Abstract: A local Riemann–Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on G-bundles and on parabolic G-bundles. The corresponding Betti data involves pairs (M, P) consisting of the local monodromy M ∈ G and a (weighted) parabolic subgroup P ⊂ G such that M ∈ P, as in the multiplicative Brieskorn–Grothendieck–Springer resolution (extended to the parabolic case). The natural quasi-Hamiltonian structures that arise on such spaces of enriched monodromy data will also be constructed.

Abstract: Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours.

Abstract: We give a presentation for the cohomology algebra of the Spaltenstein variety of all partial flags annihilated by a fixed nilpotent matrix, generalizing the description of the cohomology algebra of the Springer fiber found by De Concini, Procesi and Tanisaki.

Abstract: Let G be a finite group. Given a finite G-set \(\cal{X}\) and a modular tensor category \(\cal{C}\), we construct a weak G-equivariant fusion category \(\cal{C}^{\cal{X}}\), called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for \(\mathbb{Z}/2\)-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper.

Abstract: A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the only other known singular Dirac reduction method.

Abstract: The Danielewski hypersurfaces are the hypersurfaces X Q,n in $ {\mathbb{C}^3} $ defined by an equation of the form x n y = Q(x, z) where n ⩾ 1 and Q(x, z) is a polynomial such that Q(0, z) is of degree at least two. They were studied by many authors during the last twenty years. In the present article, we give their classification as algebraic varieties. We also give their classification up to automorphism of the ambient space. As a corollary, we obtain that every Danielewski hypersurface X Q,n with n ⩾ 2 admits at least two nonequivalent embeddings into $ {\mathbb{C}^3} $ .

Abstract: Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H ⊂ P(V) is a spherical orbit and if $$ X = \overline {G/H} $$ is its closure, then we describe the orbits of X and those of its normalization $$ \tilde{X} $$ . If, moreover, the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism $$ \tilde{X} \to X $$ is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.

Abstract: In [AB05], Alexeev and Brion have introduced the notion of invariant Hilbert schemes. We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of SL2 on \( {\left( {{\mathbb{C}^2}} \right)^{ \oplus 6}} \) as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for realizing these calculations. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.

Abstract: Let G be an almost simple reductive group with Weyl group W. Let B be a Borel subgroup of G. Let w be an elliptic element of W which has minimal length in its conjugacy class. We show that in almost all cases there exists a semisimple class in G whose intersection with BwB has dimension dim(B).

Abstract: The Birman-Murakami-Wenzl algebras (BMW algebras) of type E_ n for n = 6, 7, 8 are shown to be semisimple and free over the integral domain Z[δ^(±1),l^(±1),m]/(m1−δ)(l−l^(−1)) of ranks 1,440,585; 139,613,625; and 53,328, 069,225. We also show they are cellular over suitable rings. The Brauer algebra of type E_n is a homomorphic ring image and is also semisimple and free of the same rank as an algebra over the ring Z[δ^(±1)]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. The generalized Temperley-Lieb algebra of type E_n turns out to be a subalgebra of the BMW algebra of the same type. So, the BMW algebras of type E_n share many structural properties with the classical ones (of type A_n) and those of type D_n .

Abstract: By far the most important class of pseudo-groups, both for theory and in essentially all applications, are the Lie pseudo-groups. In this paper we propose a definition of the Lie completion of a regular pseudo-group, and establish some of its basic properties. In particular, a pseudo-group and its Lie completion have exactly the same differential invariants and invariant differential forms. Thus, for practical purposes, one can exclusively work within the category of Lie pseudo-groups.

Abstract: We study finitely-generated nondiscrete free subgroups in Lie groups. We address the following question first raised by Etienne Ghys: is it always possible to make arbitrarily small perturbations of the generators of the free subgroup in such a way that the new group formed by the perturbed generators be not free? In other words, is it possible to approximate generators of a free subgroup by elements satisfying a nontrivial relation? We prove that the answer to Ghys’ question is positive and generalize this result to certain nonfree subgroups. We also consider the question on the best approximation rate in terms of the minimal length of relation in the approximating group. We give an upper bound on the optimal approximation rate as $$ {e^{ - c{l^\kappa }}} $$ , where c > 0 is a constant, l the minimal length of relation and 0.19 < κ < 0.2.

Abstract: This paper is a follow-up to the paper I. Agol, M. Belolipetsky, P. Storm, K. Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups, Geometry, and Dynamics 2 (2008), 481–498. The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reection groups.

Abstract: We completely classify the real root subsystems of root systems of loop algebras of Kac–Moody Lie algebras. This classification involves new notions of “admissible subgroups” of the coweight lattice of a root system Ψ, and “scaling functions” on Ψ. Our results generalise and simplify earlier work on subsystems of real affine root systems.

