Quantization and unitary representations. I: Prequantization.

*(English)*Zbl 0223.53028
Lectures in modern analysis and applications. III, Lect. Notes Math. 170, 87-208 (1970).

[This article was published in the book announced in this Zbl 0213.00101.]

As the author says “this paper is the first part of a two part paper dealing with the question of setting up a unified theory of unitary representations of connected Lie groups” in the spirit of the program briefly sketched by the author in a short announcement, entitled: Orbits, symplectic structure and representation theory [Proc. United States-Japan Semin. Differ. Geom., Kyoto 1965, 71 (1966)]. This general approach to representation theory is essentially based an the use of symplectic manifolds and quantization. Without going into the matter here, it is worth mentioning that making use of this theory L. Auslander and the author have succeeded in generalizing Kirillov’s result on nilpotent groups, obtaining a complete description of the set of all unitary equivalence classes of irreducible unitary representations of a connected, simply connected type I solvable Lie group and also a criterion for being of type I, expressed in these terms. [See L. Auslander and the author, Bull. Am. Soc. 73, 692–695 (1967; Zbl 0203.03302); Invent. Math. 14, 255–354 (1971; Zbl 0233.22005)]. The present paper is devoted to the differential geometry foundations of the theory and considers only pre-quantization. We shall give in the sequel an outline of results.

Let \(\mathcal L_c(X)\) be the set of all equivalence classes \(\ell=[(L,\alpha)]\) of line bundles with connection over a manifold \(X\), which admit an \(\alpha\)-invariant Hermitian structure. The group \(\mathrm{Diff}(X)\) operates on \(\mathcal L_c(X)\) by defining: \(g\cdot \ell=(g^{-1}* \ell\) for \(g\in \mathrm{Diff}(X)\), \(\ell \in \mathcal L_c(X)\). Let \(D_\ell(X)\subseteq \mathrm{Diff}(X)\) be the isotropy group at \(\ell= [(L,\alpha)]\) and let \(E(L,\alpha)\) denote the group of all diffeomorphisms of \(L\) which are maps of line bundles with connection and preserve the Hermitian structure. The main result of §1, Theorem 1.13.2, associates to every line bundle with connection \((L,\alpha)\) an exact sequence of groups \[ 1\to T\to E(L,\alpha)\to D_\ell(X)\to 1 \] giving \(E(L,\alpha)\) as a central extension of \(D_\ell (X)\) by the circle group \(T\).

Now, for any closed 2-form \(\omega\) on \(X\) let \(\mathcal L_c(X,\omega)\) be the set of all \(\ell = [(L,\alpha)]\in\mathcal L_c(X)\) such that \(\omega= \mathrm{curv} (L,\alpha)\), i.e. is the curvature form of the connection \(\alpha\). Then \(\mathcal L_c(X,\omega)\) is not empty if and only if the de Rham cohomology class \([\omega] \in H^2(X,R)\) is integral (Proposition 2.2.1).

An important point in §2 is Theorem 2.5.1 which states that the character group \(\pi_1^*(X)\) of the fundamental group \(\pi_1(X)\) of \(X\) operates in a simply transitive way on \(\mathcal L_c(X,\omega)\).

In §3 the infinitesimal version of Theorem 1.13.1 is established. Theorem 3.3.1 explicitly gives \(\underline{e}(L,\alpha)\), the Lie algebra of connection preserving vector fields, and associates to every \([(L,\alpha)]\in \mathcal L_c(X,\omega)\) an exact sequence of Lie algebras

\[ 0\to R\to \underline{e}(L,\alpha)\to \mathcal A(X,\omega)\to 0 \] making \(\underline{e}(L,\alpha)\) a central extension of the Lie algebra of Hamiltonian vector fields, \(\mathcal A(X,\omega)\) by \(R\). Assume that \((X,\omega)\) is a symplectic manifold. Then \(R(X)\), the space of all real valued smooth functions on \(X\), is a Lie algebra under Poisson bracket and one has an exact sequence of Lie algebras \[ 0\to R\to R(X)\to \mathcal A(X,\omega)\to 0. \] Then, if in addition \([\omega]\) is integral, one has a commutative diagram

\[ *** \]

for any \([(L,\alpha)]\in \mathcal L_c(X,\omega)\).

