Blocks with Equal Height Zero Degrees

We study blocks all of whose height zero ordinary characters have the same degree. We suspect that these might be the Broue-Puig nilpotent blocks.


Introduction
The celebrated nilpotent blocks of finite groups introduced by M. Broué and L. Puig in 1980 ([8]) are locally defined in terms of the Alperin-Broué subpairs ( [1]). There is a general consensus that nilpotent blocks are the most natural blocks from the local point of view. It is not easy, however, to check if a block is nilpotent or not, and to have a global characterization of them, especially one that can be detected in the character table of the group, would be quite interesting.
Here we propose to study blocks B of a finite group G such that all of its height zero characters χ ∈ Irr 0 (B) have the same degree d. This property of blocks, that can easily be detected in the character table of G, seem to appear quite naturally in block theory, and deserves some consideration. The blocks all of whose irreducible characters have the same degree were already considered by T. Okuyama and Y. Tsushima in [35].
In a nilpotent block B all height zero degrees are equal. And we suspect that the converse might be true. In this paper, we are able to prove this in some cases, with quite different arguments.
If B is the principal block of G, or if the defect group D of B is normal in G, or if D is abelian (and we assume the Height Zero Conjecture) then the blocks with equal height zero character degrees are nilpotent. These results constitute Sections 3, 4, and 5 below.
The most difficult result in this paper, to which a large extent of it is devoted, is to prove that the blocks of simple groups with equal height zero degrees have abelian defect groups and satisfy Brauer's Height Zero Conjecture. By our previously mentioned result, this implies that equal height zero degrees blocks are also nilpotent. This certainly agrees with the recent work of J. An and C. Eaton in which they prove that nilpotent blocks of simple groups have abelian defect groups for p > 2 [2].
The study of blocks of p-solvable groups with equal height zero degrees, which we do in the last section of the paper, leads to a variation of a classical large orbit question which Date: September 23, 2009. The first author thanks the Isaac Newton Institute for Mathematical Sciences, Cambridge, for its hospitality during the preparation of part of this work.
does not seem easy to solve and which has interest in its own. (Some recent partial results are given in [14].) This new type of orbit problem has connections with delicate questions on the p ′ -character degrees of finite groups.
Finally, let us mention that the blocks B such that all character degrees χ(1) are ppowers for χ ∈ Irr(B) give another example of blocks with equal height zero characters degrees. These blocks were proved to be nilpotent by work of G. R. Robinson and the second author ( [32]).

EHZD blocks and Nilpotent Blocks
Suppose that G is a finite group, p is a prime, and B is a p-block of G. In general, we use the notation in [29]. Hence Irr(B) are the irreducible complex characters in B, IBr(B) are the irreducible Brauer characters in B, and Irr 0 (B) are the height zero characters of B.
For the sake of brevity, let us say that B is EHZD (equal height zero degrees) if there is an integer d such that χ(1) = d for all χ ∈ Irr 0 (B).
Recall that a block B is nilpotent if whenever (Q, b Q ) is a B-subpair (that is, b Q is a block of QC G (Q) such that (b Q ) G = B), then N G (Q, b Q )/C G (Q) is a p-group.
If B is nilpotent, then we know that IBr(B) = {ϕ} by Theorem (1.2) of [8]. Also, if χ ∈ Irr(B) has height zero, then by (3.11) in page 126 of [8], we have that χ(1) = ϕ (1). It then follows that all irreducible height zero characters in B have the same degree. Thus, as we mentioned in the introduction, nilpotent blocks are EHZD blocks. (We also notice here that in a nilpotent block all height zero characters are modularly irreducible. This condition, if not equivalent, seems also closely related to nilpotency as we shall point out in several places of this paper.)

Principal blocks
If B is the principal block of G, then (Q, b Q ) is B-subpair if and only if b Q is the principal block of N G (Q) (by the Third Main Theorem). Since the principal block b Q is N G (Q)-invariant, we conclude that B is nilpotent if and only if N G (Q)/C G (Q) is a p-group for every p-subgroup Q of G. Hence B is nilpotent if and only if G has a normal p-complement, by a classical theorem of Frobenius.
Theorem 3.1. Let G be a finite group, let p be a prime and let B be the principal block of G. Then the following conditions are equivalent: (a) All height zero χ ∈ Irr(B) have the same degree.
(c) B is a nilpotent block.
Proof. In Section 2 we have pointed out that (c) implies (a) and (b). Suppose now that all height zero characters in B have the same degree. Hence all non-linear characters in B have degree divisible by p. Then G has a normal p-complement by Corollary 3 of [23] and so B is nilpotent. Now, suppose that all the height zero (that is, p ′ -degree) characters in B lift an irreducible Brauer character of G. We are going to use a theorem of Pahlings that asserts that ϕ ∈ IBr(G) is linear and all nonlinear characters χ ∈ Irr(G) with decomposition number 0 = d χϕ have degrees divisible by p, then G has a normal p-complement. (See Theorem 2 of [33].) Write ϕ = 1 G ∈ IBr(G) for the trivial Brauer character of G, and suppose that χ is non-linear with d χϕ = 0. If χ has p ′ -degree, then by hypothesis, χ 0 = 1 G and therefore χ is linear. This is not possible. Hence, we conclude that p divides χ(1). It follows that G has a normal p-complement by Pahling's theorem.

Abelian Defect Groups
In this section we prove that EHZD blocks with abelian defect groups are exactly the nilpotent blocks (assuming Brauer's Height Zero Conjecture).
