Polarization and unitary representations of solvable Lie groups. Appendix by Calvin C. Moore. (English) Zbl 0233.22005

Auslander, Louis; Kostant, Bertram

Invent. Math. 14, 255-354 (1971).

In this important, long awaited and lengthy paper the authors give proofs of their assertions announced in Bull. Am. Math. Soc. 73, 692–695 (1967; Zbl 0203.03302), and expand upon the techniques they require. Their work provides a satisfactory classification of the equivalence classes of irreducible unitary representations of solvable (real) Lie groups of type I. The paper has an informative introduction of which this review is in large part a paraphrase.
If \(G\) is a (real) Lie group (not necessarily connected) \(G^{\wedge}\) denotes the set of all equivalence classes of irreducible unitary representations of \(G\). The authors’ principal concern is with the case when \(G\) is connected, simply connected and solvable. Their main results then (1) give a simple necessary and sufficient condition for \(G\) to be of type I, (2) determine \(G^{\wedge}\) when \(G\) is of type I, (3) construct elements of \(G^{\wedge}\) which, when \(G\) is of type I, exhaust \(G^{\wedge}\).
These results generalize those of A. A. Kirillov for connected and simply connected nilpotent Lie groups [Russ. Math. Surv. 17, No. 4, 53–104 (1962); translation from Usp. Mat. Nauk 17, No. 4(106), 57–110 (1962; Zbl 0106.25001)], and those of P. Bernat, and of L. Auslander and C. C. Moore. For the Lie group \(G\) let \(\mathfrak g\) be the Lie algebra; and let \(\mathfrak g'\) be the real dual of \(\mathfrak g\). Regard \(\mathfrak g'\) as a \(G\)-module with respect to the coadjoint representation (the contragredient of the adjoint representation). If \(G\) is connected, simply connected and nilpotent, Kirillov has shown that \(G^{\wedge}\) is, in a natural way, in a one-to-one correspondence with the set, \(\mathfrak g/G\), of all orbits of \(G\) operating in \(\mathfrak g'\). Moreover, given \(g\) in \(\mathfrak g'\), the element \(\Pi_0\) in \(G^{\wedge}\) corresponding to the orbit \(O = G\cdot g\) was explicitly constructed by him as an induced representation, \(\mathrm{ind}_G \xi\), where \(\xi\) is a character (a one dimensional representation) of a subgroup \(H\) of \(G\) whose Lie algebra \(\mathfrak h\) is maximal among those having the property that \(g\) vanishes on the commutator \([\mathfrak h,\mathfrak h]\) of \(\mathfrak h\). The character \(\xi\) is the unique such whose differential is the restriction of \(2\pi ig\) to \(\mathfrak h\). One of the many remarkable aspects of Kirillov’s theorem is that \(\pi_0 = \mathrm{ind}_G \xi\) is independent of the choice of \(H\) among candidates with the properties just described, and this enables one to associate \(\pi_0\) with the orbit \(O\). For example, if \(G\) is the Heisenberg group and \(g\) does not vanish on \([\mathfrak g,\mathfrak g]\) then the unique equivalence for the two choices of \(H\) is provided by the Fourier transform.
B. Kostant has developed a general theory (called a quantization theory) of obtaining unitary representations of an arbitrary Lie group \(G\) from symplectic manifolds on which \(G\) operates. The first announcement of this appeared in [Proc. United States-Japan Semin. Differ. Geom., Kyoto 1965, 71 (1966; Zbl 0141.02701)] and various unpublished courses have been given (notably at M. I. T. in 1969); part of an exposition is published in [C. T. Taam (ed.), Lectures in modern analysis and applications. III, Lect. Notes Math. 170, 87–208 (1970; Zbl 0223.53028); see also Kostant’s lecture at the International Congress of Mathematicians, Nice1970, Vol. 2, 395–400 (1971; Zbl 0233.22006)]; some similar ideas have been advanced in a different context by J.-M. Souriau [Structure des systèmes dynamiques, Maîtrises de mathématiques. Paris: Dunod (1970; Zbl 0186.58001)].
