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Cohomological test vectors

Abstract: Various types of modular symbols provide a powerful tool to study the arithmetic of special values of L-functions. The Archimedean behaviors of the modular symbols are captured by certain restriction maps of relative Lie algebra cohomology spaces. We call these restriction maps modular symbols at infinity. The modular symbols are non-zero and  of arithmetic interest only when the associated modular symbols at infinity are non-zero. Moreover, the modular symbols at infinity are non-zero if and only if certain invariant linear functionals on the cohomological representations does not vanish on the minimal K-types (in the sense of Vogan).  We will give some examples of invariant linear functionals on cohomological representations which does not vanish on the minimal K-types, including the Rankin-Selberg case GL(n)xGL(n-1).