Low-dimensional representations of finite orthogonal groups
aa r X i v : . [ m a t h . R T ] M a y LOW-DIMENSIONAL REPRESENTATIONSOF FINITE ORTHOGONAL GROUPS
KAY MAGAARD AND GUNTER MALLE
Abstract.
We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictionson the characteristic we also prove a gap result showing that the next larger irreducibleBrauer characters have a degree roughly the square of those of the smallest non-trivialcharacters. Introduction
This paper is devoted to studying low-dimensional irreducible representations of fi-nite orthogonal groups in non-defining characteristic. Our aim is a gap result show-ing that there are a few well-understood representations of very small degree, and allother irreducible representations have degree which is roughly the square of the smallestones. Knowing the low-dimensional irreducible representations of quasi-simple groups hasturned out to be of considerable importance in many applications, most notably in thedetermination of maximal subgroups of almost simple groups. More specifically we prove:
Theorem 1.
Let G = Spin ǫ n ( q ) with ǫ ∈ {±} , q odd and n ≥ . Assume that ℓ ≥ is aprime not dividing q ( q + 1) . Let ϕ be an ℓ -modular irreducible Brauer character of G ofdegree less than q n − − q n +4 . Then ϕ (1) is one of , q ( q n − ǫ q n − + ǫ q − − κ , q ( q n − − q − − κ ,
12 ( q n − ǫ q n − ± ǫ q ∓ , ( q n − ǫ q n − ± ǫ q ∓ , where κ , κ ∈ { , } . Theorem 2.
Let G = Spin n +1 ( q ) with q odd and n ≥ . Assume that ℓ ≥ is a primesuch that the order of q modulo ℓ is either odd, or bigger than n/ . Let ϕ be an ℓ -modularirreducible Brauer character of G of degree less than ( q n − − q n ) / . Then ϕ (1) is one of , q n − q − , q ( q n − q n − − q + 1 , q ( q n + 1)( q n − + 1) q + 1 , q ( q n + 1)( q n − − q − − κ , q ( q n − q n − + 1) q − − κ , q q n − q − , q n − q ± where κ , κ ∈ { , } . Date : May 23, 2019.1991
Mathematics Subject Classification.
Primary 20C33; Secondary 20D06, 20G40.
Key words and phrases. low dimensional representations, orthogonal groups, decomposition matrices.The second author gratefully acknowledges financial support by SFB TRR 195.
Observe that the character degrees listed in the theorems are of the order of magnitudeabout q n − , q n − respectively, which is only slightly larger than the square root of thegiven bound.Gap results of the form described above have already been proved for all other series offinite quasi-simple groups of Lie type. The situation for orthogonal groups is considerablyharder since the smallest dimensional representations have comparatively much larger de-gree than for the other series. For odd-dimensional orthogonal groups over fields of evencharacteristic Guralnick–Tiep [8] obtained gap results similar to ours without any restric-tion on the non-defining characteristic ℓ for which the representations are considered.Their approach crucially relies on the exceptional isomorphism to symplectic groups.Our results do not cover all characteristics ℓ as our proofs rely on unitriangularity ofa suitable part of the ℓ -modular decomposition matrix of the groups considered which inturn is proved using properties of generalised Gelfand–Graev characters. Since this hasnot been established in full generality (although it is expected to hold), the present stateof knowledge makes it necessary to impose certain restrictions on the prime numbers ℓ considered, as well as, more seriously, on the underlying characteristic having to be odd.The paper is structured as follows. In Section 2 we determine the small dimensionalcomplex irreducible characters of spin groups using Deligne–Lusztig theory. In Section 3we investigate the restriction of small dimensional Brauer characters to an end nodeparabolic subgroup. Finally, with this information we determine the precise dimensionsof the smallest Brauer characters for all three series of spin groups in Section 4 and derivethe gap results in Theorems 1 and 2 including the precise values of the κ i , see Theorem 4.5and Corollary 4.11.Kay and I started work on this paper around 2011. Sadly, he passed away very unex-pectedly shortly before the completion of the manuscript. I would like to dedicate thispaper to his memory.2. Small degree complex irreducible characters
In this section we recall the classification of the smallest degrees of complex irreduciblecharacters of the finite spin groups G by using Lusztig’s parametrisation in terms ofLusztig series E ( G, s ) indexed by classes of semisimple elements s in the dual group G ∗ ().2.1. The odd-dimensional spin groups
Spin n +1 ( q ) . Let q be a power of a prime and G = Spin n +1 ( q ) with n ≥
2. Recall that Lusztig’s Jordan decomposition (see e.g. [5,Thm. 2.6.22]) gives a bijection J s : E ( G, s ) −→ E ( C G ∗ ( s ) , C G ∗ ( s ), under which the character degreestransform by the formula χ (1) = | G ∗ : C G ∗ ( s ) | p ′ J s ( χ )(1) . We start by enumerating unipotent characters of small degree. Here, we allow forslightly larger degrees than in the general case, since this will be needed later on andmoreover we believe that this information may be of independent interest.
OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 3
Let us recall that a symbol is a pair S = ( X, Y ) of strictly increasing sequences X =( x < . . . < x r ), Y = ( y < . . . < y s ) of non-negative integers. The rank of S is thendefined to be r X i =1 x i + s X j =1 y j − $(cid:18) r + s − (cid:19) % . The symbol S ′ = ( { } ∪ ( X + 1) , { } ∪ ( Y + 1)) is said to be equivalent to S , and sois the symbol ( Y, X ). The rank is constant on equivalence classes. The defect of S is d ( S ) = || X | − | Y || , which clearly is also invariant under equivalence.The unipotent characters of the groups Spin n +1 ( q ) are parametrised by equivalenceclasses symbols of rank n and odd defect (see e.g. [5, Thm. 4.5.1]). The following resultis due to Nguyen [12, Prop. 3.1] for n ≥ Proposition 2.1.
Let G = Spin n +1 ( q ) or Sp n ( q ) . Let χ be a unipotent character of G of degree χ (1) ≤ q n − − q n − when n ≥ ,q − q when n = 5 ,q − q − q when n = 4 ,q − q when n = 3 . Then χ is as given in Table 1 where we also record the degree of χ (1) as a polynomialin q .Proof. The degree polynomials of unipotent characters for n ≤ Chevie [4]. For q <
20, a direct evaluation of these polynomials gives the claim. For q >
20, the claim then follows by easy estimates using the explicit formulas. For n ≥ (cid:3) We now enumerate the complex irreducible characters of Spin n +1 ( q ) of small degree.The irreducible character degrees of families of groups of fixed Lie type over the field F q can be written as polynomials in q . It turns out that the smallest such degree polynomialsfor groups of type B n have degree in q around 2 n , while the next larger ones have degreein q around 4 n . We list all irreducible characters whose degrees lie in the first range.Note that the complex irreducible character of smallest non-trivial degree for orthogonalgroups was determined in [13]. For n ≥
5, the following has been shown in [12, Th. 1.2].
