Introduction

In the 1980s Langlands and Shelstad proposed two conjectures which are central to the trace theoretic approach to the Langlands program (see [6, 7]). These conjectures have motivated a great deal of mathematics, and much mathematics has been constructed upon the assumption that they are valid. The deep mathematics in this book belongs largely to the first category, and it represents a giant step towards understanding the basic ideas behind the conjectures.

In the 1990s Waldspurger reduced these two conjectures to a single conjecture on the Lie algebra (see [11, 13]). This conjecture, which we call the fundamental lemma for the Lie algebra, makes a very precise prediction about how certain combinations of regular semisimple orbital integrals on the Lie algebras of a group and (any of) its unramified endoscopic groups ought to be related. Thanks to an unpublished manuscript of Waldspurger [12], the verification of these relations can be reduced to questions about stable distributions supported on the nilpotent set and the image of such distributions under endoscopic induction. As his title suggests, Waldspurger has two main accomplishments along these lines. First, for a certain class of groups, he explicitly finds a basis for the stable distributions supported on the nilpotent set (and is therefore able to verify a conjecture of Assem and Kottwitz [1]). Second, for these same groups, he determines what the image under endoscopic induction of the stable distributions supported on the nilpotent set of an elliptic endoscopic group ought to be.

In order to accurately discuss the specific results of Waldspurger’s book, it would be necessary to introduce an enormous amount of notation. Consequently, instead of reporting on the contents of Waldspurger’s book in complete and unfailing detail, I will spend the bulk of this review attempting to convey the feel of the book. Note that Waldspurger introduces most of this notation to list (and therefore keep track of) various important data – for example, nilpotent orbits and conjugacy classes of unramified maximal tori. These lists are interesting in their own right.

Finally, despite Waldspurger’s warning, “Soyons réalistes. Dans nos 450 pages de calculs, il serait bien étonnant qu’il ne se soit pas glissé quelques erreurs. Il est plus prudent de considérer les formules ci-dessus comme des approximations des formules exactes.”, it is likely that there are more errors in this review than in his entire book. Notwithstanding the fact that there are only 449 pages in the book, it is, as usual for Waldspurger, extremely carefully written.

Stability

We begin by introducing the idea of a stable distribution on the Lie algebra. For many problems in harmonic analysis, this is a more natural notion than simple invariance.

Let F denote a finite extension of ℚ_{p}, the p-adic completion of the field of rational numbers, and let F̄ denote an algebraic closure of F. Let G denote a connected reductive group defined over F. Let G denote the group of F -rational points of G and let 𝔤 denote the Lie algebra of G. In practice G will be a classical unramified group, and there will be some restrictions on p.

We begin by considering D, the space of invariant distributions on 𝔤. Since G acts on 𝔤, the space C_{c}^{∞}(𝔤) of locally constant, compactly supported, complex-valued functions on 𝔤 carries an action of G. Therefore, G acts on the space of distributions on 𝔤 (that is, the linear dual of C_{c}^{∞}(𝔤)). A distribution on 𝔤 belongs to D provided that it is fixed with respect to this action.

Suppose X belongs to 𝔤. The associated orbit carries an invariant measure (unique up to a nonzero constant). A fundamental result of Deligne and Ranga-Rao [10] says that integrating against this measure defines an element of D. We denote this element of D by O_{X} and call it an orbital integral.

Harish-Chandra showed that the elements of D could be characterized as follows [3]. If C_{c}^{∞}(𝔤)^{an} denotes all those elements of C_{c}^{∞}(𝔤) whose regular semisimple orbital integrals are zero, then a distribution on 𝔤 belongs to D if and only if it vanishes on C_{c}^{∞}(𝔤)^{an}. In other words, the regular semisimple orbital integrals are dense in D. The fact that nilpotent orbital integrals vanish on C_{c}^{∞}(𝔤)^{an} lies at the heart of Harish-Chandra’s proof.

There does not seem to be a direct way to describe the space D^{st} of stable distributions. Instead, in analogy with Harish-Chandra’s characterization of D, we define D^{st} to be the closure of the span of the stable regular semisimple orbital integrals. We recall that, for a regular semisimple element X of 𝔤, the intersection of 𝔤 with the G(F̄) orbit of X can be written as a finite union of regular semisimple orbits. After choosing measures on each of these orbits in a compatible way, we define the stable orbital integral SO_{X} associated to X to be the sum of the corresponding orbital integrals. If C_{c}^{∞}(𝔤)^{anst} denotes all those elements of C_{c}^{∞}(𝔤) whose (regular semisimple) stable orbital integrals are zero, then a distribution on 𝔤 belongs to D^{st} provided that it vanishes on C_{c}^{∞}(𝔤)^{anst}.

