Spherical Subgroups Research Articles

On the multiplicity spaces of spherical subgroups of minimal rank

AbstractLet be a complex connected semisimple Lie group and be a connected semisimple subgroup of . Consider the branching problem of decomposing the simple ‐representations as a sum of simple ‐representations . When , it is the tensor product decomposition. The multiplicity space satisfies , where the sum runs over the isomorphism classes of simple ‐representations. In the case when is spherical of minimal rank, we describe as the intersection of kernels of powers of root operators in some weight space of the dual space of . When , we recover by geometric methods a well‐known result.

Degenerations of spherical subalgebras and spherical roots

In this paper, we obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical roots of such a spherical subgroup.

Vanishing of certain equivariant distributions on spherical spaces for quasi-split groups

We prove the vanishing of z -eigen distributions on a quasi-split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup.

Parametrization, structure and Bruhat order of certain spherical quotients

Let G G be a reductive algebraic group and let Z Z be the stabilizer of a nilpotent element e e of the Lie algebra of G G . We consider the action of Z Z on the flag variety of G G , and we focus on the case where this action has a finite number of orbits (i.e., Z Z is a spherical subgroup). This holds for instance if e e has height 2 2 . In this case we give a parametrization of the Z Z -orbits and we show that each Z Z -orbit has a structure of algebraic affine bundle. In particular, in type A A , we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type A A , we show that the Bruhat order of the Z Z -orbits can be described in this way.

Stability conditions and braid group actions on affine An quivers

We study stability conditions on the Calabi–Yau-[Formula: see text] categories associated to an affine type [Formula: see text] quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order [Formula: see text]. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type [Formula: see text] based on the Khovanov–Seidel–Thomas method.

ON EXTENDED WEIGHT MONOIDS OF SPHERICAL HOMOGENEOUS SPACES

Given a connected reductive complex algebraic group $G$ and a spherical subgroup $H \subset G$, the extended weight monoid $\widehat \Lambda^+_G(G/H)$ encodes the $G$-module structures on spaces of global sections of all $G$-linearized line bundles on $G/H$. Assuming that $G$ is semisimple and simply connected and $H$ is specified by a regular embedding in a parabolic subgroup $P \subset G$, in this paper we obtain a description of $\widehat \Lambda^+_G(G/H)$ via the set of simple spherical roots of $G/H$ together with certain combinatorial data explicitly computed from the pair $(P,H)$. As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing $\widehat \Lambda^+_G(G/H)$ in the case where $H$ is strongly solvable.

An Improved Integrated Clustering Learning Strategy Based on Three-Stage Affinity Propagation Algorithm with Density Peak Optimization Theory

To better reflect the precise clustering results of the data samples with different shapes and densities for affinity propagation clustering algorithm (AP), an improved integrated clustering learning strategy based on three-stage affinity propagation algorithm with density peak optimization theory (DPKT-AP) was proposed in this paper. DPKT-AP combined the ideology of integrated clustering with the AP algorithm, by introducing the density peak theory and k-means algorithm to carry on the three-stage clustering process. In the first stage, the clustering center point was selected by density peak clustering. Because the clustering center was surrounded by the nearest neighbor point with lower local density and had a relatively large distance from other points with higher density, it could help the k-means algorithm in the second stage avoiding the local optimal situation. In the second stage, the k-means algorithm was used to cluster the data samples to form several relatively small spherical subgroups, and each of subgroups had a local density maximum point, which is called the center point of the subgroup. In the third stage, DPKT-AP used the AP algorithm to merge and cluster the spherical subgroups. Experiments on UCI data sets and synthetic data sets showed that DPKT-AP improved the clustering performance and accuracy for the algorithm.

ON THE REGULARITY OF D-MODULES GENERATED BY RELATIVE CHARACTERS

Following the ideas of Ginzburg, for a subgroup $K$ of a connected reductive $\mathbb{R}$-group $G$ we introduce the notion of $K$-admissible $D$-modules on a homogeneous $G$-variety $Z$. We show that $K$-admissible $D$-modules are regular holonomic when $K$ and $Z$ are absolutely spherical. This framework includes: (i) the relative characters attached to two spherical subgroups $H_1$ and $H_2$, provided that the twisting character $\chi_i$ factors through the maximal reductive quotient of $H_i$, for $i = 1, 2$; (ii) localization on $Z$ of Harish-Chandra modules; (iii) the generalized matrix coefficients when $K(\mathbb{R})$ is maximal compact. This complements the holonomicity proven by Aizenbud--Gourevitch--Minchenko. The use of regularity is illustrated by a crude estimate on the growth of $K$-admissible distributions which based on tools from subanalytic geometry.

The Infinitesimal Characters of Discrete Series for Real Spherical Spaces

Let $$Z=G/H$$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of G on $$L^2(Z)$$ . It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $$L^2(Z)$$ , have infinitesimal characters which are real and belong to a lattice. Moreover, let K be a maximal compact subgroup of G. Then each irreducible representation of K occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of H.

