Proof. Let $g\in G$ with $\chi (g)\neq 0$ . Then, $g_p$ belongs to a conjugate of D by [Reference Navarro24, Corollary 5.9]. Therefore, $|g|_p\leq \exp (D)$ , and we have

$$\begin{align*}\chi(g)\in{\mathbb Q}_{|g|}\subseteq {\mathbb Q}_{|g|_p|g|_{p'}}\subseteq {\mathbb Q}_{\exp(D)|G|_{p'}}.\end{align*}$$

We now have ${\mathbb Q}(\chi )\subseteq {\mathbb Q}_{\exp (D)|G|_{p'}}$ , which implies that $c(\chi )$ divides $\exp (D)|G|_{p'}$ , and the lemma follows.

Proof. This follows from Lemma 3.1. Note that ${\mathbb Q}_n={\mathbb Q}_{2n}$ for odd n, so there are no characters of $2$ -rationality level $1$ .

Proof. The first conclusion again follows from Lemma 3.1, so it remains to prove the second part. We have $\chi (1)_p=|G|_p/p$ and $\mathbf {ht}(\chi )=0$ . p-Blocks of defect one have been fully described in early work of Brauer [Reference Brauer1], [Reference Brauer2] (see also [Reference Navarro24, Chapter 11]). Let $K:={\mathbf {O}}_{p'}({\mathbf {N}}_G(D))$ . Let $b\in \mathrm {\mathrm {Bl}}({\mathbf {C}}_G(D))$ be a root of B and $\xi \in {\mathrm {Irr}}(K)$ be the restriction (to K) of the canonical character in ${\mathrm {Irr}}(b)$ of B. In fact, $\xi $ is the unique Brauer character in b. Let E be the subgroup of ${\mathbf {N}}_G(D)$ fixing b and $\overline {E}:=E/{\mathbf {C}}_G(D)$ the inertial quotient of B and $e:=|\overline {E}|$ . The block B then contains precisely $e+(p-1)/e$ ordinary irreducible characters, where e of these are p-rational and, therefore, trivially satisfy the stated equality.

Suppose that $\chi $ is one of the remaining $(p-1)/e$ other characters, a so-called exceptional character. In this case, there exists $\lambda \in {\mathrm {Irr}}(D)-\{1_D\}$ and $\epsilon \in \{\pm 1\}$ such that

$$\begin{align*}\chi(hk)=\epsilon(\lambda\times \xi)^{{\mathbf{N}}_G(D)}(hk) \end{align*}$$

for every $h\in D-\{1\}$ and $k\in K$ , by [Reference Navarro24, Chapter 11]. Note that $\chi (g)=0$ whenever $g_p$ is not conjugate to an element in D. Also, for each $h\in D-\{1\}$ , a $p'$ -element in ${\mathbf {C}}_G(h)$ must be inside K. Therefore, these elements $hk$ capture all the non-zero values of $\chi $ . It follows that ${\mathbb Q}(\chi )={\mathbb Q}(\chi _{D\times K})$ , and thus ${\mathbb Q}(\chi )={\mathbb Q}(\chi _{{\mathbf {N}}_G(D)})$ , as desired.

Proof. The case of defect zero follows from Corollary 3.2. We may assume that $|D|>1$ .

We shall follow the notation described above. If $|\Lambda |=1$ then, as mentioned already, the block B must have defect one, and we are done by Corollary 3.2 and Lemma 3.3. So we assume from now on that $|\Lambda |>1$ . If $\chi \in {\mathrm {Irr}}_{nex}(B)$ is a non-exceptional character of B, then $\chi $ is p-rational, and thus there is nothing to prove. We, therefore, assume furthermore that $\chi $ is one of the exceptional characters $X_\lambda $ for some $ \lambda \in \Lambda $ .

We claim that

$$\begin{align*}\text{if } {\mathbf{lev}}(X_\lambda)\geq 1, \text{ then } {\mathbf{lev}}(X_\lambda)={\mathbf{lev}}(\lambda). \end{align*}$$

Let $\widetilde {D}$ be the unique subgroup of D of order p and set $\widetilde {N}:={\mathbf {N}}_G(\widetilde {D})$ . Let $\widetilde {B}=(b_0)^{\widetilde {N}}=(B_0)^{\widetilde {N}}$ , which is a block of $\widetilde {N}$ that has the same defect group D and Brauer correspondent $B_0$ as B. Dade proved in [Reference Dade7, Lemma 4.9] that the exceptional characters of $\widetilde {B}$ can be labeled by ${\mathrm {Irr}}_{ex}(\widetilde {B})=\{\widetilde {X}_\lambda \mid \lambda \in \Lambda \}$ so that the bijection $X_\lambda \mapsto \widetilde {X}_\lambda $ from ${\mathrm {Irr}}_{ex}(B)$ to ${\mathrm {Irr}}_{ex}(\widetilde {B})$ satisfies $(\widetilde {X}_{\lambda _1}-\widetilde {X}_{\lambda _2})^G=\delta (X_{\lambda _1}-X_{\lambda _2})$ for some fixed $\delta \in \{\pm 1\}$ and every $\lambda _1,\lambda _2\in \Lambda $ . As mentioned in the proof of [Reference Navarro26, Theorem 3.4], Dade’s bijection commutes with the action of $\mathcal {H}_B$ , and hence preserves the p-rationality level. This allows us, for the purpose of proving the claim, to assume that $\widetilde {D}\vartriangleleft G$ .

