Representation theory of $\text{SL}(2,\mathbb{Z})$

Question

The group $\text{SL}(2,\mathbb{Z})$ is the group of two-by-two matrices with integer entries and determinant one. This is a very simple definition. Yet its representation theory seems quite wild to me (a PhD physics student) from the limited information I can find online... In particular, it seems it has uncountably many irreducible representations. I am also aware that if we restrict ourselves to the class of representations with finite image and whose kernel can be described by congruences, the representation theory becomes way more well-behaved. These are the so-called congruence subgroups.

I would like to learn more about the representation theory of $\text{SL}(2,\mathbb{Z})$ beyond the classical textbook discussions restricted on congruence subgroups.

Any good general references (papers, book, ...) I could look into?

Answer 1

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}$The reason that there are uncountably many irreducible representations is not so bad, and gets at an important point: You shouldn't think of irreducible representations of a group like $\SL_2(\mathbb Z)$ individually, but rather as points in a space of representations, i.e. objects parameterized by a geometric space.

This is simplest to see for representations sending $-I \in \SL_2(\mathbb Z)$ to the identity, i.e. representations of $\PSL_2(\mathbb Z)$. By the presentation YCor gave $\PSL_2(\mathbb Z) = \langle x,y \mid x^2 =y^3 =1 \rangle$, such a representation is given by a matrix $X$ satisfying $X^2=1$ and a matrix $Y$ satisfying $Y^3=1$, and representations up to isomorphism are given by pairs of matrices up to conjugation.

For $n$-dimensional representations, say $n$ a multiple of $6$, it's not so hard to check that the space of matrices $X \in \GL_n(\mathbb C)$ with $X^2=1$ has dimension $n^2/2$ (over $\mathbb C$), and the space of matrices $Y \in \GL_n(\mathbb C)$ with $Y^3= 1$ has dimension $2n^2/3$, so the space of pairs has dimension $n^2/2 + 2n^2/3$, isomorphism classes of $n$-dimensional representations have dimension $n^2/2 +2n^2/3 - (n^2-1) = n^2/6+1$.

One can check that irreducible representations are an open subset, and thus that the space of irreducible representations has the same dimension.

So certainly there are uncountably many, because they're parameterized by a positive-dimensional manifold!

However, it's clear from this analysis that this space should be one of the primary objects of study in the representation theory here, as it is for representations of surface groups and in some other cases of interest.

Answer 2

$\DeclareMathOperator\SL{SL}$The issue is that $\SL(2,\mathbb{Z})$ is very close to a free group, so it is not hard to map it to other groups, and in particular to produce lots of varied representations of it.

You might be interested to know that things are quite different for $\SL(n,\mathbb{Z})$ when $n$ is at least $3$. I wrote a note The representation theory of $\operatorname{SL}_n(\mathbb Z)$ describing results of Margulis and Lubotzky giving a complete and fairly simple description of its representations.

Answer 3

Far from being a specialist on the topic, section 4 of the article $\text{SL}_2(\mathbb{Z})$, by K. Conrad discusses non-congruence subgroups.