Theorem 4.1. Let B be a block with an abelian defect group, and assume that Irr(B) = Irr 0 (B). Then the following conditions are equivalent: (a) All height zero χ ∈ Irr(B) have the same degree.  [8] it is stated that blocks with abelian defect group and inertial index one are nilpotent. Now, suppose that B is nilpotent with defect group D.
is also a p-group, and we conclude that . That is, B has inertial index one.

Normal Defect Groups
In this Section we prove that EHZD blocks with a normal defect group are nilpotent.
The following should be well-known.
Lemma 5.1. Let B be a block with defect group D ⊳ G and let b D be a block of DC G (D) covered by B. LetB be the Fong-Reynolds correspondent of B over b D . IfB is nilpotent, then B is nilpotent.
For this we may replace (Q, b Q ) by any G-conjugate. Since (b Q ) G = B, we have that Q ⊆ D (Theorem (4.14) of [29]). Now let e = (b Q ) DC G (Q) . We have that e G = B by the transitivity of induction. Now, if f is a block of DC G (D) covered by e, we have that e = f DC G (Q) by Corollary (9.21) of [29]. Hence e G = B and f Hence e covers (b D ) y , and therefore (b D ) yz = b D for some z ∈ C G (Q). Since T = DC G (D), we see that yz ∈ DC G (D) and therefore y ∈ DC G (Q). Thus N G (Q, b Q )/C G (Q) is a p-group and B is nilpotent. Proof. We already know that (c) implies (a) and (b). We prove by induction on |G| that (a) implies (c). In an analog way, we could prove that (b) implies (c). Let b D be a block of DC G (D) inducing B with defect group D. Let T be the stabilizer in G of b D , and letB be the Fong-Reynolds correspondent of B over b D . If all height zero χ ∈ Irr(B) have the same degree, then the same happens inB by the Fong-Reynolds correspondence [29,Theorem (9.14)]. By Lemma (5.1), we may assume that T = G. Now by Reynolds

Quasi-Simple Groups
From now until the last section of the paper, we are devoted to proving the following result.
Theorem 6.1. Let S be a finite non-abelian simple group, G a quasi-simple group with G/Z(G) ∼ = S, and p a prime. Assume that B is a p-block of G such that all characters in Irr 0 (B) have the same degree. Then the defect group of B is abelian and thus B is nilpotent, unless possibly one of the following holds: (1) B is a faithful block for the 2-fold covering group 2.A n of the alternating group A n (n ≥ 14) (a so-called spin-block), or (2) B is a quasi-isolated block for an exceptional group of Lie type and p is a bad prime.
Theorem 5.2 shows that in order to check Theorem 6.1 it suffices to prove the following for any block B of G all of whose characters in Irr 0 (B) have the same degree: (1) the defect group of B is abelian, and (2) B satisfies the Height Zero Conjecture.
In order to do this, we invoke the classification of finite simple groups as well as the Deligne-Lusztig theory of characters of finite reductive groups and the fundamental results of Bonnafé-Rouquier and Cabanes-Enguehard on blocks. It will be given in several steps in the subsequent sections. The proof in the case of alternating groups will lead to a relative hook formula for the character degrees in p-blocks of the symmetric group.

Unipotent blocks
In this section we consider the unipotent blocks of finite groups of Lie type. We introduce the following standard setup: Any non-exceptional Schur covering group of a finite simple group of Lie type can be obtained as G := G F , where G is a simple algebraic group of simply-connected type over the algebraic closure of a finite field, and F : G → G a Frobenius map with finite group of fixed points G F , with the sole exception of the Tits group 2 F 4 (2) ′ , which will be treated later in Proposition 9.4. Let G * denote a group in duality with G and with corresponding Frobenius map F * : G * → G * and fixed points G * := G * F * . Let r denote the defining characteristic of G and q the absolute value of all eigenvalues of Frobenius on the character lattice of an F -stable torus of G, a half-integral power of r. We will then also write G = G(q) in order to indicate the corresponding value of q.
By the fundamental work of Lusztig, the irreducible characters of G can be partitioned into so called Lusztig series indexed by conjugacy classes of semisimple elements s in G * . The characters in the Lusztig series E(G) := E(G, 1) corresponding to the trivial element in G * are the so-called unipotent characters. These can be viewed as being the building blocks of the ordinary character theory of finite groups of Lie type. Again by results of Lusztig, the unipotent characters can be parametrized by a set depending only on the type of G, that is, on the Weyl group of G together with the action of F on it, not on q or r. Moreover, their degrees are given by the value at q of polynomials in one indeterminate of the form where n is either a power of 2 or a divisor of 120, and Φ i (x) denotes the ith cyclotomic polynomial over Q (see for example Chapter 13 of [10]). We write Deg(γ) ∈ Q[x] for the degree polynomial of a unipotent character γ ∈ E(G).

7.1.
Specializations of degree polynomials. We start by investigating specializations of degree polynomials of unipotent characters. We first discuss the question when two different degree polynomials f 1 , f 2 can lead to the same character degree f 1 (q) = f 2 (q).
be the degree polynomials of two unipotent characters of an exceptional group of Lie type G = G(q). If f 1 (q) = f 2 (q) for some prime power q > 1, respectively square root of some odd power of a prime for the Suzuki or Ree groups, Proof.
According to [10,Chap. 13], n j |120 for G = E 8 , and n j |24 else. Now f 1 (q) = f 2 (q) implies that q a 1 −a 2 m j i=1 Φ i (q) a i,1 −a i,2 ∈ Q has numerator and denominator a divisor of 120. Since the second factor is coprime to q, this holds in fact for both factors. Then Lemma 7.1(b) shows that q ≤ 121 for G = E 8 , and q ≤ 25 for the other types. For these finitely many values of q and finitely many types the assertion can be checked from the tables of degree polynomials. In fact, the additional restrictions in Lemma 7.1(b) allow to restrict the number of necessary computations even further.