Let $\widetilde {C}:={\mathbf {C}}_G(\widetilde {D})$ and $\widetilde {b}:=(b_0)^{\widetilde {C}}$ . By [Reference Dade6, §4], the exceptional characters of B are induced from (nontrivial) ordinary irreducible characters of $\widetilde {b}$ . In fact, ${\mathrm {Irr}}(\widetilde {b})$ consists of $|D|$ characters $\{\chi _\lambda \mid \lambda \in {\mathrm {Irr}}(D)\}$ and $(\chi _{\lambda _1})^G=(\chi _{\lambda _2})^G$ if and only if $\lambda _1=\lambda _2^z$ for some $z\in E$ , so that

$$\begin{align*}{\mathrm {Irr}}_{ex}(B)=\{(\chi_\lambda)^G\mid \lambda\in\Lambda\} \text{ and } X_\lambda=(\chi_\lambda)^G.\end{align*}$$

It was shown in [Reference Navarro26, p. 1136], using the character-valued formula of $\chi _\lambda $ in [Reference Dade6, Lemma 3.2], that a Galois automorphism $\tau \in \mathcal {H}_B$ moves the character $\chi _\lambda $ in ${\mathrm {Irr}}(\widetilde {b})$ to the character in ${\mathrm {Irr}}((\widetilde {b})^\tau )$ labeled by $\chi _{\lambda ^\tau }$ . Recall the groups $\mathcal {I}:=\mathrm {Gal}({\mathbb Q}_{|G|}/{\mathbb Q}_{|G|_{p'}})\leq \mathcal {H}_B$ and the p-subgroup $\mathcal {I}'\leq \mathcal {I}$ from §2. Note that every relevant block is point-wisely fixed by $\mathcal {I}$ . It follows that, for every $\tau \in \mathcal {I}$ , $(\chi _\lambda )^\tau =\chi _{\lambda ^\tau }, $ which implies that

$$\begin{align*}{\mathbf{lev}}(\chi_\lambda)={\mathbf{lev}}(\lambda). \end{align*}$$

Further, recall that to show ${\mathbf {lev}}(X_\lambda )={\mathbf {lev}}(\lambda )$ , it suffices to show they are stable under the same elements of $\mathcal {I}'$ , assuming that ${\mathbf {lev}}(X_\lambda )\geq 1$ if $p\neq 2$ .

For each $\tau \in \mathcal {I}'$ , we have

$$\begin{align*}X_\lambda^\tau=\left((\chi_\lambda)^G\right)^\tau=\left((\chi_\lambda)^\tau\right)^G=(\chi_{\lambda^\tau})^G. \end{align*}$$

Therefore, $(\chi _\lambda )^G$ is $\tau $ -invariant if and only if $(\chi _\lambda )^G=(\chi _{\lambda ^\tau })^G$ , which is equivalent to $\lambda ^\tau =\lambda ^z$ for some $z\in E$ . We may assume that z has $p'$ -order, because $E/C$ has $p'$ -order and C fixes every irreducible character of D. Further, note that the actions of z and of $\tau $ commute. Let $t:=|\tau |$ be the order of $\tau $ , which is a p-power. Then, $\lambda =\lambda ^{\tau ^t}=\lambda ^{z^t}$ . Thus, $z^t\in C$ as $E/C$ acts Frobeniusly on ${\mathrm {Irr}}(D)$ . But $|z|$ and t are coprime, so $z\in C$ and $\lambda ^\tau =\lambda $ . We indeed have shown that, for every $\lambda \in \Lambda $ ,

$$\begin{align*}\text{if } p=2, \text{ then } {\mathbf{lev}}(X_\lambda)={\mathbf{lev}}(\lambda) \end{align*}$$

and

$$\begin{align*}\text{if } p>2 \text{ and } {\mathbf{lev}}(X_\lambda)\geq 1, \text{ then } {\mathbf{lev}}(X_\lambda)={\mathbf{lev}}(\lambda). \end{align*}$$

The proof of the claim is completed.

Note that the desired equality ${\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})={\mathbf {lev}}(\chi )$ is obvious when $\chi $ is p-rational. By the above claim, it suffices to prove the equality for those characters $X_\lambda $ with ${\mathbf {lev}}(X_\lambda )={\mathbf {lev}}(\lambda )\geq 1$ . For convenience, let $\ell :={\mathbf {lev}}(\lambda )$ . Since ${\mathbf {lev}}(X_\lambda )=\max \{{\mathbf {lev}}(X_\lambda (g)): g\in G\}$ , there exists some $g\in G$ such that

$$\begin{align*}{\mathbf{lev}}(X_\lambda(g))={\mathbf{lev}}(X_\lambda)={\mathbf{lev}}(\lambda)=\ell. \end{align*}$$

Our job now is to show that such an element g can be chosen to be inside ${\mathbf {N}}_G(D)$ . In fact, we will see that, up to conjugation, this must be the case.

