On the group $SL(2,{Bbb R})ltimes {Bbb R}^{(m,2)}$

ON THE GROUP SL(2, R) n R ^(m,2)

Jae-Hyun Yang

Abstract. In this paper, we nd the unitary dual of the group SL(2, R)nR

and study automorphic forms related to the group SL(2, R) n R

.

1. Introduction

For any positive integer m ∈ Z ⁺ , we let

SL 2,m ( R) := SL(2, R) n R ^(m,2)

be the semidirect product of the special linear group SL(2, R) and the commutative additive group R ^(m,2) equipped with the following multi- plication law

(1.1) (g, a) ∗ (h, b) = (gh, a ^t h ⁻¹ + b ),

where g, h ∈ SL(2, R) and a, b ∈ R ^(m,2) . Here R ^(m,2) denotes the set of all m × 2 real matrices.

In this paper, we investigate some properties of the group SL 2,m ( R) and nd its unitary dual explicitly. The reason why we study the group SL 2,m ( R) is that this group plays an important role in the study of cer- tain automorphic forms. This paper is organized as follows. In Section 2, we present the basic ingredients of the group SL 2,m ( R). In Section 3, we describe the roots of SL 2,m ( R) explicitly. In Section 4, we compute the Lie derivatives explicitly for functions on SL 2,1 ( R). Similarly we can compute the Lie derivatives of SL 2,m (R) ( m ≥ 2 ) explicitly. We

Received April 4, 2002.

2000 Mathematics Subject Classication: Primary 22E47, 11F55, 32N10.

Key words and phrases: unitary representations, invariant dierential operators, automorphic forms.

This work was supported by INHA UNIVERSITY Research Grant (INHA-22067).

leave the detail to the reader. In Section 5, we discuss the actions of SL 2,m (R) on H × C ^m and SP 2 × R ^(m,2) , where H denotes the Poincare upper-half plane and S P 2 is the symmetric space consisting of all the 2 × 2 positive symmetric real matrices Y with det Y = 1. In Section 6, we make some remarks on the notion of Maass-Jacobi forms associated to the group SL 2,1 ( R). In Section 7, we compute the unitary dual of SL 2,m ( R) using the Mackey’s method.

Notations. We denote by Z, R and C the ring of integers, the eld of real numbers and the eld of complex numbers respectively. Z ⁺ denotes the set of all positive integers. The symbol “:=” means that the expression on the right is the denition of that on the left. F ^(k,l) denotes the set of all k × l matrices with entries in a commutative ring F . For a square matrix A, σ(A) denotes the trace of A. For any M ∈ F ^(k,l) , ^t M denotes the transpose of M . For A ∈ F ^(k,l) and B ∈ F ^(k,k) , we set B[A] = ^t ABA. H denotes the Poincare upper-half plane. For a eld F , we denote by F ^× the multiplicative group consisting of nonzero elements.

I n denotes the identity matrix of degree n.

2. Basic ingredients of SL 2,m ( R)

From now on, we write G = SL 2,m ( R) for brevity. We let sl(2, R) = {

X ∈ R ^(2,2) | σ(X) = 0 }

be the Lie algebra of SL(2, R). Then it is easy to see that the Lie algebra g of G is given by

(2.1) g =

{

(X, Z) | X ∈ sl(2, R), Z ∈ R ^(m,2) } equipped with the following Lie bracket

(2.2) [(X 1 , Z 1 ), (X 2 , Z 2 )] = ([X 1 , X 2 ] 0 , Z 2 t

X 1 − Z 1 t

X 2 ),

where [X 1 , X 2 ] 0 := X 1 X 2 − X 2 X 1 denotes the usual matrix bracket and (X 1 , Z 1 ), (X 2 , Z 2 ) ∈ g. The adjoint representation Ad of G is given by (2.3) Ad ((g, a))(X, Z) = (gXg ⁻¹ , (Z − a ^t X) ^t g),

where (g, a) ∈ SL 2,m ( R) and (X, Z) ∈ g. And the adjoint representation ad of g on End (g) is given by

(2.4) ad ((X, Z))((X 1 , Z 1 )) = [(X, Z), (X 1 , Z 1 )].

We easily see that the Killing form B of g is given by (2.5) B((X 1 , Z 1 ), (X 2 , Z 2 )) = (m + 4) σ(X 1 X 2 ).

Therefore the Killing form B is highly degenerate.

Let

K = { (k, 0) ∈ G | k ∈ SO(2) }

be the compact subgroup of G. Then the Lie algebra k of K is k =

{

(X, 0) ∈ g  X =

( ( 0 x

−x 0 )

, 0 )

, x ∈ R }

. We let p be the subspace of g dened by

p = {

(X, Z) ∈ g | X = ^t X ∈ R ^(2,2) , σ(X) = 0, Z ∈ R ^(m,2) } . Then we have the following relation

(2.6) [k, k] ⊂ k and [k, p] ⊂ p.

In addition, we have

(2.7) g = k ⊕ p ( direct sum ).

We note that the restriction of the Killing form B to k is negative denite and the restriction of B to the abelian subalgebra r = { (0, Z) ∈ g } is identically zero. Since r is the radical of B, B is degenerate (see (2.5)).

An Iwasawa decomposition of the group SL 2,m ( R) is given by

(2.7) G = N AK,

where

N = { ( (

1 x 0 1

) , a

)

∈ G   x ∈ R, a ∈ R ^(m,2) }

and

A = { ( (

a 0

0 a ⁻¹ )

, 0 )

∈ G   a > 0 }

.

An Iwasawa decomposition of the Lie algebra g of G is given by

(2.8) g = n + a + k,

where

n = { ( (

0 x 0 0

) , Z

)

∈ g   x ∈ R, Z ∈ R ^(m,2) }

and

a = { ( (

x 0

0 −x )

, 0 )

∈ g   x ∈ R }

.

In fact, a is the Lie algebra of A and n is the Lie algebra of N .

3. The roots of the Lie algebra of G

In this section, we will determine the roots of g explicitly.

We let F kl ( 1 ≤ k ≤ m, l = 1, 2 ) be the m × 2 matrix with 1 in the (k, l)-entry and zero elsewhere. We put

E 0 =

( ( 1 0 0 −1

)

, (0, 0) )

, E 1 =

( ( 0 1 0 0

)

, (0, 0) )

, E 2 =

( ( 0 0 1 0

)

, (0, 0) )

and

F _kl ⁰ = ( 0, F kl ), 1 ≤ k ≤ m, l = 1, 2.

