Lie Groups, Geometry, and Representation Theory. A Tribute to the Life and Work of Bertram Kostant
This volume, dedicated to the memory of the great American mathematician Bertram Kostant
(May 24, 1928 – February 2, 2017), is a collection of 19 invited papers by leading
mathematicians working in Lie theory, representation theory, algebra, geometry, and
mathematical physics. Kostant’s fundamental work in all of these areas has provided deep new
insights and connections, and has created new fields of research. This volume features the
only published articles of important recent results of the contributors with full details of their
proofs. Key topics include: Poisson structures and potentials (A. Alekseev, A. Berenstein, B.
Hoffman) Vertex algebras (T. Arakawa, K. Kawasetsu) Modular irreducible representations of
semisimple Lie algebras (R. Bezrukavnikov, I. Losev) Asymptotic Hecke algebras (A. Braverman,
D. Kazhdan) Tensor categories and quantum groups (A. Davydov, P. Etingof, D. Nikshych) Nil-
Hecke algebras and Whittaker D-modules (V. Ginzburg) Toeplitz operators (V. Guillemin, A.
Uribe, Z. Wang) Kashiwara crystals (A. Joseph) Characters of highest weight modules (V. Kac, M.
Wakimoto) Alcove polytopes (T. Lam, A. Postnikov) Representation theory of quantized Gieseker
varieties (I. Losev) Generalized Bruhat cells and integrable systems (J.-H. Liu, Y. Mi) Almost
characters (G. Lusztig) Verlinde formulas (E. Meinrenken) Dirac operator and equivariant index
(P.-É. Paradan, M. Vergne) Modality of representations and geometry of-groups (V. L. Popov)
Distributions on homogeneous spaces (N. Ressayre) Reduction of orthogonal representations (J.-
We first establish several general properties of modality of algebraic group actions. In particular,we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we classify irreducible linear representations of connected simple algebraic groups of every fixed modality < 3. Next, exploring a finer geometric structure of linear actions, we generalize to the case of any cyclically graded semisimple Lie algebra the notion of a packet (or a Jordan/decomposition class) and establish the properties of packets.
In this paper we investigate the growth with respect to p of dimensions of irreducible representations of a semisimple Lie algebra g over F¯¯¯p. More precisely, it is known that for p≫0, the irreducibles with a regular rational central character λ and p-character χ are indexed by a certain canonical basis in the K0 of the Springer fiber of χ. This basis is independent of p. For a basis element, the dimension of the corresponding module is a polynomial in p. We show that the canonical basis is compatible with the two-sided cell filtration for a parabolic subgroup in the affine Weyl group defined by λ. We also explain how to read the degree of the dimension polynomial from a filtration component of the basis element. We use these results to establish conjectures of the second author and Ostrik on a classification of the finite dimensional irreducible representations of W-algebras, as well as a strengthening of a result by the first author with Anno and Mirkovic on real variations of stabilities for the derived category of the Springer resolution.
We study the representation theory of quantizations of Gieseker moduli spaces. We describe the categories of finite dimensional representations for all parameters and categories O for special values of parameters. We find the values of parameters, where the quantizations have finite homological dimension, and establish abelian localization theorem. We describe the two-sided ideals. Finally, we determine annihilators of the irreducible objects in categories O for some special choices of one-parameter subgroups.