Selberg's trace formula

In harmonic analysis , a branch of mathematics , Selberg's trace formulas establish a connection between the trace of certain operators and a sum of geometric terms.

While the calculation of the eigenvalues ​​of a differential operator is often inaccessible, at least one statement about the sum of the eigenvalues ​​can be made with the trace formulas. In particular, the case of the Laplace-Beltrami operator of locally symmetrical spaces of rank 1 worked out by Selberg has applications in analytic number theory , representation theory and differential geometry .

The more general Arthur Selberg gauge formula plays an important role in the Langlands program .

General trace formula

Let be a co-compact lattice in a locally compact group . ${\displaystyle \Gamma }$ ${\displaystyle G}$

For a test function and a representation of defined ${\displaystyle f\in C_{unif}^{2}(G)}$ ${\displaystyle \pi }$ ${\displaystyle G}$

${\displaystyle \pi (f):=\int _{G}f(y)\pi (y)dy}$

a lane class operator on Hilbert space . ${\displaystyle L^{2}(\Gamma \backslash G)}$

Let in particular be the right-regular representation , i.e. the unitary representation of by right translations on the Hilbert space . Then the trace of this operator can be expressed by ${\displaystyle R}$ ${\displaystyle G}$ ${\displaystyle L^{2}(\Gamma \backslash G)}$

${\displaystyle \mathrm {Spur} (R(f))=\sum _{\left[\gamma \right]}\mathrm {vol} (\Gamma _{\gamma }\backslash G_{\gamma })O_{\gamma }(f)}$,

where all conjugation classes are added to the right , and the centralizers of in and are, and the orbit integral through ${\displaystyle \Gamma _{\gamma }}$ ${\displaystyle G_{\gamma }}$ ${\displaystyle \gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle G}$ ${\displaystyle O_{\gamma }(f)}$

${\displaystyle O_{\gamma }(f)=\int _{G_{\gamma }/G}f(x^{-1}\gamma x)dx}$

is defined. ( This orbit integral can be expressed in terms of characters using Fourier inversion .)

According to the theorem of Gelfand, Graev and Piatetski-Shapiro, the regular representation has a decomposition as a direct sum of irreducible representations . For an irreducible representation denote its multiplicity in . This results in the trace formula ${\displaystyle \pi }$ ${\displaystyle N_{\Gamma }(\pi )}$ ${\displaystyle L^{2}(\Gamma \backslash G)}$

${\displaystyle \sum _{\pi }N_{\Gamma }(\pi )\mathrm {Spur} (\pi (f))=\sum _{\left[\gamma \right]}\mathrm {vol} (\Gamma _{\gamma }\backslash G_{\gamma })O_{\gamma }(f)}$.

The expression on the left is referred to as the spectral side of the trace formula, the expression on the right as the geometric side of the trace formula.

Special symmetrical spaces

In order for the general trace formula to be useful, one must understand the distributions and and be able to express them in differential geometric quantities. This is especially possible for symmetric spaces of rank 1, where the Laplace operator (and its multiples) are the only invariant differential operators. ${\displaystyle \mathrm {Spur} (\pi (f))}$ ${\displaystyle O_{\gamma }(f)}$

In the following denote the eigenvalues ​​of the Laplace-Beltrami operator on a compact Riemann manifold . We use whatever is with. ${\displaystyle 0=\lambda _{0}<\lambda _{1}\leq \ldots \leq \lambda _{j}\leq }$ ${\displaystyle \rho _{j}={\sqrt {\lambda _{j}+{\frac {1}{4}}}}}$ ${\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})=\mathrm {Spur} (f({\sqrt {\Delta +{\frac {1}{4}}}}))}$

Compact groups

For a compact group with a neutral element , the dimension of a representation corresponds to its multiplicity in the regular representation and one obtains by direct application of the general trace formula ${\displaystyle K}$ ${\displaystyle e}$

${\displaystyle f(e)=\sum _{\tau \in {\widehat {K}}}\dim(\tau )\mathrm {Spur} (\tau (f))}$.

