2.1 Representations and function spaces

Most of the notation below is standard, but we point out two important points that might be unexpected. First, the function $[K,f]$ depends on the Haar measure $dg$ on $G$ , although the measure $[K,f]\,dg$ does not. Second, we have followed [Reference YuYu01, § 17] in using the notation of [Reference Bushnell and KutzkoBK93, § 4.1], which builds in a contragredient to the definition of the Hecke algebra ${\mathcal{H}}(G/\!/K,\unicode[STIX]{x1D70C})$ .

If $V$ is a vector space over any field $F$ , then we denote by $\langle \cdot ,\cdot \rangle$ the pairing between the dual space $V^{\ast }:=\operatorname{Hom}_{F}(V,F)$ and $V$ , so that, for $v^{\ast }\in V^{\ast }$ and $v\in V$ , we write $\langle v^{\ast },v\rangle$ in place of $v^{\ast }(v)$ .

If $\unicode[STIX]{x1D70C}$ is a representation of a group $G$ on the $\mathbb{C}$ -vector space $V$ such that point stabilisers are open, then we temporarily write $\unicode[STIX]{x1D70C}^{\ast }$ for the dual representation of $G$ on the complex dual space $V^{\ast }$ , given by $\langle \unicode[STIX]{x1D70C}^{\ast }(g)v^{\ast },v\rangle =\langle v^{\ast },\unicode[STIX]{x1D70C}(g)^{-1}v\rangle$ for all $v\in V$ , $v^{\ast }\in V^{\ast }$ , and $g\in G$ ; and then write $V^{\vee }$ for the space of smooth vectors (those with open stabilisers) in $V^{\ast }$ , which is a $\unicode[STIX]{x1D70C}^{\ast }$ -stable subspace of $V^{\ast }$ , and $\unicode[STIX]{x1D70C}^{\vee }$ for the restriction of $\unicode[STIX]{x1D70C}^{\ast }$ to $V^{\vee }$ , which is called the contragredient representation to $\unicode[STIX]{x1D70C}$ . If $\unicode[STIX]{x1D6FE}:G^{\prime }\rightarrow G$ is a homomorphism from some other group $G^{\prime }$ , then we write $\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}}$ for the representation $\unicode[STIX]{x1D70C}\circ \unicode[STIX]{x1D6FE}$ of $G^{\prime }$ .

If $X$ is an $l$ -space in the sense of [Reference Bernšteĭn and ZelevinskiĭBZ76, § 1.1]—i.e., a Hausdorff topological space for which every point has a neighbourhood system consisting of open and compact sets—then we write ${\mathcal{H}}(X)$ for its Hecke algebra, which is, by definition, the $\mathbb{C}$ -vector space of locally constant, compactly supported, $\mathbb{C}$ -valued functions on $X$ . If

1. – $V$ is a finite-dimensional $\mathbb{C}$ -vector space,

2. – $K_{1}$ (respectively, $K_{2}$ ) is a compact group acting on $X$ on the left (respectively, right), and

3. – $\unicode[STIX]{x1D70C}_{i}$ is a representation of $K_{i}$ on $V$ with open point stabilisers for $i\in \{1,2\}$ ,

then we write ${\mathcal{H}}((K_{1},\unicode[STIX]{x1D70C}_{1})\backslash X/(K_{2},\unicode[STIX]{x1D70C}_{2}))$ for the subspace of those functions $f\in {\mathcal{H}}(X)\otimes \operatorname{End}_{\mathbb{C}}(V^{\vee })$ such that $f(k_{1}xk_{2})=\unicode[STIX]{x1D70C}_{1}^{\vee }(k_{1})f(x)\unicode[STIX]{x1D70C}_{2}^{\vee }(k_{2})$ for all $g\in X$ , $k_{1}\in K$ , and $k_{2}\in K_{2}$ . If $\unicode[STIX]{x1D70C}_{1}$ or $\unicode[STIX]{x1D70C}_{2}$ is the trivial representation, then we may omit it, writing, for example, ${\mathcal{H}}(K_{1}\backslash X/K_{2})$ . If $K_{1}$ equals $K_{2}$ and $\unicode[STIX]{x1D70C}_{1}$ equals $\unicode[STIX]{x1D70C}_{2}$ , then we may write ${\mathcal{H}}(X/\!/K_{1},\unicode[STIX]{x1D70C}_{1})$ .

If $G$ is a topological group acting on $X$ on the left (respectively, right), then we denote the left (respectively, right) regular action of $G$ on ${\mathcal{H}}(X)$ simply by juxtaposition. Specifically, for $f\in {\mathcal{H}}(X)$ and $g_{1},g_{2}\in G$ , we define $g_{1}fg_{2}\in {\mathcal{H}}(X)$ by $(g_{1}fg_{2})(g_{1}xg_{2})=f(x)$ for all $x\in X$ (so that we have simultaneously defined left and right actions of $G$ on ${\mathcal{H}}(X)$ ).

Now suppose that $X$ is, in addition to being an $l$ -space, a regular measure space. If $K$ is a compact, open subset of $X$ and $f$ belongs to ${\mathcal{H}}(K)$ , then we write $[K,f]$ for the element of ${\mathcal{H}}(X)$ obtained by extending $\operatorname{meas}(K)^{-1}f$ by $0$ . If $f$ is the constant function $1$ , then we may omit it and write just $[K]$ . With the obvious notation, we have $g_{1}[K,f]g_{2}=[g_{1}Kg_{2},g_{1}fg_{2}]$ . We write   for $\int _{X}[K,f](x)\,dx$ .

2.2 Algebraic varieties and algebraic groups

Let $k$ be a field, and $\mathbf{G}$ a smooth group variety over $k$ . (We shall soon impose additional assumptions on both.) We say just ‘variety’ for ‘smooth variety over $k$ ’, and ‘group’ for ‘smooth group variety over $k$ ’. The exception to the implicit use of the modifier ‘over $k$ ’ is that we want to discuss mostly, but not only, complex vector spaces; so we refer to a ‘ $\mathbb{C}$ -vector space’ in that case, and, when necessary, to a ‘ $k$ -vector space’.

We denote varieties by boldface letters, and their sets of rational points by the corresponding italic letters; so, for example, $G$ is the group of rational points of $\mathbf{G}$ . We denote the identity component of an algebraic group with a following superscript circle, the Lie algebra by the corresponding Fraktur letter or by ‘Lie’, and the dual by ‘ $\operatorname{Lie}^{\ast }$ ’; so, for example,

1. – $\mathbf{G}^{\circ }$ is the identity component of $\mathbf{G}$ ,

2. – $\operatorname{Lie}(\mathbf{G})$ or $\boldsymbol{\mathfrak{g}}$ stands for the Lie algebra of $\mathbf{G}$ ,

3. – $\operatorname{Lie}^{\ast }(\mathbf{G})$ or $\boldsymbol{\mathfrak{g}}^{\ast }$ for its dual,

4. – $\operatorname{Lie}(G)$ or $\mathfrak{g}$ for its space of rational points, and

5. – $\operatorname{Lie}^{\ast }(G)$ or $\mathfrak{g}^{\ast }$ for the space of rational points of its dual (or, equivalently, the dual of its space of rational points).

We always use $G^{\circ }$ for $(\mathbf{G}^{\circ })(k)$ , not necessarily $(\mathbf{G}(k))^{\circ }$ , which is a singleton when $k$ is totally disconnected. We write

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{Int}:\mathbf{G}\rightarrow \operatorname{Aut}(\mathbf{G})^{\circ },\nonumber\\ \displaystyle & & \displaystyle \operatorname{Ad}:\mathbf{G}\rightarrow \operatorname{Aut}_{\text{Lie}}(\operatorname{Lie}(\mathbf{G})),\nonumber\\ \displaystyle & & \displaystyle \operatorname{Ad}^{\ast }:\mathbf{G}\rightarrow \operatorname{Aut}_{k}(\operatorname{Lie}^{\ast }(\mathbf{G})),\nonumber\\ \displaystyle & & \displaystyle \operatorname{ad}:\operatorname{Lie}(\mathbf{G})\rightarrow \operatorname{Der}(\operatorname{Lie}(\mathbf{G})),\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \operatorname{ad}^{\ast }:\operatorname{Lie}(\mathbf{G})\rightarrow \operatorname{End}_{k}(\operatorname{Lie}^{\ast }(\mathbf{G})) & & \displaystyle \nonumber\end{eqnarray}$$

for the interior-automorphism, adjoint, and related maps.

If $E/k$ is an extension, then we say that a torus $\mathbf{T}$ in $\mathbf{G}$ is maximally $E$ -split if it contains a subtorus $\mathbf{S}$ such that $\mathbf{S}_{E}$ is a maximal ( $E$ -)split torus in $\mathbf{G}_{E}$ . It is possible a priori that there is no torus rational over $k$ that becomes maximal split after base change, but see § 2.3 for a discussion of the specific case that we use.

We write ${\mathcal{D}}\mathbf{G}$ for the derived group of $\mathbf{G}$ ; and, if $\vec{\mathbf{G}}=(\mathbf{G}^{0},\ldots ,\mathbf{G}^{\ell }=\mathbf{G})$ is a collection of subgroups of $\mathbf{G}$ , then we write

$$\begin{eqnarray}\displaystyle & \displaystyle \operatorname{Lie}(\vec{\mathbf{G}})\quad \text{for }(\operatorname{Lie}(\mathbf{G}^{0}),\ldots ,\operatorname{Lie}(\mathbf{G}^{\ell })), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \operatorname{Lie}^{\ast }(\vec{\mathbf{G}})\quad \text{for }(\operatorname{Lie}^{\ast }(\mathbf{G}^{0}),\ldots ,\operatorname{Lie}^{\ast }(\mathbf{G}^{\ell })), & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle {\mathcal{D}}\vec{\mathbf{G}}\quad \text{for }({\mathcal{D}}\mathbf{G}^{\ell }\,\cap \,\mathbf{G}^{0},\ldots ,{\mathcal{D}}\mathbf{G}^{\ell }\,\cap \,\mathbf{G}^{\ell -1},{\mathcal{D}}\mathbf{G}^{\ell }). & & \displaystyle \nonumber\end{eqnarray}$$

We always use ${\mathcal{D}}G$ for $({\mathcal{D}}\mathbf{G})(k)$ , not necessarily ${\mathcal{D}}(\mathbf{G}(k))$ , which may be smaller.

If $\mathbf{V}$ is a representation of $\mathbf{G}$ , then we write $\operatorname{D}_{\mathbf{V}}$ for the function $\mathbf{G}\rightarrow \operatorname{GL}_{1}$ given by $g\mapsto \det _{\mathbf{V}}(g-1)$ . If $\mathbf{X}$ is a pointed, smooth variety on which $\mathbf{G}$ acts, preserving the preferred point $x$ , and $\mathbf{V}$ is the tangent space to $\mathbf{X}$ at $x$ , with the induced action of $\mathbf{G}$ , then we may write $\operatorname{D}_{\mathbf{X}}$ , or even just $\operatorname{D}_{X}$ , in place of $\operatorname{D}_{\mathbf{V}}$ . As in [Reference DeBacker and SpiceDS16, Definition 2.11], we write $\operatorname{D}_{G}^{\text{red}}$ for the functions on $G$ and its Lie algebra given by $\operatorname{D}_{G}^{\text{red}}(g)=\operatorname{D}_{G/\!\operatorname{C}_{G}(g_{\text{ss}})}(g)$ and $\operatorname{D}_{G}^{\text{red}}(Y)=\operatorname{D}_{\operatorname{Lie}(G)/\!\operatorname{Lie}(\operatorname{C}_{G}(Y_{\text{ss}}))}(Y)$ , respectively.

We write $\mathbf{G}^{\text{rss}}$ for the set of regular, semisimple elements of $\mathbf{G}$ .

2.3 Algebraic groups over non-Archimedean fields, and subgroups associated to concave functions

For the entire paper, we require that the field $k$ be complete with respect to a non-trivial discrete valuation $\operatorname{ord}$ , for which the residue field $\mathfrak{f}$ is finite. We assume throughout the paper that the residual characteristic of $\mathfrak{f}$ is odd. (This is needed anyway to use the Weil representation appearing in [Reference YuYu01], but we also need to use it again in our discussion of the Weil index in § 5.1, and, in particular, in Proposition 5.1.8.) Fix a separable closure $k^{\text{sep}}/k$ , and let $k^{\text{tame}}$ be the maximal tame extension of $k$ inside $k^{\text{sep}}$ . For any algebraic extension $E/k$ , we will denote the unique extension of $\operatorname{ord}$ to a valuation on $E$ again by $\operatorname{ord}$ .

