## Abstract

In this paper we construct a “restriction” map from the cocenter of a reductive group *G* over a local non-archimedean field *F* to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on the parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig–Spaltenstein on induced unipotent classes to all infinite fields. We also prove a group version of a theorem of Harish-Chandra about the density of the span of regular semisimple orbital integrals.

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## Notes

For every closed subgroup \(\mathbf {L}\subset \mathbf {G}\) normalized by \(\mathbf {T}\), we denote by \(\Phi (\mathbf {L},\mathbf {T})\subset \Phi (\mathbf {G},\mathbf {T})\) the set of non-zero weights in \({\text {Lie}}\,L\subset {\text {Lie}}\,G\).

## References

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**47**, 175–179 (1986)Kempf, G.: Instability in invariant theory. Ann. Math.

**108**, 299–316 (1978)Lusztig, G., Spaltenstein, N.: Induced unipotent classes. J. Lond. Math. Soc.

**19**(2), 41–52 (1979)Lusztig, G.: Character sheaves on disconnected groups, II. Represent. Theory

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*Adeles and Algebraic Groups*. In: Progress in Math, vol. 23. Birkhäuser, Boston, MA (1982)

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

*
To Iosif Bernstein with gratitude and best wishes on his birthday
*

David Kazhdan was supported by the ERC grant No. 247049-GLC, Yakov Varshavsky was supported by the ISF grant 1017/13.

## Appendices

### Appendix A: A generalization of a theorem of Lusztig–Spaltenstein

**A.1. Notation.** Let *F* be an infinite field. All algebraic varieties and all morphisms of algebraic varieties are over *F*.

(a) Let \(\mathbf {G}\) be a connected reductive group, \(\mathbf {P}\subset \mathbf {G}\) a parabolic subgroup, \({\mathbf {U}}\subset \mathbf {P}\) the unipotent radical, and \(\mathbf {M}\subset \mathbf {P}\) a Levi subgroup.

(b) For an algebraic variety \(\mathbf {X}\), we denote the set \(\mathbf {X}(F)\) by *X*. In particular, we have \(G=\mathbf {G}(F)\), \(\widetilde{G}_P=\widetilde{\mathbf {G}}_{\mathbf {P}}(F)\), etc. (compare 3.8).

(c) For an \({\text {Ad}}\, P\)-invariant subset \(D\subset P\), we set \({\text {Ind}}_P^G(D):=G\overset{P,{\text {Ad}}}{\times }D\subset \widetilde{G}_P\).

(d) For an \({\text {Ad}}\, M\)-invariant subset \(C\subset M\), we set \({\text {Ind}}_M^G(C):=G\overset{M,{\text {Ad}}}{\times }C\subset \widetilde{G}_M\), \(C_P:=U\cdot C\subset P\), \({\text {Ind}}_P^G(C_P)\subset \widetilde{G}_P\) and \(C_{P;G}:=a_{P,G}({\text {Ind}}_P^G(C_P))\subset G\), where \(a_{P,G}:\widetilde{G}_P\rightarrow G\) was defined in 1.4.

From now on, we assume that \(C\subset M\) is a unipotent *M*-conjugacy class.

**A.2. Question.** Does the set \(C_{P;G}\) depend on the choice of \(\mathbf {P}\supset \mathbf {M}\)?

**A.3. Remarks.** (a) \(C_{P;G}\) is a union of unipotent conjugacy classes in *G*.

(b)Let *F* be algebraically closed. By a theorem of Chevalley, \(C_{P;G}\subset G\) is a constructible set, whose Zariski closure \(\overline{C}_{P;G}\) is irreducible. This case was considered by Lusztig and Spaltenstein in [6], and they showed that \(\overline{C}_{P;G}\) does not depend on \(\mathbf {P}\), using representation theory. A simpler proof of this fact was given later by Lusztig [7, Lem 10.3(a)].

The goal of this appendix is to generalize the result of [6] to other fields.

