1. Introduction

Notation. Let $F$ be a $p$ -adic field, given either as a finite field extension of $\mathbb {Q}_p$ , or as the field $F=\mathbb {F}_q((t))$ . Let $\mathbb {F}_q$ (a finite field with $q$ elements and characteristic $p$ ) be the residue field of $F$ . Let $\bar F$ be a fixed algebraic closure of $F$ . Let $F^{un}$ be the maximal unramified extension of $F$ in $\bar F$ . For simplicity, we also assume that the characteristic of $F$ is not $2$ .

The fundamental lemma pertains to groups that satisfy a series of hypotheses. Here is the first.

Assumption 1.1

$G$ is a connected reductive linear algebraic group that is defined over $F$ .

The following examples give the $F$ -points of three different families of connected reductive linear algebraic groups: orthogonal, symplectic, and unitary groups.

Example 1.2

Let $M(n,F)$ be the algebra of $n$ by $n$ matrices with coefficients in $F$ . Let $J\in M(n,F)$ be a symmetric matrix with nonzero determinant. The special orthogonal group with respect to the matrix $J$ is

\operatorname {SO}(n,J,F) = \{ X \in M(n,F) {\ \vert \ }{{}^t}X J X = J,\quad \operatorname {det}(X)=1\}.

Example 1.3

Let $J\in M(n,F)$ , with $n=2k$ , be a skew-symmetric matrix ${{}^t}J = -J$ with nonzero determinant. The symplectic group with respect to $J$ is defined in a similar manner:

\operatorname {Sp}(2k,J,F) = \{ X \in M(2k,F) {\ \vert \ }{{}^t}X J X = J \}.

Example 1.4

Let $E/F$ be a separable quadratic extension. Let $\bar x$ be the Galois conjugate of $x\in E$ with respect to the nontrivial automorphism of $E$ fixing $F$ . For any $A\in M(n,E)$ , let $\bar A$ be the matrix obtained by taking the Galois conjugate of each coefficient of $A$ . Let $J\in M(n,E)$ satisfy ${{}^t}{\bar J} = J$ and have a nonzero determinant. The unitary group with respect to $J$ and $E/F$ is

U(n,J,F) = \{X \in M(n,E) {\ \vert \ }{{}^t}{\bar X} J X = J\}.

The algebraic groups $SO(n,J)$ , $Sp(2k,J)$ , and $U(n,J)$ satisfy Assumption 1.1.

Assumption 1.5

$G$ splits over an unramified field extension.

That is, there is an unramified extension $F_1/F$ such that $G\times _F F_1$ is split.

Example 1.6

In the first two examples above (orthogonal and symplectic), if we take $J$ to have the special form

\begin{equation} J = \begin{pmatrix} 0&0&*\\ 0&*&0\\ {*}&0&0 \end{pmatrix} \tag{1.6.1}\cssId{eqn:cross}{} \end{equation}

(that is, nonzero entries from $F$ along the cross-diagonal and zeros elsewhere), then $G$ splits over $F$ . In the third example (unitary), if $J$ has this same form and if $E/F$ is unramified, then the unitary group splits over the unramified extension $E$ of $F$ .

Assumption 1.7

$G$ is quasi-split.

This means that there is an $F$ -subgroup $B\subset G$ such that $B\times _F\bar F$ is a Borel subgroup of $G\times _F\bar F$ .

Example 1.8

In all three cases (orthogonal, symplectic, and unitary), if $J$ has the cross-diagonal form 1.6.1, then $G$ is quasi-split. In fact, we can take the points of $B$ to be the set of upper triangular matrices in $G(F)$ .

Assumption 1.9

$K$ is a hyperspecial maximal compact subgroup of $G(F)$ , in the sense of Definition 1.11.

Example 1.10

Let $O_F$ be the ring of integers of $F$ and let $K = GL(n,O_F)$ . This is a hyperspecial maximal compact subgroup of $GL(n,F)$ .

Definition 1.11

$K$ is hyperspecial if there exists $\mathcal {G}$ such that the following conditions are satisfied.

$\mathcal {G}$ is a smooth group scheme over $O_F$ ,
$G = \mathcal {G}\times _{O_F} F$ ,
$\mathcal {G}\times _{O_F} \mathbb {F}_q$ is reductive,
$K = \mathcal {G}(O_F)$ .

Example 1.12

In all three examples (orthogonal, symplectic, and unitary), take $G$ to have the form of Example 1.6. Assume that each cross-diagonal entry is a unit in the ring of integers. Assume further that the residual characteristic is not $2$ . Then the equations

{{}^t}X J X = J\quad (\text{or in the unitary case } {{}^t}{\bar X} J X = J)

define a group scheme $\mathcal {G}$ over $O_F$ , and $\mathcal {G}(O_F)$ is hyperspecial.