Harmonic analysis on solvable Lie groups and discrete subgroups

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Head investigator Fujiwara studied monomial representations of exponential solvable Lie groups and obtained the following results.1. On intertwining operators between equivalent monomial irreducible representations (joint work with D.Arnal and J.Ludwig of Metz university in France).(1) Explicit description of the operator for monomial representations induced from polarizations of Vergne.(2) Construction of an operator verifying the composition formula with Maslov index.(3) Local expression at the unit element of the operator in general case.2. As an application of the Penney's Plancherel formula, we gave another proof of the commutativity of the algebra of invariant differential operators associated with monomial representations of finite multiplicities. It is more direct than the original proof due to Corwin and Greenleaf.3. Under some additional condition on the support of distributions, we proved the Frobenius reciprocity in the case of nilpotent symmetric spaces with character, namely that the dimension of the space of bi-semiinvariant distributions is equal to the multiplicity in the irreducible decomposition.Investigator Kanemitsu studied the Riemann hypothesis and got the following.1. We can get a condition equivalent to the Riemann hypothesis by the evaluation of Farey series on a short interval.2. Expression in a closed form of series involving Hurwitz-Lerch zeta function.3. Calculation of series involving Hurwitz zeta function.