##
**The trace class conjecture in the theory of automorphic forms.**
*(English)*
Zbl 0701.11019

Let \({\mathcal G}\) denote a reductive group over \({\mathbb{Q}}\) and \(G={\mathcal G}({\mathbb{R}})\) the group of real points. Let \(\Gamma\subset G\) denote an arithmetic subgroup. The aim of automorphic theory is to describe the space \(L^ 2(\Gamma \setminus G)\). As G-module this space admits a decomposition:
\[
L^ 2(\Gamma \setminus G)=L^ 2_{dis}(\Gamma \setminus G)\oplus L^ 2_{con}(\Gamma \setminus G).
\]
Here \(L^ 2_{dis}\) is a direct Hilbert sum of irreducible unitary representations of G and \(L^ 2_{con}\) is a continuous Hilbert integral over irreducibles. The space \(L^ 2_{con}\) can be described by means of Eisenstein series and is well understood. The latter does not hold for \(L^ 2_{dis}.\)

One important question that is raised in connection with the Selberg trace formula is the trace class conjecture. Let \(\pi_{dis}\) denote the representation of G on \(L^ 2_{dis}\) and let \(f\in C_ c^{\infty}(G)\) be a differentiable function of compact support. The trace class conjecture states that \(\pi_{dis}(f)\) should be a trace class operator.

Under the additional assumption that f spans a finite dimensional space under left and right actions of a maximal compact subgroup K of G the article under consideration gives an affirmative answer to this question. Even more is true: f only needs to be a Harish-Chandra Schwartz-function. The assumption of K-finiteness is - at least for applications - no serious restriction and the author thinks that it can be removed.

The proof relies on a somewhat more concrete description of the space \(L^ 2_{dis}\). This space splits as \(L^ 2_{dis}=L^ 2_{cus}\oplus L^ 2_{res}\). The space \(L^ 2_{cus}\) is the space of cusp forms and the trace class conjecture was known to hold on this space before. The space \(L^ 2_{res}\) is spanned by the iterated residues of Eisenstein series. Using a detailed analysis of the constant term of Eisenstein series the author gives estimates of the dimension of the space of functions in \(L^ 2_{res}\) with Casimir-eigenvalue less than a given bound. Thus he ends with a growth estimate on Casimir- eigenvalues which gives the assertion.

A serious gap in the theory of the Trace Formula has been filled by this work. It should nevertheless be remarked that (unfortunately?) J. Arthur has circumvented these difficulties in his Trace Formula which has shown to be appropriate for arithmetical applications. On the other side the trace class conjecture was an interesting question on its own which now has found an answer.

One important question that is raised in connection with the Selberg trace formula is the trace class conjecture. Let \(\pi_{dis}\) denote the representation of G on \(L^ 2_{dis}\) and let \(f\in C_ c^{\infty}(G)\) be a differentiable function of compact support. The trace class conjecture states that \(\pi_{dis}(f)\) should be a trace class operator.

Under the additional assumption that f spans a finite dimensional space under left and right actions of a maximal compact subgroup K of G the article under consideration gives an affirmative answer to this question. Even more is true: f only needs to be a Harish-Chandra Schwartz-function. The assumption of K-finiteness is - at least for applications - no serious restriction and the author thinks that it can be removed.

The proof relies on a somewhat more concrete description of the space \(L^ 2_{dis}\). This space splits as \(L^ 2_{dis}=L^ 2_{cus}\oplus L^ 2_{res}\). The space \(L^ 2_{cus}\) is the space of cusp forms and the trace class conjecture was known to hold on this space before. The space \(L^ 2_{res}\) is spanned by the iterated residues of Eisenstein series. Using a detailed analysis of the constant term of Eisenstein series the author gives estimates of the dimension of the space of functions in \(L^ 2_{res}\) with Casimir-eigenvalue less than a given bound. Thus he ends with a growth estimate on Casimir- eigenvalues which gives the assertion.

A serious gap in the theory of the Trace Formula has been filled by this work. It should nevertheless be remarked that (unfortunately?) J. Arthur has circumvented these difficulties in his Trace Formula which has shown to be appropriate for arithmetical applications. On the other side the trace class conjecture was an interesting question on its own which now has found an answer.

Reviewer: A.Deitmar

### MSC:

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

11F12 | Automorphic forms, one variable |