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UNIVERSIT´E DE POITIERS on Representations of Simply UNIVERSITE´ DE POITIERS On representations of simply connected nilpotent and solvable Lie groups G´erardGr´elaud Universit´ede POITIERS UMR CNRS 6086 – Laboratoire de Math´ematiqueset Applications SP2MI - T´el´eport 2 - Boulevard Marie et Pierre Curie BP 30179 86962 FUTUROSCOPE CHASSENEUIL Cedex T´el: 05 49 49 69 03 e-mail :[email protected] On representations of simply connected nilpotent and solvable Lie groups Introduction In these notes following a course I gave during a visit at Pondicherry University in 1992, I write the main results of the theory of representations of simply connected nilpotent Lie groups (the Kirillov’s theory), and some generalizations to simply connected solvable Lie groups. After the basic properties of unitary representations of locally compact groups, especially construction of induced representations and “Mackey’s machine” (Sec- tion 1), I state the classical results on Lie groups and Lie algebras (section 2). In section 3, I give the constuction of polarizations in solvable Lie algebras and in section 4 the description of the the dual space of connected nilpotent Lie groups, using the famous orbit method of A.A. Kirillov. For general solvable Lie groups, I write in section 5 the construction of irreducible holomorphical induced representations. This shows the use of complex polarizations. The section 6 is devoted to a computation of the Kirillov’s character formula and the Plancherel formula for nilpotent Lie groups and also a generalization to some homogeneous spaces of nilpotent Lie groups. In the last section, I write a survey (and some proofs) of the main results of L. Pukanszky about solvable Lie groups. The results given in these notes have been discovered during the years 1960- 1980. The most important contributions are due to A.A Kirillov, J. Dixmier, P. Bernat, M. Vergne, M. Duflo, L. Pukanszky, L. Auslander, B. Kostant. It is an introduction for topical research. At the present time, mathematicians are concerned with problems about algebraic groups, homogeneous spaces of Lie groups (Plancherel formulas, differential operators ::: ). The results in these notes are well known and the proofs given are not new, except for a part of section 6. I have used lectures or books among others, M. Ra¨ıs [23], P. Bernat and Al. [3], A.A. Kirillov [14], P. Torasso [24], L. Corwin and F. Greenleaf [5]. I have given many examples in low dimensional Lie groups, and many results (more or less easy) are left in exercises. I am grateful to Miss V. Gayatri, who attended my lectures, for a carefull reading of a first version of these notes and a lot of remarks and corrections. 1. Basic facts on unitary representations In this section G is a locally compact group which is separable. We give, almost without proofs (but with references to litterature), classical results on representation theory. A good introduction for this theory is G.W. Mackey [16] or J. Dixmier [7]. For an abstract, you could read the chapter 1 of [1]. An unitary representation ¼ of G in an Hilbert space H is an homomorphism from G into the group U(H) of unitary operators on H and such that for every v 2 H the map G ¡! H g à ¼(g):v is continuous. We assume in these notes that the Hilbert spaces are separable which is not a serious restriction since it is known that an irreducible unitary representation of a connected Lie group has a separable space. It is also a well known result (see [1]), that the continuity of the above map follows from the measurability of the map g ¡!< ¼(g)v; w > for all v and w in H. SOME BASIC DEFINITIONS 1.1.– Let ¼ be an unitary representation of G in H and let V ½ H a closed subspace which is invariant by every operator ¼(x) for x 2 G. This defines a new representation (¼; V) in V. We say that it is a subrepresentation of ¼. It is immediate that V? is also ¼(G)¡invariant, so it defines an other subrepresentation of ¼. 1.2.