Galois Group Action and Jordan Decomposition of Characters of Finite Reductive Groups with Connected Center

GALOIS GROUP ACTION AND JORDAN DECOMPOSITION OF CHARACTERS OF FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER

BHAMA SRINIVASAN AND C. RYAN VINROOT

Abstract. Let G be a connected reductive group with connected center deﬁned over Fq, with Frobenius morphism F . Given an irreducible complex character χ of GF with its Jordan decomposition, and a Galois automorphism σ ∈ Gal(Q/Q), we give the Jordan decomposition of the image σχ of χ under the action of σ on its character values.

2010 AMS Subject Classiﬁcation: 20C33

1. Introduction If G is a finite group, the problem of understanding the action of the absolute Galois group on the irreducible characters of G is a natural one. The problem also has useful applications, an interesting example being a conjecture of G. Navarro [19] which is a refinement of the McKay conjecture to take into account the Galois action on characters. In particular, it is an important problem to understand the action of the Galois group on the irreducible characters of finite groups of Lie type, see [21] for example, where the conjecture of Navarro is checked to hold for certain groups of Lie type. In this paper, we describe the action of the Galois group on the irreducible characters of finite reductive groups with connected center, in terms of the Jordan decomposition of characters. This is a generalization of our results from a previous paper [24] where we accomplish this for the action of complex conjugation, and so describe the real-valued characters in terms of the Jordan decomposition. Our main result may be stated as follows. Theorem (Theorem 5.1). Let G be a connected reductive group with connected center, defined over Fq with Frobenius morphism F . Let m be the F exponent of G , and σ ∈ Gal(Q(ζm)/Q) where ζm is a primitive mth root r of unity, with σ(ζm) = ζm where r ∈ Z and (r, m) = 1. Let χ be an irreducible complex character of GF with Jordan decomposi- ∗F ∗ tion (s0, ν), where s0 ∈ G is a semisimple element in a dual group and ν F ∗ σ is a unipotent character of CG∗ (s0) . Then χ has Jordan decomposition r σ (s0, ν). We also give in Corollary 5.1 criteria to determine the field of character values of an irreducible character based on Jordan decomposition, and in 1 2 BHAMA SRINIVASAN AND C. RYAN VINROOT

Corollary 5.2 we give a particularly simple condition which implies a character is rational-valued. These results reduce the question of the image of an irreducible character of GF under a Galois automorphism to understanding conjugacy of semisimple elements, which is well understood, and understanding the fields of character values of, and the action of group automorphisms on unipotent characters, both of which are well-studied problems [9, 14, 17]. The organization of this paper is as follows. In Section 2, we establish notation for reductive groups, and in Proposition 2.1 we prove that a finite reductive group GF and its dual G∗F ∗ have the same exponent. If this common exponent is m, this allows us to work with automorphisms from F ∗F ∗ Gal(Q(ζm)/Q) which act on all irreducible characters of G , G , and all of their subgroups. While we could just as easily work with the Galois group F ∗F ∗ of Gal(Q(ζn)/Q), where n = |G | = |G |, it is nicer to work with this more refined result, and Proposition 2.1 may also be of independent interest. In Section 3, we give the basic character theory of finite reductive groups, including Lusztig series and unipotent characters, and we prove several lem- mas needed for the main result. We introduce the Jordan decomposition of characters in Section 4, including the crucial result of Digne and Michel in Theorem 4.1 that there exists a unique Jordan decomposition map with respect to a list of properties when the center Z(G) is connected. Finally, our main results are proved in Section 5.

Acknowledgements. The authors would like to thank Paul Fong, Alan Roche, Amanda Schaeﬀer Fry, Jay Taylor, Donna Testerman, and Pham Huu Tiep for helpful conversations over the course of working on this paper. The second-named author was supported in part by a grant from the Simons Foundation, Award #280496.

2. Preliminaries on Reductive Groups In this paper we follow the notation of [24, Section 2], which we now recall. Let G be a connected reductive group defined over a finite field Fq (with p = char(Fq) and fixed algebraic closure Fq), with corresponding F Frobenius morphism F : G → G. For any F -stable subgroup G1 of G, G1 will denote the group of F -fixed elements of G1. For any g ∈ G, we write g −1 G1 = gG1g . Fix a maximally split F -stable torus T of G, contained in a fixed F - stable Borel subgroup B of G. Through the root system associated with T, we define a dual reductive group G∗ with dual Frobenius morphism F ∗, and with F ∗-stable maximal torus T∗ dual to T, contained in the F ∗-stable ∗ ∗ Borel B of G . Define the Weyl group W = NG(T)/T, and the dual Weyl ∗ ∗ ∗ ∗ group W = NG∗ (T )/T . There is a natural isomorphism δ : W → W [4, Sec. 4.2], and a corresponding anti-isomorphism, w 7→ w∗ = δ(w)−1. The F F F isomorphism δ restricts to an isomorphism between W = NG(T) /T and GALOIS GROUP ACTION AND JORDAN DECOMPOSITION 3

(W ∗)F ∗ [4, Sec. 4.4]. Let l denote the standard length function on these Weyl groups. Recall that the GF -conjugacy classes of F -stable maximal tori in G may be classified by F -conjugacy classes of W as follows [3, Sec. 8.2]. For any F -stable torus T0 in G, we have T0 = gT for some g ∈ G. Then −1 −1 F g F (g) ∈ NG(T) and w = g F (g)T ∈ W . The G -conjugacy class of T0 then corresponds to the F -conjugacy class of w in W . Then we have T0F = g(TwF )g−1. We say that T0 is an F -stable torus of G of type w (noting that the reference torus T is fixed). Since T0F and TwF are isomorphic, we work with TwF instead of T0F . Similar to the case of tori, the GF -conjugacy classes of F -stable Levi subgroups are classified as follows. Let L be a Levi subgroup of a standard parabolic P, and given w ∈ W , letw ˙ denote an element in NG(T) which reduces to w in W . Then any Levi subgroup of GF is isomorphic to LwF˙ for some w ∈ W , and we work with LwF˙ instead of the Levi subgroup of GF . For precise statements, see [3, Sec. 8.2], [7, Prop. 4.3], or [18, Prop. 26.2]. If T0 is an F -stable torus of G which is type w, then the F ∗-stable maximal torus of type F ∗(w∗) in G∗ (with respect to T∗) is the dual torus (T0)∗ of T0. It follows that the finite tori TwF and T∗(wF )∗ are in duality, and there is an isomorphism, which we fix as in [3, Sec. 8.2], between T∗F ∗ and the group of characters Tˆ F of TF , ∗ T∗F ←→ Tˆ F s 7−→ θ =s. ˆ Since TwF is in duality with T∗(wF )∗ , then we may replace F with wF , and F ∗ with (wF )∗ in the correspondence above. In particular, if s ∈ T∗(wF )∗ for some w ∈ W , then we denote bys ˆ the corresponding character in Tˆ wF . ∗F ∗ Consider any semisimple element s0 ∈ G . Then s0 is contained in an ∗ ∗ ∗(wF )∗ −1 F -stable maximal torus of G , and as above, we have s0 ∈ g(T )g ∗ for some w ∈ W and g ∈ G . We may correspond to s0 (non-uniquely) the −1 ∗ ∗ element s = g s0g ∈ T , where s is (wF ) -fixed, and vice versa. We then say that s0 and s are associated semisimple elements, and we obtain that ∗F ∗ ∗ the G -conjugacy class of s0 is associated with the W -conjugacy class of ∗ s through this correspondence. Given any element s ∈ T , define WF (s) as (wF )∗ WF (s) = {w ∈ W | s = s}. Then, the semisimple elements in G∗F ∗ correspond to elements s ∈ T∗ such ∗ that WF (s) is nonempty. Given such an s ∈ T , consider CG∗ (s) and its Weyl group W ∗(s) relative to T∗, and define W (s) to be the collection of elements w ∈ W such that w∗ ∈ W ∗(s). Then, as in [7, Section 2], we may write WF (s) = w1W (s), ∗(w F )∗ where w1 ∈ WF (s) is such that T 1 is the maximally split torus inside (w ˙ F )∗ (w ˙ F )∗ of CG∗ (s) 1 . The maximal tori in CG∗ (s) 1 are then each isomorphic 4 BHAMA SRINIVASAN AND C. RYAN VINROOT

∗(wF )∗ to a torus of the form T , for w ∈ WF (s), by the same classiﬁcation of maximal tori which we applied to G∗F ∗ . We denote the exponent of a ﬁnite group H as e = e(H). Thus e(H) is the smallest positive integer e such that he = 1 for all h ∈ H. We have the following result.

Proposition 2.1. For any connected reductive group G deﬁned over Fq, with Frobenius F , and dual G∗ with dual Frobenius F ∗, we have e(GF ) = e(G∗F ∗ ).

Proof. By the Jordan decomposition of elements, we have that any g ∈ GF can be written as g = su = us where s, u ∈ GF with s semisimple and u F unipotent. If p = char(Fq), then p-power order elements in G are exactly the unipotent elements, and elements with order prime to p in GF are exactly the semisimple elements [18, Theorem 2.5]. It follows that the exponent of GF is given by the product of the maximum order of unipotent elements in GF with the least common multiple of the orders of semisimple elements. Thus in order to show GF and G∗F ∗ have the same exponent, we must show that their maximal p-power order elements have the same order, and the least common multiple of the orders of their semisimple elements are the same. F wF −1 Let s0 ∈ G be any semisimple element. Then s0 ∈ g(T )g for some g ∈ G and w ∈ W . We have TwF is in duality with T∗(wF )∗ in G∗, and ∗ ∗(wF )∗ −1 these are isomorphic as ﬁnite groups. For some g1 ∈ G , g1(T )g1 is ∗F ∗ a torus in G containing an element of the same order as s0. It follows that the least common multiple of orders of semisimple elements in GF and G∗F ∗ are equal. The notion of a regular element in a semisimple algebraic group was introduced by R. Steinberg [25] and was studied by him and others. Here, an element in G is regular if its centralizer in G has minimal dimension. We now recall a proof given by D. Testerman [27, p. 70, Proof of Corollary 0.5] that when G is a simple algebraic group, then amongst unipotent elements of G, regular unipotent elements have maximum order, and we note that this argument also holds when G is a connected reductive group. If u ∈ G is regular unipotent of order o(u), take some Borel subgroup B1 of G with unipotent radical U such that u ∈ U. If x ∈ G is any other unipotent −1 element, we have gxg ∈ U for some g ∈ G. The B1-orbit of u under conjugation is then dense in U by [4, Theorem 5.2.1], and all elements in this orbit have order o(u). However, if x (and gxg−1) has order greater than o(u), then this dense orbit must intersect the nonempty open set {v ∈ U | vo(u) 6= 1}, a contradiction. Since the regular unipotent class of G intersects GF [4, Proposition 5.1.7], and p-power order elements are always unipotent, then the orders of the maximal p-power order elements of GF and G are the same. GALOIS GROUP ACTION AND JORDAN DECOMPOSITION 5

