Geometric Analysis
on Minimal Representations
Representation Theory of Real Reductive Groups
University of Utah, Salt Lake City, USA, 27–31 July 2009
Toshiyuki Kobayashi
(the University of Tokyo)
http://www.ms.u-tokyo.ac.jp/~toshi/
Geometric Analysis on Minimal Representations – p.1/49
Minimal representations
Oscillator rep. (= Segal–Shale–Weil rep.)
Minimal rep. of Mp(n; R ) (= double cover of Sp(n; R ))
split simple group of type C
Geometric Analysis on Minimal Representations – p.2/49
Minimal representations
Oscillator rep. (= Segal–Shale–Weil rep.)
Minimal rep. of Mp(n; R ) (= double cover of Sp(n; R ))
split simple group of type C
Today:
Minimal rep. of O(p; q ), p + q : even
simple group of type D
Geometric Analysis on Minimal Representations – p.2/49
Minimal representations
Oscillator rep. (= Segal–Shale–Weil rep.)
Minimal rep. of Mp(n; R ) (= double cover of Sp(n; R ))
split simple group of type C
Today: Geometric and analytic aspects of
Minimal rep. of O(p; q ), p + q : even
simple group of type D
Geometric Analysis on Minimal Representations – p.2/49
Minimal representations
Oscillator rep. (= Segal–Shale–Weil rep.)
Minimal rep. of Mp(n; R ) (= double cover of Sp(n; R ))
split simple group of type C
Today: Geometric and analytic aspects of
Minimal rep. of O(p; q ), p + q : even
simple group of type D
(Ambitious) Project:
Use minimal reps as a guiding principle to find
new interactions with other fields of mathematics.
If possible, try to formulate a theory in a wide setting
without group, and prove it without representation theory.
Geometric Analysis on Minimal Representations – p.2/49
Minimal rep of reductive groups
Minimal representations of a reductive group G
(their annihilators are the Joseph ideal in U (g))
Loosely, minimal representations are
one of ‘building blocks’ of unitary reps.
‘smallest’ infinite dimensional unitary rep.
‘isolated’ among the unitary dual
(finitely many)
(continuously many)
‘attached to’ the minimal nilpotent orbit
matrix coefficients are of bad decay
Geometric Analysis on Minimal Representations – p.3/49
Minimal , Maximal
(Ambitious) Project:
Use minimal reps as a guiding principle to find
new interactions with other fields of mathematics.
Geometric Analysis on Minimal Representations – p.4/49
Minimal , Maximal
(Ambitious) Project:
Use minimal reps as a guiding principle to find
new interactions with other fields of mathematics.
Viewpoint:
Minimal representation (( group)
Maximal symmetries (( rep. space)
Geometric Analysis on Minimal Representations – p.4/49
Geometric analysis on minimal reps of O(p; q )
[1] Laguerre semigroup and Dunkl operators preprint, 74 pp. arXiv:0907.3749
[2] Special functions associated to a fourth order differential equation preprint, 45 pp. arXiv:0907.2608, arXiv:0907.2612
[3] Generalized Fourier transforms Fk;a
[4]
C.R.A.S. Paris (to appear)
Schrödinger model of minimal rep. Memoirs of Amer. Math. Soc. (in press), 171 pp. arXiv:0712.1769
[5] Inversion and holomorphic extension R. Howe 60th birthday volume (2007), 65 pp.
[6] Analysis on minimal representations Adv. Math. (2003) I, II, III, 110 pp.
Collaborated with
S. Ben Saïd, J. Hilgert, G. Mano, J. Möllers and B. Ørsted
Geometric Analysis on Minimal Representations – p.5/49
Indefinite orthogonal group O(p + 1; q + 1)
Throughout this talk, p; q
1, p + q: even > 2
G = O(p + 1; q + 1)
= fg 2 GL(p + q + 2; R ) : t g
Ip+1 O
O Iq+1
g=
Ip+1 O
O Iq+1
real simple Lie group of type D
Geometric Analysis on Minimal Representations – p.6/49
Minimal representation of G = O(p + 1; q + 1)
q=1
highest weight module lowest weight module
the bound states of the Hydrogen atom
Geometric Analysis on Minimal Representations – p.7/49
Minimal representation of G = O(p + 1; q + 1)
q=1
highest weight module lowest weight module
the bound states of the Hydrogen atom
p=q
spherical case
(() realized in scalar valued functions on the
Riemannian symmetric space G=K )
p = q = 3 case: Kostant (1990)
Geometric Analysis on Minimal Representations – p.7/49
Minimal representation of G = O(p + 1; q + 1)
q=1
highest weight module lowest weight module
the bound states of the Hydrogen atom
p=q
spherical case
(() realized in scalar valued functions on the
Riemannian symmetric space G=K )
p = q = 3 case: Kostant (1990)
p; q: general
non-highest, non-spherical
subrepresentation of most degenerate principal series
(Howe–Tan, Binegar–Zierau)
dual pair correspondence
(Sp(1; R ) O(p + 1; q + 1) in Sp(p + q + 2; R )) (Huang–Zhu)
Geometric Analysis on Minimal Representations – p.7/49
Two constructions of minimal reps.
1. Conformal model
Theorem B
2. L2 model
(Schrödinger model)
Theorem D
Geometric Analysis on Minimal Representations – p.8/49
Two constructions of minimal reps.
1. Conformal model
Theorem B
Group action
Hilbert structure
Clear
?
?
Clear
v.s.
2. L2 model
(Schrödinger model)
Theorem D
Clear advantage of the model
Geometric Analysis on Minimal Representations – p.8/49
Two constructions of minimal reps.
1. Conformal model
Theorem B
Group action
Hilbert structure
Clear
Theorem C
Theorem E
Clear
v.s.
2. L2 model
(Schrödinger model)
Theorem D
Clear advantage of the model
Geometric Analysis on Minimal Representations – p.8/49
Two constructions of minimal reps.
1. Conformal model
Theorem B
Group action
Hilbert structure
Clear
Theorem C
Theorem E
Clear
v.s.
2. L2 model
(Schrödinger model)
Theorem D
Clear advantage of the model
3. Deformation of Fourier transforms (Theorems F, G, H)
(interpolation, Dunkl operators, special functions)
Geometric Analysis on Minimal Representations – p.8/49
Geometric analysis on minimal reps of O(p; q )
[1] Laguerre semigroup and Dunkl operators preprint, 74 pp. arXiv:0907.3749
[2] Special functions associated to a fourth order differential equation preprint, 45 pp. arXiv:0907.2608, arXiv:0907.2612
[3] Generalized Fourier transforms Fk;a
[4]
C.R.A.S. Paris (to appear)
Schrödinger model of minimal rep. Memoirs of Amer. Math. Soc. (in press), 171 pp. arXiv:0712.1769
[5] Inversion and holomorphic extension R. Howe 60th birthday volume (2007), 65 pp.
