WEYL’S LAW FOR CUSP FORMS OF ARBITRARY $K_{\infty}$-TYPE

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 $(2π)^{−dim(M)}$. Let $\Gamma$ be an arithmetic subgroup of $SL_2(\mathbb{Z})$ and \mathbb{H}^2 be an upper-half plane. When $M = \Gamma \backslash \mathbb{H}^2$, Weyl’sasymptotic 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 $L^2_{cusp}(\Gamma \backslash G(\mathbb{R})$ be the cuspidal subspace. Let $M = \Gamma \backslash G(\mathbb{R})/K_{\infty}$ be a locally symmetric space. Suppose $d = dim(\Gamma \backslash G(\mathbb{R})/K_{\infty})$. 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.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar