Chipping away at convex sets Sydney, 2004 by Bill Casselman
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Chipping away at convex sets
Sydney, 2004
by Bill
Casselman
1
This picture has nothing whatsoever to do with the rest of my talk
Outline
Part I. Euclidean space
A homework exercise
Truncation for convex polyhedra
Bounded polyhedra
Cones
A local version
Conclusion of the proof
Langlands’ combinatorial lemma
Fourier analysis
2
Part II. Arithmetic quotients
Groups of higher rank
A simple model for Eisenstein series
Back to Arthur’s truncation
The Maass-Selberg formula
3
This talk can be found at
http://www.math.ubc.ca/˜cass/sydney/colloquium.pdf
4
Part I. Euclidean space
A homework exercise
5
Find the Fourier transform
of the characteristic
function of a triangle with vertices at , , .
B
C
A
Your answer should make manifest the inherent symmetry of the problem. Check it by verifying that evalgives the area of the triangle.
uation at
6
Truncation for convex polyhedra
7
be a closed convex polyhedron,
Let
.
If
is a face of codimension one
be the open exterior
of , let
half-plane bounded by .
is any other proper face
If
If
8
let
.
If
, replace it by its product with the vector
space perpendicular to it in these definitions.
9
be the characteristic function of
Let
.
The following theorem is due to Ishida for cones,
Brion and Vergne for nondegenerate bounded polyhedra. The result for arbitrary convex polyhedra
seems to be new, even in two dimensions.
Theorem C. We have
()'
%&
+
1
,
/.-
$
0 *
#
!
"
Minkowski or Weyl might have discovered it.
10
A few simple cases:
+1 − 1
2
Coordinate octant
or simplicial cone
2
11
Simplex
+1 − 1 + 1 − 1
+1 − 1
Another formulation:
in
3
For any
3
;
D
B
@A?
=>
:
C
;
<
9
E
otherwise.
7
4
56
8
This suggests that cohomological Euler-Poincaré
characteristics are going to play a role:
@A?
G
:
;
I
<
9
B
@A?
=>
:
C
;
E
<
9
B
;
HGF
J
if
if
if
is closed and bounded
is a closed cone
is open
The first two are equivalent, since a suitable slice
through a cone is a bounded polyhedron.
12
13
K
Two faces are allowed to have the same affine support. In effect, the Theorem is about a cell decompocompatible with its convex structure. If a
sition of
partition an open geometric face
number of faces
, their total contribution is just
L
N
M K
O
[\Z
XY
^ ]
_
O
W
` P
a
O
^_]
P
V
U T
"
SRO
O
P
Q
^ ]
b
`
^ ]
since all
here and the Euler-Poincaré
characteristic of the open face is .
O
O
P
V
14
What’s going on?
15
16
Bounded polyhedra
following Brion & Vergne
c
We want to prove that for any point
c
k
t
r
pqo
mn
j
s
k
l
i
u
otherwise
g
d
ef
h
P
c
If
is a point of
this
is not
is immediate. If
be the convex
in , let
and .
hull of
c
c
17
C
Then
is the union of all segments
with
in , and
is the
with
in
.
union of all
P
{
y
z
w v
x
{
w y
F
|
v
{
C
b
w
}
is a face of then
if and only if
.
}
Proposition. If
~
P
of is one of
the cells in the boundary of
if
and only if
.
}
Corollary. A face
w
18
C
"
"
"
"
"
"
"
"
"
19
cone
Cones
20
The formula for cones reduces to that for bounded
convex sets by taking slices.
Above the slice, the two configurations are the same.
21
A local version
22
N
¢
It possesses an obvious
.
product structure
£
¤
¢
F
¡
VFC
be the set of points in
nearest to it lies in
.