[6] Analysis on minimal representations Adv. Math. (2003) I, II, III, 110 pp.
Collaborated with
S. Ben Saïd, J. Hilgert, G. Mano, J. Möllers and B. Ørsted
Geometric Analysis on Minimal Representations – p.9/49
§1 Conformal construction of minimal reps.
Idea: Composition of holomorphic functions
holomorphic Æ holomorphic = holomorphic
Geometric Analysis on Minimal Representations – p.10/49
§1 Conformal construction of minimal reps.
Idea: Composition of holomorphic functions
holomorphic Æ holomorphic = holomorphic
=)
taking real parts
harmonic Æ conformal = harmonic
on C
' R2
Geometric Analysis on Minimal Representations – p.10/49
§1 Conformal construction of minimal reps.
Idea: Composition of holomorphic functions
holomorphic Æ holomorphic = holomorphic
=)
taking real parts
harmonic Æ conformal = harmonic
on C
' R2
make sense for general Riemannian manifolds.
Geometric Analysis on Minimal Representations – p.10/49
§1 Conformal construction of minimal reps.
Idea: Composition of holomorphic functions
holomorphic Æ holomorphic = holomorphic
=)
taking real parts
harmonic Æ conformal = harmonic
on C
' R2
make sense for general Riemannian manifolds.
But
harmonic Æ conformal 6= harmonic
in general
Geometric Analysis on Minimal Representations – p.10/49
§1 Conformal construction of minimal reps.
Idea: Composition of holomorphic functions
holomorphic Æ holomorphic = holomorphic
=)
taking real parts
harmonic Æ conformal = harmonic
on C
' R2
make sense for general Riemannian manifolds.
But
harmonic Æ conformal 6= harmonic
in general
=) Try to modify the definition!
Geometric Analysis on Minimal Representations – p.10/49
Conf(X; g) Isom(X; g)
(X; g) Riemannian manifold
' 2 Di eo(X )
Geometric Analysis on Minimal Representations – p.11/49
Conf(X; g) Isom(X; g)
(X; g) Riemannian manifold
' 2 Di eo(X )
Def.
' is isometry () ' g = g
' is conformal () 9 positive function C' 2 C 1 (X ) s.t.
' g = C'2 g
C' : conformal factor
Geometric Analysis on Minimal Representations – p.11/49
Conf(X; g) Isom(X; g)
(X; g) Riemannian manifold
' 2 Di eo(X )
Def.
' is isometry () ' g = g
' is conformal () 9 positive function C' 2 C 1 (X ) s.t.
' g = C'2 g
C' : conformal factor
Di eo(X ) Conf(X; g)
Conformal group
Isom(X; g)
isometry group
Geometric Analysis on Minimal Representations – p.11/49
Conf(X; g) Isom(X; g)
(X; g) pseudo-Riemannian manifold
' 2 Di eo(X )
Def.
' is isometry () ' g = g
' is conformal () 9 positive function C' 2 C 1 (X ) s.t.
' g = C'2 g
C' : conformal factor
Di eo(X ) Conf(X; g)
Conformal group
Isom(X; g)
isometry group
Geometric Analysis on Minimal Representations – p.11/49
Harmonic Æ conformal 6= harmonic
Modification
' 2 Conf(X n ; g),
' g = C'2 g
Geometric Analysis on Minimal Representations – p.12/49
Harmonic Æ conformal 6= harmonic
Modification
' 2 Conf(X n ; g),
pull-back
f Æ'
' g = C'2 g
twisted pull-back
C'
n
2
2
f Æ'
conformal factor
Geometric Analysis on Minimal Representations – p.12/49
Harmonic Æ conformal 6= harmonic
Modification
' 2 Conf(X n ; g),
' g = C'2 g
pull-back
f Æ'
twisted pull-back
C'
n
2
2
f Æ'
conformal factor
S ol(X ) = ff 2 C 1(X ) : X f = 0g (harmonic functions)
S ol(gX ) = ff 2 C 1(X ) : gX f = 0g
g
X
Yamabe operator
:= X + 4(nn 21) Laplacian
scalar curvature
Geometric Analysis on Minimal Representations – p.12/49
Distinguished rep. of conformal groups
(=
harmonic Æ conformal + harmonic
Modification
Geometric Analysis on Minimal Representations – p.13/49
Distinguished rep. of conformal groups
(=
harmonic Æ conformal + harmonic
Modification
(X n ; g) Riemannian mfd
n
g
=) Conf(X; g) acts on S ol(
f Æ'
X ) by f 7! C'
Theorem A ([6, Part I])
2
2
Geometric Analysis on Minimal Representations – p.13/49
Distinguished rep. of conformal groups
(=
harmonic Æ conformal + harmonic
Modification
(X n ; g) Riemannian mfd
n
g
=) Conf(X; g) acts on S ol(
f Æ'
X ) by f 7! C'
Theorem A ([6, Part I])
2
2
n 2 g
=
+
X
X 4(n 2)
g
X is not invariant by Conf(X; g ).
g
But S ol(
X ) is invariant by Conf(X; g ).
Point
Geometric Analysis on Minimal Representations – p.13/49
Distinguished rep. of conformal groups
(=
harmonic Æ conformal + harmonic
Modification
(X n ; g) Riemannian mfd
n
g
=) Conf(X; g) acts on S ol(
f Æ'
X ) by f 7! C'
Theorem A ([6, Part I])
2
2
n 2 g
=
+
X
X 4(n 2)
g
X is not invariant by Conf(X; g ).
g
But S ol(
X ) is invariant by Conf(X; g ).
Point
Di eo(X ) Conf(X; g)
Conformal group
Isom(X; g)
isometry group
Geometric Analysis on Minimal Representations – p.13/49
Distinguished rep. of conformal groups
(=
harmonic Æ conformal + harmonic
Modification
(X n ; g) pseudo-Riemannian mfd
n
g
=) Conf(X; g) acts on S ol(
f Æ'
X ) by f 7! C'
Theorem A ([6, Part I])
2
2
n 2 g
=
+
X
X 4(n 2)
g
X is not invariant by Conf(X; g ).
g
But S ol(
X ) is invariant by Conf(X; g ).
