2.1. Orbital integrals

We begin by fixing the Haar measures. Let $dg$ be the Haar measure on $G(F)$ normalized by the condition $\textrm {vol}_{dg}\big (G(\mathcal {O})\big )=1$ . For the group $T(F)$ , the definition is more involved, as there is no natural $\mathcal {O}$ -structure on T. Let $F^{\text {ur}}=\overline {\mathbf{F}}_{q}(\!(\epsilon )\!)$ ; it is the completion of the maximal unramified extension of F. Let $\sigma $ be the Frobenius automorphism of both $\overline {\mathbf{F}}_{q}/\mathbf{F}_{q}$ and $F^{\text {ur}}/F$ . We fix a separable algebraic closure $\overline {F^{\text {ur}}}$ of $F^{\text {ur}}$ and let $I_{F}=\textrm {Gal}\left (\overline {F^{\text {ur}}}/F^{\text {ur}}\right )$ – this is the inertia subgroup of $\Gamma =\textrm {Gal}\left (\overline {F}/F\right )$ . According to Kottwitz [Reference Kazhdan and Lusztig19, §7.6], we have an exact sequence

(2.1) $$ \begin{align} 1\to T(F^{\text{ur}})_{1}\to T(F^{\text{ur}})\xrightarrow{w_{T}}X_{*}(T)_{I_{F}}\to 1, \end{align} $$

which implies another exact sequence if we take the $\langle \sigma \rangle $ -invariants:

(2.2) $$ \begin{align} 1\to T(F)_{1}\to T(F) \xrightarrow{w_{T}} \left(X_{*}(T)_{I_{F}}\right)^{\langle \sigma \rangle}\to 1, \end{align} $$

with $T(F)_{1}:=T(F)\cap T(F^{\text {ur}})_{1}$ . We fix the Haar measure $dt$ on $T(F)$ by setting $\textrm {vol}_{dt}\big (T(F)_{1}\big )=1$ . The group $\Lambda $ is discrete and cocompact in $T(F)$ . The volume of the quotient $\Lambda \backslash T(F)$ is calculated in [Reference Goresky, Kottwitz and MacPherson14, §15.3]:

$$ \begin{align*} \mathrm{vol}_{dt}\big(\Lambda\backslash T(F)\big)=\frac{\left\lvert\text{coker}[X_{*}(S)_{\Gamma}\to X_{*}(T)_{\Gamma}]\right\rvert}{\left\lvert\ker[X_{*}(S)_{\Gamma}\to X_{*}(T)_{\Gamma}]\right\rvert}. \end{align*} $$

Consider the orbital integral

(2.3)

It can be interpreted as counting points on the affine Springer fiber:

Proposition 2.1 Goresky, Kottwitz, MacPherson [Reference Goresky, Kottwitz and MacPherson14]

$$ \begin{align*} I_{\gamma}^{G}=\frac{\left\lvert\ker[X_{*}(S)_{\Gamma}\to X_{*}(T)_{\Gamma}]\right\rvert}{\left\lvert\text{coker}[X_{*}(S)_{\Gamma}\to X_{*}(T)_{\Gamma}]\right\rvert}\cdot\left\lvert\Lambda\backslash \left(\mathscr{X}_{\gamma}\left(\mathbf{F}_{q}\right)\right)\right\rvert. \end{align*} $$

The $T(F)$ -action on $\mathscr {X}_{\gamma }$ can be exploited to further simplify the computations. Let $\mathscr {X}_{\gamma }^{\mathrm{reg}}$ be the open subscheme of $\mathscr {X}_{\gamma }$ consisting of the points $[g]\in \mathscr {X}_{\gamma }$ such that the image of $\textrm {Ad}\left (g^{-1}\right )\gamma $ under the reduction $\mathfrak {g}(\mathcal {O})\to \mathfrak {g}$ is regular nilpotent.

Consequently, all the irreducible components of $\mathscr {X}_{\gamma }$ are isomorphic to each other, and they are parametrized by $\pi _{0}(T(F))$ . In particular, all the connected components of $\mathscr {X}_{\gamma }$ are isomorphic and can be translated to each other under the $T(F)$ -action. In calculating the orbital integral (2.3), we can thus restrict to the central connected component of $\mathscr {X}_{\gamma }$ , which often simplifies calculations.

The calculation of $I_{\gamma }^{G}$ can be reduced to that of $I_{\gamma }^{M_{0}}$ ; it dates back at least to Harish-Chandra that

(2.4) $$ \begin{align} I_{\gamma}^{G} =q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad}(\gamma)\mid\mathfrak{g}_{F}/\mathfrak{m}_{0,F}\right)\right)}\cdot I_{\gamma}^{M_{0}}. \end{align} $$

Geometrically, this is a reflection of the existence of an affine fibration $f_{P}:\mathscr {X}_{\gamma }\to \mathscr {X}_{\gamma }^{M_{0}}$ for each $P\in \mathcal {P}(M_{0})$ . Recall that for $Q=LN_{Q}\in \mathcal {F}(A)$ , we have the retraction

$$ \begin{align*} f_{Q}:\mathscr{X}\to \mathscr{X}^{L}, \end{align*} $$

which sends $[g]=gK$ to $[h]:=hL(\mathcal {O})$ , where $g=nhk, n\in N_{Q}(F), h\in L(F), k\in K$ , is the Iwasawa decomposition. We want to point out that the retraction $f_{Q}$ is not a morphism between ind- $\mathbf{F}_{q}$ -schemes, but its restriction to the inverse image of each connected component of $\mathscr {X}^{L}$ ,

$$ \begin{align*} f_{Q}:f_{Q}^{-1}\left(\mathscr{X}^{L,\nu}\right)\to \mathscr{X}^{L,\nu},\qquad \nu\in \Lambda_{L}, \end{align*} $$

is actually a morphism over $\mathbf{F}_{q}$ between ind- $\mathbf{F}_{q}$ -schemes. Moreover, these retractions satisfy the obvious transitivity property.

Restricted to the affine Springer fibers, the retraction $f_{Q}$ sends $\mathscr {X}_{\gamma }$ to $\mathscr {X}_{\gamma }^{L}$ . To see this, for $[g]\in \mathscr {X}_{\gamma }$ , let $g=nhk$ be the Iwasawa decomposition as before. We can write $g=hn'k$ , with $n'=h^{-1}nh\in N_{Q}(F)$ . Now that $\textrm {Ad}\left (h^{-1}\right )\gamma \in \mathfrak {l}(F)$ , we have

$$ \begin{align*} \textrm{Ad}\left(n^{\prime -1}\right)\textrm{Ad}\left(h^{-1}\right)\gamma=\textrm{Ad}\left(h^{-1}\right)\gamma+n'' \end{align*} $$

for some $n''\in \mathfrak {n}_{Q}(F)$ . This implies that

$$ \begin{align*} \textrm{Ad}(h)^{-1}\gamma \in \left[\mathfrak{g}(\mathcal{O})+\mathfrak{n}_{Q,F}\right]\cap \mathfrak{l}(F)=\mathfrak{l}(\mathcal{O}), \end{align*} $$

which means that $f_{Q}([g])=[h]\in \mathscr {X}_{\gamma }^{L}$ .

Proposition 2.4 Kazhdan and Lusztig [Reference Kottwitz17, §5, Proposition 1]

For any $\nu \in \Lambda _{L}$ , the retraction

$$ \begin{align*} f_{Q}:\mathscr{X}_{\gamma}\cap f_{Q}^{-1}\left(\mathscr{X}_{\gamma}^{L,\nu}\right)\to \mathscr{X}_{\gamma}^{L,\nu} \end{align*} $$

is an iterated affine fibration over $\mathbf{F}_{q}$ of relative dimension $ \mathrm{val}(\det (\mathrm{ad}(\gamma )\mid \mathfrak {n}_{Q}(F))). $

The reader can also consult [Reference Chaudouard and Laumon11, Proposition 3.2] for a proof.

2.2. Arthur’s weighted orbital integral

2.2.1. The weight factor $\textrm {v}_{M}$

Set $M\in \mathcal {L}(M_{0})$ . Roughly speaking, the weight factor $\textrm {v}_{M}(g)$ is the volume of a polytope in $\mathfrak {a}_{M}$ generated by the point $[g]\in \mathscr {X}$ . Let $H_{M}: M(F)\to \mathfrak {a}_{M}$ be the unique mapFootnote ² satisfying

$$ \begin{align*} \chi(H_{M}(m))=\mathrm{val}(\chi(m)),\qquad \forall \chi\in X^{*}(M),m\in M(F). \end{align*} $$

Notice that it is a group homomorphism. Moreover, it is invariant under the right K-action, so it induces a map from $\mathscr {X}^{M}$ to $\mathfrak {a}_{M}$ , still denoted by $H_{M}$ . For $P=MN\in \mathcal {F}(A)$ , let $H_{P}:\mathscr {X}\to \mathfrak {a}_{M}$ be the composition

$$ \begin{align*} H_{P}:\mathscr{X}\xrightarrow{f_{P}}\mathscr{X}^{M}\xrightarrow{H_{M}}\mathfrak{a}_{M}. \end{align*} $$

As shown in [Reference Chen7, Lemma 6.1], the map $H_{M}$ is constant on each connected component of $\mathscr {X}^{M}$ , so it has a factorization $ H_{M}:\mathscr {X}^{M}\xrightarrow {\nu _{M}}\Lambda _{M}\to \mathfrak {a}_{M}$ . A simple calculation of the restriction of the map to $\mathscr {X}^{A}\subset \mathscr {X}^{M}$ shows that the map $\Lambda _{M}\to \mathfrak {a}_{M}$ is just the one induced from the natural inclusion $X_{*}(A)\hookrightarrow \mathfrak {a}_{A}= X_{*}(A)\otimes \textbf {R}$ . Hence $H_{P}$ is also the composition

$$ \begin{align*} H_{P}:\mathscr{X}\xrightarrow{f_{P}}\mathscr{X}^{M}\xrightarrow{\nu_{M}}\Lambda_{M}\to \mathfrak{a}_{M}. \end{align*} $$

The map $H_{P}$ has the following remarkable property. There is a notion of adjacency among the parabolic subgroups in $\mathcal {P}(M)$ : two parabolic subgroups $P_{1}=MN_{1}, P_{2}=MN_{2}\in \mathcal {P}(M)$ are said to be adjacent if both of them are contained in a parabolic subgroup $Q=LN_{Q}$ such that $L\supset M$ and $\textrm {rk}(L)=\textrm {rk}(M)+1$ . Given such an adjacent pair, we define an element $\beta _{P_{1},P_{2}}\in \Lambda _{M}$ in the following way: consider the collection of elements in $\Lambda _{M}$ obtained from coroots of A in $\mathfrak {n}_{1}\cap \mathfrak {n}_{2}^{-}$ . We define $\beta _{P_{1},P_{2}}$ to be the minimal element in this collection – that is, all the other elements are positive integral multiples of it. Note that $\beta _{P_{2},P_{1}}=-\beta _{P_{1},P_{2}}$ , and if $M=A$ , then $\beta _{P_{1},P_{2}}$ is the unique coroot which is positive for $P_{1}$ and negative for $P_{2}$ . We also denote its image in $\mathfrak {a}_{M}$ by $\beta _{P_{1},P_{2}}$ if no confusion is caused.

Proposition 2.5 Arthur [Reference Arthur1, Lemma 3.6]

Let $P_{1}, P_{2}\in \mathcal {P}(M)$ be two adjacent parabolic subgroups. For any $x\in \mathscr {X}$ , we have

$$ \begin{align*} H_{P_{1}}(x)-H_{P_{2}}(x)=n(x,P_{1},P_{2})\cdot \beta_{P_{1},P_{2}}, \end{align*} $$

with $n(x, P_{1}, P_{2})\in \textbf {Z}_{\geq 0}$ .

The reader can consult [Reference Chaudouard and Laumon11, Proposition 2.1] for a proof. For any point $x\in \mathscr {X}$ , we write $\mathrm{Ec}_{M}(x)$ for the convex hull in $\mathfrak {a}_{M}$ of the $H_{P}(x), P\in \mathcal {P}(M)$ . For any $Q\in \mathcal {F}(M)$ , we denote by $\mathrm{Ec}_{M}^{Q}(x)$ the face of $\mathrm{Ec}_{M}(x)$ whose vertices are $H_{P}(x), P\in \mathcal {P}(M), P\subset Q$ . When $M=A$ , we omit the subscript A to simplify the notation.

To define the volume, we need to choose a Lebesgue measure on $\mathfrak {a}_{M}^{G}$ . We fix a W-invariant inner product $\langle \cdot ,\cdot \rangle $ on the vector space $\mathfrak {a}_{A}^{G}$ . Notice that $\mathfrak {a}_{A}^{M}$ and $\mathfrak {a}_{M}$ are orthogonal to each other with respect to the inner product for any $M\in \mathcal {L}(A)$ . We fix a Lebesgue measure on $\mathfrak {a}_{M}^{G}$ normalized by the condition that the lattice generated by the orthonormal bases in $\mathfrak {a}_{M}^{G}$ has covolume $1$ .

The weight factor $\textrm {v}_{M}(g)$ is the volume of the projection $\pi _{M}^{G}\big (\mathrm{Ec}_{M}(g)\big )\subset \mathfrak {a}_{M}^{G}$ . We have to pass to $\mathfrak {a}_{M}^{G}$ because the polytope $\mathrm{Ec}_{M}(g)$ will lie in a hyperplane of $\mathfrak {a}_{M}$ if G has nontrivial connected center. The weight factor $\textrm {v}_{M}(g)$ has the following invariance properties: it is invariant under the right action of K – that is,

$$ \begin{align*} \textrm{v}_{M}(gk)=\textrm{v}_{M}(g),\qquad \forall k\in K. \end{align*} $$

This is evident from the definition of $\textrm {v}_{M}(g)$ . It is not so evident, but also true, that

$$ \begin{align*} \textrm{v}_{M}(mg)=\textrm{v}_{M}(g),\qquad \forall m\in M(F). \end{align*} $$

Indeed, for any $P\in \mathcal {P}(M)$ , we have $ f_{P}(mg)=mf_{P}(g). $ As $H_{M}$ is a group homomorphism, this implies

$$ \begin{align*} H_{P}(mg)=H_{M}(m)+H_{P}(g), \end{align*} $$

so $ \mathrm{Ec}_{M}(mg)$ is just the translation of $\mathrm{Ec}_{M}(g)$ by $H_{M}(m). $ In particular, they have the same volume.

Similar to Proposition 2.1, we can interpret Arthur’s weighted orbital integral as

That is, it is a weighted count of the rational points on the affine Springer fiber. Notice also that $J_{G}(\gamma )=I_{\gamma }^{G}$ , as $\textrm {v}_{G}(g)=1$ for all $g\in G(F)$ .

2.2.2. A variant

In their work on the weighted fundamental lemma [Reference Chen8], Chaudouard and Laumon introduce a variant of the weighted orbital integral.

Assume that G is semisimple and let $\xi \in \mathfrak {a}_{M}$ be a generic element. For $g\in G(F)$ , they introduce the weight factor

$$ \begin{align*} \textrm{w}_{M}^{\xi}(g)=\left\lvert\{\lambda\in X_{*}(M)\mid \lambda+\xi\in \textrm{Ec}_{M}(g)\}\right\rvert. \end{align*} $$

It is the number of integral points in the polytope $\mathrm{Ec}_{M}(g)-\xi $ . Similar to $\textrm {v}_{M}(g)$ , the weight factor $\mathrm{w}_{M}^{\xi }(g)$ is invariant under the right K-action and the left $M(F)$ -action. In particular, it descends to a function on $\mathscr {X}$ . Consider the following weighted orbital integral:

Remark 2.1. For a general reductive algebraic group G, $\xi \in \mathfrak {a}_{M}^{G}$ , as $G(F)=M(F)\cdot G_{\text {der}}(F)$ , we can define the weight factor $\mathrm{w}_{M}^{\xi }$ uniquely by requiring it to be invariant under the left $M(F)$ -action and the right K-action, and that as a function on $\mathscr {X}$ its restriction to $\mathscr {X}^{G_{\text {der}}}$ coincide with the given definition for $G_{\text {der}}$ . In other words, for generic $\xi \in \mathfrak {a}_{M}^{G}$ , we define

$$ \begin{align*} \textrm{w}_{M}^{\xi}(g)=\left\lvert\left\{\lambda\in X_{*}\left(M_{G_{\text{der}}}\right)\mid \lambda+\xi\in \pi_{M}^{G}(\textrm{Ec}_{M}(g))\right\}\right\rvert, \end{align*} $$

where $M_{G_{\text {der}}}=M\cap G_{\text {der}}$ . Notice that the weight factor $\textrm {v}_{M}$ satisfies these conditions as well, and this justifies our definition in the general case.