We ﬁrst assume that G is a simple algebraic group. By [4, Proposition 5.1.1], for any connected reductive G with center Z(G), the natural homo- morphism G → G/Z(G) induces a bijection between unipotent classes of G and G/Z(G), and in particular preserves orders of unipotent elements. So when G is simple, any other simple algebraic group isogenous to G has maximal p-power order elements of the same order, and this only depends on root system type. The only time G is simple and G∗ has diﬀerent root system type is when G is type Bm or Cm, and these types are dual to each other. Testerman [27, Corollary 0.5] has computed the order of the maximal p-power order elements for all types. For type Cm or Bm, the maximal p-power order is the smallest p-power larger than 2m − 1 (see [27, Proofs of Corollary 0.5 and Proposition 3.4]). It follows that when G is a simple algebraic group, then the maximal p-power order elements of GF and G∗F ∗ have the same order. Next assume that G is a semisimple algebraic group. Write G as the almost direct product of simple factors, G = G1 ··· Gk, where [Gi, Gj] = 1 when i 6= j. Then a maximal p-power order element of G is the element of maximal p-power order of all the Gi. If Φ is the root system of G, then Φ is an orthogonal disjoint union of Φi, the root systems of Gi [18, Exercise 10.33]. It follows that G∗ has root system type which is an orthogonal ∗ disjoint union of the root system types of Gi . From the case for simple algebraic groups, G and G∗ have the same order of maximal p-power order elements, and this holds for any pair of semisimple algebraic groups with dual root system types. Finally, when G is reductive, write G = [G, G]Z◦(G), where [G, G] is semisimple. Since no non-trivial unipotent elements of G are central, then the regular unipotent elements of G are contained in the semisimple factor. Since G and [G, G] have the same root system type [23, Corollary 8.1.9], it follows that [G, G] and [G∗, G∗] have dual root system types. From the semisimple algebraic group case, the orders of the maximal p-power elements are the same in [G, G] and [G∗, G∗], and so are the same in G and G∗. Thus GF and G∗F ∗ have the same order maximal p-power order elements, and F ∗F ∗ we have e(G ) = e(G ).

3. Characters of Finite Reductive Groups In this section we give some general theory and establish several prelim- inary results on the complex characters of finite reductive groups GF . We fix a prime ` which is distinct from p = char(Fq), we let Q` denote the `-adic numbers, and fix an algebraic closure Q`. We fix an abstract isomorphism ∼ of fields C = Q`, so that our characters take values in Q`. In particular we identify a fixed algebraic closure Q of the rationals with its image in Q` under this isomorphism. For any finite group H, any complex character (or Q`-valued character) η σ σ of H, and any σ ∈ Gal(Q/Q), we define the character η by η(h) = σ(η(h)). 6 BHAMA SRINIVASAN AND C. RYAN VINROOT

If ρ is the representation with character η, then σρ is the representation with character ση.

3.1. Lusztig induction. Deligne and Lusztig [6] defined certain virtual representations of finite reductive groups GF through the `-adic cohomology with compact support associated to algebraic varieties over Fq. If X is such a variety, we denote its ith `-adic cohomology space with compact support with coefficients in Q` and Q`, respectively, as i i i Hc(X, Q`) and Hc(X, Q`) = Hc(X, Q`) ⊗Q` Q`, ∗ P i i and then Hc (X) = i(−1) Hc(X, Q`) is a virtual Q`-vector space. Let L be an F -stable Levi subgroup of G of a standard parabolic P (as in Section 2) with Levi decomposition P = LU. If L : G → G is the Lang −1 −1 map, L(g) = g F (g), then L (U) = X is an algebraic variety over Fq. ∗ −1 F F Then the virtual Q`-space Hc (L (U)) may be taken to be a (G × L )- bimodule. If we identify LF with LwF˙ as in Section 2, then we may regard this as a (GF × LwF˙ )-bimodule, and it is through this structure that one GF wF˙ defines the Lusztig induction functor RLwF˙ which takes characters of L to virtual characters of GF , and which is Harish-Chandra induction when the parabolic P is F -stable. While the definition of Lusztig induction depends on the choice of parabolic P, it is known in all but very few cases that this functor is independent of this choice [1, 26]. We will need the following statement (see also [22, Lemma 2.1]). Lemma 3.1. For any Levi subgroup LwF˙ of GF , any character γ of LwF˙ , σ GF GF σ and any σ ∈ Gal(Q/Q), we have RLwF˙ (γ) = RLwF˙ ( γ). Proof. By [8, Proposition 11.2], we have for any g ∈ GF ,

F 1 X (3.1) RG (γ) (g) = Tr((g, l) | H∗(L−1(U)))γ(l−1). LwF˙ |LwF˙ | c l∈LwF˙ ∗ −1 By [8, Corollary 10.6], for example, every Tr((g, l) | Hc (L (U))) is a rational integer, so is stable under the action of σ. By applying σ to both sides of (3.1), we have

