Roe Goodman

Symmetry, representations, and invariants

Lie Groups and Algebraic Groups.- Structure of Classical Groups.- Highest-Weight Theory.- Algebras and Representations.- Classical Invariant Theory.- Spinors.- Character Formulas.- Branching Laws.- Tensor Representations of GL(V).- Tensor Representations of O(V) and Sp(V).- Algebraic Groups and Homogeneous Spaces.- Representations on Spaces of Regular Functions.

Projective unitary positive-energy representations of diff (S1)

Abstract Let D be the group of orientation-preserving diffeomorphisms of the circle S1. Then D is Frechet Lie group with Lie algebra (δ∞)R the smooth real vector fields on S1. Let δR be the subalgebra of real vector fields with finite Fourier series. It is proved that every infinitesimally unitary projective positive-energy representation of δR integrates to a continuous projective unitary representation of D . This result was conjectured by V. Kac.

Classical and quantum-mechanical systems of Toda lattice type. I

The structure of the commutant of Laplace operators in the enveloping and “Poisson algebra” of certain generalized “ax +b” groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.

Whittaker vectors and conical vectors

Abstract A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquets analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules.

Complex Fourier analysis on a nilpotent Lie group

Abstract. Let G be a simply-connected nilpotent Lie group, with complexificationGc. The functions on G which are analytic vectors for the left regular representation ofG on L2(G) are determined in this paper, via a dual characterization in terms of theiranalytic continuation to Gc, and by properties of their L2 Fourier transforms. Theanalytic continuation of these functions is shown to be given by the Fourier inversionformula. An explicit construction is given for a dense space of entire vectors for theleft regular representation. In the case G = R this furnishes a group-theoretic settingfor results of Paley and Wiener concerning functions holomorphic in a strip. Introduction. Let G be a simply-connected nilpotent Lie group. Since G is aseparable, type I, unimodular group [1], there exists by the general Planchereltheorem [3] a unique Borel measure p. on the space G of equivalence classes ofirreducible unitary representations of G with the following properties:(i) (Harmonic analysis). Fix a p. measurable cross-section C -*■ n( from G toconcrete irreducible unitary representations (ir( e £). Then a function/e L2(G) has aFourier transform/defined ii-a.e. on G, such that/(£) is a Hilbert-Schmidt operatoron the space ^(-n*-) of tt(. The map f ->/(i) is a p-measurable field of operators,

Higher-order Sugawara operators for affine Lie algebras

Let 0 be the affine Lie algebra associated to a simple Lie algebra 0 . Representations of 0 are described by current fields X(Q on the circle T (X e 0 and £ £ T ). In this paper a linear map a from the symmetric algebra 5(0) to (formal) operator fields on a suitable category of 0 modules is constructed. The operator fields corresponding to 0-invariant elements of S(

Classical and quantum mechanical systems of Toda-lattice type. III. Joint eigenfunctions of the quantized systems

) are called Sugawara fields. It is proved that they satisfy commutation relations of the form (*) [<t(h)(í), X(t})] = cx,DS{C,ln)a(Vxu)(Q + higher-order terms with the current fields, where c» is a renormalization of the central element in 0 and Dô is the derivative of the Dirac delta function. The higher-order terms in (*) are studied using results from invariant theory and finite-dimensional representation theory of 0 . For suitably normalized invariants u of degree 4 or less, these terms are shown to be zero. This vanishing is also proved for 0 = sl(n,C) and u running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever c«> is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for 0 . Introduction Let q be the affine algebra associated with a finite-dimensional simple Lie algebra g. The representation theory of g is closely related to models for quantum field theory in two space-time dimensions. In the so-called Sugawara models, the representation of g corresponds to a current field, and the energymomentum field is obtained as a quadratic function of the current fields [Fre, G-O, P-S, Sug]. In this paper we construct a linear map from the symmetric algebra S(g) to (formal) operator fields on a suitable category of g modules. We call the operator fields corresponding to g-invariant elements of S(g) Sugawara fields. Our main results concern the commutation relations between current fields and Sugawara fields. Received by the editors March 2, 1988. Presented to the 848th meeting of the AMS, Worcester, Massachusetts, April 16, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 17B35, 17B67, 22E47; Secondary 15A72, 20C30, 20G45, 81E99. Research partially supported by Rutgers University Faculty Academic Study Program, Australian National University Centre for Mathematical Analysis, and NSF Grant DMS 86-03169 (to R.G.) and by NSF Grant DMS 84-02684 (to N.R.W.). ©1989 American Mathematical Society 0002-9947/89

Holomorphic representations of nilpotent Lie groups

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Multiplicity-Free Spaces and Schur–Weyl–Howe Duality

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Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes

In a previous paper it was shown that certain Schrödinger operatorsH=Δ − V onRℓ such as the Hamiltonians for the quantized one-dimensional lattice systems of Toda type (either non-periodic or periodic) are part of a family of mutually commuting differential operatorsH=L1, ...,Lℓ onRℓ. The potentialV in these cases is associated with a finite root system of rank ℓ, and the top-order symbols of the operatorsLi are a set of functionally independent polynomials that generate the polynomial invariants for the Weyl groupW of the root system. In this paper it is proved that the spaces of joint eigenfunctions for the family of operatorsLi have dimension |W|. In the case of the periodic Toda lattices it is shown that the Hamiltonian has only bound states. An integrable holomorphic connection with periodic coefficients is constructed on a trivial |W|-dimensional vector bundle over ℂℓ, and it is shown that the joint eigenfunctions correspond exactly to the covariant constant sections of this bundle. Hence the eigenfunctions can be calculated (in principle) by integrating a system of ordinary differential equations. These eigenfunctions are holomorphic functions on ℂℓ, and are multivariable generalizations of the classical Whittaker functions and Mathieu functions. A generalization of Hills determinant method is used to analyze the monodromy of the connection.