2.1 Harmonic analysis

In this section we collect some standard results in harmonic analysis. We learned essentially all of this material from [Reference VarmaVar24]. Our notational conventions on Haar measures, Levis, Weyl groups, spaces of characters, etc. mostly follow Arthur’s paper [Reference ArthurArt96] and also coincide with the conventions in [Reference VarmaVar24]. In particular, we fix a minimal E-rational Levi $M_{0}\subset G$ , with absolute Weyl group ${W_{0}=N_{G}(M_{0})(E)/M_{0}(E)}$ , and then write $\mathcal {L}$ the set of E-rational Levi subgroups $M\subset G$ which contain $M_{0}$ equipped with its natural $W_{0}$ -action, $W(M)=N_{G}(M)(E)/M(E)$ for any $M\in \mathcal {L}$ , etc. We also fix once and for all a Haar measure $dm$ on $M(E)$ for all $M\in \mathcal {L}$ . We write $G(E)_{\mathrm {ell}}$ for the set of strongly regular elliptic elements. All representations and function spaces are defined over the complex numbers.

Inside $C_{c}(G(E))$ , we have the subspace $C_{c}(G(E))^{\mathrm {null}}$ of null functions f characterized by four equivalent conditions (see [Reference KazhdanKaz86, Theorem 0], and see also [Reference DatDat00, Section 3.1] for a nice discussion):

• • $\mathrm {tr}(f|\pi )=0$ for all irreducible representations $\pi $ .

• • $\mathrm {tr}(f|\pi )=0$ for all tempered irreducible representations $\pi $ .

• • All regular semisimple orbital integrals of f are zero.

• • f is in the subspace of commutators, i.e. the linear span of functions of the form $h(x)-h(gxg^{-1})$ .

The equivalence of these conditions is sometimes called the Kazhdan density theorem. We write $\mathcal {I}(G)=C_{c}(G(E))/C_{c}(G(E))^{\mathrm {null}}$ . Note that the action of $\mathfrak {Z}(G)$ on $C_{c}(G(E))$ preserves null functions: if $f\in C_{c}(G(E))^{\mathrm {null}}$ and $z\in \mathfrak {Z}(G)$ , then $\mathrm {tr}(z\ast f|\pi )=z_{\pi }\mathrm {tr}(f|\pi )=0$ for all irreducible $\pi $ (see e.g. [Reference HainesHai14, Corollary 3.2.1]), so $z\ast f$ is null. Therefore the action descends to an action of $\mathfrak {Z}(G)$ on $\mathcal {I}(G)$ which we also denote as $z\ast f$ .

For any Levi M, there is a canonical map $\mathfrak {Z}(G)\to \mathfrak {Z}(M)$ , denoted $z\mapsto r_{M}(z)$ or just $z\mapsto r(z)$ if M is clear from context, defined as in [Reference Bernstein, Deligne and KazhdanBDK86, Section 2.4]. There is also a canonical constant term map

$$ \begin{align*} \mathcal{I}(G) & \to\mathcal{I}(M)\\ f & \mapsto f_{M}=\delta_{P}^{1/2}(m)\int_{U(E)}\int_{K}f(kmuk^{-1})dkdu \end{align*} $$

which strictly speaking is defined on $C_{c}(G(E))$ , but it descends to the quotient $\mathcal {I}(G)$ . Here $P=MU$ is any parabolic with Levi M, $K\subset G(E)$ is any open compact with $dk$ the normalized Haar measure, and $du$ is determined by our choices of Haar measures on $G(E)$ and $M(E)$ . This map is characterized by the formula $\mathrm {tr}(f_{M}|\pi )=\mathrm {tr}(f|i_{M}^{G}\pi )$ for irreducible $\pi $ . It is easy to see from this formula that $(z\ast f)_{M}=r_{M}(z)\ast f_{M}$ .

We write $\mathcal {I}(G)^{\mathrm {cusp}}$ for the subspace of cuspidal functions f characterized by the vanishing of $f_{M}$ for all proper Levis M, or equivalently by the vanishing of all orbital integrals at nonelliptic regular semisimple elements. There is then a canonical decomposition

(1) $$ \begin{align} \mathcal{I}(G)=\bigoplus_{M\in\mathcal{L}/W_{0}}(\mathcal{I}(M)^{\mathrm{cusp}})^{W(M)} \end{align} $$

due to Arthur [Reference ArthurArt96, Section 4] (we learned this result from [Reference VarmaVar24, Remark 4.2.7. (i)]).

Dually, let $\mathrm {Dist}(G)$ be the linear dual of $\mathcal {I}(G)$ , so this is the space of all invariant distributions on G. Let $D(G)\subset \mathrm {Dist}(G)$ be the subspace of virtual characters, and let $D^{\mathrm {temp}}(G)\subset D(G)$ be the span of characters of tempered irreducible representations. The Bernstein center acts on $\mathrm {Dist}(G)$ and preserves $D(G)$ and $D^{\mathrm {temp}}(G)$ . We write this action as $z\cdot \Theta $ . This action is defined in terms of the canonical $\mathfrak {Z}(G)$ -action on $\mathcal {I}(G)$ by the tautological formula $(z\cdot \Theta )(f)=\Theta (z\ast f)$ . Of course, if $\Theta =\Theta _{\pi }$ is the character of an irreducible representation, then $z\cdot \Theta _{\pi }=z_{\pi }\Theta _{\pi }$ .

