LIFTING REPRESENTATIONS OF FINITE REDUCTIVE
GROUPS: A CHARACTER RELATION
JEFFREY D. ADLER, MICHAEL CASSEL, JOSHUA M. LANSKY, EMMA MORGAN,
AND YIFEI ZHAO
e over a finite field k, and a
Abstract. Given a connected reductive group G
e of finite order, let G denote the connected
semisimple k-automorphism ε of G
part of the group of ε-fixed points. Then there exists a natural lifting from
e
packets of representations of G(k) to packets for G(k).
In the case of DeligneLusztig representations, we show that this lifting satisfies a character relation
analogous to that of Shintani.
0. Introduction
e is a connected reductive k-group, and ε is a semisimSuppose k is a finite field, G
e
ple k-automorphism of G of finite order `. From [7, Theorem 7.5], ε must preserve
e Te) consisting of a Borel subgroup of B
e ⊆ G
e and a maximal torus
some pair (B,
e Call such a pair a Borel-torus pair for G.
e Let G be the connected part
Te ⊆ B.
e ε of ε-fixed points of G.
e Most of the following result appears in
of the group G
the statement or proof of [7, Theorem 8.2]. Other versions appear in [5, Theorem
1.1.A] and [2, Théorème 1.8]. A version that includes the rationality of G is in
[1, Proposition 3.5].
Proposition 1.
• G is a connected reductive k-group.
e Te) for G,
e one has a Borel-torus
• For every ε-invariant Borel-torus pair (B,
ε
ε
◦
ε
◦
e , (Te ) ) for G. Moreover, (Te ) = Te ∩ G.
pair (B
• For every Borel-torus pair (B, T ) for G, one has an ε-invariant Borel-torus
e Te) where Te = C e (T ), and such that B
e ε = B.
pair (B,
G
e has an
Note that by choosing T to be defined over k, we can show that G
e
e
e
ε-invariant Borel-torus pair (B, T ) whose torus T is defined over k.
e ∗ and G∗ denote the duals of G
e and G. For each semisimple element
Let G
s ∈ G, one obtains a collection Es (G(k)) of irreducible representations of G(k), and
these collections, known as Lusztig series, partition the set E(G(k)) of (equivalence
classes of) irreducible representations of G(k) [6, §14.1]. Suppose that s is regular,
and let T ∗ ⊆ G∗ be the unique maximal k-torus containing s. Then the pair
(T ∗ , s) corresponds to a pair (T, θ), where T ⊆ G is a maximal k-torus, and θ is a
character of T (k). This latter pair is uniquely determined up to G(k)-conjugacy.
Date: October 23, 2012.
1991 Mathematics Subject Classification. Primary 20C33. Secondary 20G40.
Key words and phrases. Finite reductive groups, Deligne-Lusztig representations, liftings, character relations.
1
2
ADLER, CASSEL, LANSKY, MORGAN, AND ZHAO
The Lusztig series Es (G(k)) corresponding to s is the set of irreducible components
of the Deligne-Lusztig virtual representation whose character is RG
T θ.
In an earlier work [1, Corollary 11.3], two of the authors constructed a map from
e ∗ (k), thus lifting each Lusztig
semisimple classes in G∗ (k) to semisimple classes in G
e
series for G(k) to one for G(k). The series of representations coming from ± RG
T θ
e e
e
e
e
lifts to that coming from ± RG
θ,
where
T
=
C
(T
)
and
θ
=
θ
◦N
,
and
N
:
T
−→
T
e
G
Te
is the norm map defined by
N (t) = tε(t) · · · ε`−1 (t).
e In
(Of course, one could define a similar map N on any ε-invariant torus in G.)
order to understand better this lifting of representations, one would like to have
a relation between the character RG
T θ (for θ an arbitrary character of T (k), not
necessarily associated to a regular element of G∗ (k)) and the ε-twisted character
e e
e e
θ)ε associated to the ε-invariant character RG
θ. The purpose of the present
(RG
Te
Te
paper is to prove that such a relation holds at sufficiently regular points.
e
Theorem. Suppose se ∈ G(k)
belongs to an ε-invariant, maximal k-torus and that
e
N (e
s) is regular in G. Let S denote the unique maximal torus in G containing N (e
s).
Then
X
e e
s)w−1 ).