For each face
of , let
for which the point of
¡
23
¥
¥
¦
§
For each couple of faces
let
ª
®
¯ ¬
°
ª
©_¨
ª
ª
­a¬
ª«
©_¨
Thus a point of
lies
if and only if the
in
closest to it
point of
.
lies in
© ¨
C
EF,P
±
ª
P
©¨
¬
ª«
ª
F
¥
C
EF,F
24
¦
C
EF,Q
¥
Q
²
Theorem L. For each face
³
of
²
Ä ³
²
Ä ³
Æ
´
¿À¾
¼½
´
ÄŶ
´Ã
º
 Á
*
¹
»
¸
ÂÇ
´
·
¶
´µ
´
This is one variation of Langlands’ combinatorial
lemma.
25
P
F1
F0
C
EFC0 ,P
P
F0
C
C
EC,P
+
−
F1
C
26
P
F1
F0
C
P
F0
È
F1
EFC1 ,P
−
È
C
EP,P
È
L in two dimensions is covered by these images
É
É
É
whose secret is given away by these:
−
+
−
27
C for cones implies L.
Theorem L asserts that
Ü
Ú Ý
Ü
Ú Ý
Û
Ê
ÕÖÔ
ÒÓ
Ê
ÚÅÌ
ÊÙ
Ð
Ø ×
*
Ï
Ñ
Î
ØÞ
Ê
Í
Ì
ÊË
Ê
Ü
In this,
may be replaced by its tangent cone at
.
At any face but a vertex, the tangent cone at that
face has a simple product structure, and induction
proves the claim. The formula for the full cone can
be rearranged to give it for the vertex.
Ý
28
Conclusion of the proof of C
29
C follows from L by introducing the partition
à ß
*
ã
áå
á
á
á
á
ã
ä
á
á
êë
íîì
íîì
êë
à ß
*
è
é
á
ã
áå
á
é
ç æ
"
â
à ß
*
è
"
ç æ
á
á
ä
ã
áï
ã
áå
á
á
íîì
êë
à ß
è
ã
áå
á
é
ç"æ
â
á
á
ä
ã
áï
á
ã
â
ñ
àð
30
â
à ß
*
and then rearranging the sum:
31
Langlands’ combinatorial lemma
32
ô
ò
ôó
If
is a face of , let
be the translation of
by the support of , and
its characteristic func. When
is an obtuse
tion. Thus
simplicial cone the following result is the same as
the original combinatorial lemma of Langlands.
õ
õ
ô ö
ò
õ
ø
N
ù ò
÷
ô ó
ô
õ
õ
of
ò
Theorem. For any face
÷
if
otherwise
ò
þ
õ
õ
û
ô ö
÷
þ
û
õ
ÿ
"
ý ü
õ
õ
õ
ôú
û
ú
33
The case
is trivial. The proof for other
uses the partition of
into the
, and goes by induction.
The original applied to simplicial cones and was
announced by Langlands without proof in his 1965
Boulder talk on Eisenstein series, and a result apparently equivalent to this one is contained in the
appendices to a recent paper by Goresky et al.
34
Fourier analysis
35
One curious application of the result of Brion and
Vergne is a useful formula for the Fourier transform
of the characteristic function of a bounded convex
with
. This is the entire function
polyhdron
of the complex variable
and the formula asserts that
&
%$
#
'
"
! (
'
where the right hand sum is over the vertices of ,
and the expression is taken to be the analytic continuation of the obvious integral.
36
Applying the fundamental theorem again to the exterior cones, this can be rewritten
) + 3
)
/
-.,
2
01/
-.,
+*
2
3
where is the characteristic function of the interiors
(tangent cones) of the vertices.
These integrals are easy to compute when the cones
are simplicial, but as far as I know there is no simple
formula otherwise.
37
4
The ‘formal’ Fourier transform (i.e. Laplace transand whose edges
form) of the cone with vertex
and
is
pass through
B
B
K
O D
P
M
9
R
R
B
;
A
S
;
.T7
S
.Q7
W
9
.VU
Q
etc.