For the circle , Poisson's total formula applies : for a rapidly falling function and its Fourier transform applies ${\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }$ ${\displaystyle f}$ ${\displaystyle {\widehat {f}}}$

${\displaystyle \sum _{n\in \mathbb {Z} }f(n)=\sum _{n\in \mathbb {Z} }{\widehat {f}}(n)=\sum _{n\in \mathbb {Z} }\int _{-\infty }^{\infty }f(x)e^{-2\pi inx}dx}$.

This creates a connection between the lengths of the closed geodesics and the eigenvalues ​​of the Laplace operator. This can be seen as a variant of the trace formula: there is a geometric term on the left and a spectral term on the right. ${\displaystyle n}$ ${\displaystyle 4\pi ^{2}n^{2}}$

The sphere

For are the eigenvalues ​​of the Laplace-Beltrami operator with a multiplicity , where all integers run through. With Poisson's empirical formula one gets ${\displaystyle S^{2}=SO(3)/SO(2)}$ ${\displaystyle l(l+1)}$ ${\displaystyle 2l+1}$ ${\displaystyle l}$

${\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})=\sum _{n=0}^{\infty }(-1)^{n}\int _{\mathbb {R} }\vert x\vert f(x)e^{2\pi inx}dx}$.

The right side can be interpreted geometrically as a series dependent on the lengths of closed geodesics. ${\displaystyle 2\pi n}$

Hyperbolic surfaces

To formulate the trace formula one uses any (any) analytical function with for one and . With such a function one can formulate the trace formula for co-compact lattices as follows: ${\displaystyle h\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle \vert h(x)\vert =O(\vert x\vert ^{-2-\delta })}$ ${\displaystyle \delta >0}$ ${\displaystyle x\to \pm \infty }$ ${\displaystyle G=PSL(2,\mathbb {R} )}$

${\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})={\frac {\mathrm {vol} (\Gamma \backslash \mathbb {H} ^{2})}{4\pi }}\int _{-\infty }^{\infty }x\,f(x)\tanh(\pi x)\,dx+\sum _{\{\gamma \}\not =1}{\frac {\log N(\gamma _{0})}{N(\gamma )^{\frac {1}{2}}-N(\gamma )^{-{\frac {1}{2}}}}}{\widehat {f}}(\log N(\gamma )),}$

where on the right-hand side all conjugation classes of elements are summed up, the associated primitive transformation denotes, and the norm of denotes. The right side can be interpreted geometrically, because the length of the corresponding closed geodesics is in. ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle \gamma _{0}}$ ${\displaystyle N(\gamma )}$ ${\displaystyle \gamma }$ ${\displaystyle \log N(\gamma )}$ ${\displaystyle \gamma }$ ${\displaystyle \Gamma \backslash \mathbb {H} ^{2}}$

Because of the compactness of the surface , the eigenvalues ​​of the Laplace operator form a discrete set . There is a more complicated formula for non-compact hyperbolic surfaces of finite volume that takes the continuous spectrum into account. ${\displaystyle \Sigma _{g}=\Gamma \backslash PSL(2,\mathbb {R} )/PSO(2)}$

As a consequence, one obtains, for example, Weyl's asymptotic law for the distribution of eigenvalues.