The variant $J_{M}^{\xi }(\gamma )$ has a better geometric interpretation.

Lemma 2.6. Let $T(F)^{1}=T(F)\cap \ker \left (H_{M_{0}}\right )$ . Then it is of finite volume and we have an exact sequence

$$ \begin{align*} 1\to T(F)^{1}\to T(F)\xrightarrow{H_{M_{0}}} X_{*}(M_{0})\to 1. \end{align*} $$

Proof. The first assertion is due to the fact that T is anisotropic modulo the center of $M_{0}$ . For the second assertion, only the surjectivity is nontrivial. Recall that we have the exact sequence

$$ \begin{align*} 1\to T(F)_{1}\to T(F)\xrightarrow{w_{T}} \left(X_{*}(T)_{I_{F}}\right)^{\langle \sigma\rangle}\to 1, \end{align*} $$

and that the map $w_{T}$ is defined via a map $v_{T}: T(F^{\text {ur}})\to \textrm {Hom}\left (X^{*}(T)^{I_{F}},\textbf {Z}\right )$ , similar to the definition of $H_{M_{0}}$ . Hence the morphism $H_{M_{0}}$ factors through $w_{T}$ , and the surjectivity results from those of $w_{T}$ and the homomorphism

$$ \begin{align*} \left(X_{*}(T)_{I_{F}}\right)^{\langle \sigma\rangle}=X_{*}(T)_{I_{F}}\to \textrm{Hom}(X^{*}(M_{0}),\textbf{Z})=X_{*}(M_{0}). \end{align*} $$

As a consequence, let $T(F)_{M}^{1}=T(F)\cap \,\ker (H_{M})$ and let $\Lambda ^{H_{M}}=\Lambda \cap \,\ker (H_{M})$ . Then the quotient $\Lambda ^{H_{M}}\backslash T(F)_{M}^{1}$ is of finite volume and we have an exact sequence

$$ \begin{align*} 1\to T(F)_{M}^{1}\to T(F)\xrightarrow{H_{M}} X_{*}(M)\to 1. \end{align*} $$

Proposition 2.7. We have the equality

$$ \begin{align*} J_{M}^{\xi}(\gamma)=\mathrm{vol}_{dt}\left(\Lambda^{H_{M}}\backslash T(F)_{M}^{1}\right)^{-1}\cdot \left\lvert\Lambda^{H_{M}}\backslash \left\{[g]\in \mathscr{X}_{\gamma}\left(\mathbf{F}_{q}\right)\mid \xi\in \text{Ec}_{M}(g)\right\}\right\rvert. \end{align*} $$

In particular,

$$ \begin{align*} J_{M_{0}}^{\xi}(\gamma)=\mathrm{vol}_{dt}\left(T(F)^{1}\right)^{-1}\cdot \left\lvert\left\{[g]\in \mathscr{X}_{\gamma}\left(\mathbf{F}_{q}\right)\mid \xi\in \mathrm{Ec}_{M_{0}}(g)\right\}\right\rvert. \end{align*} $$

Proof. Let

be the characteristic function of $\mathrm{Ec}_{M}(g)$ . As

$$ \begin{align*} \textrm{Ec}_{M}(tg)=\textrm{Ec}_{M}(g)+H_{M}(t),\qquad \forall t\in T(F),g\in G(F), \end{align*} $$

we have

Now we can rewrite

In particular, $J_{M}^{\xi }(\gamma )$ is a plain count of a subset of $\mathscr {X}_{\gamma }\left (\mathbf{F}_{q}\right )$ . In §4.1, we will see that the condition $\xi \in \mathrm{Ec}_{M}(g)$ behaves as a stability condition. (We believe that it is in fact a stability condition in the sense of Mumford.) In particular, there is a Harder–Narasimhan-type decomposition of $\mathscr {X}_{\gamma }$ associated with it.

Remark 2.2. It is time to explain why we have imposed the assumption that T is totally ramified over F. Without it, the Frobenius $\sigma \in \textrm {Gal}\left (\overline {\mathbf{F}}_{q}/\mathbf{F}_{q}\right )$ acts nontrivially on $X_{*}(T)_{I_{F}}$ , and the morphism $T(F)\to X_{*}(M_{0})$ in Lemma 2.6 might fail to be surjective. (Indeed, it does fail for T an unramified maximal torus in $\textrm {GL}_{n}$ .) As a consequence, the interpretation of $J_{M}^{\xi }(\gamma )$ as in Proposition 2.7 no longer holds.

For completeness, we compute the volume factors in Proposition 2.7. We have the exact sequence

$$ \begin{align*} 1\to \Lambda^{H_{M}}\to T(F)_{M}^{1}\to \Lambda\backslash T(F) \xrightarrow{H_{M}} X_{*}(M)/H_{M}(\Lambda)\to 1. \end{align*} $$

Because $\Lambda $ is of finite index in $X_{*}(M_{0})$ and the morphism $X_{*}(M_{0})\to X_{*}(M)$ is surjective, the quotient $X_{*}(M)/H_{M}(\Lambda )$ is finite, and so

(2.5) $$ \begin{align} \mathrm{vol}_{dt}\left(\Lambda^{H_{M}}\backslash T(F)_{M}^{1}\right) &= \mathrm{vol}_{dt}\left( \Lambda\backslash T(F) \right)\cdot \left\lvert X_{*}(M)/H_{M}(\Lambda)\right\rvert^{-1} \nonumber\\ &= \frac{\left\lvert\text{coker}[X_{*}(S)_{\Gamma}\to X_{*}(T)_{\Gamma}]\right\rvert}{\left\lvert\ker[X_{*}(S)_{\Gamma}\to X_{*}(T)_{\Gamma}]\right\rvert\cdot \left\lvert X_{*}(M)/H_{M}(X_{*}(S))\right\rvert}. \end{align} $$

2.2.3. Comparison of weighted orbital integrals

The weight factors $\textrm {v}_{M}$ and $\mathrm{w}_{M}^{\xi }$ are closely related, so we can compare the associated weighted orbital integrals.

Theorem 2.8 Chaudouard and Laumon [Reference Chen8]

We have the equality

$$ \begin{align*} J_{M}(\gamma)= \mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right) \cdot J_{M}^{\xi}(\gamma). \end{align*} $$

Remark 2.3. For a general reductive algebraic group G, with the definition of $\mathrm{w}_{M}^{\xi }$ as explained in Remark 2.1, the comparison theorem becomes

$$ \begin{align*} J_{M}(\gamma)= \mathrm{vol}\left(\mathfrak{a}_{M_{G_{\text{der}}}}^{G_{\text{der}}}\middle/X_{*}\left(M_{G_{\text{der}}}\right)\right) \cdot J_{M}^{\xi}(\gamma), \end{align*} $$

as can be seen from the proof below.

Chaudouard and Laumon work over the ring of adèles, but their proof carries over to the local setting. We reproduce their proof here, but to simplify the exposition, we assume moreover that G is simply connected. The key is to rewrite the convex polytope $\mathrm{Ec}_{M}(g)$ as alternating differences of translations of cones. We need some notation. For $P=MN_{P}\in \mathcal {F}(A)$ , take a Borel subgroup $B\in \mathcal {P}(A)$ contained in P. Let $\Delta _{B}$ be the simple roots of $\Phi (G,A)$ with respect to B, and let $\Delta _{B,P}=\Delta _{B}\cap \Phi (N_{P},A)$ and $\Delta _{B, P}^{\vee }$ be the associated coroots. The restriction $X^{*}(A)\to X^{*}(A_{M})$ induces a bijection from $\Delta _{B,P}$ to a subset of $X^{*}(A_{M})$ denoted $\Delta _{P}$ . Similarly, the projection $\mathfrak {a}_{A}\to \mathfrak {a}_{M}$ induces a bijection from $\Delta _{B,P}^{\vee }$ to a subset $\Delta _{P}^{\vee }$ . Obviously, the definition of $\Delta _{P}$ and $\Delta _{P}^{\vee }$ is independent of the choice of B. Moreover, they form bases of $\mathfrak {a}_{M}^{*}$ and $\mathfrak {a}_{M}$ , respectively. Let $(\varpi _{\alpha })_{\alpha \in \Delta _{P}}$ be the basis of $\mathfrak {a}_{M}^{*}$ dual to $\Delta _{P}^{\vee }$ .

For a generic element $\lambda \in \mathfrak {a}_{M}^{*}$ , let

$$ \begin{align*} \Delta_{P}^{\lambda}=\{\alpha\in \Delta_{P}\mid \langle\lambda, \alpha\rangle<0\}, \end{align*} $$

and let $\varphi _{P}^{\lambda }$ be the characteristic function of the cone

$$ \begin{align*} \left\{a\in \mathfrak{a}_{M}\mid \varpi_{\alpha}(a)>0,\ \forall \alpha\in \Delta_{P}^{\lambda};\varpi_{\alpha}(a)\leq 0,\ \forall \alpha\in \Delta_{P}\backslash \Delta_{P}^{\lambda}\right\}. \end{align*} $$

According to Arthur [Reference Arthur3], the characteristic function of the convex polytope $\mathrm{Ec}_{M}(g)$ is equal to the function

$$ \begin{align*} a\in \mathfrak{a}_{M}\longmapsto \sum_{P\in \mathcal{P}(M)} (-1)^{\left\lvert\Delta_{P}^{\lambda}\right\rvert}\varphi_{P}^{\lambda}(-H_{P}(g)+a). \end{align*} $$

The proof is best illustrated by [Reference Arthur4, Figure 11.1, p. 63]. It relies on the combinatorial identity

$$ \begin{align*} \sum_{F\subset S} (-1)^{\lvert F\rvert}=\begin{cases} 1&\text{if } S=\emptyset,\\ 0 &\text{otherwise}, \end{cases} \end{align*} $$

for any finite set S. Now we can rewrite

(2.6) $$ \begin{align} \textrm{w}_{M}^{\xi}(g) &= \sum_{\chi\in X_{*}(M)}\sum_{P\in \mathcal{P}(M)}(-1)^{\left\lvert\Delta_{P}^{\lambda}\right\rvert} \varphi^{\lambda}_{P}(-H_{P}(g)+\chi+\xi), \end{align} $$

(2.7) $$ \begin{align} \textrm{v}_{M}(g) &= \int_{\mathfrak{a}_{M}} \sum_{P\in \mathcal{P}(M)} (-1)^{\left\lvert\Delta_{P}^{\lambda}\right\rvert}\varphi_{P}^{\lambda}(-H_{P}(g)+a)da. \end{align} $$

We introduce an extra exponential factor to treat the infinite sum in equation (2.6):

$$ \begin{align*} S_{P}(\lambda)=\sum_{\chi\in X_{*}(M)} \varphi^{\lambda}_{P}(-H_{P}(g)+\chi+\xi)e^{\langle\lambda,\chi\rangle}. \end{align*} $$

The series converges absolutely for generic $\lambda $ , and hence

$$ \begin{align*} \textrm{w}_{M}^{\xi}(g)=\lim_{\lambda\to 0}\sum_{P\in \mathcal{P}(M)}(-1)^{\left\lvert\Delta_{P}^{\lambda}\right\rvert} S_{P}(\lambda), \end{align*} $$

where the limit is taken for generic $\lambda \in \mathfrak {a}_{M}^{*}$ .

We can calculate $S_{P}(\lambda )$ explicitly. Let $\xi =[\xi ]_{P}+\{\xi \}_{P}$ , with $[\xi ]_{P}\in X_{*}(M)$ and $\{\xi \}_{P}=\sum _{\alpha \in \Delta _{P}}r_{\alpha }\alpha ^{\vee }$ for some $0< r_{\alpha }<1$ . After a simple change of variables, we get

$$ \begin{align*} S_{P}(\lambda)&= e^{\left\langle \lambda, H_{P}(g)-[\xi]_{P} \right\rangle} \sum_{\chi\in X_{*}(M)} \varphi_{P}^{\lambda}(\chi+\{\xi\}_{P})e^{\langle\lambda,\chi\rangle}\\ &= e^{\left\langle \lambda, H_{P}(g)-[\xi]_{P} \right\rangle} \sum_{(m_{\alpha})_{\alpha\in \Delta_{P}}} e^{\left\langle\lambda,\sum_{\alpha}m_{\alpha}\alpha^{\vee}\right\rangle}, \end{align*} $$

where $(m_{\alpha })_{\alpha \in \Delta _{P}}$ runs over the integers satisfying $m_{\alpha }\geq 0$ for $\alpha \in \Delta _{P}^{\lambda }$ and $m_{\alpha }\leq -1$ for $\alpha \in \Delta _{P}\backslash \Delta _{P}^{\lambda }$ . The geometric series can be calculated to be

$$ \begin{align*} S_{P}(\lambda)=(-1)^{\left\lvert\Delta_{P}^{\lambda}\right\rvert}e^{\left\langle \lambda, H_{P}(g)-[\xi]_{P} \right \rangle}\prod_{\alpha\in \Delta_{P}}\frac{1}{e^{\left\langle\lambda,\alpha^{\vee}\right\rangle}-1}. \end{align*} $$

Let $\textrm {c}_{P}(\lambda )=\prod _{\alpha \in \Delta _{P}} \left (e^{\left \langle \lambda ,\alpha ^{\vee }\right \rangle }-1\right )$ . Taking everything together, we get

(2.8) $$ \begin{align} \textrm{w}_{M}^{\xi}(g)= \lim_{\lambda\to 0} \sum_{P\in \mathcal{P}(M)} \textrm{c}_{P}(\lambda)^{-1}e^{\left\langle \lambda, H_{P}(g)-[\xi]_{P} \right\rangle}. \end{align} $$

Similarly, we can rewrite equation (2.7) as

$$ \begin{align*} \textrm{v}_{M}(g) &= \lim_{\lambda\to 0} \int_{\mathfrak{a}_{M}} \sum_{P\in \mathcal{P}(M)} (-1)^{\left\lvert\Delta_{P}^{\lambda}\right\rvert}\varphi_{P}^{\lambda}(-H_{P}(g)+a) e^{\langle \lambda,a \rangle} da \\ &= \lim_{\lambda\to 0} \sum_{P\in \mathcal{P}(M)} (-1)^{\left\lvert\Delta_{P}^{\lambda}\right\rvert} \int_{\mathfrak{a}_{M}} \varphi_{P}^{\lambda}(-H_{P}(g)+a) e^{\langle \lambda,a \rangle} da \\ &= \lim_{\lambda\to 0} \sum_{P\in \mathcal{P}(M)} e^{\left\langle \lambda,H_{P}(g) \right\rangle}\cdot \mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right)\prod_{\alpha\in \Delta_{P}} \left\langle \lambda,\alpha^{\vee}\right\rangle^{-1}. \end{align*} $$

Letting $\textrm {d}_{P}(\lambda )=\textrm {vol}\left (\mathfrak {a}_{M}/X_{*}(M)\right )^{-1}\prod _{\alpha \in \Delta _{P}} \left \langle \lambda ,\alpha ^{\vee }\right \rangle $ , we get

(2.9) $$ \begin{align} \textrm{v}_{M}(g)=\lim_{\lambda\to 0} \sum_{P\in \mathcal{P}(M)} \textrm{d}_{P}(\lambda)^{-1}\cdot e^{\left\langle \lambda,H_{P}(g) \right \rangle}. \end{align} $$

To deal with limits of the form in equations (2.8) and (2.9) systematically, we need Arthur’s notion of a $(G,M)$ -family [Reference Arthur2]. It is a family of smooth functions $(\textrm {r}_{P}(\lambda ))_{P\in \mathcal {P}(M)}$ on $\mathfrak {a}_{M}^{*}$ which satisfy, for any adjacent parabolic subgroups $(P,P')$ , the property that $\textrm {r}_{P}(\lambda )=\textrm {r}_{P'}(\lambda )$ for any $\lambda $ on the hyperplane defined by the unique coroot in $\Delta _{P}^{\vee }\cap \left (-\Delta _{P'}^{\vee }\right )$ . For any such family, we define

$$ \begin{align*} \textrm{r}_{M}(\lambda)=\sum_{P\in \mathcal{P}(M)} \textrm{d}_{P}(\lambda)^{-1} \textrm{r}_{P}(\lambda), \end{align*} $$

for generic $\lambda \in \mathfrak {a}_{M}^{*}$ . Arthur has shown in [Reference Arthur2] that the function extends smoothly over all $\mathfrak {a}_{M}^{*}$ . Let

$$ \begin{align*} \textrm{r}_{M}=\lim_{\lambda\to 0} \textrm{r}_{M}(\lambda). \end{align*} $$

It generalizes equation (2.9). Indeed, the functions

$$ \begin{align*} \textrm{v}_{P}(\lambda, g)=e^{\left\langle \lambda,H_{P}(g) \right\rangle},\qquad P\in \mathcal{P}(M), \end{align*} $$

form a $(G,M)$ -family, and the resulting $\textrm {v}_{M}(g)$ is exactly Arthur’s weight factor. From this point of view, we call $\textrm {r}_{M}$ the volume of the $(G,M)$ -family $(\textrm {r}_{P}(\lambda ))_{P\in \mathcal {P}(M)}$ .