Inside $D^{\mathrm {temp}}(G)$ , we have the still smaller subspace $D^{\mathrm {ell}}(G)$ defined as the linear span of Arthur’s elliptic tempered virtual characters $\Theta (\tau )$ , $\tau \in T_{\mathrm {ell}}(G)$ (see [Reference ArthurArt93, p. 76] for the notation). Then there is a canonical decomposition

(2) $$ \begin{align} D^{\mathrm{temp}}(G)=\bigoplus_{M\in\mathcal{L}/W_{0}}(D^{\mathrm{ell}}(M))^{W(M)} \end{align} $$

where the inclusion of the M-indexed summand on the right-hand side is induced by the parabolic induction map $i_{M}^{G}:D(M)\to D(G)$ (see [Reference glin and WaldspurgerMgW18, Proposition 2.12] for a proof). In particular, any $\Theta \in D^{\mathrm {temp}}(G)$ admits a unique decomposition $\Theta =\Theta ^{\mathrm {ell}}+\Theta ^{\mathrm {ind}}$ where $\Theta ^{\mathrm {ell}}\in D^{\mathrm {ell}}(G)$ and $\Theta ^{\mathrm {ind}}$ is in the span of the M-indexed summands of (2) for proper Levis M. We will freely and crucially use the fact that the pointwise evaluation map

$$ \begin{align*} D^{\mathrm{ell}}(G) & \to C(G(E)_{\mathrm{ell}})\\ \Theta & \mapsto\Theta|_{G(E)_{\mathrm{ell}}} \end{align*} $$

is injective: for this, simply note that the kernel of the map

$$ \begin{align*} D^{\mathrm{temp}}(G) & \to C(G(E)_{\mathrm{ell}})\\ \Theta & \mapsto\Theta|_{G(E)_{\mathrm{ell}}} \end{align*} $$

consists exactly of elements $\Theta $ such that $\Theta =\Theta ^{\mathrm {ind}}$ by [Reference Hansen, Kaletha and WeinsteinHKW22, Theorem C.1.1].

These decompositions (1) and (2) are perfectly dual to each other (see [Reference VarmaVar24, Remark 4.2.7.(ii)-(iii)] for a nice discussion of this). In particular, any $f\in \mathcal {I}(G)$ admits a unique decomposition $f=f^{\mathrm {cusp}}+f^{\mathrm {nc}}$ such that $f^{\mathrm {cusp}}$ is cuspidal and $\Theta (f^{\mathrm {nc}})=0$ for all $\Theta \in D^{\mathrm {ell}}(G)$ . Note that for any $\Theta \in D^{\mathrm {temp}}(G)$ , $\Theta (f)=\Theta ^{\mathrm {ell}}(f^{\mathrm {cusp}})+\Theta ^{\mathrm {ind}}(f^{\mathrm {nc}})$ .

All of the spaces of virtual characters defined above have stable analogues, denoted $SD$ , $SD^{\mathrm {temp}}$ , $SD^{\mathrm {ell}}$ . More precisely, for $?\in \{\emptyset ,\mathrm {temp},\mathrm {ell}\}$ , $SD^{?}(G)\subset D^{?}(G)$ is the subspace of virtual characters $\Theta $ such that $\Theta (g)=\Theta (g')$ for all strongly regular semisimple elements $g,g'\in G(E)$ which are conjugate in $G(\overline {E})$ . The decomposition (2) of $D^{\mathrm {temp}}(G)$ above admits a compatible stable analogue

(3) $$ \begin{align} SD^{\mathrm{temp}}(G)=\bigoplus_{M\in\mathcal{L}/W_{0}}(SD^{\mathrm{ell}}(M))^{W(M)} \end{align} $$

as in [Reference VarmaVar24, Proposition 3.2.8]. The elliptic inner product determines a canonical projection $D^{\mathrm {ell}}(G)\to SD^{\mathrm {ell}}(G)$ splitting the obvious inclusion, as in [Reference VarmaVar24, Lemma 3.4.5] and the discussion preceding it, and the direct sum of these projections over $M\in \mathcal {L}/W_{0}$ yields an analogous projection $D^{\mathrm {temp}}(G)\to SD^{\mathrm {temp}}(G)$ .

Recall that a function $f\in \mathcal {I}(G)$ is unstable if all its stable orbital integrals vanish. It is enough to impose this vanishing at strongly regular semisimple elements. We will need the result of Arthur that an element $f\in \mathcal {I}(G)$ is unstable iff $\Theta (f)=0$ for all $\Theta \in SD^{\mathrm {temp}}(G)$ . For quasisplit groups this is explicitly proved in [Reference ArthurArt96], and for general groups it is [Reference VarmaVar24, Proposition 3.2.10]. This can be reformulated as follows.

As in the introduction, we call elements of the Bernstein center satisfying these equivalent conditions very stable.

By Kazhdan’s density theorem, it is easy to see that $D^{\mathrm {temp}}(G)$ is a faithful $\mathfrak {Z}(G)$ -module. Indeed, if $z\in \mathfrak {Z}(G)$ kills all $\Theta \in D^{\mathrm {temp}}(G)$ , then in particular $\Theta _{\pi }(z\ast f)=(z\cdot \Theta _{\pi })(f)=0$ for all f and all tempered irreducible $\pi $ , so then $z\ast f$ is a null function by Kazhdan density, and thus $z(f)=(z\ast f)(1)=0$ for all f, so $z=0$ . This result has a straightforward stable analogue.

2.2 Excursion versus Hecke

The key extra symmetry of elements $z\in \mathfrak {Z}^{\mathrm {FS}}(G)$ which will enforce their stability is their commutation with certain endomorphisms of $D(G)$ coming from Hecke operators on $\mathrm {Bun}_{G}$ . To explain this, let $\mathcal {D}(\mathrm {Bun}_{G})=\mathcal {D}_{\mathrm {lis}}(\mathrm {Bun}_{G},\overline {\mathbf {Q}_{\ell }})$ be the category of sheaves defined in [Reference Fargues and ScholzeFS24, Section VII.7]. There is an open immersion $i_{1}:[\ast /\underline {G(E)}]\to \mathrm {Bun}_{G}$ , which induces natural functors $i_{1!}:\mathcal {D}(G(E),\overline {\mathbf {Q}_{\ell }})\to \mathcal {D}(\mathrm {Bun}_{G})$ and $i_{1}^{\ast }:\mathcal {D}(\mathrm {Bun}_{G})\to \mathcal {D}(G(E),\overline {\mathbf {Q}_{\ell }})$ . Moreover, for any ireducible $\hat {G}$ -representation V, there is a natural Hecke operator $T_{V}:\mathcal {D}(\mathrm {Bun}_{G})\to \mathcal {D}(\mathrm {Bun}_{G})$ defined as in [Reference Fargues and ScholzeFS24, Section IX.2].