Throughout the paper, we fix a complex character $\unicode[STIX]{x1D6EC}$ of the additive group $k$ that is non-trivial on the ring of integers in $k$ , but trivial on its unique maximal ideal. We will denote the induced character of $\mathfrak{f}$ also by $\unicode[STIX]{x1D6EC}$ . If $V$ is a vector space and $X^{\ast }$ is an element of the dual space, then we write $\unicode[STIX]{x1D6EC}_{X^{\ast }}$ for the character $\unicode[STIX]{x1D6EC}\circ X^{\ast }$ of $V$ .

Also for the entire paper, we require that $\mathbf{G}$ be a reductive, but not necessarily connected, algebraic group over $k$ . We write ${\mathcal{B}}(G)$ for the enlarged Bruhat–Tits building of $\mathbf{G}$ over $k$ , and let $x\mapsto \overline{x}$ be the projection from the enlarged to the reduced building. We will always equip $G$ (respectively, $\operatorname{Lie}(G)$ ) with Waldspurger’s canonical Haar measures [Reference WaldspurgerWal01, § I.4], which assign mass $|\!\operatorname{Lie}(\overline{K})|^{-1/2}$ to the pro-unipotent radical $K_{+}$ of a parahoric subgroup $K$ (respectively, to the pro-nilpotent radical $\operatorname{Lie}(K_{+})$ of its Lie algebra) with reductive quotient $\overline{K}$ ; see [Reference DeBacker and ReederDR09, § 5.1, p. 835]. Our results are usually stated in such a way that we need not make any reference to a specific choice of measure, but this specific choice is important in, for example, Theorem 5.2.8.

As remarked in § 2.2, for a general extension $E/k$ , there need not exist any ( $k$ -rational) torus in $\mathbf{G}$ that becomes maximal ( $E$ -)split after base change to $E$ . By [Reference Bruhat and TitsBT84, Corollaire 5.1.12], there does exist such a torus, even a maximally ( $k$ -)split one, when $E=k^{\text{un}}$ . Since $\mathbf{G}_{k^{\text{un}}}$ is quasi-split, there also exists such a torus whenever $E$ is an extension of $k^{\text{un}}$ . Since we will need to consider maximally $E$ -split tori only for $E=k^{\text{tame}}$ (see Lemmas 3.1.1 and 3.1.2), there is no concern about whether such tori exist.

If ${\mathcal{S}}$ is a subset of $\operatorname{Lie}^{\ast }(G)$ , then we write ${\mathcal{O}}^{G}({\mathcal{S}})$ for the collection of (rational) coadjoint orbits of $G$ on $\operatorname{Lie}^{\ast }(G)$ that intersect every neighbourhood (in the analytic topology) of ${\mathcal{S}}$ . Thus, ${\mathcal{O}}^{G}(0)$ is the analogue of the set of nilpotent orbits in the Lie algebra ([Reference KempfKem78, Corollary 4.3] and [Reference Adler and DeBackerAD02, Lemma 2.5.1]). Occasionally, we find it convenient to use slightly generalised notation; for example, in Theorem 4.4.11, we refer, in the notation of that result, to ${\mathcal{O}}^{H^{\circ }}(\operatorname{Ad}^{\ast }(G)Z_{o}^{\ast })$ . Here $\operatorname{Ad}^{\ast }(G)Z_{o}^{\ast }$ is not a subset of $\operatorname{Lie}^{\ast }(H)$ , but rather of a larger space $\operatorname{Lie}^{\ast }(G)$ ; and we are simply using the abbreviated notation as shorthand for ${\mathcal{O}}^{H^{\circ }}(\operatorname{Ad}^{\ast }(G)Z_{o}^{\ast }\,\cap \,\operatorname{Lie}^{\ast }(H))$ . In particular, if $\operatorname{Ad}^{\ast }(G)Z_{o}^{\ast }$ does not intersect $\operatorname{Lie}^{\ast }(H)$ , then ${\mathcal{O}}^{H^{\circ }}(\operatorname{Ad}^{\ast }(G)Z_{o}^{\ast })$ is empty.

If $V$ is a finite-dimensional $k$ -vector space, then the Fourier transform $\hat{f}$ of $f\in {\mathcal{H}}(V)$ is the element of ${\mathcal{H}}(V^{\ast })$ given by

$$\begin{eqnarray}\hat{f}(v^{\ast })=\int _{V}f(v)\unicode[STIX]{x1D6EC}(\langle v^{\ast },v\rangle )\,dv\quad \text{for all}~v^{\ast }\in V^{\ast }.\end{eqnarray}$$

Similarly, for $f^{\ast }\in {\mathcal{H}}(V^{\ast })$ , we define

$$\begin{eqnarray}\check{f}^{\ast }(v)=\int _{V^{\ast }}f^{\ast }(v^{\ast })\overline{\unicode[STIX]{x1D6EC}(\langle v^{\ast },v\rangle )}\,dv^{\ast }\quad \text{for all}~v\in V.\end{eqnarray}$$

Note that this depends on the choice of Haar measure $dv^{\ast }$ . There is a unique choice, called the dual Haar measure to $dv$ , so that $\check{\hat{f}}$ equals $f$ for all $f\in {\mathcal{H}}(V)$ . Technically speaking, since we have not specified a choice of $dv$ , the functions $\hat{f}$ and $\check{f}^{\ast }$ are not well defined, although the measures $\hat{f}(v^{\ast })\,dv^{\ast }$ and $\check{f}^{\ast }(v)\,dv$ are. In practice, $V$ will be the Lie algebra of a reductive $p$ -adic group, equipped with its canonical measure, so that this ambiguity will not cause a problem; and then we will equip the Lie-algebra dual $V^{\ast }$ with the dual measure.

Definition 2.3.1. As in [Reference DeBacker and SpiceDS16, Definition 4.6], if $\unicode[STIX]{x1D70B}$ is an admissible representation of $G$ , then we write $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}$ for the scalar character of $\unicode[STIX]{x1D70B}$ , which is the function on $G^{\text{rss}}$ that represents the distribution character in the sense that

$$\begin{eqnarray}\operatorname{tr}\unicode[STIX]{x1D70B}(f)\quad \text{equals}\quad \int _{G}\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}(g)f(g)\,dg\end{eqnarray}$$

for all $f\in {\mathcal{H}}(G)$ [Reference Adler and KormanAK07, Proposition 13.1]; and $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70B}}$ for the function $g\mapsto {|\operatorname{D}_{G}^{\text{red}}(g)|}^{1/2}\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}(g)$ on $G^{\text{rss}}$ . Similarly, if $Z^{\ast }$ is an element of $\operatorname{Lie}^{\ast }(G)$ that is fixed by the coadjoint action of some maximal torus $\mathbf{T}$ in $\mathbf{G}$ , and ${\mathcal{O}}$ belongs to ${\mathcal{O}}^{G}(Z^{\ast })$ , then we write $\unicode[STIX]{x1D707}_{{\mathcal{O}}}^{G}$ for the orbital-integral distribution on $\operatorname{Lie}^{\ast }(G)$ given by integration against some invariant measure on ${\mathcal{O}}$ , and $\hat{\unicode[STIX]{x1D707}}_{{\mathcal{O}}}^{G}$ for the function on $\operatorname{Lie}(G)^{\text{rss}}$ that represents its Fourier transform, in the sense that

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{{\mathcal{O}}}^{G}(\hat{f})\quad \text{equals}\quad \int _{\operatorname{ Lie}(H)}\hat{\unicode[STIX]{x1D707}}_{{\mathcal{O}}}^{G}(Y)f(Y)\,dY\end{eqnarray}$$

for all $f\in {\mathcal{H}}(\operatorname{Lie}(G)^{\text{rss}})$ ; and $\widehat{O}_{{\mathcal{O}}}^{G}$ for the function $Y\mapsto {|\operatorname{D}_{G}^{\text{red}}(Y)|}^{1/2}{|\operatorname{D}_{G}^{\text{red}}(Z^{\ast })|}^{1/2}\hat{\unicode[STIX]{x1D707}}_{{\mathcal{O}}}^{G}(Y)$ on $\operatorname{Lie}(G)^{\text{rss}}$ . We sometimes write $\unicode[STIX]{x1D707}_{{\mathcal{S}}}^{G}$ in place of $\unicode[STIX]{x1D707}_{{\mathcal{O}}}^{G}$ , and similarly for $\hat{\unicode[STIX]{x1D707}}$ and $\widehat{O}$ , if ${\mathcal{S}}$ is a non-empty subset of ${\mathcal{O}}$ . As in [Reference DeBacker and SpiceDS16, Remark 2.12], we have written $\operatorname{D}_{G}^{\text{red}}(Z^{\ast })$ for the product $\prod \langle Z^{\ast },\text{d}\unicode[STIX]{x1D6FC}^{\vee }(1)\rangle$ over all weights $\unicode[STIX]{x1D6FC}$ of $\mathbf{T}_{k^{\text{sep}}}$ on $\operatorname{Lie}(\mathbf{G}_{k^{\text{sep}}})$ for which the multiplicand is non-zero. (For now we are just establishing notation, so we ignore questions about convergence of the orbital integral; but see the discussion preceding Theorem 4.4.11.)

As in [Reference Adler and SpiceAS08, § 3.1, p. 8], we put $\widetilde{\mathbb{R}}=\mathbb{R}\sqcup \mathbb{R}+\sqcup \{\infty \}$ . We define $(r+)+=r+$ for all $r\in \mathbb{R}$ , and $\pm \infty +=\pm \infty$ . We define $\tilde{r}=(-r)+$ and $\widetilde{r+}=-r$ for all $r\in \mathbb{R}$ , and $\widetilde{\pm \infty }=\mp \infty$ . We extend addition on $\mathbb{R}$ to $\widetilde{\mathbb{R}}\sqcup \{-\infty \}$ by putting $r+s+=(r+)+s=(r+s)+$ for all $r,s\in \mathbb{R}$ ; $r+(-\infty )=-\infty +r=-\infty$ and $r+\infty =\infty +r=\infty$ for all $r\in \widetilde{\mathbb{R}}$ with $r<\infty$ ; and $-\infty +(-\infty )=-\infty$ and $\pm \infty +\infty =\infty +(\pm \infty )=\infty$ . We extend subtraction on $\mathbb{R}$ to a partial operation on $\widetilde{\mathbb{R}}\sqcup \{-\infty \}$ by putting $(r+)-s=(r-s)+$ for all $r,s\in \mathbb{R}$ ; $r-(\pm \infty )=\mp \infty$ for all $r\in \widetilde{\mathbb{R}}$ with $r<\infty$ ; and $\pm \infty -(\mp \infty )=\pm \infty$ .

We follow a suggestion of Cheng-Chiang Tsai and, for $(x,r)\in {\mathcal{B}}(G)\times \mathbb{R}_{{\geqslant}0}$ , replace the usual notation $G_{x,r}$ and $G_{x,r+}$ for Moy–Prasad subgroups ([Reference Moy and PrasadMP94, §§ 2.6, 3.2, 3.5] and [Reference Moy and PrasadMP96, §§ 3.2, 3.3]) by $G_{x\geqslant r}$ and $G_{x>r}$ , respectively. For convenience, we make the convention that $G_{x>r+}$ means $G_{x>r}$ . Note that, by definition, $G_{x\geqslant r}$ equals $G_{x\geqslant r}^{\circ }$ . We use similar notation for groups associated to concave functions [Reference Bruhat and TitsBT72, § 6.4.3], as in [Reference Adler and SpiceAS08, Definition 5.14], and for the Lie algebra, where we can drop the requirement that $r$ be non-negative (respectively, that the relevant function be concave); and in Definition 3.2.4, where we define an analogous class of groups.

These conventions inevitably suggest the notation $G_{x=r}$ for $G_{x\geqslant r}/G_{x>r}$ ; we adopt this and obvious variants, even though they are a bit misleading in the case where $k$ has mixed characteristic. In particular, we write $k_{{\geqslant}r}=\{t\in k\mid \operatorname{ord}(t)\geqslant r\}$ , and similarly for other notation related to the filtration on $k$ ; so, for example, $\mathfrak{f}$ equals $k_{=0}=k_{{\geqslant}0}/k_{{>}0}$ .

2.4 Index of notation

2.5 Index of terminology

3.1 Groups associated to concave functions

In Definition 3.2.4, we define a class of compact, open subgroups related to those constructed in [Reference Adler and SpiceAS08, Definition 5.14]. For technical reasons involving the presence of algebraic subgroups that are not of full rank in $\mathbf{G}$ , we work mostly with ‘depth vectors’ (see Definition 3.2.1) and ‘depth matrices’ (see Definition 3.2.4), rather than concave functions as in [Reference Adler and SpiceAS08]; but we do need to consider such functions for two technical results.

