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Gelfand–Graev characters of the finite unitary groups
Nathaniel Thiem
University of Colorado at Boulder

C. Ryan Vinroot
College of William and Mary

Submitted: Aug 12, 2009; Accepted: Nov 25, 2009; Published: Nov 30, 2009
Mathematics Subject Classification: 20C33, 05E05
Abstract
Gelfand–Graev characters and th eir degenerate counterparts have an important
role in the rep resentation theory of finite groups of Lie type. Using a characteristic
map to translate the character theory of the finite unitary grou ps into the language
of symmetric functions, we study degenerate Gelfand–Graev characters of the finite
unitary group from a combinatorial point of view. In particular, we give the values
of Gelfand–Graev characters at arbitrary elements, recover the decomposition multi-
plicities of degenerate Gelfand–Graev characters in terms of tableau combinatorics,
and conclude with some multiplicity consequences.
1 Introduction
Gelfand–Graev modules have played an important role in the representation theory of
finite gro ups of Lie type [4, 7, 22]. In particular, if G is a finite group of Lie type, then
Gelfand–Graev modules of G both contain cuspidal representations of G as submodules,
and have a multiplicity free decomposition into irreducible G-modules. Thus, Gelfand–
Graev modules can give constructions for some cuspidal G-modules. This paper uses a
combinatorial correspondence between chara cters and symmetric functions (as described
in [23]) to examine the Gelfand–Graev character and its degenerate relatives for the finite
unitary group.
Let B
<
be a maximal unipotent subgroup of a finite group of Lie type G. Then the
Gelfand–Graev character Γ of G is the character o bta ined by inducing a generic linear

character from B
<
to G. The degenerate Gelfand–Graev characters of G are obtained by
inducing arbitrary linear characters. In the case GL(n, F
q
), Zelevinsky [27] described the
multiplicities of irreducible characters in degenerate Gelfand–Graev characters by count-
ing multi-tableaux of specified shape and weight. It is the goal of this paper to describe
the degenerate Gelfa nd–Graev characters of the finite unitary groups in a similar manner
using tableau combinatorics. In [27], Zelevinsky obtained the result that every irreducible
the electronic journal of combinatorics 16 (2009), #R146 1
character of GL(n, F
q
) appears with multiplicity one in some degenerate Gelfa nd–Gr aev
character. It is known that this multiplicity one result is not true in a general finite
group of Lie type, and in fact there are characters which do not a ppear in any degenerate
Gelfand–Graev character in the general case. This result was illustrated by Srinivasan [20]
in the case of the symplectic group Sp ( 4, F
q
), and the work of Kotlar [11] gives a geometric
description of the irreducible characters which appear in some degenerate Gelfand–G raev
character in general type. In the finite unitary case, we g ive a combinatorial descrip-
tion of which irreducible characters appear in some deg enerate Gelfand–Graev char acter,
as well as a combinatorial description of a la rge family of characters which appear with
multiplicity one.
In Section 2, we describe the main combinatorial tool which we use for calculations,
which is the characteristic map of the finite unitary group, and we follow the development
given in [23]. This map translates the Deligne-Lusztig theory of the finite unitary group
into symmetric functions, which thus translates calculations in representation theory into
algebraic combinatorics. Some of the results in this paper could be obtained, a lbeit in

a different formulation, by applying Harish-Chandra inductio n and the representation
theory of Weyl groups. However, this approach would not lead us to some of the com-
binatorics which we study here. For example, we naturally arrive at battery tableaux,
which are interesting combinatorial objects in their own right. Also, our more classical
approach gives rise to useful identities in symmetric f unction theory, such as our Lemma
4.2.
Section 3 examines the (non-deg enerate) Gelfand–Graev character. We use a remark-
able formula for the character values of the Gelfand–Graev character of GL(n, F
q
), given
in Theorem 3.2 (fo r an elementary proof see [9]), to obtain the corresponding formula
for U(n, F
q
2
) in Corollary 3.1, which states that if Γ
(n)
is the Gelfand–Graev character of
U(n, F
q
2
), and g ∈ U(n, F
q
2
), then
Γ
(n)
(g) =








(−1)
⌊n/2⌋+
(
ℓ
2
)
(q
ℓ
− (−1)
ℓ
) · · · (q + 1)
if g is unipotent,
block type (µ
1
, µ
2
, . . . , µ
ℓ
),
0 otherwise.
When compared to the original GL(n, F
q
) version of this formula given in Theorem 3.2,
Corollary 3.1 could be seen as another o ccurrence of “Ennola duality.” Although the proof
of Corollary 3.1 is a fairly straightforward application of the characteristic map, we have
not found it stated in any of the literature. We also note that we have applied Cor ollary

3.1 in another paper, to obtain [2 4, Theorem 4.4].
Section 4 computes the decomposition of degenerate Gelfand–Graev characters in a
fashion analogous to [27], using tableau combina torics. The main result is Theorem 4 .4 ,
which may be summarized as saying that the degenerate Gelfand–Graev character Γ
(k,ν)
of U(n, F
q
2
) decomposes as
Γ
(k,ν)
=

λ
m
λ
χ
λ
,
where λ is a multipartition and m
λ
is a nonnegative integer obtained by counting ‘battery
the electronic journal of combinatorics 16 (2009), #R146 2
tableaux’ of a given weight and shape. In the process of proving Theorem 4.4, we obta in
some combinatorial Pieri-type formulas (Lemma 4.2), decompositions of induced charac-
ters from GL(n, F
q
2
) to U(2n, F
q

2
) (Theorem 4.1 and Theorem 4.2), and a description of
all of the cuspidal characters of the finite unitary groups (Theorem 4.3).
Section 5 concludes with a discussion of the multiplicity implications of Section 4. In
particular, in Theorem 5.2 we give combinatorial conditions on multipartitio ns λ which
guarantee that the irreducible character χ
λ
appears with multiplicity one in some degen-
erate Gelfand–Gra ev character. Our Theorem 5.2 improves a multiplicity one result of
Ohmori [18].
Another question one might ask is how the generalized Gelfand–Graev representations
of the finite unitary group decompose. Generalized Gelfand–Graev representations, which
were defined by Kawana ka in [10], are obtained by inducing certain irreducible represen-
tations (not necessarily one dimensional) from a unipot ent subgroup. Rainbolt studies
the generalized Gelfand–Graev representations of U(3, F
q
2
) in [19], but in the general
case they seem to be significantly more complicated than the degenerate Gelfand–Graev
representations.
Acknowledgements. We would like to thank G. Malle for suggesting the questions that
led to the results in Section 5, S. Assaf for a helpful discussion regarding Section 5.1,
T. La m for helping us connect Lemma 4.2 to the literature, and anonymous referees for
helpful comments.
2 Preliminaries
2.1 Partitions
Let
P =

n0

P
n
, where P
n
= {partitions of n}.
Fo r ν = (ν
1
, ν
2
, . . . , ν
l
) ∈ P
n
, where ν
1
 ν
2
 · · ·  ν
ℓ
> 0, the length ℓ(ν) of ν is the
number of parts l, and the size |ν| of ν is the sum of the parts n. Let ν
′
denote the
conjugate of the partition ν. We also write
ν = (1
m
1
(ν)
2
m

2
(ν)
· · · ), where m
i
(ν) = |{j ∈ Z
1
| ν
j
= i}|.
We will denote the unique element of P
0
by ∅ or (0), which is the empty partition, or the
unique partition of 0. For any ν ∈ P, define n(ν) to be
n(ν) =

i
(i − 1)ν
i
.
If µ, ν ∈ P, we define µ ∪ ν ∈ P to be the partition of size |µ| + |ν| whose set of parts
is the union of the parts of µ and ν. For k ∈ Z
1
, let kν = (kν
1
, kν
2
, . . .), and if every
part of ν is divisible by k, then we let ν/k = (ν
1
/k, ν

2
/k, . . .). A partition ν is even if ν
i
is even for 1  i  ℓ(ν).
the electronic journal of combinatorics 16 (2009), #R146 3
2.2 The ring of symmetric functions
Let X = {X
1
, X
2
, . . .} be an infinite set of variables and let
Λ(X) = C[p
1
(X), p
2
(X), . . .], where p
k
(X) = X
k
1
+ X
k
2
+ · · · ,
be the graded C-algebra of symmetric functions in the variables {X
1
, X
2
, . . .}. Fo r a
partition ν = (ν

1
, ν
2
, . . . , ν
ℓ
) ∈ P, the power-sum symmetric function p
ν
(X) is
p
ν
(X) = p
ν
1
(X)p
ν
2
(X) · · · p
ν
ℓ
(X).
The irreducible characters ω
λ
of S
n
are indexed by λ ∈ P
n
. Let ω
λ
(ν) be the value of
ω

λ
on a permutation with cycle type ν.
The Schur function s
λ
(X) is given by
s
λ
(X) =

ν∈P
|λ|
ω
λ
(ν)z
−1
ν
p
ν
(X), where z
ν
=

i1
i
m
i
m
i
! (2.1)
is the order of the centralizer in S

n
of the conjugacy class corresponding to the partition
ν = (1
m
1
2
m
2
· · · ) ∈ P
n
.
Fix t ∈ C
×
. For µ ∈ P, the Hall-Littlewood symmetric function P
µ
(X; t) is given by
s
λ
(X) =

