Harmonic morphisms from the Grassmannians and their non-compact duals

In this paper we give a unified framework for the construction of complex valued harmonic morphisms from the real, complex and quaternionic Grassmannians and their non-compact duals. This gives a positive answer to the corresponding open existence problem in the real and quaternionic cases. Mathematics Subject Classifications (2000): 58E20, 53C43, 53C12The author is a member of EDGE, Research Training Network HPRN-CT-2000-00101     

Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SLn(R), SU^∗(2n) and Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO^∗(2n), SO(p,q), SU(p,q) and Sp(p,q). Applying a duality principle we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with semi-Riemannian metrics.     

Lie algebras and separable morphisms in pro-affine algebraic groups

Let be an algebraically closed field of arbitrary characteristic, and let be a surjective morphism of connected pro-affine algebraic groups over . We show that if is bijective and separable, then is an isomorphism of pro-affine algebraic groups. Moreover, is separable if and only if (its differential) is surjective. Furthermore, if is separable, then .

     

Harmonic morphisms from the compact semisimple Lie groups and their non-compact duals

In this paper we prove the local existence of complex-valued harmonic morphisms from any compact semisimple Lie group and their non-compact duals. These include all Riemannian symmetric spaces of types II and IV. We produce a variety of concrete harmonic morphisms from the classical compact simple Lie groups SO(n), SU(n), Sp(n) and globally defined solutions on their non-compact duals SO(n,C)/SO(n), SLn(C)/SU(n) and Sp(n,C)/Sp(n).     

Subelliptic estimates on compact semisimple Lie groups

In this paper, we consider a natural subelliptic structure in semisimple, compact and connected Lie groups, and estimate the constant in the so-called subelliptic Friedrichs-Knapp-Stein inequality, which has implications in the regularity theory of p-energy minimizers.     

Stabilized plethysms for the classical Lie groups

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite-dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood-Richardson coefficients.     

version of Hardy's theorem on semisimple Lie groups

We prove an analogue of the version of Hardy's theorem on semisimple Lie groups. The theorem says that on a semisimple Lie group, a function and its Fourier transform cannot decay very rapidly on an average.

     

On subsemigroups of semisimple Lie groups

In this paper we classify the subsemigroups of any connected semisimple Lie groupG which areK-bi-invariant, whereG=KAN is an Iwasawa decomposition ofG.     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Lévy processes in semisimple Lie groups and stability of stochastic flows

We study the asymptotic stability of stochastic flows on compact spaces induced by Levy processes in semisimple Lie groups. It is shown that the Lyapunov exponents can be determined naturally in terms of root structure of the Lie group and there is an open subset whose complement has a positive codimension such that, after a random rotation, each of its connected components is shrunk to a single moving point exponentially under the flow.

     

Automatic continuity of pseudocharacters on semisimple Lie groups

It is proved that an arbitrary pseudocharacter on a semisimple Lie group is continuous.     

Generating sets for compact semisimple Lie groups

Let be a compact connected semisimple Lie group. We prove that the subset of consisting of pairs which topologically generate is Zariski open.

     

Stratonovich-Weyl correspondence for compact semisimple Lie groups

Let G be a compact Lie group, M a G-homogeneous space and π a unitary representation of G realized on a Hilbert space of functions on M. We give a general presentation of the Stratonovich-Weyl correspondence associated with π. In the case when G is a compact semisimple Lie group and π _λ an irreducible representation of G with highest weight λ, we study the Stratonovich-Weyl symbol of the derived operator d π _λ (X) for X in the Lie algebra of G and its behavior as λ goes to infinity.     

Time optimal factorizations on compact Lie groups

In this paper we study the relationship between factorization problems on SU(2ⁿ) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp Fourier type and cotype with respect to compact semisimple Lie groups

Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.

     

Ergodic measures of geodesic flows on compact Lie groups

Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature (possibly strictly positive on some sections), whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.     

Sharp Gårding inequality on compact Lie groups

We establish the sharp Gårding inequality on compact Lie groups. The positivity condition is expressed in the non-commutative phase space in terms of the full matrix symbol, which is defined using the representations of the group. Applications are given to the L² and Sobolev boundedness of pseudo-differential operators.     

Asymptotics of matrix integrals and tensor invariants of compact Lie groups

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant-theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of permutation operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

     

Matrix coefficients and coadjoint orbits of compact Lie groups

Let be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the -function and the pull-back of an arbitrary unitary irreducible representation of to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups.