Let k be a finite field, kn|k the degree n extension of k, and G = GLn(k) the general linear group with entries in k. This paper studies the “generalized Steinberg” (GS) representations of G and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of G is in natu...

We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadratic twists of the Steinberg representation π of GSp4(F ), where F is a non-archimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in π to be a test vector. We apply the formula...

Let π be the automorphic representation of GSp4(A) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and τ be an arbitrary cuspidal, automorphic representation of GL2(A). Using Furusawa’s integral representation for GSp4 ×GL2 combined with a pullback formula involving the unitary group GU(3, 3), we prove that the L-functions L(s, π× τ ) are “nice”. The conve...

It is the product of my playing with beautiful geometric objects called rational cuspidal curves over the past two years. I would like to thank everyone who has contributed to this thesis. I owe so much to everyone who has ever taught me mathematics. Thank you for inspiring me and for providing me with the skills necessary to complete this thesis. To my friends and fellow students at Abel, than...

Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of “cuspidal” representations realize crystals for sub-Kac-Moody algebras. Here we put that observation on a firmer categorical footing by exhibiting a corresponding functor betwee...

We prove that any reductive group G over a non-Archimedean local field has a cuspidal complex representation.

We obtain special values results for the triple product Lfunction attached to a Hilbert modular cuspidal eigenform over a totally real quadratic number field and an elliptic modular cuspidal eigenform, both of level one and even weight. Replacing the elliptic modular cusp form by a specified Eisenstein series, we renormalize the integral defining the triple product L-function in order to obtain...

let ( r,m ) be a noetherian local ring, a an ideal of r and m a finitely generated r- module. we investigate some properties of formal local cohomology modules with respect to a serre subcategory. we provide a common language to indicate some properties of formal local cohomology modules. let ( r,m ) be a noetherian local ring, a an ideal of r and m a finitely generated r- module. we investigat...