The Trace Formula and the Proof of the Global Jacquet-Langlands Correspondence

Abstract

This paper contains the material covered in the lectures I gave at the doctoral school Introduction to Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms held at the CIRM, Luminy, 16–20 May 2016.

Appendix: The Spectral Simplification

In this Appendix I recall the proof [Fl2] of the classical Lemma which allows a simplification of the equality between the spectral sides of the trace formulae. It is a classical step in the proof of correspondences, and has been used in this paper. At the end of the Appendix, I explain how to adapt Flath’s proof in [Fl2] to quasi-split reductive groups (instead of general linear groups).

Let V  be a countable set. For almost all v ∈ V  means for all v ∈ V  but a finite set. Let {F _v}_{v ∈ V} be a set of non-Archimedean local fields. For every v ∈ V , let O _v be the ring of integers of F _v. Let G _v := GL _n(F _v) and K _v := GL _n(O _v). On G _v we fix the Haar measure giving volume 1 to the K _v. Let G be the restricted product of the G _v, v ∈ V , with respect to K _v, v ∈ V .

Let ${\mathcal H}_v^0$ be the set of functions $f:G_v\to {\mathbb C}$ with compact support and bi-invariant by K _v. Let ${\mathcal H}^0$ be the space spanned by functions on G denoted by f = ⊗_{v ∈ V}f _v with $f_v\in {\mathcal H}_v^0$ for all v ∈ V  and f _v is the characteristic function $1_{K_v}$ of K _v for almost all v ∈ V , and defined by f((g _v)_{v ∈ V})  =  ∏_{v ∈ V}f _v(g _v). (${\mathcal H}^0$ is the restricted product of the algebras ${\mathcal H}_v$ with respect to the $1_{K_v}$, where $1_{K_v}$ is the characteristic function of K _v, like in Sect. 4.2.5.)

Let J be a countable set. For each j ∈ J, for each v ∈ V , let π _j,v be a unitary unramified smooth irreducible representation of G _v.

Assume that, for every couple j ₁, j ₂ ∈ J such that j ₁ ≠ j ₂, there is at least one v ∈ V  such that $\pi _{j_1,v}$ and $\pi _{j_2,v}$ are not isomorphic.

Lemma 1

Suppose given complex numbers c _j, j ∈ J, such that for every $f=\otimes _{v\in V} f_v \in {\mathcal H}^0$ , the series

$$\displaystyle \begin{aligned} \sum_{j\in J}\ c_j\prod_{v\in V}\mathrm{tr}\pi_{j,v}(f_v) \end{aligned} $$

(4.6)

converges absolutely to zero ( in ${\mathbb C}$) . Then all the c _j are zero.

Proof

Let F be a local non-Archimedean field and let | | be the normalized norm of F. Every unramified smooth irreducible representation of GL _n(F) is a sub-quotient of a parabolic induced representation from a character $|\ ^.\ |{ }^{z_1}\otimes |\ ^.\ |{ }^{z_2}\otimes \ldots \otimes |\ ^.\ |{ }^{z_n}$, ($z_i\in {\mathbb C}$) of the diagonal torus with respect to the parabolic subgroup of upper triangular matrices. Conversely, every such induced representation has only one unramified sub-quotient. See [Car]. Then the set of isomorphism classes of unramified smooth irreducible representations of GL _n(F) is in bijection with ${\mathbb C}^n$ modulo the equivalence relations R ₁ and R ₂ defined by

• $(z_i)_{{1\leq i\leq n}}\cong _{R_1} (z^{\prime }_i)_{{1\leq i\leq n}}$ if for all i ∈{1, 2, …, n}, $z_i- z^{\prime }_i\in \frac {2\pi }{\ln |u_F|} i{\mathbb Z}$, where u _F is a prime element of F, and

• $(z_i)_{{1\leq i\leq n}}\cong _{R_2} (z^{\prime }_i)_{{1\leq i\leq n}}$ if $(z^{\prime }_i)_{{1\leq i\leq n}}$ is obtained from (z _i)_1≤i≤n by a permutation of the components.

Using Tadić classification of unitary representations [Ta2] one may show that if such an unramified representation is unitary, then, for all i, $|Re(z_i)| \leq \frac {n}{2}$, and the set with repetitions {Re(z _i)}_1≤i≤n is symmetrical with respect to zero. This last condition of symmetry is sufficient for the unramified representation to be hermitian (i.e. isomorphic to the conjugate of its contragredient).

On the set $\mathcal Y$ of n-tuples $(z_1,z_2,\ldots ,z_n)\in {\mathbb C}^n$ such that $|Re(z_i)| \leq \frac {n}{2}$ and the set with repetitions {Re(z _i)}_1≤i≤n is symmetrical with respect to zero put the induced topology by restriction from ${\mathbb C}^n$. The quotient $\mathcal Y'$ of this space by the equivalence relation R ₁ is compact. Put the induced topology on $\mathcal Y''$ which is the quotient of $\mathcal Y'$ by the equivalence relation R ₂. The set $\mathcal Y''$ is in natural bijection with the set $\mathcal W$ of isomorphism classes of unramified representations defined by elements in $\mathcal Y$. Consider the induced topology on $\mathcal W$. Then $\mathcal W$ is compact. Every class in $\mathcal W$ is unramified and hermitian; every irreducible unramified unitary class of representations of GL _n(F) is in $\mathcal W$.