Note that every (regular semisimple) stable orbital integral belongs to D^{st}. However, it is extremely difficult to determine what other distributions are stable. In particular, it is not obvious which combinations of nilpotent orbital integrals are stable. Describing these combinations (for certain groups) is one of the major achievements of this book.

Homogeneity

One of the more successful tools in the p-adic arsenal is the finiteness result of Harish-Chandra and Howe (usually referred to as Howe’s conjecture (for the Lie algebra) – note that while the fundamental lemma is a conjecture, Howe’s conjecture is a theorem [3, 4]). Howe’s conjecture states that given a compactly generated, closed, invariant subset ω of 𝔤, there exists a lattice L in 𝔤 so that

res_{C_{c}(𝔤/L)} D(ω) = res_{C_{c}(𝔤/L)} D(N).

Here D(N) denotes the span of the nilpotent orbital integrals, D(ω) denotes the subspace of distributions in D with support in ω, and if T ∈ D, then res_{C_{c}(𝔤/L)} T denotes the restriction of T to those functions in C_{c}^{∞}(𝔤) which are invariant under translation by elements of L. Although Howe’s conjecture is an example of a homogeneity result, we reserve the word homogeneity for “optimal” versions of Howe’s conjecture and its relatives. We take up this discussion in a moment.

Here is one example of how Waldspurger uses this finiteness result (Lemme IV.15). Suppose L is a lattice in 𝔤. Waldspurger shows that if T ∈ D^{st} and D is a linear combination of nilpotent orbital integrals satisfying

res_{C_{c}(𝔤/L)} T = res_{C_{c}(𝔤/L)} D,

then D is stable. (Moreover, if one looks at the summand of D involving orbits of a given fixed dimension, then this is also a stable distribution.) Thus, in order to explicitly describe the space D^{st}(N) := D^{st} ∩ D(N), one is led to consider “optimal” versions of Howe’s conjecture – that is, versions where one has a firm understanding of the left-hand side of the equality in Howe’s conjecture.

The homogeneity result Waldspurger uses in this book can be described as follows (he proved this result in [12]). Let 𝔤_{ent} denote the set of elements in 𝔤 which are integral. (For example, an element of M_{n}(F) is integral provided that each of its eigenvalues has norm less than or equal to 1.) To a facet F in the Bruhat-Tits building of G we can associate a parahoric subgroup G_{F} and its “Lie algebra” 𝔤_{F}. If F is an alcove (i.e., affine chamber), then G_{F} is an Iwahori subgroup.

Define

H := ∑_{C} C_{c}(𝔤/𝔤_{C}).

Here the sum is over all alcoves in the Bruhat-Tits building of G. Note that H is an invariant version of the Lie algebra analogue of the Iwahori-Hecke algebra. We then have the following homogeneity result

res_{H} D(𝔤_{ent}) = res_{H} D(N).

Thus, in order to describe elements in D^{st}(N), we can start by looking at stable distributions in D(𝔤_{ent}). (Recall that the restriction of the nilpotent orbital integrals to H forms a basis for res_{H} D(N) [3].)

This result is particularly useful because we have a very good understanding of where we can find a dual basis for res_{H} D(𝔤_{ent}). Namely, fix an alcove C in the Bruhat-Tits building of G. Define

H_{0} = ∑_{x≤C} C(𝔤_{x}/𝔤_{C}).

Here the sum is over the vertices contained in the closure of C and C(𝔤_{x}/𝔤_{C}) denotes the subspace of H∩C_{c}^{∞}(𝔤/𝔤_{C}) consisting of functions with support in 𝔤_{x}. In [12], Waldspurger proves that H_{0} contains a dual basis for res_{H} D(𝔤_{ent}) – this naturally leads to questions about finite groups of Lie type.

Producing a basis for res_{H} D(𝔤_{ent})

Recall that if x is a vertex in the Bruhat-Tits building with x ≤ C, then we have

G_{x}^{+} ⊂ G_{C}^{+} ⊂ G_{C} ⊂ G_{x}

where G_{x}^{+} (resp., G_{C}^{+}) is the pro-unipotent radical of the parahoric G_{x} (resp., G_{C}). We have that G_{x}/G_{x}^{+} is a finite group of Lie type, and G_{C}/G_{x}^{+} is a Borel subgroup of G_{x}/G_{x}^{+} with unipotent radical G_{C}^{+}/G_{x}^{+} and Levi component isomorphic to G_{C}/G_{C}^{+}. Similar statements apply to the lattices 𝔤_{x}^{+}, 𝔤_{C}, etc. By making appropriate choices, we can define the Fourier transform on 𝔤 so that, for all vertices x, the Fourier transform induces a (compatible) Fourier transform on 𝔤_{x}/𝔤_{x}^{+}, the Lie algebra of G_{x}/G_{x}^{+}.