Note that $X_\lambda $ takes value 0 on every element whose p-part is not conjugate to an element of D. For our purpose of analyzing the value $X_\lambda (g)$ , we may, therefore, assume that $g_p\in D$ . We next claim that $g_p\in D_0-D_1$ , so that $g_p$ generates D.

Assume, to the contrary, that $g_p\in D_1$ ; that is, $|g_p|\leq p^{a-1}$ . Then, there exists $1\leq i\leq a-1$ such that $|g_p|=p^{a-i}$ and $g_p\in D_i-D_{i+1}$ . Now $(g_p)^h\in D_i-D_{i+1}$ for every $h\in N_i$ . We have

$$\begin{align*}{\mathbf{lev}}(\lambda^h(g_p))={\mathbf{lev}}(\lambda((g_p)^{h^{-1}}))= \begin{cases} \ell-i &\text{ if } i\leq \ell,\\ 0 &\text{ if } i>\ell \end{cases}\end{align*}$$

for every $h\in N_i$ . In any case,

$$\begin{align*}{\mathbf{lev}}(\lambda^h(g_p))\leq \ell-1. \end{align*}$$

By the character-valued formula (3.1),

$$\begin{align*}X_\lambda(g)=\frac{\delta}{|C_i|} \sum_{h\in N_i} \lambda^h(g_p)(\varphi_i)^h(g_{p'})\end{align*}$$

for some $\delta \in \{\pm 1\}$ . Note that each value $(\varphi _i)^h(g_{p'})$ is p-rational. It follows that $X_{\lambda }(g)$ , being a sum of complex numbers of level at most $\ell -1$ , must have level at most $\ell -1$ , which is a contradiction.

We have shown that $g_p\in D_0-D_1$ . In other words, $g_p$ is a generator for D. Thus,

$$\begin{align*}g_{p'}\in {\mathbf{C}}_G(g_p)={\mathbf{C}}_G(D) \subseteq {\mathbf{N}}_G(D),\end{align*}$$

and it follows that

$$\begin{align*}g=g_pg_{p'}\in {\mathbf{N}}_G(D).\end{align*}$$

Let $\chi :=X_\lambda $ . We then have

$$\begin{align*}\ell={\mathbf{lev}}(\chi(g))\leq {\mathbf{lev}}(\chi_{{\mathbf{N}}_G(D)})\leq{\mathbf{lev}}(\chi)=\ell,\end{align*}$$

implying that ${\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})={\mathbf {lev}}(\chi )$ . The proof is complete.

Proof. If $|D|\leq p^2$ then either D is cyclic or $\exp (D)\leq p$ . The former case is solved by Theorem 3.4. In the latter case, ${\mathbf {lev}}(\chi )\leq 1$ by Lemma 3.1 and the result is trivial.

Proof. When S is either a sporadic simple group or $\textsf {A}_n$ with $5\leq n\leq 7$ , we have ${\mathbf {lev}}(\chi )\leq 1$ for all primes p and all $\chi \in \mathrm {Irr}(G)$ of height zero, which can be readily checked in [11]. If S is a group of Lie type with exceptional Schur multiplier or the Tits group ${}^2\operatorname {F}_4(2)'$ , we see using [11] that ${\mathbf {lev}}(\chi )\leq 1$ for all height-zero characters of G except when $p=2$ and $S={\mathrm {PSL}}_3(4)$ with $4\mid |\mathbf {Z}(G)|$ ; $S=\operatorname {B}_3(3)$ with $2\mid |\mathbf {Z}(G)|$ ; $S={\mathrm {PSU}}_4(3)$ with $4\mid |\mathbf {Z}(G)|$ ; or $S={}^2\operatorname {F}_4(2)'$ . In the latter cases, ${\mathbf {lev}}(\chi )\leq 2$ , and we in fact see using [11] that for every $\chi \in {\mathrm {Irr}}(G)$ and for every prime $p\mid |G|$ , we have ${\mathbf {lev}}(\chi )={\mathbf {lev}}(\chi _D)$ .

If $p=2$ and S is $\textsf {A}_n$ , then every irreducible character of S is p-rational (see [Reference Hung and Tiep16, §3] for instance), and we are done. Now consider the case $p=2$ and S is a simple group of Lie type defined in characteristic $2$ with nonexceptional Schur multiplier. Note that by [Reference Humphreys13], the blocks with positive defect are in fact of maximal defect. That is, the nontrivial defect groups are Sylow $2$ -subgroups in this case. Then, we have ${\mathbf {lev}}(\chi )={\mathbf {lev}}(\chi _D)$ by [Reference Navarro and Tiep28, Theorem A3].

