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).     

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.     

Large element orders and the characteristic of finite simple symplectic groups

The characteristic of a simple group of Lie type is the characteristic of the field over which this group is defined. Let G = Sp_2n (q), where q = 2 ^k . It is shown that every finite group of Lie type with the same two largest element orders as G has characteristic 2.     

Einstein metrics on compact Lie groups which are not naturally reductive

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D??Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ??? 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ??? 11), Sp(n) (n ??? 3), E ₆, E ₇, and E ₈.     

On Blattner's formula for the discrete series representations of SO(2n, 1)

Blattner's conjecture gives a formula for the multiplicity with which a unitary irreducible representation of the maximal compact subgroup K appears in any discrete series representation of a semisimple Lie group G. We give an elementary derivation of this formula from Harish Chandra's character formula for G ～- SO_e(2n, 1) (n ? 2). The idea is to regularize the character (on the Cartan subgroup), to show that the regularization is unique, and to derive the multiplicities by expanding the resulting distribution in a Fourier series. To prove uniqueness of the regularization one uses a priori constraints on the multiplicities (=Fourier coefficients) that follow from the subquotient theorem.     

Hamiltonicity of 6-connected line graphs

Let G be a graph and let D₆(G)={v∈V(G)|d_G(v)=6}. In this paper we prove that: (i) If G is a 6-connected claw-free graph and if |D₆(G)|≤74 or G[D₆(G)] contains at most 8 vertex disjoint K₄’s, then G is Hamiltonian; (ii) If G is a 6-connected line graph and if |D₆(G)|≤54 or G[D₆(G)] contains at most 5 vertex disjoint K₄’s, then G is Hamilton-connected.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).     

Central extensions of current groups

 In this paper we study central extensions of the identity component G of the Lie group C ^∞ (M,K) of smooth maps from a compact manifold M into a Lie group K which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ω(ξ,η)=[κ(ξ,dη)], where κ:𝔨×𝔨→Y is a symmetric invariant bilinear map on the Lie algebra 𝔨 of K and the values of ω lie in Ω¹(M,Y)/dC ^∞ (M,Y). For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for M=𝕊¹. If K is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of G. The groups Diff(M) and C ^∞ (M,K) act naturally on G by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group G˜ of G. Received: 11 April 2002 / Revised version: 28 August 2002 / Published online: 28 March 2003     

Equivariant K-Theory of Compact Connected Lie Groups Dedicated to Daniel Quillen on the Occasion of his Sixtieth Birthday

We compute the equivariant K-theory K _G ^* (G)for a compact connected Lie group Gsuch that ₁ (G)is torsion free (where Gacts on itself by conjugation). We prove that K _G ^* (G)is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a compact connected Lie group Gwith ₁ (G)torsion, namely PSU(3), and compute the corresponding equivariant K-theory.     

Non-existence of infinitesimally invariant measures on loop groups

Let G be a compact Lie group, L(G) the associated loop group, ω the canonical symplectic form on L(G). Set H the Hamiltonian function for which the associated ω-Hamiltonian vector field is the infinitesimal rotation. Then H generates a canonical semi-definite Riemannian structure on L(G), which induces a Riemannian structure on the free loop groupL(G)/G=L₀(G). This metric corresponds to the Sobolev norm H¹. Using orthonormal frame methodology the positivity and finiteness of the Ricci curvature of L₀(G) is proved. By studying the dissipation towards high modes of a unitary group valued SDE it is proved that the loop group does not have any infinitesimally invariant measure.     

On the second cohomology of an algebraic group and of its lie algebra in a positive characteristic

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h ? 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL₂(k), SL₃(k), SL₄(k), Sp₄(k), and G ₂, nontrivial isomorphisms of this kind exist for SL₄(k) and G ₂ only. For SL₄(k), there are two simple modules with nontrivial second cohomology and, for G ₂, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep2MatK(G) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π₀(G), π₁(G) and [α]∈H³(π₀(G),π₁(G)) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z]∈H²(π₀(G),K^∗)) of the first homotopy group π₀(G) as well as its category of representations on finite sets both live in Rep2MatK(G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep2MatK(G).     

