Showing papers in "Transformation Groups in 2020"

Braid group action and root vectors for the q-Onsager algebra

Abstract: We define two algebra automorphisms Ͳ0 and Ͳ1 of the q-Onsager algebra $$ {\mathcal{B}}_c $$, which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a PBW basis for $$ {\mathcal{B}}_c $$. We show that the root vectors satisfy q-analogs of Onsager's original commutation relations. The paper is much inspired by I. Damiani's construction and investigation of root vectors for the quantized enveloping algebra of $$ \hat{\mathfrak{s}{\mathfrak{l}}_2} $$.

The grothendieck–serre conjecture over semilocal dedekind rings

Abstract: For a reductive group scheme G over a semilocal Dedekind ring R with total ring of fractions K, we prove that no nontrivial G-torsor trivializes over K. This generalizes a result of Nisnevich–Tits, who settled the case when R is local. Their result, in turn, is a special case of a conjecture of Grothendieck–Serre that predicts the same over any regular local ring. With a patching technique and weak approximation in the style of Harder, we reduce to the case when R is a complete discrete valuation ring. Afterwards, we consider Levi subgroups to reduce to the case when G is semisimple and anisotropic, in which case we take advantage of Bruhat–Tits theory to conclude. Finally, we show that the Grothendieck–Serre conjecture implies that any reductive group over the total ring of fractions of a regular semilocal ring S has at most one reductive S-model.

Smoothness conditions in cohomogeneity one manifolds

Abstract: We present an efficient method for determining the conditions that a metric on a cohomogeneity one manifold, defined in terms of functions on the regular part, needs to satisfy in order to extend smoothly to the singular orbit.

A matsumoto–mostow result for zimmer’s cocycles of hyperbolic lattices

Abstract: Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign. As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles. In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

q -SCHUR ALGEBRAS CORRESPONDING TO HECKE ALGEBRAS OF TYPE B

Abstract: In this paper the authors investigate the q-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type Α. The authors present a coordinate algebra type construction that allows us to realize these q-Schur algebras as the duals of the dth graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the q-Schur algebra of type Α. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the 1-faithful quasi hereditary covers of the Hecke algebras of type Β. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to the category 𝒪 for rational Cherednik algebras for the Weyl group of type Β. In particular, we have introduced a Schur-type functor that identifies the type Β Knizhnik–Zamolodchikov functor.

Extremal rays in the hermitian eigenvalue problem for arbitrary types

Abstract: The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of G = SL(n). There is a polyhedral cone (the \eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups G, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of G. We give a description of the extremal rays of the eigencones for arbitrary semisimple groups G by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face. The remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize [B3] which handled the case of G = SL(n).

Inertia groups and uniqueness of holomorphic vertex operator algebras

Abstract: We continue our program on classiffication of holomorphic vertex operator algebras of central charge 24. In this article, we show that there exists a unique strongly regular holomorphic VOA of central charge 24, up to isomorphism, if its weight one Lie algebra has the type C4,10, D7,3A3,1G2,1, A5,6C2,3A1,2, A3,1C7,2, D5,4C3,2A $$ {A}_{1,1}^2 $$ , or E6,4C2,1A2,1. As a consequence, we have verified that the isomorphism class of a strongly regular holomorphic vertex operator algebra of central charge 24 is determined by its weight one Lie algebra structure if the weight one subspace is nonzero.

ADMISSIBLE LEVEL osp 1 2 $$ \mathfrak{osp}\left(1\left|2\right.\right) $$ MINIMAL MODELS AND THEIR RELAXED HIGHEST WEIGHT MODULES

Abstract: The minimal model $$ \mathfrak{osp}\left(1|2\right) $$ vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra $$ \hat{\mathfrak{osp}}\left(1\left|2\right.\right) $$ at certain rational values of the level k. We classify all isomorphism classes of ℤ2-graded simple relaxed highest weight modules over the minimal model $$ \mathfrak{osp}\left(1|2\right) $$ vertex operator superalgebras in both the Neveu–Schwarz and Ramond sectors. To this end, we combine free field realisations, screening operators and the theory of symmetric functions in the Jack basis to compute explicit presentations for the Zhu algebras in both the Neveu–Schwarz and Ramond sectors. Two different free field realisations are used depending on the level. For k < −1, the free field realisation resembles the Wakimoto free field realisation of affine $$ \mathfrak{sl}(2) $$ and is originally due to Bershadsky and Ooguri. It involves 1 free boson (or rank 1 Heisenberg vertex algebra), one βγ bosonic ghost system and one bc fermionic ghost system. For k > −1, the argument presented here requires the bosonisation of the βγ system by embedding it into an indefinite rank 2 lattice vertex algebra.

