Friedrich Knop

Friedrich-Alexander-University Erlangen-Nürnberg | FAU · Algebra and Geometry

In a previous paper, semisimple tensor categories were constructed from certain regular Mal’cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple objects. This yields also the Grothendieck ring. The main tool is the subquotient decomposition of the generating objects.

We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). The group acting is a compact connected Lie group (simply connected in the quasi-Hamiltonian case). This work is a concretization of a more general classification of multiplicity free manifolds...

A quasi-Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify compact, multiplicity free, twisted quasi-Hamiltonian manifolds for simply connected, compact Lie groups. Thereby, we recover old and find new examples of these structures.

In the previous paper arxiv:math/0610552 semisimple tensor categories were constructed out of certain regular Mal'cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple objects. This yields also the Grothendieck ring. The main tool is the subquotient decomposition of the basic obje...

To every regular category A equipped with a degree function δ one can attach a pseudo-abelian tensor category T(A,δ). We show that the generating objects of T decompose canonically as a direct sum. In this paper we calculate morphisms, compositions of morphisms and tensor products of the summands. As a special case we recover the original construct...

This paper lays the foundation for Plancherel theory on real spherical spaces Z = G / H Z=G/H , namely it provides the decomposition of L 2 ( Z ) L^2(Z) into different series of representations via Bernstein morphisms. These series are parametrized by subsets of spherical roots which determine the fine geometry of Z Z at infinity. In particular, we...

To every regular category $\mathcal{A}$ equipped with a degree function $\delta$ one can attach a pseudo-abelian tensor category $\mathcal{T}(\mathcal{A},\delta)$. We show that the generating objects of $\mathcal{T}$ decompose canonically as a direct sum. In this paper we calculate morphisms, compositions of morphisms and tensor products of the sum...

Let $G$ be a connected reductive group over a perfect field $k$ acting on an algebraic variety $X$ and let $P$ be a minimal parabolic subgroup of $G$. For $k$-spherical $G$-varieties we prove finiteness result for $P$-orbits that contain $k$-points. This is a consequence of an equality on $P$-complexities of $X$ and of any $P$-invariant $k$-dense s...

We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). Here the group acting is a compact connected Lie group (simply connected in the quasi-Hamiltonian case). This work is a concretization of the more general classification (arXiv:1612.03843) of mu...

Toric subvarieties of projective space are classified up to projective automorphisms.

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all re...

We annonce the results generalizing the Vinberg's Complexity Theorem for the action of reductive group on an algebraic variety over algebraically non-closed field. Also we give new results on the strong k-stability for the actions on flag varieties.

This paper gives a classification of all pairs gh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left(\mathfrak{g},\mathfrak{h}\right) $$\end{document} with g\documen...

We prove new results that generalize Vinberg’s complexity theorem for the action of reductive group on an algebraic variety over an algebraically nonclosed field. We provide new results on strong k-stability for actions on flag varieties are given.

Given a unimodular real spherical space $Z=G/H$ we construct for each boundary degeneration $Z_I=G/H_I$ of $Z$ a Bernstein morphism $B_I: L^2(Z_I)_{\rm disc }\to L^2(Z)$. We show that $B:=\bigoplus_I B_I$ provides an isospectral $G$-equivariant morphism onto $L^2(Z)$. Further, the maps $B_I$ are finite linear combinations of orthogonal projections...

Let $G$ be a connected reductive group. In a previous paper, arxiv:1702.08264, is was shown that the dual group $G^\vee_X$ attached to a $G$-variety $X$ admits a natural homomorphism with finite kernel to the Langlands dual group $G^\vee$ of $G$. Here, we prove that the dual group is functorial in the following sense: if there is a dominant $G$-mor...

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all re...

Let $X$ be a spherical variety for a connected reductive group $G$. Work of Gaitsgory-Nadler strongly suggests that the Langlands dual group $G^\vee$ of $G$ has a subgroup whose Weyl group is the little Weyl group of $X$. Sakellaridis-Venkatesh defined a refined dual group $G^\vee_X$ and verified in many cases that there exists an isogeny $\phi$ fr...

A (quasi-)Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify multiplicity free Hamiltonian actions for (twisted) loop groups or, equivalently, multiplicity free (twisted) quasi-Hamiltonian manifolds for simply connected compact Lie groups. As a result we recover old and...

This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum.

In this paper, we study rationality properties of reductive group actions which are defined over an arbitrary field of characteristic zero. Thereby, we unify Luna's theory of spherical systems and Borel-Tits' theory of reductive groups. In particular, we define for any reductive group action a generalized Tits index whose main constituents are a ro...

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure $L^p$-integrability of matrix coefficients on Z.

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 par...

Let G/H be a unimodular real spherical space. It is shown that every tempered representation of G/H embeds into a relative discrete series of a boundary degeneration of G/H. If in addition G/H is of wave-front type it follows that the tempered representation is parabolically induced from a discrete series representation of a lower dimensional real...