**A.4. Saturation.** Let \(\mathbf {X}\) be an algebraic variety over *F*, and let \(A\subset X\) be a subset.

(a) We denote by \({\text {sat}}'(A)={\text {sat}}'_\mathbf {X}(A)\subset X\) the union \(\cup _{(\mathbf {V},x,f)}f(x)\), taken over triples \((\mathbf {V},x,f)\), where \(\mathbf {V}\subset \mathbb A^1\) is an open subvariety, \(x\in V\), \(\mathbf {V}':=\mathbf {V}\smallsetminus \{x\}\), and \(f:\mathbf {V}\rightarrow \mathbf {X}\) is a morphism such that \(f(V')\subset A\).

(b) We say that a subset \(A\subset X\) is *saturated*, if \({\text {sat}}'(A)=A\).

(c) Let \({\text {sat}}(A)\subset X\) be the smallest saturated subset, containing *A*.

### Theorem A.5

The saturation \({\text {sat}}(C_{P;G})\) does not depend on \(\mathbf {P}\).

**A.6. Remarks.** (a) The notion of saturation is only reasonable, if the variety \(\mathbf {X}\) is rationally connected.

(b) For every closed subvariety \(\mathbf {Y}\subset \mathbf {X}\), the subset \(\mathbf {Y}(F)\subset X\) is saturated. Also, if *F* is a local field, then every closed subset of *X* is saturated.

(c) If \(\mathbf {X}=\mathbb A^1\), then a subset \(A\subset X\) is saturated if and only if either \(A=X\) or \(X\smallsetminus A\) is infinite.

(d) By (c), saturated subsets of *X* are not closed under finite unions. Therefore the set *X* does not have a topology, whose closed subsets are saturated subsets. On the other hand, our proof of Theorem A.5 indicates that in some respects saturated sets behave like closed subsets in some topology.

### Lemma A.7

Let \(\mathbf {X}\) and \(\mathbf {Y}\) be algebraic varieties.

(a) For a morphism \(f:\mathbf {X}\rightarrow \mathbf {Y}\) and a subset \(A\subset X\), we have an inclusion \(f({\text {sat}}'(A))\subset {\text {sat}}(f'(A))\).

(b) For every \(A\subset X\) and \(B\subset Y\), we have \({\text {sat}}'(A\times B)={\text {sat}}'(A)\times {\text {sat}}'(B)\).

(c) Let \(\mathbf {H}\) be an algebraic group, and let \(f:\mathbf {X}\rightarrow \mathbf {Y}\) be a principal \(\mathbf {H}\)-bundle, locally trivial in the Zariski topology. Then for every subset \(A\subset Y\) we have the equality \({\text {sat}}'(f^{-1}(A))=f^{-1}({\text {sat}}'(A))\).

(d) For an \({\text {Ad}}\, P\)-invariant subset \(A\subset P\), the corresponding subset \({\text {Ind}}_P^G(A)\subset {\text {Ind}}_P^G(X)\) satisfies \({\text {sat}}'({\text {Ind}}_P^G(A))={\text {Ind}}_P^G({\text {sat}}'(A))\).

(e) Let \(\mathbf {Y}\subset \mathbf {X}=\mathbb A^n\) be an open dense subvariety. Then \({\text {sat}}'_X (Y)=X\).

### Proof

(a) is clear.

(b) The inclusion \(\subset \) follows from (a). Conversely, assume that \(a\in {\text {sat}}'(A)\) and \(b\in {\text {sat}}'(B)\) are defined using triples \((\mathbf {V}_a,x_a,f_a)\) and \((\mathbf {V}_b,x_b,f_b)\), respectively, where \(\mathbf {V}_a\) and \(\mathbf {V}_b\) are open subsets of \(\mathbb A^1\). Then we can assume that \(\mathbf {V}_a=\mathbf {V}_b\subset \mathbb A^1\) and \(x_a=x_b\), which implies that \((a,b)\in {\text {sat}}'(A\times B)\).

(c) Since the saturation \({\text {sat}}'\) is local in the Zariski topology, we can assume that \(\mathbf {X}=\mathbf {Y}\times \mathbf {H}\). In this case the assertion follows from (b).

(d) Arguing as in 3.7(b), the assertion follows from (c).