Theorem 2.2.
Let G = Spin n +1 ( q ) with n ≥ . If χ ∈ Irr( G ) is such that χ (1) < ( q n − when n ≥ , ( q n − q n − ) / when n ∈ { , } , then χ is as given in Table 2, where G , ρ , . . . , ρ are the first five unipotent characterslisted in Table 1.Proof. For 3 ≤ n ≤
8, the complete list of ordinary irreducible characters of G and theirdegrees can be found on the website [10]. For q <
30, the claim can then be checked bycomputer, while for q >
30, an easy estimate, using the known degrees in q of the degreepolynomials, shows that the given list is complete. KAY MAGAARD AND GUNTER MALLE
Table 1.
Small unipotent characters in types B n and C n S χ S (1) deg q ( χ S (1)) conditions (cid:0) n − (cid:1) (cid:0) , ,n − (cid:1) q ( q n − q n − − q +1 n − (cid:0) , n (cid:1) q ( q n +1)( q n − +1) q +1 n − (cid:0) ,n (cid:1) q ( q n +1)( q n − − q − n − (cid:0) ,n (cid:1) q ( q n − q n − +1) q − n − (cid:0) , ,n − − (cid:1) q q n − q n − − q n − − q − n − n > (cid:0) , n − (cid:1) q q n − q n − +1)( q n − +1) q − n − n > (cid:0) ,n − (cid:1) q q n − q n − +1)( q n − − q − n − n > (cid:0) ,n − (cid:1) q q n − q n − − q n − +1)( q − n − n > (cid:0) ,n − (cid:1) q q n +1)( q n − q n − − q − n − n > (cid:0) , , ,n (cid:1) q q n − q n − − q n − − q − n − n > n, q ) = (3 , (cid:0) , , ,n (cid:1) q q n +1)( q n − − q n − +1) q − n − n > (cid:0) , ,n , (cid:1) q q n +1)( q n − − q n − − q − n − n > (cid:0) , ,n , (cid:1) q q n − q n − − q n − +1)( q − n − n > (cid:0) , (cid:1) q q − q − n = 3 Table 2.
Smallest complex characters of Spin n +1 ( q ), n ≥ χ χ (1) q odd) q even) deg q ( χ (1))1 G ρ s, ( q n − / ( q −
1) 1 0 2 n − ρ q ( q n − q n − − / ( q + 1) 1 1 2 n − ρ q ( q n + 1)( q n − + 1) / ( q + 1) 1 1 2 n − ρ q ( q n + 1)( q n − − / ( q −
1) 1 1 2 n − ρ q ( q n − q n − + 1) / ( q −
1) 1 1 2 n − ρ − t ( q n − / ( q + 1) ( q − / q/ n − ρ s,q q ( q n − / ( q −
1) 1 0 2 n − ρ + t ( q n − / ( q −
1) ( q − / q − / n − n ≥ G ∗ = PCSp n ( q ).Let s ∈ G ∗ be an isolated involution with centraliser Sp ( q ) ◦ Sp n − ( q ). The correspond-ing Lusztig series E ( G, s ) is in bijective correspondence under Jordan decomposition withthe unipotent characters of Sp ( q ) ◦ Sp n − ( q ). Thus, we obtain the semisimple character OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 5 ρ s, and the character ρ s,q corresponding to the Steinberg character in the Sp ( q )-factor(both given in Table 2), while all other characters in that series have degree at least ρ s, (1) q ( q n − − q n − − / ( q + 1) / G ∗ with centraliser Sp n − ( q ) × GL ( q ) orSp n − ( q ) × GU ( q ). There are q − t in GL ( q ) of order larger than 2,which are fused to their inverses in G ∗ . The corresponding Lusztig series contain thesemisimple characters ρ + t from Table 2. Moreover, the q − ( q ) of orderlarger than 2 give rise to the ( q − / ρ − t . All other characters inthese Lusztig series have too large degree. (cid:3) The even-dimensional spin groups
Spin ± n ( q ) . For the even-dimensional spingroups Spin +2 n ( q ) of plus-type, the unipotent characters are parametrised by symbols ofrank n and defect d ≡ d ≡ Proposition 2.3.
Let χ be a unipotent character of Spin +2 n ( q ) of degree χ (1) ≤ ( ( q n − − q n − ) / when n ≥ ,q n − − q n − when ≤ n ≤ . Then χ is as given in Table 3.Proof. The (more involved) case n ≤ n ≥ q n − . (cid:3) Similarly we obtain:
Proposition 2.4.
Let χ be a unipotent character of Spin − n ( q ) of degree χ (1) ≤ ( ( q n − − q n − ) / when n ≥ ,q n − − q n − when ≤ n ≤ .Then χ is as given in Table 4. Again, see [12, Prop. 3.3] for the list of unipotent characters of degree at most q n − .The following two results have already been shown in [12, Th. 1.3 and 1.4] when n ≥ Theorem 2.5.
Let G = Spin +2 n ( q ) . If χ ∈ Irr( G ) is such that χ (1) < q n − when n ≥ , or n = 5 and q is odd ,q − q when n = 5 and q is even , ( q − q ) / when n = 4 and q is odd ,q − q + q when n = 4 and q is even , then χ is as given in Table 5, or ( n, q ) = (4 , and χ (1) = 28 . KAY MAGAARD AND GUNTER MALLE
Table 3.
Small unipotent characters in type D n S χ S (1) deg q ( χ S (1)) conditions (cid:0) n (cid:1) (cid:0) n − (cid:1) q ( q n − q n − +1) q − n − (cid:0) ,n , (cid:1) q q n − − q − n − (cid:0) n − (cid:1) q q n − q n − − q n − +1)( q − q − n − n > (cid:0) , , ,n − − (cid:1) q q n − q n − − q n − − q n − − q +1) ( q +1) n − (cid:0) ,n − , (cid:1) q q n − q n − − q n − +1)( q n − +1)( q − n − (cid:0) ,n − , (cid:1) q q n − q n − +1)( q n − − q n − +1)( q − ( q +1) n − (cid:0) ,n − , (cid:1) q q n − q n − +1)( q n − +1)( q n − − q − n − (cid:0) , ,n , , (cid:1) q q n − − q n − − q − q − n − (cid:0) n − (cid:1) q q n − q n − − q n − − q n − +1)( q − q − q − n − n > (cid:0) (cid:1) q q − q − (2 × ) 6 n = 4 (cid:0) , , (cid:1) q q − q − (2 × ) 10 n = 4 (cid:0) , , (cid:1) q q − q − q − q − n = 5 , q > (cid:0) , , , , (cid:1) q ( q − q − q − n = 5 , q = 2 (cid:0) (cid:1) q q +1)( q − q − (2 × ) 15 n = 6 (cid:0) , , , − (cid:1) q q − q − ( q − q +1 n = 6 , q = 2 , (cid:0) , , , − (cid:1) q q − q − q − q − q +1 n = 7 (cid:0) (cid:1) q q − q − q +1)( q − q − (2 × ) 28 n = 8 (cid:0) (cid:1) q q − q +1)( q − q +1)( q − q − n = 9 , q > Proof.
