The moduli space of principally polarized Abelian varieties with real structure and with level N = 4m structure (with m≥1) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over ${\open Q}$, and to consist of finitely many copies of the quotient of the space GL(n, ${\open R}$)/O(N) (of positive definite symmetric matrices) by the principal congruence subgroup of level N in GL(n, ${\open Z}$).

Soient G = exp ${\frak g}$ un groupe de Lie résoluble exponentiel et H = exp ${\frak h}$ un sous-groupe connexe de G. Soient χ un caractère unitaire de H et τ = ${\rm Ind}_{H}^{G}$χ. Soit Dτ(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de τ est équivalente à la commutativité de Dτ(G/H). Nous répondons positivement à cette question quand H est normal dans G. Lorsque H n'est pas normal, nous préparons le terrain pour d'espaces homogènes nilpotents et nous répondons à la question dans différents cas. Nous étudions finalement l'algèbre Dπ(G)H, π∈Ĝ des opérateurs différentiels qui laissent l'espace des vecteurs C∞ de π invariant et qui commuttent avec l'action de H sur cet espace.

We prove the existence of an involution on double affine Hecke algebras. This involution plays a central role in the theory of Macdonald polynomials, being responsible for a Fourier type duality and allowing the construction of affine intertwiners.

We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst–Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given.

In the present paper we give a very short and easy proof of the speciality lemma for codimension 2 subvarieties, even those that are reducible or non-reduced, in Pn. Furthermore we give cohomological conditions that force a subcanonical surface in P4 to be a complete intersection and a rank 2 bundle to split, which generalize the classical First Theorem of Gherardelli.

It is proved that the index of a Fredholm operator between non-Archimedean Fréchet spaces is preserved under compact perturbations. A similar result is shown for Fredholm operators between non-Archimedean polar regular LF-spaces.