Highest weight harish-chandra supermodules and their geometric realizations

Abstract: In this paper we discuss the highest weight $$ {\mathfrak{k}}_r $$-finite representations of the pair (𝔤r, $$ {\mathfrak{k}}_r $$) consisting of 𝔤r, a real form of a complex basic Lie superalgebra of classical type 𝔤 (𝔤 ≠ A(n, n)), and the maximal compact subalgebra $$ {\mathfrak{k}}_r $$ of 𝔤r,0, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces Gr/Kr.

Auslander’s theorem for group coactions on noetherian graded down-up algebras

Abstract: We prove a version of a theorem of Auslander for finite group coactions on noetherian graded down-up algebras.

Tannakian classification of equivariant principal bundles on toric varieties

Abstract: Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.

Almost fixed points of finite group actions on manifolds without odd cohomology

Abstract: If X is a smooth manifold and $$ \mathcal{G} $$ is a subgroup of Diff(X) we say that (X, $$ \mathcal{G} $$ ) has the almost fixed point property if there exists a number C such that for any finite subgroup G ≤ $$ \mathcal{G} $$ there is some x ∈ X whose stabilizer Gx ≤ $$ \mathcal{G} $$ satisfies [G : Gx] ≤ C. We say that X has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if X is compact and possibly with boundary and has no odd cohomology then (X, Diff(X)) has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if Z is a non-necessarily compact smooth real affine variety, and Z has no odd cohomology, then (Z, Aut(Z)) has the almost fixed point property, where Aut(Z) is the group of algebraic automorphisms of Z lifting the identity on Spec ℝ. The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λ-stability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.

Gelfand–tsetlin degenerations of representations and flag varieties

Abstract: Our main goal is to show that the Gelfand–Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such frameworks for all Grobner degenerations intermediate between the flag variety and the GT toric variety. These degenerations are shown to induce filtrations on the irreducible representations and the associated graded spaces are acted upon by a certain associative algebra. To achieve our goal, we construct embeddings of our Grobner degenerations into the projectivizations of said associated graded spaces in terms of this action. We also obtain an explicit description of the maximal cone in the tropical flag variety that parametrizes the Grobner degenerations we consider. In an addendum we propose an alternative solution to the problem which relies on filtrations and gradings by non-abelian ordered semigroups.

Decompositions of bernstein–sato polynomials and slices

Abstract: Let G be a linearly reductive group acting on a vector space V, and f a semi-invariant polynomial on V. In this paper we study systematically decompositions of the Bernstein–Sato polynomial of f in parallel with some representation-theoretic properties of the action of G on V. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein–Sato polynomials of several classical invariants in an elementary fashion. Furthermore, we derive a “slice method” which shows that the decomposition of V as a representation of G can induce a decomposition of the Bernstein–Sato polynomial of f into a product of two Bernstein–Sato polynomials – that of an ideal and that of a semi-invariant of smaller degree. Using the slice method, we compute Bernstein–Sato polynomials for a large class of semi-invariants of quivers.

Singular riemannian foliations and their quadratic basic polynomials

Abstract: We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyl's First Fundamental Theorems, provides a characterization of the recently discovered Clifford foliations in terms of basic polynomials. This link also yields new structural results about infinitesimal foliations, such as the existence of non-trivial symmetries.

Geometry of regular Hessenberg varieties

Abstract: Let $$ \mathfrak{g} $$ be a complex semisimple Lie algebra. For a regular element x in $$ \mathfrak{g} $$ and a Hessenberg space H ⊆ $$ \mathfrak{g} $$, we consider a regular Hessenberg variety X(x, H) in the ag variety associated with $$ \mathfrak{g} $$. We take a Hessenberg space so that X(x, H) is irreducible, and show that the higher cohomology groups of the structure sheaf of X(x, H) vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.

Takiff algebras with polynomial rings of symmetric invariants

Abstract: Extending results of Rais–Tauvel, Macedo–Savage, and Arakawa–Premet, we prove that under mild restrictions on the Lie algebra $$ \mathfrak{q} $$ having the polynomial ring of symmetric invariants, the m-th Takiff algebra of $$ \mathfrak{q} $$, $$ \mathfrak{q} $$⟨m⟩, also has a polynomial ring of symmetric invariants.

Stratified hyperkähler spaces from semisimple lie algebras

Abstract: We study singular hyperkahler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkahler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie theoretic data.

Extensions of isomorphisms of subvarieties in flexible varieties

Abstract: Let X be an algebraic variety isomorphic to the complement of a closed subvariety of dimension at most n − 3 in $$ {\mathbbm{A}}_{\mathrm{k}}^n $$. We find some conditions under which an isomorphism of two closed subvarieties of X can be extended to an automorphism of X. We also study the similar problem for subvarieties of affine quadrics and SL(n, k).

