ALGEBRAIC DIFFERENTIAL OPERATORS ON ARITHMETIC AUTOMORPHIC FORMS, MODULAR DISTRIBUTIONS, 𝙋-ADIC INTERPOLATION OF THEIR CRITICAL 𝙇 VALUES VIA BGG MODULES AND HECKE ALGEBRAS

The paper extends author’s method of modular distributions (2002, [75]) to arithmetic automorphic L functions on general classical groups. Main resultat gives a p-adic interpolation of their critical L values in the form of integrals of distributions constructed from a given eigen function of Hecke algebras by applying BGG modules, (see also preprints [78] and [79].In particular, algebraic differential operators are described acting on auto-morphic forms ' on unitary groups U(n; n) over an imaginary quadratic field K=ℚ(-DK)⊂ℂ.Applications are given to Shimura’s zeta functions L(s,f) [90] attached special L-values L(s,φ) attached to φ. and normalized in accordance with Deligne’s Gamma factors rule [21]. An explicit description of Shimura’s Γ-factors is used.