Wiley

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (J Σ, ρΣ) of an open compact subgroup J Σ and its irreducible representation ρΣ which is constructed from given data Σ = (Γ, P′0, ϱ). Here, Γ is a semisimple element in the Lie algebra of G, P′0 is a parahoric subgroup in the centralizer of Γ in G, and ϱ is a cuspidal representation on the finite reductive quotient of P′0. In this paper, we explicitly describe those Hecke algebras when P′0 is a minimal parahoric subgroup, ϱ is trivial and ρΣ is a character.

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A⊗ k k[x](x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x](x).

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.

We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image Rπ* (π is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf $$\mathcal{L}_\Phi$$ , called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if Φ is rational). Using $$\mathcal{L}_\Phi$$ we define the intersection cohomology space IH(Φ). It is conjectured that a strictly convex piecewise linear function on Φ acts as a Lefschetz operator on IH(Φ). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.

The purpose of this paper is to show how generalizations of generic vanishing theorems to a ℚ -divisor setting can be used to study the geometric properties of pluritheta divisors on a principally polarized Abelian variety (PPAV for short).

Given an irreducible, modular, mod p representation p, we analyse the local components at p of newforms f which give rise to it.