For 3 ≤ n ≤
7, the complete list of ordinary irreducible characters of G and theirdegrees can be found on the website [10]. For q <
50, the claim can then be checked bycomputer, while for q >
50, an easy estimate shows that the given list is complete.For n ≥ § g , ρ , ρ denote the first three unipotentcharacters listed in Table 3. The characters ρ ± s,a , ρ ± s,b are the semisimple characters in theLusztig series of involutions with disconnected centraliser of type PCO ± n − ( q ), and thecharacters ρ ± t are the semisimple characters in the Lusztig series of semisimple elementswith (connected) centraliser of type PCSO ± n − ( q ). (cid:3) Theorem 2.6.
Let G = Spin − n ( q ) . If χ ∈ Irr( G ) is such that χ (1) < q n − when n ≥ ,q − q when n = 5 , ( q − q ) / when n = 4 and q is odd ,q − q when n = 4 and q is even , OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 7
Table 4.
Small unipotent characters in type D n S χ S (1) deg q ( χ S (1)) conditions (cid:0) ,n − (cid:1) (cid:0) ,n − − (cid:1) q ( q n +1)( q n − − q − n − (cid:0) , ,n (cid:1) q q n − − q − n − (cid:0) ,n − − (cid:1) q q n +1)( q n − − q n − − q − q − n − n > (cid:0) , ,n − (cid:1) q q n +1)( q n − +1)( q n − − q n − − q − n − (cid:0) , ,n − (cid:1) q q n +1)( q n − − q n − +1)( q n − − q − ( q +1) n − (cid:0) , ,n − (cid:1) q q n +1)( q n − − q n − − q n − +1)( q − n − (cid:0) , , n − (cid:1) q q n +1)( q n − +1)( q n − +1)( q n − +1)( q +1) ( q +1) n − (cid:0) , , ,n , (cid:1) q q n − − q n − − q − q − n − (cid:0) ,n − − (cid:1) q q n +1)( q n − − q n − − q n − − q − q − q − n − n > (cid:0) , − (cid:1) q q +1)( q +1)( q − q +1)( q − q − q − n = 9 (cid:0) , , (cid:1) q q − q +1)( q +1)( q − n = 5 Table 5.
Smallest complex characters of Spin ǫ n ( q ), n ≥ χ χ (1) q odd) q even) deg q ( χ (1))1 G ρ q ( q n − ǫ q n − + ǫ / ( q −
1) 1 1 2 n − ρ − s,a , ρ − s,b ( q n − ǫ q n − − ǫ / ( q + 1) 2 0 2 n − ρ + s,a , ρ + s,b ( q n − ǫ q n − + ǫ / ( q −
1) 2 0 2 n − ρ − t ( q n − ǫ q n − − ǫ / ( q + 1) ( q − / q/ n − ρ q ( q n − − / ( q −
1) 1 1 2 n − ρ + t ( q n − ǫ q n − + ǫ / ( q −
1) ( q − / q − / n − then χ is as given in Table 5.Proof. Again, the case n ≤ n ≥
8, werefer to [12, § G , ρ , ρ denotethe first three unipotent characters listed in Table 4. The characters ρ ± s,a , ρ ± s,b lie in theLusztig series of involutions with disconnected centraliser of type PCO ± n − ( q ), and the ρ ± t are the semisimple characters in the Lusztig series of semisimple elements with centraliserof type PCSO ± n − ( q ). (cid:3) Locating Brauer characters of low degree
Here we study the restriction of small dimensional irreducible ℓ -Brauer characters ofspin groups to an end node parabolic subgroup. This requires no assumptions on ℓ or on KAY MAGAARD AND GUNTER MALLE q . Throughout this section let G = Spin ( ± ) m ( q ) with m ≥ P = QL be a fixedmaximal parabolic subgroup of G stabilising a singular 1-space of the natural module ofSO ( ± ) m ( q ), with unipotent radical Q and Levi factor L . Observe that Q ∼ = F m − q is thenatural module for L ′ := [ L, L ] of type Spin ( ± ) m − ( q ). For χ ∈ Q ∗ := Irr( Q ) = Hom( Q, C × )we denote by L χ its inertia group in L .Let k be an algebraically closed field of characteristic ℓ not dividing q . Let W be a kG -module. Then the restriction of W to Q is semisimple and we have a direct sumdecomposition W | Q = L χ W χ into the Q -weight spaces W χ := { w ∈ W | x.w = χ ( x ) w for all x ∈ Q } for χ ∈ Q ∗ , that is, the Q -isotypic components. The following notion was introduced in [11]: a kG -module W is called Q -linear small if for all χ ∈ Q ∗ the simple L ′ χ -submodules of Soc( W χ )are trivial. A module not satisfying this property is called Q -linear large .The following is well-known: Lemma 3.1.
Let φ ∈ Hom( F q , C × ) be a non-trivial linear character. Then (a) P a ∈ F q φ ( a ) = 0 , (b) P a ∈ F × q φ ( a ) = − .Proof. Clearly (b) follows from (a). To see (a) note that the values of φ are the p -th rootsof unity, where q is a power of p . Also φ is constant on the cosets of ker( φ ), thus X a ∈ F q φ ( a ) = X k ∈ F p X a ∈ ker( φ )+ k φ ( a ) = X k ∈ F p | ker( φ ) | φ ( k ) = | ker( φ ) | X k ∈ F p φ ( k ) = 0 . (cid:3) The odd-dimensional spin groups.
