We present an explicit construction of the Macdonald polynomials and of the Heckman–Opdam generalized Jacobi polynomials associated with root systems.

Let $\pi$ be a generic irreducible representation of either a symplectic group or a split special even orthogonal group over a local field of characteristic zero. We prove that $\varepsilon(\frac12,\pi,\psi)=\pi(-1)$.

We determine exactly when a quadratic form is represented by a spinor genus of another quadratic form of three or four variables. We apply this to extend the embedding theorem for quaternion and also answer a question by Borovoi.

For the reductive dual pair (U(n, n), U(n, n)) over a p-adic field, we study the local theta correspondence for certain tempered representations in terms of R-groups. In the case we consider, the Langlands parameter is preserved, but, a twist occurs in the L-packet. Moreover this twist is determined by root numbers.

In this paper, we construct a theory of geometric Euler systems, complementary to the arithmetic theory of Rubin, Kato and Perrin-Riou. We show that geometric Euler systems can be used to prove the finiteness of certain Galois representations of weight zero and we discuss a conjectural framework for the existence of geometric Euler systems for motivic Galois representations. We give applications to adjoint Selmer groups of certain classical and Drinfeld modular forms.

Let p be a prime number and let F be a local field with finite residual field of characteristic p. The Langlands local correspondence modulo $\ell \neq p $ for GL(n, F) is known for all integers $n\geq 1$ but the case $\ell=p$ is still mysterious even when n = 2 (the case n = 1 is given by the local class field theory). Any irreducible $\overline{\bf F}_p$-representation of GL(n, F) has a non-zero vector invariant by the pro-p-Iwahori subgroup I(1) and the pro-p-Iwahori–Hecke $\overline{\bf F}_p$-algebra ${\cal H}_{ \overline{\bf F}_p}(GL(n,F),I(1))$ plays a fundamental role in the theory of $\overline{\bf F}_p$-representations of G. We get when n = 2: (i) A bijection between the irreducible $\overline{\bf F}_p$-representations of dimension 2 of the Weil group $W(\overline F/ F)$ and the simple supersingular modules of the pro-p-Iwahori–Hecke $\overline{\bf F}_p$-algebra ${\cal H}_{\overline{\bf F}_p}(GL(2,F),I(1))$. (ii) A new proof of the Barthel–Livne classification of the irreducible non-supersingular $\overline{\bf F}_p$-representations of GL(2, F) using the I(1)-invariant functor. (iii) A bijection between the irreducible $\overline{\bf F}_p$-representations of GL(2, Qp) and the simple right ${\cal H}_{\overline{\bf F}_p}(GL(2,{\bf Q}_p),I(1))$-modules given by the I(1)-invariant functor, using the recent results of Breuil.

We set up the basic theory of $\mathcal P$-adic modular forms over certain unitary PEL Shimura curves M′K′. For any PEL abelian scheme classified by M′K′, which is not ‘too supersingular’, we construct a canonical subgroup which is essentially a lifting of the kernel of Frobenius from characteristic p. Using this construction we define the U and Frob operators in this context. Following Coleman, we study the spectral theory of the action of U on families of overconvergent $\mathcal P$-adic modular forms and prove that the dimension of overconvergent eigenforms of U of a given slope is a locally constant function of the weight.

We introduce the notion of twisted jets. For a Deligne–Mumford stack $\mathscr{X}$ of finite type over $\mathbb{C}$, a twisted $\infty$-jet on $\mathscr{X}$ is a representable morphism $\mathscr{D} \to \mathscr{X}$ such that $\mathscr{D}$ is a smooth Deligne–Mumford stack with the coarse moduli space Spec$\mathbb{C}[[t]]$. We study a motivic measure on the space of the twisted $\infty$-jets on a smooth Deligne–Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology as a Hodge structure.

Fix a $\tau$-connection DL on a line bundle L over $X\times \mathbb{C}$, where X is a compact connected Riemann surface of genus at least three. Let $\mathcal{M}_X(D_L)$ denote the moduli space of all semistable $\tau$-connections of rank n, where $n\geq 2$, with the property that the induced $\tau$-connection on the top exterior product is isomorphic to (L, DL). Let $\mathcal{M}_Y(D_M)$ be the similar moduli space for another Riemann surface Y with genus(Y) = genus(X), where DM is a $\tau$-connection on a line bundle M over $Y\times \mathbb{C}$. We prove that if the variety $\mathcal{M}_X(D_L)$ is isomorphic to $\mathcal{M}_Y(D_M)$, then X is isomorphic to Y.

