Friedrich Knop

A recursion and a combinatorial formula for Jack polynomials

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.

Symmetric and non-symmetric quantum Capelli polynomials

Generalizing the classical Capelli identity has recently attracted a lot of interest ([HU], [Ok], [Ol], [Sa], [WUN]). In several of these papers it was realized, in various degrees of generality, that Capelli identities are connected with certain symmetric polynomials which are characterized by their vanishing at certain points. From this point of view, these polynomials have been constructed by Sahi [Sa] and were studied in [KS]. The purpose of this paper is twofold: we quantize the vanishing condition in a rather straightforward manner and obtain a family of symmetric polynomials which is indexed by partitions and which depends on two parameters q, t. As in [KS], their main feature is that they are non-homogeneous and one of our principal results states that the top degree terms are the Macdonald polynomials. It is an interesting problem whether these quantized Capelli polynomials are indeed connected with quantized Capelli identities (see [WUN]) as it is in the classical case. But the main progress over [KS] is the introduction of a family of non-symmetric polynomials which are also defined by vanishing conditions. They are non-homogeneous and their top degree terms turns out to be the non-symmetric Macdonald polynomials. To prove this, we introduce certain difference operators of Cherednik type of which our polynomials are a simultaneous eigenbasis. Because of these operators, the non-symmetric functions are much easier to handle than the symmetric ones. Moreover, the latter can be obtained by a simple symmetrization process. More specifically, the non-symmetric vanishing conditions are as follows: For λ ∈ Λ := N let |λ| := ∑ λi and let wλ be the shortest permutation such that w −1 λ (λ) is a partition (i.e., a non-increasing sequence). Let q and t be two formal parameters and

The Asymptotic Behavior of Invariant Collective Motion

Let G be a connected complex reductive group acting on a smooth algebraic variety X. Then the cotangent bundle T ∗ X on X carries a canonical symplectic structure, and the G-action induces a moment map Φ : T ∗ X → g . Consider the Hamiltonian vector fields attached to functions of the form f ◦ Φ with f ∈ C[g]. In this paper we study the asymptotic behavior of the associated flow (a so-called invariant collective motion) and show that it possesses a symmetry with respect to a finite reflection group WX . This is applied to the theory of equivariant embeddings of X. The approach is purely algebraic. More specifically: Choose any generic point α ∈ T ∗ X . Because the functions f ◦Φ with f ∈ C[g] are in involution (i.e., their Poisson product vanishes), the flow through α is in the orbit of an abelian group Aα. It is known (see [GS]) that this orbit is also the orbit for the connected isotropy group GΦ(α). This implies that Aα is a linear algebraic group and it turns out that it is a torus. The projection of this orbit to X is called a flat of X and just equals GΦ(α)π(α). In case, X is the complexification of a symmetric space, a flat in our sense is the complexification of a usual flat (=maximal totally geodesic, flat submanifold). Let X ⊆ X be a normal equivariant embedding. The main point of this paper is to study the closure of a generic flat in X. This will be done in two different steps. The first one is to show that a certain finite group WX acts on them. Consider the family of tori α 7→ Aα. Although every two of these groups are isomorphic to each other, the family cannot in general be trivialized globally. But we show that it can be trivialized on a finite cover T̂X of an open subset of T ∗ X . Hence, there is an action of a torus AX

On the Set of Orbits for a Borel Subgroup

Let X = G/H be a homogeneous variety for a connected complex reductive group G and let B be a Borel subgroup of G. In many situations, it is necessary to study the B-orbits in X . An equivalent setting of this problem is to analyze H-orbits in the flag variety G/B. The probably best known example is the Bruhat decomposition of G/B where one takes H = B. Another well-studied situation is the case where H is a symmetric subgroup, i.e., the fixed point group of an involution of G. Then H-orbits in G/B play a very important role in representation theory. They are the main ingredients for the classification of irreducible Harish-Chandra modules (see e.g. the surveys [Sch], [Wo]). In this paper, we introduce two structures on the set of all B-orbits. The first one is not really new, namely an action of a monoid W ∗ on the set B(X) of all B-stable closed subvarieties of X . As a set, W ∗ is the Weyl group W of G but with a different multiplication. That has already been done by Richardson and Springer [RS1] in the case of symmetric varieties and the construction generalizes easily. As an application we obtain a short proof of a theorem of Brion [Br1] and Vinberg [Vin]: If B has a open orbit in X then B has only finitely many orbits. Varieties with this property are called spherical . All examples mentioned above are of this type. The second structure which we are introducing is an action of the Weyl group W on a certain subset of B(X). Let me remark that in the most important case, X spherical, B(X) is just the set of B-orbit closures and the W -action will be defined on all of it. We give two methods to construct this action. In the first, we define directly the action of the simple reflections sα of W . This is done by reduction to the case rkG = 1 and then by a case-by-case consideration. The advantage of this method is that it is very concrete and works in general. The problem is to show that the sα-actions actually define