Let G = Spin n +1 ( q ). We first recall the L ′ ∼ = Spin n − ( q )-orbit structure on Q ∼ = F n − q and its dual. The L ′ -module Q admits an L ′ -invariant non-degenerate quadratic form F . Then two non-zero elements x , x of Q liein the same L ′ -orbit if and only if F ( x ) = F ( x ). Thus apart from the trivial orbit thereis one orbit of singular vectors of length q n − −
1, ( q − / q n − + q n − of plus-type, and ( q − / q n − − q n − of minus-type.If W is a kG -module, then for any χ ∈ Q ∗ we thus obtain a direct summand W ( ǫ,µ ) = P ψ ∈ χ L ′ W ψ of the socle of [ W, Q ], where ǫ ∈ { , ±} indicates the type of the stabiliser of χ and µ is an L ′ χ -character (a constituent of Soc( W χ ) | L ′ χ ). Denote the Brauer characterof W ( ǫ,µ ) by χ ( ǫ,µ ) . We also write W ǫ for the sum of all W ( ǫ,µ ) . Lemma 3.2.
Let G = Spin n +1 ( q ) with n ≥ and q odd, and x ∈ Q be a long rootelement of G . Then: (a) χ (0 ,µ ) ( x ) = − , and (b) χ ( ± ,µ ) ( x ) = ± q n − .Proof. We represent elements of Q by row vectors and elements of its dual Hom( Q, F q ) bycolumn vectors. Note that as Q is a self dual L ′ -module, the L ′ -orbit structure on Q andHom( Q, F q ) is identical. We call the elements of Hom( Q, F q ) functionals. So for examplea singular functional is an element of Hom( Q, F q ) on which the L -invariant quadratic form F vanishes. OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 9
We choose a basis { e , . . . , e n − , g, f n − , . . . , f } of Q and its dual basis in Hom( Q, F q )in such a way that the Gram matrix of the L ′ -invariant symmetric bilinear form withrespect to this basis is the matrix all of whose non-zero entries are 1 and appear on theanti-diagonal.Without loss we may assume that x = [1 , , . . . ,
0] as all singular vectors in Q are L ′ -conjugate and G -conjugate to a long root element. Let t = [ a, b, c ] tr ∈ Hom( Q, F q ) with a, c ∈ F q and b ∈ F n − q . Note that t ( x ) = a .Let φ ∈ Hom( F q , C × ) be a non-trivial character. Then for each χ ∈ Q ∗ there exists aunique t χ ∈ Hom( Q, F q ) such that χ ( x ) = φ ( t χ ( x )).So if C ⊆ Q ∗ , then the trace of x ∈ Q on P χ ∈ C W χ is equal to X χ ∈ C dim( W χ ) χ ( x ) = X χ ∈ C dim( W χ ) φ ( t χ ( x )) . We can now calculate the character values on W (0 ,µ ) . First observe that a functional t = [ a, b, c ] tr is singular if and only if one of the following is true:(A) F ( b ) = ac = 0, or(B) F ( b ) = − ac/ = 0.The number of t of type (B) is equal to the number of nonsingular vectors in F n − q whichis q n − − q n − times the number of non-trivial choices for a which is q −
1. By Lemma 3.1these contribute − ( q n − − q n − ) to the trace of x on W (0 ,µ ) .The elements t of type (A) come in two flavours depending on whether or not a = 0. If a = 0, then if c = 0 there are q n − choices for b , while if c = 0 there are q n − − b . In total these t contribute( q − q n − + q n − − q n − − x . Finally if a = 0, then c = 0 while there are q n − choices for b whichyields a contribution of − q n − to the trace. Thus the trace of x on W (0 ,µ ) is χ (0 ,µ ) = − ( q n − − q n − ) + ( q n − − − q n − = − x on W (+ ,µ ) . Observe that the form F evaluatesto a fixed square, say 1, on the functional t = [ a, b, c ] tr if and only if F ([ a, , c ] tr )+ F ( b ) = 1,that is, if and only if one of the following is true:(A) F ( b ) = 1 and ac = 0, or(B) 1 − F ( b ) = ac/ = 0.The contribution to the trace of x by functionals of type (A) occurs in one of two ways:If a = 0, then c = 0 and then there are q n − + q n − choices for b which yields − ( q n − + q n − ) . If a = 0, then there are q choices for c and q n − + q n − choices for b which yields q ( q n − + q n − ) . To compute the contribution by functionals of type (B) we observe that a = 0 and that forevery choice of a there are q n − − ( q n − + q n − ) choices for b after which c is determined uniquely. Thus functionals of type (B) contribute − q n − + q n − + q n − to the trace of x . Summing up the contributions yields that χ (+ ,µ ) ( x ) = − ( q n − + q n − ) + ( q n − + q n − ) − q n − + q n − + q n − = q n − . Finally we calculate the character value of x on W ( − ,µ ) . Observe that F evaluates to afixed non-square α on t = [ a, b, c ] tr , if and only if F ([ a, , c ] tr ) + F ( b ) = α , that is, if andonly if one of the following is true:(A) F ( b ) = α and ac = 0, or(B) α − F ( b ) = ac/ = 0.The contribution by functionals of type (A) occurs in one of two ways: If a = 0, then c = 0 and then there are q n − − q n − choices for b which yields − ( q n − − q n − ) . If a = 0, then there are q choices for c and q n − − q n − choices for b which yields q ( q n − − q n − ) . To compute the contribution by functionals of type (B) we observe that a = 0 and that forevery choice of a there are q n − − ( q n − − q n − ) choices for b after which c is determineduniquely. Thus functionals of type (B) contribute − q n − + q n − − q n − to the trace of x . Summing up the contributions yields that χ ( − ,µ ) ( x ) = − ( q n − − q n − ) + ( q n − − q n − ) − q n − + q n − − q n − = − q n − as claimed. (cid:3) Remark . While a similar result holds for the case of even q , we do not consider thishere as character bounds for Spin n +1 ( q ) with q even have already been obtained in [8].We next compute the trace on a long root element in the Levi factor. Lemma 3.4.
Let G = Spin n +1 ( q ) with n ≥ and q odd. If y ∈ L is a long root elementthen (a) χ (0 ,µ ) ( y ) = q n − − , and (b) χ ( ± ,µ ) ( y ) = q n − ± q n − .Proof. Let χ ∈ Q ∗ be of type ǫ . By definition W ( ǫ,µ ) = µ ↑ P ′ P ′ χ where µ is a linearcharacter of P ′ χ . The element y is unipotent and hence conjugate to an element of L ′ χ thus χ ( ǫ,µ ) ( y ) = χ ( ǫ, ( y ) for all µ . Hence it suffices to compute χ ( ǫ, ( y ).Now χ ( ǫ, | L ′ is the permutation character of L ′ on the cosets of L ′ χ . Thus χ ( ǫ, ( y ) canbe computed by counting the fixed points of y on the cosets of L ′ χ in L ′ . This amountsto counting vectors v in C Q ( y ) with F ( v ) = 0, F ( v ) = 1, and F ( v ) a fixed non-squarerespectively. To make the count we observe that C Q ( y ) is the orthogonal direct sum of atotally singular 2-space with a non-degenerate space of dimension 2 n − C Q ( y ) is equal to q q n − − q n − − C Q ( y ) with F ( y ) = 1 is equal to q ( q n − + q n − ) = q n − + q n − , OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 11 while the number of vectors with F ( y ) a fixed non-square is equal to q ( q n − − q n − ) = q n − − q n − . The claim follows. (cid:3) Proposition 3.5.
