plZ)

Journal of Algebra 396 (2013) 98–116

Contents lists available at ScienceDirect

Journal of Algebra www.elsevier.com/locate/jalgebra

Gauss sums on GL2 (Z/ pl Z) Taiki Maeda Department of Mathematics, Sophia University, Kioicho 7-1, Chiyodaku, Tokyo 102-8554, Japan

a r t i c l e

i n f o

Article history: Received 8 February 2013 Available online 13 September 2013 Communicated by Michel Broué

a b s t r a c t We determine explicitly the Gauss sum

τl (χ , e) =



χ ( X )e(Tr X )

X ∈GL2 (Z/ pl Z)

Keywords: General linear group Irreducible character Gauss sum

on the general linear group GL2 (Z/ pl Z) for every irreducible character χ of GL2 (Z/ pl Z) and a nontrivial additive character e of Z/ pl Z, where p is an odd prime and l is an integer  2. While there are several studies of the Gauss sums on finite algebraic groups defined over a finite field, this paper seems to be the first one which determines the Gauss sums on a matrix group over a finite ring. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Generalizations of Gauss sums have been considered in many ways. As for the Gauss sums on finite algebraic groups, Kondo [7] firstly determined the value of Gauss sum on the finite general linear group GLn (Fq ) for every irreducible character, where Fq denotes a finite field with q elements. Also in a series of papers starting with [6], Kim–Lee, D.S. Kim and Kim–Park described the values of Gauss sums on finite classical groups for linear characters. Saito and Shinoda [10,11] considered the Gauss sums on finite reductive groups for the Deligne–Lusztig generalized characters and applied this result to determine the value of Gauss sum on the finite symplectic group Sp(4, q) and considered Gauss sums on the Chevalley group of type G 2 corresponding to every unipotent character. For an arbitrary finite group G, Gomi, Maeda and Shinoda [4] defined the Gauss sums on G, and explicitly determined the values of Gauss sums on the complex reflection group G (m, r , n) and also on the Weyl group of any type for all irreducible characters.

E-mail address: [email protected]. 0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.08.017

T. Maeda / Journal of Algebra 396 (2013) 98–116

99

On the other hand, Gauss sums over Z/ pl Z (p: odd prime, l  2) were determined explicitly by Odoni [9] and also by Funakura [3] for p = 2. To be precise, the Gauss sum over Z/ pl Z is given as follows:



gl (μ, e ) =

μ(x)e(x),

x∈(Z/ pl Z)×

where μ is a multiplicative character of (Z/ pl Z)× and e is a nontrivial additive character of Z/ pl Z. Thus, in this vein, it would be quite natural to consider the Gauss sum on G (Z/ pl Z), where G is an algebraic group. To start to consider this problem, we deal with the case G = GL2 in this paper. Extending the definition of the sum gl (μ, e ), we define the Gauss sum on the general linear group GL2 (Z/ pl Z) as follows:



τl (χ , e) =

χ ( X )e(Tr X ),

X ∈GL2 (Z/ pl Z)

where χ is a character of GL2 (Z/ pl Z). The definition of this Gauss sum differs from that of Gauss sum in [4] in the sense that e is an additive character of Z/ pl Z. We will determine explicitly τl (χ , e ) for every irreducible character χ in the case that p is an odd prime. In order to evaluate τl (χ , e ), we need the character theory of GL2 (Z/ pl Z). The irreducible characters of GL2 (Z/ pl Z) were completely constructed by Nobs [8] and Barrington Leigh, Cliff and Wen [1]. Nobs constructed them by using the Weil representation. However he did not consider the problem of finding the character values. On the other hand, by making full use of Clifford theory, Barrington Leigh, Cliff and Wen determined the values of the irreducible characters of GL2 (Z/ pl Z). They also showed that the degrees of irreducible characters of GL2 (Z/ pl Z) which do not come from GL2 (Z/ pl−1 Z) have precisely three possibilities, namely pl−1 ( p + 1), pl−1 ( p − 1), and pl−2 ( p 2 − 1), and such irreducible characters are induced from a character ψ of the stabilizer of a linear character defined on a congruence subgroup K n . However their construction of ψ in some cases is not sufficient enough for us to compute the Gauss sum. Hence, for those cases, we give a more explicit construction and in particular, in Lemma 3.4, we construct ψ in a different way. This paper is organized as follows: in Section 2, we recall the construction of the characters of GL2 (Z/ pl Z) after [1]. In Section 3, we explicitly determine the Gauss sums on GL2 (Z/ pl Z) for all irreducible characters after constructing the ψ ’s. In Appendix A, we clarify some vague values of  P = 0c ,d< p i , p d λ( p j β d + d−1 ( pk b − c 2 )) in [1, 6.1], since we heavily use the results in [1]. 2. Preliminaries 2.1. We shall use the following usual notation in the character theory of finite groups. Let G be a G. For C-valued functions finite group. The set of complex irreducible characters of G is denoted by  f and g on G, let

 f , g G =

1 

|G |

f (x) g (x)

x∈G

be the usual hermitian inner product on the vector space of C-valued functions on G, where g (x) is the complex conjugate of g (x). Now we recall Clifford theory (cf. [5]). Let N be a normal subgroup of G. For a character φ of N and g ∈ G, we define φ g , φ g : N → C, by φ g (n) = φ( gng −1 ). Then φ g is also a character of N. Let N, then Clifford theory implies that T = StabG φ = { g ∈ G | φ g = φ} be the stabilizer of φ in G. If φ ∈  T | ψ| N , φ N > 0} onto {χ ∈  G | χ | N , φ N > 0}. the map ψ → indGT ψ is a bijection from {ψ ∈ 

100

T. Maeda / Journal of Algebra 396 (2013) 98–116

2.2. We recall the construction of the characters of GL2 (Z/ pl Z), following [1]. Let p be an odd prime, l  2 a positive integer, and let

m = l/2,

n = l/2 ,

l

Z pl = Z/ p Z,

G l = GL2 (Z pl ).

