Corollary: If X is smooth then G has either no fixed point or at least two of them.
This is an algebraic analogue of (much deeper) results of Conner-Floyd, Atiyah-Bott and others in the topological category.}, author = {Assadi, Amir and Barlow, Rebecca and Knop, Friedrich}, doi = {10.1007/BF02571787}, faupublication = {no}, journal = {Mathematische Zeitschrift}, pages = {129-136}, peerreviewed = {Yes}, title = {{Abelian} group actions on algebraic varieties with one fixed point}, volume = {210}, year = {1992} } @article{faucris.121204864, abstract = {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.}, author = {Knop, Friedrich}, doi = {10.1016/S0021-8693(02)00509-4}, faupublication = {no}, journal = {Journal of Algebra}, pages = {122-136}, peerreviewed = {Yes}, title = {{A} connectedness property of algebraic moment maps}, volume = {258}, year = {2002} } @article{faucris.121180004, abstract = {Let A be an Abelian category such that every object has only finitely many subobjects. Front A 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. This extends a construction of Deligne for symmetric groups.}, author = {Knop, Friedrich}, doi = {10.1016/j.crma.2006.05.009}, faupublication = {no}, journal = {Comptes Rendus Mathematique}, pages = {15-18}, peerreviewed = {Yes}, title = {{A} construction of semisimple tensor categories}, volume = {343}, year = {2006} } @article{faucris.120894004, abstract = {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.}, author = {Knop, Friedrich}, doi = {10.2307/2118600}, faupublication = {no}, journal = {Annals of Mathematics}, pages = {253-288}, peerreviewed = {Yes}, title = {{A} {Harish}-{Chandra} homomorphism for reductive group actions}, volume = {140}, year = {1994} } @article{faucris.116284344, abstract = {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 Jack polynomials in terms of certain tableaux. The main application is a proof of a conjecture of Macdonald stating certain integrality and positivity properties of Jack polynomials.}, author = {Knop, Friedrich and Sahi, Siddhartha}, doi = {10.1007/s002220050134}, faupublication = {no}, journal = {Inventiones Mathematicae}, pages = {9-22}, peerreviewed = {Yes}, title = {{A} recursion and a combinatorial formula for {Jack} polynomials}, volume = {128}, year = {1997} } @article{faucris.122563144, abstract = {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 this implies a conjecture of Delzant: a compact multiplicity free Hamiltonian manifold is uniquely determined by its momentum polytope and its principal isotropy group.}, author = {Knop, Friedrich}, doi = {10.1090/S0894-0347-2010-00686-8}, faupublication = {yes}, journal = {Journal of the American Mathematical Society}, pages = {567-601}, peerreviewed = {Yes}, title = {{Automorphisms} of multiplicity free {Hamiltonian} manifolds}, url = {http://www.algeo.math.fau.de/fileadmin/algeo/users/knop/papers/Delzant.html}, volume = {24}, year = {2011} } @article{faucris.120489204, abstract = {Let X=G/H be a homogeneous spherical variety and A=N_{G}(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: If A is trivial then X' is smooth.}, author = {Knop, Friedrich}, doi = {10.1090/S0894-0347-96-00179-8}, faupublication = {no}, journal = {Journal of the American Mathematical Society}, month = {Jan}, pages = {153-174}, peerreviewed = {Yes}, title = {{Automorphisms}, root systems, and compactifications of homogeneous varieties}, volume = {9}, year = {1996} } @incollection{faucris.123669304, abstract = {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.}, address = {Basel-Boston}, author = {Knop, Friedrich}, booktitle = {Algebraische Transformationsgruppen und Invariantentheorie}, editor = {H. Kraft, P. Slodowy, T. Springer}, faupublication = {no}, pages = {110-113}, peerreviewed = {No}, publisher = {Birkhäuser Verlag}, series = {DMV-Seminar}, title = {{Beweis} des {Fundamentallemmas} und des {Scheibensatzes}}, volume = {13}, year = {1989} } @article{faucris.320540258, abstract = {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.}, author = {Knop, Friedrich}, doi = {10.4310/PAMQ.2024.v20.n1.a10}, faupublication = {yes}, journal = {Pure and Applied Mathematics Quarterly}, keywords = {Multiplicity free; quasi-Hamiltonian manifolds}, note = {CRIS-Team Scopus Importer:2024-04-05}, pages = {471-523}, peerreviewed = {Yes}, title = {{Classification} of multiplicity free quasi-{Hamiltonian} manifolds}, volume = {20}, year = {2024} } @article{faucris.121213884, abstract = {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 O(V)(G) of invariants is a sub-Poisson algebra. We call V multiplicity free if O(V)(G) is Poisson commutative, i.e., if {f, g} = 0 for all invariants f and g. Alternatively, G also acts on the Weyl algebra W(V) and V is multiplicity free if and only if the subalgebra W(V)(G) of invariants is commutative. In this paper we classify all multiplicity free symplectic representations. (c) 2005 Elsevier Inc. All rights reserved.}, author = {Knop, Friedrich}, doi = {10.1016/j.jalgebra.2005.07.035}, faupublication = {no}, journal = {Journal of Algebra}, pages = {531-553}, peerreviewed = {Yes}, title = {{Classification} of multiplicity free symplectic representations}, volume = {301}, year = {2006} } @article{faucris.213971099, abstract = {If 𝔤 is a real reductive Lie algebra and 𝔥⊂𝔤 is a subalgebra, then the pair (𝔥,𝔤) is called real spherical provided that 𝔤=𝔥+𝔭 for some choice of a minimal parabolic subalgebra 𝔭⊂𝔤. This paper concludes the classification of real spherical pairs (𝔥,𝔤), where 𝔥 is a reductive real algebraic subalgebra. More precisely, we classify all such pairs which are strictly indecomposable, and we discuss (in Section 6) how to construct from these all real spherical pairs. A preceding paper treated the case where 𝔤 is simple. The present work builds on that case and on the classification by Brion and Mikityuk for the complex spherical cas}, author = {Knop, Friedrich and Krötz, Bernhard and Pecher, Tobias and Schlichtkrull, Henrik}, doi = {10.1007/s00031-019-09515-w}, faupublication = {yes}, journal = {Transformation Groups}, peerreviewed = {Yes}, title = {{Classification} of reductive real spherical pairs {II}. {The} semisimple case}, year = {2019} } @article{faucris.119830744, abstract = {This paper gives a classification of all pairs (g,h) with g a simple real Lie algebra and h<g a reductive subalgebra for which there exists a minimal parabolic subalgebra p<g such that g=h+p as vector su}, author = {Knop, Friedrich and Krötz, Bernhard and Pecher, Tobias and Schlichtkrull, Henrik}, doi = {10.1007/s00031-017-9470-5}, faupublication = {yes}, journal = {Transformation Groups}, keywords = {Classification; spherical variety; reductive group}, month = {Jan}, pages = {67-114}, peerreviewed = {Yes}, title = {{Classification} of reductive real spherical pairs {I}: the simple case}, volume = {24}, year = {2019} } @article{faucris.121195184, abstract = {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-x-fibration}, author = {Knop, Friedrich and Van Steirteghem, Bart}, doi = {10.1007/s00031-005-1116-3}, faupublication = {no}, journal = {Transformation Groups}, pages = {495-516}, peerreviewed = {Yes}, title = {{Classification} of smooth affine spherical varieties}, volume = {11}, year = {2006} } @article{faucris.116279504, abstract = {We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper 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 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.}, author = {Knop, Friedrich}, doi = {10.1016/S0021-8693(02)00633-6}, faupublication = {no}, journal = {Journal of Algebra}, pages = {194-229}, peerreviewed = {Yes}, title = {{Combinatorics} and invariant differential operators on multiplicity free spaces}, volume = {260}, year = {2003} } @article{faucris.116289184, 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.}, author = {Knop, Friedrich and Lange, Herbert}, doi = {10.1007/BF01455973}, faupublication = {yes}, journal = {Mathematische Annalen}, month = {Jan}, pages = {555-571}, peerreviewed = {Yes}, title = {{Commutative} algebraic groups and intersections of quadrics}, volume = {267}, year = {1984} } @incollection{faucris.121430144, abstract = {Macdonald defined two-parameter Kostka functions K_{λμ}(q,t) where λ, μ 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^{½} 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.}, address = {New Delhi}, author = {Knop, Friedrich}, booktitle = {Algebraic Groups and Homogeneous Spaces}, editor = {Vikram B. Mehta}, faupublication = {no}, pages = {321-352}, peerreviewed = {Yes}, publisher = {Narosa Publishing House}, series = {Tata Institute of Fundamental Research - Mumbai Studies in Mathematics}, title = {{Composition} {Kostka} functions}, year = {2007} } @article{faucris.121333784, abstract = {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 diagonalized by P. From this, Pieri type rules are derived. This generalizes results for shifted Jack polynomials.
