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We generalize the definition and properties of root systems to complex reflection groups — roots become rank one projective modules over the ring of integers of a number field k . In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes. [ABSTRACT FROM AUTHOR]

In this article we provide a new finite class of elements in any Coxeter system ( W , S ) called low elements . They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of ( W , S ) . Our first main result is to show that they form a Garside shadow in ( W , S ) , i.e., they contain S and are closed under join (for the right weak order) and by taking suffixes. These low elements are the key to prove that all finitely generated Artin–Tits groups have a finite Garside family. This result was announced in a note with P. Dehornoy in Comptes rendus mathématiques [9] in which the present article was referred to under the following working title: Monotonicity of dominance-depth on root systems and applications . The proof is based on a fundamental property enjoyed by small roots and which is our second main result; the set of small root is bipodal . For a natural number n , we define similarly n -low elements from n -small roots and conjecture that the set of n -small roots is bipodal, implying the set of n -low elements is a Garside shadow; we prove this conjecture for affine Coxeter groups and Coxeter groups whose graph is labelled by 3 and ∞. To prove the latter, we extend the root poset on positive roots to a weak order on the root system and define a Bruhat order on the root system , and study the paths in those orders in order to establish a criterion to prove bipodality involving only finite dihedral reflection subgroups. [ABSTRACT FROM AUTHOR]

We present an explicit difference equation for the Heckman–Opdam hypergeometric function associated with root systems. Via a confluent hypergeometric limit, an analogous difference equation is obtained for the class-one Whittaker function diagonalizing the open quantum Toda chain. [ABSTRACT FROM AUTHOR]

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t → ∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q -Whittaker function. These coefficients are pure q -powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebras for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G 2 , and also establish the relations of our extremal degrees to minimal q -degrees of the extremal terms of the Kostant q -partition function; they coincide with the latter only for some root systems. [ABSTRACT FROM AUTHOR]

Abstract: A series expansion for Heckman–Opdam hypergeometric functions is obtained for all . As a consequence, estimates for away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The -theory for the hypergeometric Fourier transform is developed for . In particular, an inversion formula is proved when . [Copyright &y& Elsevier]

Abstract: We establish orthogonality relations for the Baker–Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik–Macdonald–Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik–Macdonald–Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new quantum integrable models of Macdonald–Ruijsenaars type. [Copyright &y& Elsevier]

Abstract: Let G be the group or . We compute the integral of on the Hurwitz locus of curves admitting a degree 4 cover of having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the and root systems, respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds and . [Copyright &y& Elsevier]

Abstract: We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties. [Copyright &y& Elsevier]

In this article we introduce the notion of a regular partition of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi's classical result that each component of the Shi arrangement of an affine Coxeter group has a unique minimal length element. We further develop the theory of cone types in Coxeter groups by identifying the minimal set of roots required to express a cone type as an intersection of half-spaces. This set of boundary roots is closely related to the elementary inversion sets of Brink and Howlett, and also to the notion of the base of an inversion set introduced by Dyer. [ABSTRACT FROM AUTHOR]

The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type E 8. This approach is also applicable to E 8. Even if the algebra of E 8 Jacobi forms is known to be non-free, we still derive a new structure result. [ABSTRACT FROM AUTHOR]

For 25 orthogonal groups of signature (2 , n) related to the root lattices A 1 , 2 A 1 , 3 A 1 , 4 A 1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7 , D 4 , D 5 , D 6 , D 7 , D 8 , E 6 , E 7 , we prove that the algebras of modular forms on symmetric domains of type IV are freely generated by the additive lifts of some special Jacobi forms. The proof is universal and elementary. [ABSTRACT FROM AUTHOR]