Let A be the ring of adeles a number field F. Given self-dual irreducible, automorphic, cuspidal representation ? GLn(A), with trivial central character, we construct its full inverse image under weak Langlands functorial lift from appropriate split classical group G. We do this by new automorphic descent method, namely double descent. This method is derived recent generalized doubling integral...

‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...

Motivated by the study of periods automorphic forms and relative trace formulae, we develop theory descent necessary to orbital integrals arising in fundamental lemma for a general class symmetric spaces over p-adic field F. More precisely, prove that connected space F enjoys notion topological Jordan decomposition, which may be independent interest, establish version Kazhdan played crucial rol...

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 2, and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-associative simple right automorphic loops. W...

In this paper we deal with monogenic and k-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between k-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vani...

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman’s work on uniquely 2-divisible Moufang loops) and the associated L...

Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic...

In recent years, the theory of almost automorphic functions has been developed extensively see, e.g., Bugajewski and N’guérékata 1 , Cuevas and Lizama 2 , and N’guérékata 3 and the references therein . However, literature concerning pseudo-almost automorphic functions is very new cf. 4 . It is well known that the study of composition of two functions with special properties is important and bas...

These notes accompany a series of three lectures on automorphic loops to be delivered by the author at Workshops Loops ’15 (Ohrid, Macedonia, 2015). Automorphic loops are loops in which all inner mappings are automorphisms. The first paper on automorphic loops appeared in 1956 and there has been a surge of interest in the topic since 2010. The purpose of these notes is to introduce the methods ...

We study discrete almost automorphic functions sequences defined on the set of integers with values in a Banach space X. Given a bounded linear operator T defined on X and a discrete almost automorphic function f n , we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation Δu n Tu n f n . We also prove the existence of a discrete almost autom...