Nolan R. Wallach

Continuous cohomology, discrete subgroups, and representations of reductive groups

The Description for this book, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. (AM-94), will be forthcoming.

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.

A Classification of Unitary Highest Weight Modules

Let G be a simply connected, connected simple Lie group with center Z. Let K be a closed maximal subgroup of G with K/Z compact and let g be the Lie algebra of G. A unitary representation (π,H) of G such that the underlying (ℊK) — module is an irreducible quotient of a Verma module for ℊℂ is called a unitary highest weight module. Harish-Chandra ([4],[5]) has shown that G admits nontrivial unitary highest weight modules precisely when (G,K) is a Hermitian symmetric pair. In this paper we give a complete classification of the unitary highest weight modules.

Global entanglement in multiparticle systems

We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin-12 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.

On the Selberg trace formula in the case of compact quotient

It is a standard fact (see §2) that 7rr(4>) is of trace class. In particular, 7Tr(4>) is completely continuous for 4>eC7(G). This implies that L(r\G) decomposes into an orthogonal direct sum of irreducible invariant subspaces, {HjK°=i and for each i there are only a finite number of k so that H, is equivalent with Hk as a representation of G (cf. Gelfand, Graev, PyateckiïShapiro [9]). Let G denote the set of equivalence classes of irreducible representations of G. Then we have observed that

Composition series and intertwining operators for the spherical principal series. II

In this paper, we consider the connected split rank one 1 1 Lie group of real type F4 which we denote by F4. We first exhibit F4 as a group of operators on the complexification of A. A. Alberts exceptional simple Jordan algebra. This enables us to explicitly realize the symmetric space 1 16 15 F4/Spin(9) as the unit ball in R with boundary S . After decomposing the space of spherical harmonics under the action of Spin(9), we obtain the matrix of a transvection operator of F4/Spin(9) acting on a spherical principal series representation. We are then able to completely determine the Jordan Holder series of any spherical principal series representation of F4.

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.

Gelfand-Zeitlin Theory from the Perspective of Classical Mechanics. II

Let M(n) be the algebra (both Lie and associative) of n × n matrices over ℂ. Then M(n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P(n), of polynomial functions on M(n) is a Poisson algebra. In particular, if f ∈ P(n), then there is a corresponding vector field ξ f on M(n). If m ≤ n, then M(m) embeds as a Lie subalgebra of M(n) (upper left hand block) and P(m) embeds as a Poisson subalgebra of P(n). Then, as an analogue of the Gelfand-Zeitlin algebra in the enveloping algebra of M(n), let J(n) be the subalgebra of P(n) generated by P(m)Gl(m) for m = 1, . . ., n. One observes that