Difference equations and symmetric polynomials defined by their zeros

In this paper, we are starting a systematic analysis of a class of symmetric polynomials which, in full generality,was introduced in [Sa]. The main features of these functions are that they are defined by vanishing conditions and that they are nonhomogeneous. They depend on several parameters, but we are studying mainly a certain subfamily which is indexed by one parameter, r. As a special case, we obtain for r = 1 the factorial Schur functions discovered by Biedenharn and Louck [BL]. Our main result is that for general r these functions are eigenvalues of difference operators,which are difference analogues of the Sekiguchi-Debiard differential operators. Thus the functions under investigation are nonhomogeneous variants of Jack polynomials. More precisely, consider the set of partitions of length n, i.e., sequences of integers (λi) with λ1 ≥ · · · ≥ λn ≥ 0. The weight |λ| of a partition λ is the sum of its parts λi. Choose a vector ρ ∈ C which has to satisfy a mild condition. Then, for every λ, there is (up to a constant) a unique symmetric polynomial Pλ of degree at most d which satisfies the following vanishing condition:

Some remarks on multiplicity free spaces

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.

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

Mehrfach transitive Operationen algebraischer Gruppen

{\mathbb C}^{\times}

Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten

-fibrations.

A construction of semisimple tensor categories

In dieser Arbeit werden alle algebraisehen Gruppen bestimmt, die fiber einem algebraiseh abgesehlossenen K6rper k definiert sind und die auf den rationalen Punkten einer Variet~t biregul/~r und zweifaeh transitiv operieren. Diese Aufgabe ist sehon yon Tits ffir Liegruppen, insbesondere also fiir k -- C gelSst worden ([87). Daher Iiegt das Augenmerk dieser Arbeit auf den K6rpern positiver Charakteristik. Das Haupthilfsmittel ist dabei eine Invariante, die im Prinzip in [11 definiert worden ist und in der Regel gleieh der Euler-Poinear6 Charakteristik sein dfirfte. Aueh wenn sie wesentlieh nut im Punkt 9 des Beweises yon Satz 2 gebraueht wird, ist sie aueh an weiteren Stellen sehr niitzlieh. Das Endergebnis der Klassifikation ist im wesent- lichen das gleiehe wie im komplexen Fall, und nur fiir char k = 2 kommt noeh eine Gruppe hinzu. Als Teil des I-Iauptergebnisses werden alle Gruppen bestimmt, die auf einer Vektorgruppe operieren und dabei transitiv auf den yon Null versehiedenen Elementen wirken (Satz 1). Bezeichnungen. Alle Varietgten sind im folgenden fiber einem algebraiseh abge- sehlossenen K6rper k beliebiger Charakteristik p definiert. Variet/~ten brauehen nieht irreduzibel zu sein. Reduktive Gruppen sind zusammenMngend. 1. Sei M eine Varietgt und M =i= UM~ eine disjunkte Zerlegung yon M in end- lieh viele Untervariet/~ten Ml, die alle zu einem affinen Raum A n~ isomorph sind. Dann hgngt naeh [t] Th. 4.5 die Anzahl b2m der i mit dim Mi ---- ni = m nut yon M und nieht yon der Zerlegung (M dab. Ist M augerdem noeh projektiv, so gilt b2m ~ 1 fflr m =- 1, ..., dim M ([1] Prop. 4.7). Insbesondere ist r eine Invariante yon M, die im folgenden mit g (M) bezeiehnet wird. Ftir projektive M gilt die Un- gleiehung : (1) z(M) ~ dim M-k 1. Sei nun U eine unipotente zusammenh~ngende Gruppe, die auf der Variet/~t M operiert. Naeh [3] IV, w 4.3.16 ist jede Bahn yon U in M isomorph zu einem affinen Raum. Gibt es also nur endlieh viele Bahnen, so definiert dies eine Zerlegung wie oben und Z (M) ist einfaeh die Anzahl der Bahnen. Konkret ist jeder projektive homogene Raum einer reduktiven Gruppe G yon dieser Art; Seien T ein maximaler Torus, B eine T enthaltende Boreluntergruppe,