Then E 0 , E 1 , E 2 , F _kl ⁰ ( 1 ≤ k ≤ m, l = 1, 2 ) form a basis of g. Clearly a = R E 0 and n = R E 1 +

∑ m k=1

( R F k1 ⁰ + R F k2 ⁰ ).

By an easy computation, we get

[ E 0 , E 1 ] = 2 E 1 , [ E 0 , E 2 ] = − 2 E 2 , [ E 0 , F _k1 ⁰ ] = F _k1 ⁰ , [ E 0 , F _k2 ⁰ ] = − F k2 ⁰ .

Thus the roots of g relative to a are given by ± e, ± 2 e, where e is the linear functional e : a −→ C dened by e(E 0 ) := 1. The set ∑ +

= { e, 2 e } is the set of positive roots of g relative to a. We recall that for a root α, the root space g α is dened by

g α = {X ∈ g | [H, X] = α(H)X for all H ∈ a }.

Clearly we have g e =

∑ m k=1

R F k1 ⁰ , g _−e =

∑ m k=1

R F k2 ⁰ , g 2e = R E 1 , g _−2e = R E 2

and

g = g _−2e ⊕ g −e ⊕ a ⊕ g e ⊕ g 2e . So we obtain

dim g e = dim g _−e = m and dim g 2e = dim g _−2e = 1.

If we dene

ρ = 1 2

∑

α ∈ P

( dim g α ) · α,

we get ρ = ^m+2 ₂ e.

4. The Lie derivatives on G

In this section, we compute the Lie derivatives for functions on G explicitly. For a simplicity, we consider only the case m = 1. We leave the case m ≥ 2 to the reader.

We put W 1 =

( ( 0 1 0 0

) , (0, 0)

)

, W 2 =

( ( 0 0 1 0

) , (0, 0)

) ,

W 3 =

( ( 1 0 0 −1

) , (0, 0)

)

and

W 4 =

( ( 0 0 0 0

) , (1, 0)

)

, W 5 =

( ( 0 0 0 0

) , (0, 1)

) . Clearly W 1 , . . . , W 5 form a basis of g.

We dene the dierential operators L k , R k ( 1 ≤ k ≤ 5 ) on G by (4.1) L k f (˜ g) = d

dt    t=0

f (˜ g ∗ exp tW k )

and

(4.2) R k f (˜ g) = d dt    t=0

f (exp tW k ∗ ˜g), where f ∈ C ^∞ (G) and ˜ g ∈ G.

We also put H =

( ( 0 1

−1 0 )

, (0, 0) )

and V =

( ( 0 1 1 0

) , (0, 0)

) . We dene the dierential operators L H , L V , R H and R V by

L H f (˜ g) = d dt

   t=0

f (˜ g ∗ exp tH),

L V f (˜ g) = d dt

   t=0

f (˜ g ∗ exp tV ),

R H f (˜ g) = d dt

   t=0

f (exp tH ∗ ˜g)

and

R V f (˜ g) = d dt    t=0

f (exp tV ∗ ˜g).

By an easy calculation, we get

exp tW 1 =

( ( 1 t 0 1

)

, (0, 0) )

, exp tW 2 =

( ( 1 0 t 1

)

, ( 0, 0 ) )

exp tW 3 =

( ( e ^t 0 0 e ^−t

)

, ( 0, 0 ) )

, exp tW 4 =

( ( 0 0 0 0

)

, ( t, 0 ) )

and

exp tW 5 =

( ( 0 0 0 0

)

, ( 0, t ) )

. Now we use the following coordinates (g, α) in G given by

(4.3) g =

( 1 x 0 1

) ( y ^1/2 0 0 y ^−1/2

) ( cos θ sin θ

−sin θ cos θ )

and

(4.4) α = ( α 1 , α 2 ),

where x, α 1 , α 2 ∈ R, y > 0 and 0 ≤ θ < 2π. By an easy computation, we have

L 1 = y cos 2θ ∂

∂x + y sin 2θ ∂

∂y + sin ² θ ∂

∂θ − α 2

∂

∂α 1

, L 2 = y cos 2θ ∂

∂x + y sin 2θ ∂

∂y − cos ² θ ∂

∂θ − α 1

∂

∂α 2

, L 3 = − 2y sin 2θ ∂

∂x + 2 y cos 2θ ∂

∂y + sin 2θ ∂

∂θ − α 1

∂

∂α 1

+ α 2

∂

∂α 2

, L 4 = ∂

∂α 1

, L 5 = ∂

∂α 2

,

R 1 = ∂

∂x ,

R 2 = ( y ² − x ² ) ∂

∂x − 2 xy ∂

∂y − y ∂

∂θ , R 3 = 2 x ∂

∂x + 2 y ∂

∂y , R 4 = y ^−1/2 cos θ ∂

∂α 1

+ y ^−1/2 sin θ ∂

∂α 2

, R 5 = − y ^−1/2 ( x cos θ + y sin θ ) ∂

∂α 1

+ y ^−1/2 ( y cos θ − x sin θ ) ∂

∂α 2

. Since

H = W 1 − W 2 and V = W 1 + W 2 , we have

L H = L 1 − L 2 = ∂

∂θ − α 2

∂

∂α 1

+ α 1

∂

∂α 2

, L V = L 1 + L 2

= 2 y cos 2θ ∂

∂x + 2 y sin 2θ ∂

∂y − cos 2θ ∂

∂θ − α 2

∂

∂α 1 − α 1

∂

∂α 2

, R H = R 1 − R 2

= ( 1 − y ² + x ² ) ∂

∂x + 2 xy ∂

∂y + y ∂

∂θ and

R V = R 1 + R 2

= ( 1 + y ² − x ² ) ∂

∂x − 2 xy ∂

∂y − y ∂

∂θ .

For instance, we present a calculation for L 3 and R 5 . For a simplic- ity, we denote the form (4.3) by [x, y, θ]. Let (g, α) be an element of SL 2,1 ( R) with g =

( a b c d

)

= [ x, y, θ ] and α = (α 1 , α 2 ) ∈ R ^(1,2) . Then by an easy computation, we have

(4.5) d − i c = y ^−1/2 e ^iθ and d + i c = y ^−1/2 e ^−iθ . So we get

| d + i c | = y ^−1/2 and e ^iθ = d − i c

| d + i c | .

