Representations of Ui

Preliminary Lemmas

7.2 Representations of Ui

Proof o f Lemma 7.1-4- Let X = x L where x = x r. It suffices to show th a t L is transitive on X a.

Set G = (L, a) such th at ga = a(g) for g E L. T hen G acts transitively on X with Gx = L xT where T = (cr). Then N q ( T) = L T , F ix ( T ) = X a. By Lemma 7.1.5, L T is transitive on X a if and only if fl G x = T ^ x if an d only if fl L XT = T ^ x if and only if <x£cr- 1

n

L x = cr^cr-1 . Here cr^cr- 1 = {g~^cjga~x \ g E L}.

C ertainly L is a connected algebraic group. By Lemma 7.1.3, Lx is the semi-direct product of connected algebraic groups an d hence connected. So by Lemma 7.1.6, a La ~ l = L and o^cr- 1 = Lx. Therefore, th e final equality indeed holds. Hence the lemma is proved.

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Recall the ran k of x E V ~ as a m atrix is L-invariant. For each 1 ^ r ^ m , we set Irr(V, r) to be th e set of r 6 Irr(V ) identified w ith th e rank r elements in V ~ . we let r r ^G Irr(V, r) be identified w ith x r .

has I orbits on the non-trivial characters in Irr(Z ) w ith { l , r r ; l ^ r ^ 1} being a set of representatives of L-orbits on Irr(Z ), where r r is identified w ith x r in Lem ma 7.1.3.

Let p E Irr(C/) lying over r r , for some 1 ^ r ^ I. Set N = ker(<p) an d K = ker(rr ).

As Z = Z (U ), <p\z is a multiple of rr . So K ^ N . As Z is abelian, Z / K is cyclic of order p. Hence as Z £ N , it follows th a t N D Z = K . Set U = U / K and U = U /Z . As P is irreducible on U and Z and U is non-abelian, Z = = <$(£/). T hus Z = U™ = ^(t7 ) and as 1 ^ Z = , U is non-abelian.

Claim $ ( Z ( U ) ) = 1. If not, as $ (U ) = Z , \Z(U) : Qi(Z(U ))\ = p and hence Z (U )/Q i( Z ( U ) ) is centralized by some e E I? = Z ( G n~21), contradicting U = [£/, e]. Let U = U /N . As U has a faithful irreducible representation, Z (U ) is cyclic, so as Z ( U) is an homomorphic image o f Z (U ) and $ ( Z ( U ) ) = 1, Z (U ) = Z and hence U is extraspecial.

Now observe p and rr can be regarded as a character of U and Z , respectively.

B ut the irreducible representations of extraspecial p-groups are well known. See for instance (34.9) in [As], Explicitly, regarded as a character of U, p is faithful and th e unique irreducible character of U lying over r r . Equivalently p is th e unique m em ber of Irr(t/, r r ) w ith ker(<p) = N . Clearly Nz,(p) = Nz,(rr) fl Nl(N ). We have proved th at

L em m a 7 .2 .1 . Let 1 ^ r ^ I and <p E Irr(U, r r ) with N = ker(<p). Then U / N is an extraspecialp-subgroup, p is uniquely determined by N , and Nl(p) = Ni,(tt)C\Nl(N ) .

We now divide the discussion into two cases, namely r = I and r < I.

Assume r = I. We claim th a t N = K so U is extraspecial. To prove the claim, we need to show Z = Z(U). Clearly Z ^ Z (U ). If Z ^ Z(U), th en th ere exists u E U — Z w ith u E Z(U ), so [u, U] ^ K . However, Nl(ti) = Gl x Gn_2i stabilizes K and acts irreducibly on U. Therefore, U = (uNL^n \ Z ). It follows th a t [U, U] ^ K , a contradiction. Hence the claim follows.

As N l ( K ) = N l(t[), by Lemma 7.2.1, N l ( p ) = N ^ f a ) . Moreover, as |I7| =

p q 2 i ( n - 2 i ) ^ = gZ(n-2Z) have shown th a t

L e m m a 7 .2 .2 . Itt(U, t{) consists o f a unique member p . Moreover, ker(<p) = ker(rj),

<p(1) = (f^n~2L\ and

N L{p) = N L(n) = G* x Gn~21.

Now assume 1 ^ r < I. Recall th a t U = V 1 x (V n-2i)* where V k is th e natural module for G +fc, k = l , n — 21. By Lemma 7.1.3.3, Nl{tt ) = Li x Gn~2L where Gn~21 centralizes Z , and L \ stabilizes a co-dimension r subspace R of V 1 with

L x = C x (Li x L i) where

G = CcKft) n

CG[(Vl/R ) *£

Mr,z_r (* » , Li = GLl(tf) n ivGt(i*') =

Gr and

Li =

Cgi(R!)

n

Ngi{R) = G+V~rl

Here V 1 = R (B R!. Therefore, i? is the unique proper nontrivial Li-subm odule, and consequently W = R ® V n~21 is the unique proper nontrivial N l(rr )-submodule where W is th e preim age of W in U . Moreover, as Z(U ) ^ Z , it follows th a t Z(U ) = W . Let E = Z ( G n~21). Recall th a t for each e € E # , Cu(e) = Z . So Z = Cu(e) = Ccf(E), [e, W \ = [E, W] and W = Z x [LJ, W]. Set N0 = [E, W] and let Nq be the full preimage of Nq in W . So W = N q Z w ith NqC\Z = K and W = Nq x Z .