One of the ingredients of Kostant’s theory, required in order to ensure irreducibility of representations constructed, is the notion of polarization of a symplectic manifold; this may be applied to the orbits of \(G\) in \(\mathfrak g'\) which are naturally symplectic manifolds. If \(O = G\cdot g\) then a \(G\)-invariant polarization of \(O\) is provided by a polarization at \(g\). This last is an essential notion of the article. A polarization at \(g\) in \(\mathfrak g'\) is a subalgebra \(\mathfrak h\) of the complexification \(\mathfrak g_{\mathbb C}\) of \(\mathfrak g\) with the properties
(1) \(\mathfrak h\) is maximal such that \(g\) vanishes on its commutator \([\mathfrak h,\mathfrak h]\),
(2) \(\mathfrak h + \bar{\mathfrak h}\) is a subalgebra of \(\mathfrak g_{\mathbb C}\) where \(\bar{\mathfrak h}\) is the complex conjugate and,
(3) \(\mathfrak h\) is stable under \(\mathrm{Ad }G\), where \(G_g\) is the isotropy group of \(G\) at \(g\) with respect to the coadjoint representation \(\mathrm{Ad}'\).
A polarization \(\mathfrak h\) at \(g\) is said to be defined over \(\mathbb R\) in case \(\mathfrak h = \bar{\mathfrak h}\).
In the general case the authors define a real non-singular symmetric bilinear form \(S_g\) on \(\mathfrak e/ \mathfrak d\) where \(\mathfrak e = (\mathfrak h + \bar{\mathfrak h}) \cap \mathfrak g\) and \(\mathfrak d = \mathfrak h\cap \mathfrak g\) as follows. Let \(x^{\wedge}\) in \(\mathfrak e/ \mathfrak d\) denote the image of an element \(x\) of \(\mathfrak e\) by the quotient map \(\mathfrak e \to\mathfrak e/ \mathfrak d\); define the nonsingular alternating bilinear form \(B_g^{\wedge}\) on \(\mathfrak e/ \mathfrak d\) by \(B_g^{\wedge}(x^{\wedge}, y^{\wedge}) = \lbrack g,[y,x]\rbrack\), here \(\lbrack\cdot,\cdot\rbrack\) is pairing of \(\mathfrak g'\) with \(\mathfrak g\). Define \(j\) in \(\text{End}(\mathfrak e/ \mathfrak d)_{\mathbb C}\) by \(j = -i\) on \(\mathfrak h/ \mathfrak d_{\mathbb C}\) and \(j=i\) on \(\bar{\mathfrak h}/ \mathfrak d_{\mathbb C}\) so \(j^2 = - \mathrm{id}\); then define \(S_g(x^{\wedge},y^{\wedge}) = B_g^{\wedge} (jx^{\wedge},y^{\wedge})\). The polarization \(\mathfrak h\) is called positive if \(S_g\) is positive definite. The linear functional \(g\) is called integral, with respect to \(G\), if there exists a character \(\eta_g\) on \(G\) (which may not be connected even if \(G\) is connected and simply connected) whose differential is the restriction of \(2\pi ig\) to the Lie algebra of \(G_g\).