Let G = Spin n +1 ( q ) with n ≥ and q odd. If W is a Q -linear small kG -module then C W ( Q ) = { } .Proof. Let x ∈ Q and y ∈ L be long root elements of G , such that x and y are G -conjugate.As x, y are ℓ ′ -elements we can work with Brauer characters.Since W is Q -linear small, we note that W | P ′ decomposes as C W ( Q ) ⊕ W ⊕ W + ⊕ W − .Denote the sum of the multiplicities of the characters χ ( ǫ,µ ) in the character of W ǫ by a ǫ ,denote the character of the P ′ -module C W ( Q ) by χ c and the character of W by φ . Sowith our notation φ P ′ = χ c + a χ + a + χ + + a − χ − . Thus φ ( x ) = χ c (1) − a + a + q n − − a − q n − by Lemma 3.2 and φ ( y ) = χ c ( y ) + ( a + a + + a − ) q n − − a + a + q n − − a − q n − by Lemma 3.4. As x and y are G -conjugate, φ ( x ) = φ ( y ). Thus we find that χ c (1) − χ c ( y ) = ( a + a + + a − ) q n − ≥ q n − > a + a + + a − > W is faithful) and so C W ( Q ) = { } . (cid:3) Proposition 3.6.
Let G = Spin n +1 ( q ) with n ≥ and q odd and let W be an irreducible Q -linear small kG -module. Then W occurs as an ℓ -modular composition factor of theHarish-Chandra induction from L to G of one of the modules in Table 2.Proof. As W is Q -linear small Proposition 3.5 shows that C W ( Q ) = 0. An applicationof [7, Lemma 4.2(ii) and (iii)] then gives that the L -constituents of C W ( Q ) are amongstthose of [ W, Q ]. Recall that the P ′ -module [ W, Q ] is simply a sum of modules of theform W ( ǫ,µ ) . Thus the L -composition factors of the latter are precisely those occurring in W ( ǫ,µ ) . By assumption for all χ ∈ Q ∗ the L ′ χ -submodules of Soc( W χ ) are trivial, that is,any L ′ -composition factors ψ occurring in W ( ǫ,µ ) is a constituent of an induced module µ ↑ L ′ L ′ χ , where µ is a linear character of L ′ χ . In particular, ψ (1) ≤ | L : L χ | ≤ q n − − ψ is one of the modules in Table 2. (cid:3) The even-dimensional spin groups.
We now turn to the even dimensional spingroups G = Spin ǫ n ( q ), ǫ ∈ {± } , with n ≥
3. Here, the L ′ ∼ = Spin ǫ n − ( q )-orbit structureon Q ∼ = F n − q and its dual is as follows. Apart from the trivial orbit there is one orbit ofsingular vectors of length q n − + ǫ ( q n − − q n − ) − q − q n − − ǫq n − .When q is odd, then half of the q − q n − − ǫq n − are of plus type whereasthe others are of minus type (i.e., lie in distinct L -orbits). When q is even all orbits oflength q n − − ǫq n − are in the same L -orbit. As Q is a self dual L -module, the L -orbitstructures on Q and Hom( Q, F q ) are identical.We first prove the analogue of Proposition 3.5. Lemma 3.7.
Let G = Spin ǫ n ( q ) with n ≥ , and x ∈ Q be a long root element of G .Then (a) χ (0 ,µ ) ( x ) = ǫ ( q n − − q n − ) − , and (b) χ ( =0 ,µ ) ( x ) = − ǫq n − .Proof. We argue as in Lemma 3.2 and keep the same notation. Choose { e , . . . , e n − , f n − , . . . , f } ,as basis of Q and its dual basis in such a way that the Gram matrix of the L ′ -invariantsymmetric bilinear form with respect to this basis is the matrix all of whose non-zeroentries are 1 and appear on the anti-diagonal if ǫ = +, while for ǫ = − , the Gram matrixis of this form except that the middle 2 × Q by row vectors and elements of Hom( Q, F q ) bycolumn vectors. Without loss we may assume that x = [1 , , . . . ,
0] as all singular vectorsin Q are L ′ -conjugate. Let t = [ a, b, c ] tr , where a, c ∈ F q , and b is an element of F n − q .We start with the character values on W (0 ,µ ) . Observe that a functional t = [ a, b, c ] tr issingular if and only if one of the following is true:(A) F ( b ) = ac = 0, or(B) F ( b ) = − ac = 0.The number of t of type (B) is equal to the number of nonsingular vectors in F n − q whichis q n − − q n − − ǫ ( q n − − q n − ) times the number of non-trivial choices for a which is q −
1. By Lemma 3.1 these contribute − ( q n − − q n − − ǫ ( q n − − q n − ))to the trace of x on W (0 ,µ ) .The elements t of type (A) come in two flavours depending on whether or not a = 0.If a = 0, then if c = 0 there are q n − + ǫ ( q n − − q n − ) choices for b , while if c = 0 thereare q n − + ǫ ( q n − − q n − ) − b . In total these t contribute( q − q n − + ǫ ( q n − − q n − )) + ( q n − + ǫ ( q n − − q n − ) − q n − + ǫ ( q n − − q n − ) − x . Finally if a = 0, then c = 0 while there are q n − + ǫ ( q n − − q n − )choices for b which yields a contribution of − ( q n − + ǫ ( q n − − q n − )). Thus the trace of x on W (0 ,µ ) is − q n − − q n − + ǫ ( q n − − q n − )+ q n − + ǫ ( q n − − q n − − − ( q n − + ǫ ( q n − − q n − ))= ǫ ( q n − − q n − ) − x on W ( =0 ,µ ) . Note that all L ′ -orbits of non-singular vectors in Q are of length q n − − ǫq n − . We observe that the form F evaluatesto α = 0 on the functional t = [ a, b, c ] tr if and only if F ([ a, , c ] tr ) + F ( b ) = α , that is, ifand only if one of the following holds:(A) F ( b ) = α and ac = 0, or(B) α − F ( b ) = ac = 0.The contribution to the trace of x by functionals of type (A) occurs in one of two ways:If a = 0, then c = 0 and then there are q n − − ǫq n − choices for b which contributes − ( q n − − ǫq n − ) . If a = 0, then there are q choices for c and q n − − ǫq n − choices for b which contributes q ( q n − − ǫq n − ) . OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 13
To compute the contribution by functionals of type (B) we observe that a = 0 and thatfor every choice of a there are q n − − q n − + q n − choices for b after which c is determineduniquely. Thus functionals of type (B) contribute − q n − + q n − − q n − to the trace of x . Summing up the contributions yields that χ ( =0 ,µ ) ( x ) = − ( q n − − ǫq n − ) + ( q n − − ǫq n − ) − q n − + q n − − q n − = − ǫq n − as claimed. (cid:3) Lemma 3.8.
