Base Change Fundamental Lemma FOR CENTRAL ELEMENTS IN DEPTH-ZERO HECKE ALGEBRAS OVER LOCAL FUNCTION FIELDS

Let $F$ denote a local function field of characteristic $p>0$, and let $F_{r}\supset F$ denote the unique degree $r$ unramified extension of $F$ contained in a fixed separable closure $F^{s}$ of $F$, and let $\theta$ denote a generator of $\mathrm{Gal}(F_{r}/F)$. Let $G$ denote an unramified connected reductive group over $F$. The automorphism $\theta$ determines an $F$-automorphism of $G(F_{r})$, which is still denoted by $\theta$. For simplicity, we will assume that $G_{\text{der}}=G_{\text{sc}}$ in the remainder of the introduction.

From the concrete norm map

$$\begin{split}N:G(F_{r})&\to G(F_{r})\\ g&\mapsto g\theta(g)\dots\theta^{r-1}(g),\end{split}$$

Kottwitz [30] was able to define an abstract norm map $\mathcal{N}$ from the set of stable $\theta$-conjugacy classes in $G(F_{r})$ to the set of stable conjugacy classes in $G(F)$ for perfect fields $F$. We will say that $\gamma\in G(F)$ is a norm if $\gamma$ is stably conjugate to $N\delta$ for some $\delta\in G(F_{r})$, which means that there exist $g\in G(F^{s})$ such that $g^{-1}(N\delta)g=\gamma$.

Fix a semisimple element $\gamma\in G(F)$ and we let $G_{\gamma}\subset G$ denote its centralizer in $G$. For $f\in C^{\infty}_{c}(G(F))$, the algebra of locally constant and compactly supported $\mathbb{C}$-valued functions on $G(F)$, we can define the orbital integral

$$O^{G(F)}_{\gamma}(f)=\int_{G_{\gamma}(F)\backslash G(F)}f(g^{-1}\gamma g)\,dg/dt$$

depending on the choice of Haar measures $dg$ and $dt$ on $G(F)$ and $G_{\gamma}(F)$ respectively. We also consider the stable orbital integral

$$SO^{G(F)}_{\gamma}(f)=\sum_{\gamma^{\prime}}e(G_{\gamma^{\prime}})O^{G(F)}_{\gamma^{\prime}}(f),$$

where $\gamma^{\prime}$ sums over the conjugacy classes in $G(F)$ within the stable conjugacy class of $\gamma$, and $e(G_{\gamma^{\prime}})\in\{\pm 1\}$ is the sign attached by Kottwitz [31] to connected reductive groups. Notice that our assumption that $G_{\mathrm{der}}=G_{\mathrm{sc}}$ implies that the centralizers $G_{\gamma^{\prime}}$ are connected. If $\gamma$ and $\gamma^{\prime}$ are stably conjugate, their centralizers are inner forms of each form, therefore we may require the Haar measures are compatible with each other in the definition of the orbital integrals $TO^{G(F)}_{\gamma^{\prime}}(f)$ in the sense of [36] p. 631. Similarly, for an element $\delta\in G(F_{r})$ such that $N\delta$ is semisimple, we define its twisted centralizer to be the connected reductive group $G_{\delta\theta}$ such that

$$G_{\delta\theta}(F)=\{g\in G(F_{r}):g^{-1}\delta\theta(g)=\delta\}.$$

Then for $\phi\in C^{\infty}_{c}(G(F_{r}))$, similarly we can define the twisted orbital integral

$$TO^{G(F_{r})}_{\delta\theta}(\phi)=\int_{G_{\delta\theta}(F)\backslash G(F_{r})}\phi(g^{-1}\delta\theta(g))\,dg/dt$$

and its stable version

$$SO^{G(F_{r})}_{\delta\theta}(\phi)=\sum_{\delta^{\prime}}e(G_{\delta^{\prime}\theta})TO^{G(F_{r})}_{\delta^{\prime}\theta}(\phi),$$

where the sum is over $\theta$-conjugacy classes $\delta^{\prime}$ in $G(F_{r})$ inside the stable $\theta$-conjugacy class of $\delta$, or equivalently (Proposition 2.16), whose norm down to $G(F)$ is in the same stable conjugacy class as $N\delta$. If $\gamma\in G(F)$ is stably conjugate to $N\delta$, then $G_{\delta\theta}$ is an inner form of $G_{\gamma}$ (Lemma 2.5). Therefore we may also require the Haar measures on these groups to be compatible.