Symmetrical rooms of rank 1

Let be a symmetric space of non-compact type of rank 1, i. H. for the Iwasawa decomposition . Let be the centralizer of in and a parabolic subgroup . For a representation with representation space, let the Hilbert space of the functions with . For define . Then defines a representation of on . ${\displaystyle G/K}$ ${\displaystyle \mathrm {dim} (A)=1}$ ${\displaystyle G=KAN}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle P=MAN}$ ${\displaystyle \sigma \in {\widehat {M}}}$ ${\displaystyle V_{\sigma }}$ ${\displaystyle {\mathcal {H}}_{\sigma }}$ ${\displaystyle L^{2}}$ ${\displaystyle f\colon K\to V_{\sigma }}$ ${\displaystyle f(km)=\sigma (m)^{-1}f(k)}$ ${\displaystyle f\in {\mathcal {H}}_{\sigma }}$ ${\displaystyle f_{\lambda }(k\exp(tH)n)=e^{-(i\lambda +\rho )t}f(k)}$ ${\displaystyle (\pi _{\sigma ,\lambda }(g)f)(k)=f_{\lambda }(g^{-1}k)}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {H}}_{\lambda }}$

From the general trace formula it follows first for ${\displaystyle f\in C_{c}^{\infty }(G)}$

${\displaystyle \sum _{\pi }N_{\Gamma }(\pi )\mathrm {Spur} (\pi (f))=\mathrm {vol} (\Gamma \backslash G/K)f(e)+\sum _{\left\{\gamma \right\}\not =\left\{e\right\}}{\frac {1}{2\pi }}{\frac {l(\gamma )}{D(\gamma )}}\sum _{\sigma \in {\widehat {M}}}{\overline {\mathrm {Spur} (\sigma (\gamma ))}}\int _{R}\mathrm {Spur} (\pi _{\sigma ,\lambda }(f))e^{-il(\gamma )\lambda }d\lambda }$,

where the conjugation classes are totaled and defined by . ${\displaystyle \left\{\gamma \right\}}$ ${\displaystyle D(\gamma )}$ ${\displaystyle D(\gamma )=e^{-l(\gamma )\rho }\vert \det(Ad(m_{\gamma }a_{\gamma })\vert _{\mathfrak {n}}-Id)\vert }$

Then you have the trace formula

${\displaystyle \sum _{j=0}^{\infty }f({\sqrt {\lambda _{j}-\vert \rho \vert ^{2}}})=\mathrm {vol} (\Gamma \backslash G/K)\int _{\mathbb {R} }f(\lambda )\beta (\lambda )d\lambda +\sum _{\left\{\gamma \right\}\not =\left\{e\right\}}{\frac {l(0)}{D(\gamma )}}{\hat {f}}(l(\gamma ))}$.

Remarks

1. ↑ The space of test functions consists by definition of the linear combinations of functions of the form with . The space consists of uniformly integrable, continuous functions. A function is said to be uniformly integrable if there is a compact neighborhood U of 1 such that a -function is on . ${\displaystyle C_{unif}^{2}(G)}$ ${\displaystyle g*h}$ ${\displaystyle g,h\in C_{unif}(G)}$ ${\displaystyle C_{unif}(G)}$ ${\displaystyle f_{U}(y):=\sup _{x,z\in U}f(xyz)}$ ${\displaystyle L^{1}}$ ${\displaystyle G}$
2. ↑ Because is a co-compact lattice, all are hyperbolic and belong to a cyclic subgroup of . The creator of this cyclic subgroup is called the primitive element . ${\displaystyle \Gamma }$ ${\displaystyle \gamma \not =1}$ ${\displaystyle \Gamma }$ ${\displaystyle \gamma _{0}}$

literature

• A. Selberg: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , J. Indian Math. Soc. 20, 47-87, 1956. online
• D. Hejhal: The Selberg trace formula for PSL (2, R). Vol. I. Lecture Notes in Mathematics, Vol. 548. Springer-Verlag, Berlin-New York, 1976.
• J. Elstrodt: The Selberg trace formula for compact Riemann surfaces. Annual d. Deutsche Math. Verein 83, 45-77, 1981.
• D. Zagier: Eisenstein series and the Selberg trace formula , Part I ,

Web links

• Chapter 4 in Müller: Spectral Theory of Automorphic Forms
• Bump: Spectral Theory and the Trace Formula