We denote by I the identity matrix of G l and let K i = { I + p i B | B ∈ M 2 (Z pl )} for 1  i < l. Note that K i is a normal subgroup of G l for all i, and that K i is abelian if i  n. For a positive integer k, we will √ write ζk = exp(2π −1/k). Let λ be an injective additive character of Z pl defined by λ(1) = ζ pl . For A ∈ M 2 (Z pl ), we define a character φ A of K n , φ A : K n → C× , by

   φ A ( X ) = λ Tr A ( X − I ) . For

α ∈ Z pl , let μα be a multiplicative character of Z×pl , μα : Z×pl → C× , such that 







μα 1 + pn = λ pn α , where Z×l is the multiplicative group of Z pl . p

Following [1], let

 X= χ  X1 = χ  X2 = χ  X3 = χ

   l  K l−1  ker νχ for all ν ∈ Hom G l , C× , ∈G  l  χ ( I ) = pl−1 ( p + 1) , ∈G  l  χ ( I ) = pl−1 ( p − 1) , ∈G    l  χ ( I ) = pl−2 p 2 − 1 , ∈G

l − X. X4 = G Also for A ∈ M 2 (Z pl ), define

l ( A ) = G







χ ∈ Gl  χ | K n , φ A  K n > 0 .

The following properties were proven in [1, Theorem 3.1, its proof, and 3.5]. Theorem 2.1. With the notation as above, we have the following:

(1)



X1 =

l G



0α < pm , 1u ( pm −1)/2, p u

l G (2)

X2 =





     G l α + u 0  = p 2n−2 ( p − 1)2 ,  0 α 





α+u 0  l u 0 = (μα ◦ det)χ  χ ∈ G 0 0 0 α



l G

0α , < pm , ( p )=−1

l G



α+u 0 , 0 α







α , 1 α



.

       G l α  = p 2n−2 p 2 − 1 ,  1 α  



α  l 0 = (μα ◦ det)χ  χ ∈ G 1 α 1 0

where ( p ) is the Legendre symbol.



 ,

and

and

T. Maeda / Journal of Algebra 396 (2013) 98–116



X3 =

(3)

l G



0α < pm , 0β< pm−1

l G





|X1 | =

2

   G  α  l 1

p β 





 = p

α

2n−1

α pβ  l 0 p β = (μα ◦ det)χ  χ ∈ G 1 α 1 0 X = X1 X2 X3 and   1 |X2 | = p 2l−3 ( p − 1) p 2 − 1 ,

(4)

1



α pβ , 1 α

101

p 2l−3 ( p − 1)3 ,

( p − 1),

and

 .

|X3 | = p 2l−2 ( p − 1).

2

Z/ p l Z ) 3. Gauss sums on GL2 (Z Throughout this section, we fix a nontrivial additive character e of Z pl and put

3.1.

e (1) = ζ pr l . Let

χ be a character of G l . Then we define the Gauss sum on G l associated with χ by

τl (χ ) =



χ ( X )e(Tr X ) = |G l |χ , e ◦ TrG l .

X ∈G l

Let H be a subgroup of G l and ψ be a character of H . Then, by the Frobenius reciprocity, we have

1

|G l |





τl indGHl ψ = ψ, e ◦ Tr H .

The purpose of this section is to determine explicitly τl (χ ) associated with any irreducible character χ of G l . In the following four subsections, namely 3.2, 3.3, 3.4, and 3.5, we will evaluate τl (χ ) for every χ in X1 , X2 , X3 , and X4 respectively. 3.2. X1 In this subsection, we determine the Gauss sum 

0  α < pm , 1  u  ( pm − 1)/2 with p  u, A =

τl (χ )for every character χ in X1 . Let  irreducible 

α +u 0 , A0 = 0 α

u0 00

, and T = StabG l φ A 0 . Then we

have

 T = Km S =

a pm c

pm b d

    a, b , c , d ∈ Z pl ∩ G l ,

 where S =

(cf. [1, 3.2.1, 3.3.1 and the proof of Theorem 3.1]). Let

 N=

 T0 =

1 + pn a pm c a pm c

pn b d

pn b 1 + pn d

    a, b , c , d ∈ Z pl ,

    a, b , c , d ∈ Z pl ∩ G l .

a 0 0 d

    a , d ∈ Z× l p

102

T. Maeda / Journal of Algebra 396 (2013) 98–116

Then N is normal in T , and T 0 = N S [1]. Let λ be a multiplicative character of Z×l such that λ (1 + p

pn ) = λ( pn u ). Then we can extend φ A 0 to a character φ  of N by

φ





1 + pn a

pn b

m

1+ p d

p c

n

  = λ 1 + pn a .

We have Stab T φ  = T 0 [1] and we can also extend φ  to characters ψi j (0  i , j < pn−1 ( p − 1)) of T 0 by

 ψi j

a pm c

pn b d



= λ (a)1+ p i λ (d) p j . m

m

l ( A ) are By Clifford theory and Theorem 2.1 (1), the irreducible characters of G l in G





0  i , j < p n −1 ( p − 1) .

