A geometric meaning of the multiplicity of integrable discrete classes in L²(Γ＼G)

Abstract

Let F be a discrete subgroup of a group G of motions of a noncompact symmetric space M of inner type such that the quotient T\M is compact. Let K be the isotropy group at a point of M. Let co be a discrete class of G such that infinite matrix coefficients of co belong to ^(G). Depending on some parameter corresponding to co we can associate a homogeneous vector bundle over M. The Dirac operator D which is a first order elliptic G-invariant differential operator acts on the space of C°° sections of the above vector bundle. We then prove (Theorem 4, §3) that the multiplicity NJJ*) of co in the (right) regular representation of G on L(T\G) is the dimension of the space H(T; *) of sections which are annihilated by D (Dirac spinors) and which are F-invariant and obtain a formula for the same (Corollary to Theorem 4, §3). We remark that algebraic formulas for iVw(r) are already available in several cases (Langlands [6]). Also when the parameter corresponding to co satisfies some further conditions, Schmid [9] obtained geometric meaning to the multiplicity NJT) (similar to Theorem 4), working with G\T rather than the symmetric space GjK where T is a Cartan subgroup of G contained in K. Our method of proof is as follows: The space H(T; *) is the direct sum of two subspaces H(T; *) and H~(T; *). First we prove that one of these two spaces vanishes (Theorem 2, §1). Then using the Lefschetz Theorem of Atiyah and Singer [1] we obtain a formula for the difference dim H(T; *) — dim H~(T; *) (Theorem 3, §2). Let us divide our problem into two parts; namely, (1) to prove dim H(T; *)=j\Ttt(r) and (2) to compute explicitly the above number. When F has no elliptic elements other than the identity, we prove (1) by directly showing** that the expression for dim H(T; #) — dim H~(T; *) given by Theorem