While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

Let $f(x) = \Sigma a_ix^i$ be a monic polynomial of degree $n$ whose coefficients are algebraically independent variables over a base field $k$ of characteristic 0. We say that a polynomial $g(x)$ is generating (for the symmetric group) if it can be obtained from $f(x)$ by a nondegenerate Tschirnhaus transformation. We show that the minimal number ${\rm d}_k(n)$ of algebraically independent coefficients of such a polynomial is at least $[n/2]$. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert's 13th problem.

Our approach to this question (and generalizations) is based on the idea of the ‘essential dimension’ of a finite group $G$: the smallest possible dimension of an algebraic $G$-variety over $k$ to which one can ‘compress’ a faithful linear representation of $G$. We show that ${\rm d}_k(n)$ is just the essential dimension of the symmetric group ${\rm S}_n$. We give results on the essential dimension of other groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of cent ral values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex-variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.

The Hermitian function field H= K(x,y) is defined by the equation yq+y=xq +1 (q being a power of the characteristic of K). Over K=${\mirrored F}$q2 it is a maximal function field; i.e. the number N(H) of ${\mirrored F}$q2-rational places attains the Hasse–Weil upper boundN(H)=q2+1+2g(H)·q. All subfields K[subnE ] E⊆H are also maximal. In this paper we construct a large number of nonrational subfields E⊆H, by considering the fixed fields H$^{\mathcal G}$ under certain groups ${\mathcal G}$ of automorphisms of H/K. Thus we obtain many integers g[ges ]0 that occur as the genus of some maximal function field over ${\mirrored F}$q2.

Let p be a prime number and F a complete local field with residue field of characteristic p. In 1993, Barthel and Livné proved the existence of a new kind of $\overline{\bf F}_{p}$-representations of GL2(F) that they called ‘supersingular’ and on which one knows almost nothing. In this article, we determine all the supersingular representations of GL2(Qp) with their intertwinings. This classification shows a natural bijection between the set of isomorphism classes of supersingular representations of GL2(Qp) and the set of isomorphism classes of two-dimensional irreducible $\overline{\bf F}_{p}$-representations of ${\rm Gal}(\overline{\bf Q}_{p}/{\bf Q}_{p})$.

We construct a two-parameter family of actions ωk,a of the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Here k is a multiplicity function for the Dunkl operators, and a>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation of Mp(N,ℝ) and the minimal unitary representation of O(N+1,2). We prove that this action ωk,a lifts to a unitary representation of the universal covering of SL (2,ℝ) , and can even be extended to a holomorphic semigroup Ωk,a. In the k≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup studied by the second author with G. Mano (a=1) . One boundary value of our semigroup Ωk,a provides us with (k,a) -generalized Fourier transforms ℱk,a, which include the Dunkl transform 𝒟k (a=2) and a new unitary operator ℋk  (a=1) , namely a Dunkl–Hankel transform. We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for ℱk,a. We also find kernel functions for Ωk,a and ℱk,a for a=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dimA and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of ’weights‘ (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.

We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Soit $F$ un corps local non archimédien de caractéristique nulle, à corps résiduel fini. Si $M$ est un groupe réductif connexe défini sur $F$, on note par la lettre minuscule $m$ son algébre de Lie. On note $m_{\rm reg}$ l'ensemble des $X \in m$ dont le centralisateur$T_X$ dans $M$ est un tore. Si $X, X^{\prime} \in m_{\rm reg}(F)$, on dit que $X$ et $X^{\prime}$ sont conjugués, resp. stablement conjugués, s'il existe $x \in M(F)$, resp. $x \in M(\bar{F})$ où $\bar{F}$ est la clôture algébrique de $F$, tel que $({\rm Ad}\, x)(X) = X^{\prime}$. On note $C^{\infty}_c (m(F))$ l'espace des fonctions sur $m(F)$. à valeurs complexes, localement constantes et à support compact. Soient $X \in m_{\rm reg}(F)$ et $f \in C^{\infty}_c (m(F))$. Supposons $T_X(F)$ et $M(F)$ munis de mesures de Haar. On pose\[J(X, f) = D(X)^{1/2} \int_{T_X(F)\setminus M(F)} f(({\rm Ad}\,x^{-1})(X))\,{\rm d}x,\] oú \[D(X) = \vert \det ({\rm ad} X \vert m(F)/t_X (F))\vert_F,\]$\vert \cdot\vert_F$ désignant la valeur absolue usuelle de $F$.

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such as principal coefficients.

We study positivity properties of the moduli (b-)divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class.

Given a Lie n-algebra, we provide an explicit construction of its integrating Lie n-group. This extends work done by Getzler in the case of nilpotent -algebras. When applied to an ordinary Lie algebra, our construction yields the simplicial classifying space of the corresponding simply connected Lie group. In the case of the string Lie 2-algebra of Baez and Crans, we obtain the simplicial nerve of their model of the string group.

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.

we prove in this paper the green conjecture for generic curves of odd genus. that is, we prove the vanishing $k_{k,1}(x,k_x)=0$ for x a generic curve of genus $2k+1$. this completes our previous work, where the green conjecture for generic curves of genus g with fixed gonality d was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus. the case of generic curves of odd genus was considered as especially important, since hirschowitz and ramanan proved that if the conjecture is true for the generic curve of odd genus, then the locus of jumping syzygies is exactly the locus of exceptional gonality, as predicted by green's conjecture. thus, our result combined with the hirschowitz–ramanan result is a strong confirmation of green's conjecture.

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions $F(t_1, …, t_n)$ of WDVV equations of associativity polynomial in $t_1, …, t_n-1$, exp $t_n$.

Using theta correspondence, we classify the irreducible representations of Mp2n in terms of the irreducible representations of SO2n+1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n=1.

We introduce and study a category of representations of the Borel algebra associated with a quantum loop algebra of non-twisted type. We construct fundamental representations for this category as a limit of the Kirillov–Reshetikhin modules over the quantum loop algebra and establish explicit formulas for their characters. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable which are regular and non-zero at the origin but may have a zero or a pole at infinity.

We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.

We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ over 𝔽1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q=1 as expected from the infinite genus of C. Then, we develop a theory of functorial 𝔽1-schemes which reconciles the previous attempts by Soulé and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of -schemes to interpret conceptually the spectral realization of zeros of L-functions.