χi j = (μα ◦ det) indGT0 ψi j

Recall that the Gauss sum over Z pl associated with a multiplicative character

ν of Z×pl is given

by



gl (ν ) =

ν (x)e(x).

x∈Z×l p

The problem of evaluating gl (ν ) can be reduced to the case where ν is primitive and r = 1, and this value was determined explicitly by Odoni as follows (cf. [9] or [2, Theorem 1.6.2]): Theorem 3.1. Let ν be a primitive character of Z×l normalized such that ν (1 + p ) = ζ −l−11 . If r = 1, then p

p

gl (ν ) =

⎧ l/2 ⎪ if l = 2, ⎪ p ζ pl , ⎪ ⎪ ⎪ √ 2 ⎪ l/2 (1− p )/2 ( p −1)/8 ⎪ ⎨ p ζ pl −1 ζp , if l = 3, ⎪ pl/2 ζ σl , ⎪ p ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎩ pl/2 ζ σ −1(1− p )/2 , pl

if l > 3 and l is even, if l > 3 and l is odd,

where σ is a p-adic integer defined by

σ=

p



 1 − log

log(1 + p )



p log(1 + p )

.

Theorem 3.2. With the notation as above, we have

1

χi j ( I )



 1 + p m i  

τl (χi j ) = pl gl μα λ

gl

  pm j  .

μα λ

T. Maeda / Journal of Algebra 396 (2013) 98–116

103

Proof.





1

χi j ( I )

τl (χi j ) = | T 0 | (μα ◦ det)ψi j , e ◦ Tr T 0 



=

μα (ad)λ (a)1+ p i λ (d) p j e(a + d) m

m 0a,d< pl , 0b< p , 0c < pn p ad





= pl

m



μα (a)λ (a)1+ p i e(a) m

0a< pl , p a





μα (d)λ (d) p j e(d) m

0d< pl , p d

  1+ pm i      pm j  , = pl gl μα λ gl μα λ 2

which proves the assertion of the theorem. 3.3. X2

In this subsection, we determine the Gauss sum 

0  α , < pm with ( p ) = −1, A =

1

τ χ ) for every irreducible character χ in X2 . Let

l(  0 . 10

α 

α , A0 =

From the definition of φ A 0 given in 2.2, we

have

 φ A0

1 + pn a pn c

pn b 1 + pn d



  = λ pn b + pn c .

Let T = StabG l φ A 0 . Then we have

 T = Km S =

a b

b + pm c  a+p d m

  a, b , c , d ∈ Z pl ∩ G l ,



where S = (cf. [1, 3.2.2 and 3.3.2]). Let s1 =



1+ p



1+ p

, s2 =



1 p p 1

  | T | = p 2l+2n−2 p 2 − 1 ,

  a , b ∈ Z  pl ∩ G l

b 

a b

a

 . We have



K 1 ∩ S = s1  × s2 ,

p n −1 

K n ∩ S = s1

 p n −1  × s2 .

Moreover, there exists s3 ∈ S such that

S = ( K 1 ∩ S ) × s3 ,

p +1

s3

∈ Z (G l ),

where Z (G l ) denotes the center of G l (cf. [1, Lemma 5.1]). Hence the coset representatives of K n ∩ S in S are given by k

k

k

sk = s11 s22 s33

(k ∈ Ω),  where Ω = k = (k1 , k2 , k3 )  0  k1 , k2 < pn−1 , 0  k3 < p 2 − 1 . 

104

T. Maeda / Journal of Algebra 396 (2013) 98–116

Also we define δ ∈ Z pl by





pn δ = 2

1t  pn−1 , t: odd

If t  2, then

 p n −1  t

p n −1 t



pt (t +1)/2 .

pt ≡ 0 (mod pn+1 ), so δ ∈ Z×l . p

We evaluate the Gauss sum odd.

τl (χ ) for χ in X2 in two cases depending on whether l is even or

3.3.1. Even case, i.e. l = 2m The number of extensions of φ A 0 to T is | T : K m | = | K m S : K m | = | S : K m ∩ S | = |Ω|, and

 p m −1  = 1, φ A 0 s1  p m −1  = φ A0 φ A 0 s2





 0t  pm−1 , t: even

p m −1 t

 p 2 −1  = 1, φ A 0 s3







pt t /2 I +

1t  pm−1 , t: odd

  = λ pm δ = ζ pδm .

p m −1 t



pt (t −1)/2 A 0



Hence for every i ∈ Ω , there exists a unique extension ψi of φ A 0 such that δ+ pm i 2

ψi (s1 ) = ζ pi 1m−1 ,

ψ i ( s 2 ) = ζ p l −1

i

ψi (s3 ) = ζ p32 −1 .

,

l ( A ) are By Clifford theory and Theorem 2.1 (2), the irreducible characters of G l in G

χi = (μα ◦ det) indGT l ψi (i ∈ Ω). Theorem 3.3. With the notation as above, we have the following. (1) If p | r, then τl (χi ) = 0. (2) If p  r, then



1

χi ( I )

 

 where h is a unique element of Ω such that Proof. For k ∈ Ω , let sk =

1

χi ( I )



p τl (χi ) = p 2l μα det sh e Tr sh ζ pi1mh−11 ζ p(δ+ l −1

 x y  k k y k xk

−α r −1 − r −1 −r −1 −α r −1



m

i 2 )h2 i 3 h3 ζ p 2 −1 ,

∈ ( K m ∩ S )sh .

. Then





τl (χi ) = | T | (μα ◦ det)ψi , e ◦ Tr T =













 

  

μα 1 + pm a 1 + pm d μα det sk λ pm b + pm c ψi sk

0a,b,c ,d< pm k∈Ω

     a b xk · e Tr sk e pm Tr c

d

yk

yk xk



T. Maeda / Journal of Algebra 396 (2013) 98–116

=





 

  

μα det sk e Tr sk ψi sk

k∈Ω

·

105

2  

 

    λ pm αa e pm xk a

0a< pm



= p 2l



  k

2

    λ pm b e pm y k b

0b< pm

  

μα det s e Tr sk ψi sk .

k∈Ω,

α +xk r ≡0 (mod pm ), 1+ y k r ≡0 (mod pm )

Thus,

τl (χi ) = 0 if p | r.