We put

( g(t), α(t) ) = [ x _∗ , (t), y _∗ (t), θ _∗ (t) ] = (g, α) ∗ exp tW 3 , t ∈ R and

τ _∗ (t) = g(t) < i > = x _∗ (t) + i y _∗ (t), t ∈ R.

Then

g(t) =

( a e ^t b e ^−t c e ^t d e ^−t

)

and α(t) = ( α 1 e ^−t , α 2 e ^t ).

Thus

τ _∗ (t) = i a e ^t + b e ^−t

i c e ^t + d e ^−t = i a e ^2t + b

i c e ^2t + d = x _∗ (t) + i y _∗ (t).

Dierentiating τ _∗ (t) with respect to t and evaluating at t = 0, by (4.5), we have

τ _∗ ^′ (0) = 2 i y e ^2iθ . Thus

x ^′ _∗ (0) = − 2 y sin 2θ and y _∗ ^′ (0) = 2 y cos 2θ.

And since

i θ _∗ (t) = log d e ^−t − i c e ^t ( d ² e ^−2t + c ² e ^2t ) ^1/2 , we obtain

θ ^′ _∗ (0) = 2 sin θ · cos θ = sin 2θ.

We set

α(t) = ( α 1, ∗ (t), α 2, ∗ (t) ), t ∈ R.

Since

α 1, ∗ (t) = α 1 e ^−t and α 2, ∗ (t) = α 2 e ^t , we get

α ^′ _{1, ∗} (0) = − α 1 and α ^′ _{2, ∗} (0) = α 2 .

For a smooth function f on SL 2,1 ( R), using the chain rule, we get the desired formula for L 3 .

Since

exp tW 5 = ( E 2 , (0, t) ),

we have

exp tW 5 ∗ (g, α) = ( g, (0, t) ^t g ⁻¹ + α ).

We put

( g(t), α(t) ) = exp tW 5 ∗ (g, α) and

g(t) = [ x _∗ (t), y _∗ (t), θ _∗ (t) ], α(t) = ( α 1, ∗ (t), α 2, ∗ (t) ).

Then we have

g(t) = g, α 1, ∗ (t) = −b t + α 1 and α 2, ∗ (t) = a t + α 2 . From (4.3), we get

a = y ^1/2 cos θ − x y ^−1/2 sin θ and b = y ^1/2 sin θ + x y ^−1/2 cos θ.

Finally we obtain the desired explicit formula for R 5 . We leave the other formulas to the reader.

5. The actions of G on H × C ^m and S P 2 × R ^(m,2)

We recall that S P 2 is the set of all 2 × 2 positive symmetric real matrices Y with det Y = 1. Then G acts on S P 2 × R ^(m,2) transitively by

(5.1) (g, α) · (Y, V ) := ( gY ^t g, (V + α) ^t g ),

where g ∈ SL(2, R), α ∈ R ^(m,2) , Y ∈ SP 2 and V ∈ R ^(m,2) . It is easy to see that K is a maximal compact subgroup of G stabilizing the origin (I 2 , 0). Thus S P 2 ×R ^(m,2) may be identied with the homogeneous space G/K as follows:

(5.2) G/K ∋ (g, α)K 7−→ (g, α) · (I 2 , 0) ∈ SP 2 × R ^(m,2) , where g ∈ SL(2, R) and α ∈ R ^(m,2) .

We recall that SL(2, R) acts on H transitively by g < τ >:= (aτ + b)(cτ + d) ⁻¹ ,

( a b c d

)

∈ SL(2, R), τ ∈ H.

Now we dene the action of G on H × C ^m by

(5.3) (g, α) ◦ (τ, z) := ( ^t g ⁻¹ < τ >, (z + α 1 τ + α 2 )(−bτ + a) ⁻¹ ), where g =

( a b c d

)

∈ SL(2, R) and α = (α 1 , α 2 ) ∈ R ^(m,2) , and (τ, z) ∈ H×C ^m . Since the action (5.3) is transitive and K is the stabilizer of this action at the origin (i, 0), H×C ^m can be identied with the homogeneous space G/K as follows:

(5.4) G/K ∋ (g, α)K 7−→ (g, α) ◦ (i, 0).

We observe that we can express an element Y of S P 2 uniquely as (5.5) Y =

( y ⁻¹ 0

0 y

) [( 1 −x

0 1

)]

=

( y ⁻¹ −xy ⁻¹

−xy ⁻¹ x ² y ⁻¹ + y )

with x, y ∈ R and y > 0.

Proposition 5.1. We dene the mapping T : S P 2 × R ^(m,2) −→

H × C ^m by

(5.6) T (Y, V ) = (x + iy, v 1 (x + iy) + v 2 ),

where Y is of the form (5.5) and V = (v 1 , v 2 ) ∈ R ^(m,2) . Then the mapping T is a bijection which is compatible with the above two actions (5.1) and (5.3).

For any Y ∈ SP 2 of the form (5.5), we put (5.7) g Y =

( 1 0

−x 1

) ( y ^−1/2 0 0 y ^1/2

)

=

( y ^−1/2 0

−xy ^−1/2 y ^1/2 )

. and

(5.8) α Y,V = V ^t g _Y ⁻¹ . Then we have

(5.9) T (Y, V ) = ( g Y , α Y,V ) ◦ (i, 0).

Proof. It is obvious that T is a bijection. Let T 0 : S P 2 −→ H be the mapping dened by

T 0 (Y ) = x + i y,

where Y is of the form (5.5). We observe that

Y = g Y t g Y and T 0 (Y ) = ^t g _Y ⁻¹ < i >, where g Y is an element of SL(2, R) dened by (5.7).

For g ∈ SL(2, R) and Y ∈ SP 2 ,

Y _∗ = gY ^t g = g _∗ ^t g _∗ for some g _∗ ∈ SL(2, R).

Since Y _∗ = g Y

t g Y

, there is an element k ∈ K such that g Y

= g _∗ k.

So we have

T 0 (Y _∗ ) = ^t g ⁻¹ _Y

< i > = ^t (g _∗ k) ⁻¹ < i >

= ^t g ⁻¹ _∗ ^t k ⁻¹ < i > = ^t g _∗ ⁻¹ < i > .