This terminology arises because Kostant’s general theory shows that, when \(G\) is connected and simply connected, \(g\) is integral if and only if the cohomology class in \(H^2(0, \mathbb R)\) defined by the symplectic structure on \(O=G\cdot g\) lies in the image of the natural map \(H^2(0,\mathbb Z) \to H^2(0,\mathbb R)\); the symplectic structure is provided by the two-form \(\omega_f(\sigma_fx,\sigma_fy) = \lbrack f,[x,y]\rbrack = B_f(y,x)\) where \(\mathfrak g_f = \{x\in\mathfrak g\mid B_f(x,y) = 0\) for all \(y\) in \(\mathfrak g\}\) and \(0\rightarrow \mathfrak g_f\rightarrow \mathfrak g\rightarrow T_f(O)\rightarrow 0\) is an exact sequence with \(T_f(0)\) being the tangent space to the orbit \(O\) at the point \(f\). Take \(G\) to be an arbitrary Lie group, not necessarily connected, and let \(g\) in \(\mathfrak g'\) be integral. Let \(\eta_g\) be a character at \(g\) and \(\mathfrak h\) a polarization at \(g\). Let \(E_0\) and \(D_0\) be the connected Lie subgroups of \(G\) with Lie algebras \(\mathfrak e\) and \(\mathfrak d\) respectively; put \(E=G_gE_0\) and \(D=G_gD_0\). Suppose that the polarization \(\mathfrak h\) at \(g\) satisfies the Pukanszky condition, namely that the orbit \(E\cdot g\) be closed in \(\mathfrak g'\); \(\eta_g\) extends to a unique character \(\chi_g\) of \(D\) whose differential is the restriction of \(2\pi ig\) to \(\mathfrak d\). Set \(X=E/D=E_0D/\) which is clearly connected. The non-singular alternating bilinear form \(B_g\) on \(\mathfrak e/ \mathfrak d\), invariant as it is under the action of \(D\), induces a measure \(\mu_X\) on \(X\) which is invariant under the action of \(E\). Let \(\mathcal H(E,\chi_g)\) be the Hilbert space formed from the equivalence classes of all measurable functions \(\Phi\) on \(E\) such that \(\int \vert \Phi\vert^2 < \infty\) and for all \(a\) in \(E\) \(\Phi(ab) = \chi_g(b)^{-1}\Phi(a)\); it is the Hilbert space of the unitary representation \(\mathrm{ind}_E\chi\), induced in the manner of G. W. Mackey, whose action is for \(a\) and \(b\) in \(E\) and \(\Phi\) in \(\mathcal H(E,\chi_g)\), given by \(((\mathrm{ind}_E \chi(a)) \Phi)b = \Phi(a^{-1} b)\) – with the usual looseness of language.
The space \(\mathcal H(E,\eta_g,\mathfrak h)\) of all smooth functions in \(\mathcal H(E,\chi_g)\) such that for every \(z\) in \(\mathfrak h \Phi\cdot z = 2\pi i \lbrack g, z\rbrack\), where \(\Phi\cdot z= \Phi\cdot x+i\Phi\cdot y\) if \(z = x + iy\) and \((\Phi\cdot x)(e) = \frac{d}{dt} \Phi(e \exp(-tx)) \vert_{t =0}\), forms a closed subspace of the Hilbert space \(\mathcal H(E,\chi_g)\) which is stable under \(\mathrm{ind}_E\chi_g\); this defines a subrepresentation \(\mathrm{ind}_E(\eta_g,\mathfrak h\) of \(\mathrm{ind}_E\chi_g\).
The unitary representation of \(G\) corresponding to an integral \(g\) in \(\mathfrak g'\) and any polarization \(\mathfrak h\) at \(g\) satisfying the Pukanszky condition is defined as \(\mathrm{ind}_G(\eta_g,\mathfrak h) = \mathrm{ind}_G(\mathrm{ind}_E(\eta_g,\mathfrak h))\), that is the result of a further induction from \(E\) to \(G\). If \(G\) is connected, simply connected and nilpotent then any \(g\) is integral, \(\eta_g\) is unique and any polarization \(\mathfrak h\) satisfies the Pukanszky condition; further if \(h\) is defined over \(\mathbb R\) then \(\mathrm{ind}_G(\eta_g,\mathfrak h)\) is just the irreducible unitary representation of \(G\) assigned to the orbit \(\mathfrak h\) by Kirillov.
It is additionally shown by Auslander and Kostant that \(\mathrm{ind}_G(\eta_g,\mathfrak h)\) is in fact up to equivalence independent over all positive polarizations. If \(G\) is the Heisenberg group then, for suitable positive \(\mathfrak h\), \(\mathrm{ind}_G(\eta_g,\mathfrak h)\) is the Segal-Bargmann representation familiar in quantum mechanics. The equivalence instead of being defined by the Fourier kernel is given by reproducing kernels of the type discussed by V. Bargmann [Commun. Pure Appl. Math. 14, 187–214 (1961; Zbl 0107.09102)]; this is considered in detail in the paper.
