INTEGRABILITY OF GEODESIC FLOWS FOR METRICS ON SUBORBITS OF THE ADJOINT ORBITS OF COMPACT GROUPS

Let G/K be an orbit of the adjoint representation of a compact connected Lie group G, σ be an involutive automorphism of G and \( \tilde{G} \) be the Lie group of fixed points of σ. We find a sufficient condition for the complete integrability of the geodesic ow of the Riemannian metric on \( \tilde{G}/\left(\tilde{G}\cap K\right) \) which is induced by the bi-invariant Riemannian metric on \( \tilde{G} \). The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when G is the unitary group U(n) and σ is its automorphism determined by the complex conjugation.     

Principal Gelfand pairs

Let X = G/K be a connected Riemannian homogeneous space of a real Lie group G. The homogeneous space X is called commutative or the pair (G, K) is called a Gelfand pair if the algebra of G-invariant differential operators on X is commutative. We prove an effective commutativity criterion and classify Gelfand pairs under two mild technical constraints. In particular, we obtain several new examples of commutative homogeneous spaces that are not of Heisenberg type.     

Finite Group Actions on Spin Bundles

Let L be a complex line bundle over a closed, oriented, smooth 4-manifold X with c₁(L) w₂(TX) mod 2. Let a finite group G act on X as orientation preserving isometries and on L such that the projection L X is a G-map. We investigate the action of G on the Seiberg-Witten equations, and when G = Z₂ we study the G-invariant Seiberg-Witten invariants on X and the Seiberg-Witten invariants on its quotient setting.     

Coxeter groups and interpolation of operators

Let V be a finite dimensional real Euclidean space and let G be a finite irreducible group generated by orthogonal reflections across hyperplanes in V. We study interpolation of operators in G-invariant norms on V. A collection of G-invariant norms is called G-sufficient if any G-invariant norm is a strict interpolation norm for this collection. Using the general theory of sufficient collections we calculate explicitly two remarkable minimal sufficient collections and study their extremal properties.To our teacher, Professor S.G. Krein, on his 75^th birthdayThe research was supported in part by a grant from the Ministry of Absorption and the Rashi Foundation.The research was supported in part by a grant from the Ministry of Science and Maagara — a special project for absorption of new immigrants, at the Department of Mathematics, Technion     

On the sublinear functional associated to a family of invariant means

In this paper we give conditions under which we can obtain explicit analytic expressions of the fundamental sublinear functional P_K of the family of means K=[G]J where [G] is a subset of G-invariant means, G a semigroup of operators on 1 and J a saturated set of means. Such conditions allow us to assure that [G]J.We characterize the set of the almost convergent sequences related to the family K by means of the same functional p_K. We obtain also p_K in terms of p_J when p_J is known. This allow us to give different expressions of the same functional p_K when the family J changes and from which several examples are given.     

Optimization problems for KG

In this paper we are concerned with certain nonlinear type optimization problems related to the estimation of the universal constant K_G appearing in Grothendieck's fundamental Theorem in the metric theory of tensor products. Let K_G,k be the smallest constant such that Grothendieck's Theorem remains valid for m x m matrices, mk. In this paper we show K_G,2=K_G,3=2.     

Existence of homogeneous vectors on the fiber space of the tangent bundle

Let M = G/K be a homogeneous differentiable manifold. We consider the homogeneous bundle = (G, π, G/K, K) and the tangent bundle τ _G/K of M = G/K, and give some results about the existence of homogeneous vectors on the fiber space of τ _G/K, for both cases of G semisimple and weakly semisimple.      

Carter subgroups and Fitting heights of finite groups

Let G be a finite group possessing a Carter subgroup K. Denote by \(\mathbf {h}(G)\) the Fitting height of G, by \(\mathbf {h}^*(G)\) the generalized Fitting height of G, and by \(\ell (K)\) the number of composition factors of K, that is, the number of prime divisors of the order of K with multiplicities. In 1969, E. C. Dade proved that if G is solvable, then \(\mathbf {h}(G)\) is bounded in terms of \(\ell (K)\). In this paper, we show that \(\mathbf {h}^*(G)\) is bounded in terms of \(\ell (K)\) as well.     

