In the spirit of the Langlands proposal on Beyond Endoscopy, we discuss the explicit relation between the Langlands functorial transfers and AUTOMORPHICL -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 productL -functions in terms of the Langlands functoriality. This leads to the theory of the twisted auto morphic descents for cuspidal auto morphic REPRESENTATIONS of general classical groups.

This paper extends to the pro- p Iwahori subgroup of GL (2) over an unramied nite extension of Q p the presentation of the Iwasawa algebra obtained earlier by the author for the congruence subgroup of level one of SL (2; Zp). It then describes a natural base change map between the Iwasawa algebras or more correctly, as it turns out, between the global distribution algebras on the associated rigid-analytic spaces. In a forthcoming paper this will be applied top -adic representation theory.

In this work we deals with almost AUTOMORPHIC behavior of solutions of a class of semilinear evolution equations. To achieve our goal we use interpolation theory and fixed point theory. As application, we examine sufficient conditions for existence of almost AUTOMORPHIC solutions of equations of the heat conduction theory.

We rst give bounds for domains where the unitarizabile sub-quotients can show up in the parabolically induced REPRESENTATIONS of clas-sical groups over ap -adic eld of characteristic 0. Roughly, they can show up only if the central character of the inducing irreducible cuspidal repre-sentation is dominated by the square root of the modular character of the minimal parabolic subgroup. For unitarizable subquotients supported by a xed parabolic subgroup, or in a specic Bernstein component, a more precise bound is given.For the reductive groups of rank at least two, the trivial representation is always isolated in the unitary dual (D. Kazhdan). Still, we may ask if the level of isolation is higher in the case of the AUTOMORPHIC duals, as it is a case in the rank one. We show that the answer is negative to this question for symplecticp -adic groups.

Purpose: In this paper we discuss the notion of Weyl pseudo almost AUTOMORPHIC functions. These kind of functions are more general than almost AUTOMORPHIC functions and Stepanov almost AUTOMORPHIC functions. We establish the composition theorem for this class of functions.Methods: The Weyl norm has been used to prove the results. In order to prove the composition theorem, the function has been broken into different parts.Results: The concept of Weyl pseudo almost AUTOMORPHIC functions are defined. The composition theorem has been proved for this kind of functions.Conclusion: The concept is new and may be useful in the qualitative theory of differential equations.

In this study, pseudo almost AUTOMORPHIC(PAA) solutions of a Li, enard-type system with multiple delays are considered. By applying the main features of PAA, Banach , xed point theorem and some di, er-ential inequalities, su, cient conditions for the existence and uniqueness of such solutions are obtained. Since the PAA is more general than the almost periodicity(AP) and pseudo almost periodicity(PAP), this work is a new and complementary compared to previous studies. In addition, an example is given to show the correctness of the created conditions.

Let G be a tamely ramified reductive p -adic group. We study distinction of a class of irreducible admissible REPRESENTATIONS of G by the group of fixed points H of an involution of G. The REPRESENTATIONS corre-spond to G -conjugacy classes of pairs (T; f), where T is a tamely ramified maximal torus of G and f is a quasicharacter of T whose restriction to the maximal pro-p -subgroup satisfies a regularity condition.Under mild restrictions on the residual characteristic of F, we derive necessary conditions for H -distinction of a representation corresponding to (T; f), expressed in terms of properties of T and f relative to the involution.We prove that if an H -distinguished representation arises from a pair (T; f) such that T is stable under the involution and compact modulo (T Ç H) Z (here, Z is the centre of G), then the representation is H -relatively supercuspidal.