Reduced C*-Algebra of the p-Adic Group GL(n) II

THE CLASSIFICATION OF IRREDUCIBLE ADMISSIBLE MOD p REPRESENTATIONS OF A p-ADIC GLn

Let F be a finite extension of Qp. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F -split p-adic reductive group over Fp to be supersingular. We then give the classification of irreducible admissible smooth GLn(F )-representations over Fp in terms of supersingular representations. As a consequence we deduce that supersingular...

ON THE SECOND COHOMOLOGY OF THE NORM ONE GROUP OF A p-ADIC DIVISION ALGEBRA

Let F be a p-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H(SL1(D),R/Z) is a cyclic p-group whose order is bounded from below by the number of p-power roots of unity in F , unless D is a quaternion algebra over Q2. In this paper we give an ex...

SIMPLE p - ADIC GROUPS , II

0.1. For any finite group Γ, a “nonabelian Fourier transform matrix” was introduced in [L1]. This is a square matrix whose rows and columns are indexed by pairs formed by an element of Γ and an irreducible representation of the centralizer of that element (both defined up to conjugation). As shown in [L2], this matrix, which is unitary with square 1, enters (for suitable Γ) in the character for...

Group Rings over the p-Adic Integers

Isomorphisms of p-adic group rings

A long-standing problem, first posed by Graham Higman [15] and later by Brauer [4] is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG = ZH implies G 2: H? Many authors have worked on this question, but progress has been difficult [30]. Perhaps the best positive result was that of Whitcomb in 1968 [37], who showed that the implication G = H hol...