Assume p  r. Then there exists a unique element h of Ω such that For every k ∈ Ω ,





−α r −1 − r −1 −r −1 −α r −1

α + xk r ≡ 0 (mod pm ) and 1 + y k r ≡ 0 (mod pm ) if and only if

( K m ∩ S )sk , namely k = h. Hence 1

χi ( I )



 

∈ ( K m ∩ S )sh .  −α r −1 − r −1 ∈ −1 −1 −r

−α r

  

τl (χi ) = p 2l μα det sh e Tr sh ψi sh

    (δ+ pm i )h = p 2l μα det sh e Tr sh ζ pi 1mh−11 ζ pl−1 2 2 ζ pi 32h−31 , 2

which proves the assertion of the theorem. 3.3.2. Odd case, i.e. l = 2m + 1 Let

 L = K m +1 S =

a b

b + pm+1 c  a + p m +1 d

  a, b , c , d ∈ Z pl ∩ G l .

Then | L | = p 3l−3 ( p 2 − 1). The number of extensions of φ A 0 to L is | L : K m+1 | = | K m+1 S : K m+1 | = | S : K m+1 ∩ S | = |Ω|, and

 pm  φ A 0 s2 = φ A 0



 0t  pm , t: even

 pm  φ A 0 s 1 = 1,  m p pt t /2 I + t

 p 2 −1  = 1, φ A 0 s3   m 1t  pm , t: odd

p t

pt (t −1)/2 A 0



  = λ pm+1 δ = ζ pδm .

Hence for every i ∈ Ω , there exists a unique extension φi of φ A 0 such that δ+ pm i 2

φi (s1 ) = ζ pi 1m ,

φ i ( s 2 ) = ζ p l −1

,

i

φi (s3 ) = ζ p32 −1 .

Let

 N j = K j Z (G l )( K 1 ∩ S ) =

a pb

p b + p j c a + p jd

    a, b , c , d ∈ Z pl ∩ G l ,

j = m , m + 1.

Then N j is normal in T and | N m | = p 3l−1 ( p − 1), | N m+1 | = p 3l−3 ( p − 1) (cf. [1, 3.3.2]). Let φi = φi | Nm+1 .



Then φi is stable under T . Let H = N m+1 



1+ p m

1

. Then | H | = p 3l−2 ( p − 1). Since Nm / Nm+1 is

abelian, H is normal in N m . We can extend φi to a linear character

φi of H such that φi is trivial on

106

T. Maeda / Journal of Algebra 396 (2013) 98–116

  m . It is easy to find an element in Nm that does not stabilize φi and since the index of H  1+ p 1

in N m is p, we have Stab Nm φi = H . Then Clifford theory implies that ind Hm φi is irreducible. Since φi N

is stable under T , we have

(ind NHm

φi )| Nm+1 = p φi . Hence

(1/ p ) ind NNmm+1

 φi = indNHm φi ∈ N m.

Lemma 3.4. Let ψi = (1/ p ) ind TNm+1 φi − ind TL φi for i ∈ Ω . Then ψi , for i ∈ Ω , are distinct irreducible characters of T and satisfy ψi | K m+1 = p φ A 0 . Proof. Since φi is stable under T , we have





ind TNm+1 φi  N

m +1



= p 2 ( p + 1)φi ,



ind TL φi  N

m +1

= p 2 φi .

Hence we have ψi | K m+1 = p φ A 0 and by the Frobenius reciprocity



ind TNm+1 φi , ind TNm+1 φi

 T



= p 2 ( p + 1),

ind TNm+1 φi , ind TL φi

We can pick the right coset representatives of L in T to be E cd = −1

prove that if (c , d) = (0, 0) then L ∩ E cd L E cd = N m+1 . Let g =

−1 E cd g E cd =







1 pm c 0 1+ p m d

a b + p m +1 z



b a + p m +1 w

 T



= p2 . (0  c , d < p ). Let us

∈ Nm+1 , we have

a − pm bc (1 + pm d)−1

b(1 + pm d) + pm+1 z − p 2m bc 2

b(1 + pm d)−1

a + pm+1 w + pm bc (1 + pm d)−1

.

−1 Hence, E cd g E cd ∈ L if and only if





 − 1   ≡ a + pm+1 w + pm bc 1 + pm d mod pm+1 ,    − 1   b 1 + pm d + pm+1 z − p 2m bc 2 ≡ b 1 + pm d mod pm+1 ,

a − pm bc 1 + pm d

 −1

namely p | b. Therefore



−1

L ∩ E cd L E cd =

a − pm+1 bc

p b(1 + pm d) + pm+1 z

pb(1 + pm d)−1

a + pm+1 (bc + w )

    a, b , z , w ∈ Z pl ∩ G l

= N m +1 . So the number of right cosets in L E cd L is | L : N m+1 | = p + 1. Hence there exists a suitable subset Γ of {(c , d) | 0  c , d < p } with (0, 0) ∈ Γ and |Γ | = p such that E cd ((c , d) ∈ Γ ) are the double coset representatives of L in T . By the Mackey theorem, we have



ind TL φi , ind TL φi

 T

=

  (c ,d)∈Γ

=1+

 E  φi cd  L ∩ E −1 L E , φi | L ∩ E −1 L E L ∩ E −1 L E cd



cd

 E cd  φi  N

(c ,d)∈Γ, (c ,d)=(0,0)

=1+



m +1

   φi , φi N

(c ,d)∈Γ, (c ,d)=(0,0)

cd

cd

, φi

m +1

cd

 N m +1

= p.

cd

T. Maeda / Journal of Algebra 396 (2013) 98–116

107

Therefore

ψi , ψi T =

1  p2

ind TNm+1 φi , ind TNm+1 φi

 T

−

2 p

ind TNm+1 φi , ind TL φi

 T

  + ind TL φi , ind TL φi T = 1.