Equivariant models of spherical varieties

Abstract: Let G be a connected semisimple group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space of G, and let Y′ be a spherical embedding of Y. Let k0 be a subfield of k. Let G0 be a k0-model (k0-form) of G. We show that if G0 is an inner form of a split group and if the subgroup H of G is spherically closed, then Y admits a G0-equivariant k0-model. If we replace the assumption that H is spherically closed by the stronger assumption that H coincides with its normalizer in G, then Y and Y′ admit compatible G0-equivariant k0-models, and these models are unique.

Generically free representations III: extremely bad characteristic

Abstract: In parts I and II, we determined which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero, with some assumptions on the characteristic of the field. This paper settles the remaining cases, which are of a different nature because Lie(G) has a more complicated structure and there need not exist general dimension bounds of the sort that exist in good characteristic.

Adjunction for varieties with a ℂ* action

Abstract: Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

Pseudocharacters of homomorphisms into classical groups

Abstract: A GLd-pseudocharacter is a function from a group Γ to a ring k satisfying polynomial relations that make it “look like” the character of a representation. When k is an algebraically closed field of characteristic 0, Taylor proved that GLd-pseudocharacters of Γ are the same as degree-d characters of Γ with values in k, hence are in bijection with equivalence classes of semisimple representations Γ → GLd(k). Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group H over an algebraically closed field k of characteristic 0 and for any group Γ, there exists an infinite collection of functions and relations which are naturally in bijection with H(k)-conjugacy classes of semisimple homomorphisms Γ→ H(k). In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG algebra. We then define generating sets and generating relations for these objects and show that, for all H as above, the corresponding FFG-algebra is finitely presented up to radical. Hence one can always define H-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations up to radical of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of homomorphisms, following Larsen.

Local formulas for multiplicative forms

Abstract: We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. In particular, we obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis–Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids, respectively.

On the regularity of D-modules generated by relative characters

Abstract: Following the ideas of Ginzburg, for a subgroup K of a connected reductive 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 H1 and H2, provided that the twisting character χi factors through the maximal reductive quotient of Hi, for i = 1; 2; (ii) localization on Z of Harish-Chandra modules; (iii) the generalized matrix coeficients when K(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 based on tools from subanalytic geometry.

Algebraic realization for cyclic group actions with one isotropy type

Abstract: Suppose G is a cyclic group and M a closed smooth G-manifold with exactly one isotropy type. We will show that there is a nonsingular real algebraic G-variety X such that X is equivariantly diffeomorphic to M and all G-vector bundles over X are strongly algebraic.

On tensoring with the steinberg representation

Abstract: Let G be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of G and another on the existence of certain filtrations of G-modules. A key question related to these conjectures is whether the tensor product of the rth Steinberg module with a simple module with prth restricted highest weight admits a good filtration. In this paper we verify this statement (i) when p ≥ 2h − 4 (h is the Coxeter number), (ii) for all rank two groups, (iii) for p ≥ 3 when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.

SIMPLE BOUNDED WEIGHT MODULES OF sl ∞ , o ∞ , sp ∞ $$ \mathfrak{sl}\left(\infty \right),\kern0.5em \mathfrak{o}\left(\infty \right),\kern0.5em \mathfrak{sp}\left(\infty \right) $$

Abstract: We classify the simple bounded weight modules of the Lie algebras $$ \mathfrak{sl}\left(\infty \right),\kern0.5em \mathfrak{o}\left(\infty \right) $$ and $$ \mathfrak{sp}\left(\infty \right) $$ , and compute their annihilators in $$ U\left(\mathfrak{sl}\left(\infty \right)\right),\kern0.5em U\left(\mathfrak{o}\left(\infty \right)\right),\kern0.5em U\left(\mathfrak{sp}\left(\infty \right)\right) $$ , respectively.

LAGRANGIAN FIBERS OF GELFAND–CETLIN SYSTEMS OF SO( n )-TYPE

Abstract: In this paper, we study the Gelfand–Cetlin systems and polytopes of the co-adjoint SO(n)-orbits. We describe the face structure of Gelfand–Cetlin polytopes and iterated bundle structure of Gelfand–Cetlin fibers in terms of combinatorics on the ladder diagrams. Using this description, we classify all Lagrangian fibers.

The exceptional tits quadrangles

Abstract: A Tits polygon is a bipartite graph in which the neighborhood of each vertex is endowed with an “opposition relation” satisfying certain axioms. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. Every Tits polygon has a distinguished set of circuits. A Tits quadrangle is a Tits polygon in which these circuits all have length 8. There is a standard construction that produces a Tits polygon from certain pairs (∆, T), where ∆ is an irreducible spherical building and T is a Tits index of relative rank 2. We call a Tits quadrangle exceptional if it arises from such a pair (∆, T) for ∆ the spherical building associated to the group of rational points of an exceptional algebraic group. In this paper, we characterize the exceptional Tits quadrangles as extensions of orthogonal Tits quadrangles in a suitable sense.