# What is known about the cohomology of the U-duality group?

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$$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$$Let $$\Es$$ denote the split form of $$E_7$$, which is a real Lie group. It can be characterized as the subgroup of $$\mathrm{Sp}_{56}(\mathbb R)$$ preserving a certain quartic form (see, e.g., here).

Inside this is a discrete subgroup called $$\Es(\Z)$$, which is the intersection of $$\Es$$ with $$\mathrm{Sp}_{56}(\Z)$$. This group appears in theoretical physics, where it is called the U-duality group and is the symmetry group of a supergravity theory.

What is known about the group cohomology of $$\Es(\Z)$$? I am interested in knowing the ring structure of $$H^*(\Es(\Z); k)$$ where $$k = \mathbb Q$$ or $$\mathbb F_p$$, though I only need it up to about degree 6 or 7. For $$\mathbb F_p$$ coefficients, if the Steenrod action is known that would also be nice to know.

I don't know what's known about the cohomology of infinite discrete groups; as far as I know, this could be a straightforward calculation given $$H^*(B\Es;\Z)$$ (which is known), or it could be totally out of reach right now. I would also welcome an answer with that information, and/or where to read more.

This post imported from StackExchange MathOverflow at 2022-01-03 16:02 (UTC), posted by SE-user Arun Debray
retagged Jan 3, 2022
As to your last paragraph, the Cohomology of Arithmetic Groups (such as yours) is a huge subject, with a very large intersection with the Langlands program, and certainly not a consequence of cohomology of their real forms. (see e.g. under that heading the textbook by Harder or Venkatesh's Takagi lectures).

This post imported from StackExchange MathOverflow at 2022-01-03 16:02 (UTC), posted by SE-user David Ben-Zvi
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