  Kac-Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as
symmetry groups of supergravities in dimensions less than 3, and their integer
forms $G(\mathbb{Z})$ are conjecturally U-duality groups. Mathematical
descriptions of $G(\mathbb{Z})$, due to Tits, are functorial and not amenable
to computation or applications. We construct Kac-Moody groups over $\mathbb{R}$
and $\mathbb{Z}$ using an analog of Chevalley's constructions in finite
dimensions and Garland's constructions in the affine case. We extend a
construction of Eisenstein series on finite dimensional semisimple algebraic
groups using representation theory, which appeared in the context of
superstring theory, to general Kac-Moody groups. This coincides with a
generalization of Garland's Eisenstein series on affine Kac-Moody groups to
general Kac-Moody groups and includes Eisenstein series on $E_{10}$ and
$E_{11}$. For finite dimensional groups, Eisenstein series encode the quantum
corrections in string theory and supergravity theories. Their Kac-Moody analogs
will likely also play an important part in string theory, though their roles
are not yet understood.