Point
Di eo(X ) Conf(X; g)
Conformal group
Isom(X; g)
isometry group
Geometric Analysis on Minimal Representations – p.13/49
Application of Theorem A
(X; g) := (S p S q ; +
+}
| {z
p
| {z })
q
Geometric Analysis on Minimal Representations – p.14/49
Application of Theorem A
(X; g) := (S p S q ; +
+}
| {z
p
| {z })
q
Theorem B ([6, Part II])
0) Conf(X; g ) ' O(p + 1; q + 1)
g
1) S ol(
X ) 6= f0g () p + q even
2) If p + q is even and > 2, then
g
Conf(X; g) y S ol(
X ) is irreducible,
and for p + q > 6 it is a minimal rep of O(p + 1; q + 1).
Geometric Analysis on Minimal Representations – p.14/49
Application of Theorem A
(X; g) := (S p S q ; +
+}
| {z
p
| {z })
q
Theorem B ([6, Part II])
0) Conf(X; g ) ' O(p + 1; q + 1)
g
1) S ol(
X ) 6= f0g () p + q even
2) If p + q is even and > 2, then
g
Conf(X; g) y S ol(
X ) is irreducible,
and for p + q > 6 it is a minimal rep of O(p + 1; q + 1).
1) (conformal geometry) () (represntation theory)
characterizing subrep in IndG
Pmax (C ) (K -picture)
by means of differential equations
Geometric Analysis on Minimal Representations – p.14/49
Application of Theorem A
(X; g) := (S p S q ; +
+}
| {z
p
| {z })
q
Theorem B ([6, Part II])
0) Conf(X; g ) ' O(p + 1; q + 1)
g
1) S ol(
X ) 6= f0g () p + q even
2) If p + q is even and > 2, then
g
Conf(X; g) y S ol(
X ) is irreducible,
and for p + q > 6 it is a minimal rep of O(p + 1; q + 1).
"
9 a Conf(X; g)-invariant inner product, and
take the Hilbert completion
Geometric Analysis on Minimal Representations – p.14/49
Flat model
Stereographic projection
S n ! R n [ f1g
conformal map
Geometric Analysis on Minimal Representations – p.15/49
Flat model
Stereographic projection
S n ! R n [ f1g
conformal map
More generally
p Sq
S
++
-
R p+q
ds =dx ++dxp dxp+1
2
2
1
2
2
dxp+q
2
conformal embedding
Geometric Analysis on Minimal Representations – p.15/49
Flat model
Stereographic projection
S n ! R n [ f1g
conformal map
More generally
p Sq
S
++
-
R p+q
ds =dx ++dxp dxp+1
2
2
1
2
2
dxp+q
2
conformal embedding
Functoriality of Theorem A
Conf(S p S q )
y
y
S ol(e Sp Sq ) S ol(e R p;q )
- Conf(R p;q )
Geometric Analysis on Minimal Representations – p.15/49
Two constructions of minimal reps.
Group action Hilbert structure
1. Conformal construction
Theorem B
Clear
?
?
Clear
v.s.
2. L2 construction
(Schrödinger model)
Theorem D
Clear advantage of the model
Geometric Analysis on Minimal Representations – p.16/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
Unitarization of subrep (representation theory)
()
Conservative quantity (differential eqn)
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
Unitarization of subrep (representation theory)
()
Conservative quantity (differential eqn)
Unitarizability v.s. Unitarization
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
Unitarization of subrep (representation theory)
()
Conservative quantity (differential eqn)
Unitarizability v.s. Unitarization
Easy formulation
Challenging formulation
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
Problem Find an ‘intrinsic’ inner product
on (a ‘large’ subspace of) S ol(p;q )
if exists.
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
Problem Find an ‘intrinsic’ inner product
on (a ‘large’ subspace of) S ol(p;q )
if exists.
Easy: if allowed to use the integral representation of
solutions
Cf. (representation theory)
by using the Knapp–Stein intertwining formula
Challenging: to find the intrinsic formula
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
q=1
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
wave operator
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
q=1
=
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
wave operator
energy conservative quantity for wave equations
w.r.t. time translation R
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for ultra-hyperbolic eqn.
R p;q
= R p+q ; ds2 = dx21 + + dx2p
e R p;q
=
q=1
2
x21
+ +
2
x2p
2
x2p+1
dx2p+1
dx2p+q
p;q
q
2
x2p+
wave operator
(=
energy conservative quantity for wave equations
w.r.t. time translation R
?
conservative quantity for ultra-hyperbolic eqs
w.r.t. conformal group
O(p + 1; q + 1)
Geometric Analysis on Minimal Representations – p.17/49
Conservative quantity for p;q f
Fix
R p+q
=0
non-degenerate hyperplane
Geometric Analysis on Minimal Representations – p.18/49
Conservative quantity for p;q f
=0
R p+q non-degenerate hyperplane
For f 2 S ol(p;q )
Fix
(f; f ) :=
Z
Q f
1
Geometric Analysis on Minimal Representations – p.18/49
Conservative quantity for p;q f
=0
R p+q non-degenerate hyperplane
For f 2 S ol(p;q )
Fix
(f; f ) :=
Z
Q f
1
Theorem C ([6, Part III]+")
1) 1 is independent of hyperplane .
Geometric Analysis on Minimal Representations – p.18/49
Conservative quantity for p;q f
=0
R p+q non-degenerate hyperplane
For f 2 S ol(p;q )
Fix
(f; f ) :=
Z
Q f
1
Theorem C ([6, Part III]+")
1) 1 is independent of hyperplane .
2) 1 gives the unique inner product (up to scalar)
which is invariant under O(p + 1; q + 1).
Geometric Analysis on Minimal Representations – p.18/49
Conservative quantity for p;q f
=0
R p+q non-degenerate hyperplane
For f 2 S ol(p;q )
Fix
(f; f ) :=
Z
1
Q f
Theorem C ([6, Part III]+")
1) 1 is independent of hyperplane .
2) 1 gives the unique inner product (up to scalar)
which is invariant under O(p + 1; q + 1).
O(p; q)
y
R p;q
(linear)
Geometric Analysis on Minimal Representations – p.18/49
Conservative quantity for p;q f
=0
R p+q non-degenerate hyperplane
For f 2 S ol(p;q )
Fix
(f; f ) :=
Z
1
Q f
Theorem C ([6, Part III]+")
1) 1 is independent of hyperplane .
2) 1 gives the unique inner product (up to scalar)
which is invariant under O(p + 1; q + 1).
O(p; q)
O(p + 1; q + 1)
y
R p;q (linear)
(Möbius transform)
Geometric Analysis on Minimal Representations – p.18/49
Parametrization of non-characteristic hyperplane
v 2 R p;q s.t. (v; v)R p;q = 1
2R
(=
Fix
R p;q
v;
:= fx 2 R p+q : (x; v)R p;q =
non-characteristic hyperplane
g
v
v;
Geometric Analysis on Minimal Representations – p.19/49
‘Intrinsic’ inner product
Point:
f = f+ + f (idea: Sato’s hyperfunction)
Geometric Analysis on Minimal Representations – p.20/49
‘Intrinsic’ inner product
For = v; , f 2 C 1 (R p;q ) with some decay at 1
Point: f = f+ + f (idea: Sato’s hyperfunction)
Geometric Analysis on Minimal Representations – p.20/49
‘Intrinsic’ inner product
For = v; , f 2 C 1 (R p;q ) with some decay at 1
Point: f = f+ + f (idea: Sato’s hyperfunction)
f0 normal derivative of f w.r.t. v
Geometric Analysis on Minimal Representations – p.20/49
‘Intrinsic’ inner product
For = v; , f 2 C 1 (R p;q ) with some decay at 1
Point: f = f+ + f (idea: Sato’s hyperfunction)
f0 normal derivative of f w.r.t. v
1 0
Q f := f+ f+
i
f f0
v
v;
Geometric Analysis on Minimal Representations – p.20/49
Conservative quantity for p;q f
=0
= v; R p+q non-degenerate hyperplane
For f 2 S ol(p;q
)
Z
Fix
(f; f ) :=
Q f
1
Theorem C ([6, Part III]+")
1) 1 is independent of hyperplane .
2) 1 gives the unique inner product (up to scalar)
which is invariant under O(p + 1; q + 1).
Geometric Analysis on Minimal Representations – p.21/49
Conservative quantity for p;q f
=0
= v; R p+q non-degenerate hyperplane
For f 2 S ol(p;q
)
Z
Fix
(f; f ) :=
Q f
1
Theorem C ([6, Part III]+")
1) 1 is independent of hyperplane .
2) 1 gives the unique inner product (up to scalar)
which is invariant under O(p + 1; q + 1).
non-trivial even for q = 1 (wave equation)
In space-time,
average in space (i.e. time t = constant)
= average in (any hyperplane in space) R t (time)
Geometric Analysis on Minimal Representations – p.21/49
Two constructions of minimal reps.
1. Conformal construction
Theorem B
Group action
Hilbert structure
Clear
?
?
Clear
v.s.
2. L2 construction
(Schrödinger model)
Theorem D
Clear advantage of the model
Geometric Analysis on Minimal Representations – p.22/49
Two constructions of minimal reps.
Group action Hilbert structure
1. Conformal construction
Theorems A, B
Clear
conservative
quantity
?
Clear
v.s.
2. L2 construction
(Schrödinger model)
Theorem D
Clear advantage of the model
Geometric Analysis on Minimal Representations – p.23/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
Geometric Analysis on Minimal Representations – p.24/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
=
(figure for (p; q) = (2; 1))
Geometric Analysis on Minimal Representations – p.24/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
p;q f = 0 Fourier
=) Supp F f trans.
Geometric Analysis on Minimal Representations – p.24/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
p;q f = 0 Fourier
=) Supp F f trans.
F
: S 0 (R p;q )
! S 0(R p;q )
Geometric Analysis on Minimal Representations – p.24/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
p;q f = 0 Fourier
=) Supp F f trans.
F:
S 0(R p;q )
[
S ol(p;q )
! S 0(R p;q )
[
Geometric Analysis on Minimal Representations – p.24/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
p;q f = 0 Fourier
=) Supp F f trans.
F:
S 0(R p;q )
[
S ol(p;q )
! S 0(R p;q )
[
! ?
denotes the closure with respect to the inner product.
Geometric Analysis on Minimal Representations – p.24/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
p;q f = 0 Fourier
=) Supp F f trans.
F:
Theorem D ([6, Part III])
S 0 (R p;q )
[
S ol(p;q )
! S 0(R p;q )
[
! L2()
Geometric Analysis on Minimal Representations – p.24/49
Conformal model =)
2
L -model
2
2
2
2
p;q = x2 + + x2 x2
x2
p
p+q
1
p+1
:= f 2 R p+q : 12 + + p2 p2+1 p2+q = 0g
p;q f = 0 Fourier
=) Supp F f trans.
F:
Theorem D ([6, Part III])
S 0 (R p;q )
[
S ol(p;q )
conformal model
! S 0(R p;q )
[
! L2()
L2 -model
Geometric Analysis on Minimal Representations – p.24/49
Two constructions of minimal reps.
Group action Hilbert structure
1. Conformal construction
Theorems A, B
Clear
conservative
quantity
?
Clear
v.s.
2. L2 construction
(Schrödinger model)
Theorem D
Clear advantage of the model
Geometric Analysis on Minimal Representations – p.25/49
2
§2 L -model of minimal reps.
Theorem D ([6, Part III])
S ol(p;q )
conformal model
!
L2 ()
L2 -model
Geometric Analysis on Minimal Representations – p.26/49
2
§2 L -model of minimal reps.
S ol(p;q )
Theorem D ([6, Part III])
conformal model
p + q: even > 2
G = O(p + 1; q + 1)
y
L2()
!
L2 ()
L2 -model
unitary rep.
Geometric Analysis on Minimal Representations – p.26/49
2
§2 L -model of minimal reps.
S ol(p;q )
Theorem D ([6, Part III])
conformal model
p + q: even > 2
G = O(p + 1; q + 1)
Point:
y
L2()
!
L2 ()
L2 -model
unitary rep.
is too small to be acted by G.
Geometric Analysis on Minimal Representations – p.26/49
2
§2 L -model of minimal reps.
S ol(p;q )
Theorem D ([6, Part III])
!
L2 -model
conformal model
p + q: even > 2
G = O(p + 1; q + 1)
Point:
y
L2 ()
L2()
unitary rep.
is too small to be acted by G.
R p;q
R p+1;q +1
Geometric Analysis on Minimal Representations – p.26/49
2
§2 L -model of minimal reps.
!
S ol(p;q )
Theorem D ([6, Part III])
conformal model
p + q: even > 2
y
G = O(p + 1; q + 1)
Point:
L2()
L2 ()
L2 -model
unitary rep.
is too small to be acted by G.
O(p; q)
y
y
L2 ()
R p;q
R p+1;q +1
Geometric Analysis on Minimal Representations – p.26/49
2
§2 L -model of minimal reps.
!
S ol(p;q )
Theorem D ([6, Part III])
conformal model
p + q: even > 2
minimal rep.
y
G = O(p + 1; q + 1)
Point:
L2()
L2 ()
L2 -model
unitary rep.
is too small to be acted by G.