Let G = Spin ǫ n ( q ) with n ≥ . If y ∈ L is a long root element, then (a) χ (0 ,µ ) ( y ) = q n − + ǫ ( q n − − q n − ) − , and (b) χ ( =0 ,µ ) ( y ) = q n − − ǫq n − .Proof. Let χ be an element of type η ∈ { , = 0 } from Q ∗ . By definition W ( η,µ ) = µ ↑ P ′ P ′ χ where µ is a linear character of P ′ χ . As in the proof of Lemma 3.4 it suffices to compute χ ( η, ( y ).Now χ ( η, L ′ is the permutation character of L ′ on the cosets of L ′ χ . Thus χ ( η, ( y ) canbe computed by counting vectors v in C Q ( y ) with F ( v ) = 0, and with F ( v ) = α = 0. Tomake the count we observe that C Q ( y ) is the orthogonal direct sum of a totally singular2-space with a non-degenerate space of dimension 2 n − C Q ( y ) is equal to q ( q n − + ǫ ( q n − − q n − )) − q n − + ǫ ( q n − − q n − ) − v ∈ C Q ( y ) with F ( v ) = α = 0 equals q ( q n − − ǫq n − ) = q n − − ǫq n − . The claim follows. (cid:3) Proposition 3.9.
Let G = Spin ǫ n ( q ) with n ≥ . If W is a Q -linear small kG -modulethen C W ( Q ) = { } .Proof. We argue as in the proof of Proposition 3.5. Let x ∈ Q and y ∈ L be long rootelements of G .As W is Q -linear small, as a P ′ -module it decomposes as C W ( Q ) ⊕ W ⊕ W =0 . Denotethe sum of the multiplicities of the characters χ ( ǫ,µ ) in the character of W ǫ by a ǫ . Wedenote the character of the P ′ -module C W ( Q ) by χ c and the character of W by φ . So φ P ′ = χ c + a χ + a χ =0 and thus φ ( x ) = χ c (1) + a ( ǫ ( q n − − q n − ) − − ǫa q n − by Lemma 3.7 and φ ( y ) = χ c ( y ) + ( a + a ) q n − + a ( ǫ ( q n − − q n − ) − − ǫa q n − by Lemma 3.8. As x, y are G -conjugate we have φ ( x ) = φ ( y ). Noting that a + a > W is faithful we see that χ c (1) − χ c ( y ) = ( a + a ) q n − ≥ q n − > C W ( Q ) = { } . (cid:3) Proposition 3.10.
Let G = Spin ǫ n ( q ) with ǫ ∈ {±} and n ≥ and let W be an irreducible Q -linear small kG -module. Then W occurs as an ℓ -modular composition factor of theHarish-Chandra induction from L to G of one of the modules in Table 5.Proof. As W is Q -linear small Proposition 3.9 shows that C W ( Q ) = 0. As in the proof ofProposition 3.6 this implies that any constituent of W | L ′ has dimension not larger than( q n − − q n − + 1) and then we may conclude using Theorem 2.5. (cid:3) The main result
We are finally in a position to obtain the sought for gap result. For this, we keepthe notation from the previous sections. In particular P = QL is an end-node maximalparabolic subgroup of Spin ( ± ) m ( q ) with unipotent radical Q and Levi factor L . Furthermore,we keep the notation ρ i , ρ s , ρ ± t , . . . for small dimensional irreducible characters as inSection 2 with s, t certain semisimple elements in G ∗ . For an ordinary character χ wedenote by χ its ℓ -modular Brauer character, that is, its restriction to ℓ -regular classes.Throughout, for an integer m , we set κ ℓ,m := ( ℓ | m, . The odd-dimensional spin groups.
Let n ≥ G = Spin n +1 ( q ) and ℓ a prime notdividing q . We first collect some results on the decomposition numbers of low dimensionalordinary representations in non-defining characteristic. Lemma 4.1.
Let G = Spin n +1 ( q ) , n ≥ , and ǫ ∈ {±} . Then ρ ǫt remains irreduciblemodulo ℓ when ℓ ( q − ǫ or when o ( t ) = ℓ f or ℓ f , and otherwise ( ρ ǫt ) = ( ( ρ + ρ + ǫ G ) if o ( t ) = ℓ f > , ( ρ s,q + ǫρ s, ) if o ( t ) = 2 ℓ f > , ℓ = 2 . Proof.
According to the description in the proof of Theorem 2.2, the character ρ − t issemisimple in the Lusztig series indexed by an element t ∈ G ∗ of order dividing q + 1. Butby the observation in [9, Prop. 1], the semisimple characters in a Lusztig series E ( G, t )remain irreducible modulo all primes ℓ for which the ℓ ′ -part of t has the same centraliseras t . By our description of the parameters t , this is the case unless this ℓ ′ -part has orderat most 2.When o ( t ) is a power of ℓ , then by Brou´e–Michel [2, Thm. 9.12] ρ − t lies in a unipotentblock, and by its explicit description in terms of Deligne–Lusztig characters, we find that( ρ − t ) = ( ρ + ρ − G ) . Finally, if o ( t ) is twice a power of ℓ , then its 2-part is conjugate to s and hence ρ − t lies in the same ℓ -block as the Lusztig series of ρ s, . Again the claim followsfrom the explicit formula for the semisimple character ρ − t in terms of Deligne–Lusztigcharacters.The argument for the characters ρ + t is entirely similar. (cid:3) Thus, either ρ ± t remains irreducible modulo ℓ , or its ℓ -modular constituents are knownif we know them for the remaining characters in Table 2. We therefore henceforth onlyconsider the latter. OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 15
Lemma 4.2.
Let G = Spin n +1 ( q ) with q odd and n ≥ , and ℓ a prime not dividing q ( q + 1) . Assume that the ℓ -modular decomposition matrix of E ℓ ( G, s ) is unitriangular.Then the entries in its first ten rows are approximated from above by Table 6, where k := n − . In particular, both ρ s, and ρ s,q remain irreducible modulo ℓ . ρ a ρ ⊠ ⊠ . ⊠ ρ . . ⊠ ρ . . . ⊠ ρ k + 1 . . . ⊠ ρ k + 1 . . . . ⊠ ρ . . . . . . ⊠ ρ . . . . . . . ⊠ ρ . k + 1 . . . . . . ⊠ ρ . k + 1 . . . . . . . Table 6.
