Modular Forms and Root Systems

Question

In the study of semisimple Lie groups, lattices appear all over the place. In the theory of elliptic functions and modular forms, (equivalence classes of) lattices correspond to elliptic curves and to points on the modular space. I believe furthermore that it's the case that lattices corresponding to elliptic curves with extra automorphisms (equivalently, elements of the upper half plane with nontrivial stabilizer in the modular group in the one-dimensional case) tend to be the ones corresponding to root systems, e.g. $\mathfrak{sl}_3$. I don't know much about higher dimensional modular forms, but I imagine that this generalizes to higher dimensional semisimple Lie algebras. Is there a deeper connection here? Do holomorphic functions invariant under root lattices always have interesting properties? Are both part of a general theory? If this is well-known, where might I find a reference?

EDIT: Also, the Weyl chambers remind me of the fundamental domains, which appear so often in elliptic functions/modular forms.

Answer 1

The lattices corresponding to elliptic curves are not the same as the lattices appearing in Lie theory. In the theory of modular forms, it is the weight $k$ (lying in the lattice $\mathbb Z$) that is a manifestation of Lie theory. (The lattice $\mathbb Z$ is the weight lattice of $SO(2)$.)

Added: Davidac897 asks "Why are you so sure"? Of course, there could be a connection that I am missing; I am just speaking from my own experience with these objects. But in the passage from classical modular forms (say as described in Serre's Course in arithmetic) to the representation-theoretic point of view, a lattice $\Lambda$ in $\mathbb C$ becomes a lattice $\mathbb \Lambda$ in $\mathbb R^2$, which becomes a point of $\mathrm{GL}_2(\mathbb Z)\backslash \mathrm{GL}_2(\mathbb R)$. Thus the lattices corresponding to elliptic curves become points of a quotient $\Gamma\backslash G$ for some real Lie group $G$ and some cofinite volume discrete subgroup $\Gamma$. For more general real Lie groups, one can think of such quotients as being a kind of generalized "moduli of lattices of some rank with some structure". There may be points corresponding to lattices with special symmetry (e.g. the square or triangular lattice in the elliptic curve case), and, yes, you are correct that root systems are lattices with special symmetry.

But this is not the usual way that root systems intervene in the theory of automorphic forms, which is what my answer above was intended to point out.

Answer 2

This may be rephrasing of some parts of the previous answers or else may also be irrelevant for the question for all that I know, since I do not know much of the topic: I only write this from the vague feeling that it must be related.

Anyway, here is very briefly the essence of what I've been able to understand from Looijenga's Root systems and elliptic curves (Inventiones, 1976):

For any root system $R$ with lattice $Q$ and any elliptic curve $E$ Looijenga constructs a line bundle $\mathscr L$ over the abelian variety $A:=Q^\lor\otimes E$ with a lift of the action of the Weyl group $W$ of $R$ on $A$ to $\mathscr L$, such that $\mathscr L^{-1}$ is ample, $W$-invariant sections of $\mathscr L^{-k}$, $k\geqslant0$, form algebra of polynomials in $\mathrm{rank}(R)+1$ variables, while the module over this algebra formed by anti-invariant sections is free on one generator and the divisor on $A$ corresponding to this generator is roughly speaking that part of $A$ where $W$ does not act freely.

Looijenga describes all these sections in terms of theta-functions, so this gives connection between root systems and modular forms. In particular, Looijenga gives an explanation of Macdonald-Weyl formula in these terms.

The paper has had many uses since, in particular for further understanding of bundles over elliptic curves with semisimple Lie (more generally reductive algebraic) structure groups, for loop group representation theory, for elliptic cohomology, etc.

Answer 3

As others have pointed out, the word "lattice" in this context needs to be used with care. Leaving aside the use of this word to describe certain partially ordered sets, lattices in Euclidean space (such as $\mathbb{Z}$ in $\mathbb{R}$ or root lattices associated to root systems of semisimple Lie algebras) are finitely generated commutative groups. This notion of lattice gets generalized to the setting of real Lie groups in a natural way, e.g., SL$(n,\mathbb{Z})$. But here the quotient of the group by the lattice may or may not be compact, and the lattice itself may or may not be arithmetically defined.

On the other hand, root systems or their affine generalizations do come up in the study of certain modular forms, for example in the 1972 paper by I.G. Macdonald and resulting papers by people such as Victor Kac and Bram van Asch. One of van Asch's papers is titled Modular forms and root systems and is available here. I don't have an expert viewpoint to offer on any of this, but those with access to MathSciNet may find it entertaining to search simultaneously for the terms "modular form" and "root system" under Anywhere. This returns 28 items, including those I've mentioned and a fundamental paper by Richard Borcherds. But this literature goes in many directions, making it essential to formulate the original question more tightly.