(e) It suffices to show that for every \(x\in \mathbb A^n(F)\), there exists a line \(\mathbf {L}\subset \mathbb A^n\), defined over *F*, such that \(x\in \mathbf {L}\) and \(\mathbf {L}\cap \mathbf {Y}\ne \emptyset \). Consider the variety \(\mathbb P_x\) of lines \(\mathbf {L}\subset \mathbb A^n\) such that \(x\in \mathbf {L}\). Since \(\mathbf {Y}\subset \mathbb A^n\) is Zariski dense, the set of \(\mathbf {L}\in \mathbb P_x\) such that \(\mathbf {L}\cap \mathbf {Y}\ne \emptyset \), is Zariski dense. Since \(\mathbb P_x\cong \mathbb P^{n-1}\), while *F* is infinite, the subset \(\mathbb P_x(F)\subset \mathbb P_x\) is Zariski dense, and the assertion follows. \(\square \)

**A.8. Remark.** All the properties of \({\text {sat}}'\), formulated in Lemma A.7, have natural analogs for \({\text {sat}}\).

**A.9. Relative saturation.** Let \(h:\mathbf {X}\rightarrow \mathbf {Y}\) be a morphism and \(A\subset X\).

(a) Denote by \({\text {sat}}'(h;A)\subset {\text {sat}}'(h(A))\) the union \(\cup _{(\mathbf {V},x,f)}f(x)\), taken over all triples \((\mathbf {V},x,f)\) in the definition of \({\text {sat}}'(h(A))\) (see A.4(a)) such that \(f|_{\mathbf {V}'}:\mathbf {V}'\rightarrow \mathbf {Y}\) has a lift \(\widetilde{f}':\mathbf {V}'\rightarrow \mathbf {X}\) with \(\widetilde{f}'(V')\subset A\).

(b) If *h* is proper, then \({\text {sat}}'(h;A)= h({\text {sat}}'(A))\). Indeed, the valuative criterion implies that every pair \((f,\widetilde{f}')\) as in (a) defines a unique morphism \(\widetilde{f}:\mathbf {V}\rightarrow \mathbf {X}\) such that \(h\circ \widetilde{f}=f\) and \(\widetilde{f}|_{\mathbf {V}'}=\widetilde{f}'\).

(c) Let \(\mathbf {X}'\subset \mathbf {X}\) be an open subvariety such that \(A\subset X'\), and let \(h':=h|_{\mathbf {X}'}:\mathbf {X}'\rightarrow \mathbf {Y}\) be the restriction. By definition, \({\text {sat}}'(h';A)={\text {sat}}'(h;A)\).

**A.10. Notation.** (a) Let \(a_{\mathbf {M},\mathbf {G}}:\widetilde{\mathbf {G}}_\mathbf {M}^{{\text {reg}}}\rightarrow \mathbf {G}\) be the map defined in 1.4 and 1.5, let \(\nu _{\mathbf {G}}:\mathbf {G}\rightarrow \mathbf {c}_\mathbf {G}\) be the Chevalley map (see 1.1(a)), and set \(e_\mathbf {G}:=\nu _\mathbf {G}(1)\in \mathbf {c}_\mathbf {G}\).

(b) Let \(\mathbf {G}^{{\text {der}}}\subset \mathbf {G}\) be the derived group of \(\mathbf {G}\), \(\mathbf {Z}(\mathbf {M})\) the center of \(\mathbf {M}\), and set \(\mathbf {Z}_\mathbf {M}:=(\mathbf {Z}(\mathbf {M})\cap \mathbf {G}^{{\text {der}}})^0\). Then \(\mathbf {Z}_\mathbf {M}\) is a split torus over *F*. Set \(\mathbf {Z}^{{\text {reg}}}_{\mathbf {M}}:=\mathbf {Z}_\mathbf {M}\cap \mathbf {M}^{{\text {reg}}/\mathbf {G}}\). Notice that since \(\mathbf {Z}_\mathbf {M}\) acts on \({\text {Lie}}\,\mathbf {G}/{\text {Lie}}\,\mathbf {M}\) by a direct sum of non-trivial characters, the subset \(\mathbf {Z}^{{\text {reg}}}_{\mathbf {M}}\subset \mathbf {Z}_\mathbf {M}\) is open and dense.