θ)ε (e
s) =
(RG
(RG
T θ)(wN (e
Te
e
w∈Wk (G,T )\Wk (G,S,T
)
e S, T ) is defined to be the quotient Te(k)\{e
e
Here Wk (G,
g ∈ G(k)
| geSe
g −1 = T }
e S, T ) is a union of right
and Wk (G, T ) = NG(k) (T )/T (k). We see that Wk (G,
e
Wk (G, T )-cosets from the fact that NG(k) (T ) ∩ T (k) = T (k), so the index set for
the summation makes sense. We define wN (e
s)w−1 to be nN (e
s)n−1 for any lift n
e
of w to G(k),
and claim that the right-hand side does not depend on our choices of
lifts.
e e
e is an ε-invariant maximal k-torus,
θ)ε . Since Te ⊆ G
Here is what we mean by (RG
Te
e
e so it acts
e
and θ is an ε-invariant character of T (k), we have that ε preserves (Te, θ),
on the corresponding Deligne-Lusztig variety, and thus on the virtual representation
e e
θ. That is, even if this representation is reducible, we can
whose character is RG
Te
form its ε-twisted character. One can construct this character as follows. Extend θe
e = 1. Define the ε-twisted Deligne-Lusztig
to a character of Te(k) o Γ by setting θ(ε)
e e
e e
e
G
e
e
e
character (RTe θ)ε induced from θ by (RG
θ)ε (g) = (RGoΓ
θ)(gε)
for g ∈ G(k).
Te
TeoΓ
(See [2] for the definition of Deligne-Lusztig induction for nonconnected groups.)
Comparing our theorem with [2, Proposition 2.12], we see that the latter shows
e is a certain twisted character of
that the restriction to G of a twisted character of G
G, at least in the case where ε is “quasi-central.” On the other hand, our theorem
e via the norm map, and this lifting is not
concerns a lifting of characters from G to G
an inverse of restriction. As expected, the two formulas agree in those situations
where both are applicable, but such situations are rare. Moreover, while we make
regularity assumptions on our element se, we do not assume that ε is quasi-central.
Remark 2. Consider the automorphism of GL(2) given by ε(g) = t g −1 . Then
an analogous relation holds for all irreducible representations, not just those of
Deligne-Lusztig type. On the other hand, while the relation can hold for unipotent
elements, it fails if se is regular but N (e
s) is singular.
CHARACTER RELATION
3
e arises
Remark 3. Consider the special case where E/k is a finite extension, and G
e = RE/k G. Suppose that ε is the k-automorphism
from G via restriction of scalars: G
e
of G associated to the action of a generator of the Galois group Gal(E/k). Given
e
a representation π of G(k), one often has an associated representation π
e of G(k),
known as the Shintani lift of π. (See [4] for a discussion.) The character of π and the
ε-twisted character of π
e are related by the Shintani relation: Θπ (N (g)) = Θπe,ε (g).
From work of Digne [3, Cor. 3.6], one already knows that if RG
T θ has a Shintani lift,
e e
G
then it must be RTe θ. Thus, our character relation is a generalization of Shintani’s,
if one ignores the fact that the former only holds at sufficiently regular elements of
e
G(k),
while that latter holds at all elements, provided that one interprets the norm
map N correctly.
Acknowledgements. The authors were partially supported by a National
Science Foundation Focused Research Group grant (DMS-0854844). The fourthnamed author was partially supported by the Frank Cox Jones Scholarship Fund
and a Dean’s Undergraduate Research Award from the College of Arts and Sciences
of American University. We have benefited from conversations with Prof. Jeffrey
Hakim.
1. Consequences of regularity
e ε, `, G, and N be as in §0. Let Γ be the group
From now on, we let k, G,
generated by ε.
e and
Proposition 4. Suppose se belongs to an ε-invariant maximal k-torus Se in G,
e
e
s := N (e
s) is regular in G. Let S := S ∩ G. Then:
e
e
(a) S belongs to an ε-invariant Borel-torus pair for G;
e
(b) seε is regular in G o Γ; and
e
gsg −1 ∈ Im(N | e ) .
(c) N e (S) = g ∈ G(k)
G(k)
S(k)
Proof. Since Se is ε-invariant, we have that s ∈ Seε . Since Se is connected, s ∈ G.
Therefore, s ∈ S. Since S is a diagonalizable subgroup of G, it is contained in some
maximal torus S 0 of G. Let Se0 = CGe (S 0 ). From Proposition 1, Se0 is a maximal
e and belongs to an ε-invariant Borel-torus pair. But
torus in G,
e
Se0 = C e (S 0 )◦ ⊆ C e (S)◦ ⊆ C e (s)◦ = S,
G
G
G
e proving statement (a).
so Se0 = S,
e containing S,
e and let Φ+ denote
Now choose an ε-invariant Borel subgroup of G
+
the corresponding system of positive roots. For each α ∈ Φ , consider the following
e Vα := L
sum of root spaces in the Lie algebra e
g of G:
gβ . Via the adjoint
β∈Γ·α e
e
action, the element seε induces a linear transformation on g, preserving each Vα . To
see that seε is regular, it is enough to choose a positive root α ∈ Φ+ and show that
the action of seε on Vα does not have 1 as an eigenvalue.