Even though the approach is purely combinatorial, the main motivation comes from multiplicity free actions of reductive groups on vector spaces. Then the algebra of invariant differential operators has a distinguished basis, the Capelli operators, which gives rise to a basis P as above. The paper ends with a comprehensive table listing the combinatorial structure of multiplicity free actions.}, author = {Knop, Friedrich}, doi = {10.1007/PL00001395}, faupublication = {no}, journal = {Selecta Mathematica-New Series}, keywords = {Difference operators; Multiplicity free spaces}, pages = {443-470}, peerreviewed = {Yes}, title = {{Construction} of commuting difference operators for multiplicity free spaces}, volume = {6}, year = {2000} } @article{faucris.121356444, abstract = {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.}, author = {Brion, Michel and Knop, Friedrich}, faupublication = {no}, journal = {Journal of Mathematical Sciences - the University of Tokyo}, pages = {641-655}, peerreviewed = {Yes}, title = {{Contractions} and flips for varieties of small complexity}, volume = {1}, year = {1994} } @article{faucris.116291384, abstract = {We study point set topological properties of the moment map. In particular, we introduce the notion of a convex Hamiltonian manifold. This notion combines convexity of the momentum image and connectedness of moment map fibers with a certain openness requirement for the moment map. We show that convexity rules out many pathologies for moment maps. Then we show that the most important classes of Hamiltonian manifolds (e.g., unitary vector spaces, compact manifolds, or cotangent bundles) axe in fact convex., Moreover, we prove that every Hamiltonian manifold is locally convex.}, author = {Knop, Friedrich}, faupublication = {no}, journal = {Journal of Lie Theory}, month = {Jan}, pages = {571-582}, peerreviewed = {Yes}, title = {{Convexity} of {Hamiltonian} manifolds}, volume = {12}, year = {2002} } @article{faucris.108996844, author = {Knop, Friedrich and Littelmann, Peter}, doi = {10.1007/BF01163656}, faupublication = {no}, journal = {Mathematische Zeitschrift}, keywords = {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 positive dimensional isotropy group has at least codimension two. This condition is satisfied for all "generic" representations. In the paper, all pairs (G,V) with q}, month = {Jan}, pages = {211-229}, peerreviewed = {Yes}, title = {{Der} {Grad} erzeugender {Funktionen} von {Invariantenringen}}, volume = {196}, year = {1987} } @article{faucris.122820984, author = {Knop, Friedrich}, doi = {10.1016/0021-8693(89)90271-8}, faupublication = {no}, journal = {Journal of Algebra}, pages = {40-54}, peerreviewed = {Yes}, title = {{Der} kanonische {Modul} eines {Invariantenrings}}, volume = {127}, year = {1989} } @article{faucris.120489424, abstract = {Let g be a complex semisimple Lie algebra and k a reductive subalgebra of g. The paper is concerned with the centralizer U(g)^k of k in the enveloping algebra U(g). Denote by z(g), z(k) the center of U(g), U(k) respectively. Then there is a canonical homomorphism p:z(g)\otimes z(k)-->U(g)^k.
Theorem: Assume k does not contain a non-zero ideal of g. Then the following statements are equivalent:
The method is using the canonical filtration of U(g) and then proving an equivalent theorem for the associated graded algebra.}, author = {Knop, Friedrich}, faupublication = {no}, journal = {Journal für die reine und angewandte Mathematik}, month = {Jan}, pages = {5-9}, peerreviewed = {Yes}, title = {{Der} {Zentralisator} einer {Liealgebra} in einer einhüllenden {Algebra}}, volume = {406}, year = {1990} } @article{faucris.121352704, abstract = {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 be defined in terms of difference equations. As a corollary we obtain that their top homogeneous part is a Jack polynomial. This is used to give a new proof of the Pieri formula for Jack polynomials.}, author = {Knop, Friedrich and Sahi, Siddhartha}, doi = {10.1155/S1073792896000311}, faupublication = {no}, journal = {International Mathematics Research Notices}, peerreviewed = {Yes}, title = {{Difference} equations and symmetric polynomials defined by their zeros}, year = {1996} } @article{faucris.116293804, abstract = {Let V be an irreducible representation of a semisimple algebraic group G and X the orbit of an highest weight vector in the projective spave P(V). Then X is a generalized flag variety. The dual variety of X is the set of hyperplanes in P(V) which are tangent to X.
The purpose of the paper is to determine all cases in which the dual variety is not of codimension one. When G is simple there are the following cases (upto isomorphims of G and V):
G=SL(n,k), V=k^{n}; G=Sp_{n}, V=k^{n} (n even); G=SL(n,k), V=\wedge^{2}k^{n} (n odd); G=Spin(9) or Spin(10), V=k^{16}.
The classification for semisimple groups is easily obtained from that.}, author = {Knop, Friedrich and Menzel, Gisela}, doi = {10.5169/seals-47339}, faupublication = {no}, journal = {Commentarii Mathematici Helvetici}, month = {Jan}, pages = {38-61}, peerreviewed = {Yes}, title = {{Duale} {Varietäten} von {Fahnenvarietäten}}, volume = {62}, year = {1987} } @article{faucris.120558504, abstract = {Simple singularities are classified by simply-laced Dynkin diagrams (type ADE) which in turn define certain simple algebraic groups. Grothendieck conjectured and Brieskorn proved a direct connection between these structures.
In this paper, a completely different connection between simple singularities and simple algebraic groups is given. Let G be a simple algebraic group with simply-laced Dynkin diagram. Let P be the projective space attached to the adjoint representation g. Then P contains exactly one closed orbit, a certain flag variety X. Using the killing form, every non-zero x in g defines a hyperplane in P, hence a hyperplane section X{\_}x of X.
Theorem: Assume x is regular nilpotent. Then X{\_}x has exactly one singular point. This singularity is simple with the same Dynkin diagram as G. Furthermore, "perturbing x" is a versal deformation of the singularity.
