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.