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

+ 5 like - 0 dislike

$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$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
asked Oct 31, 2021 in Mathematics by Arun Debray (75 points) [ no revision ]
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
(you might want to add the tag arithmetic-groups to alert the relevant people to your question)

This post imported from StackExchange MathOverflow at 2022-01-03 16:02 (UTC), posted by SE-user David Ben-Zvi
I took the liberty to edit the tags in order to alert the relevant people to your question.

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

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