In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli–Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form. 2009 Els...

Let G be a finite group acting on a symplectic complex vector space V . Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by “symplectic reflections”, i.e. symplectomorphisms with fixed space of codimension 2 in V . Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakam...

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 [9]) of finite length over the corresponding affine Kac– Moody algebra in the case of central charge less than the critical level. Equivalent characterizations ...

In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.

We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur’s Inequality for all split and quasi-split (connected) reductive groups. These results are related to the non-emptiness of certain affine Deligne-Lusztig varieties.

We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e, u] only depend on the isomorphism type of [e, u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for sp...

We use the polynomial ring C[x1,1, . . . , xn,n] to modify the Kazhdan-Lusztig construction of irreducible Snmodules. 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. We also show that our modules are related by unitriangular transition matrices to those constructed by Claus...

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar [16] has given a framework for computing the Kazhdan-Lusztig polynomials which generally invo...

We give closed combinatorial product formulas for Kazhdan–Lusztig poynomials and their parabolic analogue of type q in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of th...