Since ψi ( I ) = p, we conclude that ψi is irreducible. Suppose ψi = ψj for i, j ∈ Ω . Restricting to N m+1 on both sides, we have φi = φj . Hence

1 = ψi , ψj  T

=

  1  1 ind TNm+1 φi , ind TNm+1 φi T − ind TNm+1 φi , ind TL φi T p

p2

−

   1 ind TNm+1 φj , ind TL φj T + ind TL φi , ind TL φj T p 

= − p + 1 + φi , φj  L +



E  φi cd  N

(c ,d)∈Γ, (c ,d)=(0,0)

m +1

, φi

 N m +1

= φi , φj  L , which yields i = j. The proof is complete.

2

l ( A ) are By this lemma, Clifford theory, and Theorem 2.1 (2), the irreducible characters of G l in G

χi = (μα ◦ det) indGT l ψi (i ∈ Ω). Theorem 3.5. With the notation as above, we have the following. (1) If p | r, then τl (χi ) = 0. (2) If p  r, then

1

χi ( I )

τl (χi ) = − p 2l−1





 



p μα det sk e Tr sk ζ pi1mk1 ζ p(δ+ l −1

m

i 2 )k2 i 3 h3 ζ p 2 −1 ,

k

and the summation is over all elements k ∈ Ω such that k1 ≡ k3 = h 3 .





−α r −1 − r −1 ∈ ( K m ∩ S )sh , −r −1 −α r −1 h1 (mod pm−1 ), k2 ≡ h2 (mod pm−1 ),

where h is a unique element of Ω which satisfies 0  h1 , h2 < pm−1 and

Proof. By the Frobenius reciprocity, we have

1

|G l |

τl (χi ) =

1 p

 (μα ◦ det)φi , e ◦ Tr N

Let us show (μα ◦ det)φi , e ◦ Tr Nm+1 = 0. Let sk = elements k of Ω with p + 1 | k3 . Since



p +1 

Z (G l )( K 1 ∩ S ) = s1  × s2  × s3

,

  − (μα ◦ det)φi , e ◦ Tr L . m +1

 x y  k k y k xk

for k ∈ Ω and let Ω0 be the set of all



pm 

K m+1 ∩ Z (G l )( K 1 ∩ S ) = s1

sk (k ∈ Ω0 ) are the coset representatives of K m+1 in N m+1 . Hence

 pm  × s2 ,

108

T. Maeda / Journal of Algebra 396 (2013) 98–116

  | Nm+1 | (μα ◦ det)φi , e ◦ Tr N m +1            = μα 1 + pm+1a 1 + pm+1d μα det sk λ pm+1 b + pm+1 c φi sk 0a,b,c ,d< pm k∈Ω0



=







· e Tr s e p

m +1



  k

k



Tr

a b c d



yk

xk yk



xk

   μα det s e Tr sk φi sk

k∈Ω0

·

 

    λ pm+1 αa e pm+1 xk a

2  

0a< pm

    λ p m +1 b e p m +1 y k b

2 .

0b< pm

Since p | y k for all k ∈ Ω0 , we have



    λ p m +1 b e p m +1 y k b = 0,

0b< pm

which yields (μα ◦ det)φi , e ◦ Tr Nm+1 = 0. Similarly we have

  | L | (μα ◦ det)φi , e ◦ Tr L        = μα det sk e Tr sk φi sk k∈Ω

·

  0a< pm

= p 2l−2

    λ pm+1 αa e pm+1 xk a

2  

    λ p m +1 b e p m +1 y k b

2

0b< pm





 

  

μα det sk e Tr sk φi sk .

k∈Ω,

α +xk r ≡0 (mod pm ), 1+ y k r ≡0 (mod pm )

Thus, τl (χi ) = 0 if p | r. Assume p  r. Let Ω1 be the set of all elements k of Ω with 0  k1 , k2 < pm−1 . Since K m ∩ S = p m −1

s1

pm−1

 × s2

, sk (k ∈ Ω1 ) are the representatives of K m ∩ S in S. Hence, there exists a  coset  −α r −1 − r −1 ∈ ( K m ∩ S )sh . For k ∈ Ω , α + xk r ≡ 0 (mod pm ) unique element h of Ω1 such that − 1 − 1 −r  −α r  −1 −1 and 1 + y k r ≡ 0 (mod pm ) if and only if −αr−1 − r −1 ∈ ( K m ∩ S )sk , namely k1 ≡ h1 (mod pm−1 ), k2 ≡ h2 (mod pm−1 ), k3 = h3 . Therefore

−r

−α r



  | L | (μα ◦ det)φi , e ◦ Tr L = p 2l−2

k∈Ω, k1 ≡h1 (mod pm−1 ), k2 ≡h2 (mod pm−1 ), k3 =h3

which proves the assertion of the theorem.

2



 



p μα det sk e Tr sk ζ pi1mk1 ζ p(δ+ l −1

m

i 2 )k2 i 3 h3 ζ p 2 −1 ,

T. Maeda / Journal of Algebra 396 (2013) 98–116

109

3.4. X3 In this subsection, we determine the   Gausssum τl (χ ) for every irreducible character α pβ 0 pβ , A0 = , and T = StabG l φ A 0 . Then we have

0  α < pm , 0  β < pm−1 , A =

1

α

 T = Km S =

χ in X3 . Let

1 0

p β b + pm c a + pm d

a b



where S =

a b

pβ b a

    a, b , c , d ∈ Z pl ∩ G l ,     a, b ∈ Z pl ∩ G l

(cf. [1, 3.2.3 and 3.3.3]). Let

 N=

1 + pm a pm c

pn b 1 + pm d

    a, b , c , d ∈ Z pl .

