arXiv subjects

Friedrich Knop

Publications and source records attributed to Friedrich Knop.

At least 37 records · Page 2Linked to original sources

arXiv2015

On the W-action on B-sheets in positive characteristic

Let G be a connected reductive group defined over an algebraically closed ground field of characteristic p, let B be a Borel subgroup of G, and let X be a G-variety. The first named author has shown that for p = 0 there is a natural action of the Weyl group W on the (finite) set of closed B-invariant subvarieties of X that are of maximal modularity, and conjectured that the same construction yields a W-action whenever p is different from 2. In the present paper we prove this conjecture.

math.AG↗

arXiv2014

The local structure theorem for real spherical varieties

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q \times_L S$. Here $Q$ is a parabolic subgroup with Levi decomposition $LU$, and $S$ is a homogeneous space for a quotient $D=L/L_n$ of $L$, where $L_n$ is normal in $L$, such that $D$ is compact modulo center.

math.RT↗

arXiv2014

Spherical subgroups in simple algebraic groups

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is called spherical provided it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer's list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, there is no classification of all such instances in positive characteristic to date. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic $2$ which has no counterpart in Krämer's classification. As one of our key tools, we prove a general deformation result for subgroup schemes allowing us to deduce the sphericality of subgroups in positive characteristic from this property for subgroups in characteristic $0$.

math.GR↗

arXiv2014

Simple compactifications and polar decomposition of homogeneous real spherical spaces

Let Z be an algebraic homogeneous space Z=G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper.

math.RT↗

arXiv2014

Spherical roots of spherical varieties

Brion proved that the valuation cone of a complex spherical variety is a fundamental domain for a finite reflection group, called the little Weyl group. The principal goal of this paper is to generalize this fundamental theorem to fields of characteristic unequal to 2. We also prove a weaker version which holds in characteristic 2, as well. Our main tool is a generalization of Akhiezer's classification of spherical rank-1-varieties.

math.RT↗

arXiv2013

Localization of spherical varieties

We prove some fundamental structural results for spherical varieties in arbitrary characteristic. In particular, we study Luna's two types of localization and use it to analyze spherical roots, colors and their interrelation. At the end, we propose a preliminary definition of a p-spherical system.

math.RT↗

arXiv2010

Automorphisms of multiplicity free Hamiltonian manifolds

We compute the sheaf of automorphisms of a multiplicity free Hamiltonian manifold over its momentum polytope and show that its higher cohomology groups vanish. Together with a theorem of Losev, arXiv:math/0612561, this implies a conjecture of Delzant: a compact multiplicity free Hamiltonian manifold is uniquely determined by its momentum polytope and its principal isotropy group.

math.SG↗

arXiv2007

Tensor envelopes of regular categories

We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for which T(A,d) is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups S_n, the hyperoctahedral groups S_n\semidir Z_2^n, or the general linear groups GL(n,F_q) over a fixed finite field. This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups S_n. It also extends (and provides proofs for) a previous paper math.CT/0605126 on the special case of abelian categories.

math.CT↗

arXiv2006

Invariant functions on symplectic representations

Let G be a connected reductive group. In this paper we are studying the invariant theory of symplectic G-modules. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration over an affine space with rational generic fibers. Of particular interest are those modules for which the generic orbit is coisotropic. We prove that they are cofree. This result has been used in another paper (math.SG/0505268) to classify all these modules. Our main tool is a symplectic version of the local structure theorem.

math.AG↗

arXiv2006

A construction of semisimple tensor categories

Starting from an abelian category A such that every object has only finitely many subobjects we construct a semisimple tensor category T. We show that T interpolates the categories Rep(Aut(p),K) where p runs through certain projective (pro-)objects of A. The main example is A=finite dimensional F_q-vector spaces. Then T can be considered as the category of representations of GL(n,F_q) where n is not a natural number. This work extends a construction of Deligne for symmetric groups.

math.CT↗

arXiv2005

Graded cofinite rings of differential operators

We classify subalgebras of a ring of differential operators which are big in the sense that the extension of associated graded rings is finite. We show that these subalgebras correspond, up to automorphisms, to uniformly ramified finite morphisms. This generalizes a theorem of Levasseur-Stafford on the generators of the invariants of a Weyl algebra under a finite group.

math.RA↗

arXiv2005

Classification of smooth affine spherical varieties

Let G be a complex reductive group. A normal G-variety X is called spherical if a Borel subgroup of G has a dense orbit in X. Of particular interest are spherical varieties which are smooth and affine since they form local models for multiplicity free Hamiltonian K-manifolds, K a maximal compact subgroup of G. In this paper, we classify all smooth affine spherical varieties up to coverings, central tori, and C*-fibrations.

math.AG↗

arXiv2005

On the Kazhdan-Lusztig basis of a spherical Hecke algebra

Lusztig proved that the Kazhdan-Lusztig basis of a spherical Hecke algebra can be essentially identified with the Weyl characters of the Langlands dual group. We generalize this result to the unequal parameter case. The new proof is pretty simple and quite different from Lusztig's.

math.RT↗

arXiv2005

Composition Kostka functions

Macdonald defined two-parameter Kostka functions K_{λμ}(q,t) where λ, μare partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture that also these more general Kostka functions are polynomials in q and t^{1/2} with non-negative integers as coefficients. If q=0 then our Kostka functions are Kazhdan-Lusztig polynomials of a special type. Therefore, our positivity conjecture combines Macdonald positivity and Kazhdan-Lusztig positivity and hints towards a connection between Macdonald and Kazhdan-Lusztig theory.

math.QA↗

arXiv2003

On Noether's and Weyl's bound in positive characteristic

In this note we generalize several well known results concerning invariants of finite groups from characteristic zero to positive characteristic not dividing the group order. The first is Schmid's relative version of Noether's theorem. That theorem compares the degrees of generators of a group with those of a subgroup. Then we prove a suitable positive characteristic version of Weyl's theorem on vector invariants: polarization works in small degrees. Using that we show that the regular representation has the "most general" ring of invariants, thereby generalizing theorems of Schmid and Smith.

math.RT↗

arXiv2003

Combinatorics and invariant differential operators on multiplicity free spaces

We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.

math.RT↗

arXiv2001

Convexity of Hamiltonian manifolds

Let K be a connected Lie group and M a Hamiltonian K-manifold. In this paper, we introduce the notion of convexity of M. It implies that the momentum image is convex, the moment map has connected fibers, and the total moment map is open onto its image. Conversely, the three properties above imply convexity. We show that most Hamiltonian manifolds occuring "in nature" are convex (e.g., if M is compact, complex algebraic, or a cotangent bundle). Moreover, every Hamiltonian manifold is locally convex. This is an expanded version of section 2 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian manifolds.

math.SG↗

arXiv2001

A connectedness property of algebraic moment maps

Let a connected reductive group G act on the smooth connected variety X. The cotangent bundle of X is a Hamiltonian G-variety. We show that its "total moment map" has connected fibers. This is an expanded version of section 6 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian manifolds.

math.AG↗