With respect to these choices, the Fourier transform of H is the subspace C_{c}^{∞}(𝔤_{tn}) of C_{c}^{∞}(𝔤) consisting of functions supported on 𝔤_{tn}, the set of topologically nilpotent elements in 𝔤. (For example, an element of M_{n}(F) is topologically nilpotent provided that its eigenvalues have positive valuation.) We can therefore restate the fact that H_{0} contains a dual basis for res_{H} D(𝔤_{ent}) as follows. We note that T ∈ D(𝔤_{ent}) vanishes on H_{0} if and only if for all vertices x ≤ C and for all nilpotent X ∈ 𝔤_{x} we have

T̂([X + 𝔤_{x}^{+}]) = 0.

Here T̂ is the Fourier transform of T and [X + 𝔤_{x}^{+}] denotes the characteristic function of the coset X + 𝔤_{x}^{+}.

Consequently, to produce a nice dual basis, we should look for functions on the Lie algebra 𝔤_{x}/𝔤_{x}^{+} which separate nilpotent orbits and have nice properties with respect to the Fourier transform. Thankfully, the (Lie algebra version of the) generalized Green functions [9] fit this bill nicely. We recall that Kazhdan [5] showed that the “usual” Green functions (which we obtain by restricting the Deligne-Lusztig character R_{T}^{G}(1) to the unipotent set) have the property that they are expressible as the restriction to the nilpotent set of the Fourier transform of an orbital integral of an element of Lie(T). Waldspurger generalizes this result, in a suitable way, to all generalized Green functions (Proposition II.8, see also [8]). For those Green functions which are not “usual” he provides a detailed study of the their associated Lusztig functions 0f (see Chapters II and V), including an explicit calculation of the scalar by which ^{0}f and its Fourier transform differ (Proposition V.8). By inflating these generalized Green functions to 𝔤 and appropriately pruning the resulting set of functions, he produces a set of functions Q_{θ},with θ belonging to an indexing set of cardinality equal to the number of nilpotent orbits in 𝔤. The Fourier transforms of these functions lie in H_{0} and form a dual basis for res_{H} D(𝔤_{ent}).

At the same time that he produces the functions Q_{θ}, Waldspurger also produces a basis φ_{θ}^{H} for res_{H} D(𝔤_{ent}). For example, for those θ for which Q_{θ} is the inflation of the Lie algebra analogue of a usual Green function (i.e., R_{T}^{G}(1)), Waldspurger takes φ_{θ}^{H} to be O_{X} where X is any regular semisimple element of 𝔤 which is a lift of a regular element of Lie(T). (It turns out that φ_{θ}^{H} is independent of this choice of a lift.) As the notation implies, the basis φ_{θ}^{H} and dual basis Q̂_{θ} are linked in a strong way (Proposition III.8).

Since the only distributions about which stability questions are readily answerable are those which involve regular semisimple orbital integrals, Waldspurger proceeds to replace the basis φ_{θ}^{H} with a wellchosen basis of this type. He denotes the elements of this new basis by D_{θ}[Γ, X_{T}]. Carrying out this program occupies a good part of the book, but, in the end, Waldspurger is able to prove (Proposition IV.3) that φ_{θ}^{H} = res_{H}(D_{θ}[Γ, X_{T}]) for all θ.

Stable combinations of nilpotent orbits and endoscopy

In order to find a basis for res_{H} D^{st}, Waldspurger first forms a new basis for res_{H} D by looking at well-chosen combinations of the D_{θ}[Γ, X_{T}]. Some of the elements of this new basis (denoted φ_{ξ,κ̄}) are seen to be stable and the remainder are not stable. In a very clever fashion (Théorème IV.13), Waldspurger shows that if T ∈ D^{st} such that the restriction of T to H can be written as a combination of the φ_{ξ,κ̄} which are not stable, then res_{H} T = 0. The proof relies in an essential way on Waldspurger’s result that the Fourier transform of a stable distribution is stable [11].

