We establish a formalism for working with incidence algebras of posets symmetries, and we develop equivariant Kazhdan–Lusztig–Stanley theory within this formalism. This gives new way thinking about the Kazhdan–Lusztig polynomial Z-polynomial matroid.

We use a quantum analog of the polynomial ringZ[x1,1, . . . , xn,n] to modify the Kazhdan-Lusztig construction of irreducible Hn(q)-modules. This modified construction produces exactly the same matrices as the original construction in [Invent. Math. 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. Our main result is dependent on new vanishing results for immanants in the quantum p...

We construct first examples of infinite groups having property (T) whose Kazhdan constants admit a lower bound independent of the choice of a finite generating set.

An infinitesimal Kazhdan constant of Sp (2,R) is computed. The methods used to prove this can also be employed to determine a quantitative estimate of the asymptotics of the matrix coefficients of Sp (n,R) in an elementary manner. An application of the result gives explicit Kazhdan constants for Sp (n,R), n ≥ 2.

Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial Pid,w(q) = 1 + q for some h.

We classify the \fully tight" simply-laced Coxeter groups, that is, the ones whose iji-avoiding Kazhdan{Lusztig basis elements are monomials in the generators B s i. We then investigate the basis of the Temperley{Lieb algebra arising from the Kazhdan{Lusztig basis of the associated Hecke algebra, and prove that the basis coincides with the usual (monomial) basis.

To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category AFF(C)κ of smooth modules (in the sense of Kazhdan and Lusztig [13]) of finite length over the corresponding affine Kac–Moody algebra in the case of central charge less than the critical level. Equivalent characterizations ...

Let H be an infinite hyperbolic group with Kazhdan property (T ) and let κ(H,X) denote the Kazhdan constant of H with respect to a generating set X . We prove that infX κ(H,X)= 0, where the infimum is taken over all finite generating sets of H . In particular, this gives an answer to a Lubotzky question.

We consider a class of relatively hyperbolic groups in the sense of Gromov and use an argument modeled after Carlsson–Pedersen to prove Novikov conjectures for these groups. This proof is related to [16, 17] which dealt with arithmetic lattices in rank one symmetric spaces and some other arithmetic groups of higher rank. Here we view the rank one lattices in this different larger context of rel...