We study automorphic Lie algebras and their applications to integrable systems. Automorphic are a natural generalisation of celebrated Kac-Moody the case when group automorphisms is not cyclic. They infinite dimensional almost graded. formulate concept graded isomorphism classify s l ( 2 , C ) based corresponding all finite reduction groups. show that hierarchies systems, Lax representations ma...

It is proved that certain types of modular cusp forms generate irreducible automorphic representations of the underlying algebraic group. Analogous Archimedean and non-Archimedean local statements are also given. Introduction One of the motivations for this paper was to show that full level cuspidal Siegel eigenforms generate irreducible, automorphic representations of the adelic symplectic sim...

‎based on an eigenvalue analysis‎, ‎a new proof for the sufficient‎ ‎descent property of the modified polak-ribière-polyak conjugate‎ ‎gradient method proposed by yu et al‎. ‎is presented‎.

The Langlands philosophy contemplates the relation between auto-morphic representations and Galois representations. A particularly interesting case is that of the non-selfdual automorphic representations of GL 3. Clozel conjectured that the L-functions of certain of these are equal to L-functions of Galois representations. Here we announce that we found an example of such an automorphic represe...

Ash's research was partially supported by NSF Grant DMS-8919696. Conjecturally, any “algebraic” automorphic representation on GL(n) should have an n-dimensional Galois representation attached. Many examples of algebraic automorphic representations come from the cohomology overC of congruence subgroups of GL(n;Z). On the other hand, the first author has conjectured that for any Hecke eigenclass ...

We propose a geometric interpretation of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program. We show that the geometric Langlands conjecture for an irreducible unramified local system E of rank n on a curve implies the existence of automorphic sheaves corresponding to the universal deformation of E. Then we calculate the ‘scalar product’ of two aut...

We review our approach to the second law of thermodynamics as a theorem asserting growth mean (Gibbs–von Neumann) entropy quantum spin systems undergoing automorphic (unitary) adiabatic transformations. Non-automorphic interactions with environment, although known produce on average strict reduction finite number degrees freedom, are proved conserve average. The results depend crucially two pro...

Let X be an even dimensional symmetric bilinear space defined over a totally real number field F with adeles A, and let σ = ⊗vσv be an irreducible tempered cuspidal automorphic representation of O(X,A). We give a sufficient condition for the nonvanishing of the theta lift Θn(σ) of σ to the symplectic group Sp(n,A) (2n by 2n matrices) for 2n ≥ dimX for a large class of X. As a corollary, we show...