(c) Set \(C^{{\text {reg}}}_Z:=C\cdot Z^{{\text {reg}}}_M\subset M\). Since \(C\subset M\) consists of unipotent elements, and \(Z_M\subset Z(M)\), we have \(C^{{\text {reg}}}_Z\subset M^{{\text {reg}}/G}\). Also \(C^{{\text {reg}}}_Z\) is \({\text {Ad}}\, M\)-invariant, so we can form a subset \({\text {Ind}}_M^G(C^{{\text {reg}}}_Z)\subset \widetilde{G}_M^{{\text {reg}}}\).

(d) Set \(C^{{\text {reg}}}_{P,Z}:=p^{-1}(C^{{\text {reg}}}_Z)=C_P\cdot Z^{{\text {reg}}}_M\subset P^{{\text {reg}}/G}\) (see Lemma 1.9(d)).

**A.11.**
*Proof of Theorem* A.5. Consider the subset \(D_P:=a_{P,G}({\text {sat}}'({\text {Ind}}_P^G(C_P)))\) of *G*. Since \(C_{P;G}\) is equal to \(a_{P,G}({\text {Ind}}_P^G(C_P))\), we have inclusions \(C_{P;G}\subset D_P\subset {\text {sat}}(C_{P;G})\) (see Lemma A.7(a)), thus \({\text {sat}}(D_P)={\text {sat}}(C_{P;G})\). It suffices to show that \(D_P\) does not depend on \(\mathbf {P}\). But this follows from the following description of \(D_P\). \(\square \)

### Claim A.12

We have the equality \(D_P={\text {sat}}'(a_{M,G};{\text {Ind}}_M^G(C^{{\text {reg}}}_Z))\cap \nu _G^{-1}(e_\mathbf {G})\).

### Proof

Recall (see 1.5(d)) that morphism \(a_{M,G}\) factors as \(\widetilde{G}_M^{{\text {reg}}}\overset{a_{M,P}}{\longrightarrow }\widetilde{G}_P\overset{a_{P,G}}{\longrightarrow } G\). Notice that \(a_{M,P}:\widetilde{G}_M^{{\text {reg}}}\rightarrow \widetilde{G}_P\) is an open embedding (use Lemma 1.9(e)) and it satisfies \(a_{M,P}({\text {Ind}}_M^G(C^{{\text {reg}}}_Z))={\text {Ind}}_P^G(C^{{\text {reg}}}_{P,Z})\). Therefore, by A.9(c), we have the equality

Next, since \(a_{P,G}\) is proper, we conclude from A.9(b) that

Thus it suffices to show the equality

Using the commutative diagram from 1.5(b) for \(\mathbf {H}=\mathbf {P}\) and equality \(\pi _{P,G}^{-1}(e_\mathbf {G})=e_\mathbf {P}\) (see Lemma 1.9(f)), we conclude that \(a_{P,G}(A)\cap \nu _G^{-1}(e_\mathbf {G})=a_{P,G}(A\cap \nu _{\widetilde{G}_P}^{-1}(e_\mathbf {P}))\) for every subset \(A\subset \widetilde{G}_P\). Thus it suffices to show the equality

Using Lemma A.7(d), it suffices to show the equality

Set \(\mathbf {P}_{{\text {un}}}:=\nu _\mathbf {P}^{-1}(e_\mathbf {P})\subset \mathbf {P}\), and \(\mathbf {P}_{\mathbf {Z}_\mathbf {M}}:=\nu _\mathbf {P}^{-1}(\nu _\mathbf {P}(\mathbf {Z}_\mathbf {M}))\subset \mathbf {P}\). Since the map \(\nu _\mathbf {P}|_{\mathbf {Z}_\mathbf {M}}:\mathbf {Z}_\mathbf {M}\rightarrow \mathbf {c}_\mathbf {P}\cong \mathbf {c}_\mathbf {M}\) is a closed embedding, the multiplication map induces an isomorphism \(\mathbf {P}_{{\text {un}}}\times \mathbf {Z}_\mathbf {M}\overset{\thicksim }{\rightarrow }\mathbf {P}_{\mathbf {Z}_\mathbf {M}}\). Moreover, it induces a bijection \(C_P\times Z^{{\text {reg}}}_M\overset{\thicksim }{\rightarrow }C^{{\text {reg}}}_{P,Z}\). Thus, formula (A.1) follows from the equality

which follows from Lemma A.7(b),(e). \(\square \)