Notice that the summands in equations (2.8) and (2.9) differ by a factor

$$ \begin{align*} \textrm{w}_{P}(\lambda,\xi)=\frac{\textrm{d}_{P}(\lambda)}{\textrm{c}_{P}(\lambda)}e^{-\left\langle \lambda,[\xi]_{P} \right\rangle}, \end{align*} $$

and that they form a $(G,M)$ -family. Letting $\mathrm{w}_{P}(\lambda ,g,\xi )=\textrm {v}_{P}(\lambda ,g)\mathrm{w}_{P}(\lambda ,\xi )$ , $P\in \mathcal {P}(M)$ , they form a $(G,M)$ -family and equation (2.8) can be rewritten as

(2.10) $$ \begin{align} \textrm{w}_{M}^{\xi}(g)=\textrm{w}_{M}(g,\xi). \end{align} $$

In other words, we have expressed the lattice point-counting weight factor $\mathrm{w}_{M}^{\xi }(g)$ as the volume of the product of two $(G,M)$ -families.

We need a result of Arthur on the volume of the product of two $(G,M)$ -families. Let $\{\textrm {r}_{P}(\lambda )\}_{P\in \mathcal {P}(M)}$ and $\{\mathrm{s}_{P}(\lambda )\}_{P\in \mathcal {P}(M)}$ be two $(G,M)$ -families. For $Q=LN_{Q}\in \mathcal {F}(M)$ , let

$$ \begin{align*} \textrm{r}_{R}^{Q}(\lambda)=\textrm{r}_{RN_{Q}}(\lambda), \qquad \forall R\in \mathcal{P}^{L}(M). \end{align*} $$

It is easy to see that $\textrm {r}_{R}^{Q}(\lambda )$ , $R\in \mathcal {P}^{L}(M)$ , form an $(L,M)$ -family. The function $\textrm {r}_{M}^{Q}(\lambda )$ and the volume $\textrm {r}_{M}^{Q}$ are defined in a similar way. From the $(G,M)$ -family $\{\mathrm{s}_{P}(\lambda )\}_{P\in \mathcal {P}(M)}$ , Arthur has defined a smooth function $\mathrm{s}_{Q}'(\lambda )$ on $\mathfrak {a}_{Q}^{*}$ . The definition is quite involved, and we refer the reader to [Reference Arthur2, §6]. Let $\mathrm{s}_{Q}'=\mathrm{s}_{Q}'(0)$ .

Lemma 2.9 Arthur [Reference Arthur2, Lemma 6.3 and Corollary 6.4]

Let $\{\textrm {r}_{P}(\lambda )\}_{P\in \mathcal {P}(M)}$ and $\{\mathrm{s}_{P}(\lambda )\}_{P\in \mathcal {P}(M)}$ be two $(G,M)$ -families, and let $\textrm {r}\cdot \mathrm{s}$ be their product. Then for any $\lambda \in \mathfrak {a}_{M}^{*}$ , we have

$$ \begin{align*} (\textrm{r}\cdot \textrm{s})_{M}(\lambda)=\sum_{Q\in \mathcal{F}(M)}\textrm{r}_{M}^{Q}(\lambda)\textrm{s}_{Q}'(\lambda). \end{align*} $$

In particular,

$$ \begin{align*} \textrm{s}_{M}(\lambda)=\sum_{P\in \mathcal{P}(M)}\textrm{s}_{P}'(\lambda). \end{align*} $$

In our situation, this implies

(2.11) $$ \begin{align} \textrm{w}_{M}^{\xi}(g)=\big(\textrm{v}(g)\cdot \textrm{w}(\xi)\big)(0)=\sum_{Q\in \mathcal{F}(M)}\textrm{v}_{M}^{Q}(g)\textrm{w}_{Q}'(\xi) \end{align} $$

and

(2.12) $$ \begin{align} \textrm{w}_{M}(\xi)=\sum_{P\in \mathcal{P}(M)}\textrm{w}_{P}'(\xi). \end{align} $$

Similar results hold for Levi subgroups L containing M:

(2.13) $$ \begin{align} \textrm{w}_{L}^{\xi_{L}}(g)&=\sum_{R\in \mathcal{F}(L)}\textrm{v}_{L}^{R}(g)\textrm{w}_{R}'(\xi_{L}), \end{align} $$

(2.14) $$ \begin{align} \textrm{w}_{L}(\xi_{L})&=\sum_{Q\in \mathcal{P}(L)}\textrm{w}_{Q}'(\xi_{L}), \end{align} $$

with $\mathrm{w}_{R}'(\xi _{L})$ deduced from the $(G,L)$ -family

$$ \begin{align*} \textrm{w}_{Q}(\lambda,\xi_{L})=\frac{\textrm{d}_{Q}(\lambda)}{\textrm{c}_{Q}(\lambda)}e^{-\left\langle \lambda,[\xi_{L}]_{Q} \right\rangle}, \qquad \forall Q\in \mathcal{P}(L). \end{align*} $$

Setting $g=e\in G$ in equation (2.13), and noting that

$$ \begin{align*} \textrm{v}_{L}^{R}(e)= \begin{cases} 1 & \text{if } R\in \mathcal{P}(L),\\ 0 & \text{otherwise}, \end{cases} \end{align*} $$

we get

(2.15) $$ \begin{align} \textrm{w}_{L}^{\xi_{L}}(e)=\sum_{Q\in \mathcal{P}(L)} \textrm{w}_{Q}'(\xi_{L})=\textrm{w}_{L}(\xi_{L}), \end{align} $$

where the second equality is just equation (2.14).

Lemma 2.10.

$$ \begin{align*} \sum_{Q\in \mathcal{P}(L)} \textrm{w}_{Q}'(\xi) =\frac{\mathrm{vol}\left(\mathfrak{a}_{L}/X_{*}(L)\right)}{\mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right)}\cdot \textrm{w}_{L}^{\xi_{L}}(e)= \begin{cases} \mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right)^{-1} & \text{if } L=G,\\ 0 &\text{otherwise}. \end{cases} \end{align*} $$

Proof. Recall that given a $(G,M)$ -family $\{\mathrm{s}_{P}(\lambda )\}_{P\in \mathcal {P}(M)}$ , we can define a $(G,L)$ -family by setting

$$ \begin{align*} \textrm{s}_{Q}(\lambda)=\textrm{s}_{P}(\lambda), \qquad \forall \lambda\in \mathfrak{a}_{L}^{*}\subset \mathfrak{a}_{M}^{*}, \end{align*} $$

for any $P\in \mathcal {P}(M)$ , $P\subset Q$ . Moreover, the function $\mathrm{s}_{Q}'(\lambda )$ deduced from the $(G,M)$ -family $\{\mathrm{s}_{P}(\lambda )\}_{P\in \mathcal {P}(M)}$ is the same as that from the $(G,L)$ -family $\{\mathrm{s}_{Q}(\lambda )\}_{Q\in \mathcal {P}(L)}$ , by [Reference Arthur2, formula 6.3]. In this way, we get the $(G,L)$ -family $\{\mathrm{w}_{Q}(\lambda ,\xi )\}_{Q\in \mathcal {P}(L)}$ and the equality

$$ \begin{align*} \sum_{Q\in \mathcal{P}(L)}\textrm{w}_{Q}'(\xi)=\textrm{w}_{L}(\xi)=\lim_{\lambda\to 0} \sum_{Q\in \mathcal{P}(L)} \textrm{d}_{Q}(\lambda)^{-1}\cdot \frac{\textrm{d}_{P}(\lambda)}{\textrm{c}_{P}(\lambda)}\cdot e^{-\left\langle \lambda,[\xi]_{P} \right \rangle}, \end{align*} $$

by the second assertion of Lemma 2.9, where for each $Q\in \mathcal {P}(L)$ we take $P\in \mathcal {P}(M)$ , $P\subset Q$ , and the limit is taken for $\lambda \in \mathfrak {a}_{L}^{*}$ generic. Now that

$$ \begin{align*} \frac{\textrm{d}_{P}(\lambda)}{\textrm{c}_{P}(\lambda)} =\frac{\mathrm{vol}\left(\mathfrak{a}_{L}/X_{*}(L)\right)}{\mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right)}\cdot \frac{\textrm{d}_{Q}(\lambda)}{\textrm{c}_{Q}(\lambda)} \end{align*} $$

and $\langle \lambda ,[\xi ]_{P} \rangle =\langle \lambda ,[\xi _{L}]_{Q} \rangle $ for any $\lambda \in \mathfrak {a}_{L}^{*}$ , we get

$$ \begin{align*} \sum_{Q\in \mathcal{P}(L)}\textrm{w}_{Q}'(\xi)=\frac{\mathrm{vol}\left(\mathfrak{a}_{L}/X_{*}(L)\right)}{\mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right)}\cdot \textrm{w}_{L}(\xi_{L}) =\frac{\mathrm{vol}\left(\mathfrak{a}_{L}/X_{*}(L)\right)}{\mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right)}\cdot \textrm{w}_{L}^{\xi_{L}}(e), \end{align*} $$

where the last equality follows from equation (2.15).

By equation (2.11), we can rewrite $J_{M}^{\xi }(\gamma )$ as

As $\textrm {v}_{M}^{Q}$ is also left $T(F)$ -invariant, we can define

Let $Q=LN_{Q}$ be the standard Levi decomposition. Let $dl$ be the Haar measure on $L(F)$ normalized by $\textrm {vol}_{dl}(L(\mathcal {O}))=1$ and let $dn $ be the Haar measure on $N_{Q}(F)$ normalized by $\textrm {vol}_{dn}\left (N_{Q}(\mathcal {O})\right )=1$ . Using Iwasawa decomposition, we can rewrite $J_{M}^{Q}(\gamma )$ as

(2.16)

where in the second and third lines we have used the equalities $\textrm {v}_{M}^{Q}(nlk)=\textrm {v}_{M}^{Q}(l)$ and $\textrm {v}_{M}^{Q}(l)=\textrm {v}_{M}^{L}(l)$ respectively, which follow directly from definitions. Notice that

where the last equality follows from Proposition 2.4. Continuing the calculation of equation (2.16), we get

Combining all the foregoing calculations, we get

$$ \begin{align*} J_{M}^{\xi}(\gamma) &= \sum_{L\in \mathcal{L}(M)}\sum_{Q\in \mathcal{P}(L)} J_{M}^{Q}(\gamma) \cdot \textrm{w}_{Q}'(\xi)=\sum_{L\in \mathcal{L}(M)}\sum_{Q\in \mathcal{P}(L)}q^{\mathrm{val}\left(\det\left(\mathrm{ad}\gamma\mid\mathfrak{n}_{Q,F}\right)\right)} J_{M}^{L}(\gamma) \cdot \textrm{w}_{Q}'(\xi)\\ &= \sum_{L\in \mathcal{L}(M)}q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad}\gamma\mid\mathfrak{g}_{F}/\mathfrak{l}_{F}\right)\right)} J_{M}^{L}(\gamma) \sum_{Q\in \mathcal{P}(L)}\textrm{w}_{Q}'(\xi)\\ &= {\mathrm{vol}\left(\mathfrak{a}_{M}/X_{*}(M)\right)}^{-1} \cdot J_{M}(\gamma), \end{align*} $$

where the last equality follows from Lemma 2.10. This finishes the proof of Theorem 2.8.

3.1. Truncations on the affine Grassmannian

Recall the following definition of Arthur [Reference Arthur1], which is a formalization of the orthogonal properties in Proposition 2.5:

Definition 3.1. A family $\Pi =(\lambda _{P})_{P\in \mathcal {P}(M)}$ of elements in $\mathfrak {a}_{M}^{G}$ is called a positive $(G,M)$ -orthogonal family if it satisfies

$$ \begin{align*} \lambda_{P_{1}}-\lambda_{P_{2}}=n_{P_{1},P_{2}}\cdot \pi_{M}^{G}\left(\beta_{P_{1},P_{2}}\right), \qquad \text{with } n_{P_{1},P_{2}}\in \textbf{R}_{\geq 0}, \end{align*} $$

for any two adjacent parabolic subgroups $P_{1}, P_{2}\in \mathcal {P}(M)$ .

Given such a positive $(G,M)$ -orthogonal family, we will denote again by $\Pi $ the convex hull of the $\lambda _{P}$ s. For $Q=LN_{Q}\in \mathcal {F}(M)$ , parallel to $\mathrm{Ec}_{M}^{Q}(x)$ , we denote by $\Pi ^{Q}$ the face of $\Pi $ whose vertices are $\lambda _{P}$ , $P\in \mathcal {P}(M), P\subset Q$ . With the projection $\pi _{M}^{L}$ , it can be seen as a positive $(L,M)$ -orthogonal family. This sets up a bijection between the set $\mathcal {F}(M)$ and the set of the faces of $\Pi $ . Moreover, we denote by $\lambda _{Q}$ or $\lambda _{Q}(\Pi )$ the element $\pi _{M,L}\left (\lambda _{P'}\right )$ for any $P'\in \mathcal {P}(M), P'\subset Q$ . One can show that $\left (\lambda _{Q}(\Pi )\right )_{Q\in \mathcal {P}(L)}$ forms a positive $(G,L)$ -orthogonal family. Later on, we also use the notation $(\lambda _{\bar {w}})_{\bar {w}\in W/W_{M}}$ for $(\lambda _{\bar {w}\cdot P})_{\bar {w}\in W/W_{M}}$ , and we use the notation $\lambda _{w}(\Pi )$ or $\lambda _{w\cdot P}(\Pi )$ to indicate the vertex of $\Pi $ indexed by $w\cdot P$ .

Following Chaudouard and Laumon [Reference Chen7], we define the truncated affine Grassmannian $\mathscr {X}(\Pi )$ to be

$$ \begin{align*} \mathscr{X}(\Pi)=\left\{x\in \mathscr{X} \mid \pi_{M}^{G}(\textrm{Ec}_{M}(x))\subset \Pi\right\}. \end{align*} $$

We want to point out that its connected components are also parametrized by $\Lambda _{G}$ , but they are not isomorphic in general. However, there is periodicity in the connected components: let $G^{\mathrm{ad}}$ be the adjoint group of G and let $c_{G}:\Lambda _{G}\to \Lambda _{G^{\mathrm{ad}}}$ be the projection induced by the natural projection $T\to T/Z_{G}$ . For $\nu ,\nu '\in \Lambda _{G}$ , we have

$$ \begin{align*} \mathscr{X}^{\nu}(\Pi)=\mathscr{X}^{\nu'}(\Pi), \qquad \text{if } c_{G}(\nu)=c_{G}(\nu'), \end{align*} $$

because they can be translated to each other by elements in $Z_{G}(F)$ .

