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nt.quantity principle – Automorphy issue and the determinant of the Jacobian matrix Answer

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nt.quantity principle – Automorphy issue and the determinant of the Jacobian matrix

There are two cocycles defining the automorphy elements.

Let $D$ breathe the bounded symmetric province (because of Harish-Chandra). Then there’s a canonical automorphy issue $J:Gtimes Dto K_{mathbb{C}}$ the place $G$ is a actual Lie group with a maximal compact subgroup $Ok$, $D=G/Ok$ and $K_{mathbb{C}}$ is the complexification of $Ok$.

Meanwhile, because of Baily and Borel, there may be too an automorphy issue $J_0:Gtimes Dto mathbb{C}^{*}$ coming from the determinant of the Jacobian matrix, i.e. the Jacobian of the motion of $g$ at $zin D$.(Baily, Walter L., and Armand Borel. “Compactification of arithmetic quotients of bounded symmetric domains.” Annals of arithmetic (1966): 442-528.)

Is there any relation between these two? The first one appears extra common to me since we are able to add the illustration of $K_{mathbb{C}}$ whereas the second is at all times a nonzero complicated quantity. And with the primary one, we are able to switch the automorphic kinds (vector-valued) on Lie teams to those outlined on the province.

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