Let G be a symmetrizable Kac–Moody group over a field of characteristic zero, let T be a split maximal torus of G. By using a completion of the algebra of strongly regular functions on G, and its restriction on T, we give a formal Chevalley restriction theorem. Specializing to the affine case, and to the field of complex numbers, we obtain a convergent Chevalley restriction theorem, by choosing the formal functions, which are convergent on the semi-groups of trace class elements Gtr⊂G resp. Ttr⊂T.

It is shown that the Fourier–Whittaker coefficients of Eisenstein series on the n-fold cover of GL(n) are L-functions, improving prior results of T. Suzuki.

In his celebrated paper ‘Generic Projections’, Mather obtained the key result that a generic projection to an affine subspace of a smooth submanifold in Euclidean space is jet-transverse to any ‘modular’ submanifold of (multi-) jet space. He also gave an explicit stratification by modular submanifolds, and used it to conclude that the projection, if in the nice dimensions, is generically (C∞-) stable. In this article, we extend the result to the semi-nice dimensions (where only C0-stability is obtained), using the stratification given in our book. We first recall the definitions of the nice and semi-nice dimensions, review the main known results which involve them, and proceed to the statement of our main results. Next we discuss the condition of modularity, and present a number of methods for establishing modularity of particular strata. Finally, we show that all the strata needed for the main result are covered by these methods.

The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,f×g) and L(s, sym2(g)× sym2(g)), where f,g are newforms, holomorphic or of Maass type, on the upper half plane, and sym2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym2(g).