For a regular element $\gamma \in \mathfrak {t}(\mathcal {O})$ , we can truncate the affine Springer fiber $\mathscr {X}_{\gamma }$ similarly by defining

$$ \begin{align*} \mathscr{X}_{\gamma}(\Pi)=\mathscr{X}_{\gamma}\cap \mathscr{X}(\Pi), \end{align*} $$

and the same observation on the connected components of $\mathscr {X}(\Pi )$ holds also for $\mathscr {X}_{\gamma }(\Pi )$ .

3.2. The intermediate fundamental domain

We generalize our construction of the fundamental domain $F_{\gamma }$ in [Reference Chaudouard and Laumon11].Footnote ³

Let $P_{1}=MN_{1}$ and $P_{2}=MN_{2}\in \mathcal {P}(M)$ be two adjacent parabolic subgroups. Let $m_{\alpha }$ be the unique positive integer such that the image of $\alpha ^{\vee }$ in $\Lambda _{M}$ is equal to $m_{\alpha }\cdot \beta _{P_{1},P_{2}}$ , and let

$$ \begin{align*} n(\gamma, P_{1},P_{2})=\sum_{\alpha\in \Phi\left(N_{1}, T_{\overline{F}}\right)\cap \Phi\left(N_{2}^{-}, T_{\overline{F}}\right)} \mathrm{val}(\alpha(\gamma))\cdot m_{\alpha}. \end{align*} $$

It can be verified that $n(\gamma , P_{1},P_{2})$ is an integer.

Notice that although Goresky, Kottwitz, and MacPherson work over the field $F=\textbf {C}(\!(\epsilon )\!)$ , their proof works for any field $F=k(\!(\epsilon )\!)$ with $\text {char}(k)>\lvert W\rvert $ . Their result motivates our definition:

Definition 3.2. Take a regular point $x_{0}\in \mathscr {X}_{\gamma }^{\mathrm{reg}}$ . Let

$$ \begin{align*} F_{\gamma}^{G,M}=\left\{x\in \mathscr{X}_{\gamma}\mid \textrm{Ec}_{M}(x)\subset \textrm{Ec}_{M}(x_{0}), \ \nu_{G}(x)=\nu_{G}(x_{0})\right\}. \end{align*} $$

We call it an intermediate fundamental domain of $\mathscr {X}_{\gamma }$ with respect to M.

We should have used the notation $F_{\gamma ,x_{0}}^{G,M}$ to indicate the dependence on $x_{0}$ , but they are isomorphic to each other for any choice of the regular point $x_{0}$ . Indeed, for any two regular points $x_{1},x_{2}$ , we can find $t\in T(F)$ such that $x_{1}=t\cdot x_{2}$ . Now that $\mathrm{Ec}_{M}(tx)=\mathrm{Ec}_{M}(x)+H_{M}(t), \ \forall x\in \mathscr {X}$ , the intermediate fundamental domain given by $x_{1}$ is just the translation by t of that given by $x_{2}$ . Notice that for $M=M_{0}$ , we recover the fundamental domain $F_{\gamma }$ . For simplicity, we assume that $\nu _{G}(x_{0})=0$ .

Unlike the fundamental domain, the intermediate $F_{\gamma }^{G,M}$ is no longer of finite type for $M\supsetneq M_{0}$ . Nonetheless, we have the following:

Proof. Recall that $\Lambda ^{H_{M}}=\Lambda \cap \ker (H_{M})$ by definition, and hence it preserves $F_{\gamma }^{G,M}$ because left translation by $m\in \ker (H_{M})$ does not change the polytope $\mathrm{Ec}_{M}(x)$ , due to the property

$$ \begin{align*} H_{P}(mx)=H_{M}(m)+H_{P}(x),\qquad \forall m\in M(F), \ x\in \mathscr{X}, \ P\in \mathcal{P}(M). \end{align*} $$

For the finiteness issue, let $\Lambda _{M_{0}}^{H_{M}}\subset \Lambda _{M_{0}}$ be the kernel of the natural projection $\Lambda _{M_{0}}\to \Lambda _{M}$ . By definition, we have

$$ \begin{align*} \pi_{M_{0},M}^{-1}(\textrm{Ec}_{M}(x_{0}))=\bigcup_{\nu\in \Lambda_{M_{0}}^{H_{M}}}\left(\nu+\textrm{Ec}_{M_{0}}(x_{0})\right), \end{align*} $$

which implies that

$$ \begin{align*} F_{\gamma}^{G,M}=\bigcup_{\nu\in \Lambda_{M_{0}}^{H_{M}}}\mathscr{X}_{\gamma}^{0}\left(\nu+\textrm{Ec}_{M_{0}}(x_{0})\right). \end{align*} $$

Now that $\Lambda \cong X_{*}(S)$ and $X_{*}(S)\hookrightarrow \Lambda _{M_{0}}$ is of finite index, the quotient $\Lambda ^{H_{M}}\backslash \Lambda _{M_{0}}^{H_{M}}$ is of finite cardinality. Hence the quotient $\Lambda ^{H_{M}}\backslash F_{\gamma }^{G,M}$ is dominated by the union of finitely many translations of $F_{\gamma }$ under the natural projection $F_{\gamma }^{G,M}\to \Lambda ^{H_{M}}\backslash F_{\gamma }^{G,M}$ . As $F_{\gamma }$ is of finite type, so is the quotient $\Lambda ^{H_{M}}\backslash F_{\gamma }^{G,M}$ .

A similar proof applies to the following:

Proposition 3.3. Let $\Pi $ be a regular positive $(G,M)$ -orthogonal family. For any $\nu \in \Lambda _{G}$ , the free discrete abelian group $\Lambda ^{H_{M}}$ acts freely on $\mathscr {X}_{\gamma }^{\nu }(\Pi )$ , and the quotient $\Lambda ^{H_{M}}\backslash \mathscr {X}_{\gamma }^{\nu }(\Pi )$ is of finite type. In particular,

$$ \begin{align*} \left\lvert\left(\Lambda^{H_{M}}\middle\backslash \mathscr{X}_{\gamma}^{\nu}(\Pi)\right)\left(\mathbf{F}_{q}\right)\right\rvert<\infty. \end{align*} $$

Remark 3.1. In the definition of the (weighted) orbital integral, we are concerned more about analogues of $\Lambda ^{H_{M}}\backslash \left (\mathscr {X}_{\gamma }^{\nu }(\Pi )\left (\mathbf{F}_{q}\right )\right )$ , but notice that there is bijection between

$$ \begin{align*} \left(\Lambda^{H_{M}}\middle\backslash \mathscr{X}_{\gamma}^{\nu}(\Pi)\right)\left(\mathbf{F}_{q}\right)\qquad \text{and} \qquad\Lambda^{H_{M}}\backslash \left(\mathscr{X}_{\gamma}^{\nu}(\Pi)\left(\mathbf{F}_{q}\right)\right), \end{align*} $$

because $\Lambda ^{H_{M}}$ acts freely on $\mathscr {X}_{\gamma }^{\nu }(\Pi )$ and the Galois group $\textrm {Gal}\left (\overline {\mathbf{F}}_{q}/\mathbf{F}_{q}\right )$ acts trivially on $\Lambda ^{H_{M}}$ . We will decompose the scheme $\Lambda ^{H_{M}}\backslash \mathscr {X}_{\gamma }^{\nu }(\Pi )$ in different ways, and the bijection given here implies that we can deduce equality of rational points over $\mathbf{F}_{q}$ from the decomposition of schemes.

In the following, we simplify the notation $\mathrm{Ec}_{M}(x_{0})$ to $\Sigma _{\gamma }^{G,M}$ . For $\nu \in \Lambda _{G}$ , let

$$ \begin{align*} F_{\gamma}^{G,M, \nu}:=\mathscr{X}_{\gamma}^{\nu}\left(\Sigma_{\gamma}^{G,M}\right). \end{align*} $$

As we have explained before, it depends only on the class $c_{G}(\nu )\in \Lambda _{G^{\mathrm{ad}}}$ . For $M=M_{0}$ , we simplify $\Sigma _{\gamma }^{G,M_{0}}$ to $\Sigma _{\gamma }$ and $F_{\gamma }^{G,M_{0}, \nu }$ to $F_{\gamma }^{\nu }$ .

3.3. The Arthur–Kottwitz reduction

Recall that we can reduce the geometry of $\mathscr {X}_{\gamma }$ to that of its fundamental domain by the Arthur–Kottwitz reduction [Reference Chaudouard and Laumon11]. The construction can be generalized to our current setting.

Let $Q_{0}$ be the unique parabolic subgroup in $\mathcal {P}(M)$ which contains $P_{0}$ . Let $\varsigma \in \mathfrak {a}_{M}^{G}$ be such that $\alpha (\varsigma )$ is positive but almost equal to $0$ for any $\alpha \in \Delta _{Q_{0}}$ . Let $D_{M}=(\lambda _{P})_{P\in \mathcal {P}(M)}$ be the $(G,M)$ -orthogonal family given by

(3.1) $$ \begin{align} \lambda_{P}=H_{P}(x_{0})+w\cdot \varsigma, \end{align} $$

where $w\in W$ is any element satisfying $P=w\cdot Q_{0}$ . For $Q=LN_{Q}\in \mathcal {F}(M)$ , define $R^{G,M}_{Q}$ to be the subset of $ \mathfrak {a}_{M}^{G}$ satisfying the conditions

$$ \begin{align*} \pi_{M}^{L}(a)&\subset D_{M}^{Q},\\ \alpha\left(\pi_{M,L}(a)\right)&\geq \alpha\left(\pi_{M,L}\left(\lambda_{Q}\right)\right),\qquad \forall \alpha\in \Delta_{Q}. \end{align*} $$

This gives us a partition which dates back at least to Arthur [Reference Arthur3]:

(3.2) $$ \begin{align} \mathfrak{a}_{M}^{G}=\bigcup_{Q\in \mathcal{F}(M)} R^{G,M}_{Q}. \end{align} $$

It induces a disjoint partition of $\Lambda _{M}$ via the map $\Lambda _{M}\to \mathfrak {a}_{M}^{G}$ , as we have perturbed $(H_{P}(x_{0}))_{P\in \mathcal {P}(M)}$ with $\varsigma $ . Figure 1 gives an illustration of the partition for the group $\textrm {GL}_{3}$ and $M_{0}=T=A$ .

Figure 1

Partition of $\mathfrak {a}_{A}^{G}$ for $\textrm {GL}_{3}$ .

Similar to [Reference Chaudouard and Laumon11, Lemma 3.1], we have the following result due to Proposition 3.1:

Lemma 3.4. For any $x\in \mathscr {X}_{\gamma }$ , there exists a unique $Q\in \mathcal {F}(M)$ such that

$$ \begin{align*} \pi_{M}^{G}\left(\textrm{Ec}_{M}^{Q}(x)\right)\subset R^{G,M}_{Q}. \end{align*} $$

The referee has suggested an equivalent form of the lemma, which is much easier to understand and to prove: let $\mathfrak {a}_{M}^{G}=\bigcup _{Q\in \mathcal {F}(M)} R^{\prime }_{Q}$ be the partition attached to the positive $(G,M)$ -orthogonal family $(H_{P}(x_{0})-H_{P}(x)+w\cdot \varsigma )_{P\in \mathcal {P}(M)}$ , then the statement is equivalent to the existence of a unique $Q\in \mathcal {F}(M)$ such that $0\in R^{\prime }_{Q}$ . Here the positiveness of the $(G,M)$ -orthogonal family is due to Proposition 3.1. Let

$$ \begin{align*} S_{Q}^{G,M}:=\left\{x\in \mathscr{X}_{\gamma}\mid \pi_{M}^{G}\left(\textrm{Ec}_{M}^{Q}(x)\right)\subset R_{Q}^{G,M}\right\}. \end{align*} $$

We thus get a disjoint partition

(3.3) $$ \begin{align} \mathscr{X}_{\gamma}=\mathscr{X}_{\gamma}(D_{M})\cup\bigcup_{\substack{Q\in \mathcal{F}(M)\\Q\neq G}}S^{G,M}_{Q}. \end{align} $$

For each parabolic subgroup $Q=LN_{Q}\in \mathcal {F}(M)$ , consider the restriction of the retraction $f_{Q}: \mathscr {X}\to \mathscr {X}^{L}$ to $S^{G,M}_{Q}$ ; its image is $S^{G,M}_{Q}\cap \mathscr {X}_{\gamma }^{L}$ . Recall that the connected components of $\mathscr {X}_{\gamma }^{L}$ are fibers of the map $\nu _{L}:\mathscr {X}_{\gamma }^{L}\to \Lambda _{L}$ . For $\nu \in \Lambda _{L}$ , let $\mathscr {X}_{\gamma }^{L,\nu }$ be its fiber at $\nu $ . Letting

$$ \begin{align*} S_{Q}^{G,M,\nu}=S_{Q}^{G,M}\cap f_{Q}^{-1}\left(\mathscr{X}_{\gamma}^{L,\nu}\right), \end{align*} $$

we have

$$ \begin{align*} S_{Q}^{G,M,\nu}\cap \mathscr{X}_{\gamma}^{L,\nu}=\mathscr{X}_{\gamma}^{L,\nu}\left(D_{M}^{Q}\right). \end{align*} $$

Proposition 3.5. The strata $S_{Q}^{G,M,\nu }$ are locally closed subschemes of $\mathscr {X}_{\gamma }$ , and the retraction $f_{Q}: S_{Q}^{G,M,\nu }\to \mathscr {X}_{\gamma }^{L,\nu }\left (D_{M}^{Q}\right )$ is an iterated affine fibration over $\mathbf{F}_{q}$ of dimension

$$ \begin{align*} \mathrm{val}\left(\det\left(\mathrm{ad}\left(\gamma\mid \mathfrak{n}_{Q,F}\right)\right)\right). \end{align*} $$

Indeed, by the bound on $\mathrm{Ec}_{M}(x)$ given by Proposition 3.1, we get

$$ \begin{align*} S_{Q}^{G,M,\nu}=\mathscr{X}_{\gamma}\cap f_{Q}^{-1}\left(\mathscr{X}_{\gamma}^{L,\nu}\left(D_{M}^{Q}\right)\right). \end{align*} $$

It is an iterated affine fibration over $\mathscr {X}_{\gamma }^{L,\nu }\left (D_{M}^{Q}\right )$ by Proposition 2.4.

The decomposition (3.3) can thus be refined to

(3.4) $$ \begin{align} \mathscr{X}_{\gamma}=\mathscr{X}_{\gamma}(D_{M})\cup \bigcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigcup_{\nu\in \Lambda_{L}\cap \pi_{L}\left(R^{G,M}_{Q}\right)} S_{Q}^{G,M,\nu}, \end{align} $$

where we have loosely used $\Lambda _{L}\cap \pi _{L}\left (R_{Q}^{G,M}\right )$ to mean elements in $\Lambda _{L}$ whose projection to $\mathfrak {a}_{L}^{G}$ lies in $\pi _{L}\left (R_{Q}^{G,M}\right )$ . Similar notations will be used later on. The decomposition (3.4) will also be called the Arthur–Kottwitz reduction. Notice that the stratum $S_{Q}^{G,M,\nu }$ is an iterated affine fibration over $\mathscr {X}_{\gamma }^{L,\nu }\left (D_{M}^{Q}\right )=F_{\gamma }^{L,M,\nu }$ , and the latter is related to $F_{\gamma }^{L,M}$ again by the Arthur–Kottwitz reduction, similar to what is explained in [Reference Chaudouard and Laumon11, Lemma 3.4].