Note that none of the exceptions is a perfect group. Unfortunately, there are infinitely many exceptions to the conclusion of the previous proposition in the case of classical groups, so we will choose a different approach for those. 7.2. e-symbols and degrees. We need to give a brief recall of the notion of e-symbols and associated degree, see [26].
Let e ≥ 1 be an integer. An e-symbol is a sequence S = (S 1 , . . . , S e ) of e strictly increasing sequences S i = (s i1 < . . . < s im ) of non-negative integers of equal length m.
The rank of an e-symbol S is defined as We define an equivalence relation on e-symbols as the reflexive, symmetric and transitive closure of the relation ∼ given by . . , s im + 1). There is a natural 1-1 correspondence between e-tuples of partitions π = (π 1 , . . . , π e ) ⊢ r of r and equivalence classes of e-symbols of rank r, as follows: by adding zeros we may assume that all π i = (π i1 ≤ . . . ≤ π im ) have the same number of parts. It is easily verified that S(π) = (S 1 , . . . , S e ), with S i := (π i1 , π i2 + 1, . . . , π im + m − 1) for 1 ≤ i ≤ e, has indeed rank r, and is well-defined up to equivalence.
Let (v; u 1 , . . . , u e ) be indeterminates over Q. For an e-symbol S we define [26, (5.12)]). It can be checked that the rational function f S only depends on the equivalence class of the e-symbol S. We shall also write f π for f S with S = S(π).
The following connection to the imprimitive complex reflection group G(e, 1, r) ∼ = C e ≀S r will be important for us. The irreducible complex characters of the wreath product G(e, 1, r) can be parametrized by e-tuples of partitions (π 1 , . . . , π e ) of r (see for example [26, (2A)]), hence by equivalence classes of e-symbols of rank r. Now let H = H(W, u) denote the cyclotomic Hecke algebra for W = G(e, 1, r) with parameters u = (v; u 1 , . . . , u e ). This carries a canonical symmetrizing form. By the main result of Geck-Iancu-Malle [18] the Schur element (with respect to this form) of the irreducible character of H indexed by the multipartition (π 1 , . . . , π e ) ⊢ r is f −1 S , where S = S(π 1 , . . . , π e ) (see Conjecture 2.20 in [26]).
In particular, specializing v to 1 and u j to the eth roots of unity ζ j := exp(2πij/e) we obtain where d S denotes the degree of the irreducible character of G(e, 1, r) indexed by S. For later use let's record the following special cases. If r = 1, so G(e, 1, r) is the cyclic group C e , then a multipartition π = (π 1 , . . . , π e ) ⊢ r is uniquely determined by the unique i such that π i = (1). The corresponding e-symbol S has S i = (1), S j = (0) for j = i, and More generally the e-symbols with S i = (r), S j = (0) for j = i parametrize linear characters ϕ i of G(e, 1, r), for 1 ≤ i ≤ e. Evaluation of the defining formula shows that then 3. d-Harish-Chandra series and cyclotomic Hecke algebras. The blocks of finite groups of Lie type are closely related to so-called d-Harish-Chandra series. Let G be as above, the group of fixed points of a simple algebraic group G under a Frobenius map. For any d ∈ N, there is a notion of d-split Levi subgroup L of G (an F -stable Levi subgroup of G), and of d-cuspidal unipotent character of L := L F , see for example [5]. A pair (L, λ) consisting of a d-split Levi subgroup L ≤ G with a d-cuspidal unipotent character λ ∈ E(L) of L is called a d-cuspidal pair. Its relative Weyl group is then defined as , the set of unipotent characters of G admits a natural partition into d-Harish-Chandra series E(G, (L, λ)), where (L, λ) runs over the d-cuspidal pairs in G modulo conjugation. Furthermore, for each d-cuspidal pair (L, λ), there is a bijection between its d-Harish-Chandra series and the irreducible characters of its relative Weyl group W G (L, λ). The degree polynomials are then given by the following d-analogue of Howlett-Lehrer-Lusztig theory: Then for any ϕ ∈ Irr(W G (L, λ)) there exists a rational function D ϕ (x) ∈ Q(x) with zeros and poles only at roots of unity or zero, but not at primitive dth roots of unity, satisfying See [28,Thm. 4.2] and the references given there. In fact, the D ϕ (x) are inverses of Schur elements of a cyclotomic Hecke algebra attached to W G (L, λ) with respect to its canonical symmetrizing form. For example, if W G (L, λ) ∼ = G(e, 1, r), then D ϕ (x) is a suitable specialization of f ϕ (u) as defined above.
We first determine for which parameters (u 1 , . . . , u e ) all Schur elements of the cyclotomic Hecke algebra for the cyclic group G(e, 1, 1) are equal: where ζ ∈ K is a primitive dth root of unity.
Proof. Equivalently we may assume that and ′ denoting the derivative with respect to x, Since a e = 1 we conclude b = e, and thus a j = 0 for j = 1, . . . , e − 1. The claim follows.
The following result will allow to show the existence of different height zero degrees in blocks of classical groups, that is, groups of type A n , B n , C n , D n , 2 A n or 2 D n .
Proof. We will show that W G (L, λ) has linear characters ϕ 1 , ϕ 2 such that D ϕ 1 (q) = D ϕ 2 (q). The claim then follows from Theorem 7.3. In groups of classical type, there are three essentially different possibilities for the structure of the relative Weyl group (see [4, (3B)]). Firstly, W G (L, λ) could be a symmetric group S n . This happens if and only if either G = SL n (q) and d = 1, or G = SU n (q) and d = 2. In both cases, all of E(G) is just one d-Harish-Chandra series and we may take the trivial and the Steinberg character, which correspond to the two linear characters of S n and have distinct degrees.
