9.1. Old-style geometric interpretations: buildings

We begin with a geometric interpretation of the fundamental lemma that was popular in the late seventies and early eighties. It was eventually discarded in favor of other approaches when the combinatorial difficulties became too great.

This approach is to use the geometry of the Bruhat-Tits building to understand orbital integrals. We illustrate the approach with the group $G=SL(2)$. The term $\chi _{G,K}(g^{-1}\gamma ' g)$ that appears in the fundamental lemma can be manipulated as follows:

$$\begin{array}{lll} \chi _{G,K}(g^{-1}\gamma ' g)\ne 0 &\Leftrightarrow g^{-1}\gamma ' g\in K\\&\Leftrightarrow \gamma ' g\in g K\\&\Leftrightarrow \gamma ' (g K) = (g K)\\&\Leftrightarrow g K \text{ is a fixed point of } \gamma ' \text{ on } G(F)/K. \end{array}$$

The set $G(F)/K$ is in bijective correspondence with a set of vertices in the Bruhat-Tits building of $SL(2)$. Thus, we may interpret the orbital integral geometrically as the number of fixed points of $\gamma '$ in the building that are vertices of a given type.

Under this interpretation, it is possible to use counting arguments to obtain explicit formulas for orbital integrals as a function of $\gamma '$. In this way, the fundamental lemma was directly verified for a few groups of small rank such as $SL(2)$ and $U(3)$.

9.2. Affine grassmannians

Until the end of Section 9, let $F = k((t))$, a field of formal Laurent series. Except for the discussion of the results of Kazhdan and Lusztig, the field $k$ will be taken to be a finite field: $k=\mathbb {F}_q$.

In 1988, Kazhdan and Lusztig showed that if $F=\mathbb {C}((t))$, then $G(F)/K$ can be identified with the points of an ind-scheme (that is, an inductive limit of schemes) [ 15]. This ind-scheme is called the affine Grassmannian. The set of fixed points of an element $\gamma$ can be identified with the set of points of a scheme over $\mathbb {C}$, known as the affine Springer fiber. The corresponding construction over $\mathbb {F}_q((t))$ is mentioned briefly in the final paragraphs of their paper. Rather than counting fixed points in the building, orbital integral can be computed by counting the number of points on a scheme over $\mathbb {F}_q$.

Based on a description of orbital integrals as the number of points on schemes over finite fields, Kottwitz, Goresky, and MacPherson give a geometrical formulation of the fundamental lemma. Furthermore, by making a thorough investigation of the equivariant cohomology of these schemes, they prove the geometrical conjecture when $\gamma$ comes from an unramified Cartan subgroup [ 5].

9.3. Geometric interpretations

Each of the terms in the fundamental lemma has a nice geometric interpretation. Let us give a brief description of the geometrical counterpart of each term in the fundamental lemma. We work with the unitary group, so that we may include various insights of Laumon.

The geometrical counterpart of cosets $gK$ are self-dual lattices in a vector space $V$ over $F$.

The counterpart of the support set, $\operatorname {SUP}=\{g{\ \vert \ }g^{-1}\gamma g \in K\}$, is the affine Springer fiber $X_\gamma$.

The counterpart of the integral of the support set $\operatorname {SUP}$ over $G$ is counting points on the scheme $X_\gamma$. The integral over all of $G$ diverges and the number of fixed points on the scheme is infinite. For that reason the orbital integral is an integral over $T\backslash G$, where $T$ is the centralizer of $\gamma$, rather than over all of $G$.

The counterpart of the integral over $T\backslash G$ is counting points on a quotient space $Z_\gamma = X_\gamma /\mathbb {Z}^\ell$. (There is a free action of a group $\mathbb {Z}^\ell$ on $X_\gamma$, and $Z_\gamma$ is the quotient.)

