Discrete spectrum of pseudo-cuspforms on GLn

The proof will show that the resolvent of (the Friedrichs extension of a restriction of) the invariant Laplacian is compact. Let Lη be the subspace of L 2 = L(PGLr(Z)\PGLr(R)/O(n,R)) with all constant terms vanishing above given fixed heights (specified by a real-valued function η on simple positive roots, described precisely below). By its construction, the resolvent of the Friedrichs extension maps continuously from L to the automorphic Sobolev space H = H(PGLr(Z)\PGLr(R)/O(n,R)) with its finer topology. Letting H η = H 1 ∩ Lη with the topology of H, it suffices to show that the injection H η → Lη is compact. To prove this compactness, we will show that the image of the unit ball of H η is totally bounded in L 2 η. Let A be the standard maximal torus consisting of diagonal elements of G = GLr, Z the center of G, and K = O(n,R). Let A be the subgroup of AR with positive diagonal entries, and Z = ZR∩A. A standard choice of positive simple roots is