Lie Groups and Algebraic Groups

Transcription

1 Lie Groups and Algebraic Groups July 2015 Faculty of Mathematics Bielefeld University Lecture Room: V3-201 This workshop is part of the DFG-funded CRC 701 Spectral Structures and Topological Methods in Mathematics at Bielefeld University Organisers: Herbert Abels and Ernest Vinberg

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3 Schedule: Wednesday July 22nd Lecture Room: V :30 11:30 Mark Sapir (Vanderbilt) On groups with Rapid Decay 11:30 12:00 Coffee Break (Room V3-201) 12:00 13:00 Hannah Bergner (Bochum) Actions on supermanifolds and automorphism groups of compact complex supermanifolds 13:00 15:30 Lunch Break 15:30 16:30 Pavel Tumarkin (Durham) Reflection groups via cluster algebras 16:30 17:00 Coffee Break (Room V3-201) 17:00 18:00 Anna Felikson (Durham) Cluster algebras via reflection groups 3

4 Schedule: Thursday July 23rd Lecture Room: V :00 11:00 Dmitri Panyushev (Moscow) Minimal inversion complete sets and maximal abelian ideals 11:00 11:30 Coffee Break (Room V3-201) 11:30 12:30 Alexander Elashvili (Tbilisi) On Lieandric numbers 12:30 15:00 Lunch Break 15:00 16:00 Matthieu Jacquemet (Fribourg) Hyperbolic truncated simplices and reflection groups 16:00 16:30 Coffee Break (Room V3-201) 16:30 17:30 Christian Lange (Cologne) Towards a generalized fixed point theorem 4

5 Schedule: Friday July 24th Lecture Room: V :00 11:00 Willem de Graaf (Trento) Real and complex nilpotent orbits of so(4,4) 11:00 11:30 Coffee Break (Room V3-201) 11:30 12:30 Ernest Vinberg (Moscow) Good reflection groups in O(2,n) 12:30 15:00 Lunch Break 15:00 16:00 Valdemar Tsanov (Göttingen) Momentum images of representations, secant varieties and invariants 5

6 Abstracts Hannah Bergner (Bochum) Actions on supermanifolds and automorphism groups of compact complex supermanifolds Let M be a supermanifold and let g be a finite-dimensional Lie subsuperalgebra of the Lie superalgebra Vec(M) of super vector fields on M. We study the question in which cases such a Lie superalgebra g of super vector fields on M is induced by an action of a Lie supergroup. Necessary and sufficient conditions for this are provided, generalizing the results of Palais in the classical case. In the special case of a compact complex supermanifold M, the Lie superalgebra Vec(M) of super vector fields on M is finite-dimensional. By a result of Bochner and Montgomery the automorphism group of a compact complex manifold M carries the structure of a complex Lie group whose Lie algebra is isomorphic to the Lie algebra of vector fields on M. We investigate how the automorphism group of a compact complex supermanifold M can be defined and prove that it carries the structure of a complex Lie supergroup with Lie superalgebra Vec(M). (Part of this work is joint with M. Kalus.) Alexander Elashvili (Tbilisi) On Lieandric numbers Lieandric numbers count the number of biparabolic Lie subalgebras of index 1 in full matrix algebras (these are examples of Frobenius Lie algebras, providing constant solutions of the Yang-Baxter equations). In the talk, a combinatoric description of these numbers will be given, a conjecture concerning their asymptotics will be formulated and some evidence for the conjecture will be presented. This is a joint work with M. Jibladze. Anna Felikson (Durham) Cluster algebras via reflection groups Cluster algebras are defined inductively via repeatedly applied operation of mutation. During the last decade it turned out that the formula of mutation appears in various contexts. We will use linear reflection groups to build a geometric model for acyclic cluster algebras, where partial reflections will play the role of mutations. Willem de Graaf (Trento) Real and complex nilpotent orbits of so(4,4) Let G(k) denote the direct product of four copies of SL(2,k). This group acts on the fourth tensor power of k 2. We consider the nilpotent orbits of this action when k is the complex and the real field. We briefly indicate the physical relevance of these orbits. Then we discuss methods to list them. These are based on the fact that the given representation of G(k) can be realized as a theta-representation in the simple complex Lie algebra of type D 4. It is well known that in the complex case there are 30 nilpotent orbits. It turns out that when k =R there are 145 nilpotent orbits (excluding 0). 6