Finally, suppose that p is odd and G is a cover of an alternating group $\textsf {A}_n$ with $n\geq 8$ or a quasisimple group of Lie type that is a quotient of ${\mathbf {G}}^F$ for some simple, simply connected algebraic group ${\mathbf {G}}$ over a field of characteristic p and a Steinberg endomorphism $F:{\mathbf {G}}\rightarrow {\mathbf {G}}$ . It was shown in [Reference Navarro and Tiep28, Theorem 6.1] that, in this situation,

$$\begin{align*}{\mathbb Q}(\chi)\subseteq {\mathbb Q}_{|G|_{p'}}(\sqrt{p}) \end{align*}$$

for every $\chi \in {\mathrm {Irr}}(G)$ . As ${\mathbb Q}_{|G|_{p'}}$ contains a primitive $4$ th root of unity and the conductor of $\sqrt {(-1)^{(p-1)/2}p}$ is p, it follows that $c(\chi )$ divides $p|G|_{p'}$ . Then, ${\mathbf {lev}}(\chi )\leq 1$ and the conjecture trivially holds in this case.

Proof of Theorem 5.1

By Theorem 5.2, we may assume that $S=M/\mathbf {Z}(M)$ is a simple group of Lie type defined in characteristic $q_0\neq p$ , and that S has non-exceptional Schur multiplier. Further, we assume that p is odd, since the statement follows from [Reference Navarro and Tiep28, Theorem A3] if $p=2$ .

Now, [Reference Hung, Tiep and Zalesski17, Theorem 4.2] gives a list of the possible $(S, r)$ in this case, where $r=\chi (1)$ is the prime for which M has an irreducible character of degree r. By our assumptions, we are not in the cases listed in (i) or (vi) of [Reference Hung, Tiep and Zalesski17, Theorem 4.2]. Then, S is one of:

$$\begin{align*}{\mathrm {PSL}}_2(q), {\mathrm {PSU}}_n(q) (n\geq 3), {\mathrm {PSL}}_n(q) (n\geq 3), \text{ or } {\mathrm {PSp}}_{2n}(q),\end{align*}$$

with specifications on $n, q, r$ , and $\chi $ in each case. Here, we have $S=G/\mathbf {Z}(G)$ and $M=G/Z$ for some $Z\leq \mathbf {Z}(G)$ and $G:={\mathbf {G}}^F$ , where ${\mathbf {G}}$ is a simple, simply connected algebraic group and $F\colon {\mathbf {G}}\rightarrow {\mathbf {G}}$ is a Frobenius endomorphism defining ${\mathbf {G}}$ over ${\mathbb F}_q$ , where q is some power of $q_0$ . We discuss each case separately.

(I) First, suppose we are in case (ii) of [Reference Hung, Tiep and Zalesski17, Theorem 4.2], so $S={\mathrm {PSL}}_2(q)$ . Then, either $r=q=q_0$ and $\chi $ is the Steinberg character, which is rational; or q is odd and $\chi (1)=r=\frac {q-\epsilon }{2}$ for some $\epsilon \in \{\pm 1\}$ ; or q is a power of $2$ and $r=q+\epsilon $ for some $\epsilon \in \{\pm 1\}$ is a Mersenne prime or Fermat prime. In the case q is odd, we have $c(\chi )\in \{q_0, 1\}$ as in [Reference Hung, Tiep and Zalesski17, §5.2.2], so ${\mathbf {lev}}(\chi )=0$ .

So, assume q is a power of $2$ . Here, we have ${\mathbb Q}(\chi )\subseteq {\mathbb Q}_{q-\epsilon }$ , so we may assume $p\mid (q-\epsilon )$ . In this case, $\chi $ is the restriction of a semisimple character $\widetilde \chi $ of $\widetilde {G}={\mathrm {GL}}_2(q)$ indexed by a semisimple element with eigenvalues $\{\zeta ^i, \zeta ^{-i}\}$ , where $\zeta $ is a primitive $(q-\epsilon )$ root of unity and $1\leq i<q-\epsilon $ . Letting $\alpha $ be a generator for ${\mathrm {Irr}}(C_{q-\epsilon })$ , we have ${\mathbb Q}(\chi )\subseteq {\mathbb Q}(\alpha ^i)$ , so that ${\mathbf {lev}}(\chi )\leq {\mathbf {lev}}(\alpha ^i)$ . Note that by [Reference Navarro and Tiep28, Lemma 4.1], ${\mathbf {lev}}(\chi )$ is the smallest positive integer e such that $\chi ^{\sigma _e}=\chi $ , since p is odd. (Recall from §2 that $\sigma _e\in \mathcal {I}'$ is the element mapping any p-power root of unity $\omega $ to $\omega ^{1+p^e}$ .)