Lemma 3.1.1 is mostly a straightforward generalisation of part of [Reference Adler and SpiceAS08, Proposition 5.40], with essentially the same proof. We use it only in the proof of Lemma 5.3.5, where we need an additional technical fact, an analogue of [Reference Adler and SpiceAS08, Lemma 5.29]; so, for convenience, we state both facts together.

Working with groups associated to arbitrary concave functions rather than depth matrices allows us to ‘move’ a group $G_{x\geqslant f}$ to another point, by writing it as $G_{y\geqslant f+(y-x)}$ . The condition on $f\vee f$ in Lemma 3.1.1 is needed to use [Reference Adler and SpiceAS08, Proposition 5.39], but it is just for maximal generality; for us, it suffices to know that it is satisfied whenever $f$ is a translate by a linear function of an everywhere positive, concave function. Lemma 3.1.1 relies on [Reference Adler and SpiceAS08, Hypothesis D], but this is automatically satisfied when $\mathbf{G}$ is $k^{\text{tame}}$ -split, which will be the case when we use the result (in Lemma 5.3.5).

In the notation $\operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})$ and $\operatorname{stab}_{G^{\prime }}(\overline{x})$ , note that the symbol $\overline{x}$ stands for a point in the reduced building of $\mathbf{G}^{\prime }(E)$ or $G^{\prime }$ , not necessarily of $\mathbf{G}(E)$ or $G$ . The lemma relies crucially on the good descent properties of full stabilisers of points in the building; it would not work, for example, if we replaced $\operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})$ and $\operatorname{stab}_{G^{\prime }}(\overline{x})$ by their parahoric subgroups $\mathbf{G}^{\prime }(E)_{x\geqslant 0}$ and $G_{x\geqslant 0}^{\prime }$ .

Lemma 3.1.1. Suppose that

1. – [Reference Adler and SpiceAS08, Hypothesis D] holds,

2. – $\mathbf{G}^{\prime }$ is the centraliser in $\mathbf{G}^{\circ }$ of a $k^{\text{tame}}$ -split torus,

3. – $\mathbf{T}$ is a maximally $k^{\text{tame}}$ -split, maximal torus in $\mathbf{G}^{\prime }$ ,

4. – $x$ is a point of ${\mathcal{B}}(T)$ ,

5. – $E/k$ is a tame extension, and

6. – $f_{1}$ and $f_{2}$ are $\widetilde{\mathbb{R}}$ -valued, Galois-invariant, concave [Reference Bruhat and TitsBT72, § 6.4.3] functions on the set $\widetilde{\unicode[STIX]{x1D6F7}}(\mathbf{G}_{k^{\text{sep}}},\mathbf{T}_{k^{\text{sep}}})$ of weights of  $\mathbf{T}_{k^{\text{sep}}}$ on $\operatorname{Lie}(\mathbf{G}_{k^{\text{sep}}})$ satisfying $f_{j}(0)>0$ and $f_{j}(\unicode[STIX]{x1D6FC})<(f_{j}\vee f_{j})(\unicode[STIX]{x1D6FC})$ [Reference Adler and SpiceAS08, Definition 3.2.6] whenever $\unicode[STIX]{x1D6FC}\in \widetilde{\unicode[STIX]{x1D6F7}}(\mathbf{G}_{k^{\text{sep}}},\mathbf{T}_{k^{\text{sep}}})$ is such that $f_{j}(\unicode[STIX]{x1D6FC})<\infty$ , for $j\in \{1,2\}$ .

Then

$$\begin{eqnarray}\displaystyle \operatorname{stab}_{G^{\prime }}(\overline{x})G_{x\geqslant f_{1}}\,\cap \,G_{x\geqslant f_{2}}\quad \text{equals}\quad G_{x\geqslant \max \{0,f_{2}\}}^{\prime }\cdot G_{x\geqslant \max \{f_{1},f_{2}\}} & & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant f_{1}}\cdot \mathbf{G}(E)_{x\geqslant f_{2}}\,\cap \,G\quad \text{equals}\quad \operatorname{stab}_{G^{\prime }}(\overline{x})G_{x\geqslant f_{1}}\cdot G_{x\geqslant f_{2}}. & & \displaystyle \nonumber\end{eqnarray}$$

Proof. The first statement is proven, just as in [Reference Adler and SpiceAS08, Lemma 5.29], by reducing to [Reference Adler and SpiceAS08, Lemmas 5.22 and A.14]; the crucial point is that $\operatorname{stab}_{G^{\prime }}(\overline{x})G_{x\geqslant f_{1}}$ is contained in the product of $G^{\prime }$ with the root groups corresponding to roots of $\mathbf{T}$ in $\mathbf{G}$ .

For the second statement, let $g_{1}$ and $g_{2}$ be any concave functions satisfying the analogues of the conditions imposed on $f_{1}$ and $f_{2}$ . By the first statement (applied to $\mathbf{G}(E)$ ), we have that

$$\begin{eqnarray}\operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant g_{1}}\,\cap \,\mathbf{G}(E)_{x\geqslant g_{2}}\quad \text{equals}\quad \mathbf{G}(E)_{x\geqslant g},\end{eqnarray}$$

where

$$\begin{eqnarray}g(\unicode[STIX]{x1D6FC})=\left\{\begin{array}{@{}ll@{}}\max \{0,g_{2}(\unicode[STIX]{x1D6FC})\}\quad & \unicode[STIX]{x1D6FC}\in \widetilde{\unicode[STIX]{x1D6F7}}(\mathbf{G}_{k^{\text{sep}}}^{\prime },\mathbf{T}_{k^{\text{sep}}}),\\ \max \{g_{1}(\unicode[STIX]{x1D6FC}),g_{2}(\unicode[STIX]{x1D6FC})\}\quad & \text{otherwise,}\end{array}\right.\end{eqnarray}$$

hence by [Reference Adler and SpiceAS08, Proposition 5.39] that the cohomology

$$\begin{eqnarray}\operatorname{H}^{1}(E/k,\operatorname{stab}_{\boldsymbol{ G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant g_{1}}\,\cap \,\mathbf{G}(E)_{x\geqslant g_{2}})\end{eqnarray}$$

is trivial. (This is where we use the hypotheses about $g_{j}\vee g_{j}$ , and also where we require [Reference Adler and SpiceAS08, Hypothesis D].) Thus, considering the short exact sequence in cohomology associated to the exact sequence of pointed sets

$$\begin{eqnarray}\displaystyle 1 & \rightarrow & \displaystyle \operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant g_{1}}\,\cap \,\mathbf{G}(E)_{x\geqslant g_{2}}\nonumber\\ \displaystyle & \rightarrow & \displaystyle \operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant g_{1}}\times \mathbf{G}(E)_{x\geqslant g_{2}}\nonumber\\ \displaystyle & \rightarrow & \displaystyle \operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant g_{1}}\cdot \mathbf{G}(E)_{x\geqslant g_{2}}\rightarrow 1,\nonumber\end{eqnarray}$$

and using [Reference Adler and SpiceAS08, Lemma 5.33], we see that the sequence

of Galois-fixed points remains exact.

Applying (∗) with $g_{1}=\infty$ , so that $\mathbf{G}(E)_{x\geqslant g_{1}}$ equals $\{1\}$ , and $g_{2}=f_{1}$ , and using that $\operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\,\cap \,G$ equals $\operatorname{stab}_{G^{\prime }}(\overline{x})$ , shows that

$$\begin{eqnarray}\operatorname{stab}_{G^{\prime }}(\overline{x})G_{x\geqslant f_{1}}\quad \text{equals}\quad \operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant f_{1}}\,\cap \,G.\end{eqnarray}$$

Then applying (∗) again, with $g_{j}=f_{j}$ for $j\in \{1,2\}$ , shows that

$$\begin{eqnarray}\operatorname{stab}_{G^{\prime }}(\overline{x})G_{x\geqslant f_{1}}\cdot G_{x\geqslant f_{2}}=(\operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant f_{1}}\,\cap \,G)G_{x\geqslant f_{2}}\end{eqnarray}$$

equals $\operatorname{stab}_{\mathbf{G}^{\prime }(E)}(\overline{x})\mathbf{G}(E)_{x\geqslant f_{1}}\cdot \mathbf{G}(E)_{x\geqslant f_{2}}\,\cap \,G$ , as desired.◻

Lemma 3.1.2 is a slightly stronger version of [Reference Adler and SpiceAS08, Lemma 5.17], adapted to take into account the depth of commutators in ${\mathcal{D}}G$ (not just in $G$ ). Again, we find it convenient to state it in terms of concave functions, for which we can use the result from [Reference Adler and SpiceAS08] to bootstrap, rather than necessarily the depth matrices appearing in Definition 3.2.4 below. Having done so, however, we use it only to prove Lemma 3.2.7 (which does concern groups and Lie algebras associated to depth matrices).

Lemma 3.1.2. Suppose that

1. – $\mathbf{T}$ is a maximally $k^{\text{tame}}$ -split, maximal torus in $\mathbf{G}$ ,

2. – $x$ is a point of ${\mathcal{B}}(T)$ ,

3. – $f_{1}$ and $f_{2}$ are $(\widetilde{\mathbb{R}}\cup \{-\infty \})$ -valued, Galois-invariant functions on the set of weights of $\mathbf{T}_{k^{\text{sep}}}$ on $\operatorname{Lie}(\mathbf{G}_{k^{\text{sep}}})$ .

If, for $j\in \{1,2\}$ ,

1. – $g_{j}$ belongs to $G_{x\geqslant f_{j}}$ ,

2. – $Y_{j}$ to $\operatorname{Lie}(G)_{x\geqslant f_{j}}$ , and

3. – $X_{j}^{\ast }$ to $\operatorname{Lie}^{\ast }(G)_{x\geqslant f_{j}}$

then

1. – $\operatorname{ad}(Y_{1})Y_{2}$ belongs to $\operatorname{Lie}({\mathcal{D}}G)_{x\geqslant f_{1}\bowtie f_{2}}$ and

2. – $\operatorname{ad}^{\ast }(Y_{1})X_{2}^{\ast }$ to $\operatorname{Lie}^{\ast }({\mathcal{D}}G)_{x\geqslant f_{1}\bowtie f_{2}}$ .

If, further, $f_{1}$ is concave, then

1. – $(\operatorname{Ad}(g_{1})-1)Y_{2}$ belongs to $\operatorname{Lie}({\mathcal{D}}G)_{x\geqslant f_{1}\rtimes f_{2}}$ and

2. – $(\operatorname{Ad}^{\ast }(g_{1})-1)X_{2}^{\ast }$ belongs to $\operatorname{Lie}^{\ast }({\mathcal{D}}G)_{x\geqslant f_{1}\rtimes f_{2}}$ .

If both $f_{1}$ and $f_{2}$ are concave, then

1. – $[g_{1},g_{2}]$ belongs to ${\mathcal{D}}G_{x\geqslant f_{1}\vee f_{2}}$ .

Here, $f_{1}\vee f_{2}$ is the function defined by

$$\begin{eqnarray}\displaystyle (f_{1}\vee f_{2})(\unicode[STIX]{x1D6FC})=\inf _{\sum a_{i}+\sum b_{j}=\unicode[STIX]{x1D6FC}}\sum f_{1}(a_{i})+\sum f_{2}(b_{j}) & & \displaystyle \nonumber\end{eqnarray}$$

in [Reference Adler and SpiceAS08, Definition B.1], and $f_{1}\bowtie f_{2}$ and $f_{1}\rtimes f_{2}$ are its analogues defined by

$$\begin{eqnarray}\displaystyle (f_{1}\bowtie f_{2})(\unicode[STIX]{x1D6FC})=\inf _{a+b=\unicode[STIX]{x1D6FC}}f_{1}(a)+f_{2}(b) & & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle (f_{1}\rtimes f_{2})(\unicode[STIX]{x1D6FC})=\inf _{\sum a_{i}+b=\unicode[STIX]{x1D6FC}}\sum f_{1}(a_{i})+f_{2}(b), & & \displaystyle \nonumber\end{eqnarray}$$

for all weights $\unicode[STIX]{x1D6FC}$ of $\mathbf{T}_{k^{\text{sep}}}$ on $\operatorname{Lie}(\mathbf{G}_{k^{\text{sep}}})$ .

Proof. We only prove the statement about commutators in the group; the others are easier.

We may, and do, assume, upon passing to a tame extension, that $\mathbf{T}$ is contained in a Borel subgroup of $\mathbf{G}$ . Then we have that any group of the form $G_{x\geqslant f}$ is generated by $T_{{\geqslant}f(0)}$ and the various $B_{x\geqslant f}$ , where $\mathbf{B}$ is a Borel subgroup of $\mathbf{G}$ containing $\mathbf{T}$ ; and similarly for ${\mathcal{D}}G_{x\geqslant f}$ . (In fact, we need only take two opposite Borel subgroups.) In particular, we have that $G_{x\geqslant f}$ is generated by $T_{{\geqslant}f(0)}$ and ${\mathcal{D}}G_{x\geqslant f}$ .

