Analysis on the Crown Domain

This paper is a further development of complex methods in harmonic analysis on semi-simple Lie groups [AG], [BeR], [KrS1,2]. We study the growth behaviour of the holomorphic extension of the orbit map of the spherical vector of an irreducible spherical representation of a real reductive group G when approaching the boundary of the crown domain of the Riemannian symmetric space G/K. As an application, we prove that Maa? cusp forms have exponential decay. Received: August 2006, Revision: June 2007, Accepted: June 2007     

On the category 𝒪 for rational Cherednik algebras

We study the category of representations of the rational Cherednik algebra A_W attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: _W-mod, where _W is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between /_tor, the quotient of by the subcategory of A_W-modules supported on the discriminant, and the category of finite-dimensional _W-modules. The standard A_W-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of cells, provided W is a Weyl group and the Hecke algebra _W has equal parameters. We prove that the category is equivalent to the module category over a finite dimensional algebra, a generalized q-Schur algebra associated to W.     

Depletion-driven four-phase coexistences in discotic systems

ABSTRACT

Free volume theory (FVT) is a versatile and tractable framework to predict the phase behaviour of mixtures of platelets and non-adsorbing polymer chains in a common solvent. Within FVT, three principal reference phases for the hard platelets are considered: isotropic (I), nematic (N) and columnar (C). We derive analytical expressions that enable us to systematically trace the different types of phase coexistences revealed upon adding depletants and confirm the predictive power of FVT by testing the calculated diagrams against phase stability scenarios from computer simulation. A wide range of multi-phase equilibria is revealed, involving two-phase isostructural transitions of all phase symmetries (INC) considered as well as the possible three-phase coexistences. Moreover, we identify the system parameters, relative disk shapes and colloid–polymer size ratios, at which four-phase equilibria are expected. These involve a remarkable coexistence of all three-phase states commonly encountered in discotics including isostructural coexistences I₁–I₂–N–C, I–N₁–N₂–C and I–N–C₁–C₂.