Let m = mα = |Γ · α|. There is a nonzero scalar µ = µα , not necessarily rational,
such that εm acts on Vα via multiplication
P by µ. By necessity, µ is a root
P of unity
of order dividing `/m. Let χ0 = χ0α = β∈Γ·α β, and let χ = χα = γ∈Γ γ(α).
` 0
Thus, χ = m
χ.
It is not hard to see that the characteristic polynomial (in the variable X) of our
transformation of Vα is X m − µ · χ0 (e
s), so it is enough to show that µ · χ0 (e
s) 6= 1.
4
ADLER, CASSEL, LANSKY, MORGAN, AND ZHAO
Since s is regular, we have that
1 6= α(s) = χ(e
s) = χ0 (e
s)`/m = (µ · χ0 (e
s))`/m ,
and therefore, µ · χ0 (e
s) 6= 1, proving statement (b).
Now consider statement (c). From statement (a) and Proposition 1, we have that
S is a maximal k-torus in G, and Se = CGe (S). To show that the right-hand-side is
e such that gsg −1 ∈ Im(N | e ) ⊆ S(k). Since s
included in the left, take any g ∈ G
S(k)
−1
and gsg are in S(k), they are both ε-invariant, so we have ε(g)sε(g)−1 = gsg −1 .
e Therefore ε(g) = ge
e
This implies g −1 ε(g) ∈ CGe (s) = S.
t for some e
t ∈ S(k).
e
e
Since g conjugates a regular element of S back into S, we must have that g
e Let t ∈ S(k). Since
normalizes S.
ε(gtg −1 ) = ge
tte
t−1 g −1 = gtg −1 ,
we have that gtg −1 ∈ Seε . But since t ∈ (Seε )◦ = S, so is gtg −1 . Thus, g ∈ NGe (S).
To prove the converse, let n ∈ NG(k)
(S). We want to show that nsn−1 ∈
e
Im(N |S(k)
). Since every element in S is ε-fixed, we have εi (n)sεi (n)−1 = nsn−1
e
e and thus commutes with
for all i, and therefore n−1 εi (n) belongs to C e (s) = S,
G(k)
e
every element of S(k).
Consider the following product:
n−1 ε(n) ε(n)−1 ε2 (n) ε2 (n)−1 ε3 (n) · · · ε`−1 (n)−1 n = 1
Since each term in the above equation commutes with εj (e
s), we have
nsn−1 = ne
sε(e
s) · · · ε`−1 (e
s)n−1
= ne
s n−1 ε(n) ε(e
s) ε(n)−1 ε2 (n) · · · ε`−1 (e
s) ε`−1 (n)−1 n n−1
= ne
sn−1 ε(n)ε(e
s)ε(n)−1 · · · ε`−1 (n)ε`−1 (e
s)εl−1 (n)−1
= ne
sn−1 ε(ne
sn−1 ) · · · ε`−1 (ne
sn−1 )
= N (ne
sn−1 ).
e we must have that
Since n conjugates a regular element (s) of Se back into S,
−1
e
e
n ∈ NGe (S), and so ne
sn ∈ S(k). Therefore
nsn−1 = N (ne
sn−1 ) ∈ Im(N |S(k)
),
e
as desired.
2. Proof of the Theorem
We start by recalling some facts about Deligne-Lusztig virtual characters.
Lemma 5.
(a) If x ∈ G(k) is a regular semisimple element, then
 X
if x is G(k)-conjugate to an


θ(vx0 v −1 )
element x0 ∈ T (k),
G
(RT θ)(x) = v∈Wk (G,T )


0
otherwise.
CHARACTER RELATION
5
e
e o Γ.
(b) Suppose that x
e ∈ G(k)
and that x
eε is a regular semisimple element of G(k)
Let θe be an ε-invariant character of Te(k), extended to trivially Te(k) o Γ. Then
e ε (e
(RG
θ)
x) =
Te
e
1
X
|Te(k) o Γ| e −1 (e
θ(h
xε)h).
xeε∈h(Te(k)oΓ)h−1
e
h∈G(k)oΓ
Proof. For each formula, see [2, Proposition 2.6]. For the second formula, note
that CGe (x) contains no nontrivial unipotent elements and that the Green function
C (e
xε)◦ (k)
QCGee (exε)◦ (k) in this proposition takes the value |CGe (e
xε)◦ (k)| at (1, 1).