Abstract: We define and study the variety of reductions for a complex reductive symmetric pair (G, θ), which is the natural compactification of the set of its Cartan subspaces. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties. We develop a theoretical basis to the study of these varieties of reductions, and relate their geometry to some problems in representation theory. A very useful result is the rigidity of semisimple elements in deformations of algebraic subalgebras of Lie algebras. We use it to show that the closure of a decomposition class is a union of decomposition classes. We apply this theory to the study of other varieties of reductions in a companion paper, which yields two new Fano varieties.

Abstract: The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each “sufficiently rich” spherical building Y of type W we associate a certain cohomology theory \( H_{BK}^*(Y) \) and verify that, first, it depends only on W (i.e., all such buildings are “homotopy equivalent”), and second, \( H_{BK}^*(Y) \) is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology “pre-ring” on Y. The convex “stability” cones in \( {\left( {{\mathbb{R}^2}} \right)^m} \) defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of [KLM1]; equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. The independence of the (co)homology theory of Y refines the result of [KLM2], which asserted that the Stability Cone depends on W rather than on Y. Quite remarkably, the cohomology ring \( H_{BK}^*(Y) \) is obtained from a certain universal algebra At by a kind of “crystal limit” that has been previously introduced by Belkale–Kumar for the cohomology of ag varieties and Grassmannians. Another degeneration of At leads to the homology theory H*(Y).

Abstract: Let V n be the SL2-module of binary forms of degree n and let $ V = {V_{{n_1}}} \oplus \cdots \oplus {V_{{n_p}}} $ . We consider the algebra $ R = \mathcal{O}{(V)^{{\text{S}}{{\text{L}}_2}}} $ of polynomial functions on V invariant under the action of SL2. The measure of the intricacy of these algebras is the length of their chains of syzygies, called homological dimension hd R. Popov gave in 1983 a classification of the cases in which hd R ≤ 10 for a single binary form (p = 1) or hd R ≤ 3 for a system of two or more binary forms (p > 1). We extend Popov’s result and determine for p = 1 the cases with hd R ≤ 100, and for p > 1 those with hd R ≤ 15. In these cases we give a set of homogeneous parameters and a set of generators for the algebra R.

Abstract: Let G be a semisimple linear algebraic group over $ \mathbb{C} $ without G 2-factors, B a Borel subgroup of G and T ⊂ B a maximal torus. The flag variety G/B is a projective G-homogeneous variety whose tangent space at the identity coset is isomorphic, as a B-module, to $ {{\mathfrak{g}} \left/ {\mathfrak{b}} \right.} $ , where $ \mathfrak{g} $ = Lie(G) and $ \mathfrak{b} $ = Lie(B). Recall that if w is an element of the Weyl group W of the pair (G, T), the Schubert variety X(w) in G/B is by definition the closure of the Bruhat cell BwB. In this paper we prove that X(w) is nonsingular if and only if: (1) its Poincare polynomial is palindromic; and (2) the tangent space TE(X(w)) to the set T-stable curves in X(w) through the identity is a B-submodule of $ {{\mathfrak{g}} \left/ {\mathfrak{b}} \right.} $ . The second condition can be interpreted as saying that the roots of (G, T) in the convex hull of a certain set of roots canonically associated to w arise as tangent weights to T-stable curves in X(w) at the identity. A corollary is that X(w) is smooth if and only if X(w -1) is smooth. Condition (2) also gives a pattern avoidance criterion for TE(X(w)) to be B-stable.

Abstract: The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of V. L. Popov. In one of his papers, the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. V. L. Popov also suggests that his arguments and methods are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain some results on the irregular locus of the component of maximal dimension of the “symmetric commuting variety”. As a by-product, we study some pairs of commuting elements specific to the symmetric case, that we call rigid pairs. These pairs allow us to find all symmetric Lie algebras whose commuting variety is reducible.

Abstract: We compute the expected degree of a randomly chosen element in a basis of weight vectors in the Demazure module Vw(Λ) of \( {\hat{\mathfrak{sl}}_2} \). We obtain en passant a new proof of Sanderson's dimension formula for these Demazure modules.

Abstract: The goal of Murnaghan-Kirillov theory is to associate to an irreducible smooth representation of a reductive p-adic group a family of regular semisimple orbital integrals in the Lie algebra with the following property: the character of π is given, on a well determined set, by an explicit combination of the Fourier transforms of these orbital integrals. Subject to certain restrictions, we adapt arguments of Waldspurger to show that, for depth-zero irreducible smooth supercuspidal representations, this problem may be reduced to a similar one for distributions associated to Lusztig functions.

