WEYL’S LAW FOR CUSP FORMS OF ARBITRARY K∞-TYPE
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)Tdim(M)/2, where C(M) is the product of the volume of M, volume of the unit ball and (2π)−dim(M). Let Γ be an arithmetic subgroup of SL2(Z) and H2 be an upper-half plane. When M = Γ\H2, 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 Q. Let G(R) be the set of R-points of G. For simplicity of this exposition let us assume that Γ ⊂ G(R) be an torsion free arithmetic subgroup. Let K∞ be the maximal compact subgroup. Let L2(Γ\G(R)) be space of square integrable Γ invariant functions on G(R). Let L2cusp(Γ\G(R)) be the cuspidal subspace. Let M = Γ\G(R)/K∞ be a locally symmetric space. Suppose d = dim(Γ\G/K∞). Then it was proved by Lindenstrauss and Venkatesh,
that number of spherical, i.e. bi-K∞ invariant cuspidal Laplacian eigenfunctions, whose eigenvalues are less than T is asymptotic to C(M)Tdim(M)/2, where C(M) is the same constant as above.
We are going to prove the same Weyl’s asymptotic estimates for K∞-finite cusp forms for the above space.