We say that $\phi\in C^{\infty}_{c}(G(F_{r}))$ and $f\in C^{\infty}_{c}(G(F))$ are associated or have matching orbital integrals (resp., at regular semisimple elements) if for every (resp., regular) semisimple element $\gamma\in G(F)$ we have

(1.1.1)

$$SO^{G(F)}_{\gamma}(f)=\left\{\begin{array}[]{rcl}SO^{G(F_{r})}_{\delta\theta}(\phi)&\mbox{if}&\gamma=\mathcal{N}\delta\\ 0&\mbox{if}&\gamma\mbox{ is not a norm}\end{array}\right.$$

The case of spherical Hecke algebras was studied first. Suppose $K_{r}\subset G(F_{r})$ and $K\subset G(F)$ are hyperspecial maximal compact subgroups associated to a hyperspecial vertex in the Bruhat-Tits building $\mathcal{B}(G(F))$ of $G(F)$, and suppose $\phi\in C^{\infty}_{c}(G(F_{r}))$ belongs to the spherical Hecke algebra $\mathcal{H}(G(F_{r}),K_{r})$ of bi-invariant functions under $K_{r}$. In particular, the spherical Hecke algebras are commutative. The Satake isomorphisms gives rise to a natural algebra homomorphism

(1.1.2)

$$b_{r}:\mathcal{H}(G(F_{r}),K_{r})\to\mathcal{H}(G(F),K)$$

which will be called the base change homomorphism. The base change fundamental lemma for spherical functions asserts that $\phi$ and $b_{r}(\phi)$ are associated. This was proved in the special cases of $\mathrm{GL}_{2}$ [45] and $\mathrm{GL}_{n}$ [1], which gave rise to global and local applications, and for general unramified reductive groups it was proved by Clozel [10] and Labesse [42].

In [18], an analogous base change fundamental lemma is proved for centers of parahoric Hecke algebras by Haines, which replaced the hyperspecial maximal compact subgroup $K_{r}$ (resp., $K$) above by a general parahoric subgroup $J_{r}$ (resp., $J$) defined as intersection of the group of elements in $G(F_{r})$ (resp. $G(F)$) that fix a facet in $\mathcal{B}(G(F))$ with the kernel of Kottwitz homomorphism(see section 2 of loc. cit.). In particular, when the facet is a hyperspecial vertex, we have recovered $J_{r}=K_{r}$. However, since the resulting parahoric Hecke algebras $\mathcal{H}(G(F_{r}),J_{r})$ and $\mathcal{H}(G(F),J)$ are in general no longer commutative, the base change homomorphism will be restricted to the center of those Hecke algebras:

$$b_{r}:\mathcal{Z}(G(F_{r}),K_{r})\to\mathcal{Z}(G(F),K)$$

Later in [19], Haines generalizes this result to so called depth-zero principal series blocks in the Bernstein decomposition. Generalizing his method, Shenghao Li[46] is able to show the base change fundamental lemma for Bernstein centers of principal series blocks (not just depth-zero ones).

All the results [10, 42, 18, 19, 46] stated above only work over groups over $p$-adic fields $F$ of $\text{char}\,F=0$. Results in positive characteristic are more ad-hoc for specific groups. In his thesis [49], Ray-Dulany proved the base change fundamental lemma for Iwahori-Hecke algebra of $\mathrm{GL}_{2}$ for any local field with characteristic not equal to $2$ by explicit calculations. In [14], Feng gave a geometric proof of the base change fundamental lemma for regular semisimple elements for parahoric Hecke algebras of $\mathrm{GL}_{n}$ over local function fields using cohomology of certain moduli stacks of shtukas and the Langlands-Kottwitz method, which generalized the previous results by [48] for spherical Hecke algebras for $\mathrm{GL}_{n}$ over local function fields. Most recently, Bartling and Ito[4] prove base change fundamental lemma for central elements in the parahoric Hecke algebras for general unramified reductive groups over local function fields, using the technique of close local fields.