To progress from nilpotent to solvable groups presents the authors with several difficulties. To begin with solvable Lie groups need not be of type I as well known examples due to Mautner and Dixmier show. Auslander and Kostant, however, prove that if \(G\) is connected, simply connected and solvable then it is of type I if and only if (1) every \(g\) in \(\mathfrak g'\) is integral and (2) every orbit \(\mathfrak h\) is the intersection of a closed and an open set in \(\mathfrak g'\). A weaker form of type theorem than this is by J. Brezin [Mem. Am. Math. Soc. 79, 122 p. (1968; Zbl 0157.36603)]. Connected, simply connected, solvable and exponential Lie groups were known to be of type I [O. Takenouchi, Math. J. Okayama Univ. 7, 151–161 (1957; Zbl 0080.02302)]; here exponential means that the exponential map is surjective or equivalently a diffeomorphism [J. Dixmier, Bull. Soc. Math. Fr. 85, 113–121 (1957; Zbl 0077.25203)]. P. Bernat extended Kirillov’s results to such groups by finding a suitable polarization \(\mathfrak h\) over \(\mathbb R\) for each orbit so that the natural one-to-one correspondence of \(\mathfrak g'/G\) with \(G^\wedge\) might be realized by an \(\mathrm{ind}_G(\eta_g,\mathfrak h)\) for each orbit [Ann. Sci. Éc. Norm. Supér. (3) 82, 37–99 (1965; Zbl 0138.07302)]. L. Pukanszky determined all possible polarizations defined over \(\mathbb R\) which yield this representation [Trans. Am. Math. Soc. 126, 487–507 (1967; Zbl 0207.33605)]. For an arbitrary solvable Lie group, even of type I, representations of the form \(\mathrm{ind}_G(\eta_g,\mathfrak h)\) with \(\mathfrak h\) defined over \(\mathbb R\) no longer exhaust \(G^\wedge\); possibly the simplest,example, the four dimensional oscillator group, was explicitly calculated by R. F. Streater [Commun. Math. Phys. 4, 217–236 (1967; Zbl 0155.32503)]. The case \(\mathfrak h \ne \bar{\mathfrak h}\) thus has to be considered.
In addition the correspondence of \(G^\wedge\) with \(\mathfrak g'/G\) is not necessarily one-to-one since \(\eta_g\) is not necessarily unique for \(G_g\) may not be connected. If \(G\) is solvable let \(\mathfrak n\) be the maximal nilpotent ideal of \(\mathfrak g\). Since \(\mathfrak n\) is stable under \(\mathrm{Ad}(G)\), its dual \(\mathfrak n'\) is a \(G\)-module by contragredience. If \(g\) is in \(g'\) and \(\mathfrak h\) is a polarization at \(g\) then \(\mathfrak h\) is called strongly admissible in case \(\mathfrak h\cap \mathfrak n_{\mathbb C}\) is a polarization at \(f = g\vert \mathfrak n\) in \(\mathfrak n'\) which is stable under \(G_f\), the isotropy group of \(G\) at \(f\), which contains \(G_g\).
The generalization of Kirillov’s results due to Auslander and Kostant is given by the compendial Theorem: Let \(G\) be a connected, simply connected and solvable Lie group. Let \(g\) be an arbitrary element in \(\mathfrak g'\). Then there exists a strongly admissible positive polarization \(\mathfrak h\) at \(g\) and any such polarization satisfies the Pukanszky condition. Thus if \(g\) is integral and \(\eta_g\) is a character at \(g\) the unitary representation \(\mathrm{ind}_G(\eta_g,\mathfrak h)\) may be formed and is both irreducible and independent of which positive strongly admissible polarization at \(g\) is used. Moreover if \(G\) is of type I, so that every \(g\) in \(\mathfrak g'\) is integral, then every irreducible unitary representation is equivalent to a representation of this form.