O(p + 1; q + 1)
X
y
y
L2 ()
R p;q
R p+1;q +1
Geometric Analysis on Minimal Representations – p.26/49
Inversion element
G = P GL(2; C )
y
Möbius transform
P1C
' C [ f1g
Geometric Analysis on Minimal Representations – p.27/49
Inversion element
G = P GL(2; C )
P=
y
Möbius transform
a b : a 2 C ; b 2 C
0 1
w = 01 01
P1C
' C [ f1g
z 7! az + b
z 7! z1
(inversion)
Geometric Analysis on Minimal Representations – p.27/49
Inversion element
G = P GL(2; C )
P=
y
Möbius transform
a b : a 2 C ; b 2 C
0 1
w = 01 01
G is generated by P and w.
P1C
' C [ f1g
z 7! az + b
z 7! z1
(inversion)
Geometric Analysis on Minimal Representations – p.27/49
Inversion element
G = P GL(2; C )
+ O(3; 1)
P=
y
Möbius transform
a b : a 2 C ; b 2 C
0 1
w = 01 01
G is generated by P and w.
P1C
' C [ f1g
+ R 2;0
z 7! az + b
z 7! z1
(inversion)
Geometric Analysis on Minimal Representations – p.27/49
Inversion element
G = P GL(2; C )
+ O(3; 1)
P=
y
Möbius transform
a b : a 2 C ; b 2 C
0 1
P1C
' C [ f1g
+ R 2;0
z 7! az + b
w = 01 01
z 7! z1
(inversion)
G is generated by P and w.
y
G = O(p + 1; q + 1)
R p;q
Möbius transform
P =
f(A; b) : A2 O(p; q) R ; b 2 R p+q g x 7! Ax + b
w = Ip I
(inversion)
q
Geometric Analysis on Minimal Representations – p.27/49
Inversion element
G = P GL(2; C )
+ O(3; 1)
P=
y
Möbius transform
a b : a 2 C ; b 2 C
0 1
P1C
' C [ f1g
+ R 2;0
z 7! az + b
w = 01 01
z 7! z1
(inversion)
G is generated by P and w.
y
G = O(p + 1; q + 1)
R p;q
Möbius transform
P =
f(A; b) : A2 O(p; q) R ; b 2 R p+q g x 7! Ax + b
w = Ip I : (x0 ; x00 ) 7! jx0 j 4jx00 j ( x0 ; x00 ) (inversion)
q
2
2
Geometric Analysis on Minimal Representations – p.27/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
minimal rep.
Geometric Analysis on Minimal Representations – p.28/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
P -action
w-action
minimal rep.
translation and multiplication
F (unitary inversion operator)
Geometric Analysis on Minimal Representations – p.28/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
P -action
w-action
minimal rep.
translation and multiplication
F (unitary inversion operator)
Problem Find the unitary operaotr F explicitly.
Geometric Analysis on Minimal Representations – p.28/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
P -action
w-action
minimal rep.
translation and multiplication
F (unitary inversion operator)
Problem Find the unitary operaotr F explicitly.
Easy:
express it as a composition of integral
transforms and a known formula for other
models (e.g. conformal model)
Challenging: to find a single and explicit formula in L2
model
Geometric Analysis on Minimal Representations – p.28/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
P -action
w-action
minimal rep.
translation and multiplication
F (unitary inversion operator)
Problem Find the unitary operaotr F explicitly.
Cf. Analogous operator for the oscillator rep.
Mp(n; R ) y L2 (R n )
unitary inversion operator coincides with
Euclidean Fourier transform FR n (up to scalar)!
Geometric Analysis on Minimal Representations – p.28/49
Fourier transform F on := f 2 R p+q : 12 + + p2
=
p2+1
p2+q = 0g
(figure for (p; q) = (2; 1))
Geometric Analysis on Minimal Representations – p.29/49
Fourier transform F on := f 2 R p+q : 12 + + p2
=
Fourier trans.
p2+1
p2+q = 0g
(figure for (p; q) = (2; 1))
FR n on R n
F
on
=
Geometric Analysis on Minimal Representations – p.29/49
Fourier transform F on := f 2 R p+q : 12 + + p2
p2+1
p2+q = 0g
(figure for (p; q) = (2; 1))
=
Fourier trans.
FR n on R n
F
on
=
Problem Define F and find its formula.
Geometric Analysis on Minimal Representations – p.29/49
‘Fourier transform’ F on Fourier trans.
FR n on R n
F
on
=
Geometric Analysis on Minimal Representations – p.30/49
‘Fourier transform’ F on Fourier trans.
FR n on R n
F
on
=
F 4 = id
Geometric Analysis on Minimal Representations – p.30/49
‘Fourier transform’ F on Fourier trans.
F 4 = id
FR n on R n
F
on
=
F2 = id
Geometric Analysis on Minimal Representations – p.30/49
‘Fourier transform’ F on FR n on R n
Fourier trans.
Qj
Pj
7! Pj
7 Qj
!
Qj = xj
Pj =
F
on
=
(multiplication by coordinate function)
1
p
1 xj
Geometric Analysis on Minimal Representations – p.30/49
‘Fourier transform’ F on FR n on R n
Fourier trans.
Qj
Pj
7! Pj
7 Qj
!
Qj = xj
Pj =
F
=
Qj 7! Rj
Rj 7! Qj
on
(multiplication by coordinate function)
1
p
1 xj
Rj = 9 second order differential op. on Geometric Analysis on Minimal Representations – p.30/49
‘Fourier transform’ F on FR n on R n
Fourier trans.
Qj
Pj
7! Pj
7 Qj
!
Qj = xj
Pj =
F
=
Qj 7! Rj
Rj 7! Qj
on
(multiplication by coordinate function)
1
p
1 xj
Rj = 9 second order differential op. on Bargmann–Todorov’s operators
Geometric Analysis on Minimal Representations – p.30/49
‘Fourier transform’ F on FR n on R n
Fourier trans.
Qj
Pj
F
7! Pj
7 Qj
!
Qj = xj
Pj =
=
Qj 7! Rj
Rj 7! Qj
on
(multiplication by coordinate function)
1
p
1 xj
Rj = 9 second order differential op. on Notice
Q1 2 + + Q 2
R1 2 + + R 2
p
p
Q +1 2
R +1 2
p
p
9
Q + 2 = 0 = on R + 2 = 0 ;
p
q
p
q
Geometric Analysis on Minimal Representations – p.30/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
P -action
w-action
minimal rep.
translation and multiplication on L2()
F (unitary inversion operator)
Problem Find the unitary operaotr F explicitly.
