Let G be a finite-dimensional Poisson algebraic, Lie or formal group. We show that the center of the quantization of G provided by an Etingof–Kazhdan functor is isomorphic as an algebra to the Poisson center of the algebra of functions on G. This recovers and generalizes Duflo’s theorem which gives an isomorphism between the center of the enveloping algebra of a finite-dimensional Lie algebra a...

In this and the next lecture we will describe an entirely different application of Hecke algebras, now to the category O. In the first section we will define the Kazhdan-Lusztig basis in the Hecke algebra of W and explain how to read the multiplicities in the category O from this basis (the Kazhdan-Lusztig conjecture proved independently by Beilinson-Bernstein and Brylinski-Kashiwara). In the r...

We give a presentation of the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan [12]. We limit our attention to a simple major subclass of V -minimal theories of the form ACVF S , that is, the theory of algebraically cl...

We show that dual canonical basis elements of the quantum polynomial ring in n variables can be expressed as specializations of dual canonical basis elements of 0-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, Rpolynomials, and certain single and double parabolic generalizations of thes...

1.a. Since its introduction by Kazhdan in [Ka], property (T ) became a fundamental concept in mathematics with a wide range of applications to such areas as: • The structure of infinite groups—finite generation and finite Abelianization of higher-rank lattices [Ka], obstruction to free or amalgamated splittings [Wa], [A], [M4], structure of normal subgroups [M2] etc.; • Combinatorics—the first ...