µ∈P
|λ|
K
λµ
(t)P
µ
(X; t), (2.2)
where K
λµ
(t) is the Kostka-Foulkes polynomial (as in [17, III.6]). For ν, µ ∈ P

n
, the
classical Green function Q
µ
ν
(t) is given by
p
ν
(X) =

µ∈P
|ν|
Q
µ
ν
(t
−1
)t
n(µ)
P
µ
(X; t). (2.3)
As a graded ring,
Λ(X) = C-span{p
ν
(X) | ν ∈ P}
= C-span{s
λ
(X) | λ ∈ P}
= C-span{P

µ
(X; t) | µ ∈ P},
with change of bases given in (2.1), (2.2), and (2.3).
We will also use several product formulas in the ring of symmetric functions. The
usual product on Schur functions
s
ν
s
µ
=

λ∈P
c
λ
νµ
s
λ
(2.4)
gives us the Littlewood-Richardson coefficients c
λ
νµ
. The plethysm of p
ν
with p
k
is
p
ν
◦ p
k

= p
kν
.
the electronic journal of combinatorics 16 (2009), #R146 4
Thus, we can consider the nonnegative integers c
γ
λ
given by
s
λ
◦ p
k
=

ν∈P
|λ|
ω
λ
(ν)
z
ν
p
kν
=

γ∈P
k|λ|
c
γ
λ

s
γ
. (2.5)
Chen, Garsia, and Remmel [2] give a combinatorial algorithm for computing the coeffi-
cients c
γ
λ
. We will use the case k = 2 in Section 4.4.
Remark. The unipotent characters χ
˜
λ
of GL(n, F
q
2
) are indexed by partitions
˜
λ of n
and the unipotent characters χ
γ
of U(2n, F
q
2
) are indexed by partitions γ of 2n. It will
follow from Theorem 4.2 that
R
U(2n,F
q
2
)
GL(n,F

q
2
)
(χ
˜
λ
) =

|γ|=2|
˜
λ|
c
γ
˜
λ
χ
γ
,
where R
G
H
is Harish-Chandra induction.
2.3 The finite unitary groups
Let
¯
G
n
= GL(n,
¯
F

q
) be the general linear group with entries in the algebraic closure o f
the finite field F
q
with q elements.
Fo r the Frobenius automorphisms
˜
F , F, F
′
:
¯
G
n
→
¯
G
n
given by
˜
F ((a
ij
)) = (a
q
ij
),
F ((a
ij
)) = (a
q
ji

)
−1
, (2.6)
F
′
((a
ij
)) = (a
q
n−j,n−i
)
−1
, where (a
ij
) ∈
¯
G
n
,
let
G
n
=
¯
G
˜
F
n
= {a ∈
¯

G
n
|
˜
F (a) = a},
U
n
=
¯
G
F
n
= {a ∈
¯
G
n
| F (a) = a},
U
′
n
=
¯
G
F
′
n
= {a ∈
¯
G
n

| F
′
(a) = a}.
(2.7)
Then G
n
= GL(n, F
q
) and U
′
n
∼
=
U
n
are isomorphic to the finite unitary group U(n, F
q
2
). In
fact, it follows from the Lang-Steinberg theorem that U
′
n
and U
n
are conjugate subgroups
of
¯
G
n
.

Fo r k ∈ Z
0
, let
˜
T
(k)
=
¯
G
˜
F
k
1
∼
=
F
×
q
k
and T
(k)
=
¯
G
F
k
1
∼
=


F
×
q
k
if k is even,
{t ∈
¯
F
q
| t
q
k
+1
= 1} if k is odd.
Fo r every partition η = (η
1
, η
2
, . . . , η
ℓ
) ∈ P
n
let
T
η
= T
(η
1
)
× T

(η
2
)
× · · · × T
(η
ℓ
)
˜
T
η
=
˜
T
(η
1
)
×
˜
T
(η
2
)
× · · · ×
˜
T
(η
ℓ
)
.
Every maximal torus of G

n
is isomorphic to
˜
T
η
for some η ∈ P
n
, and every maximal torus
of U
n
is isomorphic to T
η
for some η ∈ P
n
.
the electronic journal of combinatorics 16 (2009), #R146 5
2.4 Multipartitions
Let F :
¯
G
n
→
¯
G
n
be as in (2.6), and let T
∗
(k)
= {ξ : T
(k)

→ C
×
} be the group of
multiplicative complex-valued characters of T
(k)
=
¯
G
F
k
1
. We identify
¯
F
×
q
with
¯
G
1
=
GL(1,
¯
F
q
). Consider
Φ = {F -orbits of
¯
F
×

q
},
and note that
¯
G
1
=

f∈Φ
f =

k
T
(k)
. In particular, we may view
¯
G
1
as a direct limit of
the T
(k)
with respect to inclusion. We also have norm maps, N
m,k
, whenever k|m,
N
m,k
: T
(m)
−→ T
(k)

α →

(m/k)−1
i=0
α
(−q)
ki
, where m, k ∈ Z
1
, k|m. (2.8)
When k|m, denote by N
∗
m,k
the transpose of the map N
m,k
, which embeds T
∗
(k)
into T
∗
(m)
as follows:
N
∗
m,k
: T
∗
(k)
−→ T
∗

(m)
ξ → ξ ◦ N
m,k
(2.9)
Now, define L to be the direct limit of the groups T
∗
(k)
with respect to the maps N
∗
m,k
:
L = lim
−→
T
∗
(m)
.
Since the map F acts naturally on each T
∗
(m)
, it acts on their direct limit L. Note that we
may identify the fixed points L
F
m
with the character group T
∗
(m)
. Let Θ be the collection
of F -orbits on L:
Θ = {F -orbits o f L}.

Fo r X ∈ {Φ, Θ}, an X -partition λ = (λ
(x
1
)
, λ
(x
2
)
, . . .) is a sequence of partitions
indexed by X . The size of λ is
|λ| =

x∈X
|x||λ
(x)
|,
where |x| is the size of t he orbit x. Note that in order for |λ| to be finite, we need to
assume that λ
(x)
= ∅ for all but finitely many x ∈ X .
Let
P
X
=

n0
P
X
n
, where P

X
n
= {X -partitions of size n}.
Fo r λ ∈ P
X
, let
ℓ(λ) =

x∈X
ℓ(λ
(x)
) and n(λ) =

x∈X
|x|n(λ
(x)
).
The con j uga te of λ ∈ P
X
is the X-partition λ
′
defined by λ
′(x)
= (λ
(x)
)
′
, and if µ, λ ∈ P
X
,

then µ ∪ λ ∈ P
X
is defined by (µ ∪ λ)
(x)
= µ
(x)
∪ λ
(x)
.
the electronic journal of combinatorics 16 (2009), #R146 6
The semisimple part λ
s
of an X -partition λ is the X -partition given by
λ
(x)
s
= (1
|λ(x)|
), for x ∈ X. (2.10)
Fo r λ ∈ P
X
, define the set P
λ
s
by
P
λ
s
= {µ ∈ P
X

| µ
s
= λ
s
}.
The unipotent part λ
u
of λ is the X -partition given by
λ
({1})
u
has parts {|x|λ
(x)
i
| x ∈ X , i = 1, . . . , ℓ(λ(x))}, (2.11)
where {1} is the orbit containing 1 in Φ or the trivial character in Θ, and λ
(x)
u
= ∅ when
x = {1}.
Note that we can think of “normal” partitions as X-partitions λ that satisfy λ
u
= λ.
By a slight abuse of notat io n, we will sometimes interchange the multipartition λ
u
and
the partition λ
({1})
u
. For example, T

λ
u
will denote the torus corresponding to the partitio n
λ
({1})
u
.
Given the torus T
η
, η = (η
1
, η
2
, . . . , η
ℓ
) ∈ P
n
, there is a natural surjection
τ
Θ
: {θ = θ
1
⊗ θ
2
⊗ · · · ⊗ θ
ℓ
∈ Hom(T
η
, C
×

)} −→ {ν ∈ P
Θ
| ν
({1})
u
= η}
θ = θ
1
⊗ θ
2
⊗ · · · ⊗ θ
ℓ
→ τ
Θ
(θ),
(2.12)
where
τ
Θ
(θ)
(ϕ)
= (η
i
1
/|ϕ|, η
i
2
/|ϕ|, . . . , η
i
r

/|ϕ|), with θ
i
1
, θ
i
2
, . . . , θ
i
r
∈ ϕ.
It follows from a short calculation that if ν ∈ P
Θ
has suppo r t {ϕ
1
, ϕ
2
, . . . , ϕ
r
}, then the
preimage τ
−1
Θ
(ν) has size
r

j=1
|ϕ
j
|
ℓ(ν

(ϕ
j
)
)

i1

m
i
(ν
({1})
u
)
m
i/|ϕ
1
|
(ν
(ϕ
1
)
), m
i/|ϕ
2
|
(ν
(ϕ
2
)
), · · · , m

i/|ϕ
r
|
(ν
(ϕ
r
)
)

=

ϕ∈Θ
|ϕ|
ℓ(ν
(ϕ)
)

i1

m
i
(ν
({1})
u
)

!

ϕ∈Θ
(m

i/|ϕ|
(ν
(ϕ)
))!
. (2.13)
The conjugacy classes K
µ
of U
n
are parametrized by µ ∈ P
Φ
n
, a fact on which we
elaborate in Section 2.5. We have another natural surjection,
τ
Φ
: T
η
→ {ν ∈ P
Φ
| ν
({1})
u
= η}
t = (t
1
, t
2
, . . . , t
ℓ

) → τ
Φ
(t
1
) ∪ τ
Φ
(t
2
) ∪ · · · ∪ τ
Φ
(t
ℓ
),
(2.14)
where
τ
Φ
(t
i
) = µ
′
, if t
i
∈ K
µ
in U
η
i
.
the electronic journal of combinatorics 16 (2009), #R146 7

2.5 The characteristic map
Fo r every f ∈ Φ, let X
(f)
= {X
(f)
1
, X
(f)
2
, . . .} be an infinite set of variables, and for every
ϕ ∈ Θ, let Y
(ϕ)
= {Y
(ϕ)
1
, Y
(ϕ)
2
, . . .} be an infinite set of variables. We relate symmetric
functions in the variables X
(f)
to those in the variables Y
(ϕ)
through the transform
p
k
(Y
(ϕ)
) = (−1)
k|φ|−1


x∈T
k|ϕ|
ξ(x)p
k|ϕ|/|f
x
|
(X
(f
x
)
), where ξ ∈ ϕ, x ∈ f
x
.
The ring of s ymmetric functions Λ is
Λ =

f∈Φ
Λ(X
(f)
) =

ϕ∈Θ
Λ(Y
(ϕ)
).
Fo r µ ∈ P
Φ
, the Hall-Littlewood polynomial P
µ

is
P
µ
= (−q)
−n(µ)

f∈Φ
P
µ
(f)
(X
(f)
; (−q)
−|f|
),
and for λ ∈ P
Θ
, the power-sum symmetric function p
λ
and the Schur function s
λ
are
p
λ
=

ϕ∈Θ
p
λ
(ϕ)

(Y
(ϕ)
) and s
λ
=

ϕ∈Θ
s
λ
(ϕ)
(Y
(ϕ)
).
Fo r µ, ν ∈ P
Φ
, the Green function is
Q
µ
ν
(−q) =

f∈Φ
µ
Q
µ
(f)
ν
(f)

(−q)

|f|

,
where Φ
µ
= {f ∈ Φ | µ
(f)
= ∅}. As a graded rings,
Λ = C-span{p
ν
| ν ∈ P
Θ
}
= C-span{s
λ
| λ ∈ P
Θ
}
= C-span{P
µ
| µ ∈ P
Φ
}.
The conjugacy classes K
µ
of U
n
are indexed by µ ∈ P
Φ
n

and the irreducible characters
χ
λ
of U
n
are indexed by λ ∈ P
Θ
n
[5, 6]. Thus, the ring of class functions C
n
of U
n
is given
by
C
n
= C-span{χ
λ
| λ ∈ P
Θ
n
}
= C-span{κ
µ
| µ ∈ P
Φ
n
},
where κ
µ

: U
n
→ C is given by
κ
µ
(g) =

1 if g ∈ K
µ
0 otherwise.
the electronic journal of combinatorics 16 (2009), #R146 8
We let χ
λ
(µ) denote the value of the character χ
λ
on any element in the conjugacy K
µ
.
Fo r ν ∈ P
Θ
n
, let the Deligne-Lusztig character R
ν
= R
U
n
ν
be given by
R
ν

= R
U
n
T
ν
u
(θ)
where θ ∈ Hom(T
ν
u
, C
×
) is any homomorphism such that τ
Θ
(θ) = ν (see (2.12)).
Let C =

n1
C
n
so that
C = C-span{χ
λ
| λ ∈ P
Θ
}
= C-span{κ
µ
| µ ∈ P
Φ

}
= C-span{R
ν
| ν ∈ P
Θ
}
is a ring with multiplication given by
R
λ
R
η
= R
λ∪η
.
The next theorem follows from the results of [4, 6, 8, 10, 16, 23]. A summary of the
relevant results in these papers and how they imply the following theorem is given in [23].
Theorem 2.1 (Characteristic Map). The map
ch : C → Λ
χ
λ
→ (−1)
⌊|λ|/2⌋+n(λ)
s
λ
κ
µ
→ P
µ
R
ν

→ (−1)
|ν|−ℓ(ν)
p
ν
is an isometric ring isomorphism with respect to the natural inner products
χ
λ
, χ
η
 = δ
λη
and s
λ
, s
η
 = δ
λη
.
In the following change of basis equations, (2.15) follows from Theorem 2.1, (2.16)
follows from (2.1), and (2.17) follows from [23, Theorem 4.2].
(−1)
⌊k/2⌋+n(λ)
s
λ
=

µ∈P
Φ
k
χ

λ
(µ)P
µ
for λ ∈ P
Θ
k
, (2.15)
s
λ
=

ν∈P
Θ
k
λ
s
=ν
s


ϕ∈Θ
ω
λ
(ϕ)
(ν
(ϕ)
)
z
ν
(ϕ)


p
ν
for λ ∈ P
Θ
k
, (2.16)
(−1)
k−ℓ(ν)
p
ν
=

µ∈P
Φ
k


t∈T
ν
u
τ
Φ
(t)
s
=µ
s
θ(t)Q
µ
τ

Φ
(t)
(−q)

P
µ
for ν ∈ P
Θ
k
, τ
Θ
(θ) = ν. (2.17)
the electronic journal of combinatorics 16 (2009), #R146 9
3 Gelfand–Graev ch aracter s on arbitrary elements
3.1 G
n
= GL(n, F
q
) notation
In this Section 3, let
˜
Φ = {
˜
F -orbits in
¯
F
×
q
}.
Define norm maps

˜
N
m,k
:
˜
T
(m)
→
˜
T
(k)
, whenever k|m, the same as in (2.8), except by
replacing −q by q, and define the corresponding transpose maps
˜
N
∗
m,k
:
˜
T
∗
(m)
→
˜
T
∗
(k)
as in
(2.9), where
˜

T
∗
(m)
is the character gro up of
˜
T
(m)
. We now let
˜
L be the direct limit of the
groups
˜
T
(m)
with respect to the maps
˜
N
∗
m,k
:
˜
L = lim
−→
˜
T
(m)
,
and since
˜
F acts on

˜
L, we may consider the corresponding orbits, and we define
˜
Θ = {
˜
F -orbits in
˜
L}.
The same set-up of Sections 2.4 and 2.5 gives a characteristic map f or G
n
= GL(n, F
q
)
by replacing Φ by
˜
Φ, Θ by
˜
Θ, −q by q, T
(k)
by
˜
T
(k)
, a nd (−1)
⌊n/2⌋+n(λ)
s
λ
by s
λ
. With the

exception of the Deligne-Lusztig characters (which fo llows from the para llel argument of
[23, Theorem 4.2 ]), this can be found in [17, Chapter IV].
3.2 The Gelfand–Graev character
We will use U
′
n
= GL(n,
¯
F
q
)
F
′
(see (2.7)) to give an explicit description of the Gelfand–
Graev character. For a more general description see [4], for example.
Fo r 1  i < j  n and t ∈ F
q
, let x
ij
(t) denote the matrix with ones on the diagonal,
t in the ith row and jt h column, and zeroes elsewhere. Let
u
ij
(t) = x
ij
(t)x
n+1−j,n+1−i
(−t
q
) for 1  i < j  ⌊n/2⌋, t ∈ F

q
2
,
u
i,n+1−j
(t) = x
i,n+1−j
(t)x
j,n+1−i
(−t
q
) for 1  i < j  ⌊n/2⌋, t ∈ F
q
2
,
and for 1  k  ⌊n/2⌋, and t, a, b ∈ F
q
2
, let
u
k
(a) = x
k,n+1−k
(a) for n even, and a
q
+ a = 0,
u
k
(a, b) = x
⌈n/2⌉,n+1−k

(−a
q
)x
k,n+1−k
(b)x
k,⌈n/2⌉
(a) for n odd, and a
q+1
+ b + b
q
= 0.
Examples. In U
′
4
, we have
u
12
(t) =

1 t 0 0
0 1 0 0
0 0 1 −t
q
0 0 0 1

, u
13
(t) =

1 0 t 0

0 1 0 −t
q
0 0 1 0
0 0 0 1

, u
1
(a) =

1 0 0 a
0 1 0 0
0 0 1 0
0 0 0 1

, u
2
(a) =

1 0 0 0
0 1 a 0
0 0 1 0
0 0 0 1

,
where a
q
+ a = 0. In U
′
5
, we have

u
12
(t) =

1 t 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 −t
q
0 0 0 0 1

, u
14
(t) =

1 0 0 t 0
0 1 0 0 −t
q
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1

,
the electronic journal of combinatorics 16 (2009), #R146 10
u
1
(a, b) =

1 0 a 0 b
0 1 0 0 0

0 0 1 0 −a
q
0 0 0 1 0
0 0 0 0 1

, u
2
(a, b) =

1 0 0 0 0
0 1 a b 0
0 0 1 −a
q
0
0 0 0 1 0
0 0 0 0 1

,
where a
q+1
+ b + b
q
= 0.
Fo r 1  i < j  ⌊n/2⌋ and 1  k  ⌊n/2⌋, define the one-parameter subgro ups
X
ij
= {u
ij
(t) | t ∈ F
q

2
}
∼
=
F
+
q
2
,
X
i,n+1−j
= {u
i,n+1−j
(t) | t ∈ F
q
2
}
∼
=
F
+
q
2
,
X
k
=

{u
k

(t) | t ∈ F
q
2
, t
q
+ t = 0} if n is even,
{u
k
(a, b) | a, b ∈ F
q
2
, a
q+1
+ b + b
q
= 0} if n is odd.
so that
B
<
n
= X
ij
, X
i,n−j
, X
k
| 1  i < j  ⌊n/2⌋, 1  k  ⌊n/2⌋ ⊆ U
′
n
is the subgroup of U

′
n
of upper-t ria ng ular matrices with ones on the diagonal. Noting that
X
k
/[X
k
, X
k
]
∼
=

F
+
q
if n is even,
F
+
q
2
if n is odd,
a direct calculation gives
B
<
n
/[B
<
n
, B

<
n
]
∼
=
X
12
× X
23
× . . . × X
⌊n/2⌋−1,⌊n/2⌋
× X
⌊n/2⌋
∼
=

(F
+
q
2
)
(n/2)−1
× F
+
q
if n is even,
(F
+
q
2

)
⌊n/2⌋
if n is odd.
Similarly, let
˜
B
<
n
= x
ij
(t) | 1  i < j  n, t ∈ F
q
 ⊆ G
n
be the subgroup of unipo t ent upper-triangular matrices in G
n
.
Fix a homomorphism ψ : F
+
q
2
→ C
×
of the additive group of the field such that for all
1  k  ⌊n/2⌋, ψ is nontrivial on X
k
/[X
k
, X
k

]. Define the homomorphism ψ
(n)
: B
<
n
→ C
by
ψ
(n)




X
α
/[X
α
,X
α
]
=

ψ if α = (i, i + 1), 1  i < ⌊n/2 ⌋, or if α = ⌊n/2⌋,
1 otherwise.
The Gelfand–Graev character of U
′
n
is
Γ
′

n
= Ind
U
′
n
B
<
n
(ψ
(n)
).
Recall that U
′
n
is conjugate to U
n
in
¯
G
n
. If U
′
n
= yU
n
y
−1
, then let
Γ
n

= Ind
U
n
y
−1
B
<
n
y
(y
−1
ψ
(n)
y).
Similarly, the Gelfand–Graev character
˜
Γ
(n)
of G
n
is
˜
Γ
(n)
= Ind
G
n
˜
B
<

n
(
˜
ψ
(n)
).
the electronic journal of combinatorics 16 (2009), #R146 11
where
˜
ψ
(n)
:
˜
B
<
n
→ C is given by
˜
ψ
(n)
(x
ij
(t)) =

ψ(t) if j = i + 1,
1 otherwise.
It is well-known that the Gelfa nd–Graev character has a multiplicity free decom-
position into irreducible char acters [22, 25, 26]. The following explicit decompositions
essentially follow from [3]. Specific proofs are given in [27] in the G
n

case and in [18] in
the U
n
case.
Theorem 3.1. Let ht(λ) = max{ℓ(λ
(ϕ)
)}. Then
Γ
(n)
=

λ∈P
Θ
n
ht(λ)=1
χ
λ
and
˜
Γ
(n)
=

λ∈P
˜
Θ
n
ht(λ)=1
χ
λ

.
3.3 The character values of the Gelfand–Graev character
A unipotent conjugacy class K
µ
of U
n
or G
n
is a conjugacy class that satisfies
µ
u
= µ.
The unipotent conjugacy classes of U
n
and G
n
are thus parametrized by part itio ns µ of
n.
Lemma 3.1.
(a) Let µ ∈ P
Θ
n
, µ
({1})
u
= µ. Then
Γ
(n)
(µ) =



ν∈P
n
(−1)
n+⌊n/2⌋−ℓ(ν)
z
ν
|T
ν
|Q
µ
ν
(−q) if µ is uni potent,
0 otherwise.
(b) Let µ ∈ P
˜
Θ
n
, µ
({1})
u
= µ. Then
˜
Γ
(n)
(µ) =


ν∈P
n

(−1)
n−ℓ(ν)
z
ν
|
˜
T
ν
|Q
µ
ν
(q) if µ is unipotent,
0 otherwise.
Proof. Not e that if ht(λ)  1, then n(λ) = 0. Thus, by applying the characteristic map
and (2.16) to Theorem 3.1,
ch(Γ
(n)
) = (−1)
⌊n/2⌋

λ∈P
Θ
n
ht(λ)1

ν∈P
Θ
n
ν
s

=λ
s


ϕ∈Θ
ω
λ
(ϕ)
(ν
(ϕ)
)
z
ν
(ϕ)

p
ν
.
the electronic journal of combinatorics 16 (2009), #R146 12
Since ht(λ)  1, ω
λ
(ϕ)
is the trivial char acter for all ϕ ∈ Θ. Thus, the summand is
independent of λ, and
ch(Γ
(n)
) = (−1)
⌊n/2⌋

ν∈P

Θ
n


ϕ∈Θ
z
−1
ν
(ϕ)

p
ν
. (3.1)
By (2.13),
ch(Γ
(n)
)
= (−1)
⌊n/2⌋

ν∈P
Θ
n


ϕ∈Θ
|ϕ|
ℓ(ν
(ϕ)
)


i1

m
i
(ν
({1})
u
)

!

ϕ∈Θ
(m
i/|ϕ|
(ν
(ϕ)
))!

−1

θ∈Hom(T
ν
u
,C
×
)
τ
Θ
(θ)=ν



ϕ∈Θ
z
−1
ν
(ϕ)

p
ν
= (−1)
⌊n/2⌋

ν∈P
Θ
n

θ∈Hom(T
ν
u
,C
×
)
τ
Θ
(θ)=ν
z
−1
ν
u

p
ν
= (−1)
⌊n/2⌋

ν∈P
n

θ∈Hom(T
ν
,C
×
)
z
−1
ν
p
τ
Θ
(θ)
.
The change of basis (2.17) gives
ch(Γ
(n)
) = (−1)
⌊n/2⌋

ν∈P
n


θ∈Hom(T
ν
,C
×
)
(−1)
n−ℓ(ν)
z
ν

µ∈P
Φ
n

t∈T
ν
τ
Φ
(t)
s
=µ
s
θ(t)Q
µ
τ
Φ
(t)
(−q)P
µ
= (−1)

⌊n/2⌋

µ∈P
Φ
n

ν∈P
n
(−1)
n−ℓ(ν)
z
ν

t∈T
ν
τ
Φ
(t)
s
=µ
s

θ∈Hom(T
ν
,C
×
)
θ(t)Q
µ
τ

Φ
(t)
(−q)P
µ
.
By the o r t hogonality of characters of T
ν
, the inner-most sum is equal to zero for all t = 1.
If t = 1, then τ
Φ
(1, 1, . . . , 1)
(f)
= ∅ for f = {1} and τ
Φ
(1, 1, . . . , 1)
({1})
= ν. Thus,
ch(Γ
(n)
) = (−1)
⌊n/2⌋

µ∈P
Φ
n
µ
u
=µ

ν∈P

n
(−1)
n−ℓ(ν)
z
ν
|T
ν
|Q
µ
u
ν
(−q)P
µ
,
and in particular, if µ
({1})
u
= µ,
Γ
(n)
(µ) =


ν∈P
n
(−1)
n+⌊n/2⌋−ℓ(ν)
z
ν
|T

ν
|Q
µ
ν
(−q) if µ is unipotent,
0 otherwise.
(b) The proof is similar to (a), just using the G
n
characteristic map.
Remark. In the proof of Lemma 3.1, one may skip to (3.1) by using 10.7.3 in [3].
The values of the Gelfand–Graev character of the finite general linear group are well-
known. An elementary pro of of the following theorem is given in [9].
the electronic journal of combinatorics 16 (2009), #R146 13
Theorem 3.2. Let µ ∈ P
˜
Φ
n
with µ = µ
({1})
u
. Then
˜
Γ
(n)
(µ) =

(−1)
n−ℓ(µ)

ℓ(µ)

i=1

q
i
− 1

if µ is unipotent,
0 otherwise.
We may now apply Theorem 3.2 and Lemma 3.1 to give the values of the Gelfand–
Graev character of U
n
.
Corollary 3.1. Let µ ∈ P
Φ
n
with µ = µ
({1})
u
. Then
Γ
(n)
(µ) =

(−1)
⌊n/2⌋−ℓ(µ)

ℓ(µ)
i=1

(−q)

i
− 1

if µ is unipotent,
0 otherwise.
Proof. Combine Lemma 3.1 (b) with Theorem 3.2 to get
(−1)
ℓ(µ)
ℓ(µ)

i=1

q
i
− 1

=

ν∈P
n
(−1)
ℓ(ν)
z
ν
|
˜
T
ν
|Q
µ

ν
(q),
which implies
ℓ(µ)

i=1

1 − q
i

=

ν∈P
n
1
z
ν
ℓ(ν)

i=1
(1 − q
ν
i
)Q
µ
ν
(q).
Make the substitution q → −q to get
ℓ(µ)


i=1

1 − (−q)
i

=

ν∈P
n
1
z
ν
ℓ(ν)

i=1
(1 − (−q)
ν
i
)Q
µ
ν
(−q),
which implies
(−1)
⌊n/2⌋+ℓ(µ)
ℓ(µ)

i=1

(−q)

i
− 1

=

ν∈P
n
(−1)
⌊n/2⌋+|ν|− ℓ(ν)
z
ν
|T
ν
|Q
µ
ν
(−q).
Apply this last identity to Lemma 3.1 (a) to obtain the desired result.
4 Degener ate Gelfand–Graev characters
4.1 G
n
= GL(n, F
q
2
) notation (different from Section 3)
In this Section 4, let G
n
= GL(n, F
q
2

