Using gauge theory, we classify SU(m+1) -equivariant holomorphic maps from the projective space CP^m , equipped with the Fubini–Study metric, into the Grassmann manifold Gr_p(C^p+m+1) . We show that the moduli space of such maps, modulo image equivalence, can be regarded as a subset of the gauge equivalence classes of invariant connections on homogeneous vector bundles over CP^m satisfying semi-positivity.
Let B = ⊕ _i ∈ G B_i be a commutative integral domain of characteristic 0 graded by an abelian group G. We say that B is rigid (resp. graded-rigid) if the only locally nilpotent derivation (resp. homogeneous locally nilpotent derivation) of B is the zero derivation. Given a subgroup H of G, define B^(H) = ⊕ _i ∈ H B_i . We give results that answer or partially answer the following questions: Does non-rigidity of B imply non-rigidity of B^(H) ? When can a derivation of B^(H) be extended to one of B? What are the properties of the set of subgroups H of G such that B^(H) is not graded-rigid? We define the subgroups 𝔾̅(B) ⊆𝔾(B) of G and find that these are related to the locally nilpotent derivations of B in several interesting ways. (The definitions of 𝔾̅(B) and 𝔾(B) do not involve derivations, and these two groups are usually easy to determine.) One of our results states that if B is a normal affine G-graded domain then trdeg(B: ML(B)) ⩾rank( 𝔾(B)/𝔾̅(B) ) . We also give a result relating the rigidity of B_(x) to that of B/xB, where B is an ℕ -graded normal affine domain and x is a homogeneous prime element of B. We give some applications to Pham-Brieskorn rings.
We develop a theory of singular support for various infinite dimensional stacks and establish several functoriality properties. Then we apply this theory to prove existence theorems for singular support and compute it for various affine character sheaves.
This article proves an alcove walk description of intersections of Schubert cells and partially semi-infinite orbits (depending on a choice of parabolic subgroup) in the affine Grassmannian of a split connected reductive group (we call these intersections parabolic Mirković-Vilonen intersections). We deduce from this a parameterization of the irreducible components of the maximal possible dimension in these intersections by the alcove walks of maximal possible dimension. As a consequence we present a new combinatorial description of branching to Levi subgroups of irreducible highest weight representations, and in particular, we give a new algorithm for computing the characters of such representations.
Let A be an affine factorial domain over a field K of characteristic zero endowed with an irreducible locally nilpotent derivation ξ . Assume that ξ has the freeness property, its kernel is affine over K and its plinth ideal is generated by a power of a prime element in (ξ ) . The main result of this paper asserts that the differential algebra (A,ξ ) is K -isomorphic to the coordinate ring of a generalized Danielewski variety endowed with a Jacobian-type derivation.
We investigate the homotopy type of a certain homogeneous space for a simple complex Lie group. We calculate some of its classical topological invariants and introduce a new one. We also propose several conjectures about its topological rigidity.
Let p be a proper parabolic subalgebra of a simple Lie algebra g. Writing p = r circle plus m with r being the standard Levi factor of p and m the nilpotent radical of p, we consider the In & ouml;n & uuml;-Wigner contraction p of p with respect to this decomposition : this is the Lie algebra which is the semi-direct product r & ltimes; ma, where ma is an abelian ideal of p, isomorphic to m as an r-module. The study of the algebra of symmetric semi-invariants Sy(p) in the symmetric algebra S(p) of punder the adjoint action of pwas initiated in Fauquant-Millet (2025), wherein a lower bound for the formal character of the algebra Sy(p) was built, when the latter is well defined. Here in this paper we build an upper bound for this formal character, when p is a maximal parabolic subalgebra in a simple Lie algebra g in type B, whose Levi subalgebra is associated with the set of all simple roots without a simple root of even index, using Bourbaki notation (we call this case the even case). We show that both bounds coincide. This provides a Weierstrass section for Sy(p) and the polynomiality of Sy(p) follows. As a by-product, we obtain that the derived subalgebra p ' of p is nonsingular that is, the set of regular elements in the dual space of p ' is big.
In this paper we give a new formula for the characters of finite-dimensional irreducible 𝔤𝔩(m,n) -modules. We follow the same way as Su and Zhang did. First we give a new proof and a new formulation of the conjecture of Van der Jeugt, Hughes, King and Thierry-Mieg using weight and cap diagrams introduced by J. Brundan and C. Stroppel. Then we calculate the generating function of integer points of the corresponding polyhedron. Su and Zhang calculated that function by representing the polyhedron as the union of fundamental domains under the action of symmetric group. In our approach we calculate the generating function by means of Brion’s theorem.
