In this manuscript we make a general study of the representations realized, for a reductive Lie group of Harish-Chandra class, on the compactly supported sheaf cohomology groups of an irreducible finite-rank polarized homogeneous vector bundle defined in a generalized complex flag space. In particular, we show that the representations obtained are minimal globalizations of Harish-Chandra modules and that there exists a whole number q, depending only on the orbit, such that all cohomologies vanish in degree less than q. The representation realized on the q-th cohomology group is called a standard analytic module. Our main result is a geometric proof that a standard analytic module embeds naturally in an associated standard module defined on the full flag space of Borel subalgebras. As an application, we give geometric realizations for irreducible submodules of some principal series representations in case the group is complex.
A holomorphic discrete series can be realized as the space of the holomorphic sections of a homogeneous vector bundle on a bounded symmetric domain. We can embed it into a degenerate principal series realized as the space of hyperfunction sections of a vector bundle on the Shilov boundary by taking the boundary value. We consider an algebraic version of higher cohomological analogue of such an embedding for complex reductive groups and classical groups of type A.
Auslander and Ringel-Tachikawa have shown that for an artinian ring R of finite representation type, every R-module is the direct sum of finitely generated indecomposable R-modules. In this paper, we will adapt this result to finite representation type full subcategories of the module category of an artinian ring which are closed under subobjects and direct sums and contain all projective modules. In particular, the results in this paper hold for subspace representations of a poset, in case this subcategory is of finite representation type.
Let g' and g be isomorphic to any two of the Lie algebras gl(infty), sl(infty), sp(infty), and so(infty). Let M be a simple tensor g-module. We introduce the notion of an embedding of g' into g of general tensor type and derive branching laws for triples g', g, and M, where the embedding of g' into g is of general tensor type. More precisely, since M is in general not semisimple as a g'-module, we determine the socle filtration of M over g'. Due to the description of embeddings of classical locally finite Lie algebras given by Dimitrov and Penkov, our results hold for all possible embeddings o…
For an ideal $I_{m,n}$ generated by all square-free monomials of degree $m$ in a polynomial ring $R$ with $n$ variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this explicit embedding depends on a minimal free $R$-resolution of an ideal generated by $I_{m,n}$. Using this embedding, we give a resolution of connected sums of several copies of certain Artinian $k$-algebras where $k$ is a field.
Let $G$ be a connected reductive group. We find a necessary and sufficient condition for a quasiaffine homogeneous space of $G$ to be embeddable into an irreducible $G$-module. In addition, for an affine homogeneous space we find a criterium for a closed embedding to exist
We consider arc spaces for the compositions of Pluecker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose le…
In this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra A(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and A(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.
Invariants of persistence modules defined by order-embeddings
Départment de Mathématiques, Université de Sherbrooke（数学系，舍布鲁克大学）；Bishop’s University（主教大学）；Department of Mathematics, North Carolina State University（数学系，北卡罗来纳州立大学）
One of the most mysterious aspects of Saito's theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well a…
We study geometric representations of GL(n,R) for a ring R. The structure of the associated Hecke algebras is analyzed and shown to be cellular. Multiplicities of the irreducible constituents of these representations are linked to the embedding problem of pairs of R-modules x < y.
We define coadmissible equivariant $\mathcal{D}$-modules on smooth rigid analytic spaces and relate them to admissible locally analytic representations of semisimple $p$-adic Lie groups.
In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that ea…
We give an overview of the theory of $\wideparen{\mathcal{D}}$-modules on rigid analytic spaces and its applications to admissible locally analytic representations of $p$-adic Lie groups.
Let G --> G' be an embedding of semisimple complex Lie groups, let B and B' be a pair of nested Borel subgroups, and let f:G/B --> G'/B' be the associated equivariant embedding of flag manifolds. We study the pullbacks of cohomologies of invertible sheaves on G'/B' along the embedding f. Let O' be a G'-equivariant invertible sheaf on G'/B', and let O be its restriction to G/B. Consider the G-equivariant pullback on cohomology p : H(G'/B',O') --> H(G/B,O). The Borel-Weil-Bott theorem implies that the two cohomology spaces above are irreducible modules of G' and G respectively. By Schur's lemma,…
We prove that the category of coadmissible D-cap-modules on a smooth rigid analytic space supported on a closed smooth subvariety is naturally equivalent to the category of coadmissible D-cap-modules on the subvariety, and use this result to construct a large family of pairwise non-isomorphic simple coadmissible D-cap-modules.
We prove an Induction Equivalence and a Kashiwara Equivalence for coadmissible equivariant D-modules on rigid analytic spaces. This allows us to completely classify such objects with support in a single orbit of a classical point with co-compact stabiliser. As an application, we use the locally analytic Beilinson-Bernstein equivalence to construct new examples of large families of topologically irreducible locally analytic representations of certain compact semisimple p-adic Lie groups.
Let L_{B}(-5/2,0) (resp. L_{F}(-5/2,0)) be the simple vertex operator algebra associated to affine Lie algebra of type $B_{4}^{(1)}$ (resp. $F_{4}^{(1)}$) with the lowest admissible half-integer level -5/2. We show that L_{B}(-5/2,0) is a vertex subalgebra of L_{F}(-5/2,0) with the same conformal vector, and that L_{F}(-5/2,0) is isomorphic to the extension of L_{B}(-5/2,0) by its only irreducible module other than itself. We also study the representation theory of L_{F}(-5/2,0), and determine the decompositions of irreducible weak L_{F}(-5/2,0)-modules from the category $\mathcal{O}$ into dir…
We introduce a sheaf of infinite order differential operators D-cap on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle. We show that the sections of this sheaf over sufficiently small affinoid varieties are Fréchet-Stein algebras, and use this to define co-admissible sheaves of D-cap-modules. We prove analogues of Cartan's Theorems A and B for co-admissible D-cap-modules.
We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: t…
We develop a dimension theory for coadmissible D-cap-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic D-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinte-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves $H^i_Z(\…
Let $L_{D_8}(1, 0)$ and $L_{E_8}(1, 0)$ be the simple vertex operator algebras associated to untwisted affine Lie algebra $\widehat{\mathbf g}_{D_{8}}$ and $\widehat{\mathbf g}_{E_8}$ with level 1 respectively. In the 1980s by I. Frenkel, Lepowsky and Meurman as one of the many important preliminary steps toward their construction of the moonshine module vertex operator algebra, they use roots lattice showing that $L_{D_8}(1, 0)$ can embed into $L_{E_8}(1, 0)$ as a vertex operator subalgebra(\cite{5, 6, 8}). Their construct is a base of vertex operator theory. But the embedding they gave using…
