The homotopy type of spaces of locally convex curves in the sphere

A smooth curve \gamma: [0,1] \to \Ss^2 is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) = \gamma'(1) = e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space \cL_{-1,c} is known to be contractible. We prove that \cL_{+1} and \cL_{-1,n} are homotopy equivalent to (\Omega\Ss^3) \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots and (\Omega\Ss^3) \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots, respectively. As a corollary, we deduce the homotopy type of the components of the space \Free(\Ss^1,\Ss^2) of free curves \gamma: \Ss^1 \to \Ss^2 (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces \Free([0,1], \Ss^2) with fixed initial and final frames.Comment: 47 pages, 13 figure

An example of the derived geometrical Satake correspondence over integers

Let G^\vee be a complex simple algebraic group. We describe certain morphisms of G^\vee(\calO)-equivariant complexes of sheaves on the affine Grassmannian \Gr of G^\vee in terms of certain morphisms of G-equivariant coherent sheaves on \frakg, where G is the Langlands dual group of G^\vee and \frakg is its Lie algebra. This can be regarded as an example of the derived Satake correspondence.Comment: 16 page

Welfare Recipients and Workforce Laws

CRS_June_2004_Welfare_Recipients_and_Workforce_Laws.pdf: 1325 downloads, before Oct. 1, 2020

Tensor product structure of affine Demazure modules and limit constructions

Let \Lg be a simple complex Lie algebra, we denote by \Lhg the corresponding affine Kac--Moody algebra. Let $\Lambda_0$ be the additional fundamental weight of \Lhg. For a dominant integral \Lg--coweight \lam^\vee, the Demazure submodule V_{-\lam^\vee}(m\Lam_0) is a \Lg--module. For any partition of \lam^\vee=\sum_j \lam_j^\vee as a sum of dominant integral \Lg--coweights, the Demazure module is (as \Lg--module) isomorphic to \bigotimes_j V_{-\lam^\vee_j}(m\Lam_0). For the ``smallest'' case, \lam^\vee=\om^\vee a fundamental coweight, we provide for \Lg of classical type a decomposition of V_{-\om^\vee}(m\Lam_0) into irreducible \Lg--modules, so this can be viewed as a natural generalization of the decomposition formulas in \cite{KMOTU} and \cite{Magyar}. A comparison with the U_q(\Lg)--characters of certain finite dimensional U_q'(\Lhg)--modules (Kirillov--Reshetikhin--modules) suggests furthermore that all quantized Demazure modules V_{-\lam^\vee,q}(m\Lam_0) can be naturally endowed with the structure of a U_q'(\Lhg)--module. Such a structure suggests also a combinatorially interesting connection between the LS--path model for the Demazure module and the LS--path model for certain U_q'(\Lhg)--modules in \cite{NaitoSagaki}. For an integral dominant \Lhg--weight $\Lambda$ let V(\Lam) be the corresponding irreducible \Lhg--representation. Using the tensor product decomposition for Demazure modules, we give a description of the \Lg--module structure of V(\Lam) as a semi-infinite tensor product of finite dimensional \Lg--modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.Comment: 24 pages, in the current version we added the case of twisted affine Kac--Moody algebra