Let $\mathcal{M}^D_X$ denote the moduli space of all rank n flat connections on X. We prove that $\mathcal{M}^D_X$ determines X up to finitely many Riemann surfaces. For the very general Riemann surface X, the variety $\mathcal{M}^D_X$ determines X.

We show that the deformation functor of a maximal Cohen–Macaulay module M = coker($\phi$) over the hypersurface singularity det($\phi$) is given by deformations of the presenting matrix which keep the determinant constant. A simplified expression for an edge map in the canonical five-term exact sequence to a change of rings spectral sequence is obtained, including the tangent and obstruction spaces (H1 and H2). We relate the edge map to the Scandinavian complex$\mathcal{S}$ of $\phi$ which yields relations between the homology of $\mathcal{S}$ and Hi for i = 1, 2. This gives (infinitesimal) rigidity and non-rigidity results and a dimension estimate for the formally (mini-)versal formal hull H of the deformation functor.

Let $(P,\Xi)$ be the naturally polarized model of the Prym variety associated to the étale double cover $\pi : \tilde C\rightarrow C$ of smooth connected curves defined over an algebraically closed field k of characteristic $\ne 2$, where genus(C) = $g \ge 3$, Pic$^{(2g-2)}(\tilde C) \supset P = \{\mathcal L \in {\rm Pic}^{(2g-2)}(\tilde C) : {\rm Nm}(\mathcal L) = \omega_C$ and $h^0(\tilde C,\mathcal L)$ is even\} is the Prym variety, and $P \supset \Xi = \{\mathcal L \in P: h^0(\tilde C,\mathcal L) >0 \}$ is the Prym theta divisor with its reduced scheme structure. If $\mathcal L$ is any point on $\Xi$, we prove that ‘Riemann's singularity theorem holds at $\mathcal L$’, i.e. mult$_{\mathcal L}(\Xi) = (1/2)h^0(\tilde C,\mathcal L)$, if and only if $\mathcal L$ cannot be expressed as $\pi^*(\mathcal M)(B)$ where $B \ge 0$ is an effective divisor on $\tilde C$, and $\mathcal M$ is a line bundle on C with $h^0(C,\mathcal M) >(1/2)h^0(\tilde C,\mathcal L)$. This completely characterizes points of $\Xi$ where the tangent cone is the set theoretic restriction of the tangent cone of $\tilde {\Theta}$, hence also those points on $\Xi$ where Mumford's Pfaffian equation defines the tangent cone to $\Xi$.

We use the $A_{\infty}$-formalism to study variation in cohomology spaces under formal deformations of coherent sheaves on projective varieties. As an application we describe formal neighborhoods of some singular points in twisted Brill–Noether loci in the moduli spaces of vector bundles on a curve. Another application is a computation of the Fourier–Mukai transform of some natural line bundles on symmetric powers of a curve.

Let V be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH*(OG) and show that its product structure is determined by the ring of $\widetilde{P}$-polynomials. A ‘quantum Schubert calculus’ is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov--Witten invariants. As an application, we show that the table of three-point, genus zero Gromov–Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.

On a real algebraic variety there may exist an algebraic cycle that is algebraically equivalent to zero and whose cohomology class is non-zero. The group of such cohomology classes can be highly non-trivial. It is interesting since it allows one to detect cohomology classes, in complementary dimension, which cannot be represented by algebraic cycles.

We prove Pursell–Shanks type results for the Lie algebra $\mathcal{D}(M)$ of all linear differential operators of a smooth manifold M, for its Lie subalgebra $\mathcal{D}^1(M)$ of all linear first-order differential operators of M and for the Poisson algebra S(M) = Pol(T*M) of all polynomial functions on T*M, the symbols of the operators in $\mathcal{D}(M)$. Chiefly, however, we provide explicit formulas completely describing the automorphisms of the Lie algebras $\mathcal{D}^1(M)$, S(M) and $\mathcal{D}(M)$.

We give dimension formulas for the cotangent cohomology groups Ti for all rational surface singularities. The computations take place on the first blow-up of the singularity.