The second possibility is that W G (L, λ) ∼ = G(m, 1, r) for some m ≥ 2. This occurs for all other d-Harish-Chandra series E(G, (L, λ)) in classical groups for which λ is not parametrized by a so-called degenerate symbol. Let ϕ i denote the linear character of G(m, 1, r) parametrized by the multipartition (π 1 , . . . , π m ) with π i = (r). According to (1) for some non-zero c not depending on i, and the parameters q are certain powers of q, up to sign, as follows (see [4, Bem. 2.10, 2.14, 2.19]): (I) for G = SL n (q), d = 1, we have m = d and q = (q d ; 1, q b 1 d+1 , q b 2 d+2 , . . . , q b d−1 d+d−1 ), (I') for G = SU n (q), d = 2, we have m = d * , and q is obtained from the parameters in case (I) for d * by replacing q by −q, (II) for G of type B n , C n , D n , 2 D n and d odd we have m = 2d, e = d and q = (q e ; 1, q b 1 e+1 , . . . , q b e−1 e+e−1 , −q bee , . . . , −q b 2e−1 e+e−1 ), (II') for G of type B n , C n , D n , 2 D n and d ≡ 2 (mod 4) we have m = d, and q is obtained from the parameters in case (II) for d * by replacing q by −q, (III) for G of type B n , C n , D n , 2 D n and d ≡ 0 (mod 4) we have m = d, e = d/2 and where the b i are non-negative integers which are determined by λ. Here, for d ∈ N, d * is defined by Note that it can never happen that v k u i − u j = 0 for the above choices of parameters. Furthermore, we claim that there is at least one i with |u i | > 1. Indeed, otherwise we are necessarily in cases (II) or (II') and e = 1. But then, since λ is not parametrized by a degenerate symbol, b 1 > 0 by the definition of the b i in [4, (2B)], so |u 2 | > 1, a contradiction. By our above reductions it suffices now to show that not all f i (q) are equal. For this, we estimate the q-power in f i (q) for two choices of i. For i = 1 we have u 1 = 1, and v k u 1 − u j is at most divisible by the q-power u j (at least when q is odd), and not divisible by q if k = 0, so f 1 (q) is at most divisible by the q-power m j=2 u r−1 j . Now let i be such that |u i | is maximal among the {u 1 , . . . , u m }. Then v k u i − u j is divisible by at least the q-power u j , so f i (q) is at least divisible by By our above observation we have For fixed j, this can only happen for at most one value of k, and only when j ≡ 1 + e (mod 2e) and we are in cases (II), (II') or (III). Thus, we get an additional factor at most 2 in the q-part of f 1 (q). It follows that the q-parts of f 1 (q) and f i (q) can only agree if |u i | = q = 2 and all other u j have absolute value 1. Thus e = 1, we are in cases (II) or (II') and q = (q; 1, −q). But then (or the same with q replaced by −q).
Finally, we consider the case where λ is parametrized by a degenerate symbol, which can only happen in types D n and 2 D n . Then W G (L, λ) ∼ = G(2m, 2, r), for some m ≥ 1. We denote by ψ 1 , . . . , ψ m the m distinct linear characters contained in the restrictions of the linear characters ϕ 1 , . . . , ϕ 2m from G(2m, 1, r) to its normal subgroup G(2m, 2, r) of index 2. Evaluation of [26, (5.12)] shows that D ψ i =c g −1 i (q) for some constantc, where Clearly, unless d = 1, we can argue as before to conclude that g 1 (q) = g i (q) for a suitable index i. If d = 1 then W G (L, λ) ∼ = G(2, 2, r) is the Weyl group of type D r , and we are in the principal 1-series of G. Here instead we take the trivial and the Steinberg character, which have distinct degree.

Unipotent blocks.
After these combinatorial preparations we are ready to investigate unipotent blocks of groups of Lie type G = G(q); here a p-block of G is called unipotent if it contains at least one unipotent character of G. Proof. First assume that p is a good prime for G, odd, and not equal to 3 when G is not of type 3 D 4 . Then by Cabanes-Enguehard [9, Thm. 22.9] the intersections of unipotent pblocks with E(G) are just the d-Harish-Chandra series, where d is the multiplicative order of q modulo p. Let B be a unipotent p-block corresponding to the d-Harish-Chandra series of the d-cuspidal pair (L, λ). If L = G, so λ is a d-cuspidal character of G, then the defect group of B is trivial by [9, Thm. 22.9(ii)], whence B is of defect 0.
If L < G, then W G (L, λ) = 1. Now for γ ∈ E(G, (L, λ)), where ϕ := ρ(L, λ)(γ), and for ϕ ∈ Irr(W G (L, λ)), by Theorem 7.3(b). As p divides Φ d (q) by definition, this implies the same congruence (mod p). By the description in [9, Thm. 22.9(ii)], some unipotent character in B is of height zero. This shows that the unipotent characters in B of height 0 are precisely those γ ∈ E(G, (L, λ)) with ϕ = ρ(L, λ)(γ) of degree prime to p, for example the linear characters of W G (L, λ). For G of classical type it is shown in Proposition 7.5 that not all unipotent characters in B parametrized by linear characters of W G (L, λ) have the same degree. If G is of exceptional type and W G (L, λ) is cyclic, we may invoke Lemma 7.4 together with the parameters in [4, Tab. 8.1] to conclude. The relative Weyl groups W G (L, λ) for exceptional groups which are non-cyclic are listed in [4, Tab. 3.6] and [6, Tab. 1]. It is easy to check that these have two distinct character degrees prime to p, for all primes p which are good for G. But then the corresponding unipotent degrees must be distinct by Proposition 7.2, and of height 0 by Theorem 7.3(b).