One of the advantages of this construction is that it holds, properly modified, also in positive characteristic, even if the characteristic is bad. This predicted certain degenerations of simple singularities which are impossible in characteristic zero, e.g., E{\_}7-->A{\_}7, E{\_}8-->A{\_}8 (p=2), and E{\_}8-->A{\_}8 (p=3).}, author = {Knop, Friedrich}, doi = {10.1007/BF01389179}, faupublication = {no}, journal = {Inventiones Mathematicae}, month = {Jan}, pages = {579-604}, peerreviewed = {Yes}, title = {{Ein} neuer {Zusammenhang} zwischen einfachen {Gruppen} und einfachen {Singularitäten}}, volume = {90}, year = {1987} } @article{faucris.219417433, abstract = {Let G be a connected reductive group. Previously, it was shown that for any G-variety X one can define the dual group G(X)(V) which admits a natural homomorphism with finite kernel to the Langlands dual group G(V) of G. Here, we prove that the dual group is functorial in the following sense: if there is a dominant G-morphism X -> Y or an injective G-morphism Y -> X then there is a unique homomorphism with finite kernel G(X)(V) -> G(X)(V) which is compatible with the homomorphisms to G(V).}, author = {Knop, Friedrich}, faupublication = {yes}, journal = {Documenta Mathematica}, month = {Jan}, note = {CRIS-Team WoS Importer:2019-06-04}, pages = {47-64}, peerreviewed = {Yes}, title = {{FUNCTORIALITY} {PROPERTIES} {OF} {THE} {DUAL} {GROUP}}, volume = {24}, year = {2019} } @article{faucris.124074104, abstract = {We study subrings of a ring of differential operators such that the extension of associated graded rings is finite. We show that up to an automorphism these subrings correspond exactly 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.}, author = {Knop, Friedrich}, doi = {10.1307/mmj/1144437435}, faupublication = {no}, journal = {Michigan Mathematical Journal}, month = {Jan}, pages = {3-23}, peerreviewed = {Yes}, title = {{Graded} cofinite rings of differential operators}, volume = {54}, year = {2006} } @article{faucris.108998824, author = {Knop, Friedrich}, faupublication = {no}, journal = {Compositio Mathematica}, keywords = {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.}, pages = {77-89}, peerreviewed = {Yes}, title = {{Homogeneous} varieties for semisimple groups of rank one}, volume = {98}, year = {1995} } @article{faucris.116296664, abstract = {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 functions are polynomials with integral coefficients. The positivity part remains open.
The proof uses a non-symmetric analogue of Macdonald polynomials (also introduced by Macdonald). I derive a recursion formula for them and a formula relating the symmetric with the non-symmetric Macdonald polynomials. I also define a non-symmetric analogue of Hall-Littlewood polynomials and use them to state and prove an integrality result for the non-symmetric Macdonald polynomials. This implies integrality of Kostka functions.}, author = {Knop, Friedrich}, doi = {10.1515/crll.1997.482.177}, faupublication = {no}, journal = {Journal für die reine und angewandte Mathematik}, month = {Jan}, pages = {177-189}, peerreviewed = {Yes}, title = {{Integrality} of two variable {Kostka} functions}, volume = {482}, year = {1997} } @article{faucris.121073084, abstract = {We study invariants on symplectic representations of a connected reductive group. 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 representations where these fibers are points. We show that they are cofree. Our main tool is a symplectic version of the local structure theorem. (c) 2007 Elsevier Inc. All rights reserved.}, author = {Knop, Friedrich}, doi = {10.1016/j.jalgebra.2006.10.041}, faupublication = {no}, journal = {Journal of Algebra}, keywords = {invariant theory;symplectic representation;moment map}, pages = {223-251}, peerreviewed = {Yes}, title = {{Invariant} functions on symplectic representations}, volume = {313}, year = {2007} } @article{faucris.116263664, abstract = {We prove some fundamental structural results for spherical varieties in arbitrary characteristic. In particular, we study Luna's two types of localization and use them to analyze spherical roots, colors, and their interrelation. At the end, we propose a preliminary definition of a p-spherical system.}, author = {Knop, Friedrich}, doi = {10.2140/ant.2014.8.703}, faupublication = {yes}, journal = {Algebra & Number Theory}, keywords = {spherical varieties}, month = {Jan}, pages = {703-728}, peerreviewed = {Yes}, title = {{Localization} of spherical varieties}, url = {http://www.algeo.math.fau.de/fileadmin/algeo/users/knop/papers/localization.html}, volume = {8}, year = {2014} } @incollection{faucris.116328564, abstract = {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 projective space.
The first proof uses the language of line bundles, the second field and valuation theory. In the last section, the Picard group of G is studied.}, address = {Basel-Boston}, author = {Knop, Friedrich and Kraft, Hanspeter and Luna, Domingo and Vust, Thierry}, booktitle = {Algebraische Transformationsgruppen und Invariantentheorie}, editor = {H. Kraft, P. Slodowy, T. Springer}, faupublication = {no}, pages = {63-76}, peerreviewed = {No}, publisher = {Birkhäuser Verlag}, title = {{Local} properties of algebraic group actions}, volume = {13}, year = {1989} } @article{faucris.116300404, abstract = {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 the group of scalars, T is the group of translations, and L is either SL{\_}n(k), Sp{\_}n(k) (n even) or G{\_}2 (n=6, char k=2) acting linearly. All these actions are doubly transitive.
This paper was my Diploma thesis in Erlangen.}, author = {Knop, Friedrich}, doi = {10.1007/BF01207571}, faupublication = {yes}, journal = {Archiv der Mathematik}, month = {Jan}, pages = {438-446}, peerreviewed = {Yes}, title = {{Mehrfach} transitive {Operationen} algebraischer {Gruppen}}, volume = {41}, year = {1983} } @article{faucris.121320804, 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 particular, every semisimple group has such an action.
