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Let S0(G,X), S0(G′,X′) be
connected Shimura varieties associated to semisimple algebraic groups G, G′
defined over ℚ and Hermitian symmetric domains X, X′. Let ρ : G → G′ be a
homomorphism of algebraic groups over ℚ that induces a holomorphic map
ω : X → X′ mapping special points of X to special points of X′. Given equivariant
vector bundles 𝒥 , 𝒥′ on the compact duals X, X′ of the symmetric domains X, X′,
we can construct a mixed automorphic vector bundle ℳ(𝒥,𝒥′,ρ), on S0(G,X)
whose sections can be interpreted as mixed automorphic forms. We prove
that the space of sections of a certain mixed automorphic vector bundles
is isomorphic to the space of holomorpic forms of the highest degree on
the fiber product of a finite number of Kuga fiber varieties. We also prove
that for each automorphism τ of ℂ the conjugate τℳ(𝒥,𝒥′,ρ) of a mixed
automorphic vector bundle ℳ(𝒥,𝒥′,ρ) on a connected Shimura variety
S0(G,X) can be canonically realized as a mixed automorphic vector bundle
ℳ(𝒥1,𝒥1′,ρ1) on another connected Shimura variety S0(G1,X1) associated
to a semisimple algebraic group G1 and a Hermitian symmetric domain
X1.
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