F 1 X σRG (γ) (g) = Tr((g, l) | H∗(L−1(U)))(σγ(l−1)) LwF˙ |LwF˙ | c l∈LwF˙ GF σ = RLwF˙ ( γ) (g), proving the claim. 3.2. Lusztig series. If one takes a torus TwF for the Levi subgroup in Lusztig induction, and θ is an irreducible character of TwF , then we get GF the Deligne-Lusztig virtual character RTwF (θ), originally defined in [6]. If ∗ s ∈ T is semisimple such that WF (s) is nonempty, then the rational Lusztig GALOIS GROUP ACTION AND JORDAN DECOMPOSITION 7 series of GF corresponding to s, denoted E(GF , s), is the set of irreducible F characters χ of G such that, for some w ∈ WF (s) we have GF hχ, RTwF (ˆs)i= 6 0, where h·, ·i denotes the standard inner product on class functions. Given ∗ F F s, t ∈ T with both WF (s) and WF (t) nonempty, then E(G , s) and E(G , t) are either disjoint or equal, and are equal precisely when s and t are W ∗- ∗F ∗ conjugate. If s0 and t0 are semisimple elements of G associated with s and t, respectively, as in Section 2, then this is equivalent to s0 and t0 being G∗F ∗ -conjugate (see [8, Propositon 13.13] and its proof). We will either denote the rational Lusztig series by E(GF , s) when it is parameterized by ∗ ∗ F the W -class of some s ∈ T with WF (s) nonempty, or by E(G , s0) when it ∗F ∗ ∗F ∗ is parameterized by the G -class of some semisimple element s0 ∈ G . One may further define the geometric Lusztig series, which is parame- F terized by the G-conjugacy class of a semisimple element s0 ∈ G , and contains the associated rational Lusztig series. We only remark that when the centralizer CG∗ (s0) (or CG∗ (s)) is connected, then the geometric and rational Lusztig series coincide. Recall an irreducible character χ of GF is cuspidal if it does not appear in the truncation to any standard parabolic subgroup of GF , see [4, Section 9.1]. The set of cuspidal characters in the Lusztig series E(GF , s) will be denoted by E(GF , s)•. By Proposition 2.1, we have e(GF ) = e(G∗F ∗ ) = m, and so any irreducible character of GF , G∗F ∗ , or any of their subgroups, takes values in Q(ζm) for ζm a primitive mth root of unity. If σ ∈ Gal(Q/Q), we also denote by σ its projection to Gal(Q(ζm)/Q). So σ acting on Q(ζm) is generated r by σ(ζm) = ζm for some r ∈ Z with (r, m) = 1. We will need the following result, which is also a special case of [21, Lemma 3.4]. F Lemma 3.2. Let χ be an irreducible character of G , σ ∈ Gal(Q/Q), and r F r ∈ Z such that σ(ζm) = ζm. Then we have χ ∈ E(G , s) if and only if σχ ∈ E(GF , sr). Proof. We have χ ∈ E(GF , s) if and only if GF σ σ GF hχ, RTwF (ˆs)i = h χ, RTwF (ˆs)i= 6 0 for some w ∈ WF (s), where the first equality is obtained by the fact that the inner product is a rational integer, and so stable under σ, and by applying σ to each term of the sum defining the inner product. Each linear characters ˆ takes values in mth roots of unity, since e(TwF ) σ divides m, and since s 7→ sˆ is an isomorphism, we have sˆ = sbr. From Lemma 3.1, we thus have σ GF GF r RTwF (ˆs) = RTwF (sb ). Now χ ∈ E(GF , s) if and only if σ GF r h χ, RTwF (sb )i= 6 0, 8 BHAMA SRINIVASAN AND C. RYAN VINROOT

σ F r which is true exactly when χ ∈ E(G , s ).

3.3. Unipotent characters. The unipotent characters of GF are those irreducible characters in the Lusztig series E(GF , 1). The unipotent characters of GF may be viewed as generic objects associated with GF (see [2, Section 1B]), and there exists a canonical labeling of unipotent characters with certain uniqueness properties by a result of Lusztig [15] (see also [10, Section 4]). In this section we prove a result for unipotent characters whose proof is adapted from arguments given in [11, Corollary 3.9], and we follow the notation given there. In particular, for any w ∈ W , we let Xw be the algebraic variety over Fq given by the set of all Borel subgroups B of G which are mapped to F (B) by w (that is, the Deligne-Lusztig variety). i i We may thus consider the spaces Hc(Xw, Q`) and Hc(Xw, Q`). If d is the smallest positive integer such that F d acts trivially on W , then as in [11, d i Chapter 3] there is a natural action of F on Hc(Xw, Q`). By [11, Corollary 3.9], for any unipotent representation π of GF with character χ, there exists × d a w ∈ W , i ≥ 0, and α ∈ Q` such that α is an eigenvalue of F acting i F on Hc(Xw, Q`), and π is isomorphic to a G -submodule of the generalized d i i α-eigenspace of F on Hc(Xw, Q`), which is denoted by Hc(Xw, Q`)α. Then we say α is an eigenvalue of F d associated with χ (or π).

Lemma 3.3. Let π be a unipotent representation of GF with character F σ χ ∈ E(G , 1), and let σ ∈ Gal(Q/Q) (so χ is also unipotent). Let d be the least integer such that F d acts trivially on W . If α and α0 are eigenvalues of F d associated with χ and σχ, respectively, then αα0 is a power of qd.

0 F Proof. If w, w ∈ W , then we let G act on Xw × Xw0 by conjugation on F both factors. So we may consider the variety G \(Xw × Xw0 ) over Fq and i F the spaces Hc(G \(Xw × Xw0 ), Q`). Note that it follows from Lemma 3.2 that σπ is also a unipotent representation of GF . Thus there exists w0 ∈ W and j ≥ 0 such that σπ F j F is isomorphic to a G -submodule of Hc (Xw0 , Q`)α0 . Now the G -module σ F i+j π ⊗ π is isomorphic to a G -submodule of Hc (Xw × Xw0 , Q`)αα0 (by [8, Proposition 10.9(i)], for example). It then follows that αα0 is an eigenvalue d i+j F 0 of F on Hc (G \(Xw × Xw0 ), Q`) as in [11, pg. 24], and so αα is a power d of q by [11, Theorem 3.8].