The method is reduction to SL(2,k) and then using Schwarz's examples for that group.}, author = {Knop, Friedrich}, doi = {10.1007/BF01232264}, faupublication = {no}, journal = {Inventiones Mathematicae}, pages = {217-220}, peerreviewed = {Yes}, title = {{Nichtlinearisierbare} {Operationen} halbeinfacher {Gruppen} auf affinen {Räumen}}, volume = {105}, year = {1991} } @incollection{faucris.121423764, abstract = {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.}, address = {Providence, RI}, author = {Knop, Friedrich}, booktitle = {Invariant Theory in All Characteristics}, editor = {E. Campbell, D. Wehlau}, faupublication = {no}, pages = {175-188}, peerreviewed = {Yes}, publisher = {AMS}, series = {CRM Proceedings & Lecture Notes}, title = {{On} {Noether}'s and {Weyl}'s bound in positive characteristic}, volume = {35}, year = {2004} } @article{faucris.238895885, abstract = {Abstract: We construct the action of the restricted Weyl group on the set of principal families of orbits of a minimal parabolic subgroup over an algebraically nonclosed field. Additionally, we relate this action to the action on a polarized cotangent bundle. These results generalize the corresponding results of Knop on the action of the Weyl group on the families of Borel orbits of maximal complexity and rank.}, author = {Zhgoon, V. S. and Knop, Friedrich}, doi = {10.1134/S1064562420010093}, faupublication = {yes}, journal = {Doklady Mathematics}, keywords = {cotangent bundle; orbits of minimal parabolic subgroup; reductive group actions over algebraically nonclosed fields; Weyl group}, month = {Jan}, note = {CRIS-Team Scopus Importer:2020-06-02}, pages = {25-29}, peerreviewed = {Yes}, title = {{On} the {Action} of the {Restricted} {Weyl} {Group} on the {Set} of {Orbits} a {Minimal} {Parabolic} {Subgroup}}, volume = {101}, year = {2020} } @article{faucris.221617101, abstract = {Abstract: 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.}, author = {Zhgoon, V. S. and Knop, Friedrich}, doi = {10.1134/S1064562419020054}, faupublication = {yes}, journal = {Doklady Mathematics}, note = {CRIS-Team Scopus Importer:2019-07-02}, pages = {132-136}, peerreviewed = {Yes}, title = {{On} the {Complexity} of {Reductive} {Group} {Actions} over {Algebraically} {Nonclosed} {Field} and {Strong} {Stability} of the {Actions} on {Flag} {Varieties}}, volume = {99}, year = {2019} } @article{faucris.116320644, abstract = {Lusztig proved that the Kazhdan-Lusztig basis of a spherical algebra can be essentially identified with the Weyl characters of the Langlands dual group. We generalize this result to the unequal parameter case. Our new proof is simple and quite different from Lusztig's.}, author = {Knop, Friedrich}, doi = {10.1090/S1088-4165-05-00237-2}, faupublication = {no}, journal = {Representation Theory}, pages = {417-425}, peerreviewed = {Yes}, title = {{On} the {Kazhdan}-{Lusztig} basis of a spherical {Hecke} algebra}, volume = {9}, year = {2005} } @article{faucris.119370064, author = {Knop, Friedrich}, doi = {10.1007/BF02566009}, faupublication = {no}, journal = {Commentarii Mathematici Helvetici}, month = {Jan}, pages = {285-309}, peerreviewed = {Yes}, title = {{On} the set of orbits for a {Borel} subgroup}, volume = {70}, year = {1995} } @article{faucris.116264104, abstract = {Let G be a connected reductive group defined over an algebraically closed base field of characteristic p >= 0, let B subset of G be a Borel subgroup, and let X be a G-variety. We denote the (finite) set of closed B-invariant irreducible subvarieties of X that are of maximal complexity by B-0(X). The first named author has shown that for p = 0 there is a natural action of the Weyl group W on B-0(X) and conjectured that the same construction yields a W-action whenever p not equal 2. In the present paper, we prove this conjecture.}, author = {Knop, Friedrich and Pezzini, Guido}, doi = {10.1090/S1088-4165-2015-00464-9}, faupublication = {yes}, journal = {Representation Theory}, pages = {9-23}, peerreviewed = {Yes}, title = {{On} the {W}-action on {B}-sheets in positive characteristic}, url = {http://www.algeo.math.fau.de/fileadmin/algeo/users/knop/papers/waction.html}, volume = {19}, year = {2015} } @article{faucris.269971305, author = {Delorme, Patrick and Knop, Friedrich and Krotz, Bernhard and Schlichtkrull, Henrik}, doi = {10.1090/jams/971}, faupublication = {yes}, journal = {Journal of the American Mathematical Society}, note = {CRIS-Team WoS Importer:2022-02-25}, pages = {815-908}, peerreviewed = {Yes}, title = {{PLANCHEREL} {THEORY} {FOR} {REAL} {SPHERICAL} {SPACES}: {CONSTRUCTION} {OF} {THE} {BERNSTEIN} {MORPHISMS}}, volume = {34}, year = {2021} } @article{faucris.312691633, abstract = {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 in the special case of rank one. As a result we obtain numerous new concrete examples of multiplicity free quasi-Hamiltonian manifolds or, equivalently, Hamiltonian loop group actions.}, author = {Knop, Friedrich and Paulus, Kay}, doi = {10.1007/s00209-023-03325-3}, faupublication = {yes}, journal = {Mathematische Zeitschrift}, keywords = {Cohomogeneity; Group valued momentum map; Hamiltonian manifold; Momentum map; Multiplicity free; Quasi-Hamiltonian manifold; Spherical variety}, note = {CRIS-Team Scopus Importer:2023-10-13}, peerreviewed = {Yes}, title = {({Quasi}-){Hamiltonian} manifolds of cohomogeneity one}, volume = {305}, year = {2023} } @article{faucris.116325704, abstract = {In this paper we introduce and investigate a one-parameter family of polynomials. They are semisymmetric, i.e. symmetric in the variables with odd and even index separately. In fact, the family forms a basis of the space of semisymmetric polynomials. For two values of the parameter r, namely r=½ and r=1, the polynomials have a representation theoretic meaning. In general, they form the semisymmetric analogue of (shifted) Jack polynomials.