Now fix a conjugacy class of cocharacters $\mu :\mathbf {G}_{m,\overline {E}}\to G_{\overline {E}}$ such that $1\in B(G,\mu )$ , or equivalently such that $V_{\mu }|_{Z(\hat {G})^{\Gamma }}$ is trivial, where $V_{\mu }$ is the irreducible representation of $\hat {G}$ with highest weight  $\mu $ . Then $i_{1}^{\ast }T_{V_{\mu }^{\ast }}i_{1!}$ defines an endofunctor of $\mathcal {D}(G(E),\overline {\mathbf {Q}_{\ell }})$ , which by [Reference Hansen, Kaletha and WeinsteinHKW22, Proposition 6.4.5] preserves the finite length objects and hence induces an endomorphism on the Grothendieck group $K_{0}\mathrm {Rep}_{\mathrm {fl}}(G(E),\overline {\mathbf {Q}_{\ell }})$ . Transporting this endomorphism across our fixed isomorphism $\iota $ and $\mathbf {C}$ -linearizing, we get an endomorphism $\mathscr {T}_{\mu }:D(G)\to D(G)$ .Footnote ²

This commutation of $z\cdot $ and $\mathscr {T}_{\mu }$ is the crucial extra symmetry we will exploit.

Proof. Fix any irreducible representation $\pi $ , so then $z\cdot \pi =z_{\pi }\pi $ for some $z_{\pi }\in \mathbf {C}$ . Write $\mathscr {T}_{\mu }(\Theta _{\pi })=\sum n_{i}\Theta _{\pi _{i}}$ . Since excursion operators are built from Hecke operators, and Hecke operators commute with each other, excursion operators commute with Hecke operators, and in particular $z\cdot \pi _{i}=z_{\pi }\pi _{i}$ for all i. Therefore

$$ \begin{align*} z\cdot\mathscr{T}_{\mu}(\Theta_{\pi}) & =\sum n_{i}z\cdot\Theta_{\pi_{i}}\\ & =z_{\pi}\sum n_{i}\Theta_{\pi_{i}}\\ & =z_{\pi}\mathscr{T}_{\mu}(\Theta_{\pi})\\ & =\mathscr{T}_{\mu}(z\cdot\Theta_{\pi}), \end{align*} $$

so $\mathscr {T}_{\mu }(z\cdot -)-z\cdot \mathscr {T}_{\mu }(-)$ annihilates $\Theta _{\pi }$ for every irreducible $\pi $ . Since the $\Theta _{\pi }$ ’s form a basis of $D(G)$ , this gives the result.

We will also need some very nonformal facts about the operator $\mathscr {T}_{\mu }$ . These all follow from the main results of [Reference Hansen, Kaletha and WeinsteinHKW22], which give an explicit formula for the restriction of $\mathscr {T}_{\mu }(\Theta )$ to $G(E)_{\mathrm {ell}}$ for any $\Theta $ . We now recall this formula. Fix any $g\in G(E)_{\mathrm {ell}}$ with centralizer $T_{g}$ , and let $[[g]]$ denote the set of conjugacy classes in the stable conjugacy class of g. For any element $g'\in [[g]]$ , we defined ([Reference Hansen, Kaletha and WeinsteinHKW22, Definition 3.2.2]) a certain invariant $\mathrm {inv}(g,g')\in B(T_{g})=X_{\ast }(T_{g})_{\Gamma }$ . This invariant has the property that for each $\lambda \in X_{\ast }(T_{g})$ such that $\mathrm {dim}V_{\mu }[\lambda ]\neq 0$ , there is exactly one element $g'\in [[g]]$ such that $\overline {\lambda }=\mathrm {inv}(g,g')$ in $X_{\ast }(T_{g})_{\Gamma }$ , where $\overline {\lambda }$ is the natural projection of $\lambda $ along $X_{\ast }(T_{g})\to X_{\ast }(T_{g})_{\Gamma }$ . Indeed, the uniqueness of $g'$ is [Reference Hansen, Kaletha and WeinsteinHKW22, Remark 3.2.5], and existence follows from [Reference FuFu24, Lemma 4.2.7].

In this notation, the character formula proved in [Reference Hansen, Kaletha and WeinsteinHKW22, Theorem 6.5.2] says that

$$\begin{align*}\mathscr{T}_{\mu}(\Theta)(g)=\sum_{\substack{\lambda\in X_{\ast}(T_{g}),\,g'\in[[g]]\\ \mathrm{inv}(g,g')=\overline{\lambda} } }\mathrm{dim}V_{\mu}[\lambda]\cdot\Theta(g') \end{align*}$$

for any $\Theta \in D(G)$ . Note that the sign $(-1)^{d}$ appearing in [Reference Hansen, Kaletha and WeinsteinHKW22, Definition 3.2.7] disappears in the present situation, since this sign agrees with $e(G)e(G_{b})$ by [Reference Hansen, Kaletha and WeinsteinHKW22, Equation 3.3.3] and we are in a scenario where $G_{b}=G$ .

We record a few consequences of this result.

Proof. In general, $\Theta \in D(G)$ is parabolically induced if and only if $\Theta |_{G(E)_{\mathrm {ell}}}$ vanishes identically (see [Reference Hansen, Kaletha and WeinsteinHKW22, Theorem C.1.1]). Part i. then follows from the formula for the character of $\mathscr {T}_{\mu }(\Theta )$ at elliptic elements recalled above, since if $\Theta |_{G(E)_{\mathrm {ell}}}=0$ then visibly $\mathscr {T}_{\mu }(\Theta )|_{G(E)_{\mathrm {ell}}}=0$ . Part ii. follows similarly, since if $\Theta $ is stable then

$$ \begin{align*} \mathscr{T}_{\mu}(\Theta)(g) & =\sum_{\substack{\lambda\in X_{\ast}(T_{g}),\,g'\in[[g]]\\ \mathrm{inv}(g,g')=\overline{\lambda} } }\mathrm{dim}V_{\mu}[\lambda]\cdot\Theta(g')\\ & =\sum_{\substack{\lambda\in X_{\ast}(T_{g})} }\mathrm{dim}V_{\mu}[\lambda]\cdot\Theta(g)\\ & =\dim V_{\mu}\cdot\Theta(g) \end{align*} $$

for all $g\in G(E)_{\mathrm {ell}}$ , so $\mathscr {T}_{\mu }(\Theta )-\mathrm {dim}V_{\mu }\cdot \Theta $ vanishes identically on $G(E)_{\mathrm {ell}}$ .

We will also adapt a marvelous idea from Chenji Fu’s paper [Reference FuFu24], showing that as $\mu \to \infty $ in an appropriate sense, $\mathscr {T}_{\mu }$ implements a stable averaging. We will need a version of this result with some uniformity in g. To explain this, set $\mu _{m}=4m\rho _{G}$ for $m\geq 1$ , where $2\rho _{G}$ is the usual sum of positive roots. Now for any conjugation-invariant function $\phi \in C(G(E)_{\mathrm {ell}})^{G(E)}$ , define its stable average by the formula $\phi _{\mathrm {st}}(g)=\frac {1}{|[[g]]|}\sum _{g'\in [[g]]}\phi (g')$ for all $g\in G(E)_{\mathrm {ell}}$ .

Lemma 2.6. Fix any $\Theta \in D(G)$ , and fix $U\subset G(E)$ a compact subset whose elliptic part is stably invariant in the weak sense that for all $g\in G(E)_{\mathrm {ell}}$ , either $[[g]]\cap U=0$ or U meets every conjugacy class in $[[g]]$ . Then

$$\begin{align*}|D(g)|^{1/2}|\frac{1}{\mathrm{dim}V_{\mu_{m}}}\mathscr{T}_{\mu_{m}}(\Theta)(g)-\Theta_{\mathrm{st}}(g)| \end{align*}$$

tends to zero uniformly in $g\in G(E)_{\mathrm {ell}}\cap U$ as $m\to \infty $ .

Here $|D(g)|$ is the usual Weyl discriminant.

Proof. For any given $g\in G(E)_{\mathrm {ell}}$ , we can rearrange the character formula as

$$\begin{align*}\mathscr{T}_{\mu}(\Theta)(g)=\sum_{g'\in[[g]]}\Theta(g')\sum_{\substack{\lambda\in X_{\ast}(T_{g})\\ \mathrm{inv}(g,g')=\overline{\lambda} } }\mathrm{dim}V_{\mu}[\lambda]. \end{align*}$$

Defining the rational number

$$\begin{align*}C_{m}(g,g')=\frac{1}{\mathrm{dim}V_{\mu_{m}}}\sum_{\substack{\lambda\in X_{\ast}(T_{g})\\ \mathrm{inv}(g,g')=\overline{\lambda} } }\mathrm{dim}V_{\mu_{m}}[\lambda], \end{align*}$$

we can trivially rearrange this further to get

$$\begin{align*}\frac{1}{\mathrm{dim}V_{\mu_{m}}}\mathscr{T}_{\mu_{m}}(\Theta)(g)=\sum_{g'\in[[g]]}\Theta(g')C_{m}(g,g'). \end{align*}$$

On the other hand, Fu’s analysis (specifically, the proof of [Reference FuFu24, Theorem 4.3.1]) shows that for sufficiently large m, we have $|C_{m}(g,g')-\frac {1}{|[[g]]|}|\leq \frac {C}{m}$ for some fixed constant C which depends only on the ambient group G. Then we get that $\frac {1}{\mathrm {dim}V_{\mu _{m}}}\mathscr {T}_{\mu _{m}}(\Theta )(g)\to \Theta _{\mathrm {st}}(g)$ pointwise on $G(E)_{\mathrm {ell}}$ , and in fact that

$$\begin{align*}|\frac{1}{\mathrm{dim}V_{\mu_{m}}}\mathscr{T}_{\mu_{m}}(\Theta)(g)-\Theta_{\mathrm{st}}(g)|\leq\frac{C}{m}\mathrm{sup}_{x\in[[g]]}|\Theta(x)| \end{align*}$$

for all $g\in G(E)_{\mathrm {ell}}$ where C depends only on G. Since the Weyl discriminant is invariant under stable conjugacy, we can insert it into the above estimate, giving

$$\begin{align*}|D(g)|^{1/2}|\frac{1}{\mathrm{dim}V_{\mu_{m}}}\mathscr{T}_{\mu_{m}}(\Theta)(g)-\Theta_{\mathrm{st}}(g)|\leq\frac{C}{m}\mathrm{sup}_{x\in[[g]]}|D(x)|^{1/2}|\Theta(x)|. \end{align*}$$

Now by a deep theorem of Harish-Chandra, $|D(x)|^{1/2}|\Theta (x)|$ (extended by zero from $G(E)_{\mathrm {reg.ss}}$ to $G(E)$ ) is bounded on any compact subset of $G(E)$ . (See e.g. [Reference ClozelClo87, Theorem 1] for a proof of a more general result.) Therefore, taking a further supremum over all $g\in G(E)_{\mathrm {ell}}\cap U$ , the left side of the previous equation is bounded by $\frac {C}{m}\mathrm {sup}_{x\in U}|D(x)|^{1/2}|\Theta (x)|\leq \frac {C'}{m}$ as $m\to \infty $ . This gives the claim.

We will also need a slight generalization of some of these results, where the operator $\mathscr {T}_{\mu }$ now moves between inner forms. For this, pick any conjugacy class of cocharacters $\mu :\mathbf {G}_{m,\overline {E}}\to G_{\overline {E}}$ , and let $b\in B(G,\mu )$ be the unique basic element, with $G_{b}$ the associated extended pure inner form of G. Then by the same recipe as above, the functors $i_{1}^{\ast }T_{V_{\mu }^{\ast }}i_{b!}$ and $i_{b}^{\ast }T_{V_{\mu }}i_{1!}$ preserve finite length objects and induce linear maps $\mathscr {T}_{\mu }:D(G_{b})\to D(G)$ and $\mathscr {T}_{\mu }^{\ast }:D(G)\to D(G_{b})$ .