So far we showed that if Y 0 ∈ SP 2 such that Y 0 = g 0 t g 0 with g 0 ∈ SL(2, R), then T 0 (Y 0 ) = g 0 < i > . If Y is of the form (5.5), we take

g _∗ = g · g Y . Then g _∗ ^t g _∗ = gY ^t g. Therefore we have

T 0 (gY ^t g) = ^t g _∗ ⁻¹ < i > = ^t g ⁻¹ · ^t g _Y ⁻¹ < i > = ^t g ⁻¹ < τ >, where we put τ = x + i y.

Let (g, α) ∈ G with g =

( a b c d

)

∈ SL(2, R), α = (α 1 , α 2 ) ∈ R ^(m,2) and (Y, V ) ∈ SP 2 × R ^(m,2) such that Y ∈ SP 2 is of the form (5.5) and V = (v 1 , v 2 ) ∈ R ^(m,2) with v 1 , v 2 ∈ R ^(m,1) . Now we have, if τ = x + i y ∈ H,

(g, α) ◦ T (Y, V )

= (g, α) ◦ (τ, v 1 τ + v 2 )

= ( ^t g ⁻¹ < τ >, { ( v 1 + α 1 ) τ + ( v 2 + α 2 ) } ( − b τ + a ) ⁻¹ ).

On the other hand.

T ( (g, α) · (Y, V ) ) = T ( gY ^t g, ( V + α ) ^t g )

= ( T 0 (gY ^t g), ˜ v 1 · T 0 (gY ^t g) + ˜ v 2 )

= ( ^t g ⁻¹ < τ >, ˜ v 1 · ^t g ⁻¹ < τ > + ˜ v 2 ),

where ( ˜ v 1 , ˜ v 2 ) = (V + α) ^t g.

Since

˜

v 1 = (v 1 + α 1 ) a + (v 2 + α 2 ) b,

˜

v 2 = (v 1 + α 1 ) c + (v 2 + α 2 ) d, by a simple calculation, we get

˜

v 1 t g ⁻¹ < τ > + ˜ v 2 = { (v 1 + α 1 ) τ + (v 2 + α 2 ) } (−bτ + a ) ⁻¹ . Hence we obtain

T ((g, α) · (Y, V )) = (g, α) ◦ T (Y, V ).

Consequently T is compatible with the actions (5.1) and (5.3). It is

easily checked that (5.9) holds. 

Definition 5.2. For any (τ, z) ∈ H × C ^m with τ ∈ H and z ∈ C ^m , we dene Y (τ,z) ∈ SP 2 and V (τ,z) ∈ R ^(m,2) by

( Y (τ,z) , V (τ,z) ) = T ⁻¹ (τ, z).

Proposition 5.3. Let (g, α) be an element of G such that g = ( a b

c d )

∈ SL(2, R) and α = ( α 1 , α 2 ) ∈ R ^(m,2) with α 1 , α 2 ∈ R ^(m,1) . For (τ, z) ∈ H × C ^m with τ ∈ H, z = u + i v, u = Re z and v = Im z, we set

( τ _∗ , z _∗ ) = (g, α) ◦ (τ, z) and

τ _∗ = x _∗ + i y _∗ , z _∗ = u _∗ + i v _∗ ,

where x _∗ = Re τ _∗ , y _∗ = Im τ _∗ , u _∗ = Re z _∗ and v _∗ = Im z _∗ . Then we have

Y (τ

,z

) =

( y _∗ ⁻¹ 0 0 y _∗

) [ (

1 −x ∗

0 1

) ]

and

V (τ

,z

) = ( y _∗ ⁻¹ v _∗ , u _∗ − x ∗ y _∗ ⁻¹ v _∗ ), u _∗ , v _∗ ∈ R ^(m,1) , where

x _∗ = {

−bd|τ| ² − ac + (ad + bc) x }

| − bτ + a| ⁻² , y _∗ = y | − bτ + a| ⁻² ,

u _∗ = { (a − bx) ( u + xα 1 + α 2 ) − by (v + α 1 y) } | − bτ + a| ⁻² ,

v _∗ = { (a − bx) ( v + yα 1 ) + by ( u + xα 1 + α 2 ) } | − bτ + a| ⁻² .

Proof. It is easy to prove the above proposition and so we omit it.  Now we present a G-invariant metric on S P 2 × R ^(m,2) .

Proposition 5.4. The following Riemannian metric ds ² on S P 2 × R ^(m,2) dened by

ds ² = dx ² + dy ²

y + 1

y

∑ m k=1

{ ( x ² + y ² ) dv _k1 ² + 2 x dv k1 dv k2 + dv _k2 ² }

is invariant under the action (5.1) of G. The Laplace-Beltrami operator

m of ( S P 2 × R ^(m,2) , ds ² ) is given by

m = y ² ( ∂ ²

∂x ² + ∂ ²

∂y ² )

+ y ⁻¹

∑ m k=1

{ ∂ ²

∂v ² _k1 − 2 x ∂ ²

∂v k1 ∂v k2

+ ( x ² + y ² ) ∂ ²

∂v ² _k2 }

.

Proof. The proof of the case m = 1 can be found in [9]. In a similar way, the proof of the above proposition can be done. 

6. Maass-Jacobi forms

In this section, we make some comments on the study of Maass-Jacobi forms related to G = SL 2,1 ( R).

First we review the notion of Maass-Jacobi forms ( cf. [10] ). We let

= SL(2, Z) n Z ^(1,2)

be the discrete subgroup of G. Let D(G) be the algebra of all dieren- tial operators on G invariant under all the left translations of G. For a dieomorphism ϕ of G and an element D ∈ D(G), we dene D ^ϕ by

D ^ϕ (f ) = ( D(f ◦ ϕ) ) ◦ ϕ ⁻¹ , f ∈ C ^∞ (G).

For g ∈ G, we denote by R g the right translation of G by g. We dene D K (G) = {

D ∈ D(G) | D ^R

= D for all k ∈ K }

.

In [10], the author dened the notion of Maass-Jacobi forms. We recall this concept here. A smooth bounded function ϕ : G −→ C is called a Maass -Jacobi form if it satises the following conditions (MJ1)-(MJ3):

(MJ1) ϕ(γxk) = ϕ(x) for all γ ∈ , x ∈ G and k ∈ K.