}, author = {Knop, Friedrich}, doi = {10.1090/S1088-4165-01-00129-7}, faupublication = {no}, journal = {Representation Theory}, pages = {224-266}, peerreviewed = {Yes}, title = {{Semisymmetric} polynomials and the invariant theory of matrix vector pairs}, volume = {5}, year = {2001} } @article{faucris.122579644, abstract = {Let be an algebraic homogeneous space attached to real reductive Lie group . We assume that is real spherical, i.e., minimal parabolic subgroups have open orbits on . For such spaces, we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of which is established in this paper.}, author = {Knop, Friedrich and Krötz, Bernhard and Sayag, Eitan and Schlichtkrull, Henrik}, doi = {10.1007/s00029-014-0174-6}, faupublication = {yes}, journal = {Selecta Mathematica-New Series}, keywords = {Spherical spaces;Polar decomposition;Compactification}, pages = {1071-1097}, peerreviewed = {Yes}, title = {{Simple} compactifications and polar decomposition of homogeneous real spherical spaces}, volume = {21}, year = {2015} } @article{faucris.121492624, author = {Knop, Friedrich and Lange, Herbert}, doi = {10.1007/BF02567430}, faupublication = {yes}, journal = {Commentarii Mathematici Helvetici}, month = {Jan}, pages = {497-507}, peerreviewed = {Yes}, title = {{Some} remarks on compactifications of commutative algebraic groups}, volume = {60}, year = {1985} } @incollection{faucris.116302384, abstract = {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 differential operators in terms of a finite reflection group, the little Weyl group, and give a characterization of the spectrum of the Capelli operators. At the end, we reproduce the classification of multiplicity free representations (without proof) annotated with the same basic data.}, address = {Dortrecht}, author = {Knop, Friedrich}, booktitle = {Proc. NATO Adv. Study Inst. on Representation Theory and Algebraic Geometry}, doi = {10.1007/978-94-015-9131-7{\_}7}, editor = {A. Broer, A. Daigneault, G. Sabidussi}, faupublication = {no}, month = {Jan}, pages = {301-317}, peerreviewed = {unknown}, series = {Nato ASI Series C}, title = {{Some} remarks on multiplicity free spaces}, volume = {514}, year = {1998} } @article{faucris.122055384, abstract = {Preface to a volume dedicated to E. B. Vinberg.}, author = {Kac, Victor and Kellerhals, Ruth and Knop, Friedrich and Littelmann, Peter and Panyushev, Dmitri}, doi = {10.1016/j.jalgebra.2007.02.006}, faupublication = {yes}, journal = {Journal of Algebra}, pages = {1-3}, peerreviewed = {No}, title = {{Special} issue in honour of {Ernest} {Borisovich} {Vinberg}}, volume = {313}, year = {2007} } @article{faucris.121099704, abstract = {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 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 varieties of rank 1.}, author = {Knop, Friedrich}, doi = {10.5802/aif.2919}, faupublication = {yes}, journal = {Annales de l'Institut Fourier}, keywords = {Spherical varieties;spherical roots;homogeneous varieties;fields of positive characteristic}, month = {Jan}, pages = {2503-2526}, peerreviewed = {Yes}, title = {{Spherical} roots of spherical varieties}, url = {http://arxiv.org/abs/1303.2466}, volume = {64}, year = {2014} } @article{faucris.116265424, abstract = {Let G be a simple algebraic group. A closed subgroup H of G is said to be spherical if it has a dense orbit on the flag variety G/B of G. Reductive spherical subgroups of simple Lie groups were classified by Kramer in 1979. In 1997, Brundan showed that each example from Kramer's list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. 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 Kramer's classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.}, author = {Knop, Friedrich and Röhrle, Gerhard}, doi = {10.1112/80010437X1400791X}, faupublication = {yes}, journal = {Compositio Mathematica}, keywords = {spherical subgroup;symmetric subgroup;spherical homogeneous variety}, pages = {1288-1308}, peerreviewed = {Yes}, title = {{Spherical} subgroups in simple algebraic groups}, volume = {151}, year = {2015} } @article{faucris.120018844, abstract = {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 --> 1) they are related to Capelli identities. It is shown that the quantum Capelli polynomials form an eigenbasis for certain q-difference operators. As a corollary, we obtain that the top homogeneous part is a symmetric/non-symmetric Macdonald polynomial. Furthermore, we study the vanishing and integrality properties of the quantum Capelli polynomials.}, author = {Knop, Friedrich}, doi = {10.4171/CMH/72.1.7}, faupublication = {no}, journal = {Commentarii Mathematici Helvetici}, keywords = {symmetric polynomials;Capelli identity;Macdonald polynomials;difference operators;Hecke algebras}, month = {Jan}, pages = {84-100}, peerreviewed = {Yes}, title = {{Symmetric} and non-symmetric quantum {Capelli} polynomials}, volume = {72}, year = {1997} } @article{faucris.121534644, abstract = {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. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows:
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 on the special case of abelian categories}, author = {Knop, Friedrich}, doi = {10.1016/j.aim.2007.03.001}, faupublication = {no}, journal = {Advances in Mathematics}, keywords = {tensor categories;semisimple categories;regular categories;Mal'cev categories;Tannakian categories;Mobius function;lattices;profinite groups}, pages = {571-617}, peerreviewed = {Yes}, title = {{Tensor} envelopes of regular categories}, volume = {214}, year = {2007} } @article{faucris.108999704, abstract = {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 we prove that the invariant collective flow is symmetric under the little Weyl group W{\_}X of X.