We use the basic fact that, if ${\mathcal{G}}$ is a subgroup of $G$ that normalises ${\mathcal{D}}G_{x\geqslant f_{1}\vee f_{2}}$ , and ${\mathcal{S}}$ is a subset of $G$ , then  is a subgroup of $G$ . We temporarily introduce the notation $\operatorname{Comm}_{{\mathcal{G}}}({\mathcal{S}})$ for this subgroup.

For this paragraph, fix $j\in \{1,2\}$ . By [Reference Adler and SpiceAS08, Lemma 5.17] (applied to ${\mathcal{D}}\mathbf{G}$ ), we have that ${\mathcal{D}}G_{x\geqslant f_{j}}$ normalises ${\mathcal{D}}G_{x\geqslant f_{1}\vee f_{2}}$ , and that $T_{{\geqslant}f_{j}(0)}$ normalises $U_{x\geqslant f_{1}\vee f_{2}}$ for the unipotent radical $\mathbf{U}$ of any Borel subgroup $\mathbf{B}$ of $\mathbf{G}$ containing $\mathbf{T}$ ; so, since of course $T_{{\geqslant}f_{j}(0)}$ normalises any subgroup of $T$ , we have that $G_{x\geqslant f_{j}}$ normalises ${\mathcal{D}}G_{x\geqslant f_{1}\vee f_{2}}$ . Thus, the notation $\operatorname{Comm}_{G_{x\geqslant f_{j}}}({\mathcal{S}})$ makes sense.

For this paragraph, fix a Borel subgroup $\mathbf{B}$ of $\mathbf{G}$ containing $\mathbf{T}$ , and let $\mathbf{U}$ be its unipotent radical. By another application of [Reference Adler and SpiceAS08, Lemma 5.17], we have that the commutator of $T_{{\geqslant}f_{1}(0)}$ with $B_{x\geqslant f_{2}}$ belongs to $U_{x\geqslant f_{1}\vee f_{2}}\subseteq {\mathcal{D}}G_{x\geqslant f_{1}\vee f_{2}}$ . Thus, since $U_{x\geqslant f_{2}}$ is contained in ${\mathcal{D}}G_{x\geqslant f_{2}}$ , we have that $\operatorname{Comm}_{T_{{\geqslant}f_{1}(0)}}(B_{x\geqslant f_{2}})$ equals $T_{{\geqslant}f_{1}(0)}$ , and that $\operatorname{Comm}_{G_{x\geqslant f_{1}}}(U_{x\geqslant f_{2}})$ contains both $T_{{\geqslant}f_{1}(0)}$ and ${\mathcal{D}}G_{x\geqslant f_{1}}$ , hence equals $G_{x\geqslant f_{1}}$ . Thus

and

Symmetric results also hold.

By (∗) (applied to a pair of opposite Borel subgroups), we have that $\operatorname{Comm}_{G_{x\geqslant f_{2}}}(T_{{\geqslant}f_{1}(0)})$ equals $G_{x\geqslant f_{2}}$ , hence that $\operatorname{Comm}_{G_{x\geqslant f_{1}}}(G_{x\geqslant f_{2}})$ contains $T_{{\geqslant}f_{1}(0)}$ . By (∗∗) (or, rather, its analogue with the indices $j=1$ and $j=2$ switched), we have that $\operatorname{Comm}_{G_{x\geqslant f_{1}}}(G_{x\geqslant f_{2}})$ also contains $U_{x\geqslant f_{2}}$ whenever $\mathbf{U}$ is the unipotent radical of a Borel subgroup $\mathbf{B}$ of $\mathbf{G}$ containing $\mathbf{T}$ . It follows that $\operatorname{Comm}_{G_{x\geqslant f_{1}}}(G_{x\geqslant f_{2}})$ equals $G_{x\geqslant f_{1}}$ , as desired.◻

3.2 Groups associated to depth matrices

In Definition 3.2.4, we use Hypothesis 3.2.2 to build a class of compact, open subgroups of $G$ generalising those constructed in [Reference Adler and SpiceAS08, Definition 5.14]. Definition 3.2.1 begins to set up that generalisation.

The non-full-rank subgroups that arise in Definition 3.2.4 depend on a semisimple element $\unicode[STIX]{x1D6FE}$ of $G$ (not necessarily $G^{\circ }$ ). Choose such an element, and write $\mathbf{P}^{-}$ (respectively, $\mathbf{P}^{+}$ ) for the parabolic subgroup of $\mathbf{G}$ dilated (respectively, contracted) by $\unicode[STIX]{x1D6FE}$ [Reference DeligneDel76, § 1, p. 155], and $\mathbf{N}^{-}$ (respectively, $\mathbf{N}^{+}$ ) for its unipotent radical. Write $\mathbf{M}$ for $\mathbf{P}^{-}\,\cap \,\mathbf{P}^{+}$ .

Let $r$ be a non-negative real number, and put $s=r/2$ .

In [Reference Adler and SpiceAS08, Definition 6.4], Adler and the author defined a family $(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{\substack{ i\in \widetilde{\mathbb{R}} \\ 0\leqslant i\leqslant r}}$ of connected, full-rank subgroups of $\mathbf{G}$ associated to an element $\unicode[STIX]{x1D6FE}$ of $G^{\circ }$ satisfying certain tameness hypotheses. (The groups defined there are always connected, whereas we allow the groups in Hypothesis 3.2.2 to be disconnected; but Hypothesis 3.2.8 involves only the identity components of the various $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ , so that the difference does not matter.) It turns out to be convenient for us to extend the definition to handle also the case where $i<0$ . When we do so, we actually define two groups, a reductive group   and a parabolic subgroup $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ of it. (For $i\geqslant 0$ , these notations coincide; in that case, the parabolic subgroup $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ of   is actually the whole group.)

With an eye towards future applications, including particularly removing the requirement that $\unicode[STIX]{x1D6FE}$ belong to the identity component, and possibly the tameness requirement, we state very precisely the properties that we need in Hypothesis 3.2.2.

Hypothesis 3.2.2 always holds for $\unicode[STIX]{x1D6FE}$ an element of a tame torus satisfying [Reference Adler and SpiceAS08, Definition 6.3], as long as [Reference Adler and SpiceAS08, Hypotheses A–D] are satisfied (see [Reference Adler and SpiceAS08, Proposition 4.6, Lemmas 8.1 and 6.5]). (Actually, those results require that $\unicode[STIX]{x1D6FE}$ be compact modulo centre; but we may reduce to that case by working inside $\mathbf{M}$ .)

Although we allow the groups $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ to be disconnected, Hypothesis 3.2.2 is phrased in such a way that we may replace each $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ by its identity component.

Remark 3.2.3 discusses the descent properties of tame, twisted Levi subgroups.

By Hypothesis 3.2.2(ii), there is a canonical $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ -stable complement $\operatorname{Lie}(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))^{\bot }$ to $\operatorname{Lie}(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))$ in $\operatorname{Lie}(\mathbf{G})$ (namely, the sum of the other weight spaces for $\unicode[STIX]{x1D6FE}$ ). We identify $\operatorname{Lie}^{\ast }(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))$ with the subset of $\operatorname{Lie}^{\ast }(\mathbf{G})$ that annihilates $\operatorname{Lie}(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))^{\bot }$ , and write $\operatorname{Lie}^{\ast }(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))^{\bot }$ for the subset of $\operatorname{Lie}^{\ast }(\mathbf{G})$ that annihilates $\operatorname{Lie}(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))$ .

Definition 3.2.4 is closely related to [Reference Adler and SpiceAS08, Definition 5.14]. Our condition ‘grouplike’ is more restrictive than the analogous condition ‘admissible’ of [Reference Adler and SpiceAS08, Definition 5.8]. If we are dealing with tame, compact-modulo-centre elements $\unicode[STIX]{x1D6FE}$ in the identity component of $G$ , so that the groups $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ have full rank, then the group $\text{}_{\unicode[STIX]{x1D6FE}}\vec{G}_{x\geqslant f}$ is a special case of the groups defined in [Reference Adler and SpiceAS08, Definition 5.14]. The point of Definition 3.2.4 is to handle the case where the groups $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}^{\pm })$ may not have full rank in $\mathbf{G}$ .

Definition 3.2.4. Suppose that $\vec{\mathbf{G}}$ is a tame, twisted Levi sequence in $\mathbf{G}$ containing $\unicode[STIX]{x1D6FE}$ . Let

$$\begin{eqnarray}f=((f_{i}^{\text{n}})_{\substack{ i\in \widetilde{\mathbb{R}} \\ 0\leqslant i\leqslant r}},(f_{i}^{\text{d}},f_{i}^{\text{c}})_{\substack{ i\in \widetilde{\mathbb{R}}\cup \{-\infty \} \\ i<0}}\Bigr)\end{eqnarray}$$

be a collection of depth vectors. The entire ensemble $f$ is called a depth matrix (for $\vec{\mathbf{G}}$ ). We read the superscripts ‘n’, ‘d’, and ‘c’ as shorthand for ‘neutral’, ‘dilated’, and ‘contracted’, referring to the action of $\unicode[STIX]{x1D6FE}$ [Reference DeligneDel76, § 1, p. 155]. We occasionally write $f^{-}=(f^{\text{n}},f^{\text{d}})$ and $f^{+}=(f^{\text{n}},f^{\text{c}})$ .

For any $x\in {\mathcal{B}}(\vec{G})$ , we write $\text{}_{\unicode[STIX]{x1D6FE}}\operatorname{Lie}(\vec{G})_{x\geqslant f}$ for

$$\begin{eqnarray}\displaystyle & & \displaystyle \underset{0\leqslant i\leqslant r}{\bigoplus }\operatorname{Lie}(\operatorname{C}_{G}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}))^{\bot }\,\cap \,\operatorname{Lie}(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant f_{i}^{\text{n}}}\nonumber\\ \displaystyle & & \displaystyle \qquad \oplus \bigoplus _{i<0}\operatorname{Lie}(\operatorname{C}_{G}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}\text{}^{\pm 1}))^{\bot }\,\cap \,(\operatorname{Lie}(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant f_{i}^{\text{d}}}\oplus \operatorname{Lie}(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}^{-1}))_{x\geqslant f_{i}^{\text{c}}}),\nonumber\end{eqnarray}$$

and similarly for $\text{}_{\unicode[STIX]{x1D6FE}}\operatorname{Lie}^{\ast }(\vec{G})_{x\geqslant f}$ . (The notation $\operatorname{Lie}(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))$ here, and $\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ in the group case, is Definition 3.2.1.)

We say that $f$ is grouplike if

1. – $f_{r}^{\text{n}}$ is grouplike;

2. – for each index $i$ , the depth vectors $f_{i}^{\text{n}}$ , $f_{i}^{\text{d}}$ , and $f_{i}^{\text{c}}$ are concave;

3. – the inequality $f_{i_{-}}^{-}\geqslant f_{i_{+}}^{-}$ holds for all $i_{-}\leqslant i_{+}<0$ and all $0\leqslant i_{-}\leqslant i_{+}\leqslant r$ , and similarly for $f^{+}$ ; and

4. – the inequality $f_{i_{-}j_{-}}^{\text{d}}+f_{i_{-}j_{+}}^{\text{c}}\geqslant f_{i_{+}j_{+}}^{\text{n}}$ holds for all $i_{-}<0\leqslant i_{+}\leqslant r$ and $0\leqslant j_{-}\leqslant j_{+}\leqslant \ell$ .

In this case, for any $x\in {\mathcal{B}}(\vec{G})$ , we write $\text{}_{\unicode[STIX]{x1D6FE}}\vec{G}_{x\geqslant f}$ for

$$\begin{eqnarray}\langle \mathop{\bigcup }_{0\leqslant i\leqslant r}\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant f_{i}^{\text{n}}}\cup \mathop{\bigcup }_{i<0}(\operatorname{C}_{N^{-}\,\cap \,\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant f_{i}^{\text{d}}}\cup \operatorname{C}_{N^{+}\,\cap \,\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant f_{i}^{\text{c}}})\rangle .\end{eqnarray}$$

The element $\unicode[STIX]{x1D6FE}$ will usually be clear, and so will be omitted from the notation.

