In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given by the trace of certain functions on .

The simplest case is when is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When is the cocompact subgroup of the real numbers , the Selberg trace formula is essentially the Poisson summation formula.

The case when is not compact is harder, because there is a continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when is the group ; the extension to higher rank groups is the Arthur–Selberg trace formula.

When is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to the explicit formulas relating the zeros of the Riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced the Selberg zeta function of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula.

Early history

Cases of particular interest include those for which the space is a compact Riemann surface . The initial publication in 1956 of Atle Selberg dealt with this case, its Laplacian differential operator and its powers. The traces of powers of a Laplacian can be used to define the Selberg zeta function. The interest in this case was the analogy between the formula obtained and the explicit formulas of prime number theory. Here the closed geodesics on play the role of prime numbers.

At the same time, interest in the traces of Hecke operators was linked to the Eichler–Selberg trace formula, of Selberg and Martin Eichler, for a Hecke operator acting on a vector space of cusp forms of a given weight, for a given congruence subgroup of the modular group. Here the trace of the identity operator is the dimension of the vector space, i.e. the dimension of the space of modular forms of a given type: a quantity traditionally calculated by means of the Riemann–Roch theorem.

Applications

The trace formula has applications to arithmetic geometry, analytic number theory, spectral geometry, and the theory of automorphic forms. In the case of hyperbolic surfaces, it translates information about the spectrum of the Laplace–Beltrami operator into geometric information about closed geodesics, and conversely.

By suitable choices of the test function, the trace formula yields asymptotic information about eigenvalues of the Laplacian, including Weyl-type asymptotics. For compact hyperbolic surfaces it also shows that the Laplace spectrum determines the length spectrum of closed geodesics, so the lengths of primitive closed geodesics are spectral invariants.

For congruence subgroups, trace formulas are used to study automorphic forms and traces of Hecke operators. In the classical setting this includes the Eichler–Selberg trace formula, which gives formulas for traces of Hecke operators on spaces of cusp forms and, in particular cases, dimension formulas for spaces of modular forms. More generally, the trace formula is one of the basic tools in the spectral theory of automorphic forms and in analytic number theory.

The trace formula is also central to the analytic theory of the Selberg zeta function. It can be used to prove the meromorphic continuation and functional equation of the zeta function, and to relate its zeros to Laplace eigenvalues and to the geometry of closed geodesics.

The trace of <math>R(\phi)</math>can be expressed as the integral of the kernel <math>K(x,y)=\sum_{\gamma\in\Gamma}\phi(x^{-1}\gamma y)</math> along the diagonal, that is:

<math display=block>\operatorname{tr}R(\phi) = \int_{\Gamma\setminus G}\sum_{\gamma\in\Gamma}\phi(x^{-1}\gamma x)\,dx.</math>

Let <math>\{\Gamma\}</math> denote a collection of representatives of conjugacy classes in <math>\Gamma</math>, and <math>\Gamma^\gamma</math> and <math>G^\gamma</math> the respective centralizers of <math>\gamma</math>.

Then the above integral can, after manipulation, be written

<math display=block>\operatorname{tr}R(\phi) = \sum_{\gamma\in\{\Gamma\ a_\Gamma^G(\gamma)\int_{G^\gamma\setminus G}\phi(x^{-1}\gamma x)\,dx.</math>

This gives the geometric side of the trace formula.

The spectral side of the trace formula comes from computing the trace of <math>R(\phi)</math> using the decomposition of the regular representation of <math>G</math> into its irreducible components. Thus

<math display=block>\operatorname{tr}R(\phi) = \sum_{\pi\in\hat G}a_\Gamma^G(\pi)\operatorname{tr}\pi(\phi)</math>

where <math>\hat G</math> is the set of irreducible unitary representations of <math>G</math> (recall that the positive integer <math>a_\Gamma^G(\pi)</math> is the multiplicity of <math>\pi</math> in the unitary representation <math>R</math> on <math>L^2(\Gamma\setminus G)</math>).

The case of semisimple Lie groups and symmetric spaces

When <math>G</math> is a semisimple Lie group with a maximal compact subgroup <math>K</math> and <math>X=G/K</math> is the associated symmetric space the conjugacy classes in <math>\Gamma</math> can be described in geometric terms using the compact Riemannian manifold (more generally orbifold) <math>\Gamma \backslash X</math>. The orbital integrals and the traces in irreducible summands can then be computed further and in particular one can recover the case of the trace formula for hyperbolic surfaces in this way.

Later work

The general theory of Eisenstein series was largely motivated by the requirement to separate out the continuous spectrum, which is characteristic of the non-compact case.

The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup into an algebraic group over a field which is technically easier to work with. The case of SL<sub>2</sub>(C) is discussed in and . Gel'fand et al also treat SL<sub>2</sub>() where is a locally compact topological field with ultrametric norm, so a finite extension of the p-adic numbers Q<sub>p</sub> or of the formal Laurent series F<sub>q</sub>((T)); they also handle the adelic case in characteristic 0, combining all completions R and Q<sub>p</sub> of the rational numbers Q.

Contemporary successors of the theory are the Arthur–Selberg trace formula applying to the case of general semisimple G, and the many studies of the trace formula in the Langlands philosophy (dealing with technical issues such as endoscopy). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort.

See also

  • Jacquet–Langlands correspondence

Notes

References

  • Selberg trace formula resource page