Irreducible cuspidal representations with prescribed local behavior

Let G be a simple algebraic group defined over the global field k. In this paper, we use the simple trace formula to determine the sum of the multiplicities of the irreducible representations in the cuspidal spectrum of G, with specified local behavior at a finite set of places of k and unramified elsewhere. This sum is expressed as the product of the values of modified Artin L-functions at negative integers. Let k be a global field, with ring of adèles A. Let G be a simple (= almost absolutely simple) algebraic group over k. Then G(k) is a discrete subgroup of G(A), and the quotient G(k)\G(A) has finite Haar measure. The unitary representation of G(A) on L(G(k)\G(A)) has a discrete spectrum Ld, which contains the cuspidal spectrum L 2 0. Both decompose as a Hilbert direct sum of irreducible unitary representations π with finite multiplicities m(π) [DKV]. Furthermore, every irreducible representation π of G(A) is a restricted tensor product ⊗̂πv of local representations πv of G(kv), with πv unramified for almost all finite places v of k. In this paper, we will use the simple trace formula to calculate the sum of multiplicities m(π), for cuspidal π with specified local behavior at a finite set of places and unramified elsewhere. In the number field case, we will assume that k is totally real and that the Archimedian components πv all lie in the discrete series. The prescribed 1 components at finite places will be either Steinberg representations (for v in S) or simple supercuspidal representations [G-R] (for v in T ). The main term in our formula for the sum of multiplicities is given by a product of the values LS,T (V, 1− d) of modified Artin L-functions at negative integers, taken over the degrees d of the invariant polynomials for the Weyl group of G. This product calculates the volume of the quotient G(k)\G(A) with respect to a specific Haar measure; by our hypotheses on the sets S and T only the identity conjugacy class in G(k) has a non-zero orbital integral for the relevant test function in the trace formula. Our results are new even in the case of classical modular forms, where k = Q and G = PGL(2). Using the fact that Steinberg representations are the unique representations of conductor p and simple supercuspidal representaions are the unique representations of conductor q, we obtain the following translation of our formula into classical terminology. Let S and T denote disjoint, finite, non-empty sets of rational primes, and let N = ∏ S p. ∏ T q . Then the dimension of the space of new forms of weight 2k for the group Γ0(N) is given by the formula (2k − 1) 12 ∏