), and define
˜
Φ = {F
2
-orbits of
¯
F
×
q
}.
Note that now G
F
n
= U
n
and G
F
′
n
= U
′
n
, and also that
˜
T
2m
= T
2m
. Through the norm maps
˜

N
2m,2k
: T
(2m)
→ T
(2k)
(where k|m), defined in (2.8), and the corresponding transpose
the electronic journal of combinatorics 16 (2009), #R146 14
maps
˜
N
∗
2m,2k
: T
∗
(2m)
→ T
∗
(2k)
defined in (2.9), we let
˜
L be the direct limit of the groups
T
∗
(2m)
with respect to the maps N
∗
2m,2k
:
˜

L = lim
−→
T
∗
(2m)
.
Since F
2
=
˜
F
2
acts on
˜
L, we may consider the corresponding orbits, and define
˜
Θ = {F
2
-orbits in
˜
L}.
The same set-up of Sections 2.4 and 2.5 gives a characteristic map for G
n
by replacing
Φ by
˜
Φ, Θ by
˜
Θ, −q by q, T
(k)

by T
(2k)
, and (−1)
⌊n/2⌋+n(λ)
s
λ
by s
λ
.
4.2 The definition of degenerate Gelfand–Graev characters
Let (k, ν) b e a pair such that ν ⊢
n−k
2
∈ Z
0
, and let
ν

= (ν
1
, ν
2
, . . . , ν
ℓ
), where ν
j
= ν
1
+ ν
2

+ · · · + ν
j
.
Then the ma p ψ
(k,ν)
: B
<
n
→ C
×
, given by
ψ
(k,ν)




X
α
/[X
α
,X
α
]
=



ψ if α = (i, i + 1), 1  i < ⌊n/2 ⌋, and i /∈ ν


,
ψ if α = ⌊n/2⌋ and ⌊n/2⌋ /∈ ν

,
1 otherwise,
is a linear character of U
′
n
. Note t hat ψ
(⌈n/2⌉−⌊n/2⌋,(1
⌊n/2⌋
))
is the trivial character and
ψ
(n,∅)
= ψ
(n)
of Sectio n 3.
The degenerate Gelfand–Graev character Γ
(k,ν)
is
Γ
(k,ν)
= Ind
U
′
n
B
<
n

(ψ
(k,ν)
)
∼
=
Ind
U
n
y B
<
n
y
−1
(yψ
(k,ν)
y
−1
),
where y is an element of
¯
G
n
such that yU
′
n
y
−1
= U
n
. In particular, the Gelfand–Graev

character is Γ
(n,∅)
.
Let
L
′
(k,ν)
= L
k
, L
(1)
ν
, L
(2)
ν
, · · · , L
(ℓ)
ν
,
where
L
k
= X
ij
, X
i,n+1−j
, X
r
| |ν| < i < j  |ν| + k, |ν|  r  |ν| + k
∼

=
U(k, F
q
2
)
L
(r)
ν
= X
ij
| ν
r−1
 i < j  ν
r

∼
=
GL(ν
r
, F
q
2
).
Then
L
′
(k,ν)
∼
=
U(k, F

q
2
) ⊕ GL(ν
1
, F
q
2
) ⊕ · · · ⊕ GL(ν
ℓ
, F
q
2
)
is a maximally split Levi subgroup of U
′
n
. For example, if n = 9, k = 3, and ν = (2, 1),
then
L
′
(k,ν)
=


















A 0 0 0 0
0 B 0 0 0
0 0 C 0 0
0 0 0 F
′
(B) 0
0 0 0 0 F
′
(A)










A ∈ GL(2, F
q

2
), B ∈ GL(1, F
q
2
),
C ∈ U(3, F
q
2
)











.
the electronic journal of combinatorics 16 (2009), #R146 15
Note that since L
(i)
ν
⊆ U
′
2ν
i
∼

=
U
2ν
i
, the Levi subgroup
U
(k,ν)
= U
k
⊕ U
2ν
1
⊕ U
2ν
2
⊕ · · · ⊕ U
2ν
ℓ
⊆ U
n
contains a Levi subgroup L = U
k
⊕ L
1
⊕ · · · ⊕ L
ℓ
with L
i
⊆ U
2ν

i
such that L
∼
=
L
′
(k,ν)
.
Proposition 4.1. Let (k, ν) be such that ν ⊢
n−k
2
∈ Z
0
. Then
ch(Γ
(k,ν)
) = ch

Γ
(k)

ch

R
U
2ν
1
G
ν
1

(
˜
Γ
(ν
1
)
)

ch

R
U
2ν
2
G
ν
2
(
˜
Γ
(ν
2
)
)

· · · ch

R
U
2ν

ℓ
G
ν
ℓ
(
˜
Γ
(ν
ℓ
)
)

,
where
˜
Γ
(m)
is the Gelfand–Graev character of G
m
= GL(m, F
q
2
).
This prop osition is a consequence of Theorem 2.1 and the following lemma.
Lemma 4.1. Let (k, ν) be such that ν ⊢
n−k
2
∈ Z
0
. Then

Γ
(k,ν)
∼
=
R
U
n
U
(k,ν)

Γ
(k)
⊗ R
U
2ν
1
L
1
(
˜
Γ
(ν
1
)
) ⊗ · · · ⊗ R
U
2ν
ℓ
L
ℓ

(
˜
Γ
(ν
ℓ
)
)

.
Proof. Since L
′
(k,ν)
is maximally split,
Ind
U
n
y B
<
n
y
−1
(yψ
(k,ν)
y
−1
)
∼
=
Ind
U

′
n
B
<
n
(ψ
(k,ν)
)
∼
=
Indf
U
′
n
L
′
(k,ν)
(Γ
(k)
⊗
˜
Γ
(ν
1
)
⊗ · · · ⊗
˜
Γ
(ν
ℓ

)
).
where Indf
G
L
is Harish-Chandra induction. However,
Indf
U
′
n
L
′
(k,ν)
(Γ
(k)
⊗
˜
Γ
(ν
1
)
⊗ · · · ⊗
˜
Γ
(ν
ℓ
)
) = R
U
′

n
L
′
(k,ν)
(Γ
(k)
⊗
˜
Γ
(ν
1
)
⊗ · · · ⊗
˜
Γ
(ν
ℓ
)
),
∼
=
R
U
n
L
(Γ
(k)
⊗
˜
Γ

(ν
1
)
⊗ · · · ⊗
˜
Γ
(ν
ℓ
)
).
By transitivity of Deligne-Lusztig induction, we now have
Ind
U
n
y B
<
n
y
−1
(yψ
(k,ν)
y
−1
)
∼
=
R
U
n
U

(k,ν)

Γ
(k)
⊗ R
U
2ν
1
L
1
(
˜
Γ
(ν
1
)
) ⊗ · · · ⊗ R
U
2ν
ℓ
L
ℓ
(
˜
Γ
(ν
ℓ
)
)


.
4.3 Symplectic tableaux and domino tableaux c ombinatorics
Augment the nonnegative integers by symbols {
¯
i | i ∈ Z
>0
}, so that we have
{0,
¯
1, 1,
¯
2, 2,
¯
3, 3, . . .},
and order this set by i−1 <
¯
i < i < i + 1. Alternatively, one could identify this augmented
set with
1
2
Z
0
by
¯
i = i −
1
2
.
Let λ = (λ
1

, λ
2
, . . . , λ
r
) be a partition o f n and (m
0
, m
1
, m
2
, . . . , m
ℓ
) be a sequence
of nonnegative integers that sum to n with m
0
 λ
1
. A symplectic tableau Q of shape
λ/(m
0
) and weight (m
0
, m
1
, . . . , m
ℓ
) is a column strict filling of the boxes of λ by symbols
{0,
¯
1, 1,

¯
2, 2, . . . ,
¯
ℓ, ℓ},
the electronic journal of combinatorics 16 (2009), #R146 16
such that
m
i
=

number of 0’s in Q if i = 0,
number of
¯
i’s + number of i’s in Q if i > 0.
We write sh(Q) = λ/(m
0
) and wt(Q) = (m
0
, m
1
, . . . , m
ℓ
). For example, if
Q =
0 0
¯
1
1
¯
4

1
¯
2
¯
2
¯
3
3
, then sh(Q) = and wt(Q) = (2, 3, 2, 2, 1).
Let
T
λ
(m
0
,m
1
, ,m
ℓ
)
=

symplectic tableaux of shape λ/(m
0
)
and weight (m
0
, m
1
, . . . , m
ℓ

)

. (4.1)
A tiling of λ by dominoes is a partition of the boxes of λ into pairs of a djacent boxes.
Fo r example, if
λ = , then
is a tiling of λ by dominoes.
Let (m
0
, m
1
, . . . , m
ℓ
) be a sequence of nonnegative integers such that m
0
 λ
1
and
|λ| = m
0
+ 2(m
1
+ · · · + m
ℓ
). A domino tableau Q of shape λ/(m
0
) = sh(Q) and weight
(m
0
, m

