KP Trigonometric Solitons and an Adelic Flag Manifold

We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127-165].Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy

We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Vel\'azquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the first example of an integrable hierarchy which admits a full centerless Virasoro algebra of master symmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchies which possess only "half of" a Virasoro algebra of master symmetries, as explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863-911], Damianou [Lett. Math. Phys. 20 (1990), 101-112] and Magri and Zubelli [Comm. Math. Phys. 141 (1991), 329-351]

A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

We consider the 2-dimensional Toda lattice tau functions $\tau_n(t,s;\eta,\theta)$ deforming the probabilities $\tau_n(\eta,\theta)$ that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc $(\eta,\theta)$ of the unit circle. We show that these tau functions satisfy a centerless Virasoro algebra of constraints, with a boundary part in the sense of Adler, Shiota and van Moerbeke. As an application, we obtain a new derivation of a differential equation due to Tracy and Widom, satisfied by these probabilities, linking it to the Painleve VI equation.Comment: 15 page

Askey-Wilson Type Functions, With Bound States

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of $q$-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction

The Algebraic Complete-integrability of Geodesic-flow On So(n)

The author studies for which left invariant diagonal metrics lambda on /b SO/(/b N/), the Euler-Arnold equations /b X/ dot =[/b X/, lambda (/b X/)], /b X/=(/b x//sub ij /) isin /b so/(/b N/), lambda (/b X/)/sub ij/= lambda /sub ij//b x//sub ij /, lambda /sub ij/= lambda /sub ji/ can be linearized on an abelian variety, i.e. are solvable by quadratures. He shows that, merely by requiring that the solutions of the differential equations be single-valued functions of complex time /b t/ isin /b C/, suffices to prove that (under a non-degeneracy assumption on the metric lambda ) the only such metrics are those which satisfy Manakov's conditions lambda /sub ij/=(/b b//sub i/-/b b//sub j/) (/b a //sub i/-/b a//sub j/)/sup -1/. The case of degenerate metrics is also analyzed. For /b N/=4, this provides a new and simpler proof of a result of Adler and van Moerbeke (1982).Anglai

On a generalization of Jacobi's elegantissima

We establish a generalization of Jacobi's elegantissima, which solves the pendulum equation. This amazing formula appears in lectures by the famous cosmologist Georges Lema\^itre, during the academic years 1955-1956 and 1956-1957. Our approach uses the full power of Jacobi's elliptic functions, in particular imaginary time is crucial for obtaining the result

The spectral matrices of toda solitons and the fundamental solution of some discrete heat equations

The Stieltjes spectral matrix measure of the doubly infinite Jacobi matrix associated with a Toda $g$-soliton is computed, using Sato theory. The result is used to give an explicit expansion of the fundamental solution of some discrete heat equations, in a series of Jackson's $q$-Bessel functions. For Askey-Wilson type solitons, this expansion reduces to a finite sum