Spectral radius and Hamiltonicity of graphs

Let G be a graph of order n and μ(G) be the largest eigenvalue of its adjacency matrix. Let be the complement of G.Write K_n-1+v for the complete graph on n-1 vertices together with an isolated vertex, and K_n-1+e for the complete graph on n-1 vertices with a pendent edge.We show that:If μ(G)?n-2, then G contains a Hamiltonian path unless G=K_n-1+v; if strict inequality holds, then G contains a Hamiltonian cycle unless G=K_n-1+e.If , then G contains a Hamiltonian path unless G=K_n-1+v.If , then G contains a Hamiltonian cycle unless G=K_n-1+e.     

An explicit growth condition for middle-degree cuspidal cohomology of arithmetically defined quaternionic hyperbolic n-manifolds

Let G/\mathbb Q{G/\mathbb Q} be the simple algebraic group Sp(n, 1) and G = G(N){\Gamma=\Gamma(N)} a principal congruence subgroup of level N ≥ 3. Denote by K a maximal compact subgroup of the real Lie group G(\mathbb R){G(\mathbb R)} . Then a double quotient G\G(\mathbb R)/K{\Gamma\backslash G(\mathbb R)/K} is called an arithmetically defined, quaternionic hyperbolic n-manifold. In this paper we give an explicit growth condition for the dimension of cuspidal cohomology H²ⁿ_cusp(G\G(\mathbb R)/K,E){H^{2n}_{cusp}(\Gamma\backslash G(\mathbb R)/K,E)} in terms of the underlying arithmetic structure of G and certain values of zeta-functions. These results rely on the work of Arakawa (Automorphic Forms of Several Variables: Taniguchi Symposium, Katata, 1983, eds. I. Satake and Y. Morita (Birkh?user, Boston), pp. 1–48, 1984).     

More orthogonal double covers of complete graphs by Hamiltonian paths

An orthogonal double cover (ODC) of the complete graph K_n by a graph G is a collection G of n spanning subgraphs of K_n, all isomorphic to G, such that any two members of G share exactly one edge and every edge of K_n is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of length n-1. It is known that the existence of an ODC of K_n by a Hamiltonian path implies the existence of ODCs of K_4n and of K_16n, respectively, by almost-Hamiltonian cycles. Horton and Nonay introduced two-colorable ODCs and showed: If there are an ODC of K_n by a Hamiltonian path for some n?3 and a two-colorable ODC of K_q by a Hamiltonian path for some prime power q?5, then there is an ODC of K_qn by a Hamiltonian path. In [U. Leck, A class of 2-colorable orthogonal double covers of complete graphs by hamiltonian paths, Graphs Combin. 18 (2002) 155-167], two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths were constructed for all odd square numbers n?9. Here we continue this work and construct cyclic two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths for all n of the form n=4k²+1 or n=(k²+1)/2 for some integer k.     

Analysis of restrictions of unitary representations of a nilpotent Lie group

Let G be a connected simply connected nilpotent Lie group, K an analytic subgroup of G and π an irreducible unitary representation of G. Let DπK(G) be the algebra of differential operators keeping invariant the space of C^∞ vectors of π and commuting with the action of K on that space. In this paper, we assume that the restriction of π to K has finite multiplicities and we show that DπK(G) is isomorphic to a subalgebra of the field of rational K-invariant functions on the co-adjoint orbit Ω(π) associated to π, and for some particular cases, that DπK(G) is even isomorphic to the algebra of polynomial K-invariant functions on Ω(π). We prove also the Frobenius reciprocity for some restricted classes of nilpotent Lie groups, especially in the cases where K is normal or abelian.     

Exponential actions and maximal ?-invariant ideals

Let V be an exponential ?-module, ? being an exponential Lie algebra. Put ? = exp ?. Then every orbit of V under the action of ? admits a closed orbit in its closure. If G= exp ? is a nilpotent Lie group and ? an exponential algebra of derivations of ?, then ? = exp ? acts on G, L ¹(G), (?) and the maximal ?-invariant ideals of L ¹(G), resp. of (?) coincide with the kernels Ker Ω, resp. Ker Ω∩ (?), where Ω is a closed orbit of ?^*. Received: 6 December 1996 / Revised version: 7 December 1997