In this article, we prove the base change fundamental lemma for Bernstein centers of depth-zero principal series blocks at regular semisimple elements in unramified groups $G$ over local function fields, following and generalizing the strategies in [18, 19]. To state the theorem, let us fix some more notations. Denote the ring of integers of $F$ by $\mathcal{O}_{F}$. Let $A$ denote a maximal $F$-split torus of $G$ and set $T=Z_{G}(A)$, a maximal torus of $G$ which is defined and unramified over $F$. We can choose an Iwahori subgroup $I\subset G(F)$ which is in good position relative to $T$. Let $I^{+}$ denote the pro-unipotent radical of $I$. Furthermore, let $T(F)_{1}$ denote the maximal compact open subgroup of $T(F)$, and let $T(F)^{+}_{1}=T(F)_{1}\cap I^{+}$ denotes its pro-unipotent radical. We will denote $\chi$ to be a character on $T(F)_{1}/T(F)^{+}_{1}$. Such characters are called depth-zero characters. Via the canonical isomorphism

$$T(F)_{1}/T(F)^{+}_{1}\xrightarrow{\sim}I/I^{+}$$

we see that $\chi$ induces a character $\rho:=\rho_{\chi}$ on $I$, trivial on $I^{+}$. Then we can consider the Hecke algebra $\mathcal{H}(G,\rho)$ of our study:

$$\mathcal{H}(G,\rho):=\{f\in C^{\infty}_{c}(G):f(igi^{\prime})=\rho^{-1}(i)f(g)\rho^{-1}(i^{\prime}),\forall\,i,i^{\prime}\in I,\,\forall g\in G\},$$

where the convolution is defined using the Haar measure which gives $I$ volume $1$. Write $\mathcal{Z}(G,\rho)$ for $Z(\mathcal{H}(G,\rho))$.

Let $N_{r}:T(F_{r})_{1}\to T(F)_{1}$ denote the naive norm homomorphism. Therefore, the character $\chi_{r}:=\chi\circ N_{r}$ is a depth-zero character on $T(F_{r})_{1}$, and it gives rise to the character $\rho_{r}$ on $I_{r}$ and the Hecke algebra $\mathcal{H}(G_{r},\rho_{r})$ with center $\mathcal{Z}(G_{r},\rho_{r})$. Here $G_{r}:=G(F_{r})$ and $I_{r}\subset G_{r}$ is the Iwahori subgroup corresponding to $I$.

We will define a base change homomorphism

$$b_{r}:\mathcal{Z}(G_{r},\rho_{r})\to\mathcal{Z}(G,\rho)$$

Finally we can state the main theorem of this article:

We write $\mathcal{H}(G(F_{r}),I^{+}_{r})$ (resp., $\mathcal{H}(G(F),I^{+})$)for the Hecke algebra of locally constant compactly supported functions on $G(F_{r})$ (resp., $G(F)$) that are bi-invariant under the open compact subgroups $I^{+}_{r}$ (resp., $I^{+}$), the pro-unipotent radical of $I_{r}$ (resp., $I$), and we write $\mathcal{Z}(G(F_{r}),I^{+}_{r})$ (resp., $\mathcal{Z}(G(F),I^{+})$) for the center of these Hecke algebras. In [19], §10, a base change homomorphism

$$b_{r}:\mathcal{Z}(G(F_{r}),I^{+}_{r})\to\mathcal{Z}(G(F),I^{+})$$

was constructed and characterized using the injection

$$\mathcal{Z}(G(F_{r}),I^{+}_{r})\hookrightarrow\prod_{\chi^{\prime}}\mathcal{Z}(G(F_{r}),I_{r},\chi^{\prime}),$$

where $\chi^{\prime}$ ranges over the finite set of all depth-zero characters on $T(F_{r})_{1}$. As a result, we are able to show a base change fundamental lemma for pro-$p$ Iwahori subgroups from Theorem 1.2:

This article overlaps with [14] in the case when $G=\mathrm{GL}_{n}$. However, we are able to prove the vanishing results when $\gamma$ is not a norm, which was not done in loc. cit. Moreover, this article overlaps with [4] for central elements of Iwahori-Hecke algebras, where the results are proved for regular semisimple elements in loc. cit..

The analogs of Theorem 1.2 and Corollary 1.3 for groups over characteristic $0$ fields already could be useful in the pseudo-stabilization of the Lefschetz trace formula for certain simple Shimura varieties (see for example, [18], p.4 and [26], §13). For local function fields, similar counting points formula has been estabilished in [52], Theorem 5.9.2. We will explain how Corollary 1.3 could be used to stabilize the corresponding Lefschetz trace formula on certain moduli space of shtukas.

Let $X$ be a smooth projective geometrically irreducible curve over $\mathbb{F}_{q}$, let $F=\mathbb{F}_{q}(X)$ be the function field of $X$. For a closed point $x\in X$, we write $F_{x}$ the completion of $F$ at $x$ and $\mathcal{O}_{x}$ the corresponding valuation ring. Let $\mathbb{A}=\mathbb{A}_{F}$ be the group of adeles of the global field $F$. Let $G$ be a connected reductive group over $F$ and let $\mathcal{G}$ be a parahoric model of $G$ over $X$. We also assume that $G_{x}$ is unramified. Fix two distinct closed points $x$ and $\infty$ in $X$. Let $K\subset G(\mathbb{A})$ be a compact open subgroup of the form $K=K_{x}K^{x}\mathcal{G}(\mathcal{O}_{\infty})$, where $K^{x}$ is a sufficiently small open compact subgroup of $G(\mathbb{A}^{\infty,x})$ and $K_{x}\subset\mathcal{G}(\mathcal{O}_{x})$ is the pro-$p$ unipotent radical of some Iwahori subgroup $I_{x}\subset\mathcal{G}(\mathcal{O}_{x})$.

Consider a proper moduli space of shtukas $\mathrm{Sht}:=\mathrm{Sht}^{\underline{\mu}}_{\mathcal{G},\Xi,K}$ with two legs and with one leg fixed, over the local ring $\mathrm{Spec}\,\mathcal{O}_{y,\infty^{\prime}}$ (defined in §4.7 in [52]). We assume that the space satisfies some boundedness condition as in loc. cit.. Let $f=f_{x}\otimes f^{x}$ where $f_{x}\in C^{\infty}_{c}(\mathcal{G}(\mathcal{O}_{x}))$ and $f^{x}\in C^{\infty}_{c}(G(\mathbb{A}^{x}))$ where the $\infty$-component of $f^{x}$ is given by the characteristic function of $\mathcal{G}(\mathcal{O}_{\infty})$. The elliptic semisimple part of the alternating sum of the traces of the Hecke operator $f$ composed with a power $\sigma^{r}$ of Frobenius at $x$ is given by

(1.2.1)

$$\begin{split}\mathrm{tr}&(\sigma^{r}\times f\,|\,H^{*}(\mathrm{Sht}\otimes\overline{F}_{y,\infty^{\prime}},\bar{\mathbb{Q}}_{\ell}))^{\mathrm{ell,reg}}\\ &=\sum_{(\gamma_{0};\gamma,\delta)}|\ker^{1}(F,G_{\gamma_{0}})|\mathrm{vol}(G_{\gamma_{0}}(F)\backslash G_{\gamma_{0}}(\mathbb{A})/\Xi)O^{G(\mathbb{A}^{x,\infty})}_{\gamma}(f^{x,\infty})TO^{G(F_{x^{r}})}_{\delta_{x},\sigma}(\phi_{r,x})TO^{G(F_{\infty^{r}})}_{\delta_{\infty},\sigma}(\phi_{r,\infty})\end{split}$$

where the sum runs over the so called elliptic Kottwitz triples $(\gamma_{0};\gamma,\delta)\in G(F)\times G(\mathbb{A}^{x,\infty})\times G(F_{x^{r}}\times F_{\infty^{r}})$ attached to a fixed point with some vanishing properties. Here $F_{x^{r}}$ is the degree-$r$ unramified extension of $F_{x}$.