– A representation ¼ is said to be irreducible if it has no subrepresentation other than the trivial ones. There is an other way to say this property: let ¼ and ¼0 two representations in H and H0 respectively. Let T be a bounded linear operator from H into H0. The operator T is said to be an intertwining operator between ¼ and ¼0 if we have the relation T ± ¼(x) = ¼0(x) ± T for all x 2 G. Usualy one denotes by Hom(¼; ¼0) the space of intertwining operators and we say that ¼ and ¼0 are unitarly equivalent if there is a bijective isometry in Hom(¼; ¼0) and this is write ¼ ' ¼0. The relation with irreducibillity is described by the following lemma. Lemma 1.1 — ( Schur Lemma cf. [17] p. 14). An unitary representation ¼ in H is irreducible if and only if Hom(¼; ¼)= CIH. It is easy to see that the equivalence classes of irreducible representations is a set denoted by Gb. 1.3.– If G is an abelian group the Schur lemma proves that Gb is the set of characters of G. In fact, if ¼ 2 Gb, for x 2 G; ¼(x) 2 Hom(¼; ¼). So, by Schur On unitary representations – 3 – lemma ¼(x) = Â(x)Id where Â(x) is a complex number and it is clear that Â(x) is a character. 1.4.– If K ½ G is a closed subgroup, the operators ¼(k) for k 2 K define a representation of K in H. We say that it is the restriction of ¼ to K and note ¼jK this representation. 1.5.– Direct sum. Let ¼1; : : : ; ¼n be representations in Hilbert spaces H1;:::; Hn respectively. It is clear that one defines a new unitary representation of G in n H = ©i=1Hi by letting ¼(x)(v1 + ¢ ¢ ¢ + vn) = ¼1(v1) + ¢ ¢ ¢ + ¼n(vn) for all x 2 G and vi 2 Hi ; i = 1; : : : ; n. ¼ is said to be the direct sum of the ¼i. 1.6.– Hilbert integral. We only give a weak definition. Let X be a separable locally compact topological space, H an Hilbert space, G a locally compact group and for all x 2 X, ¼x a representation of G in H. We denote by ¹ a Radon positive 2 measureR on X. We consider the space V = L (X; H) of all functions ' on X such 2 that X k'k d¹(x) < 1. We suppose that for all g 2 G;' 2 V; à 2 V , the map x ¡!< ¼x(g)'(x) ;Ã(x) > is measurable. We define a new representation ½ by the formula £ ¤ ½(g)f (x) = ¼x(g)f(x) f 2 H; x 2 X R © x This representation is denoted ½ = X ¼ d¹(x). If º is a positive measure equivalent to ¹, we have ½¹ ' ½º . 1.7.– EXAMPLES. (a) Let G = R and H = C. Then U(H) = U = fz 2 C ; jzj = 1g. So every unitary representation of H is a continuous character  of G (see 1.3) and it is well known that there exists an y 2 R such that Â(x) = eixy, for all x 2 G. (b) Let G = O(n; R); H = Cn and let ¼ be the natural injection of G into the group U(H). Then ¼ is a unitary representation of G. (c) The regular representation. Let G be any separable locally compact group. We denote by ¹ its left Haar measure. Consider H = L2(G; ¹). For g 2 G define the operotor ¸(g) by [¸(g):f](x) = f(g¡1x). We have k¸(g):fk = kfk, so ¸(g) is a unitary operator. Moreover, the map g ¡! ¸(g):f of G into H is continuous. This is clear when f is a continuous function with compact support, from which one deduces the case of other f 2 L2(G; ¹) by an easy exercise of measure theory. The representation ¸ is called the left regular representation. The right regular 1 2 representation is defined by [½(g):f](x) = ∆(g) 2 f(xg), for f 2 L (G; ¹) where ∆ is the modular function of G and ¹ is the left Haar measure on G. – 4 – G.Gr´elaud 1.8.– INDUCED REPRESENTATIONS. We now describe the most important tool of the theory to build representations of G from representations of closed subgroups. We start with a particular case which is the only necessary for the class of groups we study in these notes. (a) Let G be a locally compact group and consider H a closed subgroup of G. Let σ be a representation of G in an Hilbert space H. Suppose that there is a left-invariant measure º on the locally compact quotient space G=H. We remark that this is always the case if G and H are unimodular. Then we can construct a new Hilbert space Hσ: we first denote by Cσ the set of all continuous functions from G into H such that (i) f(gh) = σ(h)¡1f(g) for all g 2 G and h 2 G ; Z (ii) kf(g)k2dº(g) < +1 G=H The function g ¡! kf(g)k2 is constant on each left coset of G=H, so the integral in (ii) exists and we define an inner product on Cσ by the formula Z < f; f 0 >= < f(g); f 0(g) > dº(g) (1) G=H We note Hσ the completion of Cσ for this inner product. Then, we define the G induced representation ¼ = IndH σ by left action of G on functions of Hσ ¼(x):f(y) = f(x¡1y) for all (x; y) 2 G £ G.
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