Proposition 2.7. i. If $f\in \mathfrak {Z}^{\mathrm {spec}}(G)$ is any element, with images $z=\Psi _{G}(f)$ and $z'=\Psi _{G_{b}}(f)$ , then $\mathscr {T}_{\mu }(z'\cdot \Theta )=z\cdot \mathscr {T}_{\mu }(\Theta )$ for all $\Theta \in D(G_{b})$ , and $\mathscr {T}_{\mu }^{\ast }(z\cdot \Theta )=z'\cdot \mathscr {T}_{\mu }^{\ast }(\Theta )$ for all $\Theta \in D(G)$ .

ii. If $G(E)_{\mathrm {ell}}\ni g\sim ^{\mathrm {st}}g'\in G_{b}(E)_{\mathrm {ell}}$ is any stably conjugate pair of elliptic elements and $\Theta \in SD(G_{b})$ is stable, then

$$\begin{align*}\mathscr{T}_{\mu}(\Theta)(g)=e(G)e(G_{b})(\mathrm{dim}V_{\mu})\Theta(g'). \end{align*}$$

Likewise, if $\Theta \in SD(G)$ is stable, then

$$\begin{align*}\mathscr{T}_{\mu}^{\ast}(\Theta)(g')=e(G)e(G_{b})(\dim V_{\mu})\Theta(g). \end{align*}$$

Proof. Part i. again follows from the commutation of Hecke operators with excursion operators. For part ii., we again need the character formula, but now in its most general form. Recall that for a fixed basic b, an elliptic element $g\in G(E)$ , and an elliptic element $g'\in G_{b}(E)$ which is stably conjugate to g under the canonical identification $G(\overline {E})=G_{b}(\overline {E})$ , [Reference Hansen, Kaletha and WeinsteinHKW22, Definition 3.2.2] defines an invariant $\mathrm {inv}[b](g,g')\in B(T_{g})=X_{\ast }(T_{g})_{\Gamma }$ . Let $[[g]]_{b}$ denote the set of conjugacy classes in $G_{b}(E)$ which are stably conjugate to g. In this notation, the character formula reads

$$\begin{align*}\mathscr{T}_{\mu}(\Theta)(g)=e(G)e(G_{b})\sum_{\substack{\lambda\in X_{\ast}(T_{g}),\,g'\in[[g]]_{b}\\ \mathrm{inv}[b](g,g')=\overline{\lambda} } }\mathrm{dim}V_{\mu}[\lambda]\cdot\Theta(g'). \end{align*}$$

This again follows from [Reference Hansen, Kaletha and WeinsteinHKW22, Theorem 6.5.2], using [Reference Hansen, Kaletha and WeinsteinHKW22, Equation (3.3.3)] for the signs. Moreover, if $\lambda $ is such that $\mathrm {dim}V_{\mu }[\lambda ]\neq 0$ , there is a unique $g'$ such that $\mathrm {inv}[b](g,g')=\overline {\lambda }$ , which again follows from [Reference Hansen, Kaletha and WeinsteinHKW22, Remark 3.2.5] and [Reference FuFu24, Lemma 4.2.7]. Thus if $\Theta $ is stable, the sum appearing in the character formula collapses to $(\dim V_{\mu })\Theta (g')$ , as claimed. This gives the first claim in part ii., and the second claim follows by an identical argument with the roles of G and $G_{b}$ swapped.

2.3 Stability

In this section we prove Theorem 1.1.

Let $z\in \mathfrak {Z}^{\mathrm {FS}}(G)$ be in the image of the Fargues-Scholze map $\Psi _{G}$ . We need to prove that for any unstable $f\in \mathcal {I}(G)$ , $z\ast f$ is unstable. For this, it is enough to see that $\Theta (z\ast f)=0$ for all $\Theta \in SD^{\mathrm {temp}}(G)$ as recalled in Lemma 2.2 and the discussion preceding it. We will prove this by induction on the semisimple rank of G.

First suppose $\Theta $ is parabolically induced. By the decomposition (3) recalled in Section 2.1, without loss of generality we can assume $\Theta =i_{M}^{G}\Theta _{M}$ for some $\Theta _{M}\in SD^{\mathrm {temp}}(M)$ and some proper Levi M. Then

$$ \begin{align*} \Theta(z\ast f) & =(z\cdot i_{M}^{G}\Theta_{M})(f).\\ & =i_{M}^{G}(r_{M}(z)\cdot\Theta_{M})(f) \end{align*} $$

where in the second line we used [Reference Bernstein, Deligne and KazhdanBDK86, Proposition 2.4]. Now $r_{M}(z)$ is in the image of $\Psi _{M}$ by compatibility of the Fargues-Scholze map with parabolic induction, so by induction on the semisimple rank and Lemma 2.2 we know that $r_{M}(z)\cdot \Theta _{M}$ is stable. Then also $i_{M}^{G}(r_{M}(z)\cdot \Theta _{M})$ is stable, so its evaluation on the unstable function f vanishes.

This reduces us to the case where $\Theta \in SD^{\mathrm {ell}}(G)$ . By Lemma 2.1, we can assume our unstable function f is cuspidal, in which case also $z\ast f$ is cuspidal. Now, with $\mu $ as in Lemma 2.4, consider the quantity

$$\begin{align*}C_{\mu}:=\frac{1}{\mathrm{dim}V_{\mu}}\mathscr{T}_{\mu}(\Theta)(z\ast f). \end{align*}$$

By Proposition 2.5, $\mathscr {T}_{\mu }(\Theta )=\mathrm {dim}V_{\mu }\cdot \Theta +\Theta '$ for some parabolically induced $\Theta '$ . Since $z\ast f$ is cuspidal, $\Theta '(z\ast f)=0,$ so this simplifies to $C_{\mu }=\Theta (z\ast f)$ which is evidently a constant independent of $\mu $ . Our goal is to show that this constant vanishes. Writing $\Xi =z\cdot \Theta $ , Lemma 2.4 shows that

$$ \begin{align*} C_{\mu} & =\frac{1}{\mathrm{dim}V_{\mu}}(z\cdot\mathscr{T}_{\mu}(\Theta))(f)\\ & =\frac{1}{\mathrm{dim}V_{\mu}}\mathscr{T}_{\mu}(z\cdot\Theta)(f)\\ & =\frac{1}{\mathrm{dim}V_{\mu}}\mathscr{T}_{\mu}(\Xi)(f) \end{align*} $$

for any $\mu $ . Note that although $\Theta $ is stable, $\Xi $ certainly need not be stable a priori.Footnote ³