Geometric Analysis on Minimal Representations – p.31/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
P -action
w-action
minimal rep.
translation and multiplication on L2()
F (unitary inversion operator)
Problem Find the unitary operaotr F explicitly.
it (one variable)
'
(
t
)
=
e
R
FR N f (x) = R N '(hx; yi)f (y)dy
Cf. Euclidean case
Geometric Analysis on Minimal Representations – p.31/49
Unitary inversion operator F
p + q: even > 2
G = O(p + 1; q + 1) y L2 ()
P -action
w-action
minimal rep.
translation and multiplication on L2()
F (unitary inversion operator)
Problem Find the unitary operaotr F explicitly.
it (one variable)
'
(
t
)
=
e
R
FR N f (x) = R N '(hx; yi)f (y)dy
Cf. Euclidean case
Theorem E ([4])Z Suppose p + q : even > 2
)
(F f )(x) =
"1((p;q
(hx; yi)f (y)dy
p
+
q
4)
2
Geometric Analysis on Minimal Representations – p.31/49
Mellin–Barnes type integral
Idea: Apply Mellin–Barnes type integral to distributions.
Fix m 2 N . Take a contour Lm s.t.
Lm
s
m
1
m
1
0
Geometric Analysis on Minimal Representations – p.32/49
Mellin–Barnes type integral
Idea: Apply Mellin–Barnes type integral to distributions.
Fix m 2 N . Take a contour Lm s.t.
1)
Lm starts at
i1
2) passes the real axis at s
3) ends at
+ i1
where
Lm
s
m
1
m
1
0
m 1<s< m
1< <0
Geometric Analysis on Minimal Representations – p.32/49
Explicit formula of F on Theorem E ([4])Z Suppose p + q : even > 2
"(p;q)
(F f )(x) =
1 (p+q 4) (hx; yi)f (y)dy
2
Geometric Analysis on Minimal Representations – p.33/49
Explicit formula of F on Theorem E ([4])Z Suppose p + q : even > 2
"(p;q)
(F f )(x) =
1 (p+q 4) (hx; yi)f (y)dy
2
8
>
<0
Here, "(p; q ) = 1
>
:2
min(p; q) = 1;
> 1 are both odd,
> 1 are both even.
if
if p; q
if p; q
Geometric Analysis on Minimal Representations – p.33/49
Explicit formula of F on Theorem E ([4])Z Suppose p + q : even > 2
"(p;q)
(F f )(x) =
1 (p+q 4) (hx; yi)f (y)dy
2
8
>
<0
Here, "(p; q ) = 1
>
:2
"m (t) =
8Z
>
>
>
>
>
L0
>
Z
>
<
>
Lm
>
Z
>
>
>
>
>
:
Lm
min(p; q) = 1;
> 1 are both odd,
> 1 are both even.
if
if p; q
if p; q
( )
(2t)+ d
(" = 0)
( + 1 + m)
( )
(2t)+ d
(" = 1)
( + 1 + m)
!
(2t)+
(2t)
( )
+
d (" = 2)
( + 1 + m) tan() sin()
Geometric Analysis on Minimal Representations – p.33/49
Regularity of (t)
"
m
Cf. Euclidean Fourier transform e it
2 A(R ) \ L1lo (R ) \ Geometric Analysis on Minimal Representations – p.34/49
Regularity of (t)
"
m
Cf. Euclidean Fourier transform e it
2 A(R ) \ L1lo (R ) \ Recall two distributions on R
Æ (t): Dirac’s delta function
t 1 : Cauchy’sZprincipal
value
s Z 1 1
= lim (
+
)h ; idt
s!0
t
1 s
these are not in L1lo (R )
Geometric Analysis on Minimal Representations – p.34/49
Regularity of (t)
"
m
Cf. Euclidean Fourier transform e it
2 A(R ) \ L1lo (R ) \ Prop. ([4]) 8We have the identities mod L1lo (R )
"m (t) >
0
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
m
X1
(" = 0)
( 1)l
(l) (t) (" = 1)
i
Æ
l (m l 1)!
2
l=0
m
X1
l!
l 1
i
t
(" = 2)
l
2 (m l 1)!
l=0
Geometric Analysis on Minimal Representations – p.34/49
Regularity of (t)
"
m
Cf. Euclidean Fourier transform e it
2 A(R ) \ L1lo (R ) \ Prop. ([4]) 8We have the identities mod L1lo (R )
"m (t) Cor.
>
0
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
m
X1
(" = 0)
( 1)l
(l) (t) (" = 1)
i
Æ
l (m l 1)!
2
l=0
m
X1
l!
l 1
i
t
(" = 2)
l
2 (m l 1)!
l=0
F has a locally integrable kernel if and only if
:
G is O(p + 1; 2), O(2; q + 1), or O(3; 3) (= SL(4; R )).
Geometric Analysis on Minimal Representations – p.34/49
Bessel distribution
"m (t) solves the differential equation
(2 + m + 2t)u = 0
d.
where = t dt
Prop. ([4])
Geometric Analysis on Minimal Representations – p.35/49
Bessel distribution
"m (t) solves the differential equation
(2 + m + 2t)u = 0
d.
where = t dt
Prop. ([4])
Explicit forms
p
0
2
m (t) = 2i(2t)+ Jm (2 2t+ )
m
l
X1
(
1)
(l) (t)
1m (t) = 0m (t) i
Æ
l (m l 1)!
2
l=0
m
Geometric Analysis on Minimal Representations – p.35/49
Bessel distribution
"m (t) solves the differential equation
(2 + m + 2t)u = 0
d.
where = t dt
Prop. ([4])
Explicit forms
p
0
2
m (t) = 2i(2t)+ Jm (2 2t+ )
m
l
X1
(
1)
(l) (t)
1m (t) = 0m (t) i
Æ
l (m l 1)!
2
m l=0 p
2m (t) = 2i(2t)+ 2 Ym (2 2t+ )
m
p
m
+1
2
+ 4( 1) i (2t) Km (2 2t )
m
Geometric Analysis on Minimal Representations – p.35/49
Two constructions of minimal reps.
1. Conformal construction
Theorems A, B
Group action
Hilbert structure
Clear
conservative
quantity
‘Fourier transform’
F
Clear
v.s.
2. L2 construction
(Schrödinger model)
Theorem D
Clear advantage of the model
3. Deformation of Fourier transforms (Theorems F, G, H)
Geometric Analysis on Minimal Representations – p.36/49
Two constructions of minimal reps.
Group action Hilbert structure
1. Conformal construction
Theorems A, B
Clear
Theorem C
Theorem E
Clear
v.s.