C
38
J <
C
C
NML
triangle area
=
=
where
HI
GF
E D
A
?@
= :
>
<
8
<.;
:
9
8.7
65
Jacobian
X
For the case of a triangle with vertices , ,
this
of the triangle with
is the product of twice the area
d Y
[
c Y
[
Z Y
[
b
b
e
^
]
_
]
`
\
`
`
\
`
a.\
a
_
a
`
].\
_
`
a.\
_
f
f
h
pon
f
h
, etc. Why in heck
? Why is it even an
m
bm
lkj
h
i
h.e g
]
q
39
^
^
^
^
^
].\
where
is this equal to the area at
entire function?
Part II. Arithmetic quotients
sr
t
40
To start
v
}
v
v
unipotent matrices in
~
v
Borel subgroup of upper triangular matrices
~
41
{|z
y
xw
u
upper half plane
Γ ∩ P \HY ,→ Γ\H (Y ≥ 1)
y=Y
42
Thus for
the quotient
may be iden. We have the map
tified with a subset of
|
¡
where
¢
¡
.
.£
¡
is the constant term of
43
is the characteristic function of the region
truncation at
is the operator
If
¤
The quotient
may be compactified by adding a
cusp at infinity, and truncation chops away the constant term of a function in the neighbourhood of the
cusp.
¥
¤
As a
on
-invariant function
©
¬
.ǻ
­1¬
.ǻ
©
§ ¦
¨
®
©
«
µ
¬
·¬
¶ª
´ª
¬
´ª
³
¯
±°
¯
²
44
¹ ¸
º
The most important property of
is that under a
its truncation
is
mild growth condition on
rapidly decreasing at .
»
¹ ¸
»
¼
It is for this reason that truncation plays a role in the
meromorphic continuation of Eisenstein series and
in proving the Selberg trace formula.
½
In particular, if
is the Eisenstein series of Maass
is square-integrable, and the Maassthen
is important in proving
Selberg formula for
.
properties of
¾
½
¹ ¸
¾
¿
À ¿
¾
½
¹ ¸
½
¾
All of these features occur in using the truncation
operator for groups of higher rank as well.
45
The Maass-Selberg formula:
ÇÄ
Ä
È
Å
Æ
É
à |ÂÁ
¨
Å
Æ
Ê1É
È
ÇÄ
Ä
à |ÂÁ
¨
Ã
Î
Ï
Ê
ÇÄ
Ä
È
Å
Í
Ì
Í
Ì
Ð Ï
Ñ
Ë
Ã
ÛÎ
Î
Ç
Ï
Ö
ÇÖ
È
È
Å
Å
Ú
Ù
ÓÔ
ÓÔ
Ï
Ï
Ï
Ï
Í
.ÕÌ
Í
Ø Ì
"
Ê
×
Ò
×
Ò
Ð Ï
Ë
46
const. term of
const. term of
Groups of higher rank
47
Ý
Ý
Suppose now for simplicity that
is a split group
,
. Under these condiover ,
tions, all Borel subgroups are -conjugate.
á
à
|ßÞ
Ü
Ü
ä
â
Fix one, call it
. Let
be the corresponding set of
the basic roots.
roots,
ã
â
For any rational parabolic subgroup
let
be its
,
the connected
unipotent radical,
.
component of the centre of
å
Ý
â
å
å
å
á
å
å
Given the compact subgroup
of , for any parabolic
there exists a unique copy of
in
subgroup of
stable under the Cartan involution determined by
.
â
48
æ
ç
are parametrized
è
Parabolic subgroups containing
.
by subsets
élè
For
è
è
è
ê
ê
ë
ë
49
ì
Every rational parabolic subgroup is
exactly one of these.
-conjugate to
î
í
For any rational parabolic subgroup
and
the constant term of
with respect to
í
î
ï
ð
on
is
õ ô
1
ó.ò
ñ
î
ü
ð
ô
í
ï
define (formally)
ÿ
ñ
ð
ô
ò
ó
ò
ô
î
ô
.óò
õ ô
î
ñ
ò
ö
ñ
ö
÷
ú
50
þ
ù
ÿ
ò
î
Conversely, for
on
the Eisenstein series
ý.ô
ó
ü.ò
û
øú
ù
í
ø÷
ö
ï
.
