- Struktur- und Darstellungstheorie von unendlichdimensionalen lokal endlichen Liealgebren - Verallgemeinerte Harish-Chandra Moduln - Vektorbündel von endlichem Rang auf homogenen Ind-Varietäten
Final Report Abstract
We introduce and study a new category of g = sl(∞), o(∞), sp(∞)− modules which we denote by Tg. This category is a proper analogue of the category of finite-dimensional representations of a finite-dimensional simple Lie algebra. We describe the simple objects of this category, compute the Ext’s between them, and most importantly, prove that it is Koszul. This is a major step in the study of integrable modules over the finitary infinite-dimensional Lie algebras sl(∞), o(∞), sp(∞). Further work is devoted to opening a new direction in the study of generalized Harish-Chandra modules over finite-dimensional Lie algebras. The main result is a characterization of reductive subalgebras k ⊂ g for which there exists a simple infinite-dimensional bounded (g, k)-module of finite type. A classification of maximal bounded reductive subalgebras of sl(n) is given. The above work builds on the results of the first phase of funding of the project. My joint work with E. Dan-Cohen during this period concentrates on the structure theory of the finitary Lie algebras g = sl(∞), o(∞), sp(∞), more specifically on parabolic subalgebras and Levi components of arbitrary subalgebras. We give the crucial definition of a taut pair of semi-closed generalized flags which enables us to give an explicit description of parabolic subalgebras. An unexpected aspect of this result is that a parabolic subalgebra is described in terms not of one flag, but in terms of the joint stabilizer of two (generalized) flags. The results of a second part concern Levi components of general splittable subalgebras of g = sl(∞), o(∞), sp(∞). Together, the outcomes represent significant progress in structure theory of the simple finitary Lie algebras and complete the structure theory program of the original proposal. Thus significant results about a category of integrable g-modules are established which we denote by Tensg. The adjoint and coadjoint representations, g and g∗ , are objects of Tensg. Finally the program completes understanding analogs of the classical Bott-Borel-Weil theorem for diagonal ind-groups. It strengthens results of previous work by several authors and establishes the definitive Bott-Borel-Weil theorem for diagonal indgroups. My joint work with V. Serganova has also been supported by a project within the DFG Priority Program ”Representation Theory”.
Publications
- Parabolic and Levi subalgebras of finitary Lie algebras, International Mathematics Research Notices, Article ID mp169
E. Dan-Cohen, I. Penkov
- Bott-Borel-Weil theorem for diagonal ind-groups, Canadian Journal of Mathematics
. Dimitrov, I. Penkov, A
(See online at https://dx.doi.org/10.4153/CJM-2011-032-6) - Categories of integrable sl(∞)−, so(∞)−, sp(∞)−modules, Contemporary Mathematics 557, AMS 2011, 335-357
I. Penkov, V. Serganova
- Levi compononents of parabolic subalgebras of finitary Lie algebras, Contemporary Mathematics 557, AMS 2011, 129-149
E. Dan-Cohen, I. Penkov
- A Koszul category of representation of finitary Lie algebras
E. Dan-Cohen, I. Penkov, V. Serganova
- On bounded generalized Harish-Chandra modules, Annales de l’Institut Fourier, 62(2012), 477-496
I. Penkov, V. Serganova
