Trace Formulas

A trace formula is an equation that relates two kinds of data – “spectral” data related to representations (or eigenvalues of certain operators), and “geometric” data, related to integrals along “orbits” on some space.

The name “trace formula” comes from how this equation is obtained – by expanding the “trace” of a certain operator (let’s call it $R_{f}$ . It will depend on a compactly supported “test function” $f(x)$ on a topological group $G$ ) on square-integrable functions on a compact quotient $\Gamma\backslash G$ of $G$ (which give a representation of $G$ by translation) by a discrete subgroup $\Gamma$ .

The operator $R_{f}$ takes a function $\phi(x)$ on the group $G$ , translates it by some element $y$ (recall for example that acting on functions by translation is how we defined the representation of the group $\mathbb{R}$ in Representation Theory and Fourier Analysis), multiplies it by the test function $f(x)$ , then integrates over the group $G$ (the group $G$ must have a measure called “Haar measure” to do this) to obtain a new function $(R_{f}\phi)(y)$ :

$\displaystyle (R_{f}\phi)(y)=\int_{G}\phi(xy)f(x)dx$

We can also express this as

$\displaystyle (R_{f}\phi)(y)=\int_{G}\phi(x)f(y^{-1}x)dx$

Let $\Gamma$ be a discrete subgroup of $G$ , such that the quotient $\Gamma\backslash G$ is compact (this will turn out to be important later). Instead of integrating over all of $G$ we may instead integrate over the quotient $\Gamma\backslash G$ by re-expressing the integrand as follows:

$\displaystyle (R_{f}\phi)(y)=\int_{\Gamma\backslash G}\phi(x)\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)dx$

The sum $\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)$ is called the “kernel” of the operator $R_{f}$ and is denoted by $K(x,y)$ . We have

$\displaystyle (R_{f}\phi)(y)=\int_{\Gamma\backslash G}K(x,y)\phi(x)dx$

So the operator $R_{f}$ looks like the integral of $K(x,y)\phi(x)dx$ over the quotient $\Gamma\backslash G$ . Compare this with how a matrix with entries $A_{mn}$ acts on a finite dimensional vector $v_{n}$ :

$\displaystyle v_{m}=\sum_{n}A_{mn}v_{n}$

Note that we think of integrals as analogous to sums for infinite dimensions, as functions are analogous to vectors in infinite dimensions. Now we can see that the kernel $K(x,y)$ is the analogue of the entries of some matrix!

The “trace” of a matrix is just the sum of its diagonal entries, i.e. the sum of $A_{nn}$ for all n. Therefore, the trace of the operator defined above is the integral of $K(x,x)$ (i.e. we set $x=y$ ) over $\Gamma\backslash G$ .

$\displaystyle \mathrm{tr}(R_{f})=\int_{\Gamma\backslash G} K(x,x)dx$

Now recall that the kernel $K(x,y)$ is given by the sum $\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)$ . Therefore the trace will be given by

$\displaystyle \mathrm{tr}(R_{f})=\int_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}f(x^{-1}\gamma x)dx$ .

Some analysis manipulations will allow us to re-express the trace as the sum

$\displaystyle \mathrm{tr}(R_{f})=\sum_{\gamma\in\lbrace \Gamma\rbrace}\mathrm{vol}(\Gamma_{\gamma}\backslash G_{\gamma})\int_{G_{\gamma}\backslash G} f(x^{-1}\gamma x)dx$

over representatives $\gamma$ of conjugacy classes in $\Gamma$ of the integrals of $f(x^{-1}\gamma x)$ over the quotient $G_{\gamma}\backslash G$ where $G_{\gamma}$ is the centralizer of $\gamma$ in $G$ , multiplied by some factor called the “volume” of $\Gamma_{\gamma}\backslash G_{\gamma}$ .

The integral of $f(x^{-1}\gamma x)$ over $G_{\gamma}\backslash G$ is called an “orbital integral“. This expansion of the trace is going to be the “geometric side” of the trace formula.