In §4 a Lie algebra isomorphism \(\tilde\delta: R(X)\to \underline{e}(L,\alpha)\) is constructed which preserves the commutativity of the above diagram (Theorem 4.2.1). A crucial point is that \(\tilde\delta\) gives rise to a representation \(\delta\) of \(R(X)\) on the space \(S(X,L)\) of smooth sections of the line bundle \(L\). This assignment, mapping functions to operators, which is called “pre-quantization” is the first step in the quantization procedure. The second step, involving the notion of a polarization of the symplectic manifold, is not considered in the paper under review. Now, assume that a connected, simply connected Lie group \(G\) with Lie algebra \(\mathcal G\) operates smoothly on \(X\) by a homomorphism \(\sigma: G\to \mathrm{Diff}(X)\). One says that \((X,\omega)\) is a \(G\)-symplectic space if \(\omega\) is \(G\)-invariant and is a \(G\)-strongly symplectic space if \(\text{Im}(d\sigma) \subseteq\mathcal A(X,\omega)\), so that one has a homomorphism \(d\sigma: Q\to \mathcal A(X,\omega)\). A homogeneous \(G\)-strongly symplectic space \((X,\omega)\) is called a Hamiltonian \(G\)-space if \(d\sigma: \mathcal G\to\mathcal A(X,\omega)\) can be lifted to a homomorphism \(\lambda: \mathcal G\to R(X)\), so that one has a commutative diagram

\[ *** \]

The significance of the notion of Hamiltonian \(G\)-space is pointed out by Theorem 5.1.1: Let \((X,\omega,\lambda)\) be a Hamiltonian \(G\)-space with \([\omega]\in H^2(X, R)\) integral and let \([(L,\alpha)]\in \mathcal L_c(X, \omega)\). Then, there is a unique lifting \(\tilde\sigma: G\to E(L,\alpha)\) of \(\sigma\) such that \(d\tilde\sigma = \tilde\delta\circ\lambda\).

A vital example of a Hamiltonian \(G\)-space is produced in Theorem 5.3.1: Let \(\underline{0}\) denote a \(G\)-orbit in \(\mathcal G'\), the dual vector space to \(\mathcal G\), regarded as a \(G\)-module with respect to the coadjoint representation, and let \(f\in \underline{0}\). Using the bilinear form on \(\mathcal G\), \(B_f(x,y) = \langle f,[y,x]\rangle\) one may define a 2-form \(\omega_{\underline{0}}\) on \(\underline{0}\). On the other hand the formula, \(\lambda_{\underline{0}}(x) f = \langle f,x \rangle\), \(f\in\underline{0}\), \(x\in\mathcal G\), defines a homomorphism \(\lambda_{\underline{0}}: \mathcal G\to R(\underline{0})\). The triple \((\underline{0},\omega_{\underline{0}},\lambda_{\underline{0}})\) is a Hamiltonian \(G\)-space. Moreover, any manifold which covers a \(G\)-orbit in \(\mathcal G'\) has the structure of a Hamiltonian \(G\)-space.

Conversely, Theorem 5.4.1 asserts that the set of \(G\)-orbits in \(\mathcal G'\) are universal in the sense that any Hamiltonian \(G\)-space \((X,\omega,\lambda)\) covers a unique orbit not only as a manifold but as a Hamiltonian \(G\)-space.

The paper ends up with Theorem 5.7.1, classifying all line bundles with connection over a Hamiltonian \(G\)-space \((X,\omega,\lambda)\) having \(\omega\) as curvature: Let \((X,\omega,\lambda)\) be a Hamiltonian \(G\)-space and let \(G_p\) be the isotropy group at \(p\in X\). If \(\underline{0}\) is the \(G\)-orbit in \(\mathcal G'\) associated with \((X,\omega,\lambda)\) and \(f\in \underline{0}\), let \(G_f\) be the isotropy group at \(f\), with Lie algebra \(\mathcal G_f\). \(G_p^\#\) denotes the set of all characters \(\Lambda: G_p\to T\) having as its differential \(2\pi i f| \mathcal G_f\). Then the cohomology class \([\omega]\in H^2(X,R)\) is integral if and only if \(G_p^\#\) is not empty and in this case \(\pi_1^*(X)\) acts in a simply transitive way on \(G_p^\#\). Moreover, to each \(\ell= [(L,\alpha)]\in \mathcal L_c (X,\omega)\) one associates an element \(\Delta^\ell\in G_p^\#\) and the correspondence \(\mathcal L_c (X,\omega)\to G_p^\#\) given by \(\ell\to\Lambda^\ell\) is a \(\pi_1^*(X)\) map and hence is a bijection. As a corollary, taking \((X,\omega,\lambda)=(\underline{0},\omega_{\underline{0}},\lambda_{\underline{0}})\) one obtains a result generalizing the Borel-Weil theorem in the compact case.