We now have a basis for res_{H} D^{st} ∩ res_{H} D(𝔤_{ent}) and we know that for every T ∈ D(𝔤_{ent}) there exists a unique D ∈ D(N) such that res_{H} T = res_{H} D. In principle, we therefore have enough information to write down a basis for D^{st}(N). However, before proceeding to do so, Waldspurger first recombines the stable φ_{ξ,κ̄} (see previous paragraph) to form a new basis for res_{H} D^{st} ∩ res_{H} D(𝔤_{ent}). This new basis has nice properties which Waldspurger derives via the generalized Springer correspondence. Namely, to each element in this new basis we can associate a number d such that when we write the combination of nilpotent orbital integrals corresponding to this basis element (this combination will automatically belong to D^{st}(N) – see the homogeneity section above) as ∑ c_{N}O_{N} with N ranging over N /G, we have (1) if c_{N} ≠ 0, then N belongs to a nilpotent orbit of dimension less than or equal to d, and (2) if the orbit of N has dimension d, then c_{N} is (explicitly) determined by the basis element (and these coefficients uniquely determine the basis element). These results also use arguments similar to those found in a paper of Barbasch and Moy [2]. In the end, we have a basis {ρ_{ι}} for D^{st}(N), and each ρ_{ι} is a combination of nilpotent orbital integrals for which the corresponding orbits all have the same dimension.

The remainder of the book describes what ought to be the image (in D(N)) under endoscopy of a stable distribution supported on N_{𝔥}, the set of nilpotent elements in the Lie algebra 𝔥 of an elliptic endoscopic group H. First, in this setting, Waldspurger explicitly calculates the transfer factors. Let H_{𝔥} denote the H-analogue of H. From the previous paragraph, we have a set of explicit combinations of stable (regular semisimple) orbital integrals on 𝔥 which has nice properties with respect to an explicit basis for D^{st}(N_{𝔥}). Pick one such basis element and consider the associated combination of stable orbital integrals. Waldspurger explicitly calculates the (conjectural) image of this combination of stable orbital integrals under endoscopic induction, and then relates this image to a combination of more familiar objects with which the image agrees when restricted to H. The restriction of the resulting distribution to H can be written as ∑ c_{N}O_{N} with the sum over N /G. In analogy with the situation in the above paragraph, the coefficients c_{N} are explicitly described for those N for which the orbit of N is maximal with respect to the property c_{N} ≠ 0. By using an argument similar to that which proved Lemme IV.5 (discussed in the homogeneity section above), Waldspurger shows that the summand of ∑ c_{N}O_{N} indexed by nilpotent orbits of maximal dimension with respect to the property that c_{N} ≠ 0 ought to be the image under endoscopic induction of the basis element of D^{st}(N_{𝔥}) with which we began.

References

[1] M. Assem, On stability and endoscopic transfer of unipotent orbital integrals on p-adic symplectic groups, Mem. Amer. Math. Soc. 134 (1998), no. 635.

[2] D. Barbasch and A. Moy, Local character expansions, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 5, 553–567.

[3] Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, preface and notes by Stephen DeBacker and Paul J. Sally, Jr., University Lecture Series, no. 16, American Mathematical Society, Providence, RI, 1999.

[4] R. Howe, The Fourier transform and germs of characters (case of Gl_{n} over a p-adic field), Math. Ann. 208 (1974), 305–322.

[5] D. Kazhdan, Proof of Springer’s hypothesis, Israel J. Math. 28 (1977), no. 4, 272–286.

[6] R. P. Langlands, Les débuts d’une formule des traces stable, Publications Mathématiques de l’Université Paris VII, 13. Universit de Paris VII, U.E.R. de Mathmatiques, Paris, 1983.

[7] R. P. Langlands, and D. Shelstad, On the definition of transfer factors, Math. Ann. 87 (1987), 219–271.

[8] G. Lusztig, Fourier transforms on a semisimple Lie algebra over F_{q}, Algebraic groups Utrecht 1986, 177–188, Lecture Notes in Math., 1271, Springer, Berlin, 1987.

[9] G. Lusztig, Green functions and character sheaves, Ann. of Math. (2) 131 (1990), no. 2, 355–408.

[10] R. Ranga Rao, Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505–510.

[11] J.-L. Waldspurger, Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), no. 2, 153–236.

[12] J.-L. Waldspurger, Quelques resultats de finitude concernant les distributions invariantes sur les algèbres de Lie p-adiques, preprint, 1993.

[13] J.-L. Waldspurger, Une formule des traces locale pour les algèbres de Lie padiques, J. Reine Angew. Math. 465 (1995), 41–99.