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 si...

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...

Let G G be a connected reductive group defined over an algebraically closed base field of characteristic p ≥ 0 p\ge 0 , let B ⊆ G B\subseteq G be a Borel subgroup, and let X X be a G G -variety. We denote the (finite) set of closed B B -invariant irreducible subvarieties of X X that are of maximal complexity by B 0 ( X ) \mathfrak {B}_{0}(X) . The...

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 mai...

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.

This workshop brought together, for the first time, experts on spherical varieties and experts on automorphic forms, in order to discuss subjects of common interest between the two fields. Spherical varieties have a very rich and deep structure, which leads one to attach certain root systems and, eventually, a “Langlands dual” group to them. This t...

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 a...

We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,δ) depending on a degree function δ. Assume that all objects have only finitely many subobjects. Then our results are as follows:1.Let N be the maximal proper tensor ideal of T(A,δ). We show that T(A,δ)/N is semisimple provided th...

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 generi...

Let G be a connected reductive group acting on a finite-dimensional vector space V. Assume that V is equipped with a G-invariant symplectic form. Then the ring O(V) of polynomial functions becomes a Poisson algebra. The ring OG(V) of invariants is a sub-Poisson algebra. We call V multiplicity free if OG(V) is Poisson commutative, i.e., if {f,g}=0 f...

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 cat...

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...

Let G be a connected reductive group acting on a finite dimensional vector space V. Assume that V is equipped with a G-invariant symplectic form. Then the ring C[V] of polynomial functions becomes a Poisson algebra. The ring C[V]^G of invariants is a sub-Poisson algebra. We call V multiplicity free if C[V]^G is Poisson commutative, i.e., if {f,g}=0...

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...

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.

Macdonald defined two-parameter Kostka functions K_{\lambda\mu}(q,t) where \lambda, \mu are 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 coe...

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 shifte...

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 posi...

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 o...

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.

We study root systems equipped with a basis of dominant weights such that certain axioms hold. This formalism allows to define a linear basis P of the space of Weyl group invariant polynomials. This basis is actually a family depending on at least one parameter. Our main result is the construction of difference operators which are simultaneously di...

We introduce and investigate a one-parameter family of multivariate polynomials R λ . They form a basis of the space of semisymmetric polynomials, i.e., those polynomials which are symmetric in the variables with odd and even index separately. For two values of the parameter r, namely r=1 2 and r=1, the polynomials have a representation theoretic m...

We consider a connected compact Lie group K acting on a symplectic manifold M such that a moment map m exists. A pull-back function via m Poisson commutes with all K-invariants. Guillemin-Sternberg raised the problem to find a converse. In this paper, we solve this problem by determining the Poisson commutant of the algebra of K-invariants. It is c...

We study multiplicity free representations of connected reductive groups. First we give a simple criterion to decide the multiplicity freeness of a representation. Then we determine all invariant dieren tial operators in terms of a nite reection group, the little Weyl group, and give a characterization of the spectrum of the Capelli operators. At t...

Macdonald defined a family of symmetric polynomials which depend on two parameters q and t. The coefficients of the transition matrix from Macdonald polynomials to Schur S-functions are called Kostka functions. Macdonald conjectured that they are polynomials in q and t with non-negative integers as coefficients. In the paper I prove that the Kostka...

We introduce families of symmetric and non-symmetric polynomials (the quantum Capelli polynomials) which depend on two parameters q and t. They are defined in terms of vanishing conditions. In the differential limit (q = t^\alpha and t \to 1) they are related to Capelli identities. It is shown that the quantum Capelli polynomials form an eigenbasis...

In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values ½, 1, and 2 they have an interpretation in terms of Capelli identities. First, we give explicit formulas in some special cases. Then we show that the polynomials can also b...

Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the non-symmetric ones. These formulas are then implemented by a closed expression of symmetric and non-symmetric Jac...

Let X=G/H be a homogeneous spherical variety and A=NG(H)/H its automorphism group. It is known that there is an equivariant compactification with exactly one closed orbit if and only if A is finite. In that case there is one which dominates all others: The standard (or wonderful) embedding X'. The purpose of the paper is to prove Brion's conjecture...

Without Abstract

We classify homogeneous SL2(k)-varieties where k is any algebraically closed field. Only in characteristic two, there are two series which are somewhat exceptional.

Let G be a connected reductive group and X a smooth G-variety. Theorem: Assume that X is either spherical or affine. Then the center Z(X) of the ring of G-invariant differential operators on X is a polynomial ring. More precisely, Z(X) is isomorphic to the ring of invariants of a finite reflection group.

Let G be reductive and X a smooth G-variety. Then the cotangent bundle T_X^* carries a symplectic structure and the G-action gives rise to a moment map T_X^*-->g^* (with g=Lie G). Let f be a regular function on T_X^* which is induced by an Ad G-invariant function on g^*. The associated Hamiltonian flow is called invariant collective. In this paper...