3.4. Principal series. The unipotent characters in the principal series of F GF GF G are the constituents of IndBF (1). Associated with the module IndBF (1) is the Hecke algebra

F F GF H = H(G , B ) = EndCGF IndBF (1) . GALOIS GROUP ACTION AND JORDAN DECOMPOSITION 9

F F Then we have H = eCG e, where e ∈ CG is the idempotent element 1 X e = b. |BF | b∈BF There is a natural bijection between unipotent characters in the principal series and the irreducible characters of the Hecke algebra H, which is defined as follows [5, Theorem 11.25(ii)]. Given a unipotent character χ in the F F principal series of G , extend χ linearly to a characterχ ˜ of CG , and then restrict to H to obtain an irreducible characterχ ˜|H of the Hecke algebra. Now let σ ∈ Gal(C/Q), and we consider the action of σ on χ and its effect on the bijection with irreducible characters of the Hecke algebra. Lemma 3.4. Given a unipotent character χ in the principal series of GF , and σ ∈ Gal(C/Q), we have σ −1 fχ|H = σ ◦ (˜χ|H) ◦ σ . P F Proof. Given an element g αgg ∈ CG , the action of σ is defined as ! X X σ αgg = σ(αg)g. g g F −1 Note that we then have σ(e) = e, and since H = eCG e, then σ◦(˜χ|H)◦σ is a well-defined character of H. We compute ! σ X X fχ αgg = αgσ(χ(g)) g g ! X −1 = σ σ (αg)χ(g) g !! X −1 = σ χ˜ σ (αg)g g ! −1 X = (σ ◦ χ˜ ◦ σ ) αgg . g

−1 Thus σfχ = σ ◦ χ˜ ◦ σ . Since we also have −1 −1 (σ ◦ χ˜ ◦ σ )|H = σ ◦ (˜χ|H) ◦ σ , the result follows. The Hecke algebra H(GF , BF ) has a basis indexed by the fundamental generating set for the Weyl group W F (see [4, Chapter 10], for example). There is also the Hecke algebra H∗ = H(G∗F ∗ , B∗F ∗ ), corresponding to the dual group G∗F ∗ , again with a basis indexed by the fundamental reﬂec- tions of the Weyl group W ∗F ∗ . Through the isomorphism δ between W F 10 BHAMA SRINIVASAN AND C. RYAN VINROOT and W ∗F ∗ , we identify the Hecke algebras H and H∗, and their irreducible characters.

4. Jordan Decomposition of Characters Given the connected reductive group G and a rational Lusztig series E(GF , s) of GF , a Jordan decomposition map is a bijection

∗ F (w ˙ 1F ) Js : E(G , s) −→ E(CG∗ (s) , 1), F with the property that, for any χ ∈ E(G , s) and any w ∈ WF (s), we have

∗ F (w ˙ 1F ) G l(w1) CG∗ (s) (4.1) hχ, R (ˆs)i = hJ (χ), (−1) R ∗ (1)i. TwF s T∗(wF ) The Jordan decomposition map was proved to exist in the case that the center Z(G) is connected by Lusztig [12], and in the case that the center is disconnected by Lusztig [13] and by Digne and Michel [7]. In the case that Z(G) is disconnected, then there are rational Lusztig series E(GF , s) such that CG∗ (s) is disconnected. In this case, one deﬁnes the unipotent (w ˙ F )∗ characters in the set E(CG∗ (s) 1 , 1) to be those characters which appear in the induction from unipotent characters of the group of rational points ◦ of the connected component CG∗ (s) . Lusztig found [12] that in many cases the Jordan decomposition map Js is completely determined by the property (4.1), although this is not always true. In the case that Z(G) is connected, and so CG∗ (s) is connected for any s, Digne and Michel [7, Theorem 7.1] found a list of properties which uniquely determines the Jordan decomposition map, which we now state. Theorem 4.1 (Digne and Michel, 1990). Suppose that Z(G) is connected. ∗ Given any s ∈ T such that WF (s) is nonempty, there exists a unique bijection ∗ F (w ˙ 1F ) Js : E(G , s) −→ E(CG∗ (s) , 1) which satisﬁes the following conditions: F (1) For any χ ∈ E(G , s), and any w ∈ WF (s),

∗ F (w ˙ 1F ) G l(w1) CG∗ (s) hχ, R (ˆs)i = hJ (χ), (−1) R ∗ (1)i. TwF s T∗(wF ) (2) If s = 1 then: (a) The eigenvalues of F d associated to χ are equal, up to a power d/2 ∗d of q , to the eigenvalues of F associated to J1(χ). (b) If χ is in the principal series then J1(χ) and χ correspond to the same character of the Hecke algebra. ∗F ∗ F (3) If z ∈ Z(G ) is central, and χ ∈ E(G , s), then Jsz(χ⊗zˆ) = Js(χ). ∗ (4) If L is a standard Levi subgroup of G such that L contains CG∗ (s) and such that L is wF˙ -stable, then the following diagram is commutative: GALOIS GROUP ACTION AND JORDAN DECOMPOSITION 11

F Js (w ˙ F )∗ E(G , s) −−−−→E(CG∗ (s) 1 , 1)

x F RG  LwF˙ wF˙ Js (v ˙wF˙ )∗ E(L , s) −−−−→ E(CL∗ (s) , 1) where v˙w˙ =w ˙ 1, and we extend Js by linearity to generalized characters. ∗ (5) Assume (W, F ) is irreducible, (G,F ) is of type E8, and (CG∗ (s), (w ˙ 1F ) ) 2 2 is of type E7 × A1 (respectively, E6 × A2, respectively E6 × A2). Let L be a Levi of G of type E7 (respectively E6, respectively E6) which contains the corresponding component of CG∗ (s). Then the following diagram is commutative:

F Js (w ˙ F )∗ E(G , s) −−−−→E(CG∗ (s) 1 , 1) ∗ x F x (w ˙ 1F )  G  CG∗ (s) R w˙ F R ∗  L 2  L∗(w ˙ 2F ) J ∗ E(Lw˙ 2F , s)• −−−−→s E(L∗(w ˙ 2F ) , 1)• where the superscript • denotes the cuspidal part of the Lusztig series, and w2 = 1 (respectively 1, respectively the WL-reduced element of WF (s) which is in a parabolic subgroup of type E7 of W ). (6) Given an epimorphism ϕ :(G,F ) → (G1,F1) such that ker(ϕ) is a ∗ central torus (that is, an isogeny), and semisimple elements s1 ∈ G1, ∗ ∗ s = ϕ (s1) ∈ G , the following diagram is commutative:

F Js (ϕ(w ˙ )F )∗ E(G , s) −−−−→E(CG∗ (s) 1 , 1) x  > > ∗  ϕ y ϕ

Js ∗ F1 1 (w ˙ F ) E(G , s ) −−−−→ E(C ∗ (s ) 1 1 , 1), 1 1 G1 1 where >ϕ denotes the transpose map, >ϕ(χ(g)) = χ(ϕ(g)). (7) If G is a direct product, G = Q G , then JQ = Q J . i i i si i si We will need the following result, which follows from Theorem 4.1. It has the exact same proof as [24, Lemma 3.1].

∗(w F )∗ (w ˙ F )∗ Lemma 4.1. Let s, t ∈ T 1 , so that s, t ∈ CG∗ (s) 1 , and suppose ∗ (w ˙ F )∗ −1 that there exists v˙ ∈ NG∗ (T ) 1 such that vs˙ v˙ = t. Let Js, Jt be the v˙ maps as described in Theorem 4.1. If Js(χ) = ψ, then Jt(χ) = ψ.

5. Main Results We may now prove our main result, which essentially states that the action of the Galois group on the Jordan decomposition (s, ψ) of characters is the natural one. In particular, this image may be calculated with the knowledge of the images ofs ˆ and ψ under the given Galois automorphism. 12 BHAMA SRINIVASAN AND C. RYAN VINROOT

Theorem 5.1. Suppose Z(G) is connected, and let χ be an irreducible F complex character of G . Let σ ∈ Gal(Q/Q), so σ acts on Q(ζm), where F ∗F ∗ r m = e(G ) = e(G ) and σ(ζm) = ζm with r ∈ Z and (r, m) = 1. ∗ F Let s ∈ T such that WF (s) is nonempty, where χ ∈ E(G , s) and Js(χ) = σ F r σ σ ψ. Then χ ∈ E(G , s ) and Jsr ( χ) = ψ.

Proof. It is enough to consider σ ∈ Gal(Q(ζm)/Q). By Lemma 3.2, we have σ F r σ σ χ ∈ E(G , s ), and we show that in particular Jsr ( χ) = ψ. Note that σ r (w ˙ F )∗ (w ˙ F )∗ ψ ∈ E(CG∗ (s ) 1 , 1) = E(CG∗ (s) 1 , 1). Our proof follows the same structure as the proof of [24, Theorem 4.1], replacing complex conjugation by a Galois automorphism. We prove the claim by induction on the semisimple rank of G, where the first case is when G = T is a torus, so that each Lusztig series contains exactly one character, and the statement follows immediately. Assume now that the statement holds for any group with semisimple rank smaller than (G,F ). Then we prove the statement holds for any group in the same isogeny class as (G,F ). If λ ∈ E(GF , sr), it follows from Lemma 3.2 that σ−1 λ ∈ E(GF , s). That is, every character in E(GF , sr) is of the form σχ for some χ ∈ E(GF , s). Given this fact, we have a well-defined map ∗ ∗ F r r (w ˙ 1F ) (w ˙ 1F ) µsr : E(G , s ) −→ E(CG∗ (s ) , 1) = E(CG∗ (s) , 1), σ σ where µsr ( χ) = ψ when Js(χ) = ψ. We apply the uniqueness described by Theorem 4.1 to show that µsr = Jsr , which will give the desired result. We prove the map µsr satisfies each of the properties listed in Theorem 4.1, where the induction hypothesis on the semisimple rank is employed only for properties (4) and (5). For property (1) of Theorem 4.1, we can apply Lemma 3.1. Using the GF fact that hχ, RTwF (ˆs)i ∈ Z, we have GF σ GF σ σ GF σ GF r hχ, RTwF (ˆs)i = hχ, RTwF (ˆs)i = h χ, RTwF (ˆs)i = h χ, RTwF (sb )i, and similarly, ∗ ∗ (w ˙ 1F ) (w ˙ 1F ) l(w1) CG∗ (s) σ l(w1) CG∗ (s) hψ, (−1) R ∗ (1)i = hψ, (−1) R ∗ (1)i T∗(wF ) T∗(wF ) ∗ (w ˙ 1F ) σ l(w1) CG∗ (s) = h ψ, (−1) R ∗ (1)i. T∗(wF ) Since we have ∗ F (w ˙ 1F ) G l(w1) CG∗ (s) hχ, R (ˆs)i = hψ, (−1) R ∗ (1)i, TwF T∗(wF ) then it follows we have ∗ F (w ˙ 1F ) σ G r σ l(w1) CG∗ (s) h χ, R (s )i = h ψ, (−1) R ∗ (1)i, TwF b T∗(wF ) so that µsr satisfies (1). For property (2), we take s = 1 and assume χ ∈ E(GF , 1) is unipotent 0 d and J1(χ) = ψ. In (2a), we suppose that α, α are the eigenvalues of F GALOIS GROUP ACTION AND JORDAN DECOMPOSITION 13 associated with χ and σχ, and β, β0 are the eigenvalues of F ∗d associated with ψ and σψ, respectively. We know from property (2a) that αβ−1 is a power of qd/2, and from Lemma 3.3 we know αα0 and ββ0 are both powers of qd. Thus α0β0−1 is a power of qd/2, and property (2a) holds for the map GF µ1. Now assume that χ is a constituent of IndBF (1), that is, χ is in the ∗F ∗ principal series, which means ψ = J1(χ) is in the principal series for G . By (2b) of Theorem 4.1, χ and ψ both correspond to the same character κ of the Hecke algebra H(GF , BF ) (identified with H(G∗F ∗ , B∗F ∗ ) as in Section 3.4). Note that if χ and ψ are principal series characters, then so σ σ σ σ are χ and ψ, where µ1( χ) = ψ. By Lemma 3.4, since χ and ψ both correspond to the character κ of the Hecke algebra, then σχ and σψ both correspond to the character σ ◦ κ ◦ σ−1 of the Hecke algebra. (Technically, −1 we must replace σ by any extension of σ to Gal(C/Q) for σ ◦ κ ◦ σ to be well-defined, although it is immediate that this is independent of the choice of extension.) It follows that property (2b) also holds for µ1. For property (3), let z ∈ Z(G∗F ∗ ), and recall r satisfies (r, m) = 1 and r σ σ σ(ζm) = ζm. We must show µsr ( χ) = µsrz( χ ⊗ zˆ). Let k ∈ Z such that k σ σ rk = 1 mod m. If Js(χ) = ψ, then Jszk (χ ⊗ zb ) = ψ, while µsr ( χ) = ψ. Since (szk)r = srz and σ(zbk) =z ˆkr =z ˆ, then by definition we have