2. L2 construction
(Schrödinger model)
Theorem D
Clear advantage of the model
3. Deformation of Fourier transforms (Theorems F, G, H)
Geometric Analysis on Minimal Representations – p.36/49
Deformation of Fourier transform FR N
F FR N ‘Fourier transform’ on Fourier transform on R N
R p;q
Geometric Analysis on Minimal Representations – p.37/49
Deformation of Fourier transform FR N
F FR N ‘Fourier transform’ on Fourier transform on R N
R p;q
Assume q = 1. Set p = N .
R N;1
=
!
projection
= RN
Geometric Analysis on Minimal Representations – p.37/49
Deformation of Fourier transform FR N
F FR N ‘Fourier transform’ on Fourier transform on R N
R p;q
Assume q = 1. Set p = N .
R N;1
=
F
O(N + 1; 2)
!
= RN
projection
FR N
Mp(N; R )
Geometric Analysis on Minimal Representations – p.37/49
Deformation of Fourier transform FR N
F FR N ‘Fourier transform’ on Fourier transform on R N
R p;q
Assume q = 1. Set p = N .
R N;1
=
!
= RN
projection
F FR N
deform
a=1
a=2
Geometric Analysis on Minimal Representations – p.37/49
(k; a)-deformation of exp 2 ( jxj )
2
t
Fourier transform
FR N =
exp
i
4
( jxj2)
Geometric Analysis on Minimal Representations – p.38/49
(k; a)-deformation of exp 2 ( jxj )
2
t
Fourier transform
self-adjoint op. on L2
z
FR N =
exp
phase factor
=e
iN
i
4
( }|
{
(R )
N
jxj2)
Laplacian
4
Geometric Analysis on Minimal Representations – p.38/49
(k; a)-deformation of exp 2 ( jxj )
2
t
Fourier transform
self-adjoint op. on L2
z
FR N =
exp
phase factor
=e
iN
i
4
}|
( {
(R )
N
jxj2)
Laplacian
4
Hermite semigroup
t
I (t) := exp (
2
jxj2)
R. Howe (oscillator semigroup, 1988)
Geometric Analysis on Minimal Representations – p.38/49
(k; a)-deformation of exp 2 ( jxj )
2
t
Hankel-type transform on self-adjoint op. on L2
z
i
F =
exp
phase factor
=e
i(N
2
2
}|
(R ; )
N
jxj
{
(jxj jxj)
dx
Laplacian
1)
“Laguerre semigroup” ([5], 2007 Howe 60th birthday volume)
I (t) := exp t(jxj jxj)
Geometric Analysis on Minimal Representations – p.39/49
(k; a)-deformation of exp 2 ( jxj )
2
t
(k; a)-generalized Fourier transform
self-adjoint op. on L2
z
Fk;a =
exp
phase factor
(N +2hki+a
i
2a
=e
(R ; # (x)dx)
}|
N
{
i
( jxj2 a k
2a
k;a
jxja)
Dunkl Laplacian
2)
(k; a)-deformation of Hermite semigroup ([1], 2009)
t 2 a
Ik;a (t) := exp a (jxj k jxja)
k: multiplicity on root system R, a > 0
Geometric Analysis on Minimal Representations – p.40/49
Special values of holomorphic semigroup Ik;a (t)
(k; a)-generalized Fourier transform F
k;a
! i2
!
t
a
Holomorphic semigroup Ik;a (t)
!2
a
!2
! !1
I 1(t)
k;
k;
!
k
!0
Dunkl transform Hermite semigp
t
! i2
Fourier transform
!0
t
t
! i2
! i2
F
Laguerre semigp
!
!0
!
k
k
!
! i2
!1
a
I 2(t)
t
a
k
1
k;
!0
Hankel transform
Geometric Analysis on Minimal Representations – p.41/49
Special values of holomorphic semigroup Ik;a (t)
(k; a)-generalized Fourier transform F
k;a
! i2
!
t
a
Holomorphic semigroup Ik;a (t)
!2
a
!2
I 2(t)
! !1
I 1(t)
k;
!0
k
!0
t
F
Dunkl transform Hermite semigp Laguerre semigp
Fourier transform
t
! i2
!
!0
!
k
! i2
!
k
!
! i2
!1
a
k;
t
a
k
!0
1
k;
Hankel transform
Geometric Analysis on Minimal Representations – p.41/49
Special values of holomorphic semigroup Ik;a (t)
(k; a)-generalized Fourier transform F
k;a
! i2
!
t
a
Holomorphic semigroup Ik;a (t)
!2
a
!2
I 2(t)
! !1
I 1(t)
k;
!0
k
!0
t
F
Dunkl transform Hermite semigp Laguerre semigp
t
! i2
!
!0
!
k
! i2
!
k
!
! i2
!1
a
k;
t
a
k
!0
1
k;
Fourier transform
Hankel transform
..
..
.. ( ‘unitary inversion operator’ ) ..
..
..
the Weil representation of
the metaplectic group M p(N; R )
the minimal representation of
the conformal group O(N + 1; 2)
Geometric Analysis on Minimal Representations – p.41/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
Geometric Analysis on Minimal Representations – p.42/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
P
Assume a > 0 and a + k + N 2 > 0.
Ik;a(t) := exp at (jxj2 ak jxja ) is a holomorphic semigp
on Hk;a for Re t > 0.
Thm F ([1])
Geometric Analysis on Minimal Representations – p.42/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
P
Assume a > 0 and a + k + N 2 > 0.
Ik;a(t) := exp at (jxj2 ak jxja ) is a holomorphic semigp
on Hk;a for Re t > 0.
Thm F ([1])
Ik;a(t1) Æ Ik;a(t2) = Ik;a(t1 + t2) for Re t1; t2 0
(Ik;a (t)f; g) is holomorphic for Re t > 0; for 8 f; 8 g
Geometric Analysis on Minimal Representations – p.42/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
P
Assume a > 0 and a + k + N 2 > 0.
Ik;a(t) := exp at (jxj2 ak jxja ) is a holomorphic semigp
on Hk;a for Re t > 0.
Thm F ([1])
^
(2; R )-admissible
Point: The unitary rep on Hk;a is SL
(i.e. discretely decomposable and finite multiplicities)
Geometric Analysis on Minimal Representations – p.42/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
P
Assume a > 0 and a + k + N 2 > 0.
Ik;a(t) := exp at (jxj2 ak jxja ) is a holomorphic semigp
on Hk;a for Re t > 0.
Thm F ([1])
^
(2; R )-admissible
Point: The unitary rep on Hk;a is SL
(i.e. discretely decomposable and finite multiplicities)
=) 8 Spectrum of jxj2 a k
jxja is discrete and negative
Geometric Analysis on Minimal Representations – p.42/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
P
Assume a > 0 and a + k + N 2 > 0.
Ik;a(t) := exp at (jxj2 ak jxja ) is a holomorphic semigp
on Hk;a for Re t > 0.
