In this part, we partially specialize the harmonic analysis developed in Part two to the locally compact group GL2(Fv)for a local fieldFv (short: local field), i.e., either [129]
• the fieldRof real numbers,
• the fieldCof complex numbers,
• a non-archimedean fieldFv, either
– a finite extension of the fieldQp of thep-adic rationals, or
– a finite extension of the Laurent series in one-variable over a finite field.
The treatment is to a large extent kept independent from the previous parts, despite the overlap.
We stressed the common ground for the theory in Part I. We will not stress the similarities between the harmonic analysis of the different groups GL2(R), GL2(C) and GL2(Fv)from this point forward, but it should be self-explanatory from the organization and enumeration of the material.
For GL2(C), I have omitted a full treatment. I focus instead on bi-invariant harmonic analysis. This restricts the spectral analysis to automorphic forms with complex constituents which are unramified principal series representations. There are no classical references for integral identities. The difficulties to be overcome have a purely special function theoretic origin. Although this harmonic analysis seems less interesting, because of the absence of discrete series representations, its omission is a remaining gap in a complete treatment of the GL(2) trace formula from a computational point of view.
The main goal of this part is to show the computation of all local distributions and constants, which appear in the Arthur trace formula as given in[65, page 271ff.] for a global field on a special set of test functions.
We will not describe the distributions explicitly here, but rather address them by their names only. The distributions occuring in the Arthur trace formula are
(1) the spectral distributions: the character distribution of all irreducible unitary representations of GL2(Fv)[45, page 244(7.16)]. These are
• the one-dimensional representations,
• the continuous series representations,
• the complementary series representations,
• the discrete series representations for Fv=R,
• the supercuspidal representations and Steinberg representations for Fv non-archimedean,
(2) the Eisenstein spectrum (Section 1.10),3
3Some of the values have to be computed globally.
95
(3) the Eisenstein residues (Section 1.11),4
(4) the identity distribution: this is a specialization of the Plancherel formula (Section 1.5),
(5) the parabolic distribution: the value of the local zeta integral and its derivative ats= 1(Section 1.7),
(6) the hyperbolic distribution: the orbital integral and the weighted orbital integral of a hyperbolic element (Section 1.8),
(7) the elliptic distribution: the orbital integral of an elliptic element ifFv 6=C (Section 1.9).
Note that only [65] treats the function field case, whereas the exposition in [45]
and [44] works in the algebraic number field setting only. The differences between the function field setting and the number field setting are minor. At the non-archimedean places, the local harmonic analysis only depends mildly upon the residue characteristic.
By a special set of test functions, we mean a subset of C∞c (G), which has the following two properties:
• only “very few” character distributions do not vanish
• the test functions are able to determine/distinguish automorphic represen-tations up to their factorization into local factors
Let us be precise. Consider the maximal compact subgroup K of GL2(Fv), i.e., either U(2), O(2) or GL2(ov) depending on whetherFv is complex, real, or non-archimedean. DefineK as the product ofKand the center Z(Fv)of GL(Fv). For a central unitary one-dimensional representationχ:Z(Fv)→C1, define
C∞c (GL2(Fv), χ) =n
φ:GL2(Fv)→Csmooth, compactly supported modulo Z(k) φ(zg) =χ(z)φ(g)for allz∈Z(Fv)o
.
Note that the irreducible representations ofKare unitarizable and finite-dimensional.
Definition(Distinction and separation).
• (Parametrization) We say that two infinite-dimensional, irreducible, unitary representations of GL2(Fv)areK-equivalent if they are isomorphic asK representations.
• (Distinction property) We say that a non-zero functionφ∈C∞c (GL2(Fv)) is a pseudo-coefficient of aK-equivalence class{π}if the character distribu-tion vanishes onφfor all unitary infinite-dimensional representations5but those from theK-equivalence class{π}. We say thatφis{π}-distinguishing element.6
• (Separation property) A{π}-distinguishing elementφseparates two isomor-phism classes of representationsπ1, π2∈ {π}if the character distributions differ, i.e.,
π1(φ)6=π2(φ).
4These have to be computed globally and are only treated in a global context.
5We have to exclude the one-dimensional representations here.
6Note that the Plancherel formula implies that for non-zero elements in C∞c (G), not all character distributions can vanish simultaneously.
