Polarization and unitary representations of solvable Lie groups

Auslander, Louis; Kostant, Bertram

doi:10.1007/bf01389744

Cited by 211 publications

(129 citation statements)

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“…But it is to the orbital parameters that we look for general descriptions of harmonic analysis on non-semisimple homogeneous spaces. As one knows from previous efforts in non-semisimple groups [1], [17], in order to obtain the final form of the spectral decomposition in orbital parameters, one must derive and employ the Mackey parameters as an intermediate tool.…”

Section: (G/h)mentioning

confidence: 99%

“…Nevertheless, any v e H can be realized by induction via some real polarization which satisfies the Pukanszky condition-i.e. v = Indj^/^, ^€^,ta real polarization for ψ such that K -ψ = ψ + t 1 . The bijection of Proposition 3.2, which comes from [3], is still legitimate.…”

Section: (G/h)mentioning

confidence: 99%

“…If γ is not a character, we reason as follows. We know (by [1] or [15]) that there is an //-invariant positive polarization m c n c for θ. Let D = m Π n, e = (m + m) Π n. Then we may realize the representation γ as the holomorphically induced representation via m from χ θ on Λ^.…”

Section: Let N Be Simply Connected Nilpotent θ En* C a Cartan Subgrmentioning

confidence: 99%

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Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Lipsman¹

1989

Pacific J. Math.

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One of the main results of this paper is a complete description of the spectral decomposition of the quasi-regular representation of an arbitrary exponential solvable symmetric space. Benoist had shown previously that such a representation is multiplicity-free, but he was unable to compute the precise spectrum and spectral measure. More generally, the quasi-regular representation is considered for any exponential solvable homogeneous space. In previous work of the author and Messrs. Corwin, Greenleaf and Grelaud, the analysis of these representations was carried out in the nilpotent case. The spectral decomposition arrived at was in terms of the Kirillov orbital parameters. Corresponding results are obtained here for algebraic exponential solvable homogeneous spaces in case the stability subgroup is either: a Levi component, or its nilradical is multiplicity-free in the nilradical of the homogeneous group. The description of the spectral decomposition in the Mackey parameters is also obtained for these representations.

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Section: (G/h)mentioning

confidence: 99%

Section: (G/h)mentioning

confidence: 99%

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Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Lipsman¹

1989

Pacific J. Math.

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“…It may be or may not be irreducible; even worse it may happen that ttIX) = 0. However, if G is assumed to be solvable, the results of Auslander and Kostant [1] allow us to conclude that 77/t) is irreducible and independent of the choice of the polarization. Furthermore, G acts naturally on the set of all such isomorphism classes of line bundles with connection over symplectic homogeneous Gspaces of the form (Xu, 0U) with <o running through Z2(g), and the map /r-»77/ = 7r/h is constant on the orbits of G. This construction is particularly fruitful in the case of an exponential group G, when it yields a complete parametrization of all equivalence classes of irreducible projective representations of G by the orbits of G in Z2(g).…”

Section: Henri Moscovici and Andrei Veronamentioning

confidence: 99%

“…As it is known, tj extends to a unique character x = Xe 0l T> whose differential is 27rz'j'|b. Now let p(x, b) be the holomorphically induced unitary representation of M corresponding to the polarization I) and to the character X (see [1]). There is no problem in verifying that p(x, b) is projectable so that we do not insist on this point.…”

Section: Theorem (I) I E Tg If and Only If-rrl E Gnmentioning

confidence: 99%

Quantization and Projective Representations of Solvable Lie Groups

Moscovici¹,

Verona²

1978

Transactions of the American Mathematical Society

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Abstract.Kostant's quantization procedure is applied for constructing irreducible projective representations of a solvable Lie group from symplectic homogeneous spaces on which the group acts. When specialized to a certain class of such groups, including the exponential ones, the technique exposed in the present paper provides a complete parametrization of all irreducible projective representations.

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Hamiltonian group actions and dynamical systems of calogero type

Kazhdan

Kostant

Sternberg

1978

Comm Pure Appl Math

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373

349

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In recent years there has been a renewed interest in completely integrable Hamiltonian systems, particularly in conjunction with the study of certain non-linear partial differential equations such as the Korteweg-de Vries equation and their "soliton" solutions. For example, see the paper by Moser[12] where several of these completely integrable systems are studied by the "Lax method" of relating these problems to isospectral families, that is, to curves of matrices with the same eigenvalues. In this paper we wish to establish the complete integrability of certain of these systems by a method which is more directly related to group theory and to a familiar procedure in classical mechanics. The procedure that we have in mind is the one of "quotienting out" variables associated to known integrals so as to reduce the order of the system. For example, starting out with a system of several particles in R3 for which the total linear momentum is conserved, one obtains a system with three fewer degrees of freedom by looking at the submanifold of phase space consisting of all states with a definite value of the total linear momentum and then ignoring the position of the center of mass of the system. In physical language, one usually denotes this process as introducing "co-ordinates relative to the center of mass". In more mathematical terms, we can express the procedure as follows: let "I' denote the submanifold in question. Then the restriction of the symplectic form of phase space to "I' is singular and has a three-demensional null space at each point of "I' This defines a three-dimensional foliation of "I'; the quotient space of this foliation is a symplectic manifold and the original Hamiltonian defines a Hamiltonian on this quotient symplectic manifold. Now, in the usual applications, applying this method of reduction simplifies the equations of motion. However, it is conceivable that quite the reverse might be the case-that the original equations of motion are quite transparent, but the equations of motion of the quotient system appear more complicated. Indeed, we shall show that the Calogero system (cf.[3]) of n particles on the line moving under the inverse square potential, and the corresponding Sutherland system (cf.[15]> of n particles on the circle moving under the sin-* potential are examples of mechanical systems which arise as quotients of much simpler looking mechanical systems.

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Polarization and unitary representations of solvable Lie groups

Cited by 211 publications

References 10 publications

Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Quantization and Projective Representations of Solvable Lie Groups

Hamiltonian group actions and dynamical systems of calogero type

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