We will occasionally want to enforce an inequality involving depth matrices, but only for certain arguments. See, for example, Lemma 3.2.11(ii), where we require that (in the notation of that result) the inequality $F^{-}\geqslant f^{-}+\operatorname{ord}_{\unicode[STIX]{x1D6FE}}$ holds outside $\operatorname{C}_{\mathbf{G}}^{({<}r)}(\unicode[STIX]{x1D6FE})$ . What we mean here is that the depth vector $F_{i}^{-}$ majorises the depth vector $f_{i}^{-}$ for all $i<r$ , but not necessarily for $i=r$ . Similarly, in Lemma 4.1.6(ii), where we require that (in the notation of that result) the inequality $F\geqslant f-r$ holds outside $\mathbf{G}^{\prime }$ , we mean that $F_{i\ell }^{\pm }\geqslant f_{i\ell }^{\pm }-r$ for all $i\leqslant r$ , but not necessarily that $F_{ij}^{\pm }\geqslant f_{ij}^{\pm }-r$ for any $i$ when $0\leqslant j<\ell$ . (Here, we are dealing with a tame, twisted Levi sequence $(\mathbf{G}^{0},\ldots ,\mathbf{G}^{\ell -1}=\mathbf{G}^{\prime },\mathbf{G}^{\ell }=\mathbf{G})$ .)

Remark 3.2.5. Note that, when $-\infty$ occurs in a depth vector, the resulting group or Lie algebra includes everything from the corresponding entry in the tame, twisted Levi sequence (resulting in a non-compact subgroup); and, when $\infty$ occurs in a depth vector, the resulting group or Lie algebra includes nothing from the corresponding entry in the sequence (resulting in a non-open group). Thus, for example, $\operatorname{Lie}(G^{\prime },G)_{x\geqslant (\infty ,-\infty )}$ stands for $\operatorname{Lie}(G^{\prime })^{\bot }$ . Note also that the notation is ‘left biased’; so, for example, $\operatorname{Lie}(\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE}),G^{\prime },G)_{x\geqslant (0,\infty ,-\infty )}$ stands for $\operatorname{Lie}(\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE}))_{x\geqslant 0}+\operatorname{Lie}(G^{\prime })^{\bot }$ , whereas $\operatorname{Lie}(G^{\prime },\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE}),G)_{x\geqslant (\infty ,0,-\infty )}$ stands for $\operatorname{Lie}(G^{\prime })^{\bot }\,\cap \,(\operatorname{Lie}(\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE}))_{x\geqslant 0}+\operatorname{Lie}(\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE}))^{\bot })$ .

We have that

We use certain shortcuts for Definition 3.2.4, whose meaning we hope is apparent. For example, we write $\operatorname{ord}_{\unicode[STIX]{x1D6FE}}$ (respectively, $\operatorname{ord}_{\unicode[STIX]{x1D6FE}^{\pm 1}}$ ) for the depth matrix $f$ with $f_{ij}^{-}=i$ and $f_{ij}^{\text{c}}=0$ (respectively, $f_{ij}^{-}=i$ and $f_{ij}^{+}=i$ ) for all $i$ and $j$ . (If $\unicode[STIX]{x1D6FE}$ belongs to a maximal torus $\mathbf{T}$ in $\mathbf{G}$ , then this notation is analogous to the use in [Reference DeBacker and SpiceDS16, Definition 2.10], where $\operatorname{ord}_{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D6FC})$ is defined to be $\operatorname{ord}(\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})-1)$ for $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}(\mathbf{G},\mathbf{T})$ .) Thus the notation   in Proposition 5.1.3 stands for $\operatorname{Lie}(G)_{x\geqslant f}$ , where

$$\begin{eqnarray}\displaystyle f_{r}^{\text{n}} & & \displaystyle \quad \text{equals}\quad 0+,\nonumber\\ \displaystyle f_{i}^{\text{n}} & & \displaystyle \quad \text{equals}\quad (r-i)/2~\text{for}~0\leqslant i<r,\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle f_{i}^{\text{d}}=f_{i}^{\text{c}}\quad \text{equals}\quad (r-i)/2~\text{for}~i<0; & & \displaystyle \nonumber\end{eqnarray}$$

and the notation $(\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{G^{\prime }}^{({<}0)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{G}^{({<}0)}(\unicode[STIX]{x1D6FE}))_{x\geqslant (0+,r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}},(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})/2)}$ in Proposition 4.2.4 stands for $(G^{\prime },G)_{x\geqslant f}$ , where

$$\begin{eqnarray}\displaystyle f_{r}^{\text{n}} & & \displaystyle \quad \text{equals}\quad (0+,0+),\nonumber\\ \displaystyle f_{i}^{\text{n}} & & \displaystyle \quad \text{equals}\quad (r-i,(r-i)/2)~\text{for }0\leqslant i<r,\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle f_{i}^{\text{d}}=f_{i}^{\text{c}}\quad \text{equals}\quad (\infty ,\infty )~\text{for }i<0. & & \displaystyle \nonumber\end{eqnarray}$$

Definition 3.2.6 is a modification of [Reference Adler and SpiceAS08, Definition B.1], used to describe commutator relationships among groups associated to concave functions. Our depth matrices are used to handle analogues when some of the groups involved (our $\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})$ ) do not have full rank, so that it does not make sense to speak of weights on the Lie algebra of the larger group occurring in the Lie algebra of the smaller one; but the philosophy is still that the multi-cased definition of $f_{1}\bowtie f_{2}$ is, in essence, keeping track of the many ways that a sum of weights might be a weight on $\operatorname{Lie}(\operatorname{C}_{\mathbf{G}^{j}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{k^{\text{sep}}})$ .

Definition 3.2.6. Suppose that $f_{1}$ and $f_{2}$ are depth matrices. We temporarily put

$$\begin{eqnarray}\displaystyle (f_{1}\hspace{0.0pt}\text{}_{\ast }\bowtie \,f_{2})_{ij}^{\text{n}} & = & \displaystyle \underset{\substack{ i_{-}\leqslant i\leqslant i_{+} \\ j_{-}\leqslant j\leqslant j_{+}}}{\inf }\min \{(f_{1}^{\text{n}})_{ij}+(f_{2}^{\text{n}})_{i_{+}j_{-}},(f_{1}^{\text{n}})_{ij_{-}}+(f_{2}^{\text{n}})_{i_{+}j},(f_{1}^{\text{n}})_{ij_{+}}+(f_{2}^{\text{n}})_{i_{+}j_{+}},\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad (f_{1}^{\pm })_{i_{-}j}+(f_{2}^{\mp })_{i_{-}j_{-}},(f_{1}^{\pm })_{i_{-}j_{+}}+(f_{2}^{\mp })_{i_{-}j_{+}}\}\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle (f_{1}\hspace{0.0pt}\text{}_{\ast }\bowtie \,f_{2})_{ij}^{\text{d}} & = & \displaystyle \inf _{\substack{ i_{-}\leqslant i\leqslant i_{+} \\ j_{-}\leqslant j\leqslant j_{+}}}\min \{(f_{1}^{-})_{ij}+(f_{2}^{-})_{i_{+}j_{-}},(f_{1}^{-})_{ij_{-}}+(f_{2}^{-})_{i_{+}j},(f_{1}^{-})_{ij_{+}}+(f_{2}^{-})_{i_{+}j_{+}},\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad (f_{1}^{\text{d}})_{i_{-}j}+(f_{2}^{\text{d}})_{i_{-}j_{-}},(f_{1}^{\text{d}})_{i_{-}j_{+}}+(f_{2}^{\text{d}})_{i_{-}j_{+}}\},\nonumber\end{eqnarray}$$

and similarly for . With this provisional (asymmetric) definition in place, we make the symmetric definition , and then put

and

A grouplike depth matrix $f$ satisfies $f\leqslant f\vee f$ ; and, if $f_{1}$ and $f_{2}$ are grouplike, then so is $f_{1}\vee f_{2}$ .

Lemma 3.2.7 is an adaptation to our situation of [Reference Adler and SpiceAS08, Lemma 5.17]. The proof follows from a routine application of Lemma 3.1.2 to groups of the form $\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant f_{i}}$ . We omit it, but the interested reader may see a detailed example of the relevant reasoning in the proof of Proposition 4.2.3.

Lemma 3.2.7 can be applied only when we know the depths of the elements in a commutator. We would like to apply similar results to commutators involving $\unicode[STIX]{x1D6FE}$ , but we do not have any information about its depth. The idea of Hypothesis 3.2.2 is that $\unicode[STIX]{x1D6FE}$ is supposed informally to ‘live at depth $i$ modulo $\operatorname{Z}(\operatorname{C}_{\mathbf{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))$ ’. We make this precise in Hypothesis 3.2.8 (and later state an analogous dual-Lie-algebra condition in Hypothesis 4.1.5).

In the remainder of this section, we write $\mathbf{H}$ for $\operatorname{C}_{\mathbf{G}}^{({<}r)}(\unicode[STIX]{x1D6FE})$ ; but we caution the reader that, in Proposition 5.1.3, the symbol $\mathbf{H}$ will be used for a different group.

Hypothesis 3.2.8 involves a point $x$ in ${\mathcal{B}}(G)$ (eventually, in ${\mathcal{B}}(H)$ ), which we now fix. We note that the hypothesis may hold for some points of ${\mathcal{B}}(H)$ , and not for others. The set of points for which it holds is analogous to the set ${\mathcal{B}}_{r}(\unicode[STIX]{x1D6FE})$ of [Reference Adler and SpiceAS08, Definition 9.5].

Suppose for this paragraph that [Reference Adler and SpiceAS08, Hypotheses A–D] are satisfied, and $\unicode[STIX]{x1D6FE}$ is a compact-modulo-centre element of a tame torus satisfying [Reference Adler and SpiceAS08, Definition 6.3]. Then Hypothesis 3.2.8(ii, iv) is an easy generalisation of [Reference Adler and SpiceAS08, Lemmas 5.30 and 7.2] (using Lemma 3.1.2 in place of [Reference Adler and SpiceAS08, Lemma 5.17] to make sure that commutators land in the derived group). Hypothesis 3.2.8(v) follows from [Reference Adler and SpiceAS08, Lemma 7.1, Corollary 4.8 (and Definition 6.4), and Lemma 8.1].

Lemma 3.2.11 states some consequences of Hypothesis 3.2.8, phrased in terms of groups and Lie algebras associated to depth matrices (rather than just depth vectors, as in Hypothesis 3.2.8). The only place that we need to know that the commutator in Lemma 3.2.11(i) belongs to ${\mathcal{D}}\vec{G}_{x\geqslant F}$ , rather than just $\vec{G}_{x\geqslant F}$ , is in Proposition 4.2.4. Note that Lemma 3.2.11(iii) involves $\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE}^{2})$ , not $\operatorname{C}_{G}^{({<}r)}(\unicode[STIX]{x1D6FE})$ .

As with Lemma 3.2.7, the proof of Lemma 3.2.11 follows from a routine application of Hypothesis 3.2.8 to groups of the form $\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant \vec{a}}$ , together with ‘successive approximation arguments’, as, for example, in [Reference AdlerAdl98, Lemma 2.3.2], to lift results about finite quotients to results about the ambient groups. We omit it.

The result [Reference DeBacker and SpiceDS16, Proposition 4.12] is stated for a group associated to a very particular concave function, but its proof applies much more generally. We isolate one general consequence here; the proof is identical.

Lemma 3.2.12 [Reference DeBacker and SpiceDS16, Proposition 4.12].

Suppose that $\vec{\mathbf{G}}$ is a tame, twisted Levi sequence in $\mathbf{G}$ containing $\unicode[STIX]{x1D6FE}$ , such that $x$ belongs to ${\mathcal{B}}(\vec{G})$ , and $f_{1}$ and $f_{2}$ are grouplike depth matrices satisfying $f_{1}\leqslant f_{2}$ and $(f_{1})_{i\,0}=(f_{2})_{i\,0}$ for all $0\leqslant i\leqslant r$ . Then we have that

$$\begin{eqnarray}[\vec{G}_{x\geqslant f_{1}}:\vec{G}_{x\geqslant f_{2}}]\quad \text{equals}\quad [\operatorname{Lie}(\vec{G})_{x\geqslant f_{1}}:\operatorname{Lie}(\vec{G})_{x\geqslant f_{2}}].\end{eqnarray}$$

The conditions in Lemma 3.2.12 require that we quotient out by the ‘troublesome’ part of the Moy–Prasad group, which is to say the part corresponding to a maximal torus. Thus, we may apply Lemma 3.2.12 to compute   in the proof of Lemma 5.2.1, because $\mathbf{G}^{\prime }$ there has full rank; but not to compute   in Lemma 3.2.14, because $\mathbf{H}$ need not have full rank.

Lemma 3.2.13 is used in our explicit ‘constant term (about $\unicode[STIX]{x1D6FE}$ )’ calculations in Lemma 4.4.3.