Then N is normal in T and we can extend φ A 0 to characters φi (i = (i 1 , i 2 ), 0  i 1 , i 2 < pn−m ) of N defined by

 φi

1 + pm a pm c

pn b 1 + pm d



    = λ p 2m (i 1 a + i 2 d) λ pn b + pm+1 β c .

Let T 0 = Stab T φi . Then we have

 T0 = N S =

Let



a b

p β b + pn c a + pm d

γ be a generator of Z×pl such that γ p

1 pβ 1 1

m−1



( p −1)

    a, b , c , d ∈ Z pl ∩ G l .

= 1 + pm , and let s1 =

γ



γ , s2 =



1 p2 β p 1



, s3 =

. Then we have

Z (G l )( K 1 ∩ S ) = s1  × s2 ,



p

K m ∩ S = N ∩ S = s1 p

σ σ

and s3 = s1 1 s2 2 for some integers

S = ( N ∩ S )s1 s2 s3 ,

m −1

( p −1 ) 

 p m −1  × s2 ,

σ1 and σ2 . Hence the coset representatives of N in T 0 are given by k

k

k

sk = s11 s22 s33

(k ∈ Ω),  where Ω = k = (k1 , k2 , k3 )  0  k1 < pm−1 ( p − 1), 0  k2 < pm−1 , 0  k3 < p . 

Also we define δ ∈ Z pl by

p

m +1

δ=2

m −1

1t  p t: odd

Then we have



 ,

p m −1 t



pt ( p β)(t +1)/2 .

110

T. Maeda / Journal of Algebra 396 (2013) 98–116

 p m −1 ( p −1 )       i2 = φi 1 + pm I = λ p 2m (i 1 + i 2 ) = ζ pi 1n+ φi s 1 −m ,    m −1   p m −1  p = φi pt ( p β)t /2 I + φi s 2 t

0t  pm−1 , t: even



p m −1 t

1t  pm−1 , t: odd

  = λ pm+1 δ = ζ pδn−1 .



pt ( p β)(t −1)/2 A 0



Since the number of extensions of φi to T 0 is | T 0 : N | = |Ω|, for every j ∈ Ω , there exists a unique extension ψij of φi such that i +i + pn−m j 1

ψij (s1 ) = ζ p1n−12( p −1)

δ+ pn−1 j 2

ψij (s2 ) = ζ pl−2

,

(i +i + pn−m j 1 )σ1 (δ+ pn−1 j 2 )σ2

ψij (s3 ) = ζ pn1( p −21)

ζ p l −1

,

j

ζp3 .

l ( A ) are Therefore, by Clifford theory and Theorem 2.1 (3), the irreducible characters of G l in G

χij = (μα ◦ det) indGT 0l ψij





0  i 1 , i 2 < pn−m , j ∈ Ω .

Theorem 3.6. With the notation as above, we have the following. (1) If p | r or p | α , then τl (χij ) = 0. (2) If p  r and p  α , then

1

χij ( I )



 



τl (χij ) = pl+2m μα det sh e Tr sh ζ p(in1(+p−i21+)p  ·  ·



n−m

j 1 )( ph1 +σ1 h3 ) (δ+ pn−1 j 2 )( ph2 +σ2 h3 ) j 3 h3 ζ p l −1 ζp





 

 





 

 



μα 1 + pm t 1 λ p 2m i 1t 1 e pm xht 1

0t 1 < pn−m



μα 1 + pm t 2 λ p 2m i 2t 2 e pm xht 2

,

0t 2 < pn−m

where sk =



xk p β y k y k xk



for k ∈ Ω and h is a unique element of Ω which satisfies

( K m ∩ S )sh . Proof. We define a subgroup M of G l by

 M=

1 + pn a pm c

pn b 1 + pn d

    a, b , c , d ∈ Z pl ,

and pick the coset representatives of M in N to be

 Xt = Then we have

1 + pm t 1

1 + p t2 m





t = (t 1 , t 2 ), 0  t 1 , t 2 < pn−m .



−α r −1 − p β r −1 −r −1 −α r −1



∈

T. Maeda / Journal of Algebra 396 (2013) 98–116



1

χij ( I )

111



τl (χij ) = | T 0 | (μα ◦ det)ψij , e ◦ Tr T 0 



=





μα 1 + pn a 1 + pn d



0a,b,d< pm , 0t 1 ,t 2 < pn−m , 0c < pn k∈Ω

     · μα 1 + pm t 1 1 + pm t 2 μα det sk 

n

·λ p b+ p

m +1



=

 n     2m   k p a β c λ p (i 1 t 1 + i 2 t 2 ) ψij s e Tr I + m

p c









pn b pn d

 



Xt s

k



μα 1 + pm t 1 1 + pm t 2 μα det sk λ p 2m (i 1t 1 + i 2t 2 )

0t 1 ,t 2 < pn−m , k∈Ω

    · e Tr X t sk ψij sk ·

    λ pn αa e pn xk a

0a< pm

 

    λ p m +1 β c e p m +1 y k β c

0c < pn

= pl+2m

 





0t 1 ,t 2 < pn−m

2  

    λ pn b e pn y k b



0b< pm







 



μα 1 + pm t 1 1 + pm t 2 λ p 2m (i 1t 1 + i 2t 2 )

k∈Ω,

α +xk r ≡0 (mod pm ), 1+ y k r ≡0 (mod pm )

        · μα det sk e Tr sk e pm xk (t 1 + t 2 ) ψij sk . Thus,

τl (χij ) = 0 if p | r or p | α .   −1 −1 α . Then there exists a unique element h ∈ Ω such that −αr−1 − pβ r−1 ∈

Assume p  r and p 

−r

−α r

m m ( K m ∩ S )sh . For   every k ∈ Ω , α + xk r ≡ 0 (mod p ) and 1 + y k r ≡ 0 (mod p ) if and only if −α r −1 − p β r −1 ∈ ( K m ∩ S )sk , namely k = h. Therefore −1 −1

−r

1

χij ( I )

−α r



 



 

τl (χij ) = pl+2m μα det sh e Tr sh ψij sh  ·







 

 





μα 1 + pm t 1 1 + pm t 2 λ p 2m (i 1t 1 + i 2t 2 ) e pm xh (t 1 + t 2 ) ,

0t 1 ,t 2 < pn−m

which proves the assertion of the theorem.