Now, the value of $\chi $ on a semisimple element $g_j$ of $S=G={\mathrm {SL}}_2(q)$ with eigenvalues $\{\zeta ^j, \zeta ^{-j}\}$ is $\zeta ^{ij}+\zeta ^{-ij}$ . In particular, taking $m:=(q-\epsilon )_{p'}$ and $h:=g_m$ , we have

$$\begin{align*}\chi(h)=\zeta^{im}+\zeta^{-im},\end{align*}$$

which is stable under $\sigma _e$ if and only if $\zeta ^{im}$ is, since p is odd. (This is worked out, for example, as in [Reference Peña, Pryor and Fry29, Lemma 2.1].) But this happens if and only if $\alpha ^{im}$ , and hence $\alpha ^i$ , is stable under $\sigma _e$ . It follows that

$$\begin{align*}{\mathbf{lev}}(\alpha^i)={\mathbf{lev}}(\chi(h))\leq{\mathbf{lev}}(\chi).\end{align*}$$

This establishes that

$$\begin{align*}{\mathbf{lev}}(\alpha^i)= {\mathbf{lev}}(\chi)={\mathbf{lev}}(\chi|_P),\end{align*}$$

where P is a Sylow p-subgroup of S containing h.

(II) Next, suppose we are in case (iii) of [Reference Hung, Tiep and Zalesski17, Theorem 4.2], so that $S={\mathrm {PSL}}_n(q)$ with n an odd prime, $q=q_0^f\geq 3$ with $q_0$ a prime and f odd, and $r=(q^n-1)/(q-1)$ with $(n,q-1)=1$ . Note that this means $S=M=G$ . Here, as in [Reference Hung, Tiep and Zalesski17, §5.2.1], we have $\chi $ is one of the $q-2$ irreducible Weil characters of degree $r=(q^n-1)/(q-1)$ . Note that $\chi $ extends to an irreducible Weil character $\widetilde \chi $ of $\widetilde {G}:={\mathrm {GL}}_n(q)$ . Further, $\chi $ is determined by the irreducible constituent of $\widetilde \chi |_{\mathbf {Z}(\widetilde {G})}$ .

Let $\alpha $ be a generator for ${\mathrm {Irr}}(\mathbf {Z}(\widetilde {G}))\cong C_{q-1}$ . Then, for $i=1,\ldots , q-2$ , let $\widetilde \chi _i$ be an irreducible Weil character of $\widetilde {G}$ such that $\widetilde {\chi }_i|_{\mathbf {Z}(\widetilde {G})}=\widetilde \chi _i(1)\alpha ^i$ . Then, any such choice of $\widetilde \chi _i$ has the same restriction to G, and we let $\chi _i:=\widetilde {\chi }_i|_{G}$ be that restriction. Let $\widetilde \chi :=\widetilde \chi _i$ and $\chi :=\chi _i$ . As discussed in [Reference Hung, Tiep and Zalesski17, §5.2.1], we have ${\mathbb Q}(\widetilde \chi _i)\subseteq {\mathbb Q}(\alpha ^i)$ , so that

$$\begin{align*}{\mathbb Q}(\chi)\subseteq {\mathbb Q}(\widetilde\chi)\subseteq {\mathbb Q}(\alpha^i)= {\mathbb Q}(\zeta^i)\subseteq {\mathbb Q}(\zeta),\end{align*}$$

where $\zeta $ is a primitive $(q-1)$ -root of unity in $\overline {{\mathbb Q}}^\times $ . From this, we may assume $p\mid (q-1)$ , as otherwise ${\mathbf {lev}}(\chi )=0$ .

Now, following [Reference Guralnick and Tiep12, p. 125], we see $\widetilde \chi ={\operatorname {R}}_{L}^{\widetilde {G}}(\lambda )$ , where L is a Levi subgroup of the form ${\mathrm {GL}}_1(q)\times {\mathrm {GL}}_{n-1}(q)$ of $\widetilde {G}$ , $\lambda \in \mathrm {Irr}(L)$ is the character of a module of the form

$$\begin{align*}S(s,(1))\otimes S(t, (n-1))\end{align*}$$

in the notation of [Reference Guralnick and Tiep12, p. 125] with $s\neq t\in {\mathbb F}_q^\times $ and $s/t=\alpha ^i$ , and ${\operatorname {R}}_{L}^{\widetilde {G}}$ denotes Harish–Chandra induction. Here, by an abuse of notation, we also denote by $\alpha $ a generator of ${\mathbb F}_q^\times \cong C_{q-1}$ . Since multiplying by a linear character of $\widetilde {G}$ does not affect $\chi _i$ , we may further assume that $t=1$ , so

$$\begin{align*}\lambda=S(\alpha^i,(1))\otimes S(1, (n-1)).\end{align*}$$

(Note that for $\alpha ^j\in {\mathbb F}_q^\times $ , the module $S(\alpha ^j, (k))$ affords the inflation of the linear character ${\mathrm {GL}}_k(q)/{\mathrm {SL}}_k(q)\cong {\mathbb F}_q^\times \rightarrow \overline {{\mathbb Q}}^\times $ defined by $\alpha \mapsto \zeta ^j$ .)