1
, . . . , m
ℓ
) = wt (Q) is a column strict filling of a tiling of the shape λ/(m
0
) by
dominoes, where if a domino is filled with a number, then that number occupies both
boxes covered by that domino. We put 0’s in the non- t iled boxes of λ, and m
i
is the
number of i’s which appear. For example, if
Q =
0 0 3
1
3
2
, then sh(Q) = and wt(Q) = (2, 1, 1, 2).
Let
D
λ
(m
0
,m
1
, ,m
ℓ
)
=

domino tableaux of shape λ/(m

0
)
and weight (m
0
, m
1
, . . . , m
ℓ
)

. (4.2)
In the following Lemma, (a) is a straightforward use of the usual Pieri rule, and (b) is
both similar to (and perhaps a special case of) [14, Theorem 6.3], and also related to a
Pieri formula in [12].
Lemma 4.2. Let (m
0
, m
1
, . . . , m
ℓ
) be an ℓ + 1-tuple of nonnegative integers which sum
to n. Then
(a) s
(m
0
)
ℓ

r=1
m

r

i=0
s
(i)
s
(m
r
−i)
=

λ∈P
n
|T
λ
(m
0
,m
1
, ,m
ℓ
)
|s
λ
,
(b) s
(m
0
)
ℓ


r=1
2m
r

i=0
(−1)
i
s
(i)
s
(2m
r
−i)
=

λ∈P
2n−m
0
(−1)
n(λ)
|D
λ
(m
0
,m
1
, ,m
ℓ
)

|s
λ
.
the electronic journal of combinatorics 16 (2009), #R146 17
Proof. (a) Note that
s
(m
0
)
ℓ

r=1
m
r

i=0
s
(i)
s
(m
r
−i)
=

0i
r
m
r
1rℓ
s

(m
0
)
ℓ

r=1
s
(i
r
)
s
(m
r
−i
r
)
.
Now repeated applications of Pieri’s rule implies the result.
(b) Note that
s
(m
0
)
ℓ

r=1
2m
r

i=0

(−1)
i
s
(i)
s
(2m
r
−i)
=

0i
r
2m
r
1rℓ
(−1)
i
1
+···+i
ℓ
s
(m
0
)
ℓ

r=1
s
(i
r

)
s
(2m
r
−i
r
)
.
By Pieri’s rule,

0i
r
2m
r
1rℓ
s
(m
0
)
ℓ

r=1
s
(i
r
)
s
(2m
r
−i

r
)
=

λ∈P
2n−m
0


Number of column strict fillings of λ
using m
0
0’s, and for r = 1, 2, . . . , ℓ,
using i
r
¯r’s and (2m
r
− i
r
) r’s.


s
λ
.
By observing that the sign counts the number of barred entries,
s
(m
0
)

ℓ

r=1
2m
r

i=0
(−1)
i
s
(i)
s
(2m
r
−i)
=

λ∈P
2n−m
0


Q∈T
λ
(m
0
,2m
1
, ,2m
ℓ

)
(−1)
Number of barred entries in Q

s
λ
.
(4.3)
We therefore need to determine the cancellations for a given shape λ.
Fix r ∈ {1, 2, . . . , ℓ} and λ ∈ P such that T
λ
(m
0
,2m
1
, ,2m
ℓ
)
= ∅. For a tableau Q ∈
T
λ
(m
0
,2m
1
, ,2m
ℓ
)
, let
Q

r
= skew tableaux consisting of the boxes in Q containing ¯r or r,
S
(r)
Q
= {column strict fillings of sh(Q
r
) by elements in {¯r, r}}.
Fo r example, if
Q =
0 0
¯
1
1 1
1
¯
2
¯
2
¯
3
3
then Q
1
=
¯
1
1 1
1
and

¯
1
¯
1
1
¯
1
,
1 1 1
¯
1
∈ S
(1)
Q
.
(In fact, |S
(1)
Q
| = 8).
In light of (4.3), (b) is equivalent to

Q
′
∈S
(r)
Q
(−1)
Number of ¯r’s in Q
′
=


(−1)
n(sh(Q
r
))
if sh(Q
r
) has a domino tiling,
0 otherwise.
the electronic journal of combinatorics 16 (2009), #R146 18
Note that in row j, Q
′
∈ S
(r)
Q
is of the form
d
j−1
  
d
j
  
¯r
···
¯r
?
···
?
d
j+1

  
¯r
···
¯r
?
···
?
r
···
r
?
···
?
r ··· r
←− row j − 1
←− row j
←− row j + 1
Thus, we have d
j
+ 1 choices for the values in row j. If the total number of choices is
even, then exactly half of these choices give a positive sign and half give a negative sign.
So we have

Q
′
∈S
(r)
Q
(−1)
Number of ¯r’s in Q

′
= 0,
unless d
j
is even for all rows j. In this case, the signs of all but one o f the possible tableaux
will cancel each other out, so the only tableau that we have to count has row j of the
form
d
j−1
  
d
j
   ¯r
···
¯r
r
···
r
d
j+1
  
¯r
···
¯r
r
···
r r
···
r
r

···
r r
···
r
←− row j − 1
←− row j
←− row j + 1
which can clearly be tiled by dominoes of the form
r
and
r
. For this tableau, we have
(−1)
Number of ¯r’s
= (−1)
n(sh(Q
r
))
.
Thus,
s
(m
0
)
ℓ

r=1
2m
r


i=0
(−1)
i
s
(i)
s
(2m
r
−i)
=

λ∈P
2n−m
0
(−1)
n(λ)
|D
λ
(m
0
,m
1
, ,m
ℓ
)
|s
λ
,
as desired.
Let λ ∈ P

Θ
and γ ∈ P
Θ
be such that ht(γ)  1 and |γ
(ϕ)
|  λ
(ϕ)
1
for a ll ϕ ∈ Θ. A
battery Θ-tableau Q of shape λ/γ is a sequence of tableaux indexed by Θ such that
Q
(ϕ)
=

a domino tableau of shape λ
(ϕ)
/γ
(ϕ)
if |ϕ| is odd,
a symplectic tableau of shape λ
(ϕ)
/γ
(ϕ)
if |ϕ| is even.
The weight of Q is wt(Q) = (wt(Q)
1
, wt(Q)
2
, . . .), where
wt(Q)

i
=

ϕ∈Θ
|ϕ| odd
|ϕ|wt(Q
(ϕ)
)
i
+

ϕ∈Θ
|ϕ| even
|ϕ|
2
wt(Q
(ϕ)
)
i
.
Let
B
λ
(k,ν)
= {Q battery tableaux | sh(Q) = λ/γ, γ ∈ P
Θ
k
, ht(γ)  1, wt(Q) = ν}. (4.4)
Example. If
λ =


(ϕ
1
)
,
(ϕ
2
)
,
(ϕ
3
)

where |ϕ
i
| = i,
the electronic journal of combinatorics 16 (2009), #R146 19
then B
λ
(2,(5,4))
contains

0 0
1
(ϕ
1
)
,
1 2
(ϕ

2
)
,
1
2
(ϕ
3
)

,

0 0
1
(ϕ
1
)
,
¯
1
¯
2
(ϕ
2
)
,
1
2
(ϕ
3
)


,

0 0
1
(ϕ
1
)
,
¯
1
2
(ϕ
2
)
,
1
2
(ϕ
3
)

,

0 0
1
(ϕ
1
)
,

1
¯
2
(ϕ
2
)
,
1
2
(ϕ
3
)

,

0 0
2
(ϕ
1
)
,
¯
1
¯
1
(ϕ
2
)
,
1

2
(ϕ
3
)

,

0 0
2
(ϕ
1
)
,
¯
1
1
(ϕ
2
)
,
1
2
(ϕ
3
)

,

0 0
2

(ϕ
1
)
,
1 1
(ϕ
2
)
,
1
2
(ϕ
3
)

.
Some intuition. If λ ∈ P
Θ
, we can think of the boxes in λ
(ϕ)
as being |ϕ| deep, so in
the above exa mple,
λ =













(ϕ
1
)
,












(ϕ
2
)
,

























(ϕ
3
)


.
A battery Θ-tableau is a way of stuffing the slots by numbered “batteries” where front
and back are distinguished by i and
¯
i, but the sides look generica lly like i, so




¯
i
i






i
 
?
?
?
i
i
?
?
?
?
?
?
i
.
Then a battery Θ-tableau might look like:





0


0




1


(ϕ
1
)
,



1



¯
2







(ϕ
2
)
,
1 2 2











2




1




1
``
@

@




(ϕ
3
)


,
so the weight of the tableau counts the numb er of batteries of a given type get used,
regardless of the cardinality of ϕ.
4.4 Inducing from G
n
to U
2n
Note that any maximal torus
˜
T
ν
of G
n
⊆ U
2n
becomes the maximal torus T
2ν
of U
2n
,

which gives rise to the map
i :



Pairs (
˜
T
ν
,
˜
θ
ν
) with
˜
T
ν
a
maximal torus of G
n
,
˜
θ
ν
∈ Hom(
˜
T
ν
, C
×

)