In particular, it is conjectured that we can find the test function $\phi^{\prime}_{r,x}$ (resp. $\phi^{\prime}_{r,\infty}$) inside the center $\mathcal{Z}(G(F_{x^{r}}),K_{x^{r}})$ (resp. $\mathcal{Z}(G(F_{\infty^{r}}),K_{\infty^{r}})$) of the Hecke algebra $\mathcal{H}(G(F_{x^{r}}),K_{x^{r}})$ (resp. $\mathcal{H}(G(F_{\infty^{r}}),K_{\infty^{r}})$) with the same stable twisted orbital integral as $\phi_{r,x}$ (resp. $\phi_{r,\infty}$). We may simplify the right hand side to be

(1.2.2)

$$\tau(G)\sum_{\gamma_{0}}SO^{G(\mathbb{A}^{x,\infty})}_{\gamma}(f^{x,\infty})SO^{G(F_{x^{r}})}_{\delta_{x},\sigma}(\phi^{\prime}_{r,x})SO^{G(F_{\infty^{r}})}_{\delta_{\infty},\sigma}(\phi^{\prime}_{r,\infty})$$

At the place $x$, by Corollary 1.3, we can write

(1.2.3)

$$SO^{G(F_{x^{r}})}_{\delta_{x},\sigma}(\phi^{\prime}_{r,x})=SO^{G(F_{x})}_{\gamma_{x}}(b_{r,x}(\phi^{\prime}_{r,x}))$$

where $b_{r,x}$ is the base change homomorphism $b_{r,x}:\mathcal{Z}(G(F_{x^{r}}),K_{x^{r}})\to\mathcal{Z}(G(F_{x}),K_{x})$. Similar base change identity holds for the place $\infty$ from the results in [34] since $K_{\infty}$ is hyperspecial maximal compact open subgroup.

With the extra assumptions that $\mathrm{Sht}$ has no global endoscopy and that the matching (1.2.3) can be extended to all semisimple elements, we may eventually write (1.2.2) as

(1.2.4)

$$\tau(G)\sum_{\gamma_{0}}SO^{G(\mathbb{A})}_{\gamma}(f^{x,\infty}b_{r,x}(\phi^{\prime}_{r,x})b_{r,\infty}(\phi^{\prime}_{r,\infty})),$$

which resembles the geometric side of the (stabilized) global trace formula for the regular action $R$ of $f^{\prime}:=f^{x,\infty}b_{r,x}(\phi^{\prime}_{r,x})b_{r,\infty}(\phi^{\prime}_{r,\infty})$ on the space $L^{2}(G(F)A_{G}\backslash G(\mathbb{A}))$:

(1.2.5)

$$\mathrm{tr}\,R(f)=\sum_{\pi}m(\pi)\,\mathrm{tr}\,\pi(f^{\prime}),$$

where $\pi$ ranges over irreducible representations of $L^{2}(G(F)A_{G}\backslash G(\mathbb{A}))$. Therefore, we can relate the local factor of the zeta function with automorphic representations, eventually the $L$-function. (See [26], Cor. 1.4 for more details)

In §2, we will define the abstract norm homomorphism for all semisimple elements in a quasi-split reductive group over $F$. This generalizes the classical results of Kottwitz [30] to non-perfect fields, using the results from [24]. The abstract norm homomorphism relates the stable twisted conjugacy classes of $G(E)$ with stable conjugacy classes of $G(F)$, providing the ”transfer of conjugacy classes” needed in (1.1.1). In §3, we briefly recall the necessary background for the base change fundamental lemma, including the definitions of Iwahori subgroups, Bernstein center for depth-zero principal series blocks and base change homomorphisms.

