,which starts from u(δ, 3) × u(δ, 3). At the end we have the representationL(ν1/2a(δ, 3)) × L(ν−1/2a(δ, 3)),and we can identify one irreducible subquotient which isL(ν1/2a(δ, 3) + ν−1/2a(δ, 3)).It is a unitary subquotient by (c).

Note thatν1/2a(δ, 3) + ν−1/2a(δ, 3) = a(δ, 4) + a(δ, 2)which means that L(a(δ, 4) + a(δ, 2)) is unitary. Further, this representation is asubquotient ofu(δ, 4) × u(δ, 2) = L(a(δ, 4)) × L(a(δ, 2)).Suppose again that u(δ, 4)×u(δ, 2) is irreducible.

Since it is unitary, u(δ, 4)⊗u(δ, 2)needs to be unitary by the construction (d). Therefore, u(δ, 4) will be unitary.Now it is easy to conclude that the assumptionu(δ, n) × u(δ, n −2) ∈Irrleads tou(δ, n) ∈Irru .In the case of GL(n, C), Du is equal to GL(1, C)ˆ = (C×)ˆ.Then u(δ, n) =δ ◦detn.

So, we have not obtained new unitary representations in this case.8. Scheme of unitarity for GL(n)In the last section we have seen heuristically what some simple assumptions sug-gest.

Now we shall write down some of those assumptions and their “implications”.We first recall thatu(δ, n) = L(a(δ, n)) = L((ν(n−1)/2δ, ν(n−3)/2δ, . .

. , ν−(n−1)/2δ))for δ ∈D and n ≥1.

It was also observed in §6 that R is a factorial ring.We introduce the following statements:(U0) τ, σ ∈Irru ⇒τ × σ ∈Irr. (U1) δ ∈Du, n ∈N ⇒u(δ, n) ∈Irru .

(U2) δ ∈Du, n ∈N, 0 < α < 1/2 ⇒[ναu(δ, n)] × [ν−αu(δ, n)] ∈Irru . (U3) δ ∈D, n ∈N ⇒u(δ, n) is a prime element of R.(U4) a, b ∈M(D) ⇒L(a + b) is a composition factor of L(a) × L(b).

30MARKO TADI´CHere only (U3) was not assumed or obtained in the last section. It is a strength-ening of the assumption that the u(δ, n)’s are primitive, which was present in thelast section (otherwise, we would have tried to construct new unitary representa-tions in that way).We have seen that (U0) and (U4) lead to (U1) and (U2) (i.e., unitarity of therepresentations mentioned there).

But it is interesting and surprising that withthe addition of only one assumption, namely, (U3), the preceding assumptions alsoeasily imply completeness for the unitary duals of the groups GL(n, F).8.1.Proposition. Suppose that (U0)–(U4) hold true.

SetB = {u(δ, n), [ναu(δ, n)] × [ν−αu(δ, n)], δ ∈Du, n ∈N, 0 < α < 12}.Then:(i) if τ1, . .

. , τk ∈B, we have τ1 × · · · × τk ∈Irru;(ii) if π ∈Irru, then there exist σ1, .

. .

, σm ∈B such thatπ = σ1 × · · · × σm;(iii) if σ1, . .

. , σk, τ1, .

. .

, τm ∈B and σ1 ×· · ·×σk = τ1 ×· · ·×τm, then k = mand the sequences σ1, . .

. , σk and τ1, .

. .

, τm coincide after a renumeration.From (U0), (U1), and (U2) one obtains (i) directly. Also (U3) implies (iii).

Itremains to prove only (ii). First we shall prove8.2.Lemma.

Suppose that π ∈Irr is hermitian. If (U4) holds, then there existσ1, .

. .

, σn,τ1, . .

. , τm ∈B such that π × σ1 × · · · × σn and τ1 × · · · × τm have acomposition factor in common.Proof.

Let δ ∈Du, k ∈(1/2) Z+, and 0 < β < 1/2. Then(ν−kδ, νkδ) + a(δ, 2k −1) = a(δ, 2k + 1),k > 0,and(ν−k−βδ, νk+βδ) + νβ−1/2a(δ, 2k) + ν1/2−βa(δ, 2k)= νβa(δ, 2k + 1) + ν−βa(δ, 2k + 1).Let π ∈Irr be hermitian.

Then π = L((γ1, . .

. , γs)) for some γi ∈D.

NowL((γ1, . .

. , γs))+ = L((γ+1 , .

. .

