Kac–Moody symmetric spaces: arbitrary symmetrizable complex or almost split real type

Our object of study is the ring of invariant differential operators on a hermitian symmetric space G/K of classical and noncompact type. Here, as usual, G is a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup of G. It is well-known that the ring, denoted by ?(G/K), is isomorphic to a polynomial ring of 1 variables, 1 being the rank of G/K. This is true even in the nonhermitian case. If 1 = 1, it is generated by the Laplace-Beltrami operator z/ which is essentially self-adjoint; moreover -Iz2 is nonnegative. In the general case, an easily posable problem is to find an explicitly defined set of generators. We can go one step further by focusing our attention on the nonnegativity, which necessarily limits the range of eigenvalues under a suitable integrability condition on functions. It is thus natural to ask whether there exist some canonically defined operators which generate -9(G/K) and have the property of nonnegativity. The main purpose of the present paper is to give an affirmative answer to this question. In fact, our answer applies to a somewhat more general type of rings of differential operators which includes 9(G/K) as a special case. To be explicit, we take an irreducible representation p: K -* GL(V) with a complex vector space V of finite dimension, and consider the set C'(p) of all V-valued Cx functions f on G such that f(xk -1) = p(k)f(x) for every k e K. We then denote by ?(p) the ring of left-invariant difrrential operators on G which map C'(p) into itself. Now the complexification g of the Lie algebra of G has abelian subalgebras p+ and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on G/K at the origin. For any complex vector space W let Sr(W) denote the ring of all complex-valued homogeneous polynomial functions on W of degree r, and let S(W) = Er=0Sr(W). Through the adjoint representation of G on g, K acts naturally on Sr( ?+). It is a known fact, due to Hua and Schmid, that each irreducible constituent of this representation of K has multiplicity one. Now our principal

Let $\M$ be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into $\M$ with prescribed singularities along a closed submanifold of the domain. This generalizes our previous work where such maps into the hyperbolic plane were constructed. This problem, in the case where $\M$ is the complex-hyperbolic plane, has applications to equilibrium configurations of co-axially rotating charged black holes in General Relativity.

In [8] counting complexity classes #PR and #PC in the Blum-Shub-Smale (BSS) setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of [8] is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the class FPR#PR. In this paper, we prove that the corresponding result is true over C, namely that the computation of the Euler characteristic of an affine or projective complex variety is complete in the class FPC#PC. We also obtain a corresponding completeness result for the Turing model.

In the last chapter we classified all the semi-simple algebras over the complex numbers; but each complex algebra in general has several real forms. For example, we saw in Chapter 1 that sl(2, R) and su(2) are distinct real forms of the complex algebra sl(2, C). That is, there is no real isomorphism between them, yet they have the same complexification. The task of classifying the real forms of the semi-simple algebras is more involved than that of classifying the complex forms themselves. It was first done by Cartan in 1914. Later, in 1926–28 he used this classification to determine all the symmetric spaces. A complete discussion of the real forms and their relationship to the symmetric spaces is given in the well known book of Helgason, and we shall give only a brief introduction here.

Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.

Every locally trivial action of the additive group of complex numbers on four-dimensional complex affine space that is given by a triangular derivation is conjugate to a translation. A criterion for a proper action on complex affine n n -space to be locally trivial is given, along with an example showing that the hypotheses of the criterion are sharp.

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.

Major depressive disorder (MDD) has been related to both a dysfunctional gamma-amino butyric acid (GABA) system and to hyperactivity of the hypothalamic-pituitary-adrenal axis (HPA). Although GABA has been suggested to inhibit HPA axis activity, their relationship has never been studied at the level of the central GABA(A)-benzodiazepine receptor in depressed patients or in relation to antidepressant treatment. Eleven depressed outpatients were compared, before and after treatment with citalopram, with nine age-matched healthy controls. The subjects were scanned using the positron emission tomography (PET) tracer [(11)C]flumazenil ([(11)C]FMZ). Parametric voxel-by-voxel Logan plots were compared with methods based on regions of interest (ROI), to provide volume of distribution (V(T)) and binding potential (BP(ND)) values. Plasma GABA levels were determined and a dexamethasone-corticotropin releasing hormone (DEX-CRH) test was performed. In MDD, parametric voxel-by-voxel Logan plots showed bilateral reduced [(11)C]FMZ binding in the parahippocampal gyrus and right lateral superior temporal gyrus (p uncorrected < or =0.001). In the temporal area, [(11)C]FMZ binding showed a strong inverse correlation with HPA axis activity. Plasma GABA did not discriminate MDD from controls, but correlated inversely with [(11)C]FMZ binding in the right insula. Following treatment with citalopram, voxel-based analysis revealed reduced binding in the right lateral temporal gyrus and dorsolateral prefrontal cortex. The bilateral reduction in limbic parahippocampal and right temporal [(11)C]FMZ binding found in MDD indicates decreased GABA(A)-benzodiazepine receptor complex affinity and/or number. The inverse relationship between GABA(A) binding in the temporal lobe and HPA axis activity, suggests that HPA axis hyperactivity is partly due to reduced GABA-ergic inhibition.

We present an efficient and elementary algorithm for computing the number of primes up to N N in O ~ ( N ) \tilde {O}(\sqrt N) time, improving upon the existing combinatorial methods that require O ~ ( N 2 / 3 ) \tilde {O}(N ^{2/3}) time. Our method has a similar time complexity to the analytic approach to prime counting, while avoiding complex analysis and the use of arbitrary precision complex numbers. We apply our techniques to improve the state-of-the-art complexity of elementary algorithms for computing other number-theoretic functions, such as the Mertens function (in O ~ ( N ) \tilde {O}(\sqrt N) time compared to the known O ~ ( N 3 / 5 ) \tilde {O}(N^{3/5}) ), summing Euler’s totient function, counting square-free numbers and summing primes. Implementation code is provided.

Totally geodesic submanifolds in exceptional symmetric spaces

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space \(\mathbf{CH^m}\) to a rank one symmetric space N of noncompact type with singular boundary data \(f: \mathbf{S^{2m-1}}\longrightarrow \partial_\infty N\). Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously.

Abstract. Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In [1] we proved that i(M) is bounded from below by the rank rk(M) of M, that is, rk(M) ≤ i(M). In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with i(M) ∈ {4, 5, 6}. 1.

In the 1990s, C.-L. Terng and G. Thorbergsson investigated a natural Riemannian submersion from an infinite dimensional Hilbert space onto a compact Riemannian symmetric space G/K. This map is called the parallel transport map over G/K. Later, N. Koike extended their theory to the case that G/K is a Riemannian symmetric space of non-compact type. In this paper, more generally, we define the parallel transport map over an affine symmetric space and show that it is an affine submersion with horizontal distribution in the sense of Abe and Hasegawa. Based on this result, we prove the Fredholm property of affine immersions into a Hilbertable space lifted by the parallel transport map. Furthermore, we greatly extend the author’s previous result on weakly reflective submanifolds from the case of compact Riemannian symmetric spaces to the case of affine symmetric spaces.

Scalar irreducibility of eigenspaces on the tangent space of a reductive symmetric space

A transmutation property of the generalized Abel transform associated with root system A2