PART III — LOCAL HARMONIC ANALYSIS ON GL(2) 97
We say that a subset X of the {π}-distinguishing elements in C∞c (G) separates {π} if for any two representation π1, π2 ∈ {π} there exists an elementf ∈X which separatesπ1 andπ2.
The parametrization via irreducible representations ofKis necessary and crucial for the classification of all unitary representations. The distinction property allows to specialize the trace formula to a more explicit form, such as the classical Selberg trace formula for Maass forms onΓ\H, and the Eichler-Selberg trace formula for the Hecke eigenvalues of modular forms. Actually, our specialization will be an improvement over these classical formulas, in the sense that our trace formula analyzes only one fixedK-equivalence class at every place. The separation property guarantees that the full power of the Arthur trace formula is exhausted and no information is given away.
There are three classes ofK-equivalence classes:
• The irreducible parabolic inductions include the continuous series represen-tations and the complementary series represenrepresen-tations. The construction is fairly easy, since a representationρofK will always exist which does not occur in any otherKequivalence class. Unfortunately, theseK-equivalence classes are large and have uncountably many elements. If we generalize the definition of a Hecke algebra H(G, ρ) to open subgroups K which are only compact modulo the center, then elements ofH(G, ρ)separate and distinguish thisK-equivalence class up to twist by one-dimensional characters.
• The irreducible, infinite-dimensional subquotients of the parabolic induc-tions include the discrete series representainduc-tions, and the Steinberg rep-resentations. TheK-equivalence class contains only twoG-isomorphism classes of representations, so the separation axiom is easy to satisfy. For the construction, one has to carefully study the Abel transform and ar-range suitable linear combinations of elementsφρ∈ H(G, ρ)for different irreducible representationsρofK.
• For the supercuspidal representations, we have to appeal to their classifica-tion. There will always exist a representationρofKwhich does not occur in any otherK-equivalence class of unitary representations. Additionally, theK-equivalence class contains either one or two elements, depending on whether the supercuspidal representation is associated to a ramified or unramified quadratic extension. In the first case, everything works as intended and the functionx7→trρ(x)is essentially the only option for the distinguishing function. In the second case, we have to switch to another open subgroup, which is compact modulo the center. This subgroup is the normalizer of the Iwahori subgroupΓ0(p)inside GL2(Fv). The method is the same – only in terms of this latter subgroup.
The aboveK-equivalence relation is weaker than the “local analogue” of the similarity classes. However, the only distinction is that ramified supercucpidal representation contain not one but two isomorphism classes in aK-equivalence class. The Seperation property ensures that we can seperate the two isomorphism classes, and construct pseudo coefficients in this way as well.
The results, to be quoted in this part, rely more or less directly on methods involving the use of the Lie algebra, root systems, or the Bruhat-Tits building.
These concepts are modern and powerful but require much notation. I have not
introduced these concepts here, because for the group GL(2), they seemed like overkill.
There is one technical difference between this part and the harmonic anal-ysis in the preceding part. Here, we will work with C∞c (GL2(Fv), χ) instead of C∞c (GL2(Fv)), i.e., modulo the center. This is a minor technical modification, and it is rather easy to translate results between the two settings. If the one-dimensional representation χ of Z(Fv) and the irreducible representations of K coincide on K∩Z(Fv), then everything can be generalized directly via the surjective algebra homorphism
C∞c (GL2(Fv))C∞c (GL2(Fv), χ), φ(g) = Z
Z(Fv)
χ(z)f(z)dz.
Observe that as a consequence of Schur’s Lemma, the restriction of a unitary irreducible representation to its center is a one-dimensional representation. Its character distribution on C∞c (G)factors through C∞c (GL2(Fv), χ). There is no loss of generality here.
An alternative route suggests working with GL2(Fv)1 = {g ∈ GL2(Fv) :
|detg|v= 1}instead of GL2(Fv), since the former group has a compact center. The differences in the representation theory and the harmonic analysis of GL2(Fv)and GL2(Fv)1 are not crucial, and one can translate between the settings rather easily.
This route is preferred in the papers of Arthur, cf. [5].
Note that naively switching to the group PGL2(Fv)instead of GL2(Fv)is not always possible. The representation theory of PGL2(Fv) is not as rich as that of GL2(Fv). Also understanding the representations of GL2(Fv) from those of SL2(Fv) is in general non-trivial, in particular, it is hard for the supercuspidal representations if the residue characteristic of Fv is two. I have avoided every approach which requires a case-by-case analysis depending on the characteristic or residue characteristic.
CHAPTER 7