The core technical content starts from §4. In order to prove Theorem 1.2, we will reduce the problem to the case that the group has simply connected derived group and the element is strongly regular semisimple both in the adjoint group and the group itself (see 4.10). In order to turn the identities between integrals into identities between traces of certain representations, we have to prove a version of very simple trace formulas (§5), then stabilize them (§6) in order to compare them on different groups. Finally, by producing the local data adapted to our situation (§7.3, 7.4), we are able to reduce to the cases considered in [19]. Using the same technique of Labesse elementary functions, we are finally able to prove the main theorem 1.2.

I would like to thank my advisor Thomas Haines for suggesting this topic and his continuous encouragement and interest. I would like to thank Shenghao Li for spotting a mistake in the original reduction steps. I also thank Shin Eui Song for informing me the results of Hamacher-Kim. I would like to thank Peter Dillery and Kazuhiro Ito for helpful discussions. This research was partially supported by NSF grants DMS 2200873.

We follow [30], §5. We assume $G_{\rm der}=G_{\rm sc}$ for the moment, later we will remove this assumption by using the $z$-extension of $G$.

Let $I:=R_{E/F}G_{E}$, there is a natural isomorphism $I_{E}\xrightarrow{\sim}G_{E}\times\dots\times G_{E}$, with the factors indexed by $\mathrm{Gal}(E/F)$. The element $\theta\in\mathrm{Gal}(E/F)$ determines an automorphism $s\in\text{Aut}_{F}(I)$, which takes the form

$$s:(x_{1},\dots,x_{r})\mapsto(x_{r},x_{1},\dots,x_{r-1})$$

on $E$-valued points. We can identify $G$ with $I^{s}$ by embedding $G$ into $I$ diagonally. The composition

$$G(E)=I(F)\to I(E)=G(E)\times\dots\times G(E)$$

is given by

$$x\mapsto(x^{\theta^{r-1}},\dots,x^{\theta},x).$$

We can also define the concrete norm map on $I$ by $Nx=x^{s^{r-1}}\dots x^{s}\cdot x$. It is defined over $F$ and it is the same on $G(E)=I(F)$ as the norm map $N:G(E)\to G(E)$ we defined before. We say that $x,y\in I(F^{s})$ are $F^{s}$-$\theta$-conjugate if there exists an element $g\in I(F^{s})$ such that $x=g^{-s}yg$. We say that $x,y\in G(E)$ are $F^{s}$-$\theta$-conjugate if they are $F^{s}$-$\theta$-conjugate as elements $G(E)=I(F)\subset I(F^{s})$.

We get an immediate corollary from (b) and (c) of Lemma 2.1 above.

To define stable $\theta$-conjugacy, we have to define $\theta$-centralizers.

It is an $F$-subgroup of $I$. Since we have $I_{E}=G_{E}\times\dots\times G_{E}$, where the factors are indexed by elements of $\mathrm{Gal}(E/F)$: $\theta^{r-1},\dots,\theta,1$, let $p:I_{E}\to G_{E}$ be the projection onto the last factor: $(x_{1},\dots,x_{r})\mapsto x_{r}$.

We define the notion of $\sigma$-(strongly regular) semisimplicity below.

We can show that twisted orbital integral converges for $\sigma$-regular semisimple elements. Let $\phi\in\mathcal{H}(G,\rho)$ be a function in the depth-zero Hecke algebra, where as before $\rho$ is a character on $I$, trivial on $I^{+}$. Since $\rho$ is a character on a finite abelian group, it must be unitary. Therefore, $|\phi|\in\mathcal{H}(G,I)$, the Hecke algebra of compactly supported bi-invariant functions under $I$. Using the results of Proposition 5.2 in [4], we see that the twisted orbital integrals of $\phi$ converge absolutely for $\sigma$-regular semisimple elements.