For each v ∈ V , denote ${\mathcal W}_v$ the analog space of $\mathcal W$ for the group G _v = GL _n(F _v). Let $R:=\prod _{v\in V}{\mathcal W}_v$ be the product space, which is again a compact space. The points of R are collections π = (π _v)_{v ∈ V} that have the property that every π _v is an (isomorphism class of) irreducible hermitian unramified representation(s) of G _v. Not every collection with this property is a point of R, but any collection π = (π _v)_{v ∈ V} such that every π _v is an (isomorphism class of) irreducible unitary unramified representation(s) of G _v is a point of R. For every $f\in {\mathcal H}^0$, define $F_f:R\to {\mathbb C}$ by the formula F _f(π) =∏_{v ∈ V}trπ _v(f _v). The set of all these functions verifies the Stone-Weierstrass conditions:

• it contains the constant functions (consider f equal to a scalar multiple of the characteristic function of ∏_{v ∈ V}K _v);

• it is stable by complex conjugation because, if $f=\otimes _{v\in V}f_v\in {\mathcal H}^0$, if f ^∗(g) := f(g ⁻¹), then $F_{f^*}=\overline {F_f}$ (indeed, if π _v is hermitian, then $\mathrm {tr}\pi _{v}(f_v^*)=\overline {\mathrm {tr}\pi _{v}(f_v)}$);

• it separates points, because if $\pi _{j_1,v}$ is not isomorphic to $\pi _{j_2,v}$, then there is a function in ${\mathcal H}_v^0$ such that $\mathrm {tr}\pi _{j_1,v}(f)\neq \mathrm {tr}\pi _{j_2,v}(f)$ (particular case of [Be2, Corollary 3.9 for example]).

Hence, the set of all functions F _f, $f\in {\mathcal H}^0$, is a dense subset of the set C(R) of continuous functions on R. Now, putting f = 1 in formula (4.6), the absolute convergence implies that ∑_J|c _j| converges. Then choose u in J such that |c _u| is maximal. Suppose |c _u|≠ 0. Choose a finite subset J ₀ of J such that $\sum _{J\backslash J_0} |c_j|<\frac {|c_u|}{4}$. The density result implies that we may find a function f such that

• |trπ _j(f)| < 2 for all j ∈ J,

• |trπ _u(f)| > 1 and

• $|\mathrm {tr}\pi (f)|<\frac {|c_u|}{2|J_0|}$ for all π ∈ J ₀∖{u}.

Then

$$\displaystyle \begin{aligned}|\sum_{j\in J\backslash\{u\}}c_j\mathrm{tr}\pi_j(f)|\leq \sum_{j\in J\backslash\{u\}}|c_j||\mathrm{tr}\pi_j(f)|<\end{aligned}$$

$$\displaystyle \begin{aligned}<(|J_0|-1) \frac{|c_u|}{2|J_0|} +2\frac{|c_u|}{4}<|c_u|<|c_u\mathrm{tr}\pi_u(f)|\end{aligned}$$

which contradicts the hypothesis. Hence |c _u| must be 0. □

Here is the generalization of this technique for other groups. Let G be a connected reductive quasi-split group over a local non archimedean field, let A be a maximal split torus in G, W the Weyl group of A, M the centralizer of A (so W is the quotient of the normalizer of A by M). Then M is a minimal Levi subgroup of G. Let P be a parabolic subgroup of G with Levi subgroup L. Let K be a maximal compact subgroup of G in good position with respect to P. Assume that the Hecke algebra ${\mathcal H}^0$ of left and right K-invariant functions over G with compact support is commutative. I will define a compact set ${\mathcal W}$ with the same properties as before, namely ${\mathcal W}$ parametrizes a set of irreducible hermitian unramified representations of G containing all the irreducible unitary unramified representations of G. It is known that the set X of unramified characters of M (here unramified means trivial on M ∩ K) form a complex variety endowed with the obvious action of W.

According to Cartier’s paper [Car] (I used his notation), for every χ ∈ X, the induced representation $\mathrm {ind}_P^G\chi$ has a unique irreducible quotient which is an unramified representation of G, denoted by I(χ). Every irreducible unramified representation of G is obtained this way. If χ, χ′∈ X, then I(χ) and I(χ′) are isomorphic if and only if there is w ∈ W such that χ′ = wχ. So the set of irreducible unramified representations of G is parametrized by X∕W.

Let X _u be the set of characters χ ∈ X such that $\mathrm {ind}_P^G\chi$ contains at least one irreducible quotient which has bounded coefficients (a unitary representation has bounded matrix coefficients). Tadić shows, [Ta3, Theorem 7.1], that the set X _u is relatively compact in X. It is obvious that X _u is stable by W. So the closure $\overline {X_u}$ is compact and stable by W.

A representation is hermitian if it is isomorphic to the complex conjugated of its contragredient. The contragredient representation of a character χ is $\overline {\chi (g^{-1})}$. We also know that the complex conjugation and the contragredient functor commute with parabolic induction. So using the classification, it follows that I(χ) is hermitian if and only if there is w ∈ W such that $\overline {\chi (g^{-1})}=w\chi (g)$. If w ∈ W, then the set Y _w of $\chi \in \overline {X_u}$ such that $\overline {\chi (g^{-1})}=w\chi (g)$ is closed, so it is a compact set. W is finite, so Y := ∪_{w ∈ W}Y _w is compact. The set ${\mathcal W}:=Y/W$ has the required properties.