Approximate decomposition matrices for E (Spin n +1 ( q ) , s ), n ≥ q a ρ is the precise power of q dividing ρ (1). Proof.
By a result of Geck, see [2, Thm. 14.4], the Lusztig series E ( G, s ) of the semisimpleinvolution s ∈ G ∗ forms a basic set for the union of ℓ -blocks E ℓ ( G, s ). By Lusztig’s Jordandecomposition E ( G, s ) is in bijection with E ( C, C = C G ∗ ( s ) ∼ = Sp ( q )Sp n − ( q ),hence with E (Sp ( q ) , × E (Sp n − ( q ) , E ( G, s ) by exterior tensor products of unipotent characters, so that ρ s, = 1 ⊠ ρ s,q = St ⊠ ρ s, ∈ E ( G, s ) is semisimple, so remains irreducible modulo all odd primes(see [9, Prop. 1]). We next claim that the ℓ -modular reduction of ρ s, does not occur as acomposition factor of ρ ◦ s,q . Indeed, by the known decomposition numbers for Spin ( q ) ∼ =Sp ( q ) (see [14]), ρ s,q remains irreducible unless ℓ | ( q + 1). Now assume the assertion hasalready been shown for Spin n − ( q ). Thus the upper left-hand corner of the ℓ -modulardecomposition matrix for E ℓ (Spin n − ( q ) , s ) has the form: ρ s, ρ s,q . G of the same form, and thus, by uni-triangularity, ρ ◦ s,q is irreducible. The upper bounds on the remaining entries given inTable 6 are now obtained inductively exactly as in the proof of [3, Thm. 6.3] by Harish-Chandra inducing projective characters from a Levi subgroup of an end node parabolicsubgroup. (cid:3) Proposition 4.3.
Let G = Spin n +1 ( q ) with q odd and n ≥ , and ℓ a prime not divid-ing q ( q + 1) such that the ℓ -modular decomposition matrix of G is uni-triangular Assume that ( n, q ) = (4 , , (5 , . Then any ℓ -modular irreducible Brauer character ϕ of G of de-gree less than q n − − q n is a constituent of the ℓ -modular reduction of one of the complexcharacters listed in Table 2.Proof. By assumption we have that ϕ (1) is smaller than the constant b in [11, Table 4],whence by [11, Prop. 5.3] the module W is Q -linear small, unless we are in one ofthe exceptions listed in [11, Rem. 5.4]. The only groups on that list relevant here areSpin (3) , Spin (3), which were excluded. Then Proposition 3.6 applies to show that ϕ occurs in the ℓ -modular reduction of a constituent χ of the Harish-Chandra induction ofsome character ψ of L as in Table 2. In particular χ is either unipotent, or in E ( G, s ) or E ( G, t ).If χ is unipotent, then by [3, Cor. 6.5] we obtain that ϕ is in fact a constituent of oneof the complex characters in Table 2. Now assume that χ ∈ E ( G, t ). By the main resultof [1], the union of ℓ -blocks in E ℓ ( G, t ) is Morita equivalent to the union of unipotent ℓ -blocks in E ℓ ( C, C is dual to C G ∗ ( t ) ∼ = Sp n − ( q )( q − ǫ ℓ -modular Brauer character of C is at least c := ( q n − − q n − − q ) / ( q + 1) (observe that the Weil modules are not unipotent as ℓ = 2). Hence any ϕ = ( ρ ǫt ) ◦ in E ℓ ( G, t ) has degree at least | G ∗ : C G ∗ ( s ) | q ′ · c = 12 ( q n − q n − − q n − − q )( q + 1) which is larger than our bound.Finally, assume that χ ∈ E ( G, s ). The Harish-Chandra induction of ρ s, and ρ s,q from L to G only contains the characters denoted ρ s, , ρ s,q , 1 ⊠ ρ i and St ⊠ ρ i , for i = 2 , ,
4, fromTable 6. Since we assume that the ℓ -modular decomposition matrix of G is uni-triangular,lower bounds for the degrees of the corresponding Brauer characters can be derived fromLemma 4.2. These show that ϕ must be equal to one of ρ ◦ s, , ρ ◦ s,q . (cid:3) Remark . The proof shows that in fact it suffices to assume that the ℓ -modular de-composition matrix of G has a uni-triangular submatrix for the rows corresponding tothe constituents of the Harish-Chandra induction from L to G of the complex characterslisted in Table 2. By [3, Thm. 6.3] under mild assumptions on ℓ this is known for theunipotent characters; in those cases we only need to assume it for the characters in E ( G, s )listed in Table 6.Let d ℓ ( q ) denote the order of q modulo ℓ . We then obtain Theorem 2 in the followingform: Theorem 4.5.
Let G = Spin n +1 ( q ) with q odd and n ≥ , and ℓ ≥ a prime notdividing q such that d ℓ ( q ) is either odd, or d ℓ ( q ) > n/ . Let ϕ be an ℓ -modular irreducibleBrauer character of G of degree less than ( q n − − q n ) . Then ϕ (1) is one of , q n − q − , q ( q n − q n − − q + 1 , q ( q n + 1)( q n − + 1) q + 1 , q ( q n + 1)( q n − − q − − κ ℓ,q n − , q ( q n − q n − + 1) q − − κ ℓ,q n +1 ,q q n − q − , q n − q ± . OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 17
Proof.
We claim that the assumption of Proposition 4.3 is satisfied in our situation. First,it is well-known that decomposition matrices for blocks with cyclic defect groups are uni-triangular, so we are done in that case. Now let
G ֒ → ˜ G be a regular embedding, that is,˜ G is a group coming from an algebraic group with connected centre and with the samederived subgroup as G . By the result of Gruber–Hiss [6, Thm. 8.2(c)] the decompositionmatrix of any classical group with connected centre is uni-triangular whenever ℓ > ℓ is linear for ˜ G if the order d ℓ ( q ) of q modulo ℓ isodd.) Then, the proof of Proposition 4.3 shows that all ℓ -modular Brauer characters of˜ G of degree less than q n − − q n are as claimed. Now | ˜ G/C ˜ G ( G ) | = 2, so any irreducible(Brauer) character of ˜ G restricted to G has at most two irreducible constituents. Thusour claim holds for G as well.It remains to discuss the groups Spin (3) and Spin (3) excluded in the statement ofProposition 4.3. Their Sylow ℓ -subgroups are cyclic for all primes ℓ >
5, and then allsmall-dimensional Brauer characters can easily be determined from the known ordinarycharacter degrees. For the prime ℓ = 5 we have d ℓ ( q ) = d (3) = 4, so it is excluded in ourconclusion. (cid:3) Remark . For even q we have Spin n +1 ( q ) ∼ = Sp n ( q ), and for these groups it was shownby Guralnick–Tiep [8, Thm. 1.1] that the conclusion of Theorem 4.5 continues to hold for n ≥
5, while for n = 4 the lower bound has to be replaced by q ( q − q −
1) when q > q = 2.4.2. The even-dimensional spin groups.Theorem 4.7.
