奈良教育大学学術リポジトリNEAR
Remark on the Irreducible Characters of Finite Reductive Groups of Classical Type
著者 ASAI Teruaki
journal or
publication title
奈良教育大学紀要. 自然科学
volume 42
number 2
page range 1‑8
year 1993‑11‑25
URL http://hdl.handle.net/10105/1678
Remark on the Irreducible Characters of Finite Reductive Groups of Classical Type
Teruaki Asai
{Department of Mathematics, Nara University of Education, Nara 630 , Japan) (Received April 2, 1993)
1. Introduction.
Let G be a connected reductive group over a finite held F, with the Frobenius map‑
ping F and G its F‑stable points. For simplicity, we assume that G is with connected center (i. e. the center of G is connected by the Zariski topology).
We review some of the history related with the parameterization of the irreducible characters of G over algebraicallly closed fields of characteristic 0.
In 1976, Deligne‑Lusztig [6], utilizing /‑adic cohomology spaces asssociated with certain varieties realized in G, showed that the set GF of irrducible characters is divided into the disjoint union :
(T‑LJ 」{GF, {s})
(sl
where {s} runs through the F‑stable semisimple conjugacy classes in the dual group G ofG.
This result prompted Prof. G. Lusztig [7] in 1977 more furtherly to conclude that if G is a conformal group (i.e. CSp?,,, CO孟 or CO2n+i) with char F。≠2, 0r if G is any classical type group with char F,‑2, then there exists the bijection
」(GF, {s})⊥」 *(zc*(s)*'', {1}) or equivalently
(*) eF上」 l¥」{ZG*{s)* F, {1})
s
This is so‑called Jordan decomposition of irreducible characters.
Later in 1984, utilizing the sophiscated intersection cohomology theory, Prof. G. Lusztig [8] showed that there exists the bijection (*) for any connected reductive group (with connected center).
However, if G is of classical type (but not necessarily a conformal group) with char FQi=2 then the more conventional method in [7] can also be applied. The author wants to suggest this by simply showing that the both sides of the correspondence in (*) are the same in number:
1
(**)
Teruaki Asai
leFl‑U l*(Zc*(s)*, {1})
Lsi
This is rather crucial because almost all of the irreducible characters are constructed from the inductions, so the relation (**) enables us to get the information of how many (cuspidal) characters are missing, which turns out to be at most one in number. In any way the author does not want to go in detail here.
This document was originally a part of the author's preprint [3], which is cited in Lusztig [8], nevertheless left unpublished partly because of its lengthy proof.
2. Main Teorem.
Theorem 2.1. Let G be a connected reductive group over ・ with the connected center.
We assume that char F,≠ 2, and the derived group D(G) of G is simply connected and of type Dn, Dn orBn. Then we have the following identity.
G7‑l‑∑ <?(zf;*(sr・'. {I})‑
・、:
where GF/‑ means the set of the conjugacy classes in G and the right hand side summation is over the F‑stable semisimple conjugacy classes of the dual group G
As for the statement言 *(s) , {1})I is the number of unipotent characters,
which depends only on the type of the Dynkin graph of Zc*(s) as is explicitly deter‑
mined by Lusztig [7]. SO, for the verification of the lemma, we may concentrate on the evaluation of the number of the conjugacy classes. It is relatively easy to evaluate the number of the conjugacy classes in the classical groups provided that they are conformal groups CO2ニ CSp2n or special orthogonal groups SO2n+i of odd rank. However, the reductive group G and its dual G in the theorem are slightly different from these groups. SO, we need some preparations for the proof of the theorem.
Lemma 2.2. Let G be afinitegroup and Na subgroup of the centerZ(G) ofG. Let百‑
G/N and n : G‑‑百be a canonical surjection. Then for any x∈G, there exists a bijection
tz∈N¥xz‑x] ⊥⊥ Z否OrOt))A(ZGOc))
where‑‑means the conjugacy in G.