### Corollary A.13

(a) If *F* is algebraically closed, then the closure \({\text {cl}}(C_{P;G})\subset G\) of \(C_{P;G}\) in the Zariski topology does not depend on \(\mathbf {P}\).

(b) If *F* is a local field, then the closure \({\text {cl}}(C_{P;G})\subset G\) of \(C_{P;G}\) in the analytic topology does not depend on \(\mathbf {P}\).

### Proof

In both cases, every closed subset in *G* is saturated. Therefore we have inclusions \(C_{P;G}\subset {\text {sat}}(C_{P;G})\subset {\text {cl}}(C_{P;G})\), which imply that \({\text {cl}}(C_{P;G})={\text {cl}}({\text {sat}}(C_{P;G}))\). Thus the assertion follows from Theorem A.5. \(\square \)

**A.14. Notation.** For an \({\text {Ad}}\, G\)-invariant subset \(D\subset G\), we denote by \(D^{\heartsuit }\subset D\) the union of *G*-conjugacy classes that are Zariski dense in (the Zariski closure of) *D*.

**A.15. Question.** Is it true that \(C^{\heartsuit }_{P;G}\) is independent of \(\mathbf {P}\)?

**A.16. Remarks.** (a) Let *F* be algebraically closed. Since the number of unipotent conjugacy classes in *G* is finite, we conclude that \(C^{\heartsuit }_{P;G}\) is a single conjugacy class.

(b) Let *F* be general. Then, by (a), \(C^{\heartsuit }_{P;G}\) is a union of unipotent conjugacy classes, belonging to a single conjugacy class over \(\overline{F}\).

### Lemma A.17

Let *F* be either algebraically closed or local. Then for every \({\text {Ad}}\, G\)-invariant subset \(D\subset G\), we have \(D^{\heartsuit }={\text {sat}}(D)^{\heartsuit }\).

### Proof

Let \({\text {cl}}(D)\subset G\) be the closure of *D* in the Zariski topology if *F* is algebraically closed, and in the analytic topology if *F* is local. Then, as in the proof of Corollary A.13, we have \(D\subset {\text {sat}}(D)\subset {\text {cl}}(D)\). Thus, it suffices to show that \(D^{\heartsuit }={\text {cl}}(D)^{\heartsuit }\).

Let \(O\subset {\text {cl}}(D)\) be a Zariski dense *G*-conjugacy class, and let \(\mathbf {D}\subset \mathbf {G}\) be the Zariski closure of *D*. Choose \(x\in O\). Then the morphism \(\mathbf {G}\rightarrow \mathbf {D}:g\mapsto gxg^{-1}\) is dominant. Therefore, in both cases, the corresponding map \(G\rightarrow {\text {cl}}(D)\) is open. Thus \(O\subset {\text {cl}}(D)\) is open, hence \(O\subset D\). \(\square \)

### Corollary A.18

Let *F* be either algebraically closed or local. Then the subset \(C^{\heartsuit }_{P;G}\subset G\) does not depend on \(\mathbf {P}\).

### Proof

This follows immediately from Theorem A.5 and Lemma A.17. \(\square \)

**A.19. Remark.** We do not expect that the conclusion Lemma A.17 holds for an arbitrary field *F*. We wonder whether the equality \({\text {sat}}(C_{P;G})^{\heartsuit }=C^{\heartsuit }_{P;G}\) always holds.

### Appendix B: On a theorem of Harish-Chandra

The goal of this section is to explain the proof of the following result, usually attributed to Harish-Chandra.

