Bispectral operators of prime order

The aim of this paper is to solve the bispectral problem for bispectral operators whose order is a prime number. More precisely we give a complete list of such bispectral operators. We use systematically the operator approach and in particular - Dixmier ideas on the first Weyl algebra. When the order is 2 the main theorem is exactly the result of Duistermaat-Gr\"unbaum . On the other hand our proofs seem to be simpler.Comment: 25 pages, to appear in CM

Spectral correspondences for affine Hecke algebras

We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations.Comment: 38 pages; The ordering of the material has been improved in this versio

Cuspidal representations of rational Cherednik algebras at t=0

We study those finite dimensional quotients of the rational Cherednik algebra at t=0 that are supported at a point of the centre. It is shown that each such quotient is Morita equivalent to a certain cuspidal quotient of a rational Cherednik algebra associated to a parabolic subgroup of W.Comment: Comments welcom

Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $\mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric Cremona transformations, proving also a structure theorem for these birational maps and for varieties 3-covered by twisted cubics by reinterpreting for these objects the solvability of the radical of a Jordan algebra.Comment: 30 pages, 1 figure. Corrected typo

Derived tame local and two-point algebras

We determine derived representation type of complete finitely generated local and two-point algebras over an algebraically closed field.Comment: 19 page

Universal geometric cluster algebras

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or reals. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal coefficients is equivalent to finding an R-basis for B (a "mutation-linear" analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan F_B, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between F_B and g-vectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.Comment: Final version to appear in Math. Z. 49 pages, 5 figure