The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of ${\rm gl}(\infty)$ into ${\rm sp}(\infty)$ and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a set of bilinear identities for the wave functions, which yield the existence of a sequence of "$\tau$-functions". The latter satisfy their own set of bilinear identities, which moreover characterize them.

In the semi-infinite case, the $\tau$-functions are Pfaffians, in the same way that for the Toda lattice the $\tau$-functions are Hänkel determinants; interesting examples occur in the theory of random matrices, where one considers symmetric and symplectic matrix integrals for the Pfaff lattice and Hermitian matrix integrals for the Toda lattice.

There is a striking parallel between the Pfaff lattice and the Toda lattice, and even more striking, there is a map from one to the other, mapping skew-orthogonal to orthogonal polynomials. In particular, we exhibit two maps, dual to each other, mapping Hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals.

Associated with each body $K$ in Euclidean $n$-space $\mathbb {R}\sp n$ is an ellipsoid $\Gamma\sb 2K$ called the Legendre ellipsoid of $K$. It can be defined as the unique ellipsoid centered at the body's center of mass such that the ellipsoid's moment of inertia about any axis passing through the center of mass is the same as that of the body.

In an earlier paper the authors showed that corresponding to each convex body $K\subset\mathbb {R}\sp n$ is a new ellipsoid $\Gamma\sb {-2}K$ that is in some sense dual to the Legendre ellipsoid. The Legendre ellipsoid is an object of the dual Brunn-Minkowski theory, while the new ellipsoid $\Gamma\sb {-2}K$ is the corresponding object of the Brunn-Minkowski theory.

The present paper has two aims. The first is to show that the domain of $\Gamma\sb {-2}$ can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If $K$ is a star-shaped set, then

$\Gamma\sb {-2}K\subset\Gamma\sb 2K$

with equality if and only if $K$ is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory–the Cramer-Rao inequality.

The first nontrivial zeros of the Riemann $zeta$-function are $\approx1/2\pm 14:13472i$. We investigate the question of whether or not any other $L$-function has a higher lowest zero. To do so, we try to quantify the notion that the $L$-function of a "small" automorphic representation (i.e., one with small level and archimedean type) does not have small zeros, and vice-versa. We prove that many types of automorphic $L$-functions have a lower first zero than $\zetas$'s (see Theorems 1.1 and 1.2). This is done using Weil's explicit formula with carefully chosen test functions. When this method does not immediately show that $L$-functions of a certain type have low zeros, we then attempt to turn the tables and show that no $L$-functions of that type exist. Thus, the argument is a combination of proving that low zeros exist and that certain cusp forms do not. Consequently, we are able to prove vanishing theorems and improve upon existing bounds on the Laplace spectrum on $L\sp 2({\rm SL}\sb n(\mathbb {Z})\backslash{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb n(\mathbb {R}))$. These in turn can be used to show that ${\rm SL}\sb 68(\mathbb {Z})\backslash{\rm SL}\sb 68(\mathbb {R})/{\rm SO}\sb 68(\mathbb {R})$ has a discrete, nonconstant, noncuspidal eigenvalue outside the range of the continuous spectrum on $L\sp 2({\rm SL}\sb 68(\mathbb {R})/{\rm SO}\sb 68(\mathbb {R}))$, but that this never happens for ${\rm SL}\sb n(\mathbb {Z})\backslash/{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb n(\mathbb {R})$ in lower rank. Another application is to cuspidal cohomology: we show there are no cuspidal harmonic forms on ${\rm SL}\sb n(\mathbb {Z})\backslash{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb n(\mathbb {R})$ for $n<27$.

We study the properties of level-zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by T. Akasaka and the author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremal weight. The universal extremal weight module with level-zero fundamental weight as an extremal weight is irreducible, and it is isomorphic to the affinization of an irreducible finite-dimensional module.

The purpose of this paper is to prove that the symmetric fourth power of a cusp form on ${\rm GL}(2)$, whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the Ramanujan-Petersson and Sato-Tate conjectures. In particular, we establish the bound $q\sp {1/9}\sb v$ for unramified Hecke eigenvalues of cusp forms on ${\rm GL}(2)$. Over an arbitrary number field, this is the best bound available at present.