Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics
Herbert Abels and Ernest Vinberg
The workshop is part of the conference program of the DFG-funded CRC 701 "Spectral Structures and Topological Methods in Mathematics" at the University of Bielefeld.
The program of the workshop can be found here in pdf-format. (as of July 21st)
The arrival of the participants is planned on Friday, July 23rd. For hotel information see below.
All the talks take place in the "Common Room" (V3-201).
| Herbert Abels (Bielefeld) | Less than two generators suffice (joint work with Ernest B. Vinberg) Let G be a real simple Lie group with finite center. We show that given a non–central element g in G there is an elliptic element h in G such that h and ghg−1 generate a dense subsemigroup of G. |
| Ivan V. Arzhantsev (Moscow / Tübingen) | Equivariant compactifications of commutative unipotent groups The study of equivariant compactifications of a commutative unipotent group G may be regarded as an additive analogue of toric geometry. B.Hassett and Yu.Tschinkel (Int. Math. Res. Notices 20 (1999), 1211-1230) introduced a remarkable correspondence between generically transitive G-actions and finite-dimensional local algebras. We develop their correspondence, calculate modality of generically transitive G-actions on projective spaces, classify actions of modality one, and characterize generically transitive G-actions on projective hypersurfaces of given degree. In particular, G-actions on projective quadrics are studied. This part of the talk is based on a joint work with Elena Sharoyko. In the second part we classify flag varieties which may be realized as equivariant compactifications of G. |
| Ghislain Fourier (Cologne) | Another basis and pattern for irreducible An modules There is a natural filtration on the universal enveloping algebra of a simple Lie algebra, the degree filtration. It induces the PBW filtration on any irreducible module of this Lie algebra. In this talk, we will give generators and relations for the associated graded module of an irreducible An-module. As a byproduct we obtain a new class of patterns and bases for irreducible An-modules. |
| Willem de Graaf (Trento) | Constructing semisimple subalgebras of semisimple Lie algebras The problem is to list the semisimple subalgebras of a semisimple Lie algebra g up to (linear) equivalence, to explicitly construct them in terms of a basis of g, and to obtain the inclusion relations among them. In this talk I will review previous work on this problem, mainly by Dynkin. Then I will discuss computational methods that help to approach the problem, and the results obtained by using them (lists of semisimple subalgebras of simple Lie algebras of rank at most 8). |
| Joachim Hilgert (Paderborn) | Wigner and Patterson-Sullivan distributions for locally symmetric spaces Anantharam and Zelditch observed a remarkable connection between Wigner and Patterson-Sullivan distributions on compact hyperbolic surfaces. These are distributions associated with the eigenvalues of the Laplace-Beltrami operators and satisfy invariance properties under the geodesic flow. A key tool to establish this connection is a specific pseudodifferential calculus adapted to the symmetries of the situation. Together with M. Schroeder we reformulated these results in terms of group and representation theory and generalized them to rank 1 symmetric spaces. In this talk I will sketch how this goes and explain the connection with Selberg zeta functions. |
| Werner Hoffmann (Bielefeld) | Induced conjugacy classes and applications Lusztig and Spaltenstein have defined the induction of unipotent conjugacy classes. We generalise this notion to arbitrary conjugacy classes in reductive groups. As an application, we present a conjectural expansion of the geometric side of the Arthur-Selberg trace formula, which has been checked for some groups of rank two. |
| Michael Kapovich (UCDavis) | Lie theory for noncrystallographic dihedral groups I will talk about (mostly) geometric aspects of our project with Arkady Berenstein on developing Lie theory for noncrystallographic finite Coxeter groups. I will discuss "Schubert pre-calculus" and stability inequalities for such groups. |
| Jan Möllers (Paderborn) | Minimal representations of conformal groups and generalized Laguerre functions PDF file containing the abstract for this talk can be found here. |
| D. I. Panyushev (Moscow) | Quotients by actions of the derived group of a maximal unipotent group Let G be a connected semisimple algebraic group and U a maximal unipotent subgroup of G. In my talk, I will speak on invariant-theoretic properties of actions of the derived group of U on affine G-varieties. |
| D. Poguntke (Bielefeld) | G-prime ideals in L1-convolution algebras of nilpotent Lie groups PDF file containing the abstract for this talk can be found here. |
| Andrei Rapinchuk (Charlottesville) | On division agebras having the same maximal subfields I will discuss some results concerning the following question: let D_1 and D_2 be two central quaternion division algebras over the same field K; when does the fact that D_1 and D_2 have the same maximal subfields imply that D_1 and D_2 are actually isomorphic over K? It turns out that the answer depends on the field K, and one of the results that I will present states that if the answer is positive over a field K then it is also positive over any finitely generated purely transcendental extension of K. I will also discuss some generalizations to algebras of degree > 2. This is a joint work with Igor Rapinchuk. |
| Grigorii Soifer (Tel Aviv) | The Auslander conjecture (joint work with H. Abels and G. A. Margulis) PDF file containing the abstract for this talk can be found here. |
| Ernest B. Vinberg (Moscow) | Quartic surfaces and automorphic forms on symmetric domains of type IV The theory of automorphic forms is the theory of invariants of discrete groups of holomorphic transformations, and in this sense it can be viewed as "transcendental invariant theory". It is connected with algebraic invariant theory due to the possibility to interpret the moduli varieties of some classes of algebraic varieties as arithmetic quotients of symmetric domains. A classic example: the moduli variety of cubic curves is naturally isomorphic to the quotient of the upper half-plane by the Klein modular group, which implies that the algebra of modular forms is isomorphic to the algebra of invariants of a cubic ternary form. A more involved and recent example: the moduli variety of quartic surfaces is naturally isomorphic to a certain arithmetic quotient of the 19-dimensional symmetric domain of type IV minus two divisors. The algebra of invariants of a quartic quaternary form is too complicated in order to deduce any useful consequences from this. However, considering suitable subvarieties in the above moduli variety, one can obtain concrete explicit results, in particular, to prove that certain algebras of automorphic forms on symmetric domains of type IV up to dimension 7 are free, and to determine the degrees of their generators. This implies that the corresponding arithmetic groups are generated by complex reflections. |
| O. Yakimova (Erlangen) | Good index behaviour of θ-representations PDF file containing the abstract for this talk can be found here. |
We have reserved rooms for the workshop in the hotel Tulip Inn in Bielefeld for the nights between July 23 and July 26. The reservation is valid until July 15. So please register directly in the hotel before that date. When reserving mention that you are participant/speaker of the workshop on Lie groups and algebraic groups at the department of mathematics of the University of Bielefeld.
The reservation number 96436 might also be useful.
Hotel Tulip Inn Bielefeld
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Please contact the organizers, Mr. Abels abels(at)math(dot)uni-bielefeld(dot)de or Mr. Vinberg vinberg(at)math(dot)uni-bielefeld(dot)de, for further information.