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Let G be a compact connected semisimple Lie group, let K be a closed subgroup of G, let Γ be a finite subgroup of G, and let π be a finite-dimensional representation of K. For π in the unitary dual Ĝ of G, denote by nΓ (π) its multiplicity in L2(Γ\G). We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the nΓ (π) for π in the set Ĝπ of irreducible π-spherical representations of G. More precisely, for Γ and Γ' finite subgroups of G, we prove that if nΓ(π) = nΓ'(π) for all but finitely many π ∈ Ĝπ, then Γ and Γ' are π-representation equivalent, that is, nΓ(π) = nΓ'(π) for all π ∈ Ĝπ. Moreover, when Ĝ_π can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset \widehat {Fπ of Ĝπ verifying some mild conditions, the values of the nΓ (π) for π ∈ Fπ determine the nΓ (π)'s for all π ∈ Ĝπ. In particular, for two finite subgroups Γ and Γ' of G, if nΓ (π) = nΓ'(π) for all π ∈ Fπ, then the equality holds for every π ∈ Ĝπ. We use algebraic methods involving generating functions and some facts from the representation theory of G. [ABSTRACT FROM AUTHOR]

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for $$ SL\left( {2,{\mathbb Z}} \right) $$, and let $$ {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} $$ be an orthonormal basis of Hecke-Maass cusp forms for $$ SL\left( {2,{\mathbb Z}} \right) $$. We prove an asymptotic formula for the twisted first moment of the Rankin-Selberg L-functions $$ L\left( {s,f \otimes {u_j}} \right) $$ at $$ s = \frac{1}{2} $$ as u runs over $$ {\mathcal U} $$. It follows that f is uniquely determined by the central values of the family of Rankin-Selberg L-functions $$ \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} $$. [ABSTRACT FROM AUTHOR]

Abstract: Let and be two irreducible, automorphic, cuspidal representations of . Using the logarithmic zero-free region of Rankin–Selberg L-function, Moreno established the analytic strong multiplicity one theorem if at least one of them is self-contragredient, i.e. π and will be equal if they have finitely many equivalent local components , for which the norm of places are bounded polynomially by the analytic conductor of these cuspidal representations. Without the assumption of the self-contragredient for , Brumley generalized this theorem by a different method, which can be seen as an invariant of Rankin–Selberg method. In this paper, influenced by Landau''s smooth method of Perron formula, we improved the degree of Brumley''s polynomial bound to be . [Copyright &y& Elsevier]