Finally if \(G\) is of type I and \(\mathrm{ind}_G(\eta_{g_i}, h_i)\), \(i =1,2\), are two representations of this form then they are equivalent if and only if \(g_1\) and \(g_2\) lie on the same \(G\)-orbit and \(\eta_{g_1}\) corresponds to \(\eta_{g_2}\) under the isomorphism \(G_{g_1}\to G_{g_2}\) defined by an element \(a\) of \(G\) such that \(a\cdot g_1=g_2\).
Assume \(G\) to be of type I. To each orbit \(O\) in \(g'\) the authors associate a torus \(\mathcal L_\circ(O)\) as follows. Let \(g\) be in \(O\) and let \(\mathcal L_\circ(g)\) be the set of all characters at \(g\); then \(\mathcal L_\circ(g)\) has the structure of a torus of dimension the first Betti number of \(O\), for it is in a natural way a principal homogeneous space of \(G_g/(G_g)_\circ\) where the subscript \(\circ\) denotes the identity component; \(G_g/(G_g)_\circ\) is canonically isomorphic to the fundamental group \(\Pi_1(O)\) of \(O\). The action of \(G\) on \(O\) induces for any \(f\) and \(g\) in \(O\) an isomorphism \(\sigma_{fg}\) of \(\mathcal L_\circ(f)\) with \(\mathcal L_\circ(g)\); thus they define a torus \(\mathcal L_\circ(O)\) of dimension \(B_1(O)\) and for any \(f\) in \(O\) an isomorphism \(\sigma_f: \mathcal L_\circ(O)\to \mathcal L_\circ(f)\), so that for all \(f\) and \(g\) in \(O\) \(\sigma_{f,g}\sigma_f = \sigma_g\).
Then, by the main theorem, to any in \(\mathcal L_\circ(O)\) is unambiguously associated an irreducible unitary representation (equivalence class) \(\pi_\ell\) in \(G\) by putting \(\pi_\ell = \mathrm{ind}_G(\eta_g, \mathfrak h)\), where \(\eta_g = \sigma_g(\ell)\) and \(\mathfrak h\) is any strongly admissible positive polarization at \(g\).
These constructions, which are special cases of those figuring in the general quantization theory, lead to the following generalization of Kirillov’s result on the correspondence between \(G^\wedge\) and \(\mathfrak g'/G\): If \(G\) is a connected, simply connected solvable Lie group of type I the map from the union over all orbits \(O\) in \(\mathfrak g'\) of the associated tori \(\mathcal L_\circ(O)\) to \(G^\wedge\), given by \(\ell \to \pi_{ell}: \bigcup \mathcal L_\circ(O)\to G\) is a bijection. Thus their main results do the three things claimed for them.
The detailed proofs that knit together all this structure are presented in a manner designed to make the paper fairly self-contained. Nonetheless the Newlander-Nirenberg theorem is required; results of A. Weil [Introduction à l’étude des variétés kähleriennes. Paris: Hermann (1958; Zbl 0137.41103)] are used in relation to the space of smooth functions \(H(\eta_g ,h)\); algebraic facts concerning almost algebraic Lie groups are drawn from L. Auslander and J. Brezin, J. Algebra 8, 295–313 (1968; Zbl 0197.03002)] and most importantly G. W. Mackey’s induced representation theory is essential. The account of Mackey’s techniques that the- authors quote is that in the seminal memoir of L. Auslander and C. C. Moore [Mem. Am. Math. Soc. 62, 199 p. (1966; Zbl 0204.14202)] in which was determined which class R solvable Lie groups were type I and, when \(G\) was of class R and type I, \(G^\wedge\) was found (class R means that all the eigenvalues of the adjoint action of \(G\) are of modulus 1). Indeed Moore provides in an appendix to the paper under review a strengthening of a theorem of this last memoir which is crucial to Auslander and Kostant’s theorem on type.