Proof. For anyz∈N, ifg(xz)g一m】‑x for someg∈G, then we have
g∈n‑¥Z否OrOO), z‑bcg] (聖x‑'g [xg)
Hence, for any z∈N, xz and x are conjugate in G if and only ifz belongs to the image of the following mapping.
fx‑71 1(Z百Cc))‑‑G (g‑‑ [x,g])
Since [x, gh]‑[x, h]h 」[x, g]h and Nis a subgroup of the center of G, the mapping/よis
a homomorphism. Since the kernel of/よis the centralizer Zc(x) of x, we have
tz∈N厄‑x} ‑n ‑¥Z一百(n(x))/Zc{x) ‑Z百(7rOt))A(ZGOt)) This shows our lemma.
For a connected reductive group G over F,, let GB (resp. Gs s) be the set of unipotent (resp. semisimple) elements in G.
Lemma 2.3. Let G be a connected reductive group over ・Q, N a connected closed sub‑
group ofZ(G). Let百‑G/N and n: G →否a canonical surjection. Then k induces the bijection from the set of unipotent conjugacy classes in GF to that in百F :
⊥二⊥"a/
Proof. Letu′ ∈Gニ Assume nix)‑m′ withx∈GF. Write
蝣A* mAj ^,A* Ai m¥/ *jmn^ c
with xs∈ and x,,∈Gu(the Jordan decomposition). Then n(xS)‑1, n{xu)‑u. This shows that there exists unique u∈G¥, such that n(u)‑ォ'So
n¥r,¥Gu ‑‑GニF
is bijective. Since x(GF)‑G′ for any w,, u2∈Gu, ui and u2 are conjugate in GFif and
only ifn(u,) and 7r(w2) are conjugate in G′ This is our lemma.
Lemma 2.4. Let Gyresp. G′ ) be the connected reductive group over ・ with connected
center,百‑G/Z(G) (resp.百′ ‑G'/Z(G')) and
打: G一一百(resp. e′ : G'‑,‑一一百つ
be the canonical surjection. Assume that D(G) is simply conneted and百‑百′ i. e. G and G′
have the same Dynkin graph:). Then
¥GF/‑¥‑ Z(GY B ^ZcAsT, F)
Z(G′)Fl競[zG,{s'Y‑.Zc′(s'r
where {s} runs through all the semisimple classes in G and 5i(ZG‑(s′)0,0 denotes the
number of unipotent classes in ZG′ GO"
Proof. By the Jordan decomposition,
¥G> ∑ IBtiZcisY
s
Teruaki Asai
where {s} runs through all the semisimple classesin GF. Lets∈ , ands‑7r(s). Then
zG(s) ‑‑ケZ否(s)‑
is a surjection and the kernel is the connected subgroup of the center of ZG(s). By Lemma 2.3.
zcxsYu/ ⊥二⊥Z否(s)‑u'
is bijective. On the other hand, the semisimple classes {s} in G which are mapped to
{si are
¥Z(GY
[Z百(sY : Z百(s)a Fl
Z(GY 競[Z己(育)F:Z否(s)an
G7‑I‑S
in number by 2.2. SO,
B,(Z否GOa O
where {s} runs through the semisimple classes in否 Then, in turn, we may write
Z(G)f
[Z否(7T′ (s'))F :Z否(7T′ (s'))a F]
G7‑l‑∑ [Z否(7T′ {s')Y : n'{ZG′ GOO]
z(Cy
where {s'} runs through all the semisimple classes in G. Thus
¥G' z(GY B^ZG′(s )ftO
zeey 競[方′(ZG′(s')0:Z否(7T′(s'))afl
¥z(G)*】 Bt(Zcノ(s')aO
¥Z(G′)FI競[ze(s'Y:zG′(s')ft^
This shows our lemma.
Lemma 2.5. Under the same assumption as in 2.4,
∑ c(s)* F, {1})
・、l
fi.czcCs'ro
Z(G)勺 *ォZc′(s')‑)*'F, {1})
Z(G′r¥ [Zc.(s′Y:ZG′(s')af]
where [s] (resp. W }) runs through all the semisimple classes in GF(resp. G′F).