(MJ2) ϕ is an eigenfunction for the Laplace-Beltrami operator in D K (G).

(MJ3) ϕ is slowly increasing in the sense of [1].

We observe that the notion of Maass-Jacobi forms generalizes that of Maass-wave forms. For the convenience of the reader, we review Maass wave forms. Let s ∈ C and 1 = SL(2, Z). We denote by W s ( 1 ) the vector space of all smooth bounded functions f : SL(2, R) −→ C satisfying the following conditions (a) and (b):

(a) f (γgk) = f (g) for all γ ∈ 1 , g ∈ SL(2, R) and k ∈ K.

(b) 0 f = ^{1 −s} ₄

f, where

0 = y ² ( ∂ ²

∂x ² + ∂ ²

∂y ² )

+ y ∂ ²

∂x∂θ

is the Laplace-Beltrami operator on SL(2, R) whose coordinates x, y, θ ( x ∈ R, y > 0, 0 ≤ θ < 2π ) are given by

g =

( 1 x 0 1

) ( y ^1/2 0 0 y ^−1/2

) ( cos θ −sin θ sin θ cos θ

)

, g ∈ SL(2, R) by means of the Iwasawa decomposition of SL(2, R). The elements in W s ( 1 ) are called Maass wave forms. We observe that the algebra D K (SL(2, R)) is generated by the Laplace-Beltrami operator 0 on SL(2, R). It is well known that W s ( 1 ) is nontrivial for innitely many values of s. For more detail, we refer to [2], [3], [5] and [8].

We can prove that Maass-Jacobi forms may be realized as functions on H × C or SP 2 × R ^(1,2) satisfying a certain invariance property. We explain these facts in some more detail.

Theorem 6.1. Let D(H × C) be the algebra of all dierential op- erators on H × C invariant under the action (5.3). Then the algebra D(H × C) is generated by the following dierential operators

D = y ² ( ∂ ²

∂x ² + ∂ ²

∂y ² )

+ v ² ( ∂ ²

∂u ² + ∂ ²

∂v ² ) (6.1)

+ 2 y v ( ∂ ²

∂x∂u + ∂ ²

∂y∂v )

,

(6.2) = y ( ∂ ²

∂u ² + ∂ ²

∂v ² )

,

(6.3) D 1 = 2 y ² ∂ ³

∂x∂u∂v − y ² ∂

∂y ( ∂ ²

∂u ² − ∂ ²

∂v ² )

+ (

v ∂

∂v + 1 )

and

(6.4) D 2 = y ² ∂

∂x ( ∂ ²

∂v ² − ∂ ²

∂u ² )

− 2 y ² ∂ ³

∂y∂u∂v − v ∂

∂u , where τ = x + iy and z = u + i v with x, u, v real and y > 0. Moreover, we have

[D, ] := D − D = 2y ² ∂

∂y ( ∂ ²

∂u ² − ∂ ²

∂v ² )

− 4 y ² ∂ ³

∂x∂u∂v

− 2 (

v ∂

∂v + )

.

In particular, the algebra D(H × C) is not commutative. The homoge- neous space H × C is not weakly symmetric in the sense of A. Selberg [7].

Proof. The proof can be found in [10]. 

For any right K-invariant function ϕ : G −→ C on G, we dene the function f ϕ : H × C −→ C by

(6.5) f ϕ (τ, z) = ϕ(g, α), τ ∈ H, z ∈ C,

where (g, α) is an element of G with (g, α) ◦ (i, 0) = (τ, z). Clearly it is well dened because (6.5) is independent of the choice of ( g, α) ∈ G such that (g, α) ◦ (i, 0) = (τ, z).

Proposition 6.2. Let ϕ : G −→ C be a nonzero Maass-Jacobi form.

Then the function f ϕ dened by (6.5) satises the following conditions : (MJ1) ⁰ f ϕ (˜ γ ◦ (τ, z)) = f ϕ (τ, z) for all ˜ γ ∈ .

(MJ2) ⁰ f ϕ is a joint eigenfunction for the dierential operator = D + .

(MJ3) ⁰ f ϕ has a polynomial growth.

Conversely, if a smooth function f : H × C −→ C satises the above conditions (MJ1) ⁰ − (MJ3) ⁰ , then the function ϕ f : G −→ C dened by (6.6) ϕ f (g, α) = f ((g, α) ◦ (i, 0)), (g, α) ∈ G

is a Maass-Jacobi form.

We denote by D(SP 2 × R ^(1,2) ) the algebra of all dierential operators on S P 2 × R ^(1,2) invariant under the action (5.1).

Theorem 6.3. The algebra D(SP 2 × R ^(1,2) ) is generated by the following dierential operators

(6.7) D = y ˜ ²

( ∂ ²

∂x ² + ∂ ²

∂y ² , )

(6.8) ˜= y ⁻¹ { ∂ ²

∂v ² ₁ − 2 x ∂ ²

∂v 1 ∂v 2

+ ( x ² + y ² ) ∂ ²

∂v ² ₂ }

,

D ˜ 1 = { ∂ ²

∂v ₁ ² − 2x ∂ ²

∂v 1 ∂v 2

+ (x ² − y ² ) ∂ ²

∂v ² ₂ } ∂ (6.9) ∂y

+ 2y

( ∂ ²

∂v 1 ∂v 2

− x ∂ ²

∂v ² ₂ ) ∂

∂x − and

D ˜ 2 = { ∂ ²

∂v ² ₁ − 2x ∂ ²

∂v 1 ∂v 2

+ (x ² − y ² ) ∂ ²

∂v ₂ ² } ∂

∂x (6.10)

− 2y

( ∂ ²

∂v 1 ∂v 2 − x ∂ ²

∂v ² ₂ ) ∂

∂y , where we used the coordinates

Y =

( y ⁻¹ 0

0 y

) [( 1 −x

0 1

)]

=

( y ⁻¹ −xy ⁻¹

−xy ⁻¹ x ² y ⁻¹ + y )

with x, y ∈ R, y > 0 and V = ( v 1 , v 2 ) with x, y ∈ R.

Proof. The proof of the above theorem can be found in [9]. 

