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Kac--Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur assymmetry groups of supergravity theories dimensionally reduced to dimensionsless than 3, and their integral forms $G(\mathbb{Z})$ conjecturally encodequantized symmetries. In this review paper, we briefly introduce theconjectural symmetries of Kac--Moody groups in supergravity as well as theknown evidence for these conjectures. We describe constructions of Kac--Moodygroups over $\R$ and $\Z$ using certain choices of fundamental modules that areconsidered to have physical relevance. Eisenstein series on certain finitedimensional algebraic groups are known to encode quantum corrections in the lowenergy limit of superstring theories. We describe briefly how the constructionof Eisenstein series extends to Kac--Moody groups. The constant terms ofEisenstein series on $E_9$, $E_{10}$ and $E_{11}$ are predicted to encodeperturbative string theory corrections.

Kac--Moody groups and automorphic forms in low dimensional supergravity theories