Now, let Q be a parabolic subgroup of $\widetilde {G}$ such that $L\leq Q$ is the Levi complement in Q, and let $\hat \lambda $ be the inflation of $\lambda $ to Q. Then,

$$\begin{align*}\chi={\mathrm {Res}}^{\widetilde{G}}_G{\mathrm {Ind}}_Q^{\widetilde{G}}(\hat\lambda)= {\mathrm {Ind}}_{Q\cap G}^G{\mathrm {Res}}_{Q\cap G}^Q(\hat\lambda)\end{align*}$$

since $\widetilde {G}=GQ$ . Note that $r=\chi (1)=[G:Q\cap G]$ . Then, by the first half of the proof of Theorem 4.4, we have ${\mathbf {lev}}(\chi )={\mathbf {lev}}(\chi _P)$ , as desired.

(III) Now consider case (iv) of [Reference Hung, Tiep and Zalesski17, Theorem 4.2]. Here, $S={\mathrm {PSU}}_n(q)$ with n an odd prime, $r=(q^n+1)/(q+1)$ , and $(n,q+1)=1$ . Again, this means $S=M=G$ and $\chi $ is one of the q irreducible Weil characters of degree $r=(q^n+1)/(q+1)$ (see [Reference Hung, Tiep and Zalesski17, §5.3]). Again, $\chi $ extends to an irreducible Weil character $\widetilde \chi $ of $\widetilde {G}:={\mathrm {GU}}_n(q)$ and $\chi $ is determined by its values on $\mathbf {Z}(\widetilde {G})$ . Letting $\alpha $ be a generator for ${\mathrm {Irr}}(\mathbf {Z}(\widetilde {G}))\cong C_{q+1}$ , we have Weil characters $\chi _i=\widetilde \chi _i|_{G}$ , where $\widetilde \chi _i|_{\mathbf { Z}(\widetilde {G})}=\widetilde \chi _i(1)\alpha ^i$ as in (II), now for $i=1,\ldots , q$ . Here, we similarly have ${\mathbb Q}(\chi _i)\subseteq {\mathbb Q}(\alpha ^i)\subseteq {\mathbb Q}(\xi )$ , where $\xi $ is a primitive $(q+1)$ -root of unity in $\overline {{\mathbb Q}}^\times $ . So, we assume $p\mid (q+1)$ . Further, this establishes that

$$\begin{align*}{\mathbf{lev}}(\chi_i)\leq {\mathbf{lev}}(\alpha^i).\end{align*}$$

In this case, we use the explicit formula from [Reference Tiep and Zalesski33, Theorem 4.1] for the values of $\chi _i$ . Namely, for $g\in \widetilde {G}$ , we have

$$\begin{align*}\chi_i(g)=\frac{(-1)^n}{q+1}\sum_{k=0}^q\xi^{-ik}(-q)^{\dim\mathrm{Ker}(g-\hat\xi^{-k})},\end{align*}$$

where $\hat \xi $ denotes a generator of the subgroup $C_{q+1}\leq {\mathbb F}_{q^2}^\times $ .

Now, let $m:=(q+1)_{p'}$ and let $\beta :=\hat \xi ^{m}$ be a generator of the Sylow p-subgroup of $C_{q+1}\leq {\mathbb F}_{q^2}^\times $ . Let h be a semisimple p-element of G with eigenvalues $\{\beta , \beta ^{-1}, 1,\ldots ,1\}$ . Then, since n and p are odd, we have:

$$\begin{align*}\chi_i(h)&=\frac{-1}{q+1}\left((\xi^{im}+\xi^{-im})(-q)+(-q)^{n-2}- \xi^{im}-\xi^{-im}+\sum_{k=1}^q \xi^{-ik} \right)\\&=(\xi^{im}+\xi^{-im})+\frac{1}{q+1}\left(q^{n-2}-\sum_{k=1}^q \xi^{-ik} \right)=(\xi^{im}+\xi^{-im})+\frac{q^{n-2}+1}{q+1}.\end{align*}$$

Recall again that by [Reference Navarro and Tiep28, Lemma 4.1], ${\mathbf {lev}}(\chi )$ is the smallest positive integer e such that $\chi ^{\sigma _e}=\chi $ , since p is odd. From above, we see $\chi _i(h)$ is fixed by $\sigma _e$ if and only if $(\xi ^{im}+\xi ^{-im})$ is fixed by $\sigma _e$ . From here, we conclude similar to (I). Namely, $\chi _i(h)$ is then stable under $\sigma _e$ if and only if $\xi ^{im}$ is stable under $\sigma _e$ , if and only if the character $\alpha ^{im}$ is stable under $\sigma _e$ , so also $\alpha ^i$ is. So we have ${\mathbf {lev}}(\chi _i(h))={\mathbf {lev}}(\alpha ^i)$ . Then,

$$\begin{align*}{\mathbf{lev}}(\chi_i)\geq {\mathbf{lev}}(\alpha^i),\end{align*}$$

forcing that

$$\begin{align*}{\mathbf{lev}}(\alpha^i)= {\mathbf{lev}}(\chi_i)={\mathbf{lev}}(\chi_i|_P),\end{align*}$$

where P is a Sylow p-subgroup of S containing h.