Almansi decomposition for Dunkl operators

Let Ω be a G-invariant convex domain in ℝ^N including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (Δh)ⁿf = 0 for some integer n. Here_333-01is the Dunkl Laplacian, and D_j is the Dunkl operator attached to the Coxeter group G,

$$\mathcal{D}_j f(x) = \frac{\partial }{{\partial x_j }}f(x) + \sum\limits_{v \in R_ + } {\kappa _v \frac{{f(x) - f(\sigma _v x)}}{{\left\langle {x,v} \right\rangle }}} v_j ,$$

where K_v is a multiplicity function on R and σ_v is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form

$$f(x) = f_0 (x) + \left| x \right|^2 f_1 (x) + \cdots + \left| x \right|^{2(n - 1)} f_{n - 1} (x), \forall x \in \Omega ,$$

where f_j are Dunkl harmonic functions, i.e. Δ_hf_j = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.

     

Examples of homogeneous vectors on the fiber space of the associated and the tangent bundles

The paper considers the associated bundle ξ = (G × KG/K, ρ _ξ , G/K, G/K) and the tangent bundle τ _G/K = (T _G/K , π _G/K , G/K, R ^m ), and gives special examples of odd dimensional solvable Lie groups equipped with left invariant Riemannian metric. Some conditions about existence of homogeneous geodesic vectors on the fiber space of ξ and τ _G/K are proved.     

A note on lattices in semi-stable representations

Let p be a prime, K a finite extension over \mathbb Q_p{{\mathbb Q}_p} and G = Gal([`(K)] /K){G = {\rm Gal}(\overline K /K)} . We extend Kisin’s theory on j{\varphi} -modules of finite E(u)-height to give a new classification of G-stable \mathbb Z1_p{{\mathbb Z}1_p} -lattices in semi-stable representations.     

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric g_Q and second fundamental form L_Q on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data g_Q and L_Q; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.     

Garlands containing the general linear group over an intermediate field

Let K/k be a finite extension of fields with an intermediate subfield L, and let H = GL_L(K) be the general linear group of all L-linear invertible mappings of the vector space of the field K over L. It is proved that the subgroups lying between GL_K(K)^H and the normalizer of H in G, where G = GL_k(K), form a garland. Bibliography: 4 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 34–41.     

On Fiber Coefficients of Fiber Cones †

Let (R,𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R, and K an ideal containing I. When depth G(I) ≥ d ? 1, depth F_K(I) ≥ d ? 2, and r(I|K) < ∞, we calculate the fiber coefficients f_i(I). Under the above assumptions on depth G(I) and r(I|K), we give an upper bound for f₁(I), and also provide a characterization, in terms of f₁(I), of the condition depth F_K(I) ≥ d ? 2.     

On the conjugate locus of pseudo-Riemannian manifolds

Let expm :T_mM → M be the exponential map of a Riemannian manifold M at a point m ∈ M. Warner proved that in any neighbourhood of a conjugate point in T_mM, the map exp_m is not injective. Moreover, he described the exponential map in a suitable coordinate system in a neighbourhood of a regular conjugate point, these points build an open dense set in the conjugate locus. We will investigate in the pseudo-Riemannian case such subsets, where the results of Warner generalize. For the definition of these subsets of the conjugate locus we use a bilinear form on ker(T_v exp_m), where v is a conjugate point, which will defined by the geodesic flow and the pseudo-Riemannian metric tensor.     

Fixed point iterations for certain classes of nonlinear mappings

Let X = L_p or L_p, 2≤p<∞, and let K be a nonempty closed convex bounded subset of X. It is proved that for some classes of nonlinear mappings T:K → K (more precisely, for T P₂ or C in the terminology of F.E. Browder and W.V. Pretryshyn; and B.E. Rhoades), the iteration process: x₁ ?K,X_n+1 = (1-C_n)x_n+C_n Tx_n, n ≥1,under suitable conditions on K and the real sequence {C_n}_n=1 ^∞ converges strongly to a fixed point of T. While our thorems generalize serveral known results, our method is also of independent interest     

Stability of the Graph of a Convex Function

Let x: M → A ⁿ⁺¹ be the graph of some strongly convex function x ⁿ+1= ?( x₁,…,x_n) defined on a domain Ω ? A ⁿ in a real affine space. We consider the relative metric G, defined by $ G=\sum{\partial^{2}f\over\partial x_{i}\partial x_{j}}dx_{i}dx_{j}$ .In this paper, we calculate the second variation of the area integral with respect to the relative metric G. We prove that the parabolic affine hyperspheres are stable.     

Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive

Let K be a field of characteristics 0 complete with respect to a discrete valuation v, with a perfect residue field of characteristic p>0. Let be an algebraic closure of K and K_nr its maximal unramified subextension. Let E be an elliptic curve over K with an integral modular invariant. The curve E has potentially good reduction at v, and there exists a smallest extension L of K_nr over which E has good reduction at v. The Galois group, Gal (L/K_nr) is known in the case p≥5. In this paper we give receipts to determine this group in the cases p=2 and p=3.