LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical

Let P = M N be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group G over a p-adic field F. Assume that there exists w0 ∊ G(F) that normalizes M and conjugates P to an opposite parabolic subgroup. When N has a Zariski dense Int M-orbit, F. Shahidi and X. Yu described a certain distribution D on M(F), such that, for irreducible unitary supercuspidal representations π of M(F) with is irreducible if and only if D( f )≠ 0 for some pseudocoefficient f of π. Since this irreducibility is conjecturally related to π arising via transfer from certain twisted endoscopic groups of M, it is of interest to realize D as endoscopic transfer from a simpler distribution on a twisted endoscopic group H of M. This has been done in many situations where N is abelian. Here we handle the standard examples in cases where N is nonabelian but admit a Zariski dense Int M-orbit.

The Ado theorem for finite Lie conformal algebras with Levi decomposition

We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.

SYMMETRY CLASSIFICATION OF NEWTONIAN INCOMPRESSIBLE FLUID'S EQUATIONS FLOW IN TURBULENT BOUNDARY LAYERS

Lie-group method is applicable to both linear and nonlinear partial dierential equations, which leads to nd new solutions for partial dierential equations. Lie symmetry group method is applied to study Newtonian incompressible uid’s equations ow in turbulent boundary layers. (Flow and heat transfer of an incompressible viscous uid over a stretching sheet appear in several manufacturing processes of industry such as the extrusion of polymers, the cooling of metallic plates, the aerodynamic extrusion of plastic sheets, etc. In the glass industry, blowing, oating or spinning of bres are processes, which involve the ow due to a stretching surface.) The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained [9, 10]. Finally the structure of the Lie algebra such as Levi decomposition, radical subalgebra, solvability and simplicity of symmetries is given.

The local structure theorem for real spherical varieties

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q\times _{L}S$. Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.

On the Lie structure of zero row sum and related matrices

We study the structure of zero row sum matrices as an algebra and as a Lie algebra in the context of groups preserving a given projection in the algebra of matrices. We find the structure of the Lie algebra of the group that fixes a given projection. Details for the zero row sum matrices are presented. In particular, we find the Levi decomposition and give an explicit unitary equivalence with the affine Lie algebra. An orthonormal basis for zero row sum matrices appears naturally.