Mixed automorphic vector bundles on Shimura varieties

Cohomology of Complex Torus Bundles Associated to Cocycles

“…Thus j ρ,τ : Γ × D → GL(n, C) is an automorphy factor, and automorphic forms involving the determinant of such an automorphy factor have been studied in a number of papers (see e.g. [1] and [6]).…”

Poisson transforms and mixed automorphic forms on semisimple Lie groups

“…On the other hand, holomorphic forms on the fibre product of elliptic surfaces correspond to mixed automorphic forms of higher weights (see, for example [7]). Mixed automorphic forms of several variables and their connection with holomorphic forms on families of Abelian varieties have also been studied recently (see [8,10,9,11]). In this section we describe holomorphic mixed automorphic forms on semisimple Lie groups.…”

5. Mixed Automorphic Forms on Semisimple Lie Groups

“…[10], [15]). Holomorphic mixed automorphic forms of several variables were also introduced in [17] and [18], and it was proved that a certain class of such automorphic forms can be interpreted as holomorphic forms on some families of abelian varieties over an arithmetic variety.…”

“…This assumption implies that D and D' are equivalent to bounded symmetric domains (e.g., see [8]). Let J: G x D GL(V) and J': G' x D' GL(V') be automorphy factors, and let r: D D' be aholomorphic map satisfying z(gz) p(g)z(z) for all g e G and z e D. Then we can define a mixed automorphic vector bundle AA (see [18]) on the Shimura variety X ['\D whose sections can be considered as holomorphic mappings f" D V (R) V' satisfying f (gz) (J(g, z) (R) J'(p(g), r(z)))f (z) for all g e G and z e D. Given such a bundle A4 associated to J, J', p and r, we define mappings tr" G --GL(V) and ix': G' GL(V') by r (k) ] (k, 0), r'(k') J'(k', 0'),…”