### Theorem B.1

Let *F* be a local non-archimedean field of characteristic zero, and let \(h\in \mathcal {H}(G)_G\) be such that \(O_{\gamma }(h)=0\) for every \(\gamma \in G^{{\text {rss}}}\). Then \(h=0\).

**B.2. The Lie algebra analog.** Let \(\mathfrak {g}\) be the Lie algebra of \(\mathbf {G}\), equipped with the adjoint action of *G*, and let \(h\in \mathcal {H}(\mathfrak {g})_G\) be such that \(O_{x}(h)=0\) for every \(x\in \mathfrak {g}^{{\text {rss}}}\). Then the original theorem of Harish-Chandra ([3, Thm 3.1]) asserts that \(h=0\).

The goal of this section is to deduce Theorem B.1 from its Lie algebra analog.

**B.3.**
*G*
**-domains.** (a) Let *X* be a smooth analytic variety over *F* equipped with an action of *G*, let \(\mathcal {H}(X)\) be the space of locally constant measures with compact support (see 3.4) and let \(\mathcal {H}(X)_G\) be the space of *G*-coinvariants.

(b) By a *G*
*-domain* in *X*, we mean an open and closed *G*-invariant subset \(U\subset X\). Then \(\mathcal {H}(U)\subset \mathcal {H}(X)\) is a *G*-invariant subspace, and the map \(h\mapsto 1_U\cdot h\) is a *G*-equivariant projection \(\mathcal {H}(X)\rightarrow \mathcal {H}(U)\). Taking *G*-coinvariants, we get an inclusion \(\mathcal {H}(U)_G\hookrightarrow \mathcal {H}(X)_G\) and a projection \(\mathcal {H}(X)_G\rightarrow \mathcal {H}(U)_G\subset \mathcal {H}(X)_G:h\mapsto h|_U\).

### Lemma B.4

Let \(f:X\rightarrow Y\) be a proper, surjective *G*-equivariant local isomorphism between smooth analytic varieties. Then the pullback map \(f^*:\mathcal {H}(Y)_G\rightarrow \mathcal {H}(X)_G\) (see 3.5(b)) is injective.

### Proof

For every \(m\in \mathbb N\), we denote by \(Y_m\subset Y\) the set of all \(y\in Y\) such that the cardinality of \(f^{-1}(y)\) is *m*. The assumptions on *f* imply that every \(Y_m\subset Y\) is a *G*-domain, and that *Y* is the disjoint union of the \(Y_m\)’s. Then every \(X_m:=f^{-1}(Y_m)\subset X\) is a *G*-domain as well, and it suffices to show that the induced map \(f^*:\mathcal {H}(Y_m)_G\rightarrow \mathcal {H}(X_m)_G\) is injective. Since for every \(h\in \mathcal {H}(Y_m)\) we have \(f_!f^*(h)=mh\), we are done. \(\square \)

### Proof of Theorem B.1

We carry out the proof in five steps.

**Step 1.** There exists a *G*-domain \(U\ni 1\) in *G* such that \(h|_U=0\).

### Proof

Observe first that there exist *G*-domains \(\mathfrak {u}\ni 0\) in \(\mathfrak {g}\) and \(U\ni 1\) in *G* such that the exponential map induces an \({\text {Ad}}\, G\)-equivariant analytic isomorphism \(\epsilon :\mathfrak {u}\overset{\thicksim }{\rightarrow }U\). Namely, if \(\mathbf {G}=\mathbf {GL}_n\), the assertion is straightforward, and the general case follows from it.

We claim that this *U* satisfies the required property. Indeed, consider the pullback \(h':=\epsilon ^*(h|_U)\in \mathcal {H}(\mathfrak {u})_G\subset \mathcal {H}(\mathfrak {g})_G\). It suffices to show that \(h'=0\). For \(x\in \mathfrak {g}^{{\text {rss}}}\) we have \(O_{x}(h')=0\) if \(x\notin \mathfrak {u}\), since \(h'\in \mathcal {H}(\mathfrak {u})_G\), and \(O_{x}(h')=O_{\epsilon (x)}(h)=0\) if \(x\in \mathfrak {u}\) by our assumption on *h*. Then \(h'=0\) by [3, Thm 3.1] (see B.2). \(\square \)