Next, if G is of classical type and p = 2, then all unipotent characters of G lie in the principal p-block of G, by [9, Th. 21.14]. Here, the trivial character and the Steinberg character have p-height 0 and different degrees.
It remains to consider the case where G is of exceptional type and p is a bad prime for G (including the case of 3 D 4 with p = 3). There is no bad prime for 2 B 2 . The 2-blocks for 2 G 2 and the 3-blocks of 2 F 4 have been determined by Fong [17] resp. Malle [25, Bem. 1]: unipotent characters lie either in the principal block or are of defect zero. In the principal block the trivial and the Steinberg character have different degree. Table 1. Non-principal p-blocks of positive defect for bad p For the other types of exceptional groups, we use the description of unipotent blocks for bad primes p obtained by Enguehard [15, Th. A]. Here, again any unipotent p-block is either of defect 0, or it contains at least one non-trivial d-Harish-Chandra series. According to loc. cit. and the tables in [15, pp. 347-358], the non-principal unipotent blocks not of defect zero are as listed in Table 1 (up to Ennola duality and algebraic conjugacy; the notation is as in loc. cit.) In each case either the relative Weyl group has two distinct character degrees prime to p, in which case we may conclude as above, or we list two unipotent characters γ 1 , γ 2 in the corresponding d-Harish-Chandra series which are of p-height 0 and have distinct r-parts in their degrees (see [6,Tab. 2] for the list of d-Harish-Chandra series and [10,Ch. 13] for the degrees of unipotent characters).
This completes the proof of Theorem 7.6. Note that the results hold even when the finite group G is not perfect, or even solvable.
We now extend the result to unipotent blocks of arbitrary finite connected reductive groups.
Theorem 7.7. Let G be a connected reductive group with a Frobenius map F : G → G and group of fixed points G := G F . Let B be a unipotent p-block of G, where p is not the defining characteristic r of G. Then either B is of central defect, and all characters of B have the same degree, or B contains two height 0 characters of different degrees. Moreover, these two degrees have different r-parts unless possibly if r = 2.
Proof. The derived group [G, G] is semisimple, hence a central product G 1 • . . . • G r of simple algebraic groups. We assume the G i ordered such that G 1 , . . . , G s , for some s ≤ r, is a system of representatives for the F -orbits on {G 1 , . . . , G r }. Then, where m i is the size of the F -orbit of G i . Note that, in general, G ′ will be larger than the commutator subgroup of G. Modulo Z([G, G]) F = Z(G ′ ) we obtain a direct product Since unipotent characters restrict irreducibly to the F -fixed points of the derived group [24] B covers a unique block B ′ of G ′ . Furthermore unipotent characters have the center in their kernel, so the same holds for unipotent blocks. Thus B ′ corresponds to a unique blockB of the direct productḠ. This is a direct productB =B 1 × . . . ×B s of blocksB i ofḠ i . Now assume that one of theB i is not of defect 0 forḠ i . SinceḠ i is a central factor group of a group as in Theorem 7.6,B i then contains two height 0 unipotent characters of different degrees, with different r-parts if r = 2. Thus, the same is true forB, hence also for B ′ . By the above-mentioned irreducibility of restrictions, this then also holds for B.
On the other hand, if allB i are of defect 0, then so isB, so B ′ is of central defect, contained in Z([G, G]) F . But Z([G, G]) ⊆ Z(G) as G = [G, G]Z(G), so the block B is also of central defect in G, as claimed. Moreover, as eachB i contains a unique ordinary character, we also have Irr(B ′ ) = {χ ′ } for some ordinary (unipotent) character χ ′ of G ′ . Since this is the restriction of an irreducible character of G, and G/G ′ is abelian, all characters of G above χ ′ have the same degree, hence the height zero conjecture holds for B in this case. Proof. The first assertion is well-known. For the second, let ν i be a character of b lying above χ i , for i = 1, 2. Since p does not divide the index |G : N|, ν i is again of height 0. Furthermore, the assumption that gcd(r, |G : N|) = 1 and G/N is cyclic or Klein four implies by Clifford theory that ν 1 (1) = ν 2 (1).

Blocks of groups of Lie type
Proposition 8.1. The assertion of Theorem 6.1 holds when S is a simple group of Lie type and p is the defining characteristic.
Proof. By the result of Humphreys [21] the covering group G of S has exactly one p-block of defect zero, consisting of the Steinberg character, and all other p-blocks are of full defect, in one-to-one correspondence with the irreducible characters of Z(G). For the principal block, it is clear that there exist two height 0 characters of distinct degree (viz. the trivial character and at least one further non-linear character). For the remaining blocks, some more work is needed. By the above we may now assume that Z(G) = 1, so in particular p is odd for classical groups not of types A n or 2 A n .