As the author says “this paper is the first part of a two part paper dealing with the question of setting up a unified theory of unitary representations of connected Lie groups” in the spirit of the program briefly sketched by the author in a short announcement, entitled: Orbits, symplectic structure and representation theory [Proc. United States-Japan Semin. Differ. Geom., Kyoto 1965, 71 (1966)]. This general approach to representation theory is essentially based an the use of symplectic manifolds and quantization. Without going into the matter here, it is worth mentioning that making use of this theory L. Auslander and the author have succeeded in generalizing Kirillov’s result on nilpotent groups, obtaining a complete description of the set of all unitary equivalence classes of irreducible unitary representations of a connected, simply connected type I solvable Lie group and also a criterion for being of type I, expressed in these terms. [See L. Auslander and the author, Bull. Am. Soc. 73, 692–695 (1967; Zbl 0203.03302); Invent. Math. 14, 255–354 (1971; Zbl 0233.22005)]. The present paper is devoted to the differential geometry foundations of the theory and considers only pre-quantization. We shall give in the sequel an outline of results.

Let \(\mathcal L_c(X)\) be the set of all equivalence classes \(\ell=[(L,\alpha)]\) of line bundles with connection over a manifold \(X\), which admit an \(\alpha\)-invariant Hermitian structure. The group \(\mathrm{Diff}(X)\) operates on \(\mathcal L_c(X)\) by defining: \(g\cdot \ell=(g^{-1}* \ell\) for \(g\in \mathrm{Diff}(X)\), \(\ell \in \mathcal L_c(X)\). Let \(D_\ell(X)\subseteq \mathrm{Diff}(X)\) be the isotropy group at \(\ell= [(L,\alpha)]\) and let \(E(L,\alpha)\) denote the group of all diffeomorphisms of \(L\) which are maps of line bundles with connection and preserve the Hermitian structure. The main result of §1, Theorem 1.13.2, associates to every line bundle with connection \((L,\alpha)\) an exact sequence of groups \[ 1\to T\to E(L,\alpha)\to D_\ell(X)\to 1 \] giving \(E(L,\alpha)\) as a central extension of \(D_\ell (X)\) by the circle group \(T\).

Now, for any closed 2-form \(\omega\) on \(X\) let \(\mathcal L_c(X,\omega)\) be the set of all \(\ell = [(L,\alpha)]\in\mathcal L_c(X)\) such that \(\omega= \mathrm{curv} (L,\alpha)\), i.e. is the curvature form of the connection \(\alpha\). Then \(\mathcal L_c(X,\omega)\) is not empty if and only if the de Rham cohomology class \([\omega] \in H^2(X,R)\) is integral (Proposition 2.2.1).

An important point in §2 is Theorem 2.5.1 which states that the character group \(\pi_1^*(X)\) of the fundamental group \(\pi_1(X)\) of \(X\) operates in a simply transitive way on \(\mathcal L_c(X,\omega)\).

In §3 the infinitesimal version of Theorem 1.13.1 is established. Theorem 3.3.1 explicitly gives \(\underline{e}(L,\alpha)\), the Lie algebra of connection preserving vector fields, and associates to every \([(L,\alpha)]\in \mathcal L_c(X,\omega)\) an exact sequence of Lie algebras

\[ 0\to R\to \underline{e}(L,\alpha)\to \mathcal A(X,\omega)\to 0 \] making \(\underline{e}(L,\alpha)\) a central extension of the Lie algebra of Hamiltonian vector fields, \(\mathcal A(X,\omega)\) by \(R\). Assume that \((X,\omega)\) is a symplectic manifold. Then \(R(X)\), the space of all real valued smooth functions on \(X\), is a Lie algebra under Poisson bracket and one has an exact sequence of Lie algebras \[ 0\to R\to R(X)\to \mathcal A(X,\omega)\to 0. \] Then, if in addition \([\omega]\) is integral, one has a commutative diagram

\[ *** \]

for any \([(L,\alpha)]\in \mathcal L_c(X,\omega)\).

