Invariant differential operators on a real semisimple Lie algebra and their radial components
Authors: Pazirandeh Mohsen 
Published: July 2010

Let

S
(

g

C

)

S({\mathfrak {g}_C})

be the symmetric algebra over the complexification

g

C

{\mathfrak {g}_C}

of the real semisimple Lie algebra

g

\mathfrak {g}

. For

u

ϵ

S
(

g

C

)
,
∂
(
u
)

u\;\epsilon \;S({\mathfrak {g}_C}),\partial (u)

is the corresponding differential operator on

g

\mathfrak {g}

.

D

(

g

)

\mathcal {D}(\mathfrak {g})

denotes the algebra generated by

∂
(
S
(

g

C

)
)

\partial (S({\mathfrak {g}_C}))

and multiplication by polynomials on

g

C

{\mathfrak {g}_C}

. For any open set

U
⊂

g

,

Diff

(
U
)

U \subset \mathfrak {g},{\text {Diff}}(U)

is the algebra of differential operators with

C
∞

{C^\infty }

-coefficients on U. Let

h

\mathfrak {h}

be a Cartan subalgebra of

g

,

h

′

\mathfrak {g},\mathfrak {h}’

the set of its regular points and

π
=

Π

α
ϵ
P

α

\pi = {\Pi _{\alpha \epsilon P}}\alpha

, P some positive system of roots. Let

W
=

(

h

′

)
G

W = {(\mathfrak {h}’)^G}

, G the connected adjoint group of

g

\mathfrak {g}

. Harish-Chandra showed that, for each

D

ϵ

Diff

(
W
)

D\;\epsilon \;{\text {Diff}}(W)

, there is a unique differential operator

δ

h

′

(
D
)

\delta {’_\mathfrak {h}}(D)

on

h

′

\mathfrak {h}’

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper D f right-parenthesis StartAbsoluteValue prime Subscript German h Baseline equals delta prime Subscript German h Baseline left-parenthesis upper D right-parenthesis left-parenthesis f EndAbsoluteValue Subscript German h Baseline right-parenthesis">

(
D
f
)

|

h

′

=
δ

h

′

(
D
)
(
f

|

h

)

(Df){\left | {_\mathfrak {h}’ = \delta {’_\mathfrak {h}}(D)(f} \right |_\mathfrak {h}})

for all G-invariant

f
ϵ

C
∞

(
W
)

f\epsilon \;{C^\infty }(W)

, and that if

D

ϵ

D

(

h

)

D\;\epsilon \mathcal {D}(\mathfrak {h})

, then

δ

h

′

(
D
)
=

π

−
1

∘

D
¯

∘
π

\delta {’_\mathfrak {h}}(D) = {\pi ^{ - 1}} \circ \bar D \circ \pi

for some

D
¯

ϵ

D

(

g

)

\bar D\epsilon \mathcal {D}(\mathfrak {g})

. In particular
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove partial-differential left-parenthesis u right-parenthesis With bar equals partial-differential left-parenthesis u vertical-bar Subscript German h Baseline right-parenthesis comma u epsilon upper S left-parenthesis German g Subscript upper C Baseline right-parenthesis">

∂
(
u
)

¯

=
∂
(
u

|

h

)
,
u

ϵ

S
(

g

C

)

\overline {\partial (u)} = \partial (u{|_\mathfrak {h}}),u\;\epsilon \;S({\mathfrak {g}_C})

and invariant. We prove these results by different, yet simpler methods. We reduce evaluation of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta prime Subscript German h Baseline left-parenthesis partial-differential left-parenthesis u right-parenthesis right-parenthesis left-parenthesis u epsilon upper S left-parenthesis German g Subscript upper C Baseline right-parenthesis">

δ

h

′

(
∂
(
u
)
)

(
u

ϵ

S
(

g

C

)

\delta {’_\mathfrak {h}}(\partial (u))\;(u\;\epsilon \;S({\mathfrak {g}_C})

, invariant) via Weyl’s unitarian trick, to the case of compact G. This case is proved using an evaluation of a family of G-invariant eigenfunctions on:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi left-parenthesis upper H right-parenthesis pi left-parenthesis upper H prime right-parenthesis integral Underscript upper G Endscripts exp upper B left-parenthesis upper H Superscript x Baseline comma upper H Superscript prime Baseline right-parenthesis d x equals c sigma-summation Underscript upper S epsilon upper W left-parenthesis German g Subscript upper C Baseline comma German h Subscript upper C Baseline right-parenthesis Endscripts epsilon left-parenthesis s right-parenthesis exp upper B left-parenthesis s upper H comma upper H Superscript prime Baseline right-parenthesis comma upper H comma upper H prime epsilon German g comma c greater-than 0 period">

π
(
H
)
π
(

H
′

)

∫
G

exp
⁡
B
(

H
x

,

H
′

)
d
x
=
c

∑

S
ϵ
W
(

g

C

,

h

C

)

ϵ
(
s
)
exp
⁡
B
(
s
H
,

H
′

)
,
H
,

H
′

ϵ

g

,
c
>
0.

\pi (H)\pi (H’)\int _G {\exp B({H^x},H’)dx = c\sum \limits _{S\epsilon W({\mathfrak {g}_C},{\mathfrak {h}_C})} {\epsilon (s)\exp B(sH,H’),H,H’\epsilon \;\mathfrak {g},c > 0.} }

\]
For G-invariant

D

ϵ

D

(

g

)

D\;\epsilon \;\mathcal {D}(\mathfrak {g})

, we prove

π

−
1

∘

δ
′

(
D
)
∘
π

ϵ

D

(

h

)

{\pi ^{ - 1}} \circ \delta ’(D) \circ \pi \;\epsilon \;\mathcal {D}(\mathfrak {h})

using properties of derivations

E
→

[

∂
(
u
)
,
E

]

E \to \left [ {\partial (u),E} \right ]

of

D

(

g

)

\mathcal {D}(\mathfrak {g})

induced by

∂
(
u
)

(
u

ϵ

S
(

g

C

)
)

\partial (u)\;(u\;\epsilon \;S({\mathfrak {g}_C}))

and of the algebra of polynomials on

h

C

{\mathfrak {h}_C}

invariant under the Weyl group.