At this point we use Fu’s method. More precisely, taking $\mu =\mu _{m}$ with $m\to \infty $ as in Section 2.2, we will use that the operator $\frac {1}{\mathrm {dim}V_{\mu }}\mathscr {T}_{\mu }(\Xi )$ effects a stable averaging as discussed there. To implement this, for any $\Theta \in D^{\mathrm {temp}}(G)$ , let $\Theta ^{\mathrm {st}}\in SD^{\mathrm {temp}}(G)$ be its stable projection. Writing $\frac {1}{\mathrm {dim}V_{\mu _{m}}}\mathscr {T}_{\mu _{m}}(\Xi )=\Xi ^{\mathrm {st}}+\Phi _{\mu _{m}}$ , it is clear that $\Xi ^{\mathrm {st}}(f)=0$ since f is unstable, so $C_{\mu _{m}}=\Phi _{\mu _{m}}(f)$ . We will now show that as $m\to \infty $ , $\Phi _{\mu _{m}}(f)\to 0$ .

To proceed further, we exploit the cuspidality of f to rewrite $\Phi _{\mu _{m}}(f)$ via a simple form of the Weyl integration formula. More precisely, fix a Haar measure $da$ on the split center $A_{G}(E)$ , and set $O_{\gamma }(f)=\int _{A_{G}(E)\backslash G(E)}f(x^{-1}\gamma x)dx$ as a function on $G(E)_{\mathrm {ell}}$ , where $dx=dg/da$ in the usual manner. Then for any $\Theta \in D(G)$ and any $f\in \mathcal {I}(G)^{\mathrm {cusp}}$ , the Weyl integration formula can be written as

$$\begin{align*}\Theta(f)=\sum_{T}\frac{1}{|W(G,T)(E)|}\int_{T(E)}\Theta(t)O_{t}(f)|D(t)|dt. \end{align*}$$

Here the sum runs over a (finite) set of representatives for the $G(E)$ -conjugacy classes of elliptic maximal tori in G, and $dt$ is the Haar measure on $T(E)$ determined by the chosen Haar measure on $A_{G}(E)$ and the normalized Haar measure on the compact group $T(E)/A_{G}(E)$ . We briefly recall some facts about convergence. For each T, the set of elements $t\in T(E)\cap G(E)_{\mathrm {ell}}$ such that $O_{t}(f)\neq 0$ has compact closure $C_{T}$ in $T(E)$ . Now, by fundamental results of Harish-Chandra, the function $|D(g)|^{1/2}\Theta (g)$ (extended by zero from the regular semisimple locus) is locally bounded on $G(E)$ (as recalled in Section 2.2), and $|D(\gamma )|^{1/2}O_{\gamma }(f)$ is a bounded function on $G(E)_{\mathrm {ell}}$ (see e.g. [Reference ArthurArt91, Section 4]). In particular, for a fixed cuspidal f and varying $\Theta $ , we can replace each integral above by an integral over the fixed compact subset $C_{T}\subset T(E)$ , and the integrand is a bounded function on that compact subset and is locally constant on a dense open subset thereof.

Now substituting in $\Phi _{\mu _{m}}$ for $\Theta $ in the Weyl integration formula above, we are reduced to showing that $|D(t)|^{1/2}\Phi _{\mu _{m}}(t)\to 0$ uniformly on $C_{T}$ as $m\to \infty $ . Here again, $|D(t)|^{1/2}\Phi _{\mu _{m}}(t)$ is defined a priori as a bounded function on $C_{T}\cap G(E)_{\mathrm {ell}}$ and extended by zero to $C_{T}$ . Recall that by definition, $\Phi _{\mu _{m}}=\frac {1}{\mathrm {dim}V_{\mu _{m}}}\mathscr {T}_{\mu _{m}}(\Xi )-\Xi ^{\mathrm {st}}$ . First we compute the restriction of $\Xi ^{\mathrm {st}}$ to $G(E)_{\mathrm {ell}}$ . This follows from some general theory: by [Reference VarmaVar24, Lemma 3.4.5], $\Theta ^{\mathrm {st}}|_{G(E)_{\mathrm {ell}}}=(\Theta |_{G(E)_{\mathrm {ell}}})_{\mathrm {st}}$ for any $\Theta \in D(G)$ , where $f\mapsto f_{\mathrm {st}}$ is the naive stable averaging discussed immediately before Lemma 2.6. In particular, this applies to $\Xi $ , so we get

$$\begin{align*}\Phi_{\mu_{m}}(g)=\frac{1}{\mathrm{dim}V_{\mu_{m}}}\mathscr{T}_{\mu_{m}}(\Xi)(g)-\Xi_{\mathrm{st}}(g) \end{align*}$$

for any $g\in G(E)_{\mathrm {ell}}$ . Now choose a compact subset $U\subset G(E)$ whose elliptic part is stably invariant and which moreover contains $C_{T}$ for each T. Then by Lemma 2.6, $|D(g)|^{1/2}\Phi _{\mu _{m}}(g)\to 0$ uniformly as $m\to \infty $ for all $g\in U\cap G(E)_{\mathrm {ell}}$ , and in particular for all $t\in C_{T}\cap G(E)_{\mathrm {ell}}$ . But then this immediately extends to the same statement for all $t\in C_{T}$ since $|D(g)|^{1/2}\Phi _{\mu _{m}}(g)$ is extended by zero from the regular semisimple locus.