Let G be a connected reductive group and let X be a projective, unirational, normal G-variety of complexity at most one. Then we show that some of the basic problems of Mori theory have a positive solution for X: Every face of NE(X) can be contracted, flips exist, and every sequence of directed (or inverse) flips terminates.

LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely dete...

Let G be a connected reductive group with Borel subgroup B acting on a normal variety X. The complexity of X is lowest codimension of a B-orbit. Theorem: If X is unirational (e.g. quasi-homogeneous) of complexity at most one then the ring k[X] of global functions is finitely generated. This result is sharp: If X is not unirational or of complexity...

Without Abstract

The paper studies reductive groups acting (algebraically) on an affine space. Gerald Schwarz found the first examples which are not linearizable, i.e., where the action is not conjugate to a linear action. The main result of the paper is that every connected, non-abelian, reductive group admits a non-linearizable action on some affine space. In par...

In this paper we prove the Fixed Point Conjecture for odd order abelian groups, and we construct algebraically Smith equivalent representations. Furthermore, we show that certain nonlinear phenomena in smooth transformation groups also occur in real algebraic transformation groups.

This note contains the very short proof of the Hochster-Roberts Theorem by Hochster-Huneke.

the importance of a very distinguished class of homogeneous varieties G/H, those which are now called spherical. Such varieties are homogeneous for a connected reductive group G and are characterized by many equivalent properties, the most important being (see [BLV]): — Any Borel subgroup B of G has an open orbit in G/H. — Every equivariant complet...

Let G be a connected, reductive group defined over an algebraically closed field of characteristic zero. We assign to any G-variety X a finite cristallographic reflection group W X by means of the moment map on the cotangent bundle. This generalizes the “little Weyl group” of a symmetric space. The Weyl group W X is related to the equivariant compa...

Let g be semisimple Lie algebra und h a reductive subalgebra. In this paper we are concerned with the relationship of the centralizer U(g)h of h in the enveloping algebra and the centers of U(g) and U(h). Let for example g be a real semisimple algebra and h be a maximal compact subalgebra. Then U(g)h is canonically isomorphic to U(g)g U(h)h if and...

This is my thesis for Habilitation in Basel. It consists of three parts. The first one is identical with the paper "Weylgruppe und Momentabbildung". The last two parts were later strongly revised and were published as "Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind" and "The asymptotic behavior of invariant collective motion"...

We calculate the canonical module of a ring of invariants R of a reductive group acting on an affine variety. From that we derive a criterion for the Gorenstein property to hold for R. In case R is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of R.

Let G be a connected linear algebraic group acting on a normal variety X. This note contains two proofs of a basic theorem of Sumihiro which we hope are more transparent than the original one. Theorem: 1. Every point of X has a G-stable open quasi-projective neighborhood. 2. If X is quasi-projective then it can be equivariantly embedded into a pro...

We give a short proof of Luna's slice theorem. It is based on an (unpublished) idea of Luna. This note was as appendix to an elementary introduction to the slice theorem by Peter Slodowy.

Let G be connected linear algebraic group acting on a variety X. In this note, the group of G-linearized line bundles on X is studied. For once, this is needed for one of the proofs of Sumihiro's Theorem. On the other hand, it has interesting applications to the Picard group of the categorical quotient X//G, in case X is affine.

We describe a new construction to obtain a simple hypersurface singularity from the corresponding simple complex Lie-groupG. LetX be the closed orbit in the projective space attached to the Lie algebra\(\mathfrak{g}\) ofG. Consider a regular nilpotent element\(y_0 \in \mathfrak{g}\) and denote byH y 0 the hyperplane orthogonal toy 0 with respect to...

Let G be a semisimple group and V a finite-dimensional faithful representation of G. The ring of invariants is graded which gives rise to a generating function h(t). This function satisfies a functional equation h(1/t)=±t^q h(t). In an earlier paper, it was shown that q<=dim V. Furthermore, equality holds if and only if the set of all v in V with p...

This little note fixes two gaps in M. Demazure: Automophismes et déformations de variétés de Borel. Invent. Math. 39 (1977), 179-186

Let G be a semisimple algebraic group acting on a factorial Gorenstein algebra S. Let X:=Spec S, Y:=Spec SG and π:X→Y be the quotient map. The main results are:1. Let x be a smooth point of X whose orbit has maximal dimension and such that π(x) is a smooth point of Y. Then π is smooth at x. 2. Let S be positively graded and let χS(t) be its generat...

Without Abstract

We construct certain compactifications of commutative (non-linear) algebraic groups and study their embeddings into projective space. This work was motivated by questions coming from transcendental number theory.

All d-fold (d>1) transitive actions of algebraic groups defined over an algebraic closed field k are classified. The result: (G acts on X) 1. X is the n-dimensional projective space and G=PGL(n+1,k). This action is always doubly transitive and triply transitive for n=1. 2. X is the n-dimensional affine space and G=S.L.T, where S is a subgroup of th...