Thm F ([1])
^
(2; R )-admissible
Point: The unitary rep on Hk;a is SL
(i.e. discretely decomposable and finite multiplicities)
=) automorphisms of the ring of operators.
a = 1 =) SL(2; Z ) action on degenerate DAHA (Cherednik)
Geometric Analysis on Minimal Representations – p.42/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
P
Assume a > 0 and a + k + N 2 > 0.
Ik;a(t) := exp at (jxj2 ak jxja ) is a holomorphic semigp
on Hk;a for Re t > 0.
Thm F ([1])
Fk;a :=
|{z}
Ik;a( i2 )
phase factor
Geometric Analysis on Minimal Representations – p.42/49
(k; a)-deformation of Hermite semigp
k = (k ): multiplicity ofY
root system R in R N
Hk;a := L2(R N ; jxja 2 jhx; ijk dx)
2R
P
Assume a > 0 and a + k + N 2 > 0.
Ik;a(t) := exp at (jxj2 ak jxja ) is a holomorphic semigp
on Hk;a for Re t > 0.
Thm F ([1])
Fk;a :=
|{z}
Ik;a( i2 )
phase factor
e
i
(N +2hki+a
2a
2)
Geometric Analysis on Minimal Representations – p.42/49
Generalized Fourier transform Fk;a
i
Fk;a = Ik;a ( 2 )
Geometric Analysis on Minimal Representations – p.43/49
Generalized Fourier transform Fk;a
i
i 2 a
Fk;a = Ik;a ( 2 ) = exp 2a (jxj k
Thm G 1)
jxja )
Fk;a is a unitary operator
Geometric Analysis on Minimal Representations – p.43/49
Generalized Fourier transform Fk;a
i
i 2 a
Fk;a = Ik;a ( 2 ) = exp 2a (jxj k
Thm G 1)
2)
3)
4)
jxja )
Fk;a is a unitary operator
F0;2 = Fourier transform on R N
Fk;a = Dunkl transform on R N
F0;1 = Hankel transform on L2( )
Fk;a is of finite order () a 2 Q
Fk;a intertwines jxja and jxj2 ak
Geometric Analysis on Minimal Representations – p.43/49
Generalized Fourier transform Fk;a
i
i 2 a
Fk;a = Ik;a ( 2 ) = exp 2a (jxj k
Thm G 1)
2)
3)
4)
jxja )
Fk;a is a unitary operator
F0;2 = Fourier transform on R N
Fk;a = Dunkl transform on R N
F0;1 = Hankel transform on L2( )
Fk;a is of finite order () a 2 Q
Fk;a intertwines jxja and jxj2 ak
=) generalization of classical identities such as Hecke identity,
Bochner identity, Parseval–Plancherel formulas,
Weber’s second exponential integral, etc.
Geometric Analysis on Minimal Representations – p.43/49
Application to special functions
Minimal reps (( group)
Geometric Analysis on Minimal Representations – p.44/49
Application to special functions
Minimal reps (( group)
Maximal symmetries (( space)
)
Geometric Analysis on Minimal Representations – p.44/49
Application to special functions
Minimal reps (( group)
Maximal symmetries (( space)
)
‘Special functions’, ‘orthogonal polynomials’
associated to 4th order differential eqn [2a, 2b]
Geometric Analysis on Minimal Representations – p.44/49
Application to special functions
Minimal reps (( group)
Maximal symmetries (( space)
)
‘Special functions’, ‘orthogonal polynomials’
associated to 4th order differential eqn [2a, 2b]
with 4 parameters
( p;
q ; l;| {zm} )
| {z}
dimension branching laws (multiplicity-free)
Special case q = 1: Laguerre polynomials 4 = 2 2
Geometric Analysis on Minimal Representations – p.44/49
Heisenberg-type inequality
Thm H (Heisenberg inequality)
kjxj f (x)kk kj j (Fk;af )( )kk 2hki+N2 +a 2 kf (x)k2k
a
2
a
2
k 0, a = 2
Weyl–Pauli–Heisenberg inequality
for Fourier transform FR N
k: general, a = 2 Heisenberg inequality for Dunkl
transform Dk (Rösler, Shimeno)
k 0, a = 1, N = 1 Heisenberg inequality for Hankel
transform
Geometric Analysis on Minimal Representations – p.45/49
Special values of holomorphic semigroup Ik;a (t)
(k; a)-generalized Fourier transform F
k;a
! i2
!
t
a
Holomorphic semigroup Ik;a (t)
!2
a
!2
I 2(t)
! !1
I 1(t)
k;
!0
k
!0
t
F
Dunkl transform Hermite semigp Laguerre semigp
t
! i2
!
!0
!
k
! i2
!
k
!
! i2
!1
a
k;
t
a
k
!0
1
k;
Fourier transform
Hankel transform
..
..
.. ( ‘unitary inversion operator’ ) ..
..
..
the Weil representation of
the metaplectic group M p(N; R )
the minimal representation of
the conformal group O(N + 1; 2)
Geometric Analysis on Minimal Representations – p.46/49
(
2
N
Hidden symmetries in L R ; #k;a
(x)dx)
O(N + 1; 2)
!! 1
a
Coxeter group
k!0
^
SL(2; R )
C
(k; a : general)
(2; R )
! O(N ) SL^
! !2
a
Mp(N; R )
Geometric Analysis on Minimal Representations – p.47/49
Bessel functions
J (z ) =
1
z X
2
2j
z
j
1) 2
(
j
!
(
j
+
+
1)
j =0
p 1
p 1 J e
z
I (z ) := e
J (z ) os J (z )
Y (z ) :=
sin K (z ) :=
(I (z ) I (z ))
2 sin 2
2
(second kind)
(third kind)
Geometric Analysis on Minimal Representations – p.48/49
Geometric analysis on minimal reps of O(p; q )
[1] Laguerre semigroup and Dunkl operators preprint, 74 pp. arXiv:0907.3749
[2] Special functions associated to a fourth order differential equation preprint, 45 pp. arXiv:0907.2608, arXiv:0907.2612
[3] Generalized Fourier transforms Fk;a
[4]
C.R.A.S. Paris (to appear)
Schrödinger model of minimal rep. Memoirs of Amer. Math. Soc. (in press), 171 pp. arXiv:0712.1769
[5] Inversion and holomorphic extension R. Howe 60th birthday volume (2007), 65 pp.
[6] Analysis on minimal representations Adv. Math. (2003) I, II, III, 110 pp.
Collaborated with
S. Ben Saïd, J. Hilgert, G. Mano, J. Möllers and B. Ørsted
Geometric Analysis on Minimal Representations – p.49/49