, γ+s ))(see §6). This implies that we can write π in the formπ = L n1Xi=1(ν−kiδi, νkiδi) +n2Xi=n1+1(ν−ki−βiδi, νki+βiδi) +n3Xi=n2+1(δi)!where δi ∈Du,ki ∈(1/2) Z+,0 < βi < 1/2 and all k1, .

. .

, kn1 > 0. Heren1, n2, n3 ∈Z+ and n2 ≥n1, n3 ≥n2 (i.e., not all three sums need to show up

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS31in the above formula). The two relations at the beginning of the proof and (U4)imply that bothπ ×" n1Yi=1u(δi, 2ki −1)#×"n2Yi=n1+1(νβi−1/2u(δi, 2ki) × ν1/2−βiu(δi, 2ki))#and" n1Yi=1u(δi, 2ki + 1)#×"n2Yi=n1+1(νβiu(δi, 2ki + 1) × ν−βiu(δi, 2ki + 1))#×"n3Yi=n2+1u(δi, 1)#haveL n1Xi=1a(δi, 2ki + 1)+n2Xi=n1+1(νβia(δi, 2ki + 1) + ν−βia(δi, 2ki + 1)) +n3Xi=n2+1(δi)!as a composition factor.

This completes the proof of the lemma.□End of proof of Proposition 8.1. Let π ∈Irru.

Then, by the preceding lemma thereexist σi, τj ∈B so that π ×σ1 ×· · ·×σn and τ1 ×· · ·×τm have a composition factorin common. Since (U0), (U1), and (U2) imply that both sides are irreducible, wehaveπ × σ1 × · · · × σn = τ1 × · · · × τm.Since R is factorial and the u(δ, n)’s are prime by (U3), π is a product of someu(δ, k)’s, δ ∈D.But the fact that π is hermitian implies that π is actually aproduct of elements of B.

So, we have proved (ii).□8.3.Remark. For GL(n) over a central simple division F-algebra, we expect thata scheme of this type should work too.

In that case a slight modification in thedefinitions of the u(δ, n)’s and lengths of complementary series is necessary (see[Td7]).9. On proofsWe have seen that the fulfillment of (U0)–(U4) implies a complete solution ofthe unitarizability problem for GL(n, F), as was described in Proposition 8.1.Let us remark that (U0)–(U4) were expected to hold for F = C (except maybe(U3) because such questions were not considered).

Statements (U1) and (U2) wereknown, (U0) was expected even by Gelfand and Naimark, while (U4) is easy toprove. A simple consequence of (U0)–(U4), namely, the description of the unitarydual of GL(n, C), was not generally expected to hold.

Now we shall make a fewremarks on the history and proofs of (U0)–(U4).

32MARKO TADI´CWe start with the statement (U4), which belongs to the theory of the nonunitarydual. This fact was proved by Zelevinsky in the case of nonarchimedean F for hisclassification of GL(n, F).

His proof uses induction on Gelfand-Kazhdan derivatives[Ze, Proposition 8.4]. Rodier noticed in [Ro] that Zelevinsky’s proof implies (U4)for the Langlands classification in the nonarchimedean case.

We proved (U4) in asimple manner for the archimedean case [Td2, Proposition 3.5. and 5.6]. Such aproof is outlined for nonarchimedean F in Remark A.12(iii) of [Td3].

Sometimesone can conclude the equality in (U4), using the Zelevinsky’s proof of Proposition8.5 in [Ze].9.1.Proposition. Let ai = (δi1, .

. .

, δini) ∈M(D), i = 1, 2. If δ1k × δ2m ∈Irr forall 1 ≤k ≤n1 and 1 ≤m ≤n2, thenL(a1) × L(a2) = L(a1 + a2).The above proposition may be helpful in constructing complementary series.Let us now consider (U1).

Certainly if F = C, then there is nothing to provesince as we already mentioned, Du = (C×)ˆ and for χ ∈Du, u(χ, n) = χ ◦detn.For F = R, Du ⊆GL(1, R)ˆ ∪GL(2, R)ˆ. Again, (U1) is evident if χ ∈(R×)ˆ.Speh considered the remaining case of u(δ, n),δ ∈GL(2, R)ˆ ∩Du [Sp2].

Sheproved unitarity using adelic methods.Surprisingly, it seems that Gelfand andGraev were already aware of this series of representations in the 1950s (see [Sp2,Remark 1.2.2.] about [GfGr]).For nonarchimedean F, we have determined in [Td3] the representations u(δ, n)through the ideas presented in §7.