(IV) Finally, assume we are in case (v) of [Reference Hung, Tiep and Zalesski17, Theorem 4.2]. Then, $S={\mathrm {PSp}}_{2n}(q)$ and either

1. (a) $r=(q^n+1)/2$ with $n=2^a\geq 2$ and $q=q_0^{2^k}$ with $q_0$ odd and $k\geq 0$ ; or

2. (b) $r=(3^n-1)/2$ , where n is an odd prime and $q=3$ .

In case (a), we have as in [Reference Hung, Tiep and Zalesski17, §5.4.1] that either q is a square and hence $\chi $ is rational-valued, or $k=0$ so $q=q_0$ and ${\mathbb Q}(\chi )\subseteq {\mathbb Q}(\zeta _{q})$ , where $\zeta _{q}$ is a primitive qth root of unity in $\overline {{\mathbb Q}}^\times $ . Then, $\chi $ is almost p-rational (hence has ${\mathbf {lev}}(\chi )\leq 1$ ) if $p=q_0$ and is p-rational (so ${\mathbf {lev}}(\chi )=0$ ) if $p\neq q_0$ . In case (b), we similarly have ${\mathbb Q}(\chi )\subseteq {\mathbb Q}(\zeta _3)$ with $\zeta _{3}$ a primitive $3$ rd root of unity. Again, ${\mathbf {lev}}(\chi )\leq 1$ if $p=3$ and ${\mathbf {lev}}(\chi )=0$ if $p\neq 3$ .

Proof. By Theorems 5.2 and 3.4, we may assume that p is not the defining characteristic for G and that the Sylow p-subgroups of G are non-cyclic. Then, we are left to consider the case that $G={}^2\operatorname {F}_4(q^2)$ with $q^2=2^{2n+1}$ and p is an odd prime dividing $ (q^2-1)$ , $(q^2+1)$ , or $(q^4+1)$ .

If $p\nmid (q^2-1)$ [Reference Malle22, Bemerkung 1] yields that each $\mathcal {E}_p(G, s)$ for $s\in G^{\ast }$ a semisimple $p'$ -element contains a unique block of positive defect, which, therefore, has as defect groups a Sylow p-subgroup of $\mathbf {C}_{G^{\ast }}(s)$ , using [Reference Kessar and Malle21, Lemma 2.6]. If instead $p\mid (q^2-1)$ , we have by [Reference Malle22, Bemerkung 1] that each $\mathcal {E}_p(G, s)$ has one or three blocks of positive defect, but only one of these is noncyclic. In either case, a block B with non-cyclic defect groups has maximal defect for $p\neq 3$ , completing the proof by our assumption that Conjecture B holds.

Proof. Let q be a power of a prime $q_0$ . In this case, the Sylow p-subgroups of M are cyclic unless $p\in \{2,q_0\}$ . So, we may assume by Theorems 3.4 and 5.2 that $p=2$ and by [Reference Navarro and Tiep28, Theorem A3] we need only consider blocks of M with non-maximal defect. We have $M\in \{G, S\}$ , where $G={\mathrm {SL}}_2(q)$ and $S={\mathrm {PSL}}_2(q)$ . Let $\widetilde {G}={\mathrm {GL}}_2(q)$ and let $\bar B$ be a block of M with positive, non-maximal defect dominated by a block B of G. Let $\widetilde {B}$ be a block of $\widetilde {G}$ covering B. Then, as discussed above, there is some odd-order, semisimple element $\widetilde s$ of $\widetilde {G}^{\ast }\cong \widetilde G$ such that $\widetilde {B}=\mathcal {E}_2(\widetilde {G}, \widetilde s)$ and a Sylow $2$ -subgroup of $\mathbf {C}_{\widetilde {G}^{\ast }}(\widetilde s)$ gives a defect group for $\widetilde {B}$ . Then, with our assumption that $\bar B$ is not of maximal defect, we see that any such defect group has cyclic intersection with G, so that B (hence $\bar B$ ) has cyclic defect groups. Then, we again apply Theorem 3.4.

Proof. Let D be a defect group for $\widetilde {B}$ , such that $D=\prod D_i$ with $D_i\in \mathrm {Syl}_2({\mathrm {GL}}_{m_i}(q^{d_i}))$ . Note that in this case, $q^{d_i}\equiv -1\pmod 4$ for each factor ${\mathrm {GL}}_{m_i}(q^{d_i})$ of $C:=\mathbf { C}_{\widetilde {G}^{\ast }}(\widetilde s)$ .