2

3.5. X4 In this subsection, we consider the Gauss sum τl (χ ) for every irreducible character χ in X4 . Let ν0 be an injective multiplicative character such that ν0 (1 + pl−1 ) = ζ p , and let ν = ν0 ◦ det. By the

l . The derived subgroup of G l is G l−1 as a subset of G natural homomorphism G l → G l−1 , we regard  × SL2 (Z pl ), and G l /SL2 (Z pl ) is isomorphic to Z l . Thus the group Hom(G l , C× ) is generated by ν . For p

θ, θ  ∈  G l−1 and i , j ∈ Z, ν i θ = ν j θ  if and only if ν i − j ∈ Hom(G l−1 , C× ) = ν p , namely i ≡ j (mod p ). Hence the irreducible characters of G l in X4 are

112

T. Maeda / Journal of Algebra 396 (2013) 98–116

ν i θ (0  i < p , θ ∈  G l −1 ) (cf. [1, the proof of Theorem 3.1]). Let us evaluate τl (ν i θ) for 0  i < p and θ ∈  G l−1 .









τl ν i θ =



νiθ

   a b c

0a,b,c ,d< pl−1 , 0x, y , z, w < p p ad−bc

    a b x · e Tr I + p l −1 c

d



=



νiθ

 a b c

0a,b,c ,d< pl−1 , p ad−bc

·

 

  ζ pix e pl−1ax

0x< p

·

 

  e Tr

d

 

a b c d



e p

l −1

cy

e p

0 z < p



 l −1

0a,b,c ,d< p , p ad−bc , br ≡cr ≡0 (mod p ), i +ar ≡i +dr ≡0 (mod p )



νiθ

x z

y w

i





  ζ pi w e pl−1 dw

0 w < p

  

0 y < p

= p4

d





y w

z



ν I + p l −1

l −1

bz



    a b a b c

d

e Tr

c

d

.

Thus, τl (ν i θ) = 0 if either (i) 1  i < p and p | r or (ii) i = 0 and p  r. If i = 0 and p | r, then we have

τl (θ) = p 4 τl−1 (θ, e), where e is regarded as an additive character of Z pl−1 on the right hand side. If 1  i < p and p  r, then



i



τl ν θ = p



4



i

νθ

0a,b,c ,d< pl−1 , b≡c ≡0 (mod p ), a≡d≡−ir −1 (mod p )

  a b c

d

e (a + d).

It seems likely that we cannot simplify this sum any further. Appendix A Let i, j, k be three integers such that 1  i  m, 1  j  i and 0  k  i, λ an injective additive character of Z p i , and β, b ∈ Z×i . Barrington Leigh, Cliff and Wen considered the following sum P to p

determine the values of characters in X3 in Section 6.1 of [1, p. 1314]:

P=

 0c ,d< p i , p d

   λ p j β d + d −1 p k b − c 2 .

T. Maeda / Journal of Algebra 396 (2013) 98–116

113

However their evaluation of this sum seems to be mistaken. The purpose of this section is to give the correct value of this sum. Since λ is an additive character, we have

P=

   −1 2   λ p j β d + pk bd−1 λ −d c .



0c < p i

0d< p i , p d

The second sum



0c < p i

λ(−d−1 c 2 ) is the quadratic Gauss sum. By the standard evaluation of this

sum (cf. [2, Theorem 1.5.2]), we have



  λ −d−1 c 2 =



−rd

i 

p

0c < p i

−1

δi /2

p

p i /2 ,

where

  1, if i is odd, δi = 1 − (−1)i /2 = 0, if i is even, and λ(1) = ζ r i . p

Hence

 i  P=

Put P 1 = cases:

r

−1

p

p



d i j 0d< p i , p d ( p ) λ( p β d

i +δi /2

  d i   λ p j β d + pk bd−1 .

p i /2

0d< p i , p d

p

+ pk bd−1 ). In order to evaluate P 1 , we consider the following five

(i) j = k = i. In this case, we have

 i



P1 =

d

0d< p i , p d

=

p

p i −1 ( p − 1),

if i is even,

0,

if i is odd.

(ii) j < k  i. Let λ1 be an additive character of Z p i− j defined by λ1 (1) = λ( p j ). Then

 i



P1 = p j

d

0d< p i − j , p d

= pj

 i β p

p

  λ1 β d + pk− j bd−1  i



d

0d< p i − j , p d

p

  λ1 d + pk− j β bd−1 .