For a right K-invariant function ϕ : G −→ C, we dene another function h ϕ : SP 2 × R ^(1,2) −→ C by

(6.11) h ϕ (Y, V ) = ϕ(g, V ^t g ⁻¹ ), Y ∈ SP 2 , V ∈ R ^(1,2) , where Y = g ^t g with some g ∈ SL(2, R). The denition (6.11) is well dened because it is independent of the choice of g with Y = g ^t g.

Indeed, such g is unique up to K.

Now we can realize Maass-Jacobi forms as functions on S P 2 × R ^(1,2) satisfying a certain invariance property.

Proposition 6.4. Let ϕ : G −→ C be a nonzero Maass-Jacobi form.

Then the function h ϕ dened by (6.7) satises the following conditions : (MJ1) ^∗ h ϕ (γY ^t γ, (V + δ) ^t γ ) = h ϕ (Y, V ) for all (γ, δ) ∈ 2,1 with γ ∈ SL(2, Z) and δ ∈ Z ^(1,2) .

(MJ2) ^∗ h ϕ is a joint eigenfunction for the dierential operator ˜=

D + ˜ ˜ ψ.

(MJ3) ^∗ h ϕ has a suitable polynomial growth.

Conversely, if a smooth function h : S P 2 × R ^(1,2) −→ C satises the conditions (MJ1) ^∗ − (MJ3) ^∗ , the function ϕ h : G −→ C dened by (6.12) ϕ h (g, α) = h(g ^t g, α ^t g), g ∈ SL(2, R), α ∈ R ^(m,2) is a Maass-Jacobi form.

The proof of Theorem 6.2 and 6.4. can be found in [10].

7. The unitary dual of SL 2,m ( R)

For brevity, we set H = R ^(m,2) . Then we have the split exact sequence 0 −→ H −→ G −→ SL(2, R) −→ 1.

We see that the unitary dual ˆ H of H is isomorphic to R ^(m,2) . The unitary character χ (λ,µ) of H corresponding to (λ, µ) ∈ R ^(m,2) with λ, µ ∈ R ^(m,1) is given by

χ _(λ,µ) (x, y) = e ^2πi(

^λx+

^µy) , (x, y) ∈ H.

G acts on H by conjugation and hence this action induces the action of G on ˆ H as follows.

(7.1) G × ˆ H −→ ˆ H, (g, χ) 7→ χ ^g , g ∈ G, χ ∈ ˆ H,

where the character χ ^g is dened by

χ ^g (a) = χ(g ⁻¹ ag), a ∈ H.

If g = (g 0 , α) ∈ G with g 0 ∈ SL(2, R) and α ∈ H, it is easy to check that for each (λ, µ) ∈ R ^(m,2) ,

(7.2) χ ^g _(λ,µ) = χ _(λ,µ)g

. Indeed, if a ∈ H, then

g ⁻¹ ag = (g 0 , α) ⁻¹ (I 2 , a)(g 0 , α)

= (g ₀ ⁻¹ , −α ^t g 0 )(g 0 , a ^t g ₀ ⁻¹ + α)

= (I 2 , a ^t g ₀ ⁻¹ )

and therefore we have for each c ∈ R ^(m,2) and a ∈ H, χ ^g _c (a) = χ c (g ⁻¹ ag) = χ c (a ^t g ⁻¹ ₀ )

= e ^{2πi σ(}

^{c a}

^g

⁾

= e ^{2πi σ(}

^(cg

^)a)

= χ _cg

(a),

where σ(A) denotes the trace of a square matrix A.

We know that H is of type I. We denote by χ the G-orbit of χ ∈ ˆ H and let

G χ = {g ∈ G | χ ^g = χ } be the stabilizer of G at χ. The mapping

G/G χ −→ χ , g · G χ 7→ χ ^g

is a homeomorphism, in other words, H is regularly embedded. Obvi- ously H is a subgroup of G χ . We dene the subset G ^∗ _χ of the unitary dual ˆ G χ of G χ by

G ^∗ _χ = {

τ ∈ ˆ G χ  τ | A is a multiple of χ }

.

According to Mackey [6], we obtain

Theorem 7.1 (Mackey). For any τ ∈ G ^∗ χ , the induced representation Ind ^G _G

τ is an irreducible unitary representation of G. And the unitary dual ˆ G of G is given by

G = ˆ ∪

[χ] ∈G\ ˆ H

{

Ind ^G _G

τ  τ ∈ G ^∗ χ

} .

Case I. m = 1.

We see easily from (7.2) that the G-orbits in ˆ H ∼ = R ² consist of two orbits 0 , 1 given by

0 = {(0, 0)}, 1 = R ² − {(0, 0)}.

We observe that 0 is the G-orbit of (0, 0) and 1 is the G-orbit of any element (λ, µ) ̸= 0.

Now we choose the element δ = χ _(1,0) of ˆ H. That is, δ(x, y) = e ^2πix for all (x, y) ∈ R ² . It is easy to check that the stabilizer of χ _(0,0) is G and the stabilizer G δ of δ is given by

G δ =

{(( 1 0 c 1

) , α ) 

 c ∈ R, α ∈ R ^(1,2) } .

In order to nd the unitary dual G of G, we need the following fact. ˆ Lemma 7.2. Let (τ, V ) be an irreducible unitary representation of G δ

whose restriction to H is a multiple of δ. Then (τ, V ) is a one-dimensional representation of G δ .

Proof. Let W be a nonzero subspace of V invariant under the sub- group

G ⁰ _δ =

{(( 1 0 c 1

) , 0

)

∈ G δ  c ∈ R }

of G δ .

Since ((

1 0 c 1

) , α

)

=

(( 1 0 c 1

) , 0

)

(I 2 , α), we have

τ

((( 1 0 c 1

) , α

)) W = τ

((( 1 0 c 1

) , 0

))

τ ((I 2 , α))W

⊆ τ

((( 1 0 c 1

) , 0

))

W ⊆ W.

Therefore W is a nonzero subspace of V invariant under G δ . By the assumption on the irreducibility of V, W = V. Since (τ, V ) is an irre- ducible unitary representation of G ⁰ _δ and G ⁰ _δ is an abelian group, (τ, V ) must be an one-dimensional representation of G ⁰ _δ and hence G δ .  From now on, we denote by R ^× and C ^× 1 the set of all nonzero real numbers and the set of all complex numbers z with |z| = 1 respectively.