Recall that Z(G) is naturally isomorphic to the commutator factor group G * /[G * , G * ] of the dual group G * of G. If s ∈ G * is semsimple, the corresponding semisimple character χ s ∈ Irr(G) is of p ′ -degree given by χ s (1) = |G : C G * (s)| p ′ (see for example [28, (2.1)]; note that C G * (s) is not necessarily connected). So we are done if we can find two semisimple elements s 1 , s 2 ∈ G * whose centralizer orders have different p ′ -part. Table 2. Tori and Zsigmondy primes for classical groups A n (n ≥ 3 odd) (q n+1 − 1)/(q + 1) q n + 1 l(n + 1) l(2n) B n , C n (n ≥ 2 even) q n + 1 (q n−1 + 1)(q + 1) l(2n) l(2n − 2) B n , C n (n ≥ 3 odd) q n + 1 q n − 1 l(2n) l(n) D n (n ≥ 4 even) (q n−1 − 1)(q − 1) (q n−1 + 1)(q + 1) l(n − 1) l(2n − 2) D n (n ≥ 5 odd) q n − 1 (q n−1 + 1)(q + 1) l(n) l(2n − 2) 2 D n (n ≥ 4) q n + 1 (q n−1 + 1)(q − 1) l(2n) l(2n − 2) In Table 2 we have listed two maximal tori T 1 , T 2 of G * for each type of classical group G (by giving their orders, which determines them uniquely). Except for types B n , C n with n even this is Table 3.5 in Malle [27]. We write l(m) for a Zsigmondy prime divisor of q m − 1. Then |T i | is divisible by the Zsigmondy prime ℓ i as indicated in the table, which exists unless G is of type A 1 , or G is of type A 2 , 2 A 2 or B 2 and i = 2. (Note that the case that q = 2 and G of type A 5 , A 6 or 2 A 6 does not concern us here, since then the center of G is trivial.) If |T i | is divisible by a Zsigmondy prime ℓ i , then there exist regular semisimple elements s i of order ℓ i in G * , that is, elements with centralizer order |C G * (s i )| = |T i |. If both Zsigmondy primes exist, this yields two semisimple characters of different degrees, and we are done.
The only exceptional simply-connected groups with non-trivial center are those of types E 6 , 2 E 6 and E 7 . For these, we may argue as above using the maximal tori and Zsigmondy primes listed in Table 3. The proof is complete. Table 3. Tori T 1 and T 2 We now turn to the non-defining primes for groups of Lie type. According to the work of Broué-Michel [7], for any p-block B of G there exists a unique G * -conjugacy class [s] of semisimple p ′ -elements of G * , such that some irreducible representation of B is in the rational Lusztig series attached to [s]. Let's write E p (G, s) for the union of all p-blocks of G associated with the class of the p ′ -element s ∈ G * . The blocks in E p (G, 1) are called unipotent. More generally, if G is disconnected, then a block of G F is called unipotent if it covers a unipotent block of (G • ) F . We need the following crucial result of Enguehard [16, Th. 1.6]: Theorem 8.2 (Enguehard). Assume that p is good for G, and different from 3 if F induces a triality automorphism on G.
Let s ∈ G * be a semisimple p ′ -element, and B a p-block in E p (G, s). Then there exists a reductive group G(s) defined over F r , with corresponding Frobenius map again denoted by F , and a unipotent p-block b of G(s) := G(s) F , such that the defect groups of B and b are isomorphic and there is a height-preserving bijection Irr(B) → Irr(b). Here, G(s) • is a group in duality with C G * (s) • , and G(s)/G(s) In the case of p = 2 for classical groups, he proves [16, Prop. 1.5]: Theorem 8.3 (Enguehard). Assume that G is of classical type in odd characteristic. Let s ∈ G * be a semisimple p ′ -elements. Then all 2-blocks in E 2 (G, s) have defect group isomorphic to a Sylow 2-subgroup of C G * (s) • . If moreover G is of type B n , C n or D n , then E 2 (G, s) is a single 2-block.
Proposition 8.4. The assertion of Theorem 6.1 holds if G is quasi-simple of Lie-type.
Proof. By Proposition 8.1 we may assume that p is not the defining characteristic for G, and by Proposition 9.4 we have that S ∼ = 2 F 4 (2) ′ . Furthermore, by the remarks at the beginning of Section 7 we have that G = G F for some simple, simply connected algebraic group G with Frobenius map F : G → G.
Let B be a p-block of G and s ∈ G * semisimple such that B ⊆ E p (G, s) (see above). First assume that s is not quasi-isolated in G * , that is, C G * (s) is a Levi subgroup of G * . Then by the result of Bonnafé-Rouquier [9, Th. 10.1] the block B is Morita-equivalent to a block b ⊆ E p (L, 1) where L is a Levi subgroup of G in duality with C G * (s), and Jordan decomposition gives a height preserving bijection from B to b. We may then conclude by Theorem 7.7.
Next assume that p is good for G, different from 3 if G is of type 3 D 4 . Then by Theorem 8.2 there is a group G(s) in duality with the centralizer C := C G * (s) of s in G * and a height preserving bijection between B and a unipotent block b of G(s) := G(s) F with the same defect group as B. By [3,Cor. 2.9] the order a(s) := |C : C • | of the component group of C is prime to the defining characteristic r and divides the order of s. As s is a p ′ -element, this implies that a(s) is prime to p as well. Moreover, by loc. cit. C/C • is isomorphic to a subgroup of the fundamental group of G, hence either cyclic or a Klein four group. Now let b ′ be a p-block of the normal subgroup N := (C • ) F of C F = C G * (s) lying below b. We showed in Theorem 7.7 that any unipotent block of the connected group N with non-abelian defect group contains two height 0 characters which are divisible by different powers of the defining prime r. Thus, Proposition 7.8 applies in this case and the claim follows. Now assume that p = 2 and G is of classical type B n , C n , D n or 2 D n . Then E 2 (G, s) is a single 2-block by Theorem 8.3. By Jordan decomposition the character degrees in E 2 (G, s) are obtained from those in E 2 (C G * (s), 1) by multiplication with a common constant. If C G * (s) • is not a torus, the trivial character and the Steinberg character in E 2 (C G * (s), 1) have distinct degrees prime to p and the claim follows. On the other hand, if C G * (s) • is a torus, that is, s is a regular element in G * , then again by the Theorem 8.3 of Enguehard the defect group of B is isomorphic to a Sylow 2-subgroup of C G * (s) • , hence abelian. Moreover, by the result of Lusztig [24], all characters in B have the same degree, whence B satisfies the height zero conjecture.