The map d → d + pk− j β bd−1 is a bijection on Z×i− j and preserves squares. Hence p

114

T. Maeda / Journal of Algebra 396 (2013) 98–116

P1 = p j

 i β p

 i



d

λ1 (d)

p

0d< p i − j , p d

⎧ i −1 ⎪ if j = i − 1 and i is even, ⎨ −p , β r −1 1/2 i −1/2 = ( p )( p ) p , if j = i − 1 and i is odd, ⎪ ⎩ 0, otherwise. (iii) k < j  i. As in case (ii), we have

⎧ i −1 ⎪ if k = i − 1 and i is even, ⎨ −p , br P 1 = ( p )( −p1 )1/2 p i −1/2 , if k = i − 1 and i is odd, ⎪ ⎩ 0, otherwise. (iv) j = k < i and i is even. Let λ1 be an additive character defined by λ1 (1) = λ( p j ). Then



P1 = p j

  λ1 β d + bd−1 = p j K ,

0d< p i − j , p d



+ β bd−1 ) is the Kloosterman sum. While we cannot give an explicit determination of K in the case i − j = 1, there is an evaluation of K in the case i − j > 1 as follows

where K =

0d< p i − j , p d λ1 (d

(cf. [12, §2]): If β b is nonsquared, K = 0. If β b is squared, then

 K=

ur

j

p

−1

δ j /2 p

p

(i − j )/2



 λ1 (2u ) +

−1

j

p

λ1 (−2u ) ,

where u 2 = β b. (v) j = k < i and i is odd. Let h = i − j and λ1 be an additive character of Z ph defined by λ1 (1) = λ( p j ). Then





P1 = p j

d

0d< ph , p d

= pj

Put K  =



 β p

p

  λ1 β d + bd−1    d λ1 d + β bd−1 .

 0d< ph , p d

p

+ β bd−1 ). Since K  = ( βpb ) K  , K  = 0 if β b is nonsquared. Assume β b is squared, say u 2 = β b. Let f be a C-valued function on Z ph defined by d 0d< ph , p d ( p )λ1 (d

f (a) =

 0d< ph , p d

   d λ1 d + a2 d−1 . p

T. Maeda / Journal of Algebra 396 (2013) 98–116

115

For c ∈ Z, we have



p

0a< ph 0d< ph , p d





=

d

0d< ph , p d



=



=

p

 d

 j −1 

p

r

−1

p

p



λ1 (d)

p

   λ1 d−1 a2 − cda

0a< ph



0d< ph , p d



λ1 (d)

d

0d< ph , p d

=

   d λ1 d + a2 d−1 λ1 (−ca)







p h f , λc1 Z = ph

  2  2  λ1 d−1 a − 2−1 cd − 2−1 cd

0a< ph

      λ1 d − 1 a 2 λ1 1 − 4 − 1 c 2 d 0a< ph

δ j−1 /2



p h/2

0d< ph , p d

 j d

p

   λ1 4 − c 2 d .

Put

 j



g (c ) =

d

0d< ph , p d

p

   λ1 4 − c 2 d .

Then

⎧ h −1 p ( p − 1), ⎪ ⎪ ⎪ ⎨ − p h −1 , g (c ) = ⎪ ( c1 r )( −1 )1/2 ph−1/2 , ⎪ p ⎪ ⎩ p 0,

if p h | 4 − c 2 and j is even, if 4 − c 2 = p h−1 c 1 for some p  c 1 and j is even, if 4 − c 2 = p h−1 c 1 for some p  c 1 and j is odd, otherwise.

Hence, if j is even, we have

 

K  = f (u ) =

  =

r

−1

p

p

  =

0c < ph

r

−1

p

p

1/2

1/2



f , λc1 Z λ1 (cu ) ph







p −h/2 p h−1 ( p − 1) λ1 (2u ) + λ1 (−2u ) − p h−1



λ1 (cu )

0c < ph , ph−1 4−c 2





p (i − j )/2 λ1 (2u ) + λ1 (−2u ) ,

and if j is odd, we have

K  = f (u ) =

  =

  0c < ph

r

−1

p

p



f , λc1 Z λ1 (cu ) ph

1/2 p

(h−1)/2

   −e       e h −1 h −1 λ1 u 2 + p e + λ1 u −2 + p e 1e < p

p

p

116

T. Maeda / Journal of Algebra 396 (2013) 98–116

  =

r

−1

p

p

 =

u

p

1/2 

p (h−1)/2



p (i − j )/2 λ1 (2u ) +

−u



p



−1 p



λ1 (2u ) +

     u e λ1 (−2u ) λ1 ph−1 e

λ1 (−2u ) .

p

1e < p

p

This completes the evaluation of P 1 in all cases. References [1] R. Barrington Leigh, G. Cliff, Q. Wen, Character values for GL(2, Z/ pl Z), J. Algebra 323 (2010) 1288–1320. [2] B.C. Berndt, R.J. Evans, K.S. Williams, Gauss and Jacobi Sums, Canad. Math. Soc. Ser. Monogr. Adv. Texts, vol. 21, John and Wiley & Sons, Inc., 1998. [3] T. Funakura, A generalization of the Chowla–Mordell theorem on Gaussian sums, Bull. Lond. Math. Soc. 24 (1992) 424–430. [4] Y. Gomi, T. Maeda, K. Shinoda, Gauss sums on finite groups, Tokyo J. Math. 35 (2012) 165–179. [5] I. Martin Isaacs, Character Theory of Finite Groups, Dover, New York, 1976. [6] D.S. Kim, I. Lee, Gauss sums for O + (2n, q), Acta Arith. 78 (1996) 75–89. [7] T. Kondo, On Gaussian sums attached to the general linear groups over finite fields, J. Math. Soc. Japan 15 (1963) 244–255. [8] A. Nobs, Die irreduziblen Darstellungen von GL2 (Z p ) insbesondere GL2 (Z2 ), Math. Ann. 229 (1974) 113–133. [9] R. Odoni, On Gauss sums (mod pn ), n  2, Bull. Lond. Math. Soc. 5 (1973) 325–327. [10] N. Saito, K. Shinoda, Character sums attached to finite reductive groups, Tokyo J. Math. 23 (2000) 373–385. [11] N. Saito, K. Shinoda, Some character sums and Gauss sums over G 2 (q), Tokyo J. Math. 24 (2001) 277–289. [12] H. Salié, Über die Kloostermanschen Summen S (u , v ; q), Math. Z. 34 (1932) 91–109.