σ σ k σ µsrz( χ ⊗ zˆ) = Jszk (χ ⊗ zb ) = ψ, and property (3) for µsr follows. For property (4), since the Levi subgroup L has semisimple rank strictly smaller than G, we may apply the induction hypothesis. So for any ξ ∈ F wF˙ r σ σ G E(L , s), if Js(ξ) = ψ, then Js ( ξ) = ψ. Also, if RLwF˙ (ξ) = χ, then GF σ σ RLwF˙ ( ξ) = χ by Lemma 3.1. It follows that the diagram in property (4) commutes when we replace Js with µsr in the top row and Js with Jsr in the bottom row, as desired. The proof that the map µsr satisﬁes property (5) is very similar to the proof for (4) above. We may again apply the induction hypothesis to the w˙ F • ∗(w ˙ F )∗ • Levi subgroup L, and so if ξ ∈ E(L 2 , s) and Js(ξ) = λ ∈ E(L 2 , 1) , σ σ GF GF σ σ then J r ( ξ) = λ. If R (ξ) = χ, then R ( ξ) = χ, and if s Lw˙ 2F Lw˙ 2F

(w ˙ F )∗ (w ˙ F )∗ CG∗ (s) 1 CG∗ (s) 1 σ σ R ∗ (λ) = ψ, then R ∗ ( λ) = ψ. L∗(w ˙ 2F ) L∗(w ˙ 2F )

By applying the commutativity of the diagram in the case of Js, we know σ σ Js(χ) = ψ, and thus µsr ( χ) = ψ. The diagram therefore commutes when Jsr is on the bottom row and µsr is on the top row, and it follows that property (5) holds for µsr . ∗ For property (6), let ϕ :(G,F ) → (G1,F1) be an isogeny and ϕ : ∗ ∗ ∗ ∗ F1 > (G1,F1 ) → (G ,F ) the dual isogeny. Let χ1 ∈ E(G1 , s1), ϕ(χ1) = χ ∈ F E(G , s), with Js(χ) = ψ and Js1 (χ1) = ψ1. By applying property (6) to Js > ∗ σ σ > ∗ σ ∗ and Js1 , we have ϕ (ψ) = ψ1. We have ψ1 = ( ϕ (ψ)) = ψ(ϕ ). Thus, > ∗ σ σ r r > σ σ σ ϕ ( ψ) = ψ1. We also have ϕ(s ) = s1, and ϕ( χ1) = χ1(ϕ) = χ. 14 BHAMA SRINIVASAN AND C. RYAN VINROOT

