Let \(X=\text{SL}(n,\mathbb R)/\text{SO}(n)\) be the symmetric space of positive-definite quadratic forms in \(n\geq2\) variables up to homothety and \(\Gamma\) be a principal congruence subgroup of level \(N\geq3\) in \(\text{SL}(n,\mathbb Z)\). The commutative algebra of invariant differential operators on \(\text{SL}(n,\mathbb R)\) acts on the cuspidal part of \(L^2(\Gamma\backslash X)\), with joint cuspidal spectrum \(\Lambda_{\text{cusp}}(\Gamma)\) identified to a discrete \(W\)-invariant subset of \( \mathfrak{a}^*_{\mathbb C}\) where \(\mathfrak{a}\) is the Lie algebra of the group \(A\subset \text{SL}(n,\mathbb R)\) of diagonal matrices with positive real diagonal entries and \(W\) the associated Weyl group to the pair \((\text{SL}(n,\mathbb R),A)\).
Counting eigenvalues with multiplicity, the authors prove the asymptotic estimates when \( t\to+\infty\) \[ \begin{gathered} \left|\Lambda_{\text{cusp}}(\Gamma)\cap t\Omega\right| = {\text{Vol}(\Gamma\backslash X)} \text{Vol}({t\Omega}) +\mathcal{O}(t^{d-1}(\log t)^{\max(n,3)}),\\ \left|\Lambda_{\text{cusp}}(\Gamma)\cap (tB_1\setminus\text{i}\mathfrak{a}^*) \right| =\mathcal{O}(t^{d-2}). \end{gathered} \tag{\(W\)} \] Here, the set \(\Omega\) is any \(W\)-invariant bounded regular domain in \(\text{i}\mathfrak{a}^*\) and \(B_1\) is the unit ball in \(\mathfrak{a}^*_\mathbb C\), the space \(\text{i}\mathfrak{a}^*\) is endowed with a well normalized Plancherel measure and \(d\) is the dimension of the symmetric space \(X\). These results can be formulated in adelic terms, the convenient framework for the formulation of the Langlands program and the Arthur’s trace formula.
The Weyl’s law \((W)\) with remainder estimate for the cuspidal spectrum extends the classical result for arithmetical lattices of \(\text{SL}(2,\mathbb R)\) due to [A. Selberg, Collected Papers. Volume II. Berlin etc.: Springer-Verlag (1991; Zbl 0729.11001)], the uniform cases in higher rank proved by J. J. Duistermaat, J. A. C. Kolk and V. S. Varadarajan [[DKV], Invent. Math. 52, 27–93 (1979; Zbl 0434.58019)] and the Weyl’s law proved by W. Müller [Ann. Math. (2) 165, No. 1, 275–333 (2007; Zbl 1119.11027)] for the cuspidal spectrum of the Laplace-Beltrami operator on \(\Gamma\backslash\text{SL}(n,\mathbb R)/\text{SO}(n)\). The Ramanujan conjecture for \(\text{GL}(n)\) states that \(\Lambda_{\text{cusp}}(\Gamma)\subset i\mathfrak{a}^*\), so that the second bound is empty if the Ramanujan conjecture is true.
For the rank one case (eg arithmetic lattices of \(\text{SL}(2,\mathbb R)\)), the authors sketch a proof based on the Selberg’s trace formula and Hörmander’s method on spectral asymptotics for elliptic operators. In the higher rank case which is the heart of the paper, they follow the [DKV] method, taking an appropriate family of test functions in the Arthur’s trace formula (which forces to work in the adelic setting) so that the contributions of the identity class on the geometrical side and of the cuspidal spectrum in the spectral side are dominant. Analysis of the unipotent conjugacy classes terms on the geometrical side represents the main new technical difficulties, worked out by stationary phase arguments. Arguments in Müller (2007) are used to control the continuous spectrum contribution on the spectral side of the trace formula. It requires knowledge on general Eisenstein series and automorphic \(L\)-functions associated to cusp forms on lower rank groups: such detailed information for general reductive algebraic \(\mathbb Q\)-group \(G\) is for the time being missing, which is the main obstacle to the proof of analogue Weyl’s law for such general \(G\), beyond the case of \(\text{SL}(n,\mathbb R)\) as solved in this paper and a few other low dimensional cases.