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# 1. Introduction. Let λ be a nontrivial additive character of the finite field F

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(1)

q

w∈Sp(2n,q)

r

q

w∈GSp(2n,q)

r

q

γ∈Fq

r

[155]

(2)

q

w∈Sp(2n,q)

r

γ∈Fq

r

n2−1

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l

n−2b+2−2l

r

l−1

ν=1

jν−2ν

n2−1

n

j=1

2j

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l−1

n−2b+2−2l

l−1

ν=1

jν−2ν

1

l−1

l−1

1

m

r

m

r

w∈GSp(2n,q)

r

γ∈Fq

r

(3)

n−2b+2−2l

r

α∈Fq

n−2b+2−2l

r

q

Sp(2n,q)

n2−1

n j=1

2j

n2−1

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l

−1

n−2b+2−2l

l−1

ν=1

jν−2ν

q

GSp(2n,q)

n2−1

n j=1

2j

−1

w∈GSp(2n,q)

n2−1

n j=1

2j

−1

w∈GSp(2n,q)

q

q

d

q

a

(4)

q

q

a

p

pd−1

t

q

n

n−1

j=0

n

j

n2

n

j=1

j

q

t

n

n

n2

n j=1

2j

t

−1

n

n

t

n b=0

b

b

b

b

b

−1b

b

b

n−b

b

n−b

b

b+12

q

(5)

b

q

b

(b/2)(b/2−1)

b/2 i=1

2i−1

q

t

×q

n2

n j=1

2j

t

−1

n

n

×q

t

n b=0

b

b

q

γ∈Fq

r

e−1

j=1

j

q

j

j

γ∈F×q

j

q

q

γ∈F×q

−1

q

b−1

j=0

n−j

b−j

(6)

n

n−1

n b=0

q

b

2b

b

n

q

q

GL(t,q)

r

w∈GL(t,q)

−1

r

−1

t−2

×q

q

GL(t,q)

r

w∈GL(t,q)

−1

r

×q

GL(t,q)

r

−1

r

−1

r

(7)

w∈P (t−1,1;q)

−1

r

w∈P (t−1,1;q)

−1

r

w∈P (t−1,1;q)

−1

r

A,B,d

−1

−1

r

t−1

d∈F×q

GL(t−1,q)

r

−1

11

12

1

21

22

2

11

12

21

22

−1

11

12

21

22

11

11

22

22

1

22

22

−1

11

1

−1

12

2

r

11

12

21

22

1

2

12

12

12

12

12

## (3.8) X

A with A126=0 B1,d

B2

1

22

22

−1

11

1

−1

12

2

r

(8)

12

1

12

1

t−2

2

t

1

t−2

k

×q

q

γ∈Fq

r

γ∈Fq

r

all βi with i6=k

βk

−1

k

k

r

t−3

γ∈Fq

r

12

t−1

t−2

t−2

t−2

γ∈Fq

r

12

−1

−111

1

22

−122

r

A11 0

A21 A22

B1

B2

−1

−111

−1

−111

−1

−111

−1

−111

−1

−111

2t−4

t−2

γ∈Fq

r

−1

−111

2t−3

GL(t−2,q)

r

t−1

GL(t,q)

r

×q

q

(9)

GL(t,q)

r

t2

t−1

t−1

j=1

j

γ∈Fq

r

2t−2

t−1

GL(t−2,q)

r

t−1

γ∈F×q

GL(t−1,q)

r

−1

GL(0,q)

r

r

q

m

r

m

r

γ1,...,γm∈F×q

1

−11

m

m−1

r

0

r

r

m

m

m

r

m

r

×q

q

GL(t,q)

r

GL(t,q)

r

(t+1)(t−2)/2

t

j=1

j

[(t+2)/2]

l=1

l−1

t+2−2l

l−1

ν=1

jν−2ν

γ∈Fq

r

(t+1)(t−2)/2

[(t+2)/2]

l=1

l

t+2−2l

r

l−1

ν=1

jν−2ν

(10)

1

l−1

l−1

l−2

1

w∈GSp(2n,q)

r

q

n b=0

b

w∈Q

b

r

b

b

b

w∈Q

b

r

w∈Q

b

r

w∈Q

b

r

n

n

t

−1

n

n

11

12

21

22

t

−1

11

12

21

22

11

12

t

12

22

t

11

11

t

22

22

11

12

21

22

t

−1

w∈Q

b

r

11

11

12t

12

22

22

r

11

12

22

11

22

12

12

(11)

12

11

A126=0

## . . . + X

A12=0 A11not alternating

## . . . + X

A12=0 A11alternating

n−b+12

## × X

A with A126=0 B11

B12

11

11

12t

12

22

22

r

b(n−b)−1

γ∈Fq

r

(n−1)(n+2)/2

n

b

n−b

b(n−b)

γ∈Fq

r

12

A21,B12,B22

A11,A22,B11

11

11

22

−122

r

n−b+12

+2b(n−b)

A11,A22,B11

11

11

22

−122

r

11

ij

11

ij

11

11

1≤i≤j≤b

ij

ij

ij

ii

ij

ji

11

ij

11

## (4.11) X

A11not alternating A22

B11

11

11

22

−122

r

11

st

b+12

−1

γ∈Fq

r

(12)

(n−1)(n+2)/2

b(n−b)

n−b

b

b

γ∈Fq

r

b

q

11

## (4.14) X

A11alternating B11

α

A22

22

−122

r

b

b+12

α∈F×q

GL(n−b,q)

r

GL(n−b,q)

r

(n−1)(n+2)/2

b(n−b)+1

b

α∈F×q

GL(n−b,q)

r

(n−1)(n+2)/2

n

b(n−b)

n−b

b

γ∈Fq

r

b(n−b)+1

b

α∈F×q

GL(n−b,q)

r

w∈Q

r

12

(n−1)(n+2)/2

n

n

γ∈Fq

r

n

0

GL(0,q)

r

0

GL(0,q)

r

0

r

(13)

w∈Q

−1

r

n+12

α∈F×q

GL(n,q)

r

0

0

(n−1)(n+2)/2

n

b=0

b

n

b(n−b)

n−b

b

γ∈Fq

r

n+12

n

b=0

b

b(n−b)

b

α∈F×q

GL(n−b,q)

r

GL(t,q)

r

q

w∈GSp(2n,q)

r

n2−1

n

j=1

2j

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l−1

n−2b+2−2l

l−1

ν=1

jν−2ν

e−1

j=1

j

n2−1

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l

α∈F×q

n−2b+2−2l

r

l−1

ν=1

jν−2ν

1

l−1

l−1

1

q

m

r

(14)

α

1

n 0 0 α1n

α∈F×q

α

w∈Sp(2n,q)

r

α∈F×q

n−2b+2−2l

r

n−2b+2−2l

r

q

w∈Sp(2n,q)

r

n2−1

n

j=1

2j

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l−1

n−2b+2−2l

l−1

ν=1

jν−2ν

e−1

j=1

j

n2−1

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l

n−2b+2−2l

r

l−1

ν=1

jν−2ν

m

r

q

q

G(q)

q

G(q)

α∈F×q

w∈G(q)

(15)

q

q

α∈F×q

m

m

1

m

×q

m

1

−11

m

−1m

w∈Sp(2n,q)

×q

q

Sp(2n,q)

n2−1

n j=1

2j

n2−1

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l

−1

n−2b+2−2l

l−1

ν=1

jν−2ν

1

l−1

l−1

1

×q

GSp(2n,q)

−1

−1

w∈GSp(2n,q)

α∈F×q

(16)

q

GSp(2n,q)

n2−1

n j=1

2j

−1

w∈GSp(2n,q)

n2−1

n j=1

2j

−1

w∈GSp(2n,q)

w∈GSp(2n,q)

n2−1

[n/2]

b=0

b(b+1)

q

b j=1

2j−1

[(n−2b+2)/2]

l=1

l

α∈F×q

n−2b+2−2l

l−1

ν=1

jν−2ν

1

l−1

l−1

1

α∈F×q

t

q

α∈F×q

t

α∈F×q

1

t

α∈F×q

1

t

2

t−1

t−1

t−1

t

1

t

×q

1

t

1−1

−1t

0

1

## However, this assumption is not necessary and it holds true for any q.

(17)

2

2

+

### and in revised form on 23.10.1998 (3334)

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