Cusp Forms on GL(2n) with GL(n) × GL (n) Periods,and Simple Algebras

The notion of a period of a cusp form on GL(2,D(??)), with respect to the diagonal subgroup D(??)^X × D(??)^X, is defined. Here D is a simple algebra over a global field F with a ring ?? of adeles. For D^x = GL(1), the period is the value at 1/2 of the L-function of the cusp form on GL(2, ??). A cuspidal representation is called cyclic if it contains a cusp form with a non zero period. It is investigated whether the notion of cyclicity is preserved under the Deligne - Kazhdan correspondence, relating cuspidal representations on the group and its split form, where D is a matrix algebra. A local analogue is studied too, using the global technique. The method is based on a new bi-period summation formula. Local multiplicity one statements for spherical distributions, and non - vanishing properties of bi - characters, known only in a few cases, play a key role.     

Bi-invariant Integrals on GL(n) with Applications

In this paper measures and functions on GL(n) are called bi-invariant if they are invariant under left and right multiplication of their arguments. If v is any bi-invariant Borel measure on GL(n), then there exists a unique Borel measure v^* on D _{+ ≥}(n), the set of all diagonal matrices of rank n with positive non-increasing diagonal entries, such that holds for each v-integrable bi-invariant function f:GL(n) → IR. An explicit formula for v^* will be derived in case v equals the Lebesgue measure on GL(n) and the above integral formula will be applied to concrete integration problems. In particular, if v is a probability measure, then v^* can be interpreted as the distribution of the singular value vector. This fact will be used to derive a stochastic version of a theorem from perturbation theory concerning the numerical computation of the polar decomposition.     

Farrell cohomology of GL(n,Z)

The Tate-Farrell cohomology of GL(n,Z) with coefficients inZ/p is computed forp an odd prime andp−1 ≦n ≦ 2p−3. Its size depends on the Galois structure of the class group of the cyclotomic fieldQ(p√1) and is shown to be quite large in general. Research partially supported by NSF Grant No. DMS-8701758.     

The unitary dual of GL(n) over an archimedean field

Inventiones mathematicae -     

Special bases forS N and GL(n)

The special basis in spaces of finite dimensional representation ofS _N and GL(n) is constructed and its properties are studied. Both authors are partially supported by the National Science Foundation.     

Tame Supercuspidal Representations of GL(n) Distinguished by a Unitary Group

This paper analyzes the space Hom_H(, 1), where is an irreducible, tame supercuspidal representation of GL(n) over a p-adic field and H is a unitary group in n variables contained in GL(n). It is shown that this space of linear forms has dimension at most one. The representations which admit nonzero H-invariant linear forms are characterized in several ways, for example, as the irreducible, tame supercuspidal representations which are quadratic base change lifts.Research supported in part by NSA grant #MDA904-99-1-0065.Research supported in part by NSERC     

The reduced C1-algebra of the p-adic group GL(n)

The reduced C¹-algebra of the p-adic group GL(n) is Morita equivalent to an abelian C¹-algebra. The structure of this abelian C¹-algebra is described in terms of unramified unitary characters of Levi subgroups. The K-groups K₀ and K₁ are both free abelian of infinite rank. Generators are essentially parametrized by two items of Langlands data.     

Discrete Series Characters for GL(n, q)

The discrete series characters of the finite general linear group GL(n, q) are expressed as uniquely defined integral linear combinations of characters induced from linear characters on certain subgroups Hd, n of GL(n, q). The coefficients in these linear combinations are determined (for all n, q) by a family of polynomials r(T) Z[T] indexed by the set of all partitions .     

The unitary representation theory of GL(n) of an infinite discrete field

IfK is an infinite field and ifG=GL(n, K) with the discrete topology, then all principal-series representations ofG are irreducible, and any two such with the same central character ψ are weakly equivalent to one another and to the ψ-regular representation. In addition, every irreducible unitary representation ofG which is not one-dimensional weakly contains a representation of the principal series. We deduce that every maximal ideal ofC*(G) is either of codimension 1 or else a kernel of a principal-series representation. In particular, except in the exceptional case whereK is an infinite algebraic extension of a finite field, the reducedC*-algebra of PGL(n, K) is simple, as was also shown by de la Harpe in many cases. Partially supported by NSF Grant DMS-85-06130. It is a pleasure also to acknowledge the hospitality of the Institute for Advanced Studies, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, from January to August, 1988. Partially supported by NSF Grants DMS-84-00900 and DMS-87-00551. Much of this work was done while visiting at, and partially supported by, the Department of Mathematics and Computer Science, Bar-Ilan University, 52100 Ramat Gan, Israel.     

A formalization of Sambins's normalization for GL

Sambin [6] proved the normalization theorem (Hauptsatz) for GL, the modal logic of provability, in a sequent calculus version called by him GLS. His proof does not take into account the concept of reduction, commonly used in normalization proofs. Bellini [1], on the other hand, gave a normalization proof for GL using reductions. Indeed, Sambin's proof is a decision procedure which builds cut-free proofs. In this work we formalize this procedure as a recursive function and prove its recursiveness in an arithmetically formalizable way, concluding that the normalization of GL can be formalized in PA. MSC: 03F05, 03B35, 03B45.     

The local Rankin-Selberg convolution for GL(n): divisibility of the conductor

Let F be a non-Archimedean local field and an integer. Let be irreducible supercuspidal representations of GL with . One knows that there exists an irreducible supercuspidal representation of GL, with , such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) are distinct. In this paper, we show that, when is an unramified twist of , one may here takem dividingn and , for a prime divisor ofn depending on and the order of : in particular, , where is the least prime divisor of . This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations. Received: 11 November 2000 / Accepted: 15 January 2001 / Published online: 23 July 2001