Lemma 3.2.14 states a useful analogue of [Reference DeBacker and SpiceDS16, Proposition 4.12], also proven using Lemma 3.2.12. It accounts for the appearance of factors involving discriminants in Proposition 5.2.6. As mentioned after Lemma 3.2.12, the necessity to deal with a quotient of indices, rather than a single index, comes from the fact that $\mathbf{H}$ need not have full rank. If it happened that $\mathbf{H}$ did have full rank, then reduction to the Lie algebra would allow us to compute $[(H,M,G)_{x\geqslant (0+,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})/2,s)}:(H,M,G)_{x\geqslant (0+,s,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}^{\pm 1}})/2)}]$ itself, giving the expected answer.

Lemma 3.2.14. If $\mathbf{G}^{\prime }$ is a tame, twisted Levi subgroup of $\mathbf{G}$ containing $\unicode[STIX]{x1D6FE}$ , with $x$ in ${\mathcal{B}}(G^{\prime })$ , and $r$ is positive, then we have that

$$\begin{eqnarray}\displaystyle \frac{[(H,M,G)_{x\geqslant (0+,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})/2,s)}:(H,M,G)_{x\geqslant (0+,s,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})/2)}]}{[(H^{\prime },M^{\prime },G^{\prime })_{x\geqslant (0+,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})/2,s)}:(H^{\prime },M^{\prime },G^{\prime })_{x\geqslant (0+,s,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})/2)}]} & & \displaystyle \nonumber\end{eqnarray}$$

equals

$$\begin{eqnarray}\displaystyle \frac{{|(H,G)_{x=(0+,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})/2)}|}^{-1/2}}{{|(H^{\prime },G^{\prime })_{x=(0+,(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})/2)}|}^{-1/2}}\cdot \frac{{|\operatorname{D}_{G/H}(\unicode[STIX]{x1D6FE})|}^{-1/2}\unicode[STIX]{x1D6FF}_{P^{-}}(\unicode[STIX]{x1D6FE})^{1/2}}{{|\operatorname{D}_{G^{\prime }/H^{\prime }}(\unicode[STIX]{x1D6FE})|}^{-1/2}\unicode[STIX]{x1D6FF}_{P^{\prime \,-}}(\unicode[STIX]{x1D6FE})^{1/2}}\cdot \frac{[G_{x=s}:H_{x=s}]^{-1/2}}{[G_{x=s}^{\prime }:H_{x=s}^{\prime }]^{-1/2}}. & & \displaystyle \nonumber\end{eqnarray}$$

Proof. For the proof, we write $s_{\unicode[STIX]{x1D6FE}^{\pm 1}}$ in place of $(r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}^{\pm 1}})/2$ .

Lemma 3.2.12 gives that

$$\begin{eqnarray}\displaystyle \frac{[G_{x=s}:H_{x=s}]}{[G_{x=s}^{\prime }:H_{x=s}^{\prime }]}=[G_{x\geqslant s}:(G^{\prime },H,G)_{x\geqslant (s,s,s+)}] & & \displaystyle \nonumber\end{eqnarray}$$

equals

$$\begin{eqnarray}\displaystyle [\operatorname{Lie}(G)_{x\geqslant s}:\operatorname{Lie}(G^{\prime },H,G)_{x\geqslant (s,s,s+)}], & & \displaystyle \nonumber\end{eqnarray}$$

and similarly for

$$\begin{eqnarray}\frac{[(H,M,G)_{x\geqslant (0+,s_{\unicode[STIX]{x1D6FE}}^{-},s)}:(H,M,G)_{x\geqslant (0+,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}}^{-})}]}{[(H^{\prime },M^{\prime },G^{\prime })_{x\geqslant (0+,s_{\unicode[STIX]{x1D6FE}}^{-},s)}:(H^{\prime },M^{\prime },G^{\prime })_{x\geqslant (0+,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}}^{-})}]}.\end{eqnarray}$$

Thus we may, and do, work on the Lie algebra.

We have that $[\operatorname{Lie}(H,M,G)_{x\geqslant (0+,s_{\unicode[STIX]{x1D6FE}},s)}:\operatorname{Lie}(H,M,G)_{x\geqslant (0+,s,s_{\unicode[STIX]{x1D6FE}^{\pm 1}})}]$ equals both

$$\begin{eqnarray}\displaystyle [\operatorname{Lie}(H,M,G)_{x\geqslant (0,s_{\unicode[STIX]{x1D6FE}},s)}:\operatorname{Lie}(H,M,G)_{x\geqslant (0,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}] & & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle & & \displaystyle [\operatorname{Lie}(H,M,G)_{x\geqslant (0+,s_{\unicode[STIX]{x1D6FE}},s)}:\operatorname{Lie}(H,M,G)_{x>(0,s_{\unicode[STIX]{x1D6FE}},s)}]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,[\operatorname{Lie}(H,M,G)_{x>(0,s_{\unicode[STIX]{x1D6FE}},s)}:\operatorname{Lie}(H,M,G)_{x>(0,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,[\operatorname{Lie}(H,M,G)_{x\geqslant (0+,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}:\operatorname{Lie}(H,M,G)_{x>(0,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}]^{-1}\nonumber\\ \displaystyle & & \displaystyle \quad =|\operatorname{Lie}(H,M,G)_{x=(0+,s_{\unicode[STIX]{x1D6FE}},s)}|\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,[\operatorname{Lie}(H,M,G)_{x>(0,s_{\unicode[STIX]{x1D6FE}},s)}:\operatorname{Lie}(H,M,G)_{x>(0,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,|\operatorname{Lie}(H,M,G)_{x=(0+,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}|.\nonumber\end{eqnarray}$$

Since

$$\begin{eqnarray}\displaystyle |\operatorname{Lie}(H,M,G)_{x=(0+,s_{\unicode[STIX]{x1D6FE}},s)}|\cdot |\operatorname{Lie}(H,M,G)_{x=(0+,s,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}| & & \displaystyle \nonumber\end{eqnarray}$$

equals

$$\begin{eqnarray}\displaystyle & & \displaystyle |\operatorname{Lie}(H,M,G)_{x=(0+,s_{\unicode[STIX]{x1D6FE}},s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}|\cdot |\operatorname{Lie}(H,M,G)_{x=(0+,s,s)}|\nonumber\\ \displaystyle & & \displaystyle \quad =|\operatorname{Lie}(H,G)_{x=(0+,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}|\cdot |\operatorname{Lie}(H,G)_{x=(0+,s)}|,\nonumber\end{eqnarray}$$

and similarly for $G^{\prime }$ , and since

$$\begin{eqnarray}\frac{\operatorname{Lie}(H,G)_{x=(0+,s)}}{\operatorname{Lie}(H^{\prime },G^{\prime })_{x=(0+,s)}}\quad \text{equals}\quad \frac{[\operatorname{Lie}(G)_{x=s}:\operatorname{Lie}(H)_{x=s}]}{[\operatorname{Lie}(G^{\prime })_{x=s}:\operatorname{Lie}(H^{\prime })_{x=s}]},\end{eqnarray}$$

we have by [Reference DeBacker and SpiceDS16, Corollary 3.15] that the left-hand side equals

$$\begin{eqnarray}\frac{|\operatorname{Lie}(H,G)_{x=(0+,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}|^{-1/2}}{|\operatorname{Lie}(H^{\prime },G^{\prime })_{x=(0+,s_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}|^{-1/2}}\cdot \frac{[\operatorname{Lie}(G)_{x=s}:\operatorname{Lie}(H)_{x=s}]^{-1/2}}{[\operatorname{Lie}(G^{\prime })_{x=s}:\operatorname{Lie}(H^{\prime })_{x=s}]^{-1/2}}\end{eqnarray}$$

times

$$\begin{eqnarray}\displaystyle & & \displaystyle \underbrace{\frac{[\operatorname{Lie}(H,M)_{x\geqslant (0,r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})}:\operatorname{Lie}(H,M)_{x\geqslant (0,r)}]^{1/2}}{[\operatorname{Lie}(H^{\prime },M^{\prime })_{x\geqslant (0,r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})}:\operatorname{Lie}(H,M)_{x\geqslant (0,r)}]^{1/2}}}_{(\text{I})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\underbrace{\frac{[\operatorname{Lie}(N^{+})_{x\geqslant (0,r)}:\operatorname{Lie}(N^{+})_{x\geqslant (0,r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}^{-1}})}]^{1/2}}{[\operatorname{Lie}(N^{\prime \,+})_{x\geqslant (0,r)}:\operatorname{Lie}(N^{\prime \,+})_{x\geqslant (0,r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}^{-1}})}]^{1/2}}}_{(\text{II}^{+})}\nonumber\end{eqnarray}$$

times

$$\begin{eqnarray}\underbrace{\frac{[\operatorname{Lie}(N^{-})_{x\geqslant (0,r)}:\operatorname{Lie}(N^{-})_{x\geqslant (0,r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})}]^{1/2}}{[\operatorname{Lie}(N^{\prime \,-})_{x\geqslant (0,r)}:\operatorname{Lie}(N^{\prime \,-})_{x\geqslant (0,r-\operatorname{ord}_{\unicode[STIX]{x1D6FE}})}]^{1/2}}}_{(\text{II}^{-})}.\end{eqnarray}$$

We have that the numerator of (I) equals ${|\det _{\operatorname{Lie}(M)/\!\operatorname{Lie}(H)}(\unicode[STIX]{x1D6FE}-1)|}^{-1/2}={|\operatorname{D}_{M/H}(\unicode[STIX]{x1D6FE})|}^{-1/2}$ , while the numerator of ( $\text{II}^{\pm }$ ) equals ; and similarly for the denominator. Since   equals $|\operatorname{D}_{G/H}(\unicode[STIX]{x1D6FE})|$ , and similarly for $G^{\prime }$ , we are done.◻

4.1 Good, and nearly good, elements

The main result of § 4, Theorem 4.4.11, is the analogue of [Reference Kim and MurnaghanKM03, Theorem 5.3.1]. That result is stated in terms of a so-called good minimal K-type, which in turn is defined in terms of an element $\unicode[STIX]{x1D6E4}$ satisfying properties analogous to the genericity assumptions [Reference YuYu01, § 8, p. 596, GE].

We fix an element $Z_{o}^{\ast }$ in $\operatorname{Lie}^{\ast }(G)$ , which will be the analogue of Kim and Murnaghan’s element $\unicode[STIX]{x1D6E4}$ . We require that it satisfy similar genericity assumptions, slightly upgraded to handle the possible disconnectedness of $\mathbf{G}$ , which we state as Hypothesis 4.1.1. Recall that we fixed a non-negative real number $r$ in § 3.2.

This section analyses the properties of $Z_{o}^{\ast }$ and nearby elements. Note that the subgroup $\mathbf{G}^{\prime }$ in Hypothesis 4.1.1 is uniquely determined, and that, if $Z_{o}^{\ast }$ has depth $-r$ (which it need not; for example, it could be $0$ ), then it is good of that depth, in the sense of [Reference Kim and MurnaghanKM03, Definition 2.1.1(2)].

We now recall, in addition to the non-negative real number $r$ , the

1. – element $\unicode[STIX]{x1D6FE}\in G$ , with its associated groups $\mathbf{P}^{\mp }=\operatorname{C}_{\mathbf{G}}^{({<}-\infty )}(\unicode[STIX]{x1D6FE}^{\pm 1})$ , $\mathbf{N}^{\mp }$ , $\mathbf{M}=\operatorname{C}_{\mathbf{G}}^{({<}0)}(\unicode[STIX]{x1D6FE})$ , and $\mathbf{H}=\operatorname{C}_{\mathbf{G}}^{({<}r)}(\unicode[STIX]{x1D6FE})$ , and

2. – point $x\in {\mathcal{B}}(H)$ ,

satisfying Hypotheses 3.2.2 and 3.2.8, from § 3.2. These hypotheses say nothing about the relationship between $\unicode[STIX]{x1D6FE}$ and $Z_{o}^{\ast }$ . Lemmas 4.1.3 and 4.1.4 discuss conditions under which some such relationship can be deduced.

Lemma 4.1.3. If $o\in {\mathcal{B}}(G^{\prime })$ and $X^{\ast }\in \operatorname{Lie}^{\ast }(H)_{x\geqslant -r}$ are such that

$$\begin{eqnarray}\displaystyle (X^{\ast }+\operatorname{Lie}^{\ast }(G)_{x>-r})\,\cap \,(Z_{o}^{\ast }+\operatorname{Lie}^{\ast }(G)_{o>-r}) & & \displaystyle \nonumber\end{eqnarray}$$

is non-empty, then $x$ belongs to ${\mathcal{B}}(G^{\prime })$ and $Z_{o}^{\ast }$ to $\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}$ , and there is an element $g$ of $(G_{x\geqslant 0}\,\cap \,G_{o>0})G_{x>0}$ such that

$$\begin{eqnarray}\displaystyle (X^{\ast }+\operatorname{Lie}^{\ast }(G^{\prime })_{x>-r})\,\cap \,\operatorname{Ad}^{\ast }(g)^{-1}(Z_{o}^{\ast }+(\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}\,\cap \,\operatorname{Lie}^{\ast }(G^{\prime })_{o>-r})) & & \displaystyle \nonumber\end{eqnarray}$$

is non-empty.