As U / N is extraspecial, W = N Z . As Z fl N — K , it follows th a t W = N x Z.

So as <p is uniquely determ ined by N, Irr(t/, r r ) is in 1-1 correspondence w ith th e set X of complements of Z in W as $(W") = 1. Moreover, X is in 1-1 correspondence w ith

Hom(7Vo, Z ) = Hom(iVo,Fp) = Irr(iVo)

via a m ap Y 4>y w ith k e r ( ^ ) = Y D N 0. Now ATz,(rr ) centralizes r r and hence also Z , so these bijections are all N p(rr )-equivariant. Let (p correspond to 0 E Irr(iV'o).

Observe

(1) Cxt (i?) = C x L i acts trivially on N , and CLl( V L/ R ) = C x L" acts trivially on U = U /N . So CLl(U) = Z (G )C Ll( V l/ R ) . Moreover, |*7| = pq2^ n~2l\ So 9?(1) = gr(n~20 .

(2) Nq = V i-r x V n_2i is a tensor module for L" x Gn~21.

(3) If 0 is the trivial character, then N = N 0 N L^ xGn-2i(4>) = L'{ x Gn~21. Conse­

quently Nz,(<p) = N L(rr) and

V i M / C i l<yl/R) =

Li X c r - M. (7.5)

(4) If 0 is non-trivial, then iVi»xc?‘- Ji(0) is described in Proposition 6.1.4.5. As­

sume th e radicals of 0 are (J?1, R 2) as in th e proposition. T hen in this case, NU<p) / C Li (V ‘/ R ) n N l (V ) at L[ x L , (7.6)

w ith i/2 = N Gn-2l(R2).

In summary,

L e m m a 7 .2 .3 . Let r < I and P ' = N p (rr). There is a P'-equivariant bijection from Irr{U,Tr) to Itt{Nq), such that if tp corresponds to 0, then Np(jp) = Np>{4>).

Moreover, <p(l) = gr(n_20 fo r all ip £ Irr[U,rr).

Observe th a t in th e above lemma, if r = I, th en N = 1. And hence Lemma 7.2.3 coincides w ith Lemma 7.2.2.

L e m m a 7 .2 .4 . Let 1 ^ r ^ I, and ip £ IrriJJ, r r ). Then cp is extendable to Np(ip).

That is, there exists <p £ Irr{Np{<p)) such that <p\u = <P-

T he proof is an application of the following result due to Dade (see [D3]).

L em m a 7 .2 .5 . Let E be an extraspecial p-group and G = E x H with Z (E ) ^ Z(G ).

Assume that fo r each normal pf-subgroup K o f H , [E, K] = E . Let <p £ Irr(E) with

<p{1) > 1. Then ip is extendable to G.

Proof o f Lemma 7.2.4- Let ker(<p) = N and U = U /N . We may regard <p as a character of U. To prove the lemma, it suffices to show th a t <p can be extended to H U where H = N[,(cp). Recall from Lemma 7.2.1 th a t U is extraspecial.

Assume r = I. T hen by Lemma 7.2.2, H — Gl x Gn~21. As H is irreducible on U / Z = U /Z , [X, U] = U for any normal p'-subgrup of H w ith X ^ Z{G). Therefore, [K, U] = U for any norm al p'-subgroup of H /Z { G ) . Hence by Lemma 7.2.5, <p is extendable to (H /Z { G ) ) U . B ut [Z(G), U] = 1, so <p can be extended to Np(<p).

Assume r < I. A dopt the notation of th e proof of Lemma 7.2.3. Let <p correspond to <f> G Irr(iVo). If (j) = 1, then from equation 7.5,

H / H n CLl( V l/ R ) = G r x Gn_2i.

B ut U is an extraspecial p-group. So as in th e preceeding case, <p is extendable to U ( H / H D C ^ i y 1 /R ))- B ut [H fl Cl^{V1 / R ), U] = 1, so <p can be extended to HU , or equivalently to Np(tp) in this case.

Assume 0 ^ 1 and th a t the radicals of <f> axe ( R l , R 2). Then H / H fl C£t ( V l/ R ) is described in equation (7.6). If R 2 is nondegenerate, then L2 is the p roduct of two general linear groups, and W = [W, L 2] © C ^r(L2), so there is a (yj)-invariant autom orphism of U m appping Nq to N . T hus U / N 0 is Nl ((p)-isomorphic to U / N , so we can take N = No, a case already handled (i.e., <f = 1).

If R 2 is degenerate, then

L 2 = CGn-2z(Rad(R2)) fl ^{C Gn -2 i} (T)

where T is a complement to Rad(i?2) to R 2, so F * (L 2) = Op(L2) and the only normal p'-subgroups of Au.tff(U) are the normal p'-subgroup X of L'x. B ut as L[ x G n~21 is irreducible on U / W = U /Z (U ), U = [U, X ], so again by Lemma 7.2.5, <p can be extended to Np(<p). Done.

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