Unitarizability is proved there essentially alongthose lines. It is also possible to prove unitarizability by the method of Speh as wasdone in the appendix of [Td3].

Note that here Du ∩GL(n, F)ˆ ̸= ∅for all n ≥1.For F = C, (U2) was proved by Stein in [St]. In general, (U2) follows from (U0),using the irreducibility of representations ναu(δ, n) × ν−αu(δ, n), 0 < α < 1/2,obtained from Proposition 9.1 and from the analytic properties of intertwiningoperators.

There is also another method in the nonarchimedean case presented in[Bn2].To prove (U3), one considers the u(δ, n)’s as polynomials and proves the irre-ducibility of these polynomials. Here one uses the fact that R is a graded ring andthat u(δ, n)’s are homogeneous elements.

In the proof, one uses basic facts aboutthe composition series of generalized principal series representations (one does notneed more detailed information, such as that obtained from Kazhdan-Lusztig typemultiplicity formulas). It is a bit surprising that, although we do not know how towrite down the polynomials u(δ, n), we can nevertheless carry out the proof.

Forproofs of (U3) see [Td3] and [Td2]. The statement (U3) is obvious for u(δ, 1) byProposition 6.1.Finally, let us return to (U0).

Let Pn denote the subgroup of GL(n, F) of allmatrices with bottom row equal to (0, . .

. , 0, 1).Already Gelfand and Naimarknoticed the importance of the statement(I) If π ∈GL(n, F)ˆ, then π|Pn is irreducible (n ∈N).Actually, they proved the above statement for F = C, for the representationsthat they expected to exhaust the unitary dual of GL(n, C).

Several people wereaware that the above statement implies (U0). For a written proof see [Sh].

Proofof the implication is based on Mackey theory and Gelfand-Naimark models.

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS33Kirillov stated (I) as a theorem in [Ki1] for F an archimedean field. There hesketched a proof.

Vahutinskii’s classification of representations of GL(3, R)ˆ wasbased on the proof. Having in mind a correspondence obtained by Mackey theorybPn →GL(n −1, F)ˆ∪GL(n −2, F)ˆ∪· · ·∪GL(2, F)ˆ∪GL(1, F)ˆ∪GL(0, F)ˆ,the statement (I) would imply that one would have simpler realizations for rep-resentations from GL(n, F)ˆ.

It is from this setup that the name Kirillov modelappears. In [Ki1] Kirillov’s intention was to prove that π|Pn is operator irreducible(i.e., the commutator consists only of scalars), which is enough by Schur’s lemmato see the irreducibility of π|Pn.

One takes any T from the commutator of π|Pnand considers a distributionΛT : ϕ 7→Trace (T π(ϕ))on GL(n, F) which is invariant for conjugations with elements from Pn, since Tis Pn-intertwining. Now if ΛT is GL(n, F)-invariant, then using the irreducibil-ity of π, it is not difficult to obtain that T must be a scalar.

Kirillov indicatedthat he proved in [Ki1] that ΛT is GL(n, F)-invariant (however, see below). Thisproperty of the distribution is especially easy to see when π is a continuous finite-dimensional representation.Then the distribution ΛT is given by a continuousfunction which must be constant on Pn-conjugacy classes (ΛT is Pn-invariant).Since in GL(n, F), GL(n, F)-conjugacy classes contain dense Pn-conjugacy classes,ΛT must be GL(n, F)-invariant.Bernstein proved in [Bn2] that for F nonarchimedean, each Pn-invariant distri-bution is GL(n, F)-invariant.

He proved (I) using essentially the Kirillov’s strategy.Besides proving (U0) in the nonarchimedean case, he gave a different proof of theimplication (I) ⇒(U0).Bernstein states in [Bn2] that Kirillov’s proof of (I) for F archimedean in [Ki1] isincorrect, and he wrote that he himself had an almost complete proof (see [Bn2, p.55]). In any case, there is no written complete proof of (I) in the archimedean casenow.

We would say that Kirillov’s proof is incomplete rather than incorrect. Hefailed to give a complete argument that the distribution ΛT is GL(n, F)-invariant.As was noted by Bernstein, the tools used in Kirillov’s paper do not seem to besufficient for proving (I).

The distribution ΛT is a very special one. Actually (I)would imply that it is a multiple of an irreducible character; so by Harish-Chandra’sregularity theorem, it is locally L1 and analytic on regular semisimple elements.So if one proves that the (eigen) distribution ΛT is locally L1 and analytic onregular semisimple elements, one could apply finite-dimensional argument.