As in [Reference Chaudouard and Laumon11], the existence of Arthur–Kottwitz reduction implies the following:

We can restrict the Arthur–Kottwitz reduction to the truncated affine Springer fibers. A positive $(G,M)$ -orthogonal family $\Pi =(\mu _{P})_{P\in \mathcal {P}(M)}$ is said to be regular with respect to $D_{M}$ if $\mu _{P}\in R^{G,M}_{P}, \ \forall P\in \mathcal {P}(M)$ . In this case, each $S_{Q}^{G,M,\nu }$ is either contained in $\mathscr {X}_{\gamma }^{\nu }(\Pi )$ or disjoint from it. So we have

(3.5) $$ \begin{align} \mathscr{X}_{\gamma}(\Pi)=\mathscr{X}_{\gamma}(D_{M})\cup \bigcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigcup_{\substack{\nu\in \Lambda_{L}\cap \pi_{L}\left(R^{G,M}_{Q}\right) \\ \cap \pi_{L}(\Pi)}} S_{Q}^{G,M,\nu}. \end{align} $$

The reduction can be further restricted to each connected component of $\mathscr {X}_{\gamma }(\Pi )$ :

(3.6) $$ \begin{align} \mathscr{X}_{\gamma}^{\nu_{0}}(\Pi)=\mathscr{X}_{\gamma}^{\nu_{0}}(D_{M})\cup \bigcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigcup_{\substack{\nu\in \Lambda^{\nu_{0}}_{L}\cap \pi_{L}\left(R^{G,M}_{Q}\right) \\ \cap \pi_{L}(\Pi)}} S_{Q}^{G,M, \nu}. \end{align} $$

As we have explained, left translation by elements in $\ker (H_{M})$ does not change the polytope $\mathrm{Ec}_{M}(x)$ , and hence the group $\Lambda ^{H_{M}}$ acts on each item of equation (3.6). Now that we have finiteness results – Propositions 3.2 and 3.3, combined with Proposition 3.5 and the periodicity of $F_{\gamma }^{L,M,\nu }$ in $\nu \in \Lambda _{L}$ – equation (3.6) implies equality of counting points:

Corollary 3.7. We have the equality

$$ \begin{align*} \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}_{\gamma}^{\nu_{0}}(\Pi)\right)\left(\mathbf{F}_{q}\right)\right\rvert &= \left\lvert\left(\Lambda^{H_{M}}\middle\backslash F_{\gamma}^{G,M,\nu_{0}}\right)\left(\mathbf{F}_{q}\right)\right\rvert+\sum_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\Q\neq G}}\sum_{\mu\in \Lambda_{L^{\mathrm{ad}}}} q^{\mathrm{val}\left(\det\left(\mathrm{ad}\gamma\mid\mathfrak{n}_{Q,F}\right)\right)} \\ & \quad \cdot \left\lvert\left(\Lambda^{H_{M}}\middle\backslash F_{\gamma}^{L,M,\mu}\right)\left(\mathbf{F}_{q}\right)\right\rvert\cdot \left\lvert\Lambda^{\nu_{0}}_{L}\cap \pi_{L}\left(R_{Q}^{G,M}\right)\cap \pi_{L}(\Pi)\cap c_{L}^{-1}(\mu)\right\rvert. \end{align*} $$

Notice that the term $\left \lvert \Lambda ^{\nu _{0}}_{L}\cap \pi _{L}\left (R_{Q}^{G,M}\right )\cap \pi _{L}(\Pi )\cap c_{L}^{-1}(\mu )\right \rvert $ counts the number of lattice points in a polytope. Well-known techniques from toric geometry tells us that the counting result depends quasi-polynomially on the size of the polytope.

Remark 3.2. As the foregoing constructions rely ultimately on the bound of $\mathrm{Ec}_{M}(x)$ given by Proposition 3.1, they continue to work if we replace $\Sigma _{\gamma }^{G,M}$ from the beginning by any integral positive $(G,M)$ -orthogonal family $\Sigma $ which satisfies

(3.7) $$ \begin{align} \lambda_{P_{1}}(\Sigma)-\lambda_{P_{2}}(\Sigma)=n_{P_{1},P_{2}}\cdot\beta_{P_{1},P_{2}},\qquad \text{with } n_{P_{1},P_{2}}\geq n(\gamma,P_{1},P_{2}), \end{align} $$

for any two adjacent parabolic subgroups $P_{1},P_{2}\in \mathcal {P}(M)$ . The resulting decomposition will also be called the Arthur–Kottwitz reduction.

3.4. Counting points on the intermediate fundamental domains

Although the intermediate fundamental domains $F_{\gamma }^{G,M}$ look like something new, it turns out that counting points of $\Lambda ^{H_{M}}\backslash F_{\gamma }^{G,M}$ can be reduced to counting points of the fundamental domains.

As explained in the proof of Proposition 3.2, we have

$$ \begin{align*} \pi_{M_{0},M}^{-1}\left(\Sigma_{\gamma}^{G,M}\right)=\bigcup_{\nu\in \Lambda_{M_{0}}^{H_{M}}}\left(\nu+\Sigma_{\gamma}\right) \end{align*} $$

and

$$ \begin{align*} F_{\gamma}^{G,M}=\bigcup_{\nu\in \Lambda_{M_{0}}^{H_{M}}}\mathscr{X}_{\gamma}^{0}\left(\nu+\Sigma_{\gamma}\right). \end{align*} $$

Let $P_{0}^{M}=P_{0}\cap M$ , and let $\beta _{1}^{\vee },\dotsc ,\beta _{r'}^{\vee }\in \Lambda _{M_{0}}^{H_{M}}$ be a basis of $\mathfrak {a}_{M_{0}}^{M}$ which is positive with respect to $P_{0}^{M}$ . For $\mu _{1},\mu _{2}\in \Lambda _{M_{0}}^{H_{M}}$ , we say that $\mu _{1}\preccurlyeq _{P_{0}^{M}} \mu _{2}$ if $\mu _{2}-\mu _{1}$ is a linear combination of $\beta _{i}^{\vee }$ s with positive coefficients. This defines a partial order $\preccurlyeq _{P_{0}^{M}}$ on $\Lambda _{M_{0}}^{H_{M}}$ . For $\mu \in \Lambda _{M_{0}}^{H_{M}}$ , define

$$ \begin{align*} \Lambda_{M_{0},\preccurlyeq \mu}^{H_{M}}:=\left\{\nu\in \Lambda_{M_{0}}^{H_{M}}\mid \nu\preccurlyeq_{P_{0}^{M}} \mu\right\} \qquad \text{and} \qquad \Pi_{\gamma,\preccurlyeq \mu}^{G,M}:=\bigcup_{\nu\in \Lambda_{M_{0},\preccurlyeq \mu}^{H_{M}}}\left(\nu+\Sigma_{\gamma}\right). \end{align*} $$

Then $\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}$ is a semi-infinite polytope in $\mathfrak {a}_{M_{0}}^{G}$ and $\Lambda _{M_{0},\preccurlyeq \mu }^{H_{M}}$ is the integral points in it. Similar definitions hold for $\Lambda _{M_{0},\prec \mu }^{H_{M}}$ and $\Pi _{\gamma ,\prec \mu }^{G,M}$ . But notice that $\Pi _{\gamma ,\prec \mu }^{G,M}$ is not a semi-infinite polytope: it is the union of finitely many semi-infinite polytopes of the form $\Pi _{\gamma ,\preccurlyeq \mu '}^{G,M}$ , $\mu '\in \Lambda _{M_{0}}^{H_{M}}$ . Define

$$ \begin{align*} F_{\gamma,\preccurlyeq \mu}^{G,M}:=\mathscr{X}_{\gamma}^{0}\left(\Pi_{\gamma,\preccurlyeq \mu}^{G,M}\right) =\bigcup_{\nu\in \Lambda_{M_{0},\preccurlyeq \mu}^{H_{M}}}\mathscr{X}_{\gamma}^{0}\left(\nu+\Sigma_{\gamma}\right) \end{align*} $$

and similarly

$$ \begin{align*} F_{\gamma,\prec \mu}^{G,M}:=\bigcup_{\nu\in \Lambda_{M_{0},\prec \mu}^{H_{M}}}\mathscr{X}_{\gamma}^{0}\left(\nu+\Sigma_{\gamma}\right). \end{align*} $$

It is the union of finitely many $F_{\gamma ,\preccurlyeq \mu '}^{G,M}$ , $\mu '\in \Lambda _{M_{0}}^{H_{M}}$ . Define

$$ \begin{align*} F_{\gamma,\mu}^{G,M}:=F_{\gamma,\preccurlyeq \mu}^{G,M}\Big\backslash F_{\gamma,\prec \mu}^{G,M}. \end{align*} $$

Being a difference of closed subschemes, $F_{\gamma ,\mu }^{G,M}$ is locally closed in $\mathscr {X}_{\gamma }^{0}$ . As $F_{\gamma ,\preccurlyeq \mu }^{G,M}$ is semi-infinite unions of translations of the fundamental domains, they are all isomorphic, and similarly for $F_{\gamma ,\prec \mu }^{G,M}$ . Hence $F_{\gamma ,\mu }^{G,M}$ are all isomorphic. Moreover,

$$ \begin{align*} F_{\gamma,\preccurlyeq \mu}^{G,M}=\bigsqcup_{\nu\in \Lambda_{M_{0},\preccurlyeq \mu}^{H_{M}}} F_{\gamma,\nu}^{G,M} \end{align*} $$

by induction, and

$$ \begin{align*} F_{\gamma}^{G,M}=\lim_{\mu\to \infty} F_{\gamma,\preccurlyeq \mu}^{G,M}=\bigsqcup_{\nu\in \Lambda_{M_{0}}^{H_{M}}} F_{\gamma,\nu}^{G,M}. \end{align*} $$

From all these we conclude the following:

Proposition 3.8. $F_{\gamma ,\mu }^{G,M}$ is isomorphic to each other for all $\mu \in \Lambda _{M_{0}}^{H_{M}}$ . In particular,

$$ \begin{align*} \left\lvert\left(\Lambda^{H_{M}}\middle\backslash F_{\gamma}^{G,M}\right)\left(\mathbf{F}_{q}\right)\right\rvert=\left\lvert\Lambda^{H_{M}}\middle\backslash \Lambda_{M_{0}}^{H_{M}}\right\rvert\cdot \left\lvert F_{\gamma,\mu}^{G,M}\left(\mathbf{F}_{q}\right)\right\rvert. \end{align*} $$

Counting points on $F_{\gamma ,\mu }^{G,M}$ can be reduced to the fundamental domains via a process similar to the Arthur–Kottwitz reduction. To begin with, notice that $\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}$ is bounded only in directions that are positive with respect to $P_{0}^{M}$ . Indeed, its vertices are indexed by $P\in \mathcal {P}(M_{0})$ satisfying $P\cap M=P_{0}^{M}$ , and its faces by $Q\in \mathcal {F}(M_{0})$ such that $Q\cap M\supset P_{0}^{M}$ . Then we define a semi-infinite polytope $\Pi _{\gamma ,\preccurlyeq \mu ^{-}}^{G,M}$ which is a translation of $\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}$ : let $\Delta ^{M}$ be the set of simple roots in $\Phi (M,A)$ with respect to $B_{0}\cap M$ , and let $\Delta ^{M}_{P_{0}^{M}}=\Delta ^{M}\cap \Phi \left (N_{P_{0}^{M}},A\right )$ , with $N_{P_{0}^{M}}$ the unipotent radical of $P_{0}^{M}$ . For $\alpha \in \Delta ^{M}_{P_{0}^{M}}$ , let $\left \{\omega _{\alpha }^{\vee }\right \}$ be the corresponding fundamental coweights. Let

(3.8) $$ \begin{align} \mu^{-}=\mu-\pi_{2}\left(\textstyle\sum_{\alpha\in \Delta^{M}_{P_{0}^{M}}}\omega_{\alpha}^{\vee}\right), \end{align} $$

where $\pi _{2}$ is the projection to the second factor in the orthogonal decomposition $\mathfrak {a}_{A}^{G}=\mathfrak {a}_{A}^{M_{0}}\oplus \mathfrak {a}_{M_{0}}^{M}\oplus \mathfrak {a}_{M}^{G}$ . Then $\Pi _{\gamma ,\preccurlyeq \mu ^{-}}^{G,M}$ is a translation of $\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}$ by the same vector. Now let $\varsigma \in \mathfrak {a}_{M_{0}}^{G}$ be a generic element such that $\alpha (\varsigma )$ is positive but almost equal to $0$ for any $\alpha \in \Delta _{P_{0}}$ . We perturb the semi-infinite polytope $\Pi _{\gamma ,\preccurlyeq \mu ^{-}}^{G,M}$ to a similar one $\Pi '$ , with vertices

$$ \begin{align*} \lambda_{P}(\Pi')=\lambda_{P}\left(\Pi_{\gamma,\preccurlyeq \mu^{-}}^{G,M}\right)+w\cdot \varsigma, \qquad \forall P\in \mathcal{P}(M_{0}), \ P\cap M=P_{0}^{M}, \end{align*} $$

where $w\in W$ is any element satisfying $P=w\cdot P_{0}$ . Both $\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}$ and $\Pi '$ can be seen as limits of positive $(G,M_{0})$ -orthogonal families containing $\Sigma _{\gamma }$ , hence we can apply an analogue of the Arthur–Kottwitz reduction to get a decomposition of the complement $\mathscr {X}_{\gamma }^{0}\left (\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}\right )\Big \backslash \mathscr {X}_{\gamma }^{0}(\Pi ')$ . For $Q=LN_{Q}\in \mathcal {F}(M_0)$ satisfying $Q\cap M\supset P_{0}^{M}$ , define $R_{\Pi ',Q}$ to be the subset of $ \mathfrak {a}_{M_0}^{G}$ satisfying conditions

$$ \begin{align*} \pi_{M_{0}}^{L}(a)&\subset {\Pi'}^{Q},\\ \alpha\left(\pi_{M_{0},L}(a)\right)&\geq \alpha\left(\pi_{M_{0},L}\left(\lambda_{Q}(\Pi')\right)\right),\qquad \forall \alpha\in \Delta_{Q}. \end{align*} $$

This gives us a partition

(3.9) $$ \begin{align} \mathfrak{a}_{M_0}^{G}=\Pi'\cup \bigcup_{\substack{Q\in \mathcal{F}\left(M_0\right), Q\neq G\\ Q\cap M\supset P_{0}^{M}}} R_{\Pi',Q}. \end{align} $$

It induces a disjoint partition of $\Lambda _{M_{0}}$ . For $G=\textrm {GL}_{3}$ , $\gamma $ split, and $M=M_{\alpha _{12}}$ , we get Figure 2.

Figure 2

Arthur–Kottwitz reduction for $\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}$ .