−→



Pairs (T
2ν
, θ
ν
) with T
2ν
a
maximal torus of U
2n
,
θ
ν
∈ Hom(T
2ν
, C
×
)



(
˜

T
ν
,
˜
θ
ν
) → (T
2ν
,
˜
θ
ν
).
the electronic journal of combinatorics 16 (2009), #R146 20
To translate the combinatorics between G
n
and U
2n
, we define the map
ι : P
˜
Θ
n
−→ P
Θ
2n
˜
λ → ι
˜
λ

where for ϕ ∈ Θ, ι
˜
λ
(ϕ)
=

2
˜
λ
( ˜ϕ)
if ϕ = ˜ϕ,
˜
λ
( ˜ϕ)
∪
˜
λ
(F ( ˜ϕ))
if ϕ = ˜ϕ ∪ F ˜ϕ,
which has the property that τ
Θ
◦i = ι◦τ
˜
Θ
(see (2.12)). The map ι is neither surjective nor
injective. We note that F ˜ϕ = ˜ϕ implies that | ˜ϕ| is odd, and if F ˜ϕ = ˜ϕ, then ϕ = ˜ϕ ∪ F ˜ϕ
implies |ϕ| is even (see [5]). Thus, the image of ι is the set of even Θ-partitions,
Image(ι) = {λ ∈ P
Θ
n

| |ϕ|λ
(ϕ)
is even for ϕ ∈ Θ}.
Theorem 4.1.
R
U
2n
G
n
(
˜
Γ
(n)
) =

λ∈P
Θ
2n
ht(λ)2
|B
λ
(0,(n))
|χ
λ
.
Proof. Not e that by Theorem 3.1, (2.16), and the characteristic map fo r G
n
,
˜
Γ

(n)
=

˜
λ∈P
˜
Θ
n
ht(
˜
λ)=1
χ
˜
λ
=

˜
λ∈P
˜
Θ
n
ht(λ)=1

˜
ν∈P
˜
λ
s
(−1)
n−ℓ(

˜
ν)
z
˜
ν
R
G
n
˜
ν
.
By transitivity of induction, and the fact that τ
Θ
◦ i = ι ◦ τ
˜
Θ
, we have R
U
2n
G
n
(R
G
n
˜
ν
) = R
U
2n
ι

˜
ν
,
and so
R
U
2n
G
n
(
˜
Γ
(n)
) =

˜
λ∈P
˜
Θ
n
ht(
˜
λ)=1

˜
ν∈P
˜
λ
s
(−1)

n−ℓ(
˜
ν)
z
˜
ν
R
U
2n
ι
˜
ν
.
We now change the second sum to a sum over ν = ι
˜
ν ∈ P
ι
˜
λ
s
, and we obtain
R
U
2n
G
n
(
˜
Γ
(n)

) =

˜
λ∈P
˜
Θ
n
ht(
˜
λ)=1

ν∈P
ι
˜
λ
s


˜
ν∈P
˜
λ
s
ι
˜
ν=ν
1
z
˜
ν


(−1)
n−ℓ(ν)
R
U
2n
ν
=

ν∈P
Θ
2n
ν even


˜
ν∈P
˜
Θ
n
ι
˜
ν=ν
1
z
˜
ν

(−1)
n−ℓ(ν)

R
U
2n
ν
.
Recall that F ˜ϕ = ˜ϕ implies that | ˜ϕ| is odd, and F ˜ϕ = ˜ϕ implies that ϕ = ˜ϕ ∪ F ˜ϕ where
|ϕ| is even. Apply the characteristic map, factor, and then reindex to obtain
ch(R
U
2n
G
n
(
˜
Γ
(n)
)) = (−1)
n

ν∈P
Θ
2n
ν even


˜
ν∈P
˜
Θ
ι

˜
ν=ν
1
z
˜
ν

p
ν
= (−1)
n

ν∈P
Θ
2n
ν even

ϕ∈Θ
|ϕ| odd
1
z
ν
(ϕ)
/2
p
ν
(ϕ)
(Y
(ϕ)
)


ϕ∈Θ
|ϕ| even


ν,µ∈P
η∪µ=ν
(ϕ)
1
z
η
z
µ

p
ν
(ϕ)
(Y
(ϕ)
)
the electronic journal of combinatorics 16 (2009), #R146 21
= (−1)
n

γ∈P
Θ
2n
ht(γ)=1
γ even


ϕ∈Θ
|ϕ| odd


ν∈P
|ν|=|γ
(ϕ)
|
ν even
1
z
ν/2
p
ν
(Y
(ϕ)
)


ϕ∈Θ
|ϕ| even


η,µ∈P
|η|+|µ|=|γ
(ϕ)
|
1
z
η

z
µ
p
η∪µ
(Y
(ϕ)
)

.
Note that by (2.1),

η,µ∈P
|η|+|µ|=|γ|
1
z
η
z
µ
p
η∪µ
=
|γ|

i=0


|η|=i
z
−1
η

p
η


|µ|=|γ|−i
z
−1
µ
p
µ

=
|γ|

i=0
s
(i)
s
(|γ|−i)
.
A computation similar to [17, I.2.14] shows that

|ν|=|γ|
ν even
1
z
ν/2
p
ν
=

|γ|

i=0
(−1)
i
s
(i)
s
(|γ|−i)
.
Thus,
ch(R
U
2n
G
n
(
˜
Γ
(n)
)) = (−1)
n

γ∈P
Θ
n
ht(γ)=1
γ even

ϕ∈Θ

|γ
(ϕ)
|

i=0
(−1)
|ϕ|i
s
(i)
(Y
(ϕ)
)s
(|γ
(ϕ)
|−i)
(Y
(ϕ)
). (4.5)
Lemma 4.2 (a) and (b), respectively, imply that
k

i=0
s
(i)
s
(k−i)
=

λ∈P
k

|T
λ
(0,k)
|s
λ
, and
k

i=0
(−1)
i
s
(i)
s
(k−i)
=

λ∈P
k
(−1)
n(λ)
|D
λ
(0,k/2)
|s
λ
.
Since |D
λ
(0,k/2)

| = |T
λ
(0,k)
| = 0 unless ht(λ)  2,
ch(R
U
2n
G
n
(
˜
Γ
(n)
))
= (−1)
n

γ∈P
Θ
2n
ht(γ)=1
γ even

ϕ∈Θ
|ϕ| odd

|λ
(ϕ)
|=|γ
(ϕ)

|
(−1)
n(λ
(ϕ)
)


D
λ
(ϕ)
(0,|γ
(ϕ)
|/2)


s
λ
(ϕ)
(Y
(ϕ)
)
·

ϕ∈Θ
|ϕ| even

|λ
(ϕ)
|=|γ
(ϕ)

|


T
λ
(ϕ)
(0,|γ
(ϕ)
|)


s
λ
(ϕ)
(Y
(ϕ)
)
=

λ∈P
Θ
2n
ht(λ)2
(−1)
n+n(λ)
|B
λ
(0,(n))
|s
λ

.
Apply ch
−1
to get the result.
the electronic journal of combinatorics 16 (2009), #R146 22
Corollary 4.1. Fo r n ∈ Z
1
,
ch(R
U
2n
G
n
(
˜
Γ
(n)
)) = (−1)
n

ν∈P
Θ
2n
ht(ν)=1
ν even

ϕ∈Θ
|ν
(ϕ)
|


i=0
(−1)
i|ϕ|
s
(i)
(Y
(ϕ)
)s
(|ν
(ϕ)
|−i)
(Y
(ϕ)
).
Proof. This is (4.5) in the proof of Theorem 4.1.
Using similar techniques, we can prove a result for arbitrary irreducible characters of
G
n
. For λ ∈ P
˜
Θ
and γ ∈ P
ι
˜
λ
s
, let
c
γ

˜
λ
=

ϕ∈Θ
c
γ
˜
λ
(ϕ), where c
γ
˜
λ
(ϕ) =





c
γ
(ϕ)
λ
( ˜ϕ)
if ϕ = ˜ϕ ∈
˜
Θ,
c
γ
(ϕ)

˜
λ
( ˜ϕ)
˜
λ
(F ˜ϕ)
if ϕ = ˜ϕ ∪ F ˜ϕ and F ˜ϕ = ˜ϕ ∈
˜
Θ,
where c
λ
νµ
is as in (2.4), and c
γ
λ
is as in (2.5).
Theorem 4.2. Let
˜
λ ∈ P
˜
Θ
n
. Then
R
U
2n
G
n
(χ
˜

λ
) =

γ∈P
ι
˜
λ
s
(−1)
n(γ)
c
γ
˜
λ
χ
γ
.
Proof. By (2.16) and the characteristic map for G
n
,
χ
˜
λ
=

˜
ν∈P
˜
λ
s



˜ϕ∈
˜
Θ
ω
˜
λ
( ˜ϕ)
(
˜
ν
( ˜ϕ)
)
z
˜
ν
( ˜ϕ)

(−1)
n−ℓ(
˜
ν)
R
G
n
˜
ν
.
By transitivity of induction, and the fact that τ

Θ
◦ i = ι ◦ τ
˜
Θ
, we have R
U
2n
G
n
(R
G
n
˜
ν
) = R
U
2n
ι
˜
ν
,
and so
R
U
2n
G
n
(χ
˜
λ