Proof. For any s∈Gfs,,
e(zG(sy F, {i}) ⊥㌧ i<SZ百(7T(S))‑)* F, {1}) SO, we get our lemma by the same argument as in 2.4.
As we have proved Lemma 2.4 and 2.5, it is a matter of simple computation to show Theorem 2.1. We sketch it in the following in the case when G isoftypeDn or Dn
For a formal power seriesf(t) ‑∑n≧oantn, we define
/(O+‑ 」ォ***・
れ≧O
Lemma 2.6. Assume char ・Qi= 2. Fore‑+or‑ , let Gn‑CO%,‑ and 5,(Zc, (S)ft O
レ"廿[Z,.(s/:Z,.(s)af]
whrere {s} runs through all the semisimple classes in G^ Let
z{t)‑¥ n
∑
(¥+t2n‑iy
zo(0‑yz(O+‡nq書芸ni‑t"o
z,(0‑z(f)+nT n>ii吉‑2
" ' : ∑(cn++c‑)t2"‑ ∑ fi十(t) where
/i+(0‑( n
〟(∫)‑(
/3+a) ‑(且
lこさi≦5
‑1)×2,
)zo(o x2,
){zo(ty+(jz(tト‡nql等)2) ×211
・4+ォ‑(n音‑1)×2,
・5+(0‑( n音)zl(t2)×2×21
胃の 胃
Teruaki Asai
∑(.c:‑c‑)t2サ‑ ∑ /, (f) where
り I .蝣:‑ .蝣
/r(0‑{n
/rォ‑{n ff(t)‑{黒l f41t) ‑ (nq、
f51(i) ‑ fnql
(W2") (W")
l‑qt*
a ‑^xl‑f4")
¥‑qV
(W2") (l14n)
¥‑qt"
(l‑t'2")(l+t4n) 1‑qt*
(l‑t2n)(l+t4n) 1‑gr
‑1)×2,
}{2(n三戸‑01×2, 日且1寺12×4×21
‑1)×2,
MS,丁三戸司×4×2 】
Since the generating functions of the numbers of coniugacy classes in the orthogonal groups are described in [10], it is fairly easy to modify the formulae for our conformal groups. SO, we may omit the proof of the lemma. As for the statement, we should remark that
Zo(f) ‑ i+∑GB,(SO2:‑0 +5i(SO2‑ 0)*2"
・1 I:
Zl{t)‑¥+∑(fil{Gn+‑F)+Bl{G‑‑F))ti
,: A .
and吉(t) (1≦i≦5) are the contributions from the semisimple classes {s} with the
following properties.
(i) /,±(t), fit(t), h (t) (resp.flit), h {i))¥ the multiplicator A of s is a square (resp. non‑square),
(近) fl⊥(0; the minimal polynomialm(x) ofs is not divided byx‑a(witha‑X2),
(iii) fi上(t)¥ the minimal polynomial m(jc) is divided by x‑a or x+a but not by
both,
(iv) fa¥t), /,±(i)', the minimal polynomial mix) is divided by x一入
It should be also noted that the terms such as 2×2ーin the formulae of Lemma 2.6 have ansed by counting the number of conjugacy classes with some multiplicities divided by some indices and are left untouched intensionally for easy rechecking.
Lemma 2.7. Under the assumption of2.6, foγe‑+ or ‑, let
・?((zG,(s)‑y‑F, {i}) W'E 芯 [znXsY:Z%n
where {s} runs through all the semisimple classes in Gr,,. Let
d*‑∑
∑ a2)2‑+ ∑ (‑D'Cf2)2"
くnくco
5(0‑
Then
・ I
v I
2 ∑(dn++d‑)t2n‑ ∑ g{ (i) where
*,+(0‑ n gl‑(*)‑{n
・ヾ蝣' 111
(i‑t2ny
l‑qt*
(¥‑t2ny
¥‑qtl
(1‑t2ny
l‑qtl
1≦i≦5
‑1)×2,
)β(〜) ×2,
¥B{tyx2 1, giL‑,i音‑1)×2,
g5+Q)‑n音}B{t2)×2×211
(近)(旦訪n(d;‑d‑)t2サ‑
n>¥1黒5g了(t)where g,‑(t)‑fi‑(.i)
foranyl≦i≦5withfrit)asin2.6.
