Our principal interest in this paper is to study higher order degenerate Hermite-Bernoulli polynomials arising from multivariate p-adic invariant integrals on Zp. We give interesting identities and properties of these polynomials that are derived using the generating functions and p-adic integral equations. Several familiar and new results are shown to follow as special cases. Some symmetry identities are also established.

We study R -groups for p -adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-SteinR -group and the Langlands-Arthur R -group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-SteinR -group within L -packets and between inner forms. This work is applied to transferring known results in the second-named author's earlier work for quasi-split special unitary groups to their non-quasi-split inner forms.

The fi eld Qp of p-adic numbers is defi ned as the completion of the fi eld of the rational numbers Q with respect to the p-adic norm j: jp. In this paper, we study the continuous and discrete padic shearlet systems on L 2 (Q 2 p). We also suggest discrete padic shearlet frames. Several examples are provided.

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.

Let G be a reductive p -adic group. We consider the general question of whether the reducibility of an induced representation can be detected in a \co-rank one" situation. For smooth complex representa-tions induced from supercuspidal representations, we show that a suffi-cient condition is the existence of a subquotient that does not appear as a sub representation. An important example is the Langlands' quotient.In addition, we study the same general question for continuous principal series onp -adic Banach spaces. Although we do not give an answer in this case, we describe a related ltration on the corresponding Iwasawa modules.

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.

In The endoscopic classication of representations, J. Arthur has proved the Langlands' classication for discrete series of p-adic clas-sical groups. This uses endoscopy and twisted endoscopy. In this very short note, we remark that the normalization a la Langlands-Shahidi of the intertwining operators, allows to avoid endoscopy. This is based on the intertwining relation which is a very important point of this book.