Proof. By [Reference Kim and MurnaghanKM03, Lemma 2.3.3], we have that $x$ belongs to ${\mathcal{B}}(G^{\prime })$ . Then, by Hypothesis 4.1.1(i, ii), we have that $Z_{o}^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}$ . It follows that

$$\begin{eqnarray}(X^{\ast }+\operatorname{Lie}^{\ast }(G)_{x>-r})\,\cap \,(Z_{o}^{\ast }+(\operatorname{Lie}^{\ast }(G)_{x\geqslant -r}\,\cap \,\operatorname{Lie}^{\ast }(G)_{o>-r}))\end{eqnarray}$$

is non-empty, so

$$\begin{eqnarray}X^{\ast }\quad \text{belongs to}\quad Z_{o}^{\ast }+(\operatorname{Lie}^{\ast }(G)_{x\geqslant -r}\,\cap \,\operatorname{Lie}^{\ast }(G)_{o>-r})+\operatorname{Lie}^{\ast }(G)_{x>-r}.\end{eqnarray}$$

The result now follows from the dual-Lie-algebra analogue of [Reference Adler and SpiceAS08, Lemma 7.4]. ◻

Lemma 4.1.4. If $\unicode[STIX]{x1D6FE}$ centralises $\mathbf{H}^{\circ }$ and

$$\begin{eqnarray}\operatorname{Lie}^{\ast }(H)\,\cap \,(Z_{o}^{\ast }+\operatorname{Lie}^{\ast }(G^{\prime })_{{>}-r})\end{eqnarray}$$

is non-empty, then $\unicode[STIX]{x1D6FE}$ belongs to $G^{\prime }$ , and $Z_{o}^{\ast }$ to $\operatorname{Lie}^{\ast }(\operatorname{C}_{G^{\prime }}^{({<}r)}(\unicode[STIX]{x1D6FE}))$ .

Until § 4.2, we assume that $\unicode[STIX]{x1D6FE}$ belongs to $G^{\prime }$ . (Lemma 4.1.4 shows that this is automatic if we require $\unicode[STIX]{x1D6FE}$ to centralise $\mathbf{H}^{\circ }$ , as we do in Hypothesis 4.4.2.) We use primes (the punctuation marks, not the integers) to denote the analogues in $\mathbf{G}^{\prime }$ of constructions in $\mathbf{G}$ , so, for example, $\mathbf{H}^{\prime }$ stands for $\operatorname{C}_{\mathbf{G}^{\prime }}^{({<}r)}(\unicode[STIX]{x1D6FE})$ (subject to the proviso in Remark 3.2.3, that we may refer directly only to the identity component of $\mathbf{H}^{\prime }$ ). We remind the reader that, in Proposition 5.1.3, the symbol $\mathbf{H}$ will be used for a different group.

In order to prove the existence of asymptotic expansions in Theorem 4.4.11, we need to consider, not just good elements such as $Z_{o}^{\ast }$ , but elements of $\operatorname{Lie}^{\ast }(H)$ close to them. These ‘nearly good’ elements $X^{\ast }$ enjoy weakened versions of many of the same properties as $Z_{o}^{\ast }$ , which nonetheless suffice for the explicit calculations in § 5. See Remark 4.1.7. We isolate these properties in Hypothesis 4.1.5, so that they may be used without explicit reference to $Z_{o}^{\ast }$ . Note, however, that the hypothesis does depend on the element $\unicode[STIX]{x1D6FE}\in G$ , the point $x\in {\mathcal{B}}(H)$ , and the subgroup $\mathbf{G}^{\prime }$ .

Hypotheses 3.2.8 and 4.1.5 bear a close resemblance, the idea being that $\mathbf{G}^{\prime }$ plays the role of $\operatorname{C}_{\mathbf{G}}^{({\leqslant}-r)}(X^{\ast })$ ; but they are not literal translations. For example, the similar-appearing Hypotheses 3.2.8(v) and 4.1.5(v) are subtly different. We genuinely need the extra strength of the latter, for example in Lemma 5.2.2, and can get away with assuming it because $\mathbf{G}^{\circ }\,\cap \,\mathbf{G}^{\prime }$ is automatically connected (Remark 3.2.3), whereas $\mathbf{G}^{\circ }\,\cap \,\mathbf{H}$ need not be.

Proof. As with Lemmas 3.2.7 and 3.2.11, it suffices simply to apply Hypothesis 4.1.5 repeatedly to groups of the form $\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant \vec{a}}$ , once we show the claimed heredity. Hypotheses 4.1.5(i, ii, v) are automatic, so we need only consider Hypotheses 4.1.5(iv, iii). The calculations below are lengthy, but routine. The idea is that, if we know something about $(\operatorname{Ad}^{\ast }(g)-1)X^{\ast }$ relative to  $\unicode[STIX]{x1D6FE}$ , then we can use it to learn something about $(\operatorname{Ad}^{\ast }([g,\unicode[STIX]{x1D6FE}])-1)X^{\ast }$ , so that Hypothesis 4.1.5(iv) (for $\mathbf{G}$ ) gives us information about $[g,\unicode[STIX]{x1D6FE}]$ , and then Lemma 3.2.11(ii) gives us information about $g$ itself. Similarly, if we know something about $\operatorname{ad}^{\ast }(Y)X^{\ast }$ relative to $\unicode[STIX]{x1D6FE}$ , then we can use it to learn something about $\operatorname{Ad}^{\ast }((\operatorname{Ad}(\unicode[STIX]{x1D6FE})-1)Y)X^{\ast }$ .

For Hypothesis 4.1.5(iii), note that $\operatorname{Lie}(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}^{\pm 1}))_{x=\vec{a}}/\!\operatorname{Lie}(\operatorname{C}_{G^{\prime }\,\cap \,\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}^{\pm 1}))_{x=\vec{a}}$ sits naturally inside $\vec{G}_{x=\vec{a}}/(G^{\prime }\,\cap \,\vec{G})_{x=\vec{a}}$ , and similarly for $\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}^{\pm 1}))_{x=\vec{a}-r}/\!\operatorname{Lie}^{\ast }(\operatorname{C}_{G^{\prime }\,\cap \,\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}^{\pm 1}))_{x=\vec{a}-r}$ , and that the image of the former is contained in the latter (all by Hypotheses 3.2.2(iv) and 4.1.5). Thus, it suffices to show that, if $Y\in \vec{G}_{x\geqslant \vec{a}}$ is such that $\operatorname{ad}^{\ast }(Y)X^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r}$ , then $Y$ belongs to $\operatorname{Lie}(G^{\prime },\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}),\vec{G})_{x\geqslant (\vec{a},\vec{a},\vec{a}+)}$ . In this case, with the clumsy notation $X_{\unicode[STIX]{x1D6FE}^{-1}}^{\ast }:=(\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE})^{-1}-1)X^{\ast }$ , which belongs to $\operatorname{Lie}^{\ast }(H)_{x\geqslant 0}$ (by Hypothesis 4.1.5(ii) and Remark 3.2.10), and $X_{Y}^{\ast }:=\operatorname{ad}^{\ast }(Y)X^{\ast }$ , which belongs to $\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r}$ (by assumption), we have by Lemmas 3.2.7(i) and 3.2.11(i) that

$$\begin{eqnarray}\operatorname{ad}^{\ast }((\operatorname{Ad}(\unicode[STIX]{x1D6FE})-1)Y)X^{\ast }=((\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE})-1)\operatorname{ad}^{\ast }(Y)+\operatorname{ad}^{\ast }(Y))X_{\unicode[STIX]{x1D6FE}^{-1}}^{\ast }+(\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE})-1)X_{Y}^{\ast }\end{eqnarray}$$

belongs to

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{Lie}^{\ast }(H\,\cap \,\vec{G},N^{+}\,\cap \,\vec{G},\vec{G})_{x\geqslant (\vec{a}+r,\vec{a},\vec{a}+\operatorname{ord}_{\unicode[STIX]{x1D6FE}})}+\operatorname{Lie}^{\ast }(\vec{G})_{x\geqslant \vec{a}}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r+i}\nonumber\\ \displaystyle & & \displaystyle \quad =\operatorname{Lie}^{\ast }(H\,\cap \,\vec{G},M\,\cap \,\vec{G},\vec{G})_{x\geqslant (\vec{a}+r,\vec{a},\vec{a}+\operatorname{ord}_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r+i}\nonumber\\ \displaystyle & & \displaystyle \quad \subseteq \operatorname{Lie}^{\ast }(M\,\cap \,\vec{G},\vec{G})_{x\geqslant (\vec{a}-r+i,\vec{a}+\operatorname{ord}_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})}.\nonumber\end{eqnarray}$$

Then Hypothesis 4.1.5(iii) gives that $(\operatorname{Ad}(\unicode[STIX]{x1D6FE})-1)Y$ belongs to

$$\begin{eqnarray}\operatorname{Lie}(G^{\prime },M\,\cap \,\vec{G},\vec{G})_{x\geqslant (-\infty ,\vec{a}+i,\vec{a}+r+\operatorname{ord}_{\unicode[STIX]{x1D6FE}\text{}^{\pm 1}})},\end{eqnarray}$$

and Lemma 3.2.11(ii) gives that $Y$ belongs to

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{Lie}(G^{\prime },\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}),M\,\cap \,\vec{G},N^{+}\,\cap \,\vec{G},\vec{G})_{x\geqslant (-\infty ,-\infty ,\vec{a},\vec{a}+,\vec{a}+r+\operatorname{ord}_{\unicode[STIX]{x1D6FE}^{-1}},\vec{a}+r)}\nonumber\\ \displaystyle & & \displaystyle \quad \subseteq \operatorname{Lie}(G^{\prime },\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}),N^{+},\vec{G})_{x\geqslant (-\infty ,-\infty ,\vec{a},-\infty ,\vec{a}+)}.\nonumber\end{eqnarray}$$

If we write $Y=Y^{-}+Y^{+}$ , with $Y^{-}\in \operatorname{Lie}(G^{\prime },\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}),\vec{G})_{x\geqslant (\vec{a},\vec{a},\vec{a}+)}$ and $Y^{+}\in \operatorname{Lie}(N^{+})$ , then we see that $\operatorname{ad}^{\ast }(Y)X^{\ast }-\operatorname{ad}^{\ast }(Y^{-})X^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(P^{-})$ by assumption, but, since it equals $\operatorname{ad}^{\ast }(Y^{+})X^{\ast }$ , also to $\operatorname{Lie}^{\ast }(P^{+})$ ; so that it equals $0$ , and hence (by Hypothesis 4.1.5(iii) again) that $Y^{+}$ belongs to $\operatorname{Lie}(G^{\prime })$ . Thus, we have shown that $Y\in \operatorname{Lie}(\vec{G})_{x\geqslant \vec{a}}$ belongs to $\operatorname{Lie}(G^{\prime },\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}),N^{+},\vec{G})_{x\geqslant (-\infty ,-\infty ,\vec{a},\infty ,\vec{a}+)}$ , hence to $\operatorname{Lie}(G^{\prime },\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}),\vec{G})_{x\geqslant (\vec{a},\vec{a},\vec{a}+)}$ , as desired.

Similarly, we reduce Hypothesis 4.1.5(iv) to showing that, if $\vec{a}$ is grouplike, $i\in \mathbb{R}_{{\geqslant}0}$ satisfies $i<r$ , and $g\in \operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant \vec{a}}$ is such that $(\operatorname{Ad}^{\ast }(g)-1)X^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r}$ , then $g$ belongs to $(M^{\prime },\operatorname{C}_{G}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),M)_{x\geqslant (\vec{a},\vec{a},\vec{a}+)}$ .

