Preliminary Lemmas
Chapter 3 Chapter 3 Refinement and General Discussion
3.1 A Refinement for The Finite Groups of Lie Type
Let G = G{q) be a finite group of Lie type of rank n over a finite field of order q = pe. T h a t is, G is either a n ordinary Chevalley group defined over Fg or the subgroup of fixed points of an ordinary Chevalley group defined over F qi«ri or F ? by some autom orphism a. Let (G, B , N , S ) be a T its system for some root system E , H — B fl N , and B = U H where U = 0 P{B). Set I = [n]. Let {P i,i £ I} be the set of maximal parabolic subgroups of G over B and P j = C\jeJ f°r J Q I- In particular P i = B . By convention Pq, = G. For J C I , set U j = Op(P j). Let P j be th e parabolic subgroup opposite to P j and U j = Op{ P j) th e opposite unipotent radical. Clearly P j = UjLj where L j = Pj C \ P J is a Levi factor of P j. The following is a n example (when th e base field F is finite), which we will refer to from tim e to tim e.
E x a m p le 3 .1 .1 . Let V be an n-dimensional vector space over a field F and G = G L (V ). T hen G is a group of ran k n — 1 and in this case I = [n —1]. We fix a basis { e j] j € [n]} for V . Set V0 = 0, Vj = (e*-; 1 ^ ^ j ) for 1 ^ j ^ n — 1, and Vn = V . For
j i < j 2 < - - - < ja, let c j be th e flag
0 < V J l < V j 2 < . - - < V j'.
We may choose B = iVG(cy) and P j = Ng(c j). Consequently for 1 ^ r ^ n —1, P ~ = N ’g (V ~ ) where V ~ = (e*;r + 1 ^ i ^ n).
Let A be the set of chains of the poset on {Uj] J C 1} ordered by inclusion.
L em m a 3 .1 .2 . A is a set o f representatives o f the G-orbits on radical p-chains o f G. The stabilizer in G o f c E A is the stabilizer o f the fina l term of c.
Proof. If 0 < Uj-t < - • • < Ujm is a chain in V , then N ciJJji) = P jt ^ Pjs for i ^ j , so th e second statem ent holds. Let c : J70 C I7i C - • • C {/r be a radical p-chain of G. So Uq = Op{G) = 1. By definition, Ui is a radical p-subgroup of G. However, it is well known th a t the radical p-subgroups are precisely the unipotent radicals of th e parabolics and each parabolic subgroup is conjugate to a unique member of {Pj-, J c I } . Therefore, replacing c by a G-conjugate if necessary, we m ay assume U\ = Ujl for some J i, J i Q I . Set P — P jl = N g(U i).
Next by definition U% is a radical p-subgroup of P and therefore, U<ijU\ is a radical p-subgroup of P /U i- B ut P = P /U x is a finite group of Lie type, possibly with a disconnected diagram, and the radical p-subgroups of P are th e images in P of the radical p-subgroups of G contained in P , which are conjugates of U j for J xC J C I.
Therefore, replacing c by a P-conjugate if necessary, we may assume U2 = Op(Pj2) for some J2, J \ Q J2 I-
By definition Ui is a radical p-subgroup of C \\^j^i-XNG{Ui) for all i. We may proceed by induction to conclude th at c can be conjugated to a chain of P . As distinct parabolics over B are not conjugate, no two chains in V axe conjugate. The
proof is complete. □
L em m a 3 .1 .3 . S et Z = Z (G ).
(1) G has [ G /G ^ l p-blocks o f defect 0 and \Z\ p-blocks o f fu ll defect and no other blocks. There is a 1-1 correspondence between Irr{Z) and the set o f p-blocks o f G o f fu ll defect given by p Sp, such that cp £ Irr(G ) lies in S p if and only if (p lies over p and |<p(l)|p < |G|P.
(2) Let 0 7^ J C I . P j has only p-blocks o f fu ll defect. Moreover, the B rauer map gives a 1-1 correspondence between the set o f p-blocks o f P j and the set o f p- blocks o f G o f fu ll defect. I f S is a p-block o f P j and the Brauer correspondent S G corresponds to p £ Itt( Z) in (1), then (p £ Irr{P j) lies in S i f a nd only i f p lies over p.
These axe either well known results (see [H]) on the block theory of the finite groups of Lie type or direct consequence of Lemma 2.1 in [OU].
For d ^ 0, recall Irr<i(G) is the set of characters p £ Irr(G ) whose 5-height is d and kd{G) = |Irrd(G)|.
P ro p o s itio n . 3 .1 .4 . Let p £ Irr{Z), \G\P = 5* . The following are equivalent:
(1) D O C holds fo r (G , p , Sp) and i > 0. Here S p is the p-block o f fu ll defect o f G corresponding to p.
(2) E x c / ( - D |J|fcd(Px,p) = 0, where d = d o ~ i/e .
Proof. By p art (1) of Lemma 3.1.3, th e p-blocks of G of positive defect axe of full defect and param eterized by the central characters of G. Fix our choice o f p-block S = Sp corresponding to p £ Irr(Z ). Let A be as in Lemma 3.1.2. So A is a set of representatives of the G-orbits on the set of p-radical chains. For J Q I , let A (J) be the set of chains in A whose final term is U j. By Lemma 3.1.2, Gc = P j for all c £ A (J). As |.P/|p = |G|P and i > 0, by Lemma 3.1.3 we have k (G c, S , i) = kd (P j,p ). Finally the Euler characteristic x(AT) of the complex AT of all chains of proper nonempty subsets of J is 1 + (—1)IJ '. T hen as
A(<7) = { Um < • • • < < U j I J i < • • • < *7|c|- i E AT U {0} } ,
it follows th a t
E (-1 )w=Ec -1)'61"1 -1=x(*o -1 = c-i)w-
c€A(J] b€K
Therefore, (1) is tru e if and only if
0 = £ ( - l ) M f c ( C „ S ,i ) = J 2 E ( - l ) wk ( e / , p) = S ( - l ) w * a(P /, p),
c € A J C I C 6 A (J ) j c i
i.e., if and only if (2) is true. Done.
□
R e m a rk 3 .1 .5 . (1) T he reason th a t we don’t have a reformulation when i = 0 is because k(G , S p, 0) ^ k ^ G , p). T he p-defect 0 case will be handled below.
(2) We consider th e g-height of characters of G rath er th an p-height to simplify notation. In th e case of G = GUn (q), it follows from Proposition 4.2.2 th a t the p-part of th e degree o f <p(l) for <p € Irr(G) is an integral power of q, so th a t th e g-heights involved are all integers. In general, according to Deligne-Lusztig theory, the p -p art of <p( 1) is always an integral power of q (including th e case G = G Ln(q)) except in some cases when p is small.
L em m a 3 .1 .6 . ( I ) D O C holds fo r G at p when i — 0.
(2) D O C holds fo r G at p i f and only i f
5 3 ( - l){J[k d(P j,p ) = 0 , Vp € Irr(Z ),V d < do- (3.1)
J C I
Proof. A n irreducible character <p of G lies in a p-block of defect 0 if and only <p(l) is divisible by the p -p art of |G |, or equivalently if and only <p has p-defect 0. So as S p is of positive defect, k(G , Sp, 0) = 0. O n th e o th er hand, for all 0 ^ J C / , Op(P j) ^ 1. So it is easy to show th a t k (P j, 5 P, 0) = 0, i.e., |<p(l)|p < g4*0 for cp6 Irr(P j). Therefore, D O C holds for G a t p when i = 0. P a rt (2) th en follows imm ediately from p a rt (1) and Proposition 3.1.4.