The main application is: Let Z(X) be the set of all G-invariant valuations of k(X) which are trivial on k(X)^B, B=Borel subgroup. Then Z(X) is canonically in bijection to a Weyl chamber of W{\_}X.}, author = {Knop, Friedrich}, doi = {10.1007/BF01231563}, faupublication = {no}, journal = {Inventiones Mathematicae}, pages = {309-328}, peerreviewed = {Yes}, title = {{The} asymptotic behavior of invariant collective motion}, volume = {116}, year = {1994} } @article{faucris.111992804, abstract = {Let X be a spherical variety for a connected reductive group G. Work of Gaitsgory-Nadler strongly suggests that the Langlands dual group G^{v} of G has a subgroup whose Weyl group is the little Weyl group of X. Sakellaridis-Venkatesh defined a refined dual group G^{v}_{X} and verified in many cases that there exists an isogeny φ from G^{v}_{X} to G^{v}. In this paper, we establish the existence of φ in full generality. Our approach is purely combinatorial and works (despite the title) for arbitrary G-varieties.}, author = {Knop, Friedrich and Schalke, Barbara}, doi = {10.1090/mosc/270}, faupublication = {yes}, journal = {Transactions of the Moscow Mathematical Society}, keywords = {spherical variety; dual group; Langlands program}, pages = {187-216}, peerreviewed = {Yes}, title = {{The} dual group of a spherical variety}, volume = {78}, year = {2017} } @article{faucris.122861464, abstract = {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 x S-L. Here Q is a parabolic subgroup with Levi decomposition L (sic) U, and S is a homogeneous space for a quotient D = L/L-n of L, where L-n subset of L is normal, such that D is compact modulo center.}, author = {Knop, Friedrich and Krötz, Bernhard and Schlichtkrull, Henrik}, doi = {10.1112/S0010437X15007307}, faupublication = {yes}, journal = {Compositio Mathematica}, keywords = {spherical varieties;homogeneous spaces;real reductive groups}, pages = {2145-2159}, peerreviewed = {Yes}, title = {{The} local structure theorem for real spherical varieties}, volume = {151}, year = {2015} } @incollection{faucris.121422444, abstract = {Let X be a homogeneous for a connected reductive group G. Luna and Vust have classified all equivariant open embeddings of X into a larger normal G-variety. This paper contains a survey of this result together with a full proof valid in any characteristic. Moreover, many other results relating to spherical embeddings are mentioned and most of them proved.}, address = {Madras}, author = {Knop, Friedrich}, booktitle = {Proceedings of the Hyderabad Conference on Algebraic Groups}, editor = {S. Ramanan}, faupublication = {no}, pages = {225-249}, peerreviewed = {No}, publisher = {Manoj Prakashan}, title = {{The} {Luna}-{Vust} theory of spherical embeddings}, year = {1991} } @incollection{faucris.116329224, abstract = {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.}, address = {Basel-Boston}, author = {Knop, Friedrich and Kraft, Hanspeter and Vust, Thierry}, booktitle = {Algebraische Transformationsgruppen und Invariantentheorie}, editor = {H. Kraft, P. Slodowy, T. Springer}, faupublication = {no}, pages = {77-88}, peerreviewed = {No}, publisher = {Birkhäuser Verlag}, series = {DMV-Seminar}, title = {{The} {Picard} group of a {G}-variety}, volume = {13}, year = {1989} } @article{faucris.268129061, abstract = {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 construction of Deligne's category RepSt.}, author = {Knop, Friedrich}, doi = {10.1016/j.indag.2021.12.008}, faupublication = {yes}, journal = {Indagationes Mathematicae}, keywords = {Mal'cev categories; Möbius functions; Pseudo-abelian categories; Regular categories; Symmetric categories}, note = {CRIS-Team Scopus Importer:2022-01-14}, peerreviewed = {Yes}, title = {{The} subobject decomposition in enveloping tensor categories}, year = {2022} } @article{faucris.119830524, abstract = {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 spherical space.}, author = {Knop, Friedrich and Krötz, Bernhard and Schlichtkrull, Henrik}, doi = {10.4310/ACTA.2017.v218.n2.a3}, faupublication = {yes}, journal = {Acta Mathematica}, keywords = {spherical variety; tempered spectrum; Lie groups}, pages = {319-383}, peerreviewed = {Yes}, title = {{The} tempered spectrum of a real spherical space}, volume = {218}, year = {2017} } @article{faucris.116304584, abstract = {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.