where the unordered pair {αv , βv} defines the diagonal conjugacy class in GL2(C) attached to πv. Even at a ramified (resp. archimedean) place v, one has by the local Langlands correspondence a 2-dimensional representation σv of the extended Weil groupWFv ×SL(2,C) (resp. of the Weil group WFv), and the v-factor of the symmetric m-th power L-function is associated to sym(σv). A basic conjecture ...

For compact Hausdorff admissible right topological (CHART) group G, we prove w(G)=πχ(G). This equality is well known for groups. implies the criteria metrizability of CHART groups: if G first-countable (2013, Moors, Namioka) or Fréchet Glasner, Megrelishvili), has countable π-character (2022, Reznichenko) then metrizable. Under continuum hypothesis (CH) assumption, a sequentially Namioka's theo...

This paper is intended to prepare new antimicrobial complexes with proven efficiency. The Schiff base, through the condensation process of salicylaldehyde and p-toludine, its Cu Zn were successfully synthesized. base characterized using molar conductivity, Ultraviolet-visible (UV-Vis), atomic absorption spectroscopy (AAS), Fourier transform infrared (FTIR) techniques. Accordingly, these charact...

We have seen that irreducible representations of a compact Lie group G can be constructed starting from a highest weight space and applying negative roots to a highest weight vector. One crucial thing that this construction does not easily tell us is what the character of this irreducible representation will be. The character would tell us not just which weights occur in the representation, but...

Let π be a free group of rank 2. Its outer automorphism group Out(π) acts on the space of equivalence classes of representations ρ ∈ Hom(π,SL(2, C)). Let SL − (2, R) denote the subset of GL(2, R) consisting of matrices of determinant −1 and let ISL(2, R) denote the subgroup SL(2, R) q iSL − (2, R) ⊂ SL(2, C). The representation space Hom(π, ISL(2, R)) has four connected components, three of whi...