Thus we may assume that G is of exceptional type, p is a bad prime and s is quasiisolated. There are no quasi-isolated elements for 2 B 2 . The p-blocks for 2 G 2 , G 2 , 2 F 4 and 3 D 4 have been determined by Fong [17], Hiss-Shamash [19,20], Malle [25], Deriziotis-Michler [12] respectively. The claim can be easily checked from those results. The remaining cases are the possible exceptions mentioned in the theorem.

Alternating and sporadic groups
In order to prove our main result for the alternating groups, we first derive a similar statement for blocks of the symmetric group.
Recall that the irreducible characters of S n as well as the unipotent characters of GL n (q), where q is any prime power, are parametrized by partitions λ ⊢ n. We write χ λ resp. γ λ for the corresponding character of S n , resp. of GL n (q). The following important connection between their degrees is well-known: χ λ is obtained by specializing q to 1 in the degree polynomial for γ λ (see for example the formula in [10, 13.8] and compare to the hook formula for χ λ (1)). This is sometimes referred to by saying that S n is 'the general linear group over the field with one element'.
Furthermore, χ λ and χ µ for two partitions λ, µ ⊢ n lie in the same p-block of S n if and only if λ and µ have the same p-core, which in turn happens if and only if γ λ and γ µ lie in the same d-Harish-Chandra series of Irr(GL n (q)), where d = p. Thus, the degrees of irreducible characters of S n in a fixed p-block are specializations at q = 1 of degree polynomials of unipotent characters in a fixed p-Harish-Chandra series.
Let S = (S 1 , . . . , S d ) be a d-symbol. A hook of S is a pair h = (s, t) where for some 1 ≤ i, j, ≤ d. We then also write i(h) := i, j(h) := j, and l(h) := s − t. For 1-symbols, that is, β-sets of partitions, this is just the usual notion of hook. We can now formulate the following relative hook formula for characters in a fixed p-block of a symmetric group which seems to be new: Theorem 9.1. Let p be a prime. Let π ⊢ n be a partition with p-core µ ⊢ r and p-quotient (ν 1 , . . . , ν p ) ⊢ w, with corresponding p-symbol S. Let b i denote the number of beads on the ith runner of the p-abacus diagram for µ, and c i := pb i + i − 1. Then and where ψ ν denotes the irreducible character of C p ≀ S w parametrized by ν.
Proof. Let γ be the unipotent character of GL n (q) parametrized by π, for q a prime power. Set d := p. Then γ lies in the d-Harish-Chandra series above (L, λ), where L ∼ = GL r (q) × GL 1 (q d ) w , with λ parametrized by µ ⊢ r and n = r + dw. Let S = (S 1 , . . . , S d ) be the d-symbol corresponding to (ν 1 , . . . , ν d ). According to Theorem 7.3, [26, (2.19) with (v; u 1 , . . . , u d ) = (q d ; 1, q c 2 , . . . , q c d ) (see (I) in Sect. 7.3 for the parameter values). By our above remarks, specialization at q = 1 gives the corresponding character degrees for S n . Note that numerator and denominator of the expression for Deg(γ)/Deg(λ) are indeed divisible by the same power of (q − 1), viz. n + w + me 2 , so that the specialization makes sense. We obtain as claimed.
Let's note the following special case of p-quotients (ν 1 , . . . , ν p ) ⊢ w such that the corresponding p-symbol S has S i = (w), S j = (0) for j = i. Since these correspond to linear characters of the relative Weyl group in GL n (q), they parametrize characters of height 0 in B by the congruence in Theorem 9.1. We obtain A p-blocks B of S n labelled by a p-core µ ⊢ n − wp is said to be of weight w. So w denotes the number of p-hooks which must be removed from any partition π indexing a character in B to obtain its core µ. The block is said to be self-dual if µ is a self-dual partition.
Proposition 9.2. Let G = S n , n ≥ 5, p a prime, and B a p-block of G. Then one of the following occurs: (a) B is of weight (and hence defect) 0, (b) p = 2 and B is of weight 1, (c) p = 3, B is of weight 1 and self-dual, or (d) B contains two height 0 characters of different degrees d 1 < d 2 , either both indexed by non-self-dual partitions or with d 2 = 2d 1 .
Proof. We use the relative hook formula in (4) for the character degrees of S n for certain height 0 characters in B. We may assume that the weight w of B is positive. Let µ ⊢ n − pw denote the p-core associated to B, let 0 = e 1 < e 2 . . . < e p be the ordered set of the c i as in Theorem 9.1, and f i := w−1 k=0 j =i |pk + e i − e j |, for 1 ≤ i ≤ p. Note that by [34,Prop. 3.5] none of the partitions λ i corresponding to the p-quotients S i is self-dual, unless w = 1 in which case at most one of them is. Clearly, f p > f p−1 unless p = 2 and w = 1 (which is case (b)), which yields two distinct height 0 degrees d 1 , d 2 . If both corresponding partitions are self-dual, then w = 1. But by Theorem 9.1 we have d i ≡ ±1 (mod p), and then d 1 = d 2 /2 implies that p = 3. Corollary 9.3. Let p be a prime, B a p-block of A n . Then one of the following holds: (a) B is of defect 0, (b) p = 3, B is of weight 1 (hence with cyclic defect group C 3 ), self-dual, and all χ ∈ Irr(B) have the same degree, or (c) B contains two height 0 characters of different degrees. In particular, the assertion of Theorem 6.1 holds when S is an alternating group.