In §4 a Lie algebra isomorphism \(\tilde\delta: R(X)\to \underline{e}(L,\alpha)\) is constructed which preserves the commutativity of the above diagram (Theorem 4.2.1). A crucial point is that \(\tilde\delta\) gives rise to a representation \(\delta\) of \(R(X)\) on the space \(S(X,L)\) of smooth sections of the line bundle \(L\). This assignment, mapping functions to operators, which is called “pre-quantization” is the first step in the quantization procedure. The second step, involving the notion of a polarization of the symplectic manifold, is not considered in the paper under review. Now, assume that a connected, simply connected Lie group \(G\) with Lie algebra \(\mathcal G\) operates smoothly on \(X\) by a homomorphism \(\sigma: G\to \mathrm{Diff}(X)\). One says that \((X,\omega)\) is a \(G\)-symplectic space if \(\omega\) is \(G\)-invariant and is a \(G\)-strongly symplectic space if \(\text{Im}(d\sigma) \subseteq\mathcal A(X,\omega)\), so that one has a homomorphism \(d\sigma: Q\to \mathcal A(X,\omega)\). A homogeneous \(G\)-strongly symplectic space \((X,\omega)\) is called a Hamiltonian \(G\)-space if \(d\sigma: \mathcal G\to\mathcal A(X,\omega)\) can be lifted to a homomorphism \(\lambda: \mathcal G\to R(X)\), so that one has a commutative diagram

\[ *** \]

The significance of the notion of Hamiltonian \(G\)-space is pointed out by Theorem 5.1.1: Let \((X,\omega,\lambda)\) be a Hamiltonian \(G\)-space with \([\omega]\in H^2(X, R)\) integral and let \([(L,\alpha)]\in \mathcal L_c(X, \omega)\). Then, there is a unique lifting \(\tilde\sigma: G\to E(L,\alpha)\) of \(\sigma\) such that \(d\tilde\sigma = \tilde\delta\circ\lambda\).

A vital example of a Hamiltonian \(G\)-space is produced in Theorem 5.3.1: Let \(\underline{0}\) denote a \(G\)-orbit in \(\mathcal G'\), the dual vector space to \(\mathcal G\), regarded as a \(G\)-module with respect to the coadjoint representation, and let \(f\in \underline{0}\). Using the bilinear form on \(\mathcal G\), \(B_f(x,y) = \langle f,[y,x]\rangle\) one may define a 2-form \(\omega_{\underline{0}}\) on \(\underline{0}\). On the other hand the formula, \(\lambda_{\underline{0}}(x) f = \langle f,x \rangle\), \(f\in\underline{0}\), \(x\in\mathcal G\), defines a homomorphism \(\lambda_{\underline{0}}: \mathcal G\to R(\underline{0})\). The triple \((\underline{0},\omega_{\underline{0}},\lambda_{\underline{0}})\) is a Hamiltonian \(G\)-space. Moreover, any manifold which covers a \(G\)-orbit in \(\mathcal G'\) has the structure of a Hamiltonian \(G\)-space.

Conversely, Theorem 5.4.1 asserts that the set of \(G\)-orbits in \(\mathcal G'\) are universal in the sense that any Hamiltonian \(G\)-space \((X,\omega,\lambda)\) covers a unique orbit not only as a manifold but as a Hamiltonian \(G\)-space.

The paper ends up with Theorem 5.7.1, classifying all line bundles with connection over a Hamiltonian \(G\)-space \((X,\omega,\lambda)\) having \(\omega\) as curvature: Let \((X,\omega,\lambda)\) be a Hamiltonian \(G\)-space and let \(G_p\) be the isotropy group at \(p\in X\). If \(\underline{0}\) is the \(G\)-orbit in \(\mathcal G'\) associated with \((X,\omega,\lambda)\) and \(f\in \underline{0}\), let \(G_f\) be the isotropy group at \(f\), with Lie algebra \(\mathcal G_f\). \(G_p^\#\) denotes the set of all characters \(\Lambda: G_p\to T\) having as its differential \(2\pi i f| \mathcal G_f\). Then the cohomology class \([\omega]\in H^2(X,R)\) is integral if and only if \(G_p^\#\) is not empty and in this case \(\pi_1^*(X)\) acts in a simply transitive way on \(G_p^\#\). Moreover, to each \(\ell= [(L,\alpha)]\in \mathcal L_c (X,\omega)\) one associates an element \(\Delta^\ell\in G_p^\#\) and the correspondence \(\mathcal L_c (X,\omega)\to G_p^\#\) given by \(\ell\to\Lambda^\ell\) is a \(\pi_1^*(X)\) map and hence is a bijection. As a corollary, taking \((X,\omega,\lambda)=(\underline{0},\omega_{\underline{0}},\lambda_{\underline{0}})\) one obtains a result generalizing the Borel-Weil theorem in the compact case.

Reviewer: H. Moscovici