The Jantzen conjecture for singular characters

Geometry of the Jantzen region in Lusztig's conjecture

The Lusztig Conjecture expresses the character of a finite-dimensional irreducible representation of a reductive algebraic group G in prime characteristic as a linear combination of characters of Weyl modules for G. The representations described by the conjecture are in one-to-one correspondence with the (finitely many) alcoves in the intersection of the dominant cone and the so-called Jantzen ...

Rogawski’s Conjecture on the Jantzen Filtration for the Degenerate Affine Hecke Algebra of Type A

The functors constructed by Arakawa and the author relate the representation theory of gln and that of the degenerate affine Hecke algebra Hl of GLl. They transform the Verma modules over gln to the standard modules over Hl. In this paper we prove that they transform the simple modules to the simple modules (in more general situations than in the previous paper). We also prove that they transfo...

Adelic Singular Series and the Goldbach Conjecture

The purpose of this paper is to show how adelic ideas might be used to make progress on the Goldbach Conjecture. In particular, we present a new Schwartz function which is able to keep track of the number of prime factors of an integer. We then use this, along with the Ono/Igusa adelic methods for Diophantine equations, to present an infinite sum whose evaluation would prove or disprove the ver...

The Auslander-Reiten Conjecture for Group Rings

This paper studies the vanishing of $Ext$ modules over group rings. Let $R$ be a commutative noetherian ring and $ga$ a group. We provide a criterion under which the vanishing of self extensions of a finitely generated $Rga$-module $M$ forces it to be projective. Using this result, it is shown that $Rga$ satisfies the Auslander-Reiten conjecture, whenever $R$ has finite global dimension and $ga...

The Cyclotomic Jantzen{schaper Theorem