ñ
a function on
In
lie two naturally defined cones, one obtuse
be the characteristic function
and one acute. Let
that of the acute one.
of the obtuse one,
τP
51
χP
There is a canonical projection
and
,
,
be also their lifts back to
their shifts by in .
Let
Fix
in the positive Weyl chamber in
far away
from the walls. Arthur’s definition of truncation is
this:
&
$%#
( '
)
&
$%#
*
,
+
"
!
The sum evaluated at any given element is finite.
52
:
897
/
<);
:
897
=
4
A0
1
>
5
2
2.
?
6
2
3
6
3
!
0 /
.-
@
2
B
If you are familiar with the geometry of
you will
likely find this definition puzzling, because there is
no longer any obvious relationship between truncation and the geometry of a compactification of .
C
For groups of rational rank greater than one, Arthur’s
truncation is not local on any Satake compactification.
53
D
FGE
D
Nonetheless, there is no doubt that Arthur’s definition is the correct one. It is again true, but not so
simple to prove, that under a mild growth condition
the truncation
is rapidly decreasing at
on
infinity.
Truncation is a projection operator, too.
H
It does not affect functions whose constant term
support lies inside a well defined compact subset
. In particular it does not affect cusp forms.
of
I
54
J
Truncation is defined on every
as well as
itself. There is an equivalent recursive definition of
in terms of truncation
truncation that defines it for
.
on the other
J
Theorem. We have an orthogonal decomposition
U
W
O N
P
Y
VUT
J
SRQ
J
\[
J
K
X
MLK
Z
J
This is proven by means of a purely geometric lemma
about obtuse simplicial cones, originally due to Langlands.
55
The rest of this talk will try to explain why the definition of truncation is reasonable, and why this theorem holds. Without, however, proving either of them!
56
A simple model for Eisenstein series
57
When trying to understand Arthur’s calculations, I
find it helpful to see what’s going on in a much simpler situation, one where combinatorial difficulties
are isolated from analytical ones.
58
and a root
]
Suppose given a split algebraic torus
associated to it.
system
]
Let
be the algebraic group generated by the torus
of . In some sense, this is
and the Weyl group
a reductive group in which the unipotent groups are
reduced to shadows.
The parabolic subgroups in this scheme are parametrized
by the faces of Weyl chambers.
^
Given a face , the associated group of rational
points is the subgroup generated by the torus and
of
whose elements fix the points
the subgroup
on the face.
_
59
`
The group
is just
.
The space
may be identified with the quotient of
by its torsion subgroup, which I will identify with
in which the (co)roots live.
the vector space
Automorphic functions are the characters of
that
-invariant, and the analogue of an Eisenstein
are
series is the finite sum
e
i g
j
h
g
Sna
g
Sha
c fed
ba
k
k
l
m
to one in
k
l
60
k
that maps a function in
.
p
o
The natural definition of truncation associated to a
in
is multiplication of a function
by the
point
characteristic function of the convex hull of .
o
T
o
If
is non-singular,
then the faces of
its convex hull
are parametrized by
the ‘parabolic subgroups’.
q
r
CT
61
s
The orthogonal decomposition is that corresponding
according to the nearest face of
to the partition of
the convex set
.
t
62
That this agrees with Arthur’s definition is not obvious.
The agreement of the two definitions is a actually a
special case of a much more general result about
convex polyhedra.
63
Back to Arthur’s truncation
64
Arthur’s truncation is, as I believe I have worked
out this week, a generalization of this theory to the
building of the rational group .
65
References
66
uv
J. Arthur, A trace formula for reductive groups I.
, Duke MathTerms associated to classes in
ematics Journal 45 (1978), 911–952.
, A trace formula for reductive groups II.
Applications of a truncation operator, Compositio
Mathematica 40 (1980), 87–121.
M. Brion and M. Vergne, Lattice points in simple
polytopes, Journal of the American Mathematical Society 10 (1997), 371–392.
M. Goresky, R. Kottwitz, and R. MacPherson, Discrete series characters and the Lefschetz formula
for Hecke operators, Duke Mathematics Journal 89
67
(1997), 