**Step 2.** For every \(s\in Z(G)\) there exists a *G*-domain \(U\ni s\) in *G* such that \(h|_U=0\).

### Proof

Since the map \(g\mapsto gs:G\rightarrow G\) is \({\text {Ad}}\, G\)-equivariant, the assertion follows from the \(s=1\) case shown in Step 1. More precisely, if \(U\ni 1\) is the *G*-domain constructed in Step 1, then \(sU\ni s\) is the *G*-domain such that \(h|_U=0\). \(\square \)

**Step 3.** For every semisimple \(s\in G\smallsetminus Z(G)\) there exists a *G*-domain \(U\ni s\) in *G* such that \(h|_U=0\).

### Proof

Let \(\mathbf {H}:=\mathbf {G}_s^0\) be the connected centralizer of *s*. Then \(\mathbf {H}\subsetneq \mathbf {G}\), and by induction, we may assume that Theorem B.1 is valid for \(\mathbf {H}\).

Let \(\mathbf {H}^{{\text {reg}}/\mathbf {G}}\subset \mathbf {H}\) and \(\mathbf {c}_{\mathbf {H}}^{{\text {reg}}/\mathbf {G}}\subset \mathbf {c}_\mathbf {H}\) be the open subschemes defined in 1.2(b). Note that \(\mathbf {H}\subset \mathbf {G}\) is an equal rank subgroup (see 1.6(a)), and \(s\in \mathbf {H}^{{\text {reg}}/\mathbf {G}}\). Indeed, let \(\mathbf {T}\ni s\) be a maximal torus of \(\mathbf {G}\). Then \(s\in \mathbf {T}\subset \mathbf {H}\), and \(\mathbf {Z}_{\mathbf {G}}(s)^0=\mathbf {H}=\mathbf {Z}_{\mathbf {H}}(s)^0\). Hence \(s\in \mathbf {H}^{{\text {reg}}/\mathbf {G}}\) by 1.6(b).

Let \(\nu _{\mathbf {H}}:\mathbf {H}^{{\text {reg}}/\mathbf {G}}\rightarrow \mathbf {c}_{\mathbf {H}}^{{\text {reg}}/\mathbf {G}}\) be the Chevalley map (see 1.1(a) and 1.2(b)), and we denote by \(\nu _H:H^{{\text {reg}}/G}\rightarrow c_{H}^{{\text {reg}}/G}\) the induced map on *F*-points (compare 3.8).

Choose an open and compact neigbourhood \(V\subset c_{H}^{{\text {reg}}/G}\) of \(\nu _H(s)\), and consider its preimage \(U':=\nu _{H}^{-1}(V)\subset H^{{\text {reg}}/G}\subset H\). Then \(U'\subset H\) is an *H*-domain, so we can form the induced space \({\text {Ind}}_H^G(U')\) (see 3.7) and the *G*-equivariant morphism \(f:=a_{H,G}|_{{\text {Ind}}_H^G(U')}:{\text {Ind}}_H^G(U')\rightarrow G:[g,x]\mapsto gxg^{-1}\) (compare 1.4(c)).

Recall that the subset \({\text {Ind}}_H^G(U')\subset {\text {Ind}}_{\mathbf {H}}^{\mathbf {G}}(\mathbf {H}^{{\text {reg}}/\mathbf {G}})(F)\) is open and closed (by 5.2(a)). Since \(a_{\mathbf {H},\mathbf {G}}:{\text {Ind}}_{\mathbf {H}}^{\mathbf {G}}(\mathbf {H}^{{\text {reg}}/\mathbf {G}})\rightarrow \mathbf {G}\) is étale (see Corollary 2.6), we conclude that *f* is a local isomorphism. On the other hand, since both morphisms \(\iota _{\mathbf {H},\mathbf {G}}:{\text {Ind}}_{\mathbf {H}}^{\mathbf {G}}(\mathbf {H}^{{\text {reg}}/\mathbf {G}})\rightarrow \mathbf {G}\times _{\mathbf {c}_\mathbf {G}} \mathbf {c}_\mathbf {H}^{{\text {reg}}/\mathbf {G}}\) (see Corollary 1.11) and \(\pi _{\mathbf {H},\mathbf {G}}:\mathbf {c}_{\mathbf {H}}\rightarrow \mathbf {c}_\mathbf {G}\) (see 1.1(b)) are finite, and \(V\subset c_H\) is a compact subset, the composition

is proper. Therefore \(U:={\text {Im}}\,f\) is a *G*-domain containing *s*, and we claim that \(h|_U=0\).