However much of the technique of this paper of Auslander and Kostant will itself be of wider interest. The only criticism of the article possible appears to be that there are rather numerous small misprints; but since the paper necessarily has to be read closely perhaps they cause little trouble. An introductory lecture by B. Kostant has appeared in [Group Represent. Math. Phys., Lect. Notes Phys. 6, 237–253 (1970; Zbl 0229.20048)]. Extension of Auslander and Kostant’s theory to solvable Lie groups not necessarily of type I has been carried out by L. Pukanszky [Ann. Sci. Éc. Norm. Supér. (4) 4, 457–608 (1971; Zbl 0238.22010)]. It is even so that a book on the subject of solvable Lie groups’ representations has lately been written “Représentations des groupes de Lie résolubles” par P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Rais, et M. Vergne, Paris: Dunod (1972; Zbl 0248.22012).

Reviewer: Patrick D. F. Ion

MSC:

22E27	Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
81R05	Finite-dimensional groups and algebras motivated by physics and their representations
22D10	Unitary representations of locally compact groups
22E70	Applications of Lie groups to the sciences; explicit representations

Citations:

Zbl 0203.03302; Zbl 0106.25001; Zbl 0141.02701; Zbl 0223.53028; Zbl 0233.22006; Zbl 0186.58001; Zbl 0107.09102; Zbl 0157.36603; Zbl 0080.02302; Zbl 0077.25203; Zbl 0138.07302; Zbl 0207.33605; Zbl 0155.32503; Zbl 0137.41103; Zbl 0197.03002; Zbl 0204.14202; Zbl 0229.20048; Zbl 0238.22010; Zbl 0248.22012

References:

[1]	Auslander, L., Kostant, B.: Quantization and representations of solvable Lie groups. Bull. A.M.S.73, 692-695 (1967). · Zbl 0203.03302 · doi:10.1090/S0002-9904-1967-11829-9
[2]	Auslander, L., Moore, C. C.: Unitary representations of solvable Lie groups. Memoirs A.M.S.62 (1966). · Zbl 0204.14202
[3]	?, Brezin, J.: Almost algebraic Lie algebras. J. of Algebra8, 295-313 (1968). · Zbl 0197.03002 · doi:10.1016/0021-8693(68)90061-6
[4]	Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform I. Comm. Pure and Applied Math.14, 187-214 (1961). · Zbl 0107.09102 · doi:10.1002/cpa.3160140303
[5]	Bernat, M. P.: Sur les representations unitaires des groupes de Lie resolubles. Ann. Sci. Ecole Norm. Sup.82, 37-99 (1965). · Zbl 0138.07302
[6]	Brezin, J.: Unitary representation theory for solvable Lie groups. Memoirs A.M.S.79 (1968). · Zbl 0157.36603
[7]	Kirillov, A. A.: Unitary representations of nilpotent Lie groups. Uspehi, Mat. Nauk.17, 57-110 (1962). · Zbl 0106.25001
[8]	Kostant, B.: Quantization and unitary representations, p. 87-207, Lecture Notes in Mathematics.170. Berlin-Heidelberg-New York: Springer 1970. · Zbl 0223.53028
[9]	Pukanszky, L.: On the theory of exponential groups. Trans. A.M.S.126, 487-507 (1967). · Zbl 0207.33605 · doi:10.2307/1994311
[10]	?: Lecons sur les representations des groupes. Monographes Soc. Math. de France. Paris: Dunod 1967.
[11]	Rosenlicht, M.: A remark on quotient spaces. Anais da Academic Brasileina de Ciencias35, 487-489 (1963). · Zbl 0123.13804
[12]	Streater, R.F.: The representations of the oscillator group. Comm. Math. and Phys.4, 217-236 (1967). · Zbl 0155.32503 · doi:10.1007/BF01645431
[13]	Weil, A.: Varietes Kahleriennes, Actualites Scientific et Industrielle, 1267. Paris: Hermann 1958.
[14]	Effros, E.: A decomposition theorem for representations of aC *-algebra. Trans. A.M.S.107, 83-106 (1963). · Zbl 0113.09602