We claim that in this situation, each $D_i/D_i'$ has exponent at most $2$ , and hence so does $D/D'$ . We can see this from the description of Sylow $2$ -subgroups of ${\mathrm {GL}}_{m}(q^d)$ in [Reference Carter and Fong5]. Indeed, by the description in [Reference Carter and Fong5], such a group is a direct product of Sylow $2$ subgroups of ${\mathrm {GL}}_{2^j}(q)$ for various powers $2^j$ of $2$ . Hence, it suffices to prove the claim for m a power of $2$ . A Sylow $2$ -subgroup $P_{2^j}$ of ${\mathrm {GL}}_{2^j}(q^d)$ is an iterated wreath product $P_2\wr C_2\wr C_2\cdots \wr C_2$ , where $P_2\in {\mathrm {Syl}}_2({\mathrm {GL}}_2(q^d))$ . Since the latter is semidihedral, we know $\exp (P_2/P_2')\leq 2$ . For $j\geq 2$ , we have $P_{2^j}=P_{2^{j-1}}\wr C_2$ . Then, $P_{2^j}/P_{2^{j}}'\cong P_{2^{j-1}}^2/\langle (P_{2^{j-1}}')^2, [P_{2^{j-1}}^2, C_2]\rangle \times C_2$ , and we can see inductively that $\exp (P_{2^j}/P_{2^j}')\leq 2$ . This proves our claim. (See also [Reference Isaacs, Liebeck, Navarro and Tiep19, Proposition 4.3]).

Hence, by [Reference Isaacs, Liebeck, Navarro and Tiep19, Theorem D], each odd-degree character in the principal block $B_0(C)$ of C is $2$ -rational. But then the same is true for the height-zero characters of $\widetilde {B}$ by the main Theorem of [Reference Srinivasan and Vinroot32], since these correspond to the height-zero characters in $B_0(C)$ by Jordan decomposition using [Reference Fong and Srinivasan9, Theorem (7A)]. (See also [Reference Broué3], which shows the results of [Reference Fong and Srinivasan9] continue to hold for $p=2$ .)

Proof. We keep the above notation. First, the AMN conjecture implies that there exists an $\mathcal {H}_B$ -equivariant bijection $^{*}: {\mathrm {Irr}}_0(B) \rightarrow {\mathrm {Irr}}_0(b)$ such that ${\mathbf {lev}}(\chi )={\mathbf {lev}}(\chi ^{\ast })$ . Conjecture A then implies that ${\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})={\mathbf {lev}}(\chi ^{\ast })$ for all $\chi \in {\mathrm {Irr}}_0(B)$ of level at least 2.

For the converse statement, let $\chi $ be a height-zero character in a block B and assume that there exists a bijection $^{*}: {\mathrm {Irr}}_0(B) \rightarrow {\mathrm {Irr}}_0(b)$ that commutes with the action of $\mathcal {H}_B$ such that ${\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})={\mathbf {lev}}(\chi ^{\ast })$ . As mentioned, we then have ${\mathbf {lev}}(\chi )={\mathbf {lev}}(\chi ^{\ast })$ , and it follows that ${\mathbf {lev}}(\chi )={\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})$ , as wanted.

Consequence 7.4. Let $\chi $ be a height-zero character of a finite group G and D a defect group of the p-block of G containing $\chi $ . Assume Conjecture A holds. Then, $\chi $ is almost p-rational if and only if $\chi _{{\mathbf {N}}_G(D)}$ is almost p-rational. In particular, when $p=2$ , $\chi $ is $2$ -rational if and only if $\chi _{{\mathbf {N}}_G(D)}$ is $2$ -rational.

Proof. It is clear that if $\chi $ is almost p-rational then so is $\chi _{{\mathbf {N}}_G(D)}$ . Conversely, if $\chi $ is not almost p-rational then ${\mathbf {lev}}(\chi )\geq 2$ , and it follows from Conjecture A that ${\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})\geq 2$ .

Consequence 7.5. Let G be a finite group with abelian Sylow p-subgroups. Assume that Conjecture A holds. Then, ${\mathbf {lev}}(\chi )={\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})$ for every $\chi \in {\mathrm {Irr}}(G)$ of p-rationality level at least $2$ and D a defect group of the p-block of G containing $\chi $ .

Proof. This follows from the solution of the “if” implication of Brauer’s height zero conjecture [Reference Kessar and Malle21].

Consequence 7.6. Let $\chi $ be an irreducible $2$ -height zero character of a finite group G. Suppose that ${\mathbb Q}(\chi )={\mathbb Q}(\sqrt {d})$ is a quadratic number field, where $d\not \equiv 1 (\bmod \, 4)$ is a square-free integer. Assume that Conjecture A holds. Then, ${\mathbb Q}(\chi )={\mathbb Q}(\chi _{{\mathbf {N}}_G(D)})$ , where D a defect group of the p-block of G containing $\chi $ .

Proof. Note that $c(\sqrt {d})=4|d|$ when $d\not \equiv 1 (\bmod 4)$ is a square-free integer. Therefore, by the hypothesis, ${\mathbf {lev}}(\chi )\geq 2$ . By Conjecture A, we then have ${\mathbf {lev}}(\chi _{{\mathbf {N}}_G(D)})\geq 2$ . In particular, $\chi _{{\mathbf {N}}_G(D)}$ is not rational, implying that ${\mathbb Q}(\chi )={\mathbb Q}(\chi _{{\mathbf {N}}_G(D)})$ .