Running the same construction as in §3.3, for $Q=LN_{Q}\in \mathcal {F}(M_0)$ , $Q\neq G$ and $Q\cap M\supset P_{0}^{M}$ , define

$$ \begin{align*} S_{\Pi',Q}:=\left\{x\in \mathscr{X}_{\gamma}\mid \pi_{M_{0}}^{G}\left(\textrm{Ec}_{M_{0}}^{Q}(x)\right)\subset R_{\Pi', Q}\right\}, \end{align*} $$

and let

$$ \begin{align*} S_{\Pi',Q}^{\nu}=S_{\Pi',Q}\cap f_{Q}^{-1}\left(\mathscr{X}_{\gamma}^{L,\nu}\right),\qquad \forall \nu\in \Lambda_{L}. \end{align*} $$

We get a disjoint partition

(3.10) $$ \begin{align} \mathscr{X}_{\gamma}^{0}\left(\Pi_{\gamma,\preccurlyeq \mu}^{G,M}\right) = \mathscr{X}_{\gamma}^{0}(\Pi')\cup \bigcup_{\substack{Q\in \mathcal{F}\left(M_0\right),Q\neq G\\ Q\cap M\supset P_{0}^{M}}}\bigcup_{\substack{\nu\in \Lambda^{0}_{L}\cap \pi_{L}\left(R_{\Pi',Q}\right) \\ \cap \pi_{L}\left(\Pi_{\gamma,\preccurlyeq \mu}^{G,M}\right)}} S_{\Pi', Q}^{\nu}. \end{align} $$

The strata $S_{\Pi ', Q}^{\nu }$ are locally closed subschemes of $\mathscr {X}_{\gamma }$ , and the retraction $f_{Q}: S_{\Pi ',Q}^{\nu }\to \mathscr {X}_{\gamma }^{L,\nu }(\Pi ^{'Q})$ is an iterated affine fibration over $\mathbf{F}_{q}$ of dimension $ \mathrm{val}\left (\det \left (\mathrm{ad}\left (\gamma \mid \mathfrak {n}_{Q,F}\right )\right )\right ). $

Proposition 3.9. We have the equality

$$ \begin{align*} F_{\gamma,\mu}^{G,M}=\bigcup_{\substack{Q\in \mathcal{F}\left(M_0\right),Q\neq G\\ Q\cap M=P_{0}^{M}}}\bigcup_{\substack{\nu\in \Lambda^{0}_{L}\cap \pi_{L}\left(R_{\Pi',Q}\right) \\ \cap \pi_{L}\left(\Pi_{\gamma,\preccurlyeq \mu}^{G,M}\right)}} S_{\Pi', Q}^{\nu}. \end{align*} $$

Moreover, the index set $\Lambda ^{0}_{L}\cap \pi _{L}\left (R_{\Pi ',Q}\right ) \cap \pi _{L}\left (\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}\right )$ consists of at most one element, and is nonempty if and only if Q is not contained in any $Q'\in \mathcal {F}(M)$ .

Proof. For the first assertion, by construction, the points $x\in F_{\gamma ,\,\mu }^{G,M}$ are characterized by the property

$$ \begin{align*} \textrm{Ec}_{M_{0}}^{Q}(x)\subset \left(\mu+\Sigma_{\gamma}\right)^{Q} \end{align*} $$

for some $Q\in \mathcal {F}(M_0)$ , $Q\neq G, Q\cap M=P_{0}^{M}$ . Since this is also the property characterizing points on the right-hand side of the equality, we get the equality as claimed. The second assertion follows from the observation that $\Pi '$ is a slight expansion of $\Pi _{\gamma ,\preccurlyeq \mu ^{-}}^{G,M}$ , and hence the regions $R_{\Pi ',Q}$ , Q contained in some maximal parabolic subgroup in $\mathcal {F}(M)$ , contain no elements in $\Lambda _{L}^{0}\cap \pi _{L}\left (\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}\right )$ .

If the index set $\Lambda ^{0}_{L}\cap \pi _{L}\left (R_{\Pi ',Q}\right ) \cap \pi _{L}\left (\Pi _{\gamma ,\preccurlyeq \mu }^{G,M}\right )$ is nonempty, we denote by $\nu _{Q}$ the unique element in it. Let $\mu _{Q}\in \Lambda _{L}^{0}$ be the unique element such that $\alpha \left (\mu _{Q}\right )=1$ for all $\alpha \in \Delta _{Q}$ ; then we have

$$ \begin{align*} \mathscr{X}_{\gamma}^{L,\nu_{Q}}\left(\Pi^{'Q}\right)\cong F_{\gamma}^{L,\mu_{Q}}. \end{align*} $$

Combining with the fact that

$$ \begin{align*} f_{Q}: S_{\Pi',Q}^{\nu_{Q}}\to \mathscr{X}_{\gamma}^{L,\nu_{Q}}\left(\Pi^{'Q}\right) \end{align*} $$

is an iterated affine fibration, we get the following:

Corollary 3.10. For $M\in \mathcal {L}(M_{0})$ , $M\neq M_{0}$ , we have the equality

$$ \begin{align*} \left\lvert F_{\gamma,\mu}^{G,M}\left(\mathbf{F}_{q}\right)\right\rvert= \sum_{\substack{Q=LN_{Q}\in \mathcal{F}\left(M_0\right),Q\neq G\\ \text{satisfying }(*)}} q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad}\gamma\mid\mathfrak{g}_{F}/\mathfrak{l}_{F}\right)\right)}\cdot \left\lvert F_{\gamma}^{L,\mu_{Q}}\left(\mathbf{F}_{q}\right)\right\rvert, \end{align*} $$

where $(*)$ refers to the condition that $Q\cap M=P_{0}^{M}$ and $Q\nsubseteq Q', \forall Q'\in \mathcal {F}(M)$ .

Notice that the equation does not involve the fundamental domain $F_{\gamma }$ . Together with Corollary 3.7 and Proposition 3.8, we get an expression of $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash \mathscr {X}_{\gamma }^{\nu _{0}}(\Pi )\right )\left (\mathbf{F}_{q}\right )\right \rvert $ in terms of $\left \lvert F_{\gamma }^{L,\mu _{Q}}\left (\mathbf{F}_{q}\right )\right \rvert $ , $L\in \mathcal {L}(M_{0}), L\neq G$ . Recalling that counting points on $F_{\gamma }^{L,\mu _{Q}}$ can be reduced to counting points on $F_{\gamma }^{L'}$ , $L'\in \mathcal {L}(M_{0}), L'\subset L$ , by the Arthur–Kottwitz reduction we get an expression of $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash \mathscr {X}_{\gamma }^{\nu _{0}}(\Pi )\right )\left (\mathbf{F}_{q}\right )\right \rvert $ in terms of $\left \lvert F_{\gamma }^{L'}\left (\mathbf{F}_{q}\right )\right \rvert $ , $L'\in \mathcal {L}(M_{0})$ .

4.1. Harder–Narasimhan reduction on the affine Springer fibers

We have introduced a notion of $\xi $ -stability on the affine Grassmannian and constructed the associated Harder–Narasimhan reduction in [Reference Chen10]. In this section, we generalize it to a broader setup. The following lemma is an analogue of [Reference Chen8, Proposition 5.6.1]. Let S be an affine $\mathbf{F}_{q}$ -scheme and set $x\in \mathscr {X}(S)$ . For every point $s\in S$ , let $x_{s}\in \mathscr {X}(k(s))$ be the base change of x to the residue field $k(s)$ of S at s. Let $\mathcal {C}_{x}$ be the map on S which sends every point $s\in S$ to the convex polytope $\mathrm{Ec}(x_{s})$ .

Lemma 4.1. Suppose that S is noetherian. The map $\mathcal {C}_{x}$ from S to the set of convex polytopes in $\mathfrak {a}_{A}^{G}$ ordered by inclusion is lower semicontinuous. In other words, for any convex polytope $\Xi $ , the set

$$ \begin{align*} \{s\in S\mid\mathcal{C}_{x}(s)\supset \Xi\} \end{align*} $$

is open.

Proof. To begin with, we show that $\mathcal {C}_{x}$ is constructible and takes only finitely many values. Passing to the irreducible components of S, we can suppose that S is irreducible. Let $\eta $ be the generic point of S, and let $g_{\eta }\in G\big (k(\eta )(\!(\epsilon )\!)\big )$ be a representative of $x_{\eta }$ . For $B=AN\in \mathcal {P}(A)$ , we have the Iwasawa decomposition

$$ \begin{align*} g_{\eta}=n_{\eta}a_{\eta}k_{\eta}, \end{align*} $$

where $n_{\eta }\in N\big (k(\eta )(\!(\epsilon )\!)\big )$ , $a_{\eta }\in A\big (k(\eta )(\!(\epsilon )\!)\big )$ , and $k_{\eta }\in G\big (k(\eta )[\![\epsilon ]\!]\big )$ . Because $\eta $ is the generic point and the map $\nu _{A}:\mathscr {X}^{A}\to X_{*}(A)$ is essentially the valuation map, there exists an open subscheme U of S such that $H_{B}(x_{s})=\nu _{A}\left (a_{\eta }\right )$ for any $x\in U$ . As $\mathrm{Ec}(x_{s})$ is the convex hull of $H_{B}(x_{s})$ , $B\in \mathcal {P}(A)$ , the map $\mathcal {C}_{x}$ takes the constant value $\mathrm{Ec}\left (x_{\eta }\right )$ on the intersection of all such open subschemes U. This proves the constructibility of $\mathcal {C}_{x}$ . By noetherian induction, the map $\mathcal {C}_{x}$ takes only finitely many values.

To finish the proof, we only need to show that the map $\mathcal {C}_{x}$ decreases under specialization. In other words, let S be the spectrum of a discrete valuation ring and let s be its special point and $\eta $ its generic point. Then

$$ \begin{align*} \textrm{Ec}(x_{s})\subset \textrm{Ec}\left(x_{\eta}\right). \end{align*} $$

This is equivalent to the assertion that

(4.1) $$ \begin{align} f_{B}(x_{s})\prec_{B} f_{B}\left(x_{\eta}\right), \qquad \forall B\in \mathcal{P}(A), \end{align} $$

where $\prec _{B}$ is the order on $X_{*}(A)$ such that $\mu _{1}\prec _{B}\mu _{2}$ if and only if $ \mu _{2}-\mu _{1}$ is a positive linear combination of positive coroots with respect to B.

Let $\mu =f_{B}\left (x_{\eta }\right )\in X_{*}(A)$ . By definition, we have

$$ \begin{align*} x_{\eta}\in U_{B}(\!(\epsilon)\!)\epsilon^{\mu}G[\![\epsilon]\!]/G[\![\epsilon]\!], \end{align*} $$

where $U_{B}$ is the unipotent radical of B. So

$$ \begin{align*} x_{s}\in \overline{x_{\eta}}\subset \overline{U_{B}(\!(\epsilon)\!)\epsilon^{\mu}G[\![\epsilon]\!]/G[\![\epsilon]\!]}=\bigcup_{\substack{\lambda\in X_{*}(A)\\ \lambda\prec_{B}\mu}} U_{B}(\!(\epsilon)\!)\epsilon^{\lambda}G[\![\epsilon]\!]/G[\![\epsilon]\!], \end{align*} $$

which implies the relation (4.1).

As $\mathrm{Ec}_{M}(x)=\pi _{M}(\mathrm{Ec}(x))$ , the subset

$$ \begin{align*} \mathscr{X}^{\xi}=\left\{x\in \mathscr{X}\mid \xi\in \pi^{G}(\textrm{Ec}_{M}(x))\right\} \end{align*} $$

is an open sub-ind- $\mathbf{F}_{q}$ -scheme of $\mathscr {X}$ by Lemma 4.1. This being shown, all the other constructions of [Reference Chen10] generalize.

Harder–Narasimhan reduction works as well in this setting. For $Q=LN_{Q}\in \mathcal {F}(M)$ , let $\Phi _Q(G,L)$ be the image of $\Phi \left (N_{Q},A\right )$ in $\left (\mathfrak {a}_L^G\right )^{*}$ . For any point $a\in \mathfrak {a}_L^G$ , we define a cone in $\mathfrak {a}_L^G$ ,

$$ \begin{align*} D_Q(a)=\left\{y\in\mathfrak{a}_L^G\mid \alpha(y-a)\geq 0, \ \forall \alpha\in \Phi_Q(G,L)\right\}. \end{align*} $$

Definition 4.2. For any geometric point $x\in \mathscr {X}$ , we define a semicylinder $C_Q(x)$ in $\mathfrak {a}_{M}^G$ by

$$ \begin{align*} C_Q(x)=\pi_{M}^{L,-1}\left(\textrm{Ec}_{M}^{L}\left(f_{Q}(x)\right)\right)\cap \pi_{M,L}^{-1}\left(D_Q\left(H_{Q}(x)\right)\right). \end{align*} $$

By definition, we get a partition

$$ \begin{align*} \mathfrak{a}_{M}^{G}=\pi^{G}(\textrm{Ec}_{M}(x))\cup\bigcup_{\substack{Q\in \mathcal{F}(M)\\Q\neq G}}C_Q(x), \end{align*} $$

for which the interior of any two parts does not intersect. The picture is similar to Figure 1. Hence for any $x\notin \mathscr {X}^{\xi }$ , there exists a unique parabolic subgroup $Q\in \mathcal {F}(M)$ such that $\xi \in C_Q(x)$ , as $\xi $ is generic. In this case, $f_Q(x)\in \mathscr {X}^L$ is $\xi ^{L}$ -stable, where $\xi ^{L}=\pi _{M}^{L}(\xi )\in \mathfrak {a}_{M}^L$ . Let

$$ \begin{align*} X_Q=\{x\in \mathscr{X}\mid \xi\in C_{Q}(x)\}. \end{align*} $$

We have the decomposition of the affine Grassmannian

(4.2) $$ \begin{align} \mathscr{X}=\mathscr{X}^{\xi}\sqcup \bigsqcup_{\substack{Q\in \mathcal{F}(M)\\Q\neq G}}X_Q. \end{align} $$

For $Q\in \mathcal {P}(L)$ , let $Q^{-}$ be the parabolic subgroup opposite to Q with respect to L. Let $\Lambda ^{\xi }_{L,Q}=\left (\pi _{L}^{G}\right )^{-1}\left (D_{Q^{-}}(\xi _{L})\right )\cap \Lambda _{L}$ , we have the disjoint partition

$$ \begin{align*} \Lambda_L=\bigsqcup_{Q\in \mathcal{P}(L)}\Lambda^{\xi}_{L,Q}. \end{align*} $$

For $\lambda \in \Lambda _{L}$ , let $ \mathscr {X}^{L,\lambda ,\xi ^L}=\mathscr {X}^{L,\xi ^L}\cap \mathscr {X}^{L,\lambda }. $ The stratum $X_{Q}$ can be further decomposed into $N_{Q}(\!(\epsilon )\!)$ -orbits

$$ \begin{align*} X_{Q}=\bigsqcup_{\lambda\in \Lambda^{\xi}_{L,Q}} N_{Q}(\!(\epsilon)\!)\mathscr{X}^{L,\lambda,\xi^{L}}. \end{align*} $$

Each orbit is locally closed in $\mathscr {X}$ , and they are infinite-dimensional homogeneous affine fibrations on $\mathscr {X}^{L,\lambda ,\xi ^{L}}$ under the retraction $f_{Q}$ . The foregoing discussions can be summarized as follows:

Theorem 4.2. The affine Grassmannian can be decomposed as

$$ \begin{align*} \mathscr{X}=\mathscr{X}^{\xi}\sqcup \bigsqcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigsqcup_{\lambda\in \Lambda^{\xi}_{L,Q}} N_{Q}(\!(\epsilon)\!)\mathscr{X}^{L,\lambda,\xi^{L}}. \end{align*} $$

Each stratum $N_{Q}(\!(\epsilon )\!)\mathscr {X}^{L,\lambda ,\xi ^{L}}$ is an infinite-dimensional homogeneous affine fibration over $\mathscr {X}^{L,\lambda ,\xi ^{L}}$ .

Now that $\gamma \in \mathfrak {m}(F)$ , we can restrict these constructions to $\mathscr {X}_{\gamma }$ . Let $\mathscr {X}_{\gamma }^{\xi }=\mathscr {X}_{\gamma }\cap \mathscr {X}^{\xi }$ ; it is an open subscheme of $\mathscr {X}_{\gamma }$ . As $T(F)\xrightarrow {H_{M}} X_{*}(M)$ is surjective, the connected components of $\mathscr {X}_{\gamma }^{\xi }$ can be translated to each other by elements of $T(F)$ . Moreover, for different choices of generic element $\xi $ , $\xi '\in \mathfrak {a}_{M}^{G}$ , the corresponding $\mathscr {X}_{\gamma }^{\xi },\mathscr {X}_{\gamma }^{\xi '}$ can be translated to each other by elements of $T(F)$ . Hence $\mathscr {X}_{\gamma }^{\xi }$ does not depend on the choice of $\xi $ .