This paper studies automorphisms and monomorphisms of direct products Γ =Γ _1×⋯×Γ _r of finitely generated virtually solvable minimax groups, a class containing all virtually polycyclic groups. Under an indecomposability assumption on the ℚ -algebraic hulls, we prove that every monomorphism of Γ factorizes uniquely as φ =θ·ζ , where θ sends each factor into a permuted factor with ℚ -isomorphic hull and ζ is central and off-diagonal. Conversely, every such pair defines a monomorphism of Γ , and φ is an automorphism if and only if θ is. This indecomposability assumption is sharp: we show it cannot be weakened to direct indecomposability of the factors. The proof proceeds in three steps: first by establishing the corresponding central mixing property for finite-dimensional Lie algebras and algebraic Lie algebras, then for connected linear algebraic groups, and finally by transferring these results to minimax groups via ℚ -algebraic hulls. This extends the previously known nilpotent case both from automorphisms to monomorphisms and from finitely generated torsion-free nilpotent groups to the broader class of finitely generated virtually solvable minimax groups. As applications, we characterize co-Hopfian direct products and derive formulas for Reidemeister numbers and Reidemeister spectra.
Approximate lattices are a class of approximate subgroups (i.e. subsets of groups closed under multiplication up to a finite error) that generalise lattices of locally compact groups. We provide and motivate in this paper a natural framework for the study of approximate lattices. Namely, we consider approximate lattices in so-called S-adic linear groups and define relevant notions of arithmeticity involving Pisot numbers. We also adapt to this framework classical results of the theory of lattices and Meyer sets. Results from this paper will play a role in the proof of a structure theorem for approximate lattices in S-adic linear groups which is the subject of a companion paper. We extend a theorem of Schreiber’s concerning the coarse structure of approximate subgroups in Euclidean spaces to approximate subgroups of unipotent S-adic groups. We generalise Meyer’s structure theorem for approximate lattices in locally compact abelian groups to a precise structure theorem for approximate lattices in unipotent S-adic groups. Finally, we study intersections of approximate lattices of S-adic linear groups with certain subgroups such as the nilpotent radical and Levi subgroups in the spirit of a theorem of Bieberbach. We furthermore show that the framework of S-adic linear groups enables us to provide statements more precise than earlier results.
Let k be a field with char(k) 2 . We prove that all maximal flags of composition algebras over k, appear as the k-rational Sp_6 -orbits in a Zariski-dense Sp_6 -invariant subset V^ss⊂ V=∧ ^3V_6 , where V_6 is the standard 6-dimensional irreducible representation of Sp_6 . This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces (Sp_6× GL_1^2,V) and (GSp_6× GL_1^2,V) . We also get all reduced Freudenthal algebras of dimensions 6 and 9, represented by the same orbit spaces.
The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let A be a regular domain and let K be its field of fractions. Let G⊆ GL_n(A) be a finite group. Let G act linearly on A[X_1,X_2,… , X_n] (fixing A). Assume that |G| is invertible in A. We prove that G⊆ GL_n(K) is generated by pseudo-reflections if and only if (A[X_1,X_2,… , X_n])^G is regular.
We provide exact integral formulas for hyperbolic and spherical volumes of cone-manifolds whose underlying space is the $3$-sphere and whose singular set belongs to three infinite families of two-bridge knots: $C(2n,2)$ (twist knots), $C(2n,3)$, and $C(2n,-2n)$ for any non-zero integer $n$. Our formulas express volumes as integrals of explicit rational functions involving Chebyshev polynomials of the second kind, with integration limits determined by roots of algebraic equations. This extends previous work where only implicit formulas requiring numerical approximation were known.
We investigate Hamiltonian actions of non-compact Lie groups on a homogeneous bounded domain X. As a main result, we point out a Lie-theoretical condition for a closed Lie group H of the automorphism group of X which ensures that the symplectic reduction μ ^-1(0)/H with respect to the momentum map μ at hand, is a Stein manifold. Moreover, for the class of connected subgroups of translations the quotient (H^ℂ· X)/H^ℂ is realized as a Siegel domain and we show that the symplectic reduction μ ^-1(0)/H is biholomorphic to such a Stein quotient.
We consider twisted conjugacy classes of continuous automorphisms φ of a Lie group G . We obtain a necessary and sufficient condition on φ for its Reidemeister number, the number of twisted conjugacy classes, to be infinite when G is connected and solvable or compactly generated and nilpotent. We also show for a general connected Lie group G that the number of conjugacy classes is infinite. We prove that for a connected non-nilpotent Lie group G , there exists n∈ℕ such that the Reidemeister number of φ ^n is infinite for every φ . We say that G has topological R_∞ -property if the Reidemeister number of every φ is infinite. We obtain conditions on a connected solvable Lie group under which it has topological R_∞ -property; which, in particular, enables us to prove that the group of invertible n×n upper triangular real matrices and its quotient group modulo its center have topological R_∞ -property for every n≥2 . We also prove that the Walnut group also has this property. We show that 𝐒𝐋(2, ℝ) and GL(2, ℝ) have topological R_∞ -property, and construct many examples of Lie groups with this property.
We define the notion of a Lie superalgebra over a field k of characteristic 2 which unifies the two pre-existing ones – ℤ/2 -graded Lie algebras with a squaring map and Lie algebras in the Verlinde category Ver_4^+(k) , and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, we discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect k), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero – the notion of a mixed Lie superalgebra over a ramified quadratic extension R of the ring of Witt vectors W(k).