Putting all of this together, we get that

$$\begin{align*}\Phi_{\mu_{m}}(f)=\sum_{T}\frac{1}{|W(G,T)(E)|}\int_{C_{T}}|D(t)|^{1/2}\Phi_{\mu_{m}}(t)\cdot|D(t)|^{1/2}O_{t}(f)dt \end{align*}$$

where $C_{T}\subset T(E)$ is compact, both halves of the integrand are bounded on $C_{T}$ , and $|D(t)|^{1/2}\Phi _{\mu _{m}}(t)\to 0$ uniformly on $C_{T}$ for each T as $m\to \infty $ . Therefore $\Phi _{\mu _{m}}(f)\to 0$ as $m\to \infty $ . This gives the result.

2.4 Inner forms

In this section we deal with Theorem 1.4. To construct the map $\tau _{G}$ , we proceed by induction on the semisimple rank. More precisely, fix some element $f\in \mathfrak {Z}^{\mathrm {spec}}(G^{\ast })$ such that $z:=\Psi _{G}(f)\neq 0$ . We need to show that $z^{\ast }:=\Psi _{G^{\ast }}(f)\neq 0$ .

By Lemma 2.3, $SD^{\mathrm {temp}}(G)$ is a faithful $\mathfrak {Z}^{\mathrm {vst}}(G)$ -module, thus a faithful $\mathfrak {Z}^{\mathrm {FS}}(G)$ -module by Theorem 1.1. In particular, the endomorphism $z\cdot $ of $SD^{\mathrm {temp}}(G)$ is not identically zero. Suppose first that $z\cdot \Theta \neq 0$ for some parabolically induced $\Theta =i_{M}^{G}\Theta _{M}$ with $\Theta _{M}\in SD^{\mathrm {temp}}(M)$ . Let $M^{\ast }\subset G^{\ast }$ be the Levi subgroup corresponding to M. We have a commutative diagram

where $\tau _{M}$ exists and is surjective by induction on the semisimple rank. Now by [Reference Bernstein, Deligne and KazhdanBDK86, Proposition 2.4], the linear map $z\cdot i_{M}^{G}(-)$ coincides with the linear map $i_{M}^{G}(r(z)\cdot -)$ . In particular, since $z\cdot i_{M}^{G}\Theta _{M}\neq 0$ , we get that $r(z)\neq 0$ . But then

$$ \begin{align*} r(z) & =\Psi_{M}(r^{\mathrm{spec}}(f))\\ & =\tau_{M}(\Psi_{M^{\ast}}(r^{\mathrm{spec}}(f)))\\ & =\tau_{M}(r^{\ast}(\Psi_{G^{\ast}}(f)))\\ & =\tau_{M}(r^{\ast}(z^{\ast})) \end{align*} $$

using the commutativity of the diagram, so $z^{\ast }\neq 0$ .

It remains to deal with the case where z annihilates all parabolically induced elements of $SD^{\mathrm {temp}}(G)$ . By Lemma 2.3, we may choose some $\Theta \in SD^{\mathrm {ell}}(G)$ such that $z\cdot \Theta \neq 0$ . Pick some $\mu $ such that $b\in B(G^{\ast },\mu )$ , and let $\mathscr {T}_{\mu }:D(G)\to D(G^{\ast })$ be the linear map induced by $i_{1}^{\ast }T_{V_{\mu }^{\ast }}i_{b!}$ (and our choice of isomorphism $\mathbf {C}\simeq \overline {\mathbf {Q}_{\ell }}$ ) as in the discussion preceding Proposition 2.7. By the commutation of Hecke operators with excursion operators, we get that $z^{\ast }\cdot \mathscr {T}_{\mu }(\Theta )=\mathscr {T}_{\mu }(z\cdot \Theta )$ , as in Proposition 2.7.i. By assumption, $z\cdot \Theta \in SD^{\mathrm {ell}}(G)$ is nonzero, so it is not identically zero on $G(E)_{\mathrm {ell}}$ . Now the character formula, Proposition 2.7.ii, shows that for all matching stably conjugate pairs $G^{\ast }(E)_{\mathrm {ell}}\ni g^{\ast }\sim ^{\mathrm {st}}g\in G(E)_{\mathrm {ell}}$ , we have an equality

$$\begin{align*}\mathscr{T}_{\mu}(z\cdot\Theta)(g^{\ast})=e(G)\mathrm{dim}V_{\mu}(z\cdot\Theta)(g), \end{align*}$$

so $\mathscr {T}_{\mu }(z\cdot \Theta )(g^{\ast })$ is nonzero for some $g^{\ast }\in G^{\ast }(E)_{\mathrm {ell}}$ . Therefore $\mathscr {T}_{\mu }(z\cdot \Theta )=z^{\ast }\cdot \mathscr {T}_{\mu }(\Theta )\neq 0$ , so $z^{\ast }\neq 0$ as desired.

Next, for the compatibility with parabolic induction, note that we already have a diagram

where everything commutes except possibly the trapezoid spanned by $r,r^{\ast },\tau _{G},\tau _{M}$ . But the surjectivity of $\Psi _{G^{\ast }}$ immediately implies that this trapezoid commutes as well.

It remains to show compatibility with the transfer map. We first recall some properties of this map, referring to [Reference VarmaVar24, Section 3.2] for details.Footnote ⁴ Fixing an inner twist and all other data as in [Reference VarmaVar24, Section 3.2], we get a canonical injection $\mathcal {L}/W_{0}\to \mathcal {L}^{\ast }/W_{0}^{\ast }$ . The transfer map $\mathrm {Trans}_{G}:SD^{\mathrm {temp}}(G^{\ast })\to SD^{\mathrm {temp}}(G)$ is then compatible with the grading

$$\begin{align*}SD^{\mathrm{temp}}(G^{\ast})=\bigoplus_{M^{\ast}\in\mathcal{L}^{\ast}/W_{0}^{\ast}}(SD^{\mathrm{ell}}(M^{\ast}))^{W(M^{\ast})} \end{align*}$$

and its analogue for G in the following very strong sense.

• • Its restriction to the summand $SD^{\mathrm {ell}}(G^{\ast })$ factors over an isomorphism $\mathrm {Trans}_{G}^{\mathrm {ell}}:SD^{\mathrm {ell}}(G^{\ast })\overset {\sim }{\to }SD^{\mathrm {ell}}(G)$ characterized by the equality $\mathrm {Trans}_{G}^{\mathrm {ell}}(\Theta )(g)=e(G)\Theta (g^{\ast })$ for all matching stably conjugate pairs $G^{\ast }(E)_{\mathrm {ell}}\ni g^{\ast }\sim ^{\mathrm {st}}g\in G(E)_{\mathrm {ell}}$ .