The geometric counterpart of $\kappa (a_\tau )$ is somewhat more involved. For elliptic endoscopic groups of unitary groups $\kappa$ has order $2$. The character $\kappa$ has the form.

$$\kappa :H^1(\operatorname {Gal}(\bar F/F),T)\cong (\mathbb {Z}/2\mathbb {Z})^\ell \to \{\pm 1\}.$$

The character $\kappa$ pulls back to a character of $\mathbb {Z}^\ell$. The rational points of $X_\gamma$ are identified with self-dual lattices: $A^{\perp } = A$. The points of the quotient space $Z_\gamma$ are lattices that are self-dual modulo the group action: $A^{\perp } =\lambda \cdot A$, for some $\lambda \in \mathbb {Z}^\ell$. The character $\kappa$ then partitions the points of $Z_\gamma$ into two sets, depending on the sign of $\kappa (\lambda )$:

$$Z_\gamma ^{\pm } = \{A \ |\ A^{\perp } = \lambda A;\quad \kappa (\lambda ) = \pm 1\}.$$

(In a more sophisticated treatment of $\kappa (a_\tau )$, it gives rise to a local system on $Z_\gamma$; and counting points on varieties gives way to Grothendieck's trace formula.)

The counterpart of the $\kappa$-orbital integral $\Lambda _{G,H}(\gamma )$ is the number

$$\#Z_\gamma ^+ - \#Z_\gamma ^-.$$

The counterpart of the stable-orbital integral $\Lambda _H^{st}(\gamma )$ is the number

$$\#Z_\gamma ^{H,st}$$

for a corresponding variety constructed from the endoscopic group.

The factors $\prod _{\Phi } |\alpha (\gamma )-1|^{1/2}$ that appear on the two sides of the fundamental lemma can be combined into a single term

$$\prod _{\alpha \in \Phi _G\setminus \Phi _H}|\alpha (\gamma )-1|^{1/2}.$$

This has the form $q^{-d}$ for some value $d=d(\gamma )$. The factor $q^{-d}$ has been interpreted in various ways. We mention that [ 24] interprets $q^d$ as the points on an affine space of dimension $d$. That paper expresses the hope that it might be possible to find an embedding $Z_\gamma ^-\to Z_\gamma ^+$ such that the complement of the embedded $Z_\gamma ^-$ in $Z_\gamma ^+$ is a rank $d$ fiber bundle over $Z_\gamma ^{H,st}$. The realization of this hope would give an entirely geometric interpretation of the fundamental lemma. Laumon and Rapoport found that this construction works over $\mathbb {F}_{q^2}((t))$, but not over $\mathbb {F}_q((t))$. In more recent work of Laumon, the constant $d$ is interpreted geometrically as the intersection multiplicity of two singular curves.

9.4. Compactified Jacobians

Laumon, in the case of unitary groups, has made the splendid discovery that the orbital integrals – as they appear in the fundamental lemma – count points on the compactification of the Jacobians of a singular curve associated with the semisimple element $\gamma$. (In fact, $Z_\gamma$ is homeomorphic to and can be replaced with the compactification of a Jacobian.) Thus, the fundamental lemma may be reformulated as a relation between the compactified Jacobians of these curves. By showing that the singular curve for the endoscopic group $H$ is a perturbation of the singular curve for the group $G$, he is able relate the compactified Jacobians of the two curves, and prove the fundamental lemma for unitary groups (assuming a purity hypothesis related to the cohomology of the schemes).

Figure 9-1. Figure 1.

The singular curve on the left can be deformed into the singular curve on the right by pulling up on the center ring. The curve on the left controls $\Lambda ^{st}_H(\gamma )$, and the curve on the right controls $\Lambda _{G,H}(\gamma )$. This deformation relating the two curves is a key part of Laumon's work on the fundamental lemma for unitary groups.

The origin of the curve $C$ is the following. The ring $O_F[\gamma ]$ is the completion at a point of the local ring of a curve $C$. In the interpretation in terms of Jacobians, the self-dual lattices $A^{\perp } = A$ that appear in the geometric interpretation above are replaced with $O_C$-modules, where $O_C$ is the structure sheaf of $C$.

The audio recording of Laumon's lecture at the Fields Institute on this research is highly recommended [ 23].

9.5. Final remarks