By Lemma B.4, the induced map \(f^*:\mathcal {H}(U)_G\rightarrow \mathcal {H}({\text {Ind}}_{H}^{G}(U'))_G\) is injective. Let \(\phi :\mathcal {H}(U)_G\rightarrow \mathcal {H}(U')_H\) be the composition of \(f^*\) and the isomorphism \(\varphi _H^G:\mathcal {H}({\text {Ind}}_{H}^{G}(U'))_G\overset{\thicksim }{\rightarrow }\mathcal {H}(U')_H\) from 3.7(c). Then \(\phi \) is injective, thus it remains to show that \(h':=\phi (h|_U)\in \mathcal {H}(U')_H\subset \mathcal {H}(H)_H\) is zero.

Since Theorem B.1 is valid for *H*, it suffices to show that \(0_{\gamma }(h')=0\) for all \(\gamma \in H^{{\text {rss}}}\). This is clear for \(\gamma \notin U'\), since \(h'\in \mathcal {H}(U')_H\). By construction, for every \(\gamma \in U'\cap G^{{\text {rss}}}\) we have \(O_{\gamma }(h')=O_{\gamma }(h)\). Hence \(O_{\gamma }(h')=0\) by the assumption on *h*. This shows that \(0_{\gamma }(h')=0\) for all \(\gamma \in H^{G-{\text {rss}}}:=H\cap G^{{\text {rss}}}\).

Finally, since \(S\cap H^{G-{\text {rss}}}\subset S\cap H^{{\text {rss}}}\) is dense for every maximal torus \(\mathbf {S}\subset \mathbf {H}\), the equality \(0_{\gamma }(h')=0\) for every \(\gamma \in H^{{\text {rss}}}\) now follows from 6.10. \(\square \)

**Step 4.** Let \(U\subset G\) be a *G*-domain, and let \(g=su\) be the Jordan decomposition of \(g\in G\). Then \(g\in U\) is and only if \(s\in U\).

### Proof

Set \(H:=G_s^0\). It suffices to show the closure of the \({\text {Ad}}\, H\)-orbit of *u* contains 1, hence the closure of the \({\text {Ad}}\, G\)-orbit of *g* contains *s*.

Since *u* is a unipotent element of *H*, the Zariski closure of the \({\text {Ad}}\, \mathbf {H}\)-orbit of *u* contains 1. Hence the assertion follows from a theorem of Kempf [5, Cor. 4.3]. \(\square \)

**Step 5: Completion of the proof.** By Steps 3 and 4, for every \(g\in G\) there exists a *G*-domain \(U\ni g\) in *G* such that \(h|_U=0\). From this the assertion follows. Indeed, choose a lift \(\widetilde{h}\in \mathcal {H}(G)\) of *h*, and let \(K\subset G\) be the support of \(\widetilde{h}\). Since *K* is compact, there is a finite collection of *G*-domains \(U_i,i=1,\ldots ,n\) such that \(K\subset \cup _{i}U_i\) and each \(h_i:=h|_{U_i}\) is zero. Moreover, replacing \(U_j\) by \(U_j{\smallsetminus }(\cup _{i=1}^{j-1}U_i)\) we can assume that the \(U_i\)’s are disjoint. Then \(h=\sum _{i=1}^n h_i=0\). \(\square \)

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Kazhdan, D., Varshavsky, Y. Geometric approach to parabolic induction.
*Sel. Math. New Ser.* **22**, 2243–2269 (2016). https://doi.org/10.1007/s00029-016-0275-5

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DOI: https://doi.org/10.1007/s00029-016-0275-5