In what follows, we use Lemmas 3.2.7(ii) and 3.2.11(i) repeatedly, without explicit mention. Note that $g$ normalises $\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r}$ (since we have the inequality $\vec{a}\,\vee \,(\vec{a}-r)\geqslant \vec{a}-r$ ), so that $\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE}g)^{-1}X^{\ast }$ belongs to

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{Ad}^{\ast }(g)^{-1}(X^{\ast }+\operatorname{Lie}^{\ast }(H)_{x\geqslant 0})\nonumber\\ \displaystyle & & \displaystyle \quad \subseteq X^{\ast }+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r}+\operatorname{Lie}^{\ast }(H,\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant (0,\vec{a})}\nonumber\\ \displaystyle & & \displaystyle \quad =X^{\ast }+\operatorname{Lie}^{\ast }(H)_{x\geqslant 0}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant (\vec{a}-r,\vec{a})}.\nonumber\end{eqnarray}$$

Thus $(\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE})-1)\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE}g)^{-1}X^{\ast }$ belongs to

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{Lie}^{\ast }(H)_{x\geqslant 0}+\operatorname{Lie}^{\ast }(H)_{x\geqslant r}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant ((\vec{a}-r+i)+,\vec{a}+i)}\nonumber\\ \displaystyle & & \displaystyle \quad \subseteq \operatorname{Lie}^{\ast }(H)_{x\geqslant 0}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x>\vec{a}-r+i},\nonumber\end{eqnarray}$$

and $(\operatorname{Ad}^{\ast }(g)-1)\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE}g)^{-1}X^{\ast }$ belongs to

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}-r}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant (\vec{a}-r,\vec{a})}\nonumber\\ \displaystyle & & \displaystyle \quad \subseteq \operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x>\vec{a}-r},\nonumber\end{eqnarray}$$

so

$$\begin{eqnarray}(\operatorname{Ad}^{\ast }([g,\unicode[STIX]{x1D6FE}])-1)X^{\ast }= [\operatorname{Ad}^{\ast }(g)-1,\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE})-1]\operatorname{Ad}^{\ast }(\unicode[STIX]{x1D6FE}g)^{-1}X^{\ast }\end{eqnarray}$$

belongs to

$$\begin{eqnarray}\displaystyle & & \displaystyle (\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant \vec{a}}+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x>\vec{a}-r+i})+\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x>\vec{a}-r+i}\nonumber\\ \displaystyle & & \displaystyle \quad =\operatorname{Lie}^{\ast }(\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x>\vec{a}-r+i}.\nonumber\end{eqnarray}$$

Now we have by Hypothesis 4.1.5(iv) that $[g,\unicode[STIX]{x1D6FE}]$ , which certainly belongs to $M_{x>0}$ , in fact belongs to $(G^{\prime },M)_{x>(0,\vec{a}+i)}$ ; hence by Lemma 3.2.11(ii) that $g$ belongs to $(G^{\prime },\operatorname{C}_{G}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),M)_{x\geqslant (0+,\vec{a},\vec{a}+)}$ . Since we already know that $g$ belongs to $\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE})_{x\geqslant \vec{a}}$ , it belongs to $(G^{\prime },\operatorname{C}_{G}^{({\leqslant}i)}(\unicode[STIX]{x1D6FE}),\operatorname{C}_{\vec{G}}^{({<}i)}(\unicode[STIX]{x1D6FE}))_{x\geqslant (\vec{a},\vec{a},\vec{a}+)}$ , as desired.◻

So far, we have discussed the consequences of Hypothesis 4.1.5 abstractly, without any reference to the element $Z_{o}^{\ast }$ . Although we still do not require that the element $X^{\ast }$ of Hypothesis 4.1.5 is close to $Z_{o}^{\ast }$ , we observe in Remark 4.1.7 that choosing such nearby elements is one way to satisfy the hypothesis.

Remark 4.1.7. Suppose that $X^{\ast }$ is an element of $\operatorname{Lie}^{\ast }(H)\,\cap \,(Z_{o}^{\ast }+\operatorname{Lie}^{\ast }(G^{\prime })_{{>}-r})$ . In particular, $X^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(G^{\prime })$ .

Hypothesis 4.1.5(i) follows from [Reference Kim and MurnaghanKM03, Lemma 2.3.3]. Hypothesis 4.1.5(ii) is built into our choice of $X^{\ast }$ . By Lemma 4.1.4, we have that $Z_{o}^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}$ , so, since $X^{\ast }$ does as well, we have that $X^{\ast }-Z_{o}^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}\,\cap \,\operatorname{Lie}^{}(G^{\prime })_{{>}-r}$ . Since $(\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}\,\cap \,\operatorname{Lie}^{\ast }(G^{\prime })_{{>}-r})+\operatorname{Lie}^{\ast }(G^{\prime })_{x>-r}$ is contained in $\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}\,\cap \,\operatorname{Lie}^{\ast }(G^{\prime })_{{>}-r}$ , we have that

$$\begin{eqnarray}X^{\ast }+\operatorname{Lie}^{\ast }(G^{\prime })_{x>-r}\quad \text{is contained in}\quad Z_{o}^{\ast }+(\operatorname{Lie}^{\ast }(G^{\prime })_{x\geqslant -r}\,\cap \,\operatorname{Lie}^{\ast }(G^{\prime })_{{>}-r}).\end{eqnarray}$$

In particular, Hypothesis 4.1.5(iii, iv) is proven as in [Reference Kim and MurnaghanKM03, Lemma 2.3.4], and Hypothesis 4.1.5(v) follows from the dual-Lie-algebra analogue of [Reference Kim and MurnaghanKM03, Lemma 2.3.6].

In Lemma 4.1.8, we lay the groundwork for the proof in Lemma 4.4.14 that only certain orbits need to be considered when checking the correctness of a possible asymptotic expansion.

Lemma 4.1.8. If $X^{\ast }$ belongs to $\operatorname{Lie}^{\ast }(H)\,\cap \,(Z_{o}^{\ast }+\operatorname{Lie}^{\ast }(G^{\prime })_{{>}-r})$ and $g\in G$ is such that the intersection

$$\begin{eqnarray}{\mathcal{O}}^{H^{\prime \,\circ }}(\operatorname{Ad}^{\ast }(H^{\circ })(X^{\ast }+\operatorname{Lie}^{\ast }(H)_{x>-r}))\,\cap \,{\mathcal{O}}^{H^{\prime \,\circ }}(\operatorname{Ad}^{\ast }(g)^{-1}(Z_{o}^{\ast }+\operatorname{Lie}^{\ast }(G^{\prime })_{{>}-r}))\end{eqnarray}$$

is non-empty, then $G^{\prime }gH^{\circ }$ is the trivial $(G^{\prime },H^{\circ })$ -double coset, and the intersection equals ${\mathcal{O}}^{H^{\prime \,\circ }}(X^{\ast }+\operatorname{Lie}^{\ast }(H^{\prime })_{x>-r})$ .

4.2 A group analogue of a quadratic form

In § 4.4, we express distributions on the group (eventually, characters, in Theorem 4.4.11) in terms of distributions on the Lie algebra (Fourier transforms of orbital integrals). This requires some way of passing between the two. In Hypothesis 4.3.1, we choose a ‘mock’ exponential map that we will use to move from a neighbourhood of $0$ in $\operatorname{Lie}(H)$ to a neighbourhood of the identity in $H$ , but we still face the problem of extracting information about distributions on $G$ . The ‘perpendicular group’ $(H,G)_{x\geqslant (r+,r)}$ in Lemma 4.2.5 is an important tool, but a bit of machinery is necessary before we can define it.

We recall the

1. – non-negative real number $r$ ,

2. – element $\unicode[STIX]{x1D6FE}\in G$ , with associated groups $\mathbf{M}=\operatorname{C}_{\mathbf{G}}^{({<}0)}(\unicode[STIX]{x1D6FE})$ and $\mathbf{H}=\operatorname{C}_{\mathbf{G}}^{({<}r)}(\unicode[STIX]{x1D6FE})$ , and

3. – point $x\in {\mathcal{B}}(H)$

from § 3.2. Although we will soon (in Proposition 4.2.4) need $r$ to be positive, there is no harm in avoiding that assumption for a little bit. We do not need explicitly to mention the element $Z_{o}^{\ast }$ or the group $\mathbf{G}^{\prime }$ of § 4.1 in this section.

Notation 4.2.1 defines $\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}$ and $\mathfrak{B}_{\unicode[STIX]{x1D6FE}}$ , which are multiplicative analogues of a quadratic and a bilinear form, respectively. Indeed, we have chosen the notation to parallel the quadratic form $q_{X^{\ast },\unicode[STIX]{x1D6FE}}$ and the bilinear form $b_{X^{\ast },\unicode[STIX]{x1D6FE}}$ that are introduced in Notation 5.1.2. The main use of our multiplicative analogues is in Proposition 5.1.8, where we show that a certain integral appearing in a Frobenius-type formula (see Proposition 5.2.6) may be rewritten as a Weil index. We have moved these results from §§5.1 to 4.2 so that they can be used in Lemma 4.2.5.

Notation 4.2.1. Put

$$\begin{eqnarray}\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(v)= [v,\unicode[STIX]{x1D6FE}]\quad \text{and}\quad \mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{1},v_{2})= [v_{1},\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(v_{2})]\end{eqnarray}$$

for all $v,v_{1},v_{2}\in G$ .

As preparation for the bi-multiplicativity results of Proposition 4.2.3, we note in Lemma 4.2.2 a few basic algebraic identities involving $\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}$ and $\mathfrak{B}_{\unicode[STIX]{x1D6FE}}$ .

Lemma 4.2.2. We have that

(1) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(v_{1}v_{2})\quad \text{equals}\quad \mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{1},v_{2})\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(v_{2})\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(v_{1}), & \displaystyle\end{eqnarray}$$

(2) $$\begin{eqnarray}\displaystyle & \displaystyle \mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{1}w_{1},v_{2})\quad \text{equals}\quad \operatorname{Int}(v_{1})\mathfrak{B}_{\unicode[STIX]{x1D6FE}}(w_{1},v_{2})\cdot \mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{1},v_{2}), & \displaystyle\end{eqnarray}$$

(3) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{1},v_{2}w_{2})\mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{2},w_{2})\quad \text{equals}\quad \nonumber\\ \displaystyle & & \displaystyle \quad \operatorname{Int}(v_{1})\mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{2},w_{2})\cdot \mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{1},w_{2})\cdot \operatorname{Int}(\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(w_{2}))\mathfrak{B}_{\unicode[STIX]{x1D6FE}}(v_{1},v_{2}),\end{eqnarray}$$

and

(4) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathfrak{B}_{\unicode[STIX]{x1D6FE}}(\operatorname{Int}(\unicode[STIX]{x1D6FE})v_{1},v_{2})\quad \text{equals}\quad \nonumber\\ \displaystyle & & \displaystyle \quad \operatorname{Int}(\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(v_{1}))^{-1}\mathfrak{B}_{\unicode[STIX]{x1D6FE}}([v_{1},v_{2}]\cdot v_{2},v_{1})\cdot \mathfrak{B}_{\unicode[STIX]{x1D6FE}}([v_{1},v_{2}],v_{2})\operatorname{Int}(\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}(v_{2}))\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}([v_{1},v_{2}]),\end{eqnarray}$$

for all $v_{1},w_{1},v_{2},w_{2}\in G$ .

Proof. These are all direct computations. The last is the Hall–Witt identity

$$\begin{eqnarray}[[\unicode[STIX]{x1D6FE},v_{1}],\operatorname{Int}(v_{1})v_{2}]\cdot [[v_{1},v_{2}],\operatorname{Int}(v_{2})\unicode[STIX]{x1D6FE}]\cdot [[v_{2},\unicode[STIX]{x1D6FE}],\operatorname{Int}(\unicode[STIX]{x1D6FE})v_{1}]=1,\end{eqnarray}$$

rewritten as

$$\begin{eqnarray}[\operatorname{Int}(\unicode[STIX]{x1D6FE})v_{1},[v_{2},\unicode[STIX]{x1D6FE}]]=\operatorname{Int}([v_{1},\unicode[STIX]{x1D6FE}])^{-1}[\operatorname{Int}(v_{1})v_{2},[v_{1},\unicode[STIX]{x1D6FE}]]\cdot [[v_{1},v_{2}],[v_{2},\unicode[STIX]{x1D6FE}]\cdot \unicode[STIX]{x1D6FE}]\end{eqnarray}$$

and then translated to our notation. ◻

We think informally of $\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}$ and $\mathfrak{B}_{\unicode[STIX]{x1D6FE}}$ as being something like a quadratic and a bilinear form, respectively. We justify the latter claim in Proposition 4.2.3, by showing that $\mathfrak{B}_{\unicode[STIX]{x1D6FE}}$ is bi-multiplicative, up to an error term, on an appropriate domain; and then the former in Proposition 4.2.4. Note that Proposition 4.2.3 concerns the values of $\mathfrak{Q}_{\unicode[STIX]{x1D6FE}}$ and $\mathfrak{B}_{\unicode[STIX]{x1D6FE}}$ only on $M$ , not on all of $G$ . The restriction comes mainly from the fact that our reasoning requires at least that each $(f_{2})_{i}+i$ is concave, and that is true ‘for free’ only if $i$ is non-negative.