We study the image of the model subspace $K_\theta$ under the composition operator $C_\varphi$, where $\varphi$ and $\theta$ are inner functions, and find the smallest model subspace which contains the linear manifold $C_\varphi K_\theta$. Then we characterize the case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to the study of the inner functions $\varphi$ and $\psi$ such that the composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of inner functions.

In this paper, we study the reduced loci of special cycles on local models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

Let $\BF_q$ be a finite field of $q$ elements, $\CF$ a $p$-adic field, and $D$ a quaternion division algebra over $\CF$. This paper proves uniqueness of Shalika models for $\GL_{2n}(\BF_q) $ and $\GL_{2n}(D)$, and re-obtains uniqueness of Shalika models for $\GL_{2n}(\CF)$ for any $n\in \BN$.

This paper shows the existence and uniqueness of Klyachko models for irreducible unitary representations of $\GL_5(\CF)$, where $\CF$ is a $p$-adic field. It is an extension of the work of Heumos and Rallis on $\GL_4(\CF)$.

We explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QM curves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our $j$-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using $j$, we construct the fields of moduli and definition for some moduli problems associated to the Atkin--Lehner group actions.

The Casselman--Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $GL_n$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y. Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use $S^1$-equivariant minimal models to give a number of examples of $S^1$-spaces illustrating the properties of the various definitions.

This paper is a continuation of \cite{HM02I}. We consider the model subspaces $K_\Theta=H^2\ominus\Theta H^2$ of the Hardy space $H^2$ generated by an inner function $\Theta$ in the upper half plane. Our main object is the class of admissible majorants for $K_\Theta$, denoted by $\Adm \Theta$ and consisting of all functions $\omega$ defined on $\mathbb{R}$ such that there exists an $f \ne 0$, $f \in K_\Theta$ satisfying $|f(x)|\leq\omega(x)$ almost everywhere on $\mathbb{R}$. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any $K_\Theta$ generated by a meromorphic inner function. In contrast with \cite{HM02I}, we consider the generating functions $\Theta$ such that the unit vector $\Theta(x)$ winds up fast as $x$ grows from $-\infty$ to $\infty$. In particular, we consider $\Theta=B$ where $B$ is a Blaschke product with ``horizontal'' zeros, {\it i.e.}, almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$. It is shown, among other things, that for any such $B$, any even $\omega$ decreasing on $(0,\infty)$ with a finite logarithmic integral is in $\Adm B$ (unlike the ``vertical'' case treated in \cite{HM02I}), thus generalizing (with a new proof) a classical result related to $\Adm\exp(i\sigma z)$, $\sigma>0$. Some oscillating $\omega$'s in $\Adm B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to $\Adm\exp(i\sigma z)$, $\sigma>0$, and to de~Branges' space $\mathcal{H}(E)$.

A model subspace $K_\Theta$ of the Hardy space $H^2 = H^2 (\mathbb{C}_+)$ for the upper half plane $\mathbb{C}_+$ is $H^2(\mathbb{C}_+) \ominus \Theta H^2(\mathbb{C}_+)$ where $\Theta$ is an inner function in $\mathbb{C}_+$. A function $\omega \colon \mathbb{R}\mapsto[0,\infty)$ is called {\it an admissible majorant\/} for $K_\Theta$ if there exists an $f \in K_\Theta$, $f \not\equiv 0$, $|f(x)|\leq \omega(x)$ almost everywhere on $\mathbb{R}$. For some (mainly meromorphic) $\Theta$'s some parts of $\Adm\Theta$ (the set of all admissible majorants for $K_\Theta$) are explicitly described. These descriptions depend on the rate of growth of $\arg \Theta$ along $\mathbb{R}$. This paper is about slowly growing arguments (slower than $x$). Our results exhibit the dependence of $\Adm B$ on the geometry of the zeros of the Blaschke product $B$. A complete description of $\Adm B$ is obtained for $B$'s with purely imaginary (``vertical'') zeros. We show that in this case a unique minimal admissible majorant exists.

We settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, \ie, to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra.

In a previous paper, we gave a correspondence between certain exact solutions to a \((2+1)\)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions. Dans un article pr\'{e}c\'{e}dent, nous avons d\'{e}montr\'{e} que les solutions d'un mod\`{e}le chiral int\'{e}grable en dimension \( (2+1) \) correspondent aux fibr\'{e}s vectoriels holomorphes sur une surface compacte. Ici, nous employons la g\'{e}om\'{e}trie alg\'{e}brique dans une construction explicite des solutions. Nous donnons une formule matricielle et illustrons avec trois exemples la signification des invariants alg\'{e}briques pour le comportement physique des solutions.

Let $M(X,Y)$ denote the space of all continous functions between $X$ and $Y$ and $M_f(X,Y)$ the path component corresponding to a given map $f: X\rightarrow Y.$ When $X$ and $Y$ are classical flag manifolds, we prove the components of $M(X,Y)$ corresponding to ``simple'' maps $f$ are classified up to rational homotopy type by the dimension of the kernel of $f$ in degree two cohomology. In fact, these components are themselves all products of flag manifolds and odd spheres.