G
e containing se. From Proposition
Let Se denote an ε-invariant maximal k-torus in G
e
e
4(a) and Proposition 1, S = S ∩ G and S = CGe (S).
e
Suppose that for all e
h ∈ G(k)
o Γ, we have that e
h(e
sε)e
h−1 ∈
/ Te(k)ε. Then
−1
−1
e
e
e
e
he
sε(h ) ∈
/ T (k), so in particular ge
sg
∈
/ T (k) for all g ∈ G(k). This implies
e −1 6= Te, and thus from Proposition 1 that gSg −1 6= T .
that for all g ∈ G(k), g Sg
Then the regularity of N (e
s) implies that gN (e
s)g −1 ∈
/ T (k). For every lift n of w to
e
G(k), we have that the summand corresponding to w in the right-hand side of the
equation in the theorem is zero. But our assumption on se implies that the left-hand
side is zero, too. Thus, the theorem holds in this case.
e
Now suppose that e
h(e
sε)e
h−1 ∈ Te(k)ε for some e
h = geεi ∈ G(k)
o Γ. Then
−1
−1
−1
e
ge · se · ε(e
g ) ∈ T (k) so geN (e
s)e
g = N (e
g · se · ε(e
g ) ) ∈ T (k). Thus geSe
g −1 = T .
Rewriting each index w in the summation in the theorem in the form w0 ge, where
e T ) (where Wk (G,
e T ) is defined to
w0 is a coset representative for Wk (G, T )\Wk (G,
e T, T )), we see that the right-hand side is equal to
be Wk (G,
X
g N (e
s)e
g −1 w−1 ).
(RG
T θ)(we
e )
w∈Wk (G,T )\Wk (G,T
Thus, we may replace se by its twisted conjugate e
h · se · ε(e
h−1 ), and N (e
s) by its
−1
conjugate geN (e
s)e
g . Having done so, we may now assume that S = T and Se = Te.
From Proposition 4(b), we have that seε is regular, so we can analyze the righthand side of the equation (LHS) using Lemma 5(b):
LHS =
X
X
1
1
e −1 (e
θ(h
sε)h) =
θe h−1 seε(h)ε ,
e
e
`|T (k)|
`|T (k)|
e
e
where each sum is over the set h ∈ G(k)
o Γ seε ∈ h Te o Γ h−1 . If g ∈ G(k),
then gεi belongs to the index set if and only if g −1 seε(g) ∈ Te(k). Thus,
`−1
X
1
`|Te(k)|
X
θe (gεi )−1 seε(gεi )ε
i=0
g−1 seε(g)∈Te(k)
e
g∈G(k)
X
1
e −1 seε(g))
=
θ(g
e
|T (k)| g−1 seε(g)∈Te(k)
e
g∈G(k)
LHS =
6
ADLER, CASSEL, LANSKY, MORGAN, AND ZHAO
Letting s = N (e
s), we hence have
LHS =
1
X
|Te(k)| e
t∈Te(k)
=
1
|Te(k)| e
1
t∈Te(k)
=
g−1 seε(g)=et
e
g∈G(k)
X
gesε(g)−1 =et
e
g∈G(k)
X
gesε(g)−1 =et
e
g∈G(k)
X
|Te(k)| e
1
|Te(k)| e
e −1 seε(g))
θ(g
X
t∈Te(k)
=
X
e sε(g)−1 )
θ(ge
θ(N (e
t))
X e
ge
t))
sε(g)−1 = e
t θ(N (e
g ∈ G(k)
t∈Te(k)
=
=
1
X e
ge
sε(g)−1 ∈ Te and gsg −1 = t θ(t)
g ∈ G(k)
|Te(k)| t∈T (k)
X
1
|Te(k)| t∈Im(N | e
e
gsg −1 = t θ(t)
g ∈ G(k)
T (k) )
=
1
X
|Te(k)| θ(gsg −1 )
gsg−1 ∈Im(N |Te (k) )
e
g∈G(k)
X
θ(gsg −1 )
(by Proposition 4(c))
1
e
|T (k)| g∈N e (T )
G(k)
X
=
θ(wsw−1 )
=
e )
w∈Wk (G,T
=
X
X
θ(vwsw−1 v −1 ),
e ) v∈Wk (G,T )
w∈Wk (G,T )\Wk (G,T
where the first summation in the final line is over a set of representatives for the
e T ). But from Lemma 5(a), the final expression
right-coset space Wk (G, T )\Wk (G,
above is equal to the right-hand side of the relation in the Theorem.
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CHARACTER RELATION
7
[7] Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
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(Adler, Cassel, Lansky, and Morgan) Department of Mathematics and Statistics, American University, Washington, DC 20016-8050
E-mail address, Adler: jadler@american.edu
E-mail address, Cassel: michael.cassel@american.edu
E-mail address, Lansky: lansky@american.edu
E-mail address, Morgan: emmamorgan90@gmail.com
(Zhao) Department of Mathematics, Columbia University, New York, NY 10027
E-mail address: yz2427@columbia.edu