Now, the diagram from property (6) is commutative when we have µ r in s1 the bottom row and µsr in the top row, as desired. Q Q For the final property (7), suppose χ = χi and Js(χ) = ψ = ψi, Q i i with s = i si and Jsi (χi) = ψi. Then σ σ σ Y σ Y σ µQ r ( χ) = µ r ( χ) = ψ = ψ = µ r ( χ ). i si s i s i i i Since all properties from Theorem 4.1 hold for the map µsr , we must have σ σ µsr = Jsr , and it follows that if Js(χ) = ψ, then Jsr ( χ) = ψ. The following is our main application of Theorem 5.1, which allows us to reduce the problem of computing the field of character values of an irreducible character of GF to the question of conjugacy of powers of semisimple elements, and the computation of the actions of group and Galois automorphisms on unipotent characters. F ∗F ∗ Corollary 5.1. Let F be any subfield of C, let m = e(G ) = e(G ), and K = Q(ζm) ∩ F . For any σ ∈ Gal(Q(ζm)/K), let rσ ∈ Z such that rσ (rσ, m) = 1 and σ(ζm) = ζm . Suppose Z(G) is connected, and let χ be an irreducible character of GF , ∗F ∗ F where s0 ∈ G is semisimple, χ ∈ E(G , s0), and Js0 (χ) = ν. Then Q(χ) ⊆ F if and only if the following hold for every σ ∈ Gal(Q(ζm)/K): ∗ ∗F rσ (i) The element s0 is G -conjugate to s0 . ∗ ∗F −1 rσ h0 σ (ii) If h0 ∈ G satisfies h0s0h0 = s0 , then ν = ν. Proof. In the notation of Theorem 5.1, we suppose that χ ∈ E(GF , s) with ∗ s ∈ T and WF (s) nonempty (so s corresponds to s0). First, consider ∗(w ˙ F )∗ (w ˙ F )∗ some h ∈ G 1 which normalizes CG∗ (s) 1 . The automorphism of (w ˙ F )∗ CG∗ (s) 1 given by conjugation by h permutes the unipotent characters (w ˙ F )∗ −1 r E(CG∗ (s) 1 , 1), by [2, (1.27)] for example. If hsh = s for some r ∈ Z (w ˙ F )∗ r (w ˙ F )∗ such that (r, m) = 1, then h normalizes CG∗ (s) 1 = CG∗ (s ) 1 . Any ∗ ∗(w ˙ 1F ) −1 r other element h1 ∈ G satisfying h1sh1 = s must have the property (w ˙ F )∗ (w ˙ F )∗ that h1 ∈ hCG∗ (s) 1 . This implies that the action on E(CG∗ (s) 1 , 1) given by conjugation by an element h such that hsh−1 = sr, is independent of the choice of h. σ Suppose that Q(χ) ⊆ F , which is equivalent to χ = χ for each σ ∈ σ F rσ Gal(Q(ζm)/K) since Q(χ) ⊆ Q(ζm). Then for each σ, χ = χ ∈ E(G , s ) by Lemma 3.2. So E(GF , s) = E(GF , srσ ), which is equivalent to condition ∗ (i). Since s, srσ ∈ T∗(w1F ) , then by [24, Lemma 2.2] we havevs ˙ v˙−1 = srσ ∗ (w ˙ F )∗ for somev ˙ ∈ NG∗ (T ) 1 . By Lemma 4.1, if Js(χ) = ψ, then we have v˙ σ v˙ rσ v˙ σ Jv˙ s(χ) = ψ. Since Jsrσ (χ) = ψ and s = s , then ψ = ψ. By the previous paragraph, we also then have hψ = σψ, which gives condition (ii). Conversely, suppose that conditions (i) and (ii) hold for each σ. That is, ∗ −1 rσ ∗(w ˙ F ) for each σ we have hsh = s for some h ∈ G 1 , and if Js(χ) = ψ, h σ ∗ (w ˙ F )∗ then ψ = ψ. Again by [24, Lemma 2.2], there is somev ˙ ∈ NG∗ (T ) 1 v˙ rσ v˙ σ σ σ such that s = s , and so also ψ = ψ. We have Jsrσ ( χ) = ψ by GALOIS GROUP ACTION AND JORDAN DECOMPOSITION 15

v˙ σ Theorem 5.1, and Jv˙ s(χ) = ψ by Lemma 4.1. Now Jsrσ ( χ) = Jsrσ (χ), σ and thus χ = χ. Since this holds for every σ ∈ Gal(Q(ζm)/K), we have Q(χ) ⊆ K ⊆ F . We conclude with a criterion for an irreducible character χ of GF to be rational-valued. Recall that an element g of a ﬁnite group G is rational if, r for every r ∈ Z such that (|g|, r) = 1, g and g are conjugate in G, see [20, Section 5] for example. It follows from Lemma 3.2 that if χ has Jordan ∗F ∗ decomposition (s0, ν), and s0 is not rational in G , then χ is not rational- valued. The following gives a partial converse to this statement. Note that there are many unipotent characters which are rational-valued [14] and many which are invariant under group automorphisms [16, Proposition 3.7]. Corollary 5.2. Suppose Z(G) is connected, and χ is an irreducible char- F ∗F ∗ acter of G with Jordan decomposition (s0, ν). If s0 is rational in G , and ν is both rational-valued and invariant under the automorphism group F ∗ of CG∗ (s0) , then χ is rational-valued.

Proof. If s0 is rational, then for every r ∈ Z such that (|s0|, r) = 1, we have −1 r ∗F ∗ σ hs0h = s0 for some h ∈ G . For any σ ∈ Gal(Q/Q), we have ν = ν since ν is rational-valued, and since ν is invariant under automorphisms F ∗ ∗F ∗ of CG∗ (s0) , which includes conjugation by elements in G , we have h σ ν = ν = ν. It follows from Corollary 5.1 that χ is rational-valued. Remark. Through conversations with Jay Taylor, it appears that the uniqueness statement for the Jordan decomposition map in Theorem 4.1 F might be generalized to Lusztig series E(G , s) such that CG∗ (s) is connected, while Z(G) is not necessarily connected. If this holds, then the results of this section immediately generalize to this situation.

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Department of Mathematics, Statistics, and Computer Science (MC 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60680-7045 E-mail address: [email protected]

Department of Mathematics, College of William and Mary, P. O. Box 8795, Williamsburg, VA 23187-8795 E-mail address: [email protected]