) =

˜
ν∈P
˜
λ
s


˜ϕ∈
˜
Θ
ω
˜
λ
( ˜ϕ)
(
˜
ν
( ˜ϕ)
)
z
˜
ν
( ˜ϕ)

(−1)
n−ℓ(
˜
ν)

R
U
2n
ι
˜
ν
.
We now change the sum to a sum over ν = ι
˜
ν ∈ P
ι
˜
λ
s
, and using the image of the map ι,
we obtain
R
U
2n
G
n
(χ
˜
λ
) =

ν∈P
ι
˜
λ

s
ν even


˜
ν∈P
˜
λ
s
ι
˜
ν=ν


ϕ∈Θ
ω
˜
λ
(ϕ)
(
˜
ν
(ϕ)
)
z
˜
ν
(ϕ)



(−1)
n−ℓ(ν)
R
U
2n
ν
.
Apply the characteristic map, and rewrite the inner sum and product, to get
ch

R
U
2n
G
n
(χ
˜
λ
)

= (−1)
n

ν∈P
ι
˜
λ
s
ν even



˜
ν∈P
˜
λ
s
ι
˜
ν=ν


˜ϕ∈
˜
Θ
ω
˜
λ
( ˜ϕ)
(
˜
ν
( ˜ϕ)
)
z
˜
ν
( ˜ϕ)


p

ν
the electronic journal of combinatorics 16 (2009), #R146 23
= (−1)
n

ν∈P
ι
˜
λ
s
ν even

˜ϕ∈
˜
Θ
F ˜ϕ= ˜ϕ
ω
˜
λ
( ˜ϕ)
(ν
( ˜ϕ)
/2)
z
ν
( ˜ϕ)
/2

˜ϕ∈
˜

Θ
F ˜ϕ= ˜ϕ


|γ|=|
˜
λ
( ˜ϕ)
|
|µ|=|
˜
λ
(F ˜ϕ)
|
γ∪µ=ν
( ˜ϕ∪F ˜ϕ)
ω
˜
λ
( ˜ϕ)
(γ)ω
˜
λ
(F ˜ϕ)
(µ)
z
γ
z
µ


p
ν
.
Recall that F ˜ϕ = ˜ϕ implies that | ˜ϕ| is odd, and F ˜ϕ = ˜ϕ implies that ϕ = ˜ϕ ∪ F ˜ϕ where
|ϕ| is even. Thus, for every ϕ ∈ Θ such that ν
(ϕ)
= ∅, if |ϕ| is odd then ϕ = ˜ϕ for
some ˜ϕ ∈
˜
Θ, and if |ϕ| is even then ϕ = ˜ϕ ∪ F ˜ϕ for some ˜ϕ ∈
˜
Θ. Factor our expression
accordingly as
ch

R
U
2n
G
n
(χ
˜
λ
)

= (−1)
n

ν∈P
ι

˜
λ
s
ν even

ϕ∈Θ
ϕ= ˜ϕ
ω
˜
λ
( ˜ϕ)
(ν
(ϕ)
/2)
z
ν
(ϕ)
/2
p
ν
(ϕ)

ϕ∈Θ
ϕ= ˜ϕ∪F ˜ϕ


|γ|=|
˜
λ
( ˜ϕ)

|
|µ|=|
˜
λ
(F ˜ϕ)
|
γ∪µ=ν
(ϕ)
ω
˜
λ
( ˜ϕ)
(γ)ω
˜
λ
(F ˜ϕ)
(µ)
z
γ
z
µ

p
ν
(ϕ)
= (−1)
n

ϕ∈Θ
ϕ= ˜ϕ


|ν|=|
˜
λ
( ˜ϕ)
|
ω
˜
λ
( ˜ϕ)
(ν)
z
ν
p
2ν
(Y
(ϕ)
)
·

ϕ∈Θ
ϕ= ˜ϕ∪F ˜ϕ

|ν|=|
˜
λ
( ˜ϕ)
|+|
˜
λ

(F ˜ϕ)
|


|γ|=|
˜
λ
( ˜ϕ)
|
|µ|=|
˜
λ
(F ˜ϕ)
|
γ∪µ=ν
ω
˜
λ
( ˜ϕ)
(γ)ω
˜
λ
(F ˜ϕ)
(µ)
z
γ
z
µ

p

ν
(Y
(ϕ)
).
The first product is the case that |ϕ| is odd, and the second product is the case that |ϕ|
is even. For the sum in the first product, note that

|ν|=|λ|+|η|


|γ|=|λ|
|µ|=|η|
γ∪µ=ν
ω
λ
(γ)ω
η
(µ)
z
γ
z
µ

p
ν
=


|γ|=|λ|
ω

λ
(γ)
z
γ
p
γ


|µ|=|η|
ω
η
(µ)
z
µ
p
µ

= s
λ
s
η
.
Fo r the sum in the product for |ϕ| even, we have

|ν|=|λ|
ω
λ
(ν)
z
ν

p
2ν
=

|ν|=|λ|
ω
λ
(ν)
z
ν
p
ν
◦ p
(2)
= s
λ
◦ p
(2)
where ◦ is the plethysm product (2.5). Thus, from the definition of the coefficients c
γ
˜
λ
, we
have
ch

R
U
2n
G

n
(χ
˜
λ
)

= (−1)
n

γ∈P
ι
˜
λ
s
c
γ
˜
λ
s
γ
,
as desired.
It is perhaps worth noting that since we know Ha r ish-Chandra induction R
U
2n
G
n
(χ
˜
λ

)
gives a character, then the sign of the coefficient c
γ
˜
λ
must be (−1)
n(γ)
.
the electronic journal of combinatorics 16 (2009), #R146 24
4.5 Cuspidal characters
A cuspidal character of U
n
is an irreducible character χ
λ
such that, for any Levi subgroup
L contained in a proper parabolic subgroup P ⊂ U
n
, and any character χ of L, χ
λ
satisfies
R
U
n
L
(χ), χ
λ
 = 0 (see [4, Proposition 6.3 ]). Note that because of the existence of P , the
functor R
U
n

L
will always be Harish–Chandra induction.
It follows from the description of Levi subgroups in Section 4.2 that every Levi of U
n
contained in a parabolic is isomorphic to
L
∼
=
G
k
1
× G
k
2
× · · · × G
k
ℓ
× U
m
,
where 2(k
1
+ · · · + k
ℓ
) + m = n. Thus, the irreducible character of L are indexed by
{(λ
1
, λ
2
, . . . , λ

ℓ
, µ) | λ
j
∈ P
˜
Θ
k
j
, µ ∈ P
Θ
m
}.
The characteristic of the corresponding induced character is given by
ch

R
U
2k
1
G
k
1
(χ
λ
1
)

· · · ch

R

U
2k
ℓ
G
k
ℓ
(χ
λ
ℓ
)

ch(χ
µ
).
Suppose λ ∈ P
˜
Θ
k
. D efine a subgroup G
λ
⊆ G
k
by
G
λ
=

˜ϕ∈
˜
Θ

G
( ˜ϕ)
λ
, where G
( ˜ϕ)
λ
= G
|λ
( ˜ϕ)
|
| ˜ϕ|
.
Note that G
λ
is a Levi of U
2k
contained in a parabo lic. This parabolic has a distinguished
irreducible character χ
[λ]
given by
χ
[λ]




G
( ˜ϕ)
λ
= χ


( ˜ϕ)
⊗ · · · ⊗ χ

( ˜ϕ)
  
|λ
( ˜ϕ)
| terms
.
Fo r example, if k = 17, and
λ =

( ˜ϕ
1
)
,
( ˜ϕ
2
)
,
( ˜ϕ
3
)

,
where | ˜ϕ
1
| = 2, | ˜ϕ
2

| = | ˜ϕ
3
| = 3. Then G
λ
= G
2
× G
3
× G
3
× G
3
× G
3
× G
3
, and
[λ] =

(
( ˜ϕ
1
)
), (
( ˜ϕ
2
)
), (
( ˜ϕ
2

)
), (
( ˜ϕ
2
)
), (
( ˜ϕ
3
)
), (
( ˜ϕ
3
)
)

.
The characteristic map for G
k
sends the Harish–Chandra induced character R
G
k
G
λ
(χ
[λ]
)
to

˜ϕ∈
˜

Θ
(s
(1)
(Y
( ˜ϕ)
))
|λ
( ˜ϕ)
|
. It follows from Pieri’s rules that the inner product
R
G
k
G
λ
(χ
[λ]
), χ
λ
 =

˜ϕ∈
˜
Θ
K
λ
( ˜ϕ)
,(1
|λ
( ˜ϕ)

|
)
= 0,
where K
λ
( ˜ϕ)
,(1
|λ
( ˜ϕ)
|
)
is a Kostka number given by the number of sta ndard tableaux of shape
λ
( ˜ϕ)
(see [17, Section I.5, Example 2(a)]). Thus, to understand which characters are in
the decomposition of characters induced from the split Levis, it suffices to consider the
case where L = G
k
1
× G
k
2
× · · · × G
k
ℓ
× U
m
and χ
(λ
1

,λ
2
, ,λ
ℓ
,µ)
satisfies
λ
j
= (
( ˜ϕ
j
)
), for some ˜ϕ
j
∈
˜
Θ with | ˜ϕ
j
| = k
j
.
the electronic journal of combinatorics 16 (2009), #R146 25