# KOECHER-MAASS SERIES OF A CERTAIN HALF-INTEGRAL WEIGHT MODULAR FORM RELATED TO THE DUKE-IMAMOGLU-IKEDA LIFT

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KOECHER-MAASS SERIES OF A CERTAIN HALF-INTEGRAL WEIGHT MODULAR FORM RELATED TO THE DUKE-IMAMOG¯LU-IKEDA LIFT HIDENORI KATSURADA AND HISA-AKI KAWAMURA Abstract. Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k−n/2+1/2 for Γ0(4), let f be the corresponding primitive form of weight 2k − n for SL2(Z) under the Shimura correspondence, and In(h) the Duke-Imamog¯lu-Ikeda lift of h to the space of cusp forms of weight k for Spn(Z). Moreover, let φIn(h),1 be the first Fourier-Jacobi coefficient of In(h) and σn−1(φIn(h),1) be the cusp form in the generalized Kohnen plus space of weight k−1/2 corresponding to φIn(h),1 under the Ibukiyama isomorphism. We then give an explicit formula for the Koecher-Maass series L(s, σn−1(φIn(h),1)) of σn−1(φIn(h),1) expressed in terms of the usual L-functions of h and f . 1. Introduction Let l be an integer or a half integer, and let F be a modular form of weight l for the congruence subgroup Γ (m) 0 (N) of the symplectic group Spm(Z). Then the Koecher-Maass series L(s, F ) of F is defined as L(s, F ) = ∑ A cF (A) e(A)(detA)s , whereA runs over a complete set of representatives for the SLm(Z)-equivalence classes of positive definite half-integral matrices of degree m, cF (A) is the A-th Fourier coefficient of F, and e(A) denotes the order of the special orthogonal group of A. We note that L(s, F ) can also be obtained by the Mellin transform of F, and thus, its analytic properties are relatively known. (As for this, see Maass [19] and Arakawa [1, 2, 3].) Now we are interested in an explicit form of the Koecher-Maass series for a specific choice of F . In particular, whenever F is a certain lift of an elliptic modular form h of either integral or half-integral weight, we may hope to express L(s, F ) in terms of certain Dirichlet series related to h. Indeed, this expectation is verified in the case where F is a lift of h such that the weight l is an integer (cf. [8, 9, 10]). In

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