Triple product p-adic L-functions for balanced weights

We construct $p$-adic triple product $L$-functions that interpolate (square roots of) central critical $L$-values in the balanced region. Thus, our construction complements that of M. Harris and J. Tilouine. There are four central critical regions for the triple product $L$-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three $p$% -adic $L$-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region. An especially interesting feature of our construction is that we get three different $p$-adic triple product $$-functions with the same (balanced) region of interpolation. To the best of the authors' knowledge, this is the first case where an interpolation problem is solved on a single critical region by different $p$-adic $L$% -functions at the same time. This is possible due to the structure of the Euler-like factors at $p$ arising in the interpolation formulas, the vanishing of which are related to the dimensions of certain Nekovar period spaces. Our triple product $p$-adic $L$-functions arise as specializations of $p$-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of $p$-adic period integrals is showing that these branching laws vary in a $$-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.