Here we will consider the behaviour of the outer automorphisms γ and φ of the unitary groups. Let q “ pe. Let d :“ pq `1, nq “ |δ| and let Unpqq “Unpq, Inq. Recall from Section 3.2.6 that
OutpUnpqqq “ xδ, φ, γ|δd“φ2e“γ2 “1, δγ “δ´1, φe“γ, δφ“δpy. Note that by [7] the isomorphism type ofxUnpq, Bq, φydepends on the choice of formBpreserved by the unitary group whennis even andqis odd. Recall that in Section 3.2 we defined σ and φfor groups preserving our standard unitary form matrixIn. The problem is that even thoughσ “φeand φare automorphisms of SUnpq, Inqand stabilise this group, they do not necessarily stabilise SUnpq, Bq for every non-degenerate unitary formB. One reason is that these automorphisms might not fix the form at all and even if they fix B, thenxSUnpq, Inq, φy(or xSUnpq, Inq, σy) is not necessarily isomorphic to xSUnpq, Bq, φy (orxSUnpq, Bq, σy).
To find the action of the field automorphism for any non-degenerate unitary form B we find a way to map GρďSUnpq, Bq to some isomorphic groupHďSUnpq, Inq. LetAPGLnpq2qsuch thatpGρqA“H ďSUnpq, Inq. Since ρA is equivalent toρ it is sufficient for our purpose to determine the action of the outer automorphisms of Unpq, Inq onH.
If we do not need to use any specific generators for Gρ however than we can assume without loss of generality that Gρ preserves our standard unitary form matrixIn.
Assume throughout that each C“CGUnpqqconjugacy class ofGρsplits into d conjugacy classes in SUnpqq. By [8, Lemma 4.6.3, p.190] there are
two OutpUnpqqq-conjugacy classes containing elements of the formφδiwhen dis even and one such conjugacy class when dis odd. We will first consider the case whendis odd.
Lemma 4.5.1. Let ρ:GÑΩ“SUnpqq be an absolutely irreducible repre- sentation. Suppose thatdis odd and that there exists αPOutpGq such that αρis equivalent toρφ. Then anΩ-class ofGρis stabilised byxφyinOut
p¯Ωq. Proof. This follows from [8, Lemma 4.6.3, p.190] and Lemma 4.3.7.
Now we will look at the case whendis even. The following lemma gives us a way of deciding whether a subgroup of a unitary group is stabilised by φor by φδ.
Lemma 4.5.2 ([8, Lemma 4.6.5, p.191]). Let ρ : G Ñ GLnpq2q be an absolutely irreducible representation such that Gρ ď SUnpq, Bq – SUnpqq, where B is some non-degenerate unitary form. Assume that d“ pq`1, nq is even. Also assume that there exist α P OutpGq and x P GLnpq2q such that x´1pgρqφx “ gαρ for all g P G and xBxσT “ λBφ with λP
Fˆq. Let A P GLnpq2q such that pGρqA ď SUnpqq and let l “ adetpxq. Then a conjugacy class of pGρqA in SUnpqq is stabilised in Unpqq by φ if and only if either φ“γ and 1 λn{2l 1`σdetpBq “1 or p 1 λn{2ql 1`σ pdetpBqq1´p2 “1.
Since the representation ρA is equivalent to ρ, we will just say that an SUnpqq-conjugacy class of Gρ is stabilised byφin OutpUnpqqq.
Remark 4.5.3. Again we have to consider characteristic 0 representations ˆ
ρ of G preserving a unitary form. Let R be the character ring of ˆρ and let tp1, . . . , pku be the set of exceptional primes of Gρˆ. Let ˆB be the positive
definite σ-Hermitian form preserved by ˆρ. Then there exists a complex matrix ˆAsuch that ˆAAˆσT “Bˆ by [27]. Thep-modular reductionµB of ˆµBˆ is a non-degenerate unitary form if the entries of both ˆµBˆ and pµˆBqˆ ´1 lie
inRrp1
1, . . . ,
1
pksfor some scalar ˆµPC. Even if such a ˆµexists, however, we may not be able to find a suitable ˆAwith entries only inRrp1
1, . . . ,
1
pks. From this it follows that ˆAcannot necessarily be reduced modulo pand hence we may not be able to use Lemma 4.5.2.
However we can find an ˆxPGLnpCqsuch that ˆx´1pgρqˆφxˆ“ pgα
qρˆfor all g PG. If the entries of λxˆ and pλxqˆ ´1 lie in Rr1
p1, . . . ,
1
λ and if φ “ γ, then we do not have to find ˆA explicitly, as the following lemma shows.
Lemma 4.5.4 ([8, Prop 4.6.6, p.193]). Let ρˆ: GÑ SUnpB,ˆ Cq be a char- acteristic 0 representation preserving some unitary formBˆ such thatGhas an absolutely irreducible representation ρ with Gρ ď SUnpq, Bq that arises as ap-modular reduction of ρˆ. Furthermore, let S“Rrp1
1, . . . ,
1
pss, whereR
is the character ring ofρˆand the pis are the exceptional primes withpi‰p for all i. Assume that:
(i) φ“γ;
(ii) there exist α PAutpGq and xˆPGLnpCq such that xˆ´1pgρqˆ φxˆ“ pgα qρˆ for all gPG;
(iii) B,ˆ Bˆ´1,xˆ and xˆ´1 have entries in S; and
(iv) ˆrνˆ2 withrˆPRgives a factorisation of detpxqˆ in S.
Now letrbe thep-modular reduction ofˆrand let“1if?rPFˆq and“ ´1 otherwise. LetAPGLnpq2q such thatpGρqAďSUnpqq. If sgnprq “ˆ 1 then an SUnpqq-conjugacy class of pGρqA is stabilised by φ in OutpUnpqqq. If sgnprq “ ´ˆ 1 then a conjugacy class is stabilised by φδ.