We consider another way to expand the trace. Recall that to define the operator $R_f$ we needed to act by translation. In this case that the quotient $\Gamma\backslash G$ is compact, as we stated earlier, this representation (let us call it $R$ ) by translation decomposes into a direct sum of irreducible representations $\pi$ , with multiplicities $m(\pi,R)$ . So we decompose first before getting the trace!

This other expansion is called the “spectral side“. Since we have now expanded the same thing, the trace, in two ways, we can equate the two expansions:

$\displaystyle \sum_{\gamma\in\lbrace \Gamma\rbrace}\mathrm{vol}(\Gamma_{\gamma}\backslash G_{\gamma})\int_{G_{\gamma}\backslash G}f(x^{-1}hx)dx=\sum_{\pi} m(\pi,R)\mathrm{tr}(\int_{G}f(x)\pi(x)dx)$

This equation is what is called the “trace formula”. Let us test it out for $G=\mathbb{R}$ , $H=\mathbb{Z}$ , like in Representation Theory and Fourier Analysis.

In the geometric side, $f(x^{-1}\gamma x)=f(\gamma)$ , since $\mathbb{R}$ is abelian. $\mathbb{Z}$ is also abelian, so the conjugacy classes are just elements of $\mathbb{Z}$ . We have $G_{\gamma}=G$ and $\Gamma_{\gamma}=\Gamma$ . One can check that the volume is $1$ and the orbital integral is just $f(\gamma)$ . Replacing $\gamma$ by $n$ for notational convenience, we see that the geometric side is just a sum of $f(n)$ over each integer $n$ in $\mathbb{Z}$ .

Let us now look at the spectral side. Recall that the representation decomposes into irreducible representations, each with multiplicity $1$ , which are given by multiplication by $e^{2 \pi i k x}$ . We consider the operator $R_{f}$ now.

Recall that we let our representation act, then multiply it with the test function f, then integrate. We broke it up into irreducible representations, which act by multiplication by $e^{2 \pi i k x}$ . What is multiplication of a function of the form $e^{2 \pi i k x}$ and integrating over $x$ ?

This is just the Fourier transform of the test function $f$ ! Since we have an irreducible representation for every integer $k$ , we sum over those. So we have an equality between the sum of $f(n)$ where $n$ is an integer, and the corresponding sum of its Fourier transforms!

This is actually a classical result in Fourier analysis known as Poisson summation:

$\displaystyle \sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}} e^{2\pi i k x}f(x)dx$

Atle Selberg famously applied the trace formula to the representation of $G=\mathrm{SL}_{2}(\mathbb{R})$ on functions on a double quotient $H\backslash\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2)$ . Note that the quotient $\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2)$ is the upper half-plane. H is chosen by Selberg so that the double quotient is a Riemann surface of genus $g\geq 2$ .

Selberg used the trace formula to relate lengths of geodesics (given by orbital integrals) to eigenvalues of the 2D Laplacian. Note that the Laplacian already appears in our example of Poisson summation, because $e^{2 \pi i k x}$ is also an eigenfunction of the 1D Laplacian.

This may be why the spectral side is called “spectral”. The trace formula is fascinating on its own, but very commonly used with it is to study representations of certain groups via more familiar representations of other groups.

To do this, note that the spectral side contains information related to representations. If we could only somehow find a way to relate the geometric sides of trace formulas of two different representations, then we can relate their spectral sides!

This is an approach to the part of representation theory known as Langlands functoriality, which studies how representations are related given that the respective groups have “Langlands duals” that are related. Relating the geometric sides involves proving difficult theorems such as “smooth transfer” and the “fundamental lemma”.

Finally, it is worth noting that the spectral side is also used to study special values of L-functions. This is inspired by the work of Hecke expressing completed L-functions as Mellin transforms of modular forms. But that is for another time!

References:

Arthur-Selberg trace formula on Wikipedia

Poisson summation formula on Wikipedia

An introduction to the trace formula by James Arthur

Selberg’s trace formula: an introduction by Jens Marklof

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