The Harder–Narasimhan reduction restricts to

(4.3) $$ \begin{align} \mathscr{X}_{\gamma}=\mathscr{X}_{\gamma}^{\xi}\sqcup \bigsqcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigsqcup_{\lambda\in \Lambda^{\xi}_{L,Q}}\left(\mathscr{X}_{\gamma}\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}}\right). \end{align} $$

By Proposition 2.4, the retraction

$$ \begin{align*} f_{Q}:\mathscr{X}_{\gamma}\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}}\to \mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}} \end{align*} $$

is an iterated affine fibration over $\mathbf{F}_{q}$ of relative dimension $ \mathrm{val}\left (\det \left (\mathrm{ad}(\gamma )\mid \mathfrak {n}_{Q}(F)\right )\right ). $

Coming back to the weighted orbital integrals, with the definition for general reductive algebraic groups as explained in Remark 2.1, Proposition 2.7 can be reformulated as follows:

Proposition 4.3. Let $\xi \in \mathfrak {a}_{M}^{G}$ be a generic element. Then

$$ \begin{align*} J_{M}^{\xi}(\gamma)=\mathrm{vol}_{dt}\left(\Lambda^{H_{M}}\middle\backslash T(F)_{M}^{1}\right)^{-1}\cdot \left\lvert\Lambda^{H_{M}}\middle\backslash \left(\left(\mathscr{X}^{G_{\text{der}}}\cap\mathscr{X}_{\gamma}^{\xi}\right)\left(\mathbf{F}_{q}\right)\right)\right\rvert. \end{align*} $$

In particular, let $\xi _{0}\in \mathfrak {a}_{M_{0}}^{G}$ be a generic element. Then

$$ \begin{align*} J_{M_{0}}^{\xi_{0}}(\gamma)=\mathrm{vol}_{dt}\left(T(F)^{1}\right)^{-1}\cdot \left\lvert\left(\mathscr{X}^{G_{\text{der}}}\cap\mathscr{X}_{\gamma}^{\xi}\right)\left(\mathbf{F}_{q}\right)\right\rvert. \end{align*} $$

Proof. When G is semisimple, the proposition is a reformulation of Proposition 2.7. The complexity arises when G has nontrivial connected center.

As T is totally ramified over F, with the exact sequence (2.2) we see that the morphism $T(F)\xrightarrow {\nu _{G}} \Lambda _{G}$ is surjective, hence $G(F)=T(F)G_{\text {der}}(F)$ , and so

with $T_{G_{\text {der}}}=T\cap G_{\text {der}}$ and $M_{G_{\text {der}}}=M\cap G_{\text {der}}$ . Following calculations in Proposition 2.7, we get a result similar to what we claim, with $\Lambda ^{H_{M}}$ replaced by $\Lambda \cap G_{\text {der}}(F)\cap \ker \left (H_{M_{G_{\text {der}}}}\right )$ and $T(F)_{M}^{1}$ replaced by $T_{G_{\text {der}}}(F)_{M_{G_{\text {der}}}}^{1}$ . Noting that $\ker (H_{M})=M_{\text {der}}(F)\cdot M(\mathcal {O})$ by [Reference Chen7, Lemma 6.1], we have

$$ \begin{align*} \Lambda\cap G_{\text{der}}(F)\cap \ker\left(H_{M_{G_{\text{der}}}}\right)=\Lambda\cap \ker(H_{M})=\Lambda^{H_{M}} \end{align*} $$

and

$$ \begin{align*} T_{G_{\text{der}}}(F)_{M_{G_{\text{der}}}}^{1}=T(F)\cap G_{\text{der}}(F)\cap \ker\left(H_{M_{G_{\text{der}}}}\right)=T(F)\cap \ker(H_{M})=T(F)_{M}^{1}, \end{align*} $$

and the proposition is proved.

The volume factors have been calculated in equation (2.5).

4.2. Harder–Narasimhan reduction for the truncated affine Springer fibers

In contrast to the Arthur–Kottwitz reduction, the Harder–Narasimhan reduction does not work well on the truncated affine Springer fiber $\mathscr {X}_{\gamma }(\Pi )$ . We need to cut it into two parts, the tail and the main body. The Harder–Narasimhan reduction works well on the main body, and we can handle the tail with the Arthur–Kottwitz reduction.

For $Q\in \mathcal {F}(M)$ , $Q\neq G$ , we define the positive $(G,M)$ -orthogonal family $ E_{Q}(\Pi ) $ , which as a polytope is the union of the translations $\pi ^{G}\left (\Sigma _{\gamma }^{G,M}+\lambda \right )$ , $\lambda \in \Lambda _{M}$ , such that $ \pi ^{G}\left (\Sigma _{\gamma }^{G,M}+\lambda \right )^{Q}\subset \Pi ^{Q}. $ Let

$$ \begin{align*} {}^{t}\mathscr{X}_{\gamma}(\Pi)=\bigcup_{\substack{Q\in \mathcal{F}(M)\\ Q\neq G}}\mathscr{X}_{\gamma}\left(E_{Q}(\Pi)\right), \qquad {}^{m}\mathscr{X}_{\gamma}(\Pi)=\mathscr{X}_{\gamma}(\Pi)\backslash {}^{t}\mathscr{X}_{\gamma}(\Pi). \end{align*} $$

We call them the tail and the main body of $\mathscr {X}_{\gamma }(\Pi )$ , and they are respectively closed and open subschemes of $\mathscr {X}_{\gamma }(\Pi )$ . Figure 3 gives an example of $E_{Q}(\Pi )$ for the group $G=\textrm {GL}_{3}$ when $M=A$ .

Figure 3

$E_{Q}(\mathrm{I{\kern0.5pt}I})$ for $\textrm {GL}_{3}$ when $M_{0}=A$ .

Before proceeding, we make precise the condition of $\Pi $ being sufficiently regular. We would like it to satisfy the following conditions:

1. (1) $\Pi $ is $\Sigma _{\gamma }$ -regular.

2. (2) For all $P,Q\in \mathcal {F}(M)$ , $E_{P}(\Pi )\cap E_{Q}(\Pi )=E_{P\cap Q}(\Pi )$ .

3. (3) The complement $\Pi \backslash \bigcup _{\substack {Q\in \mathcal {F}(M)\\ Q\neq G}}E_{Q}(\Pi )$ is a polytope associated to a positive $(G,M)$ -orthogonal family; let $\Pi _{0}$ be a slight shrinking of it (the definition is similar to equation (3.1), with plus sign replaced by a minus). We require that $\Pi _{0}$ be sufficiently large: for all $Q=LN_{Q}\in \mathcal {F}(M)$ , the face $\Pi _{0}^{Q}$ contains the translations of $\Sigma _{\gamma }^{Q}$ in $\mathfrak {a}_{M}^{L}$ which have $\xi ^{L}$ as one of their vertices.

Remark 4.2. As $\Pi _{0}$ is convex, condition (3) implies that for any $\nu \in \Lambda _{L,Q}^{\xi }\cap \pi _{L}(\Pi _{0})$ , the intersection $\Pi _{0}\cap \pi _{M_{0},L}^{-1}(\nu )$ contains translations of $\Sigma _{\gamma }^{Q}$ in the hyperplane $\pi _{M_{0},L}^{-1}(\nu )$ which have $\xi ^{L}$ as one of their vertices. By the definition of $\xi $ -stability, this implies

$$ \begin{align*} \mathscr{X}_{\gamma}^{L,\nu,\xi^{L}}\subset \mathscr{X}_{\gamma}^{L,\nu}\left(\Pi_{0}\cap \pi_{M_{0},L}^{-1}(\nu)\right),\qquad \forall \nu\in \Lambda_{L,Q}^{\xi}\cap \pi_{L}(\Pi_{0}). \end{align*} $$

Actually, this is the reason to impose condition (3).

4.2.1. The main body

By definition, a Harder–Narasimhan stratum

$$ \begin{align*} N_{Q}(\!(\epsilon )\!)\mathscr {X}_{\gamma }^{L,\nu ,\xi ^{L}}\cap \mathscr {X}_{\gamma },\ \nu \in \Lambda _{L,Q}^{\xi }, \end{align*} $$

intersects nontrivially with $\mathscr {X}_{\gamma }(\Pi )$ if and only if $\nu \in \Lambda _{L,Q}^{\xi }\cap \pi _{L}(\Pi )$ . So, after restriction, the Harder–Narasimhan reduction becomes

$$ \begin{align*} \mathscr{X}_{\gamma}(\Pi)=\mathscr{X}_{\gamma}^{\xi}\sqcup \bigsqcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigsqcup_{\lambda\in \Lambda^{\xi}_{L,Q}\cap \pi_{L}(\Pi)}\left(\mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}}\right). \end{align*} $$

The problem is that the retraction

$$ \begin{align*} f_{Q}: \mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}} \to \mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}} \end{align*} $$

is not necessarily an iterated affine fibration. This problem disappears on the main body ${}^{m}\mathscr {X}_{\gamma }(\Pi )$ . We begin by analyzing the polytope $\mathrm{Ec}(x)$ , $x\in \mathscr {X}_{\gamma }(\Pi )$ .

Lemma 4.4. For $x\in \mathscr {X}_{\gamma }(\Pi )$ , suppose that

$$ \begin{align*} \pi^{G}(\textrm{Ec}_{M}(x))\subset \bigcup_{\substack{Q\in \mathcal{F}(M)\\ Q\neq G}} E_{Q}(\Pi). \end{align*} $$

Then $\pi ^{G}(\mathrm{Ec}_{M}(x))\subset E_{Q}(\Pi )$ for some $Q\in \mathcal {F}(M),\,Q\neq G$ .

Proof. By Proposition 3.1, it is enough to prove the lemma for $x\in \mathscr {X}_{\gamma }^{\mathrm{reg}}$ . In this case, the polytope $\pi ^{G}(\mathrm{Ec}_{M}(x))$ is a translation of $\Sigma _{\gamma }^{G,M}$ . As $\bigcup _{\substack {Q\in \mathcal {F}(M)\\ Q\neq G}} E_{Q}(\Pi )$ is the union of translations of $\Sigma _{\gamma }^{G,M}$ along the facets of $\Pi $ , there must be a maximal parabolic subgroup $Q\in \mathcal {F}(M)_{\max }$ such that $\pi ^{G}(\mathrm{Ec}_{M}^{Q}(x))\subset \Pi ^{Q}$ . By definition, this means that $\pi ^{G}(\mathrm{Ec}_{M}(x))\subset E_{Q}(\Pi )$ .

Lemma 4.5. Let $Q=LN\in \mathcal {F}(M)$ , $\nu \in \Lambda _{L,Q}^{\xi }$ . Suppose that

$$ \begin{align*} {}^{m}\mathscr{X}_{\gamma}(\Pi)\cap N(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\nu}\neq \emptyset. \end{align*} $$

Then $\nu \in \Lambda _{L,Q}^{\xi }\cap \pi _{L}(\Pi _{0})$ .

Proof. We only need to show that $\nu \in \pi _{L}(\Pi _{0})$ . Set $x\in {}^{m}\mathscr {X}_{\gamma }(\Pi )\cap N(\!(\epsilon )\!)\mathscr {X}_{\gamma }^{L,\nu }$ . As $x\in {}^{m}\mathscr {X}_{\gamma }(\Pi )$ , we have

$$ \begin{align*} \pi^{G}(\textrm{Ec}_{M}(x))\nsubseteq E_{Q'}(\Pi),\qquad \forall Q'\in \mathcal{F}(M), \ Q'\neq G. \end{align*} $$

By Lemma 4.4, this is equivalent to

(4.4) $$ \begin{align} \pi^{G}(\textrm{Ec}_{M}(x))\nsubseteq \bigcup_{\substack{Q'\in \mathcal{F}(M)\\Q'\neq G}}E_{Q'}(\Pi). \end{align} $$

Suppose that $\nu \notin \pi _{L}(\Pi _{0})$ . Then $\nu \in \pi _{L}\left (E_{Q_{0}}(\Pi )\right )$ for some $Q_{0}\in \mathcal {F}(M), Q_{0}\supset L$ . As $\nu \in \Lambda _{L,Q}^{\xi }$ , the parabolic subgroup $Q_{0}$ needs to satisfy $Q_{0}\supset Q^{-}$ . Now that $x\in N(\!(\epsilon )\!)\mathscr {X}_{\gamma }^{L,\nu }$ and $\mathrm{Ec}_{L}(x)$ is a positive $(G,L)$ -orthogonal family, we have

$$ \begin{align*} \alpha\left(H_{Q'}(x)-\nu\right)\geq 0, \qquad \forall \alpha\in \Delta_{Q}, \ Q'\in \mathcal{P}(L). \end{align*} $$

As $Q_{0}\supset Q^{-}$ , this implies that

$$ \begin{align*} H_{Q'}(x)\subset \pi_{L}\left(E_{Q_{0}}(\Pi)\right),\qquad \forall Q'\in \mathcal{P}(L). \end{align*} $$

Hence $\pi _{L}(\mathrm{Ec}_{M}(x))\subset \pi _{L}\left (E_{Q_{0}}(\Pi )\right )$ , so

$$ \begin{align*} \textrm{Ec}_{M}(x)\subset \pi_{L}^{-1}\left(\pi_{L}\left(E_{Q_{0}}(\Pi)\right)\right)\subset\bigcup_{\substack{Q'\subset \mathcal{F}(M),Q'\neq G\\Q'\cap Q_{0}\neq \emptyset}} E_{Q'}(\Pi). \end{align*} $$

This is in contradiction to the relation (4.4), hence $\nu $ must lie in $\pi _{L}(\Pi _{0})$ .

Restricting the Harder–Narasimhan reduction (4.3) to the main part ${}^{m}\mathscr {X}_{\gamma }(\Pi )$ , we have

$$ \begin{align*} {}^{m}\mathscr{X}_{\gamma}(\Pi)=\mathscr{X}_{\gamma}^{\xi}\sqcup \bigsqcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigsqcup_{\lambda\in \Lambda^{\xi}_{L,Q}\cap \pi_{L}(\Pi_{0})} \left({}^{m}\mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}}\right). \end{align*} $$

The retraction $f_{Q}$ behaves much better on the stratum ${}^{m}\mathscr {X}_{\gamma }(\Pi )\cap N_{Q}(\!(\epsilon )\!)\mathscr {X}_{\gamma }^{L,\lambda ,\xi ^{L}}$ :

Proposition 4.6. Let $Q=LN_{Q}\in \mathcal {F}(M)$ , $\nu \in \Lambda _{L,Q}^{\xi }\cap \pi _{L}(\Pi _{0})$ . We have

$$ \begin{align*} {}^{m}\mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\nu,\xi^{L}}=\mathscr{X}_{\gamma}\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\nu,\xi^{L}}. \end{align*} $$

Hence the retraction

$$ \begin{align*} f_{Q}: {}^{m}\mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\nu,\xi^{L}}\to \mathscr{X}_{\gamma}^{L,\nu,\xi^{L}} \end{align*} $$

is an iterated affine fibration over $\mathbf{F}_{q}$ .

Proof. Notice that the second assertion is the corollary of the first one, as follows from Proposition 2.4. It is thus enough to show the first one. In particular, it is enough to show

$$ \begin{align*} \mathscr{X}_{\gamma}\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\nu,\xi^{L}}\subset {}^{m}\mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\nu,\xi^{L}}, \end{align*} $$

as the inclusion in the other direction is obvious.

Let $x\in \mathscr {X}_{\gamma }\cap N_{Q}(\!(\epsilon )\!)\mathscr {X}_{\gamma }^{L,\nu ,\xi ^{L}}$ . We claim that $\mathrm{Ec}_{M}(x)\subset \Pi $ . According to Remark 4.2, condition (3) of $\Pi $ being sufficiently regular implies

(4.5) $$ \begin{align} \textrm{Ec}_{M}^{L}\left(f_{Q}(x)\right)\subset \Pi_{0}\cap \pi_{L}^{-1}(\nu), \end{align} $$

because $f_{Q}(x)\in \mathscr {X}_{\gamma }^{L,\nu ,\xi ^{L}}$ . This implies that $\mathrm{Ec}_{M}(x)\subset \Pi $ by Proposition 3.1, because of the inclusion

$$ \begin{align*} \bigcup_{\substack{\lambda\in \Lambda_{M}\\ \text{satisfying }(*)}}\left(\lambda+\Sigma_{\gamma}^{G,M}\right)\subset \Pi, \end{align*} $$

where the condition $(*)$ refers to

$$ \begin{align*} \left(\lambda+\Sigma_{\gamma}^{G,M}\right)\cap \Pi_{0}\neq \emptyset. \end{align*} $$

The inclusion (4.5) also implies that

$$ \begin{align*} \textrm{Ec}_{M}(x)\nsubseteq \bigcup_{\substack{Q\in \mathcal{F}(M)\\Q\neq G}}E_{Q}(\Pi). \end{align*} $$

So $x\in {}^{m}\mathscr {X}_{\gamma }(\Pi )$ , and the proof is concluded.