Thismay provide a strategy to prove (I) in the archimedean case. This would be alonger proof, and there are also some disadvantages in proving (U0) through (I).We shall say a few words about these disadvantages.

Before that, observe thatthere is an implicit proof of (U0) in [Vo3].We have mentioned that the approach to the unitary dual of GL(n, F) is ex-pected to be applicable to GL(n) over a central division F-algebra A. As far aswe understand, Vogan’s description of the unitary dual of GL(n, H) confirms this.Here (U0) cannot be proved through (I), simply because (I) is false in general inthis case.

The simplest example can be obtained for GL(2, A) (A ̸= F). It is not

34MARKO TADI´Cdifficult to see that there exist (irreducible) tempered representations which arereducible when restricted to a nontrivial parabolic subgroup. Thus, it may be morereasonable to search for proof of (U0) which works also for division algebras.

Thereare some candidates for it (see the last remark at the end of this section).After all, we have the following:Theorem. Let F be a locally compact nondiscrete field.

SetB = {u(δ, n), [ναu(δ, n)] × [ν−αu(δ, n)], δ ∈Du, n ∈N, 0 < α < 12}.Then(i) if σ1, . .

. , σk ∈B, then σ1 × · · · × σk ∈Irru;(ii) if π ∈Irru, then there exist σ1, .

. .

, σm ∈B, unique up to a permutation,such thatπ = σ1 × · · · × σm.We remind the reader once more that there is no written complete proof yetof (U0) in the archimedean case, but there is a complete proof [Vo3] of the abovetheorem in this case.Remarks. (1) We give in [Td4] a concrete realization of the topological spaceGL(n, F)ˆ when F is nonarchimedean.

(2) The last theorem, together with [Ka], implies that π ∈SL(n, C)ˆ is isolatedif and only if π is the trivial representation and n ̸= 2. (3) Let F be nonarchimedean.

Let ρ ∈Irru be a representation having a non-trivial compactly supported modulo center matrix coefficient. Then for k ∈N, therepresentationν(k−1)ρ × ν(k−3)ρ × · · · × ν−(k−1)ρhas a unique square integrable subquotient which will be denoted by δ(ρ, k).

Inthis way one obtains all Du (see [Ze] and [Jc1]). In [Td4] we have proved thatπ ∈GL(n, F)ˆ is isolated modulo center if and only if π equals some u(δ(ρ, k), m)with k ̸= 2 and m ̸= 2.

(4) One could try to prove (U0) for GL(n) over local division algebras by provingfirst the following conjecture: Let A be a central local simple algebra, let S be thesubgroup of the diagonal matrices in GL(2, A), let N be the subgroup of uppertriangular unipotent elements in GL(2, A), and let σ be an irreducible unitaryrepresentation of S. Then IndGL(2,A)SN(σ) should be irreducible. (5) A proof of (U3) and (U4) for GL(n) over a local nonarchimedean divisionalgebra is contained in [Td7].ReferencesThe references that we include here are directed more to an inexperienced readerin the field of the representation theory than to the experienced one.

This is thereason that we have classified them into several groups (see below). We have triedto avoid too many very technical references, which are very common in the field.We include a number of expository and survey papers (they also omit many techni-calities).

A more demanding reader can also find a choice of relevant references forfurther reading. We have omitted a number of important references in order not toconfuse a reader who is not very familiar with the field.

AN EXTERNAL APPROACH TO UNITARY REPRESENTATIONS35Let us explain our classification of the references into several groups. The clas-sification is not very rigid.

To each reference we have attached a superscript thatindicates the group where it belongs. Here is a description of the groups.sur denotes the group of survey and expository papers.gen denotes the group of general references.GL (n) denotes the group of papers that are directly related to the topic of ourpaper.

They are very useful for the further understanding of the topicsdiscussed in our paper. The complete proofs that are omitted in this papercan be found in the papers from this group.his denotes the group of the historically important references for the topic ofthis paper.Lan denotes the group of references directly related to the Langlands programand groups GL(n).

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(4) 13 (1980),165–210.Department of Mathematics, University of Zagreb, Bijeniˇcka 30, 41000 Zagreb,CroatiaCurrent address: Sonderforschungsbereich 170, Geometrie und Analysis, Mathematisches In-stitut Bunsenstr. 3-5, D-3400 G¨ottingen, GermanyE-mail address: tadic@cfgauss.uni-math.gwdg.de

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