homogeneous zonal distributions

Convolution on homogeneous spaces
Capelle, Johan
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Capelle, J. (1996). Convolution on homogeneous spaces Groningen: s.n.
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246
Glossary of Symbols
>
<
Italics indicate that the symbol is not in the
<
long list of fixed notations in Harmonic
Analysis
a~
a®
A
acting bilaterally
character of A
140
Lie algebra of A
convolution) operators in the
distributions
B
normalizer of
equal to MAN
(.␣|␣.)
´æ(≈)
´æ(≈
)␣%␣␣␣
139
algebra of G–invariant (or
å
‹
©␣≈
Ì
H
sesquilinear
75
duality brackets, bilinear
73
quotient of modular functions
117
compactly supported ç°–functions
on a manifold ≈
∂æ(≈)
dual of ´(≈)
H
31
zonal distributions with inversely
91
Â*
202
diffeomorphisms
106
group of G–invariant
%
usually denotes involution of zonal
sets
anti-duality brackets, sesquilinear 75
çµ
∂(≈)
40
compactly supported distributions,
character of
108
140
132
80
~
78, 92
128
compact supports
inner product in a Hilbert space,
<.␣,␣.>
<.␣|␣.>
<␣␣␣␣
smooth functions on ≈
140, 165, 196
139, 193
a
in H >: stabilizer of p in G >
<␣␣␣␣
´(≈)
␣special character of A
>
in G : extended group G*Ì,
20
or of zonal distributions
Ó
usually denotes a reproducing
Ó°
Ó–°
operator
35
¥
67
∂¥(≈) distributions on ≈ concentrated on
ç°– vectors
co–ç°–vectors
Hilb˚µ˚˚∂æ(≈)
G
distributions on the manifold ≈, dual
of ∂(≈)
93
89, 92
75
174
174
cone of †µ–invariant ␣ ␣Hilbert
G
subspaces of ∂æ(≈)
1
76
identity map, or identity character
æ
∂æ(≈)H zonal (i.e.: or H–invariant)
distributions, on ≈=G/H
∂æ(™)~,␣N
*
80
homogeneous zonal distributions
≈
Ω–homogeneous zonal distributions
213
indicator function of the set A
232
K
maximal compact subgroup 139, 193
k
139
Lie algebra of K
left and right regular translations in
the distributions on a group
209
]
*
ÈA
L,R
≈
∂æ(™)[¬,N
1,185
LG␣“E¡,E™‘ intertwining space
¬
symmetrizer map
Ò␣≈
family of homogeneous
65
74
13, 65
~
distributions
207
247
— Glossary of Symbols —
M
M*
fl
MAp
M
centralizer of A in K
normalizer of A in K
unitary dual of M
realization of the group
submanifold of
µ␣␣␣␣␣␣␣␣␣␣␣␣␣
©
140, 193
G/N
140, 194
≈
˜
µ
µ
148, 152
MA as
tu
¤™
141
operator, in a space of distributions,
of multiplication by the function
˜~
†g
†xU
©
space of matrix coefficients
153
the two-point group ”±1’
73
zonal
83
transpose of u
84
equals †gU␣, when
x=gp;
ambiguity resolved by U being
(complete) Hilbert tensor product of
two Hilbert spaces
61, 165
associated to ≈¤a~¤1
fl
unitarized version of †g
185
V¤∂
1¤T
ͤTº
Ÿ
201
202
202
usually denotes the anti-linear
relatively invariant measure on
N
ì
N
involution in the zonal distributions,
≈=G/H
72
maximal nilpotent subgroup139, 193
nilpotent group opposite to N
®
®µ
®Ì
#
Í
µ
ßs
*
%
Ë(g)
Lie algebra of N
139, 196
special weight
139, 196
algebra g
≈
203
generalized Verma module
148
140
modular function
117
≈
usually denotes a homogeneous
80
element group, or of Â*
185, 197
Sobolev space
228, 229
in W␣ *␣V
83
218, 219
: push-forward by ^
in ^* : pull-back by ^
in Â* : Â without the origin
in Ó* : anti-dual of Ó
in h* : adjoint of h
179
unitarized push-forward
101
55
56
75
75
operation of g∑G in the distributions
on ≈
space
ÛT
signum character of either a two-
73
13
unitarized right action of Â*
is usually the Weyl group
n
248
√~
W
in ^*
*
†g
v©
73
convolution product
127
universal algebra over the
modular function
Ÿ
û
U
complexification of the real Lie
U␣
ß␣
Ÿ=ì
U
141, 196
n
191
Ω
spectrum of the distribution T
228
algebra of central convolution
operators in ∂æ(≈)
134
INDEX
Italics indicate that the term or its particular
use is specific to the thesis
bi-invariant 131
bilaterally invariant
Hilbert subspace 128
convolution operator 128
Bruhat
Lemma (decomposition) 140
theory 169, 238n.
central
compactly supported zonal
distributions 134
convolution operators 134
contragredient 72
convolution
operator 83 &c.
algebra 19-20, 31, 88, 92, 134
°
co–ç –vector 174, 215
direct integral 76, 121, 129, 169, 170, 190
distribution vector 215
effective (action) 100
exchange property 102, 104, 183
fundamental solution 112-116, 159-164, 186187
Generalized Gelfand Pair 77, 105, 131, 172,
188
Hermitian
character 135
symmetry 98
Hilbert subspace 74, &c.
homogeneous
distribution 205-211
space 72 &c
induced
convolution operators
110-112, 120, 147
Hilbert subspace 120ff., 185
module 10-11, 30-31
inverselybounded or compact 91
Kelvin-inversion 183
kernel (theorem) 80
lattice cone 189, 223
Malgrange-Ehrenpreis 113, 160
multiplicity free
decomposition 129, 130, 170
representation 77, 172
Plancherel decomposition and measure 170,
190, 223
of positive type:
operators 75, 116-117
zonal distributions 116-117, 237
principal fibre bundle 108
propagator83
pull-back (of a distribution) 79, 101, 195
push-forward (of a distribution) 55, 63, 85
relatively invariant measure 73
reproducing operator 75
Schur’s Lemma
(for closeable operators) 137
Sobolev space 228
spectrum (of a distribution on K) 228
strongly zonal 128, 133, 156
symmetric (convolution operators) 211
symmetry 103
(Ë(g),B)–module 70, 148
unitarized
push-forward 101, 221
representation 73, 123
249
— Index —
Verma module 149
weakly symmetric 97-98, 103
Weil formula 81, 118
Weyl
group 140, 194
type 202-205
zonal 80, 83, 91
Ω–homogeneous 211
250