We summarize the foregoing discussions in a proposition.

Proposition 4.7. The main body has a decomposition

$$ \begin{align*} {}^{m}\mathscr{X}_{\gamma}(\Pi)=\mathscr{X}_{\gamma}^{\xi}\sqcup \bigsqcup_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\bigsqcup_{\lambda\in \Lambda^{\xi}_{L,Q}\cap \pi_{L}\left(\Pi_{0}\right)} \left({}^{m}\mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\lambda,\xi^{L}}\right), \end{align*} $$

and the retraction $f_{Q}$ on each stratum

$$ \begin{align*} f_{Q}: {}^{m}\mathscr{X}_{\gamma}(\Pi)\cap N_{Q}(\!(\epsilon)\!)\mathscr{X}_{\gamma}^{L,\nu,\xi^{L}}\to \mathscr{X}_{\gamma}^{L,\nu,\xi^{L}} \end{align*} $$

is an iterated affine fibration over $\mathbf{F}_{q}$ of dimension $ \mathrm{val}\left (\det \left (\mathrm{ad}(\gamma )\mid \mathfrak {n}_{Q,F}\right )\right ). $

Of course, we can restrict the decomposition to each connected component ${}^{m}\mathscr {X}_{\gamma }^{\nu _{0}}(\Pi )$ , $\nu _{0}\in \Lambda _{G}$ . Let $\Lambda _{L,Q}^{\nu _{0},\xi }=\Lambda _{L,Q}^{\xi }\cap \Lambda _{L}^{\nu _{0}}$ . The decomposition implies

$$ \begin{align*} &\left\lvert\left(\Lambda^{H_{M}}\middle\backslash {}^{m}\mathscr{X}_{\gamma}^{\nu_{0}}(\Pi)\right)\left(\mathbf{F}_{q}\right)\right\rvert\\ & \quad = \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}_{\gamma}^{\nu_{0},\xi}\right)\left(\mathbf{F}_{q}\right)\right\vert+\sum_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}}\sum_{\lambda\in \Lambda^{\nu_{0},\xi}_{L,Q}\cap \pi_{L}\left(\Pi_{0}\right)} q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad}(\gamma)\mid \mathfrak{g}_{F}/\mathfrak{l}_{F}\right)\right)} \nonumber \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad \cdot \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}^{L,\lambda,\xi^{L}}_{\gamma}\right)\left(\mathbf{F}_{q}\right)\right\rvert \\ & \quad = \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}_{\gamma}^{0,\xi}\right)\left(\mathbf{F}_{q}\right)\right\rvert +\sum_{\substack{Q=LN_{Q}\in \mathcal{F}(M)\\ Q\neq G}} q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad} \gamma\mid\mathfrak{g}_{F}/\mathfrak{l}_{F}\right)\right)}\cdot \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}^{L,0,\xi^{L}}_{\gamma}\right)\left(\mathbf{F}_{q}\right)\right\rvert\nonumber \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad \cdot \left\lvert\Lambda^{\nu_{0},\xi}_{L,Q}\cap \pi_{L}(\Pi_{0})\right\rvert \\ & \quad = \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}_{\gamma}^{0,\xi}\right)\left(\mathbf{F}_{q}\right)\right\vert +\sum_{\substack{L\in \mathcal{L}(M)\\ L\neq G}} q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad} \gamma\mid\mathfrak{g}_{F}/\mathfrak{l}_{F}\right)\right)}\cdot \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}^{L,0,\xi^{L}}_{\gamma}\right)\left(\mathbf{F}_{q}\right)\right\rvert\nonumber \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad \cdot\sum_{Q\in \mathcal{P}(L)} \left\lvert\Lambda^{\nu_{0},\xi}_{L,Q}\cap \pi_{L}(\Pi_{0})\right\rvert \\ & \quad = \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}_{\gamma}^{0,\xi}\right)\left(\mathbf{F}_{q}\right)\right\rvert +\sum_{\substack{L\in \mathcal{L}(M)\\ L\neq G}} q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad} \gamma\mid\mathfrak{g}_{F}/\mathfrak{l}_{F}\right)\right)}\cdot \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}^{L,0,\xi^{L}}_{\gamma}\right)\left(\mathbf{F}_{q}\right)\right\rvert\nonumber \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \cdot\left\lvert\Lambda^{\nu_{0}}_{L}\cap \pi_{L}(\Pi_{0})\right\rvert. \end{align*} $$

Here for the second equality we have used the fact that all the connected components of $\mathscr {X}_{\gamma }^{L,\xi _{L}}$ are isomorphic. Moreover, the last term in the equation counts the number of lattice points in a polytope; it can be calculated effectively with methods from toric geometry. The following theorem summarizes:

Theorem 4.8. For any $\nu _{0}\in \Lambda _{G}$ , the number of rational points on the main body is

$$ \begin{align*} \left\lvert\left(\Lambda^{H_{M}}\middle\backslash {}^{m}\mathscr{X}_{\gamma}^{\nu_{0}}(\Pi)\right)\left(\mathbf{F}_{q}\right)\right\rvert&=\left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}_{\gamma}^{0,\xi}\right)\left(\mathbf{F}_{q}\right)\right\rvert +\sum_{\substack{L\in \mathcal{L}(M)\\ L\neq G}} q^{\frac{1}{2}\mathrm{val}\left(\det\left(\mathrm{ad} \gamma\mid\mathfrak{g}_{F}/\mathfrak{l}_{F}\right)\right)}\\ & \qquad \cdot \left\lvert\left(\Lambda^{H_{M}}\middle\backslash\mathscr{X}^{L,0,\xi^{L}}_{\gamma}\right)\left(\mathbf{F}_{q}\right)\right\rvert\cdot\left\lvert\Lambda^{\nu_{0}}_{L}\cap \pi_{L}(\Pi_{0})\right\rvert. \end{align*} $$

4.2.2. The tail

As the polytope $\Pi $ satisfies

$$ \begin{align*} E_{P}(\Pi)\cap E_{Q}(\Pi)=E_{P\cap Q}(\Pi),\qquad \forall P,Q\in \mathcal{F}(M), \end{align*} $$

by the inclusion-exclusion principle we have

(4.6) $$ \begin{align} \left\lvert\left(\Lambda^{H_{M}}\middle\backslash {}^{t}\mathscr{X}_{\gamma}^{\nu_{0}}(\Pi)\right)\left(\mathbf{F}_{q}\right)\right\rvert=\sum_{\substack{Q\in \mathcal{F}(M)\\ Q\neq G}} (-1)^{\textrm{rk}(G)-\textrm{rk}(Q)-1}\left\lvert\left(\Lambda^{H_{M}}\middle\backslash \mathscr{X}_{\gamma}^{\nu_{0}}\left(E_{Q}(\Pi)\right)\right)\left(\mathbf{F}_{q}\right)\right\rvert, \end{align} $$

where the notation $\textrm {rk}$ means the semisimple rank. Although the polytope $E_{Q}(\Pi )$ is not $\Sigma _{\gamma }^{G,M}$ -regular, we can use the general Arthur–Kottwitz reduction, as explained in Remark 3.2, repeatedly to decompose $\mathscr {X}_{\gamma }\left (E_{Q}(\Pi )\right )$ into locally closed subschemes which are iterated affine fibrations over $F_{\gamma }^{L, M}$ , $L\in \mathcal {L}(M)$ . This gives a formula for $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash \mathscr {X}_{\gamma }^{\nu _{0}}\left (E_{Q}(\Pi )\right )\right )\left (\mathbf{F}_{q}\right )\right \rvert $ in terms of the $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash F_{\gamma }^{L,M}\right )\left (\mathbf{F}_{q}\right )\right \rvert $ s, which can be further reduced to counting points on the fundamental domains by Proposition 3.8 and Corollary 3.10. This process applies to a large family of truncated affine Springer fibers.

We introduce a family of operators on the set of all positive $(G,M)$ -orthogonal families. Recall that $Q_{0}=MU_{0}$ is the unique parabolic subgroup in $\mathcal {P}(M)$ which contains $P_{0}$ . For $L\in \mathcal {L}(M)$ , let $Q_{0}^{L}=Q_{0}\cap L$ . For a positive $(L,M)$ -orthogonal family, we say that two faces of it are conjugate if their associated parabolic subgroups are conjugate to each other by the Weyl group $W_{L}/W_{M}$ . In particular, the edges of the polytope are parametrized by minimal elements in $\mathcal {F}^{L}(M)\backslash \{M\}$ . An edge is said to be of type $\alpha \in \Delta _{A_{M}}^{Q_{0}^{L}}:=\Phi (L,A)\cap \Phi (U_{0},A)\cap \Delta $ if it is conjugate to the edge having vertices $\lambda _{Q^{L}_{0}},\lambda _{s_{\alpha }Q^{L}_{0}}$ , where $s_{\alpha }$ is the simple reflection associated to $\alpha $ . Let $A_{M,\alpha }^{G,L}$ be the operator on the set of positive $(G,M)$ -orthogonal families defined as follows: as a polytope, it increases by $1$ the length of all the edges whose images in $\mathfrak {a}_{M}^{L}$ under the projection $\pi _{M}^{L}$ are of type $\alpha $ , and keeps the lengths of all the others invariant. To check that it actually sends a positive $(G,M)$ -orthogonal family to another one, it suffices to verify for the faces of dimension $2$ , but this is clear. Then we set the vertex

$$ \begin{align*} \lambda_{Q_{0}}\left(A_{M,\alpha}^{G,L}(\Pi)\right)=\lambda_{Q_{0}}(\Pi)+\frac{1}{2}\pi_{M}\left(\varpi_{\alpha}^{\vee}\right), \end{align*} $$

to make $A_{M,\alpha }^{G,L}(\Pi )$ symmetric with respect to $\Pi $ . Here $\varpi _{\alpha }^{\vee }$ is the fundamental coweight corresponding to $\alpha $ . By definition, we see that the operators $A_{M,\alpha }^{G,L}$ commute with each other. When $G=L$ , we simplify the notation $A_{M,\alpha }^{G,G}$ to $A_{M,\alpha }^{G}$ .

Given a tuple of nonnegative integers $\underline {n}=(n_{\alpha })$ , $\alpha \in \Delta _{A_{M}}^{Q_{0}^{L}}$ , let

(4.7) $$ \begin{align} \Sigma_{\gamma}^{\underline{n}} =\prod_{\alpha\in \Delta_{A_{M}}^{Q_{0}^{L}}}\left(A_{M,\alpha}^{G,L}\right)^{n_{\alpha}}\left(\Sigma_{\gamma}\right). \end{align} $$

It is easy to see that the polytopes $E_{Q}(\Pi )$ can be made by iterating this process. For $\alpha \in \Delta _{A_{M}}^{Q_{0}^{L}}$ , let $1_{\alpha }$ be the tuple taking value $1$ at $\alpha $ and $0$ otherwhere. By Remark 3.2, the Arthur–Kottwitz reduction works for the complement $\mathscr {X}_{\gamma }\left (\Sigma _{\gamma }^{\underline {n}+1_{\alpha }}\right ) \Big \backslash \mathscr {X}_{\gamma }\left (\Sigma _{\gamma }^{\underline {n}}\right )$ . The process is completely the same as explained in §3.3, so we do not repeat it here. The resulting strata are iterated affine fibrations over truncated affine Springer fibers of the form $\mathscr {X}_{\gamma }^{L,\nu '}\left (\Sigma _{\gamma }^{L,(\underline {n}')}\right )$ , $L\in \mathcal {L}(M),\nu '\in \Lambda _{L}$ . Iterating this process, $\mathscr {X}_{\gamma }\left (\Sigma _{\gamma }^{\underline {n}}\right )$ can be decomposed as a disjoint union of locally closed subschemes, which are iterated affine fibrations over $F_{\gamma }^{L,M,\mu }$ , $L\in \mathcal {L}(M),\mu \in \Lambda _{L}$ . In particular, counting points on $\Lambda ^{H_{M}}\big \backslash \mathscr {X}_{\gamma }\left (\Sigma _{\gamma }^{\underline {n}}\right )$ can be reduced to counting points on $\Lambda ^{H_{M}}\big \backslash F_{\gamma }^{L,M,\mu }$ , which can be further reduced to counting points on the fundamental domains $F_{\gamma }^{L'}$ , $L'\in \mathcal {L}(M_{0}),$ as we have explained in §3.2. This process applies to counting points on $ \Lambda ^{H_{M}}\big \backslash \mathscr {X}_{\gamma }^{\nu _{0}}\left (E_{Q}(\Pi )\right ). $ By equation (4.6), it gives an expression of $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash {}^{t}\mathscr {X}_{\gamma }^{\nu _{0}}(\Pi )\right )\left (\mathbf{F}_{q}\right )\right \rvert $ in terms of $\left \lvert F_{\gamma }^{L}\left (\mathbf{F}_{q}\right )\right \rvert $ , $L\in \mathcal {L}(M_{0})$ .

4.3. Application to Arthur’s weighted orbital integral

By Theorem 2.8 and Proposition 4.3, Arthur’s weighted orbital integral $J_{M}(\gamma )$ calculates essentially $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash \mathscr {X}_{\gamma }^{0,\xi }\right )\left (\mathbf{F}_{q}\right )\right \rvert $ , as $\mathscr {X}^{G_{\text {der}}}\cap \mathscr {X}_{\gamma }^{\xi }$ is the union of $\left \lvert \Lambda _{G_{\text {der}}}\right \rvert $ -copies of $\mathscr {X}_{\gamma }^{0,\xi }$ . The two approaches in §3 and §4 to calculating $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash \mathscr {X}_{\gamma }^{0}(\Pi )\right )\left (\mathbf{F}_{q}\right )\right \rvert $ give us a recurrence relation involving $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash F_{\gamma }^{L, M,\mu }\right )\left (\mathbf{F}_{q}\right )\right \rvert $ and $\left \lvert \left (\Lambda ^{H_{M}} \big\backslash \mathscr {X}_{\gamma }^{L,0,\xi ^{L}}\right )\left (\mathbf{F}_{q}\right )\right \rvert $ , for $L\in \mathcal {L}(M), \mu \in \Lambda _{L^{\mathrm{ad}}}$ . If we are able to solve this recurrence relation, we will get an expression for $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash \mathscr {X}_{\gamma }^{0,\xi }\right )\left (\mathbf{F}_{q}\right )\right \rvert $ in terms of $\left \lvert \left (\Lambda ^{H_{M}}\middle \backslash F_{\gamma }^{L, M,\mu }\right )\left (\mathbf{F}_{q}\right )\right \rvert $ s, which can be further reduced to counting points on fundamental domains, as explained in §3.2.

5.1. Split elements

We can take T to be the maximal torus of G of the diagonal matrices and $\gamma \in \mathfrak {t}(\mathcal {O})$ a regular element. Let

$$ \begin{align*} n=\mathrm{val}(\alpha_{12}(\gamma)), \end{align*} $$

which we call the root valuation of $\gamma $ . The dimension of the affine Springer fiber $\mathscr {X}_{\gamma }$ is known to be

$$ \begin{align*} \dim\left(\mathscr{X}_{\gamma}\right)=n. \end{align*} $$

In the remainder of this section, we assume that $n\geq 1$ , as the case $n=0$ reduces to the group $\textrm {GL}_{1}$ . Recall that we have calculated $F_{\gamma }$ in [Reference Chaudouard and Laumon11]. Let $X_{*}(T)\cong \textbf {Z}^{2}$ be the usual identification, set $(n,0)\in \textbf {Z}^{2}$ , and let