Abstract: Let \( \mathfrak{a} \) be an algebraic Lie subalgebra of a simple Lie algebra \( \mathfrak{g} \) with index \( \mathfrak{a} \) ≤ rank \( \mathfrak{g} \). Let \( Y\left( \mathfrak{a} \right) \) denote the algebra of \( \mathfrak{a} \) invariant polynomial functions on \( {\mathfrak{a}^*} \). An algebraic slice for \( \mathfrak{a} \) is an affine subspace η + V with \( \eta \in {\mathfrak{a}^*} \) and \( V \subset {\mathfrak{a}^*} \) subspace of dimension index \( \mathfrak{a} \) such that restriction of function induces an isomorphism of \( Y\left( \mathfrak{a} \right) \) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which \( h \in \mathfrak{a} \) is ad-semisimple, η is a regular element of \( {\mathfrak{a}^*} \) which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to \( \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta \) in \( {\mathfrak{a}^*} \). The classical case is for \( \mathfrak{g} \) semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \( \mathfrak{g} \). Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \( \mathfrak{a} \) be a truncated biparabolic of index one. (This only arises if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \( \mathfrak{a} \) is the restriction of a particularly carefully chosen regular nilpotent element of \( \mathfrak{g} \). A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.

Abstract: Smale proved that the orientation-preserving diffeomorphism group of \({\mathbb{S}^{2}}\) has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.

Abstract: Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ and G k (resp. H k ) the set of k-rational points of G (resp. H). The variety G k /H k is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space L 2(G k /H k ) of square integrable functions decomposes into several series, one for each H k -conjugacy class of Cartan subspaces of G k /H k . In this paper we give a characterization of the H k -conjugacy classes of these Cartan subspaces in the case that there exists a splitting extension of order 2 and (G, σ) is (σ, k)-split conjugate (see Subsection 3.8). This condition is satisfied for k the real numbers and several other fields for which the symmetric k-variety has a splitting extension of order 2. For $$ k = \mathbb{R} $$ we prove a number of additional results as well.

Abstract: Let \( \mathfrak{g} \) be a reductive Lie algebra and \( \mathfrak{k} \subset \mathfrak{g} \) be a reductive in \( \mathfrak{g} \) subalgebra. A (\( \mathfrak{g},\mathfrak{k} \))-module M is a \( \mathfrak{g} \)-module for which any element m ∈ M is contained in a finite-dimensional \( \mathfrak{k} \)-submodule of M. We say that a (\( \mathfrak{g},\mathfrak{k} \))-module M is bounded if there exists a constant CM such that the Jordan-Holder multiplicities of any simple finite-dimensional \( \mathfrak{k} \)-module in every finite-dimensional \( \mathfrak{k} \)-submodule of M are bounded by CM. In the present paper we describe explicitly all reductive in \( \mathfrak{s}{\mathfrak{l}_n} \) subalgebras \( \mathfrak{k} \) which admit a bounded simple infinite-dimensional (\( \mathfrak{s}{\mathfrak{l}_n},\mathfrak{k} \))-module. Our technique is based on symplectic geometry and the notion of spherical variety. We also characterize the irreducible components of the associated varieties of simple bounded (\( \mathfrak{g},\mathfrak{k} \))-modules.

Abstract: It is well known that the category of real Lie supergroups is equivalent to the category of the so-called (real) Harish-Chandra pairs, see [DM], [Kost], [Kosz]. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra. In this paper we give a proof of this result in the complex-analytic case. Furthermore, if (G, \( \mathcal{O} \)G) is a complex Lie supergroup and H ⊂ G is a closed Lie subgroup, i.e., it is a Lie subsupergroup of (G, \( \mathcal{O} \)G) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H, \( \mathcal{O} \)G/H) is split. In particular, any complex Lie supergroup is a split supermanifold.

Abstract: We give a simplified proof of the Tits classification of semisimple algebraic groups that remains valid over semilocal rings. We also give a new proof of the existence of all indices of exceptional inner type using the notion of canonical dimension of projective homogeneous varieties.

Abstract: We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric Tn−1 action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of \( \mathbb{R}{P^2} \), which is not a manifold.

Abstract: In this note we study the equivariant cohomology with compact supports of the zeroes of the moment map for the cotangent bundle of a linear representation of a torus and some of its notable subsets, using the theory of the infinitesimal index, developed in [8]. We show that, in analogy to the case of equivariant K-theory dealt with in [7] using the index of transversally elliptic operators, we obtain isomorphisms with notable spaces of splines studied in [2], [3].