• • If $M^{\ast }\in \mathcal {L}^{\ast }/W_{0}^{\ast }$ is irrelevant in the sense that it is not the image of some $M\in \mathcal {L}/W_{0}$ , $\mathrm {Trans}_{G}$ is identically zero on the $M^{\ast }$ -indexed summand.

• • If $M^{\ast }$ is the image of some M, then $\mathrm {Trans}_{G}i_{M^{\ast }}^{G^{\ast }}\Theta _{M^{\ast }}=i_{M}^{G}\mathrm {Trans}_{M}^{\mathrm {ell}}\Theta _{M^{\ast }}$ for all $\Theta _{M^{\ast }}\in (SD^{\mathrm {ell}}(M^{\ast }))^{W(M^{\ast })}$ , compatibly with the Weyl equivariance via the appropriate identification $W(M)=W(M^{\ast })$ .

Now let $z\in \mathfrak {Z}^{\mathrm {FS}}(G^{\ast })$ be any element, and pick any $\Theta \in SD^{\mathrm {temp}}(G^{\ast })$ . We need to show that $\mathrm {Trans}_{G}(z\cdot \Theta )=\tau _{G}(z)\cdot \mathrm {Trans}_{G}(\Theta )$ . First suppose $\Theta $ is parabolically induced, say of the form $i_{M^{\ast }}^{G^{\ast }}\Xi $ for some $\Xi \in SD^{\mathrm {ell}}(M^{\ast })$ . Then by the remarks on the grading above and compatibility of the Bernstein center action with parabolic induction, we compute that

$$\begin{align*}\mathrm{Trans}_{G}(z\cdot i_{M^{\ast}}^{G^{\ast}}\Xi)=\mathrm{Trans}_{G}(i_{M^{\ast}}^{G^{\ast}}(r^{\ast}(z)\cdot\Xi)). \end{align*}$$

If $M^{\ast }$ is irrelevant, this is identically zero, as is $\mathrm {Trans}_{G}(\Theta )$ , so there is nothing to prove. If $M^{\ast }$ is relevant, we compute further that

$$ \begin{align*} \mathrm{Trans}_{G}(i_{M^{\ast}}^{G^{\ast}}(r^{\ast}(z)\cdot\Xi)) & =i_{M}^{G}\mathrm{Trans}_{M}^{\mathrm{ell}}(r^{\ast}(z)\cdot\Xi)\\ & =i_{M}^{G}\tau_{M}(r^{\ast}(z))\cdot\mathrm{Trans}_{M}^{\mathrm{ell}}\Xi\\ & =i_{M}^{G}r(\tau_{G}(z))\cdot\mathrm{Trans}_{M}^{\mathrm{ell}}\Xi\\ & =\tau_{G}(z)\cdot i_{M}^{G}\mathrm{Trans}_{M}^{\mathrm{ell}}\Xi\\ & =\tau_{G}(z)\cdot\mathrm{Trans}_{G}(\Theta), \end{align*} $$

where the second equality follows by induction on the semisimple rank.

This reduces us to the case that $\Theta \in SD^{\mathrm {ell}}(G^{\ast })$ . Here we just need to see that $(\tau (z)\cdot \mathrm {Trans}_{G}^{\mathrm {ell}}(\Theta ))(g)=\mathrm {Trans}_{G}^{\mathrm {ell}}(z\cdot \Theta )(g)$ for all $g\in G(E)_{\mathrm {ell}}$ . Now we exploit [Reference Hansen, Kaletha and WeinsteinHKW22] one more time. More precisely, defining $\mathscr {T}_{\mu }^{\ast }:D(G^{\ast })\to D(G)$ as the map induced by $i_{b}^{\ast }T_{V_{\mu }}i_{i!}$ (with b and $\mu $ as above), the character formula as in Proposition 2.7.ii shows that

$$\begin{align*}\mathscr{T}_{\mu}^{\ast}(\Theta)(g)=\mathrm{dim}V_{\mu}\cdot\mathrm{Trans}_{G}^{\mathrm{ell}}(\Theta)(g) \end{align*}$$

for all $g\in G(E)_{\mathrm {ell}}$ . But again, Hecke operators commute with excursion operators, so we know that $\mathscr {T}_{\mu }^{\ast }(z\cdot \Theta )=\tau _{G}(z)\cdot \mathscr {T}_{\mu }^{\ast }(\Theta )$ . Evaluating both sides of this equality on any $g\in G(E)_{\mathrm {ell}}$ and invoking the character formula, we get the result.

Finally, suppose $f^{\ast }$ and f are any matching functions. Fix any $z\in \mathfrak {Z}^{\mathrm {FS}}(G^{\ast })$ , and let $(\tau _{G}(z)\ast f)^{\ast }$ be a function on $G^{\ast }(E)$ matching $\tau _{G}(z)\ast f$ . We need to see that $h:=z\ast f^{\ast }-(\tau _{G}(z)\ast f)^{\ast }$ is unstable. For this, pick any $\Theta \in SD^{\mathrm {temp}}(G^{\ast })$ . We then compute

$$ \begin{align*} \Theta(z\ast f^{\ast}) & =(z\cdot\Theta)(f^{\ast})\\ & =\mathrm{Trans}_{G}(z\cdot\Theta)(f)\\ & =(\tau_{G}(z)\cdot\mathrm{Trans}_{G}\Theta)(f)\\ & =\mathrm{Trans}_{G}(\Theta)(\tau_{G}(z)\ast f)\\ & =\Theta\left((\tau_{G}(z)\ast f)^{\ast}\right) \end{align*} $$

where the first and fourth equalities are trivial, the second and fifth equalities follow from the definition of the transfer map, and the third equality follows from our results so far. Therefore $\Theta (h)=0$ , and since $\Theta \in SD^{\mathrm {temp}}(G^{\ast })$ is arbitrary, this implies that h is unstable, as desired.