Proof. Let B be a p-block of S n , containing all characters χ λ for which λ has fixed p-core µ ⊢ (n − pw). According to [34,Prop. 12.2], for example, if w > 0 then B covers a unique block B 1 of A n . First assume that p is odd. Let χ 1 , χ 2 ∈ B be two height zero characters of different degrees, parametrized by non self-dual partitions, according to Proposition 9.2. These restrict irreducibly to characters of A n in B 1 of height 0. Similarly, if χ 1 , χ 2 ∈ B have different degrees d 1 < d 2 with d 2 = 2d 1 , then the restrictions of χ 1 , χ 2 to A n contain characters of B 1 of height 0 and of different degrees.
If p = 3, B is of weight 1 and self-dual, then two characters of B have the same irreducible restriction and one splits into two constituents for A n . We obtain a block B 1 with defect group of order 3 and three equal character degrees.
For p = 2, restriction of characters from G to B 1 either preserves heights or decreases it, by [34,Prop. 12.5]. Thus, we may conclude by Proposition 9.2 unless w = 1. Here, the two irreducible characters in B have the same restriction to A n , so B 1 is a block with a unique ordinary character, that is, a block of defect zero.
Note that case (b) of Corollary 9.3 occurs if and only if there is a self-dual 3-core for n − 3. The conditions for this to occur have been worked out in [2, Lemma 3.1].
It can be checked from the known character tables that the assertion of Theorem 6.1 remain true for the faithful blocks of 2.A n when n ≤ 13.
We complete our investigation of blocks of quasisimple groups by showing: Proposition 9.4. The assertion of Theorem 6.1 holds when S is sporadic or a simple group of Lie type with exceptional Schur multiplier, or S = 2 F 4 (2) ′ .
Proof. The ordinary character tables of all quasi-simple groups such that S is as in the assumption are contained in the Atlas [11]. From this, or using the electronic tables available in GAP, it can be checked that whenever B is a p-block of G with all height zero characters of the same degree then the defect group satisfies |D| ≤ p 2 , hence must be abelian.

p-Solvable Groups
Our main result in this section is to reduce the study of EHZD blocks of general psolvable groups to groups with p ′ -lenght one. This latter case naturally leads us to consider a variation of a classical large orbit problem.
Question 10.1. Suppose that V is a finite faithful completely reducible F G-module, where F has characteristic p and G has a normal p-complement K > 1. Let P ∈ Syl p (G). Does there exists v ∈ C V (P ) such that |C K (v)| 2 < |K|? Question 10.1 is not trivial, even if P = 1. In this case, it has an affirmative answer if K is solvable (by [13]). Also, Question 10.1 has an affirmative answer if K is nilpotent, and this constitutes the main result of [14]. In some sense, it is unfortunate that our only way to prove that EHZD blocks of p-solvable groups are nilpotent is via large orbits. On the other hand, Question 10.1 has interest in its own and it is closely related to the study of p ′ -degrees of p-solvable groups, so it might deserve some consideration.
Our main result in this Section is the following. Theorem 10.2. Suppose that Question 10.1 has an affirmative answer. If B is an EHZD block of a p-solvable group G, then B is nilpotent. Now we have that Irr(V ) is a completely reducible, finite, and faithfulḠ-module. By using the affirmative answer to Question (10.1), there exists β ∈ Irr(V ) centralized by P such that |K β /U| 2 < |K/U| , where K β is the stabilizer in K of β. In other words, |K : K β | 2 > |K/U| . Now, since K β /V is a p ′ -group, there exists a unique extensionβ ∈ Irr(K β ) of β, by using Corollary (8.16) of [22], which has p-power order. In particular, this linear character has Z in its kernel, and by uniqueness is P -invariant (because β is P -invariant). Letλ = 1 V × λ ∈ Irr(U). Since K β /U is a p ′ -group andλ is P -invariant, then we may find some γ ∈ Irr(K β |λ) which is P -invariant (this is becauseλ K β has p ′ -degree). Now we have that γβ ∈ Irr(K β ) (becauseβ is linear) lies over β. By the Clifford correspondence, we have that ρ = (γβ) K ∈ Irr(K). This character ρ is P -invariant, has p ′ -degree |K : K β |γ(1) and lies over λ. Also ρ K 0 ∈ Irr(K 0 ) is P -invariant, has p ′ -degree, and therefore it has an extension χ ∈ Irr(G) with χ K 0 = ρ K 0 , by using Corollary (8.16) of [22] and the fact that K 0 = O p (G). Hence d = |K : K β |γ(1) ≥ |K : K β | . Therefore, d 2 ≥ |K : K β | 2 > |K/U| . Now, let H be a p-complement of G. Hence HV = K and H ∩ V = 1. Finally, using thatλ is P -invariant and (λ) K has p ′ -degree, we can find a P -invariant ξ ∈ Irr(K|λ) of p ′ -degree. Arguing as before, we have that ξ extends to G, and therefore ξ(1) = d. However, ξ H ∈ Irr(H|λ). Hence, d 2 ≤ |H : Z| by elementary character theory. However, |H : Z| = |K : U| and this is a contradiction.
(A similar argument gives the same conclusion of Lemma 11.3 if we assume that χ 0 ∈ IBr(G) for all χ ∈ Irr(G|λ) of p ′ -degree.) Proof of Theorem 10.2. We argue by induction on |G|. Let Z = O p ′ (G), and let λ ∈ Irr(Z) be covered by B. If T is the stabilizer of λ in G and b is the block of T which corresponds to B via , Theorem (9.14)), then b is a EHZD block. If T < G, then b is nilpotent by induction. Thus B is nilpotent by Lemma 1 of [30], for instance. Hence, we may assume that T = G. In this case, Irr(B) = Irr(G|λ) by Theorem (10.20) of [29]. Now we conclude that G has a normal p-complement by Lemma 10.3.