Multiplicity Conjecture For Discrete Series And Stable Trace Formula

We will prove the Kottwitz multiplicity conjecture for discrete series. When the test function is in the Schwartz space, we will directly stabilize the spectral side of the local trace formula in the Archimedean case. In particular, we will construct the explicit form of the spectral side of stable local trace formula in the archimedean case, when one component of the test function is cuspidal.This paper has two goals. One is to solve the Kottwitz multiplicity conjecture, and the other is to directly stabilize the spectral side of the local trace formula. In my work, stable trace formula is the main tool to solve the Kottwitz conjecture. In this paper, we will use both global and local stable trace formulas. The stabilization of the general trace formula is in Arthur’s work [A9],[A10],[A11], which assumed the Fundamental Lemma. In2008, Ngo solved this difficult problem in [N]. So we have the complete stabilization of invariant global trace formula. The main method of the stabilization of global trace formula is induction. The terms of the stable trace formula are not explicit. The geometric side of the stable local trace formula is relatively explicit. In particular, in the p-adic case, if the test function is cuspidal, Arthur gave a concrete formula in [A7], which is just an inner product. However, as in the global case, the spectral side of stable local trace formula is not explicit. We will directly stabilize the spectral side of the local trace formula in the Archimedean case, where many things have to be done. Langlands classified the irreducible representations of real algebraic groups in [L2]. Shelstad classified the tempered representations, directly constructed the spectral transfer factors, and gave the inverse adjoint relations in [S3],[S4]. Those are the foundation for the present work. The Kottwitz multiplicity conjecture (see [SP]) is a classical problem in automorphic representation theory. The conjecture is about computing the multiplicity of discrete series representation πR of G(R) on L2(Γ\G), where Γ is a discrete subgroup of G, and G is a K-group over Q. Many cases were solved, The simplest case is when G is anisotropic over Q, then G(Q)\G(A) is compact. Langlands gave a formula for mdisc(πR,K0) in [L1]. The first result for noncompact quotient for G=SL(2), appeared in Selberg’s paper [S]. More generally, if G has R-rank one, there is a formula for mdisc(πR,K0) in [OW]. If G is of general R-rank, Arthur proved a formula for the sum of multiplicities Σπ∈Πdisc(μ)mdisc(πR,K0) in [A3]. A formula for each single multiplicity mdisc(πR, K0) was conjectured by Kottwitz in [K4]. Steven Spallone checked two special cases for this conjecture in [Sp]. We will solve this conjecture completely, and extend this conjecture to more general case, where the reductive group is a K-group.In chapter1, I will introduce the K-group, which is union of the disconnected reductive group. When F is an Archimedean field, the K-group is not connected, and in this case, K-group is a good object for stabilizing the trace formula. Any connected group G1is a component of an unique K-group G. We will give an example for understanding the K-group, where the invariant and stable distribution can be extended to K-group as in [A8].In chapter2, we obtain the relation between multiplicity and invariant trace for-mula. But when the test function is a pseudo-coefficient, we can not give a concrete invariant trace formula, since in this case, the cuspidal part is not stable. So we need to stabilize the invariant trace formula to overcome this obstacle. In general, the invariant trace formula is the identity obtained from two different expansions of a certain linear form I(f). One expansion is the geometric expansion, which is a linear combination of distributions parametrized by conjugacy classes γ in Levi subgroups M(Fs). The other expansion is the spectral expansion which is a linear combination of distributions parametrized by representations π of Levi subgroup M(FS). Here f∈H(G, V) is in the Hecke algebra of G(FV)(see [A1],[A2]). Arthur has stabilized the invariant trace formula in [A11]. So we can expand the multiplicity by using the geometric side of stable global trace formula in chapter3. We can use the splitting formula to reduce the local component of global trace formula to Archimedean case. Of course, compared to the local trace formula, the local component of global trace formula is more complicated. Because it contains the contribution of distribution of unipotent element. But when the test function is stable cuspidal, the distribution coming from the unipotent elements vanishes, and then the local trace formula and the local component of global trace formula are the same. Moreover, the pseudo-coefficient of representation can be transfered to stable cuspidal function under the Shelstad transfer mapping. So it is enough for us to study the stabilization of local trace formula. In general, the geometric side of local trace formula concerns the semisimple regular elements, and the spectral side of the local trace formula concerns the tempered representations.The spectral side of the invariant local trace formula contains a natural object, which is the virtual character. A simple invariant local trace formula can be given by the virtual character in [A5]. Therefore, in chapter4, we have to introduce the virtual character in the Archimedean case, and define the transfer factors△(τ,φ) and△(φ,τ), which includes Shelstad’s work in [S3]. Then we can stabilize the spectral side of the local trace formula. When one of the test function is cuspidal, we just need to consider the elliptic representations. In chapter5, we obtain a formula about the spectral side of local trace formulaIn chapter6, we will directly stabilize the spectral side of the local trace formula in general case, which combines Arthur’s work in [All]. In chapter7, we can obtain the main term SMG(δ,φ). And we need to stabilize the Weyl integral formula, which connects the geometric side and spectral side of the local trace formula. We can then compare the stable local trace formula with stable Weyl integral formula, from where the main term of the formula will appear.In chapter8, we establish the relation between SMG(δ,f) and the invariant main term ΦM(r,f),then we can overcome the key obstacle. We have SMG(δ,fφμ)=0,if δ is not semisimple, where the fφμ is a stable cuspidal function which is transfered by a pseudo-coefficient of πμ. We shall collect various terms, and they will be combined as one of our main formula in Theorem8.0.18.