}, author = {Dovermann, Karl Heinz and Knop, Friedrich and Suh, Dong Youp}, doi = {10.1016/0166-8641(91)90049-R}, faupublication = {no}, journal = {Topology and its Applications}, keywords = {REAL ALGEBRAIC ACTIONS;EQUIVARIANT NASH CONJECTURE}, pages = {171-188}, peerreviewed = {Yes}, title = {{Topological} invariants of real algebraic actions}, volume = {40}, year = {1991} } @article{faucris.123076844, abstract = {Let G be a reductive group defined over an algebraically closed field k and let X be a G-variety. In this paper we study G-invariant valuations v of the field K of rational functions on X. These objects are fundamental for the theory of equivariant completions of X. Let B be a Borel subgroup and U the unipotent radical of B. It is proved that v is uniquely determined by its restriction to K(U). Then we study the set of invariant valuations having some fixed restriction v0 to K(B). If V0 is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection group W(X). Thus, the classification of invariant valuations is almost reduced to the classification of valuations of K(B).}, author = {Knop, Friedrich}, doi = {10.1007/BF01444891}, faupublication = {no}, journal = {Mathematische Annalen}, pages = {333-363}, peerreviewed = {Yes}, title = {{Über} {Bewertungen}, welche unter einer reduktiven {Gruppe} invariant sind}, volume = {295}, year = {1993} } @article{faucris.116303264, author = {Knop, Friedrich}, doi = {10.1007/BF01168503}, faupublication = {no}, journal = {Manuscripta Mathematica}, month = {Jan}, pages = {419-427}, peerreviewed = {Yes}, title = {{Über} die {Glattheit} von {Quotientenabbildungen}}, volume = {56}, year = {1986} } @article{faucris.116296224, abstract = {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 at least two then k[X] might not be finitely generated.}, author = {Knop, Friedrich}, doi = {10.1007/BF03025706}, faupublication = {no}, journal = {Mathematische Zeitschrift}, pages = {33-35}, peerreviewed = {Yes}, title = {{Über} {Hilberts} vierzehntes {Problem} für {Varietäten} mit {Kompliziertheit} eins}, volume = {213}, year = {1993} } @article{faucris.117983404, abstract = {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}, author = {Knop, Friedrich and Krötz, Bernhardt and Sayag, Eitan and Schlichtkrull, Henrik}, doi = {10.1016/j.jfa.2016.04.001}, faupublication = {yes}, journal = {Journal of Functional Analysis}, pages = {12-36}, peerreviewed = {Yes}, title = {{Volume} growth, temperedness and integrability of matrix coefficients on a real spherical space}, url = {http://arxiv.org/abs/1407.8006v2}, volume = {271}, year = {2016} } @misc{faucris.121422884, abstract = {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 completely controlled by the image of m and a certain subquotient W_{M} of the Weyl group of K. The group W_{M} is also a reflection group and forms a symplectic analogue of the little Weyl group of a symmetric space. The proof rests ultimately on techniques from algebraic geometry. In fact, a major part of the paper is of independent interest: it establishes connectivity and reducedness properties of the fibers of the (complex algebraic) moment map of a complex cotangent bundle.}, author = {Knop, Friedrich}, faupublication = {no}, peerreviewed = {automatic}, title = {{Weyl} groups of {Hamiltonian} manifolds, {I}}, year = {1997} } @misc{faucris.121418704, abstract = {This is my thesis for Habilitation in Basel. It consists of three parts. The first one is identical with the paper
The last two parts were later strongly revised and were published as
Habilitationsschrift, Universität Basel, 1990}, author = {Knop, Friedrich}, faupublication = {no}, peerreviewed = {automatic}, title = {{Weylgruppe}, {Momentabbildung} und äquivariante {Einbettung}}, year = {1990} } @article{faucris.116307224, abstract = {Let G be a connected, reductive group defined over an alebraically 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 compactification theory of X. We determine the closure of the image of the moment map and the generic isotropy group of the action of G on the cotangent bundle. As a byproduct we determine the ideal of elements of U(g) which act trivially on X as a differential operator.}, author = {Knop, Friedrich}, doi = {10.1007/BF01234409}, faupublication = {no}, journal = {Inventiones Mathematicae}, month = {Jan}, pages = {1-23}, peerreviewed = {Yes}, title = {{Weylgruppe} und {Momentabbildung}}, volume = {99}, year = {1990} }