Let q be odd. Let either G = Spin +2 n ( q ) with n ≥ , and ℓ ≥ a prime notdividing q ( q + 1) , or let G = Spin − n ( q ) with n ≥ , and ℓ ≥ is a prime not dividing q .Then any ℓ -modular irreducible Brauer character ϕ of G of degree less than q n − − q n +4 isa constituent of the ℓ -modular reduction of one of the complex characters listed in Table 5.Proof. By comparing we see that ϕ (1) is smaller than the constant b in [11, Table 4],whence by [11, Prop. 5.3] the module W is Q -linear small, unless we are in one of theexceptions listed in [11, Rem. 5.4]. Then Proposition 3.10 applies to show that ϕ is aconstituent of the ℓ -modular reduction of the Harish-Chandra induction of some character ψ of L as in Table 5. But in fact, the only exception relevant here is Spin +10 (3), and therethe smallest degree of a non-trivial character of L ′ χ = Spin +6 (3) = SL (3) is 26 and [ L : L χ ] = (3 − + 1) = 2240, while the bound in the statement is 3 − − = 39366,so the conclusion holds here as well.We consider the various possibilities. If ψ is unipotent, so one of 1 L , ρ or ρ , then itsHarish-Chandra induction only contains characters occurring in Table 6 or 7 of [3]. Ourclaim in this case follows from [3, Cor. 5.8].Next assume that ψ is one of ρ − t or ρ + t . Then its Harish-Chandra induction lies in theLusztig series E ( G, t ). By the main result of [1] the ℓ -blocks in E ℓ ( G, t ) are Morita equiv-alent to the unipotent ℓ -blocks of a group dual to C G ∗ ( t ) ∼ = CSO ǫ n − ( q ). In particular,the decomposition matrices are the same. For the latter we may apply [3, Prop. 5.7] tosee that all Brauer characters in that series apart from ρ ± t have degree at least( q n − ǫ q n − − ǫ q + 1) (cid:16) q ( q n − − ǫ q n − + ǫ q − − (cid:17) which is larger than our bound. A similar argument applies to the constituents of theHarish-Chandra induction of ρ ± s,a and ρ ± s,b . In this case, [1] yields a Morita equivalence be-tween the blocks in E ℓ ( G, s ) and the unipotent blocks of the disconnected group CO ǫ n − ( q ),with connected component of index 2. Another application of [3, Prop. 5.7] shows ourassertion in this last case. (cid:3) In order to make the previous result more explicit, we determine the ℓ -modular re-ductions of some of the low-dimensional Spin ± n ( q )-modules in Table 5. The first resultextends [3, Thm. 5.5]: Proposition 4.8.
Let G = Spin − n ( q ) with q odd and n ≥ , and ℓ ≥ a prime dividing q + 1 . Then the first eight rows of the decomposition matrix of the unipotent ℓ -blocks of G are approximated from above by Table 7, where k := n − . ρ a ρ (cid:0) ,n − (cid:1) (cid:0) ,n − − (cid:1) k (cid:0) , ,n (cid:1) k (cid:0) ,n − − (cid:1) (cid:0) k (cid:1) k . (cid:0) , ,n − (cid:1) (cid:0) k +12 (cid:1) k k (cid:0) , ,n − (cid:1) (cid:0) k (cid:1) (cid:0) k +12 (cid:1) k . . (cid:0) , ,n − (cid:1) (cid:0) k +12 (cid:1) k k . . (cid:0) , , n − (cid:1) . . . . . . ps ps ps ps A ps ps . Table 7.
Approximate decomposition matrices for Spin − n ( q ), n ≥ Proof.
This is proved along the very same lines as [3, Thm. 5.5]. We start with the case n = 6. Here, the six principal series PIMs are obtained from the decomposition matrixof the Hecke algebra H ( B ; q ; q ). The projective character in the A -series comes byHarish-Chandra induction from a PIM of a Levi subgroup of type A , while the projectivecharacter in the “.2”-series is obtained from a Levi subgroup of type D × A . This showsthe claim for n = 6 (with k = 0). Then Harish-Chandra induction of these eight projectivecharacters yields projective characters of G with the stated decompositions for all n ≥ a -value at most 3 by[3, Prop. 5.3]. (cid:3) Remark . For n = 5 there is at least one unipotent PIM of D ( q ) in the Harish-Chandra series of type A of a -value 2, and we do not see how to rule out that theremight be several of them. Lemma 4.10.
Let G = Spin ǫ n ( q ) , ǫ ∈ {±} and n ≥ . Then ( ρ ǫt ) = ( ( ǫρ + ρ + 1 G ) if o ( t ) = ℓ f > , ( ρ ǫs,a + ρ ǫs,b ) if o ( t ) = 2 ℓ f > , ℓ = 2 , OW-DIMENSIONAL REPRESENTATIONS OF FINITE ORTHOGONAL GROUPS 19 and ρ ǫt remains irreducible modulo ℓ otherwise. Furthermore, ρ ± s,a , ρ ± s,b remain irreduciblemodulo all primes ℓ = 2 .Proof. According to the description in the proof of Theorem 2.5, the characters ρ ± s,a and ρ ± s,b are semisimple in Lusztig series indexed by elements of order 2, so we may argue asin the proof of Lemma 4.2 using [9, Prop. 1]. The proof for ρ ± t is completely analogousto the one of Lemma 4.1. (cid:3) Corollary 4.11.
Keep the assumptions on n, q and ℓ from Theorem 4.7. If ϕ is an ℓ -modular irreducible Brauer character of Spin ǫ n ( q ) of degree ϕ (1) < q n − − q n +4 then ϕ (1) is one of , q ( q n − ǫ q n − + ǫ q − − κ ℓ,q n − + ǫ , q ( q n − − ǫ q − − κ ℓ,q n − ǫ ,
12 ( q n − ǫ q n − ± ǫ q ∓ , ( q n − ǫ q n − ± ǫ q ∓ . Proof.
This follows directly from Theorem 4.7 with the partial decomposition matrix forthe unipotent characters in [3, Prop. 5.7] and the statement of Lemma 4.10. (cid:3)
This implies Theorem 1.
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