7 Matthieu Jacquemet (Fribourg) Hyperbolic truncated simplices and reflection groups A hyperbolic truncated simplex is obtained as polarly truncated finite-volume part of a so-called total simplex in the extended hyperbolic space. The class of hyperbolic truncated simplices contains interesting polytopes, such as the hyperbolic Coxeter pyramid P 17 H 17 related to the orientable hyperbolic arithmetic orbifold of absolute minimal volume. In this talk, we shall discuss geometric, algebraic and arithmetic features of Coxeter hyperbolic truncated simplices. Christian Lange (Cologne) Towards a generalized fixed point theorem It is well known that isotropy groups of finite real reflection groups are generated by the reflections they contain. The same is true for isotropy groups of unitary reflection groups due to Steinberg s fixed-point theorem. In the talk we explain that these results are special cases of a more general fixed point theorem, whose proof, however, still relies on a classification and on cumbersome computations. We sketch what is known and discuss illustrating examples. Dmitri Panyushev (Moscow) Minimal inversion complete sets and maximal abelian ideals Let g be a simple Lie algebra, b a fixed Borel subalgebra, and W the Weyl group of g. I am going to speak about a relationship between the maximal abelian ideals of b and the minimal inversion complete sets of W. The latter have been recently introduced by Malvenuto, Moseneder Frajria, Orsina, and Papi. Mark Sapir (Vanderbilt) On groups with Rapid Decay The property of Rapid Decay (RD) of groups and group actions was introduced by Haagerup and is very important for analytic application of groups. The property is also very geometric and so it is of interest to geometric group theorists. I will survey results and methods in this area of group theory. Valdemar Tsanov (Göttingen) Momentum images of representations, secant varieties and invariants We address the following question: What is the momentum image of an irreducible unitary representation of a compact Lie group? Despite the extensive general theories on momenta and representations, where the given case takes a central place, there lacks, to the best of our knowledge, a computable explicit description of the momentum image in terms of the invariants of the representation, say, the highest weight. Many cases are known, but many important ones still escape. I will discuss an approach based on works of Wildberger, Sjamaar and Heinzner. I will also discuss relations to secant varieties of embedded flag varieties and degrees of invariant polynomials. This is joint work with E. Hristova and T. Maciazek. Pavel Tumarkin (Durham) Reflection groups via cluster algebras I will describe a construction of presentations of finite and affine Weyl groups arising from cluster algebras. In particular, this leads to presentations of Weyl groups as quotients of various Coxeter groups. I will also discuss some generalizations. 7

8 Ernest Vinberg (Moscow) Good reflection groups in O(2,n) According to a classical result of Shephard Todd Chevalley, finite linear complex reflection groups are characterized by the property that their algebra of polynomial invariants is free. If we consider these groups as acting on the corresponding projective spaces (which are Hermitian symmetric spaces of positive curvature), a natural infinite analogue of them are cofinite discrete reflection groups in symmetric domains (Hermitian symmetric spaces of negative curvature), the analogue of polynomials being automorphic forms. The only symmetric domains admitting reflections are complex balls B(n)=U(1,n)/(U(1) U(n)) (of rank 1) and domains of Cartan type IV D(n)=O + (2,n)/(SO(2) O(n)) (of rank 2). Many examples of cofinite discrete reflection groups in B(n) and D(n) for small n are known. For some of them it is known that the algebra of automorphic forms is free. However, due to a general result of Margulis for symmetric spaces of rank >1, any cofinite discrete group in D(n) containing a reflection, has a finite index subgroup generated by reflections. This means that there are a lot of cofinite discrete reflection groups in D(n) for any n, and it is not likelyhood that all of them share any good properties. To distinguish good reflection groups Γ O(2,n), one can require that dimh 2 (Γ,Q)=1. Under this condition, there exists a semi-automorphic form (possibly, of fractional weight) vanishing (with multiplicity 1) exactly at the mirrors of reflections contained in Γ (an analogue of the Vandermonde determinant). Hopefully, this will permit to prove that, for such good reflection groups in O(2,n), the algebra of automorphic forms is free. In particular, let O d is the ring of integers of the quadratic field Q( d), and σ be its involution. The extended Hilbert modular group Γ d = PSL(2,O d ),σ is a cofinite discrete group in the domain D(2) (which is the direct product of two copies of the hyperbolic plane). It is often generated by reflections. One can try to calculate H 2 (Γ d,q), making use of a presentation of Γ d obtained in a geometric way. This program was realised for d=2 with the result that the group Γ 2 is good in the above sense. 8

9 Participant List Herbert Abels Hannah Bergner Stephanie Cupit-Foutou Alexander Elashvili Anna Felikson Willem de Graaf Werner Hoffmann Matthieu Jacquemet Matthias Kalus Christian Lange Dmitri Panyushev Detlev Poguntke Mark Sapir Gregory Soifer Valdemar Tsanov Pavel Tumarkin Ernest Vinberg (Bielefeld) (Bochum) (Bochum) (Tbilisi) (Durham, Great Britain) (Trento, Italy) (Bielefeld) (Fribourg, Switzerland) (Bochum) (Cologne) (Moscow) (Bielefeld) (Vanderbilt University, USA) (Bar-Ilan, Israel) (Bochum) (Durham, Great Britain) (Moscow) 9