TheverificationofthelemmaismoreorlessasinLemma2.6andmaybeomittedin anyway.Weshouldremarkthatg,±(0(1≦i≦5)arethecontributionsfromthesemi‑
simpleclassesasinLemma2.7.
ProofofTheorem2.1.Foranyintegern>Oande‑+or一,letC≡n+(resp.G√)bethe connectedreductivegroupwithconnectedcentersuchthat刀(eJ)(resp.刀(∂㌃))is simplyconnectedandoftypeDn(resp.Dn).Fore‑+or‑,letGnJbeasinLemma2.6.
Thenthefollowingconditionsareequivalent.
(i)IGn」,F/‑l‑∑i(Z,c(s)*‑F,{1}
{sin
where{s}runsthroughallthesemisimpleclassesin(登nE,'(」‑+,‑).
害叩g Z(GOf 5,(zc;(s)ao ̲ ¥z(Qyy T tf.CCZc.Cs)o)*^
z{GcnyiZG,(s)F:ZG孟(s)aT¥Z(GZ)f宣[Zc.tsy‑.Zc.is)o^
where in the first and second summation, {s} runs through all the semisimple classes in G*(s‑+, ‑).
(UII
Teruaki Asai
5,(Zc; (s)aO 5,((Zc:(s)‑)*‑0 [zG;(s)f:zG,(s)af] 号[zc;(s)F:zc;(S)af]
where {s} runsasin (ii) (.」‑+, ‑).
(iv) (‑ ‑(In
where cnJ and df, are as in Lemma 2.6 and 2.7 (e‑ ‑).
( i) and (ii) are equivalent by Lemma 2.4 and 2.5 Note that Lemma 2.5 is applied with G‑G^ *. (ii) and (in) are equivalent since ￨Z(Gn」)Fl‑¥Z{G*, ) I. The equivalence of (iii) and (iv) is just the definition of c^ and d%.
So to show the proposition, it suffices to show the following equations.
∑{c;+cn‑)t2n‑ ∑(dn++d ‑)t*
n≧1 n≧1
∑(cS‑c ‑)t2サ‑∑(dn+ ‑d㌃)t2n
〃≧1 〃≧1
The second one has already been proved in Lemma 2.7. As for the first one, we need very popular identities between (formal) products and series such as Jacobi's triple pro‑
duct identity and Euler's pentagonal number theorem, which are discussed in detail by G. E. Andrews ([1], [2]). In any way, the verification is more or less a routine one and is omitted.
References
[ 1 ] G. E. Andrews, Number Theory, Saunders, Philadelphia (1971)
[ 2] G. E. Andrews, The Theory of Partitions, Encyclopedia of Math, and its Applications Vol. 2, Addison‑Wesley, Massachusetts (1976)
[ 3] T. Asai, Endomorphism algebras of reductive groups over F, of classical type, preprint [4] A. Borel and J. Tits, Groupes Reductifs, Publ. Math., I.H.E.S. 27, 55‑150 (1965)
5] N. Boubaki, Groupes et Alg色bre de Lie, Chap. 4. 5 et 6, Hermann, Paris (1968)
[6] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103, 103‑ 161 (1976)
[7] G. Lusztig, Irreducible representations of finite chassical groups, Inv. Math. 43, 125‑175 (1977)
[8 ] G. Lusztig, Characters of reductive groups over a finite Field, Annals of Math. Studies 107, Princeton University Press (1984)
[9 ] I. Satake, Classification theory of semi‑simple algebraic groups, MarceLDekker Inc. New York,1971
[10] G.E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J.
Austr. Math. Soc. 3, 1‑62 (1963)
[11] A. Weil, Ad色Ies and algebraic groups, Lecture Notes, Institute for Advanced Study, Princeton, 1961