An Application of Spherical Geometry to Hyperkähler Slices

AbstractThis work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$, defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$. This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$, $n\geqslant 3$), prompting us to seek necessary and sufficient conditions for non-emptiness.We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$-regularity of $(G,H)$. This $\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$. We also provide a classification of the $\mathfrak{a}$-regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields

Let G be a split Kac-Moody group over a non-Archimedean local field. We define a completion of the Iwahori-Hecke algebra of G, then we compute its center and prove that it is isomorphic (via the Satake isomorphism) to the spherical Hecke algebra of G. This is thus similar to the situation for reductive groups. Our main tool is the masure ℐ associated to this setting, which plays here the same role as Bruhat-Tits buildings do for reductive groups. In a second part, we associate a Hecke algebra to each spherical face F of type 0, extending a construction that was only known, in the Kac-Moody setting, for the spherical subgroup and for the Iwahori subgroup.

Periods and harmonic analysis on spherical varieties

Given a spherical variety X for a group G over a non-archimedean local field k, the Plancherel decomposition for L^2(X) should be related to distinguished Arthur parameters into a dual group closely related to that defined by Gaitsgory and Nadler. Motivated by this, we develop, under some assumptions on the spherical variety, a Plancherel formula for L^2(X) up to discrete (modulo center) spectra of its boundary degenerations, certain G-varieties with more symmetries which model X at infinity. Along the way, we discuss the asymptotic theory of subrepresentations of C^{infty}(X), and establish conjectures of Ichino-Ikeda and Lapid-Mao. We finally discuss global analogues of our local conjectures, concerning the period integrals of automorphic forms over spherical subgroups.

Wonderful symmetric varieties and Schubert polynomials

Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of the general linear group on the flag variety. Combining this with a slight extension of results of Can, Joyce and Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms.

Orbits of strongly solvable spherical subgroups on the flag variety

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup \(H \subset B\) which acts with finitely many orbits on the flag variety G / B, and we classify the H-orbits in G / B in terms of suitable root systems. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G / B, and we give a combinatorial model for this action in terms of weight polytopes.

Spherical subgroups of Kac–Moody groups and transitive actions on spherical varieties

We define and study spherical subgroups of finite type of a Kac–Moody group. In analogy with the standard theory of spherical varieties, we introduce a combinatorial object associated with such a subgroup, its homogeneous spherical datum, and we prove that it satisfies the same axioms as in the finite-dimensional case. Our main tool is a study of varieties that are spherical under the action of a connected reductive group L, and come equipped with a transitive action of a group containing L as a Levi subgroup.

Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein–Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group \({{G}}\) defined over \(\mathbb {R}\), the space of \({{G(\mathbb {R})}}\)-equivariant distributions on the manifold of real points of any algebraic \({{G}}\)-manifold \({{X}}\) is finite-dimensional if \({{G}}\) has finitely many orbits on \({{X}}\).

Erratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle $${\mathcal {E}}$$ on X. Let $$\mathfrak {h}$$ be the Lie algebra of H. Let $$\mathcal {S}(X,{\mathcal {E}})$$ be the space of Schwartz sections of $${\mathcal {E}}$$ . We prove that $$\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})$$ is a closed subspace of $$\mathcal {S}(X,{\mathcal {E}})$$ of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $$\pi $$ be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants $$V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}$$ is an isomorphism if and only if any linear $$\mathfrak {h}$$ -invariant functional on V is continuous in the topology induced from $$\pi $$ . The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.

On the sphericity of the subgroup PSL2(ℝ) in the group of diffeomorphisms of the circle

We show that the group $PSL(2,R)$ is a spherical subgroup in the group of $C^3$-diffeomorphisms of the circle. Also, the group of automorphisms of a Bruhat--Tits tree is a spherical subgroup in the group of hierarchomorphisms of the tree.

Satake diagrams and real structures on spherical varieties

With each antiholomorphic involution [Formula: see text] of a connected complex semisimple Lie group [Formula: see text] we associate an automorphism [Formula: see text] of its Dynkin diagram. The definition of [Formula: see text] is given in terms of the Satake diagram of [Formula: see text]. Let [Formula: see text] be a self-normalizing spherical subgroup. If [Formula: see text] then we prove the uniqueness and existence of a [Formula: see text]-equivariant real structure on [Formula: see text] and on the wonderful completion of [Formula: see text].

ANTI-HOLOMORPHIC INVOLUTIONS AND SPHERICAL SUBGROUPS OF REDUCTIVE GROUPS

We study the action of an anti-holomorphic involution σ of a connected reductive complex algebraic group G on the set of spherical subgroups of G. The results are applied to σ-equivariant real structures on spherical homogeneous G-spaces admitting a wonderful embedding. Using combinatorial invariants of these varieties, we give an existence and uniqueness criterion for such real structures. We also investigate the associated real parts of the wonderful varieties.