Let us assume $G$ is a general quasi-split reductive group defined over $F$, we need to define yet another subgroup $I^{\circ}_{sx}$. We define $I^{\circ}_{sx}$ as the inverse image under the $E$-isomorphism $p:I_{sx}\to G_{Nx}$ of the subgroup $G^{\circ}_{Nx}$ of $G_{Nx}$. We will see that $I^{\circ}_{sx}$ is defined over $F$ in the following lemma. It is clear that $I^{\circ}_{sx}=I_{sx}$ when $G_{\text{der}}$ is simply connected. In general, we consider a $z$-extension $\alpha:G^{\prime}\to G$, whose definitions are given below. We will also get a corresponding $z$-extension $\gamma:I^{\prime}\to I$ where $I^{\prime}=R_{E/F}((G^{\prime})_{E})$.

The existence of $z$-extensions is proved in [13] Prop 3.1.

Now since $x,y\in G(E)=I(F)$, we have $(g^{-s}xg)^{\tau}=g^{-\tau s}xg^{\tau}=g^{-s}xg$, hence $g^{\tau}g^{-1}\in I_{sx}$ for all $\tau\in\Gamma$. Therefore stable $\theta$-conjugacy and $F^{s}$-$\theta$-conjugacy coincide whenever $I^{\circ}_{sx}=I_{sx}$, in particular whenever $G_{\text{der}}$ is simply connected.

We introduce a special kind of $z$-extension $\alpha:G^{\prime}\to G$.

To show that for any $\delta\in G(E)$, $N_{r}(\delta)$ is stably conjugate to an element in $G(F)$, therefore defines a stable conjugacy class in $G(F)$, we will have to use the classical results from [33] and recent progress from [24]. Denote $\breve{F}$ to be the maximal unramified extension of $F$, in particular we have $E\subset\breve{F}$.

Now we use Lemma 8.1 from [24].

Now we consider an element $\delta\in G(E)$ and whose concrete norm $N_{r}(\delta)$ is semisimple. Since $\delta\,\theta(N_{r}\delta)=(N_{r}\delta)\delta$, by the above proposition, with $G^{*}=G$, $H^{*}=M$ and $H=J$. We see that $N_{r}\delta\in H(F)$, and the stable conjugacy class of $N_{r}\delta$ contains an element of $G(F)$. Combined with Corollary 2.2, this finishes the definition of the abstract norm map 2.0.1 in the case when $G_{\text{der}}$ is simply connected.

For the next step, we need to follow the argument after Lemma 5.1 in [30] to define the abstract norm map for all quasi-split reductive groups. To be specific, we find a $z$-extension $\alpha:G^{\prime}\to G$ adapted to $E$. Then we can define the abstract norm map on $G^{\prime}(E)$ by the previous results, and we have the following commutative diagram

and we claim that there exists a unique homomorphism, still denoted by $\mathcal{N}_{r}$, mapping stable-$\theta$-conjugate classes in $G(E)$ to stable conjugate classes in $G(F)$. Indeed, uniqueness of the map comes from the fact that $\alpha:G^{\prime}(E)\to G(E)$ is surjective. For existence, we have to show that if $x,y\in G^{\prime}(E)$ and $\alpha(x)=\alpha(y)$, then $\alpha(\mathcal{N}_{r}(x))=\alpha(\mathcal{N}_{r}(y))$ as stable classes. We can find $z\in\ker(\alpha)(E)$ such that $x=yz$. Since $\ker(\alpha)$ is central, we have $N_{r}(x)=N_{r}(y)N_{r}(z)$ with $N_{r}(z)\in\ker(\alpha)(F)$. Hence $\alpha(N_{r}(x))=\alpha(N_{r}(y))$ as desired.

Next, we show that the definition of $\mathcal{N}_{r}$ does not depend on the choice of the $z$-extension. As in the [30], this reduces to show that the following diagram is commutative

for $G_{1},G_{2}$ two $z$-extensions of $G$. But this comes from the definition of $\mathcal{N}_{r}$ when the derived group is simply connected.