We obtain the following

Theorem 7.3. Let m = 1. Then the irreducible unitary representa- tions of G are the following:

(a) The irreducible unitary representations π, where the restriction of π to H is trivial and the restriction of π to SL(2, R) is an irreducible unitary representation of SL(2, R). For the unitary dual of SL(2, R), we refer to [4], p.123.

(b) The representations π (r) := Ind ^G _G

σ r (r ∈ R) induced from the unitary character σ r of G δ dened by

σ r

((( 1 0 c 1

)

, (λ, µ) ))

= δ(rc + λ) = e ^2πi(rc+λ) , c, λ, µ ∈ R.

Proof. The G-orbits in ˆ H consist of two orbits 0 = { χ (0,0)

} and

1 = {

χ (λ,µ) | λ, µ ∈ R, (λ, µ) ̸= (0, 0) }

. We choose a representative δ := χ (1,0) of 1 . Since G is the stabilizer of χ (0,0) , an element of G ^∗ _χ

= G ^∗ is of the form

π ρ ((g, α)) = ρ · π(g), g ∈ SL(2, R), α ∈ R ^(1,2) ,

where π is an irreducible unitary representation of SL(2, R) and ρ ∈ C ^× 1 . We observe that π ρ is unitarily equivalent to π = π 1 for any ρ ∈ C ^× 1 . On the other hand, since the stabilizer G δ of δ = χ (1,0) is given by

G δ =

{(( 1 0 c 1

) , α ) 

 c ∈ R, α ∈ R ^(1,2) } ,

according to Lemma 7.2, we see that an element of G ^∗ _δ is of the form σ r,ρ

((( 1 0 c 1

)

, (λ, µ) ))

= ρ · e ^2πi(rc+λ) , c, λ, µ ∈ R,

where r is a xed real number and ρ ∈ C ^× 1 . We observe that σ r,ρ is unitarily equivalent to σ r := σ r,1 for any ρ ∈ C ^× 1 . By Theorem 7.1, we

complete the proof. 

Case II. m = 2.

In this case, G = SL(2, R) n R ^(2,2) and ˆ H ∼ = R ^(2,2) . From now on, we identify ˆ H with R ^(2,2) .

Lemma 7.4. Let m = 2. Then the G-orbits in ˆ H consist of the following orbits

0 =

{( 0 0 0 0

)}

,

1 = {( A

0 )

∈ R ^(2,2)   A ∈ R ^(1,2) , A ̸= 0 }

,

2 = {( 0

A )

∈ R ^(2,2)   A ∈ R ^(1,2) , A ̸= 0 }

,

3 (δ) = {( A

δA )

∈ R ^(2,2)   A ∈ R ^(1,2) , A ̸= 0 }

( δ ∈ R ^× ) and

4 (θ) = {

M ∈ R ^(2,2)   det M = θ }

(θ ∈ R ^× ).

0 is the G-orbit of 0=

( 0 0 0 0

)

, 1 is the G-orbit of ( α

0 )

with 0 ̸=

α ∈ R ^(1,2) , 2 is the G-orbit of ( 0

β )

with 0 ̸= β ∈ R ^(1,2) , 3 (δ) is the G-orbit of

( α δα

)

with 0 ̸= α ∈ R ^(1,2) and 4 (θ) is the G-orbit of M ∈ R ^(2,2) with det M = θ.

Proof. We recall that G acts on ˆ H ∼ = R ^(2,2) via (7.2). If we identify H with ˆ R ^(2,2) , we know that G acts on R ^(2,2) by

(g 0 , α) · X := Xg ⁻¹ 0 , (g 0 , α) ∈ G, g 0 ∈ SL(2, R), X ∈ R ^(2,2) . Then we see without diculty that the G-orbits 0 , 1 , 2 , 3 (δ) (δ ∈ R ^× ) and 4 (θ) (θ ∈ R ^× ) form all the G-orbits in ˆ H ∼ = R ^(2,2) . 

We put

e =

( 1 0 0 0

)

and f =

( 0 0 1 0

) . Obviously e ∈ 1 and f ∈ 2 .

Then we may prove the following.

Lemma 7.5. (a) The stabilizer of 0 is G.

(b) The stabilizer G e of e is given by G e =

{(( 1 0 c 1

) , α

)   c ∈ R, α ∈ R ^(2,2) } .

For each x ∈ 1 , the stabilizer G(x) of x is conjugate to G e . Precisely if x = eg 0 with g 0 ∈ SL(2, R), then G(x) = (g 0 , 0) ⁻¹ G e (g 0 , 0).

(c) The stabilizer G f of f is given by G f =

{(( 1 0 c 1

) , α

)   c ∈ R, α ∈ R ^(2,2) } . For each y ∈ 2 , the stabilizer G(y) of y is conjugate to G f .

(d) The stabilizer G δ of

( 1 0 δ 0

)

(δ ∈ R ^× ) is given by

G δ =

{(( 1 0 c 1

) , α

)   c ∈ R, α ∈ R ^(2,2) } . For each z ∈ 3 (δ), the stabilizer G(z) of z is conjugate to G δ .

(e) The stabilizer G θ of M θ ∈ 4 (θ) (θ ∈ R ^× ) is given by G θ =

{

(I 2 , α) | α ∈ R ^(2,2) }

∼ = R ^(2,2) ,

where M θ =

( 1 0 0 θ

)

. Therefore H is regularly embedded.

Proof. (a) is obvious. Since the stabilizer G e of e is G e = { (g 0 , α) ∈ G | e = eg 0 , g 0 ∈ SL(2, R) } , by an easy computation, we get

G e =

{(( 1 0 c 1

) , α ) 

 c ∈ R, α ∈ R ^(2,2) } .

For x ∈ 1 , we let x = eg 0 with some g 0 ∈ SL(2, R). Let G(x) be

the stabilizer of x. If g = (g 1 , α) ∈ G(x) with g 1 ∈ SL(2, R), then

x = xg 1 and so eg 0 = eg 0 g 1 . So (g 0 , 0)(g 1 , α)(g 0 , 0) ⁻¹ ∈ G e , in other

words, (g 1 , α) ∈ (g 0 , 0) ⁻¹ G e (g 0 , 0). Similarly we obtain (c) and (d). Let

M θ ∈ R ^(2,2) such that det M θ = θ, θ ∈ R ^× . Let G θ be the stabilizer of M θ . If (g 0 , α) ∈ G θ with g 0 ∈ SL(2, R), then M θ g 0 = M θ . Since M θ is nonsingular, g 0 = I 2 . Therefore we obtain (e). 

Now we obtain the following

Theorem 7.6. Let m = 2. Then the irreducible unitary representa- tions of G are the following:

(a) The irreducible unitary representations π, where the restriction of π to H is trivial and the restriction of π to SL(2, R) is an irreducible unitary representation of SL(2, R).

(b) The representations π e;r := Ind ^G _G

σ r (r ∈ R) induced from the unitary character σ r of G e dened by

σ r

((( 1 0 c 1

) ,

( α 1 α 2

α 3 α 4

)))

= e ^2πi(rc+α

⁾ , c, α 1 , . . . , α 4 ∈ R.

(c) The representations π f ;r := Ind ^G _G

τ r (r ∈ R) induced from the unitary character τ r of G f dened by

τ r

((( 1 0 c 1

) ,

( α 1 α 2

α 3 α 4

)))

= e ^2πi(rc+α

⁾ , c, α 1 , . . . , α 4 ∈ R.

(d) The representations π δ,r := Ind ^G _G

θ r (δ ∈ R ^× , r ∈ R) induced from the unitary character θ r of G δ dened by

θ r

((( 1 0 c 1

) ,

( α 1 α 2

α 3 α 4

)))

= e ^2πi(rc+α

^+α

^δ) , c, α 1 , . . . , α 4 ∈ R.

(e) The representations π θ := Ind ^G _G

ψ θ (θ ∈ R ^× ) induced from the unitary character ψ θ of G θ dened by

ψ θ

((

I 2

( α 1 α 2

α 3 α 4

)))

= e ^2πi(α

^+α

^θ) , α 1 , . . . , α 4 ∈ R.

Proof. According to Lemma 7.4 and Lemma 7.5, 0 is the G-orbit of 0=

( 0 0 0 0

)

and G is the stabilizer of 0. We note that the representation χ 0 of H corresponding to 0 is the trivial representation of H. Thus an element of G ^∗ is of the form

π ρ ((g, α)) = ρ · π(g), g ∈ SL(2, R), α ∈ R ^(2,2) ,

where π is an irreducible unitary representation of SL(2, R) and ρ ∈ C ^× 1 . We observe that π ρ is unitarily equivalent to π 1 for any ρ ∈ C ^× 1 .

From Lemma 7.4 and Lemma 7.5, we see that 1 is the G-orbit of e =

( 1 0 0 0

)

and the stabilizer G e of e is given by

G e =

{(( 1 0 c 1

) , α ) 

 c ∈ R, α ∈ R ^(2,2) } .

We note that the representation χ e of H corresponding to e is given by χ e (α) = e ^2πiσ(

^eα) , α ∈ H, where σ( ^t eα) denotes the trace of a 2 × 2 matrix ^t eα. Precisely,

χ e

(( α 1 α 2

α 3 α 4

))

= e ^2πiα

,

( α 1 α 2

α 3 α 4

)

∈ H.

Since Lemma 7.2 holds in this case, we see that an element of G ^∗ _e is of the form

σ r,ρ

((( 1 0 c 1

) ,

( α 1 α 2

α 3 α 4

)))

= ρ · e ^2πi(rc+α

⁾ , c, α 1 , . . . , α 4 ∈ R where r is a xed real number and ρ ∈ C ^× 1 . We observe that σ r,ρ is unitarily equivalent to σ r := σ r,1 for any ρ ∈ C ^× 1 . Similarly 2 is the G-orbit of f =

( 0 0 1 0

)

and the stabilizer G f of f is given by

G f =

{(( 1 0 c 1

) , α ) 

 c ∈ R, α ∈ R ^(2,2) } .

We see easily that the representation χ f of H corresponding to f is given by

χ f

(( α 1 α 2

α 3 α 4

))

= e ^2πiα

,

( α 1 α 2

α 3 α 4

)

∈ H.

Since Lemma 7.2 holds in this case, we see that an element of G ^∗ _f is of the form

τ r,ρ

((( 1 0 c 1

) ,

( α 1 α 2

α 3 α 4

)))

= ρ · e ^2πi(rc+α

⁾ , c, α 1 , . . . , α 4 ∈ R where r is a xed real number and ρ ∈ C ^× 1 . We observe that τ r,ρ is unitarily equivalent to τ r := τ r,1 for any ρ ∈ C ^× 1 .

According to Lemma 7.4 and Lemma 7.5, 3 (δ) (δ ∈ R ^× ) is the G-orbit of

( 1 0 δ 0

)

and the stabilizer G δ of

( 1 0 δ 0

)

is given by

G δ =

{(( 1 0 c 1

) , α ) 

 c ∈ R, α ∈ R ^(2,2) } .

We see easily that the representation χ δ of H corresponding to δ is given by

χ δ

(( α 1 α 2

α 3 α 4

))

= e ^2πi(α

^+α

^δ) , α 1 , . . . , α 4 ∈ R.

Since Lemma 7.2 holds in this case, we see that an element of G ^∗ _δ is of the form

θ r,ρ

((( 1 0 c 1

) ,

( α 1 α 2

α 3 α 4

)))

= ρ·e ^2πi(rc+α

^+α

^δ) , c, α 1 , . . . , α 4 ∈ R

where r is a xed real number and ρ ∈ C ^× 1 . We observe that θ r,ρ is unitarily equivalent to θ r := θ r,1 for any ρ ∈ C ^× 1 .

It follows from Lemma 7.4 and Lemma 7.5 that 4 (θ) (θ ∈ R ^× ) is the G-orbit of M θ :=

( 1 0 0 θ

)

and the stabilizer G θ of M θ is given by

G θ = {

(I 2 , α) ∈ G | α ∈ R ^(2,2) } .

We see that the representation χ θ of H corresponding to M θ is given by χ θ

(( α 1 α 2

α 3 α 4

))

= e ^2πi(α

^+α

^θ) , α 1 , . . . , α 4 ∈ R.

We see easily that an element of G ^∗ _θ is of the form ψ θ,ρ

((

I 2 ,

( α 1 α 2

α 3 α 4

)))

= ρ · e ^2πi(α

^+α

^θ) , α 1 , . . . , α 4 ∈ R ,