| Abstract: Let $M$ be a compact Riemannian manifold. It was proved by Weyl
that number of Laplacian eigenvalues less than T, is asymptotic
to $C(M) T^{\dim(M)/2}$, where $C(M)$ is the product of the
volume of $M$, volume of the unit ball and $(2\pi)^{-\dim(M)}$.
Let $\Gamma$ be an arithmetic subgroup of $SL_2(\mathbb Z)$
and let $\mathbb H^2$ be an upper-half plane. When
$M = \Gamma \backslash \mathbb H^2$, Weyl’s asymptotic holds true
for the discrete spectrum of Laplacian. It was proved by
Selberg, who used his celebrated trace formula. Let $G$ be a semisimple algebraic group of Adjoint and split
type over $\mathbb Q$. Let $G(\mathbb R)$ be the set of
$\mathbb R$-points of $G$. For simplicity of this exposition
let us assume that $\Gamma \subset G(\mathbb R)$ be an torsion
free arithmetic subgroup. Let $K_\infty$ be the maximal compact
subgroup. Let $L^2(\Gamma \backslash G(\mathbb R))$ be space
of square integrable $\Gamma$ invariant functions on
$G(\mathbb R)$. Let
$L^2_{\rm cusp}(\Gamma \backslash G(\mathbb R))$ be the
cuspidal subspace. Let
$M = \Gamma \backslash G(\mathbb R) \slash K_\infty$ be a
locally symmetric space. Suppose
$d = \dim(\Gamma \backslash G \slash K_\infty)$.
Then it was proved by Lindenstrauss and Venkatesh, that number
of spherical, i.e. bi-$K_\infty$ invariant cuspidal Laplacian
eigenfunctions, whose eigenvalues are less than $T$ is asymptotic
to $C(M) T^{\dim(M)/2}$, where $C(M)$ is the same constant as
above. We are going to prove the same Weyl’s asymptotic estimates for
$K_\infty$-finite cusp forms for the above space. |