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The convergence performance of nonlinear acceleration methods for the method of characteristics (MOC) with flat source (FS) approximation (FS MOC) or linear source (LS) approximation (LS MOC) is nu...

The impact of various parameters in the coarse mesh finite difference (CMFD) acceleration method on overall convergence behavior is investigated through numerical calculations using the method of c...

We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.

We prove a determinant formula for a parabolic Verma module of a Lie superalgebra, previously conjectured by the second author. Our determinant formula generalizes the previous results of Jantzen for a parabolic Verma module of a (non-super) Lie algebra, and of Kac concerning a (non-parabolic) Verma module for a Lie superalgebra. The resulting formula is expected to have a variety of applications in the study of higher-dimensional supersymmetric conformal field theories. We also discuss irreducibility criteria for the Verma module., 24 pages

For a symmetric pair $(G,H)$ of reductive groups we construct a family of intertwining operators between spherical principal series representations of $G$ and $H$ that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. We further discuss uniqueness of intertwining operators, and for the rank one cases $$ (G,H)=(SU(1,n;\mathbb{F}),S(U(1,m;\mathbb{F})\times U(n-m;\mathbb{F}))), \qquad \mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}, $$ and for the pair $$ (G,H)=(GL(4n,\mathbb{R}),GL(2n,\mathbb{C})) $$ we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners., 44 pages, added another detailed example in Section 5

This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not necessarily "maximally degenerating". Our results also give a proof of Kontsevich-Soibelman [KS04, Conjecture 1] (cf., [GW00, Conjecture 6.2]) in the case of K3 surfaces as a byproduct., Comment: 12 pages

In this article, we study the restriction of Zuckerman's derived functor (g ,K )-modules Aq(λ) to gfor symmetric pairs of reductive Lie algebras (g,g � ). When the restriction decomposes into irreducible (g � ,K � )-modules, we give an upper bound for the branching law. In particular, we prove that each (g � ,K � )-module occurring in the restriction is isomorphic to a submodule of Aq� (λ � ) for a parabolic subalgebra qof g � , and determine their associated varieties. For the proof, we realize Aq(λ) on complex partial flag varieties by using D-modules. 1. Introduction. Our object of study is branching laws of Zuckerman's derived functor modules Aq(λ) with respect to symmetric pairs of real reductive Lie groups. Let G0 be a real reductive Lie group with Lie algebra g0. Fix a Cartan invo- lution θ of G0 so that the fixed set K0 : =( G0) θ is a maximal compact subgroup of G0. Write K for the complexification of K0, g0 = k0 ⊕p0 for the Cartan de- composition with respect to θ and g := g0 ⊗RC for the complexification. Similar notation will be used for other Lie algebras. The cohomologically induced module Aq(λ) is a (g ,K )-module defined for a θ-stable parabolic subalgebra q of g and a character λ .T he(g ,K )-module Aq(λ) is unitarizable under certain conditions on the parameter λ and therefore plays a large part in the study of the unitary dual of real reductive Lie groups. One of the fundamental problems in representation theory is to decompose a given representation into irreducible constituents. To begin with, we consider the restriction of (g ,K )-modules to K, or equivalently, to the compact group K0 .I n this case, it is known that any irreducible (g ,K )-module decomposes as a direct sum of irreducible representations of K and each K-type occurs with finite multi- plicity. For Aq(λ), the following formula gives an upper bound for the multiplici- ties.

We demonstrate a close connection between observed field-induced antiferromagnetic (AFM) order and quantum critical fluctuation (QCF) in the Zn7%-doped heavy-fermion superconductor ${\mathrm{CeCoIn}}_{5}$. Magnetization, specific heat, and electrical resistivity at low temperatures all show the presence of new field-induced AFM order under the magnetic field $B$ of 5--10 T, whose order parameter is clearly distinguished from the low-field AFM phase observed for $Bl5\phantom{\rule{4pt}{0ex}}\mathrm{T}$ and the superconducting phase for $Bl3\phantom{\rule{4pt}{0ex}}\mathrm{T}$. The 4f electronic specific heat divided by the temperature, ${C}_{e}/T$, exhibits $\ensuremath{-}lnT$ dependence at $B\ensuremath{\sim}10\phantom{\rule{4pt}{0ex}}\mathrm{T}$ $(\ensuremath{\equiv}{B}_{0})$, and furthermore, the ${C}_{e}/T$ data for $B\ensuremath{\ge}{B}_{0}$ are well scaled by the logarithmic function of $B$ and $T$: $ln[(B\ensuremath{-}{B}_{0})/{T}^{2.7}]$. These features are quite similar to the scaling behavior found in pure ${\mathrm{CeCoIn}}_{5}$, strongly suggesting that the field-induced QCF in pure ${\mathrm{CeCoIn}}_{5}$ originates from the hidden AFM order parameter equivalent to high-field AFM order in Zn7%-doped ${\mathrm{CeCoIn}}_{5}$.

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $., 47 pages, 3 tables; add all the details of the classification

We give a complete classification of reductive symmetric pairs (g, h) with the following property: there exists at least one infinite-dimensional irreducible (g,K)-module X that is discretely decomposable as an (h,H \cap K)-module. We investigate further if such X can be taken to be a minimal representation, a Zuckerman derived functor module A_q(\lambda), or some other unitarizable (g,K)-module. The tensor product $\pi_1 \otimes \pi_2$ of two infinite-dimensional irreducible (g,K)-modules arises as a very special case of our setting. In this case, we prove that $\pi_1 \otimes \pi_2$ is discretely decomposable if and only if they are simultaneously highest weight modules., Comment: To appear in Crelles J. (19 pages)

We give a classification of the triples ( g , g ′ , q ) such that Zuckerman’s derived functor ( g , K ) -module A q ( λ ) for a θ -stable parabolic subalgebra q is discretely decomposable with respect to a reductive symmetric pair ( g , g ′ ) . The proof is based on the criterion for discretely decomposable restrictions by the first author and on Berger’s classification of reductive symmetric pairs.

We give a geometric realization of cohomologically induced (g,K)-modules. Let (h,L) be a subpair of (g,K). The cohomological induction is an algebraic construction of (g,K)-modules from a (h,L)-module V. For a real semisimple Lie group, the duality theorem by Hecht, Milicic, Schmid, and Wolf relates (g,K)-modules cohomologically induced from a Borel subalgebra with D-modules on the flag variety of g. In this article we extend the theorem for more general pairs (g,K) and (h,L). We consider the tensor product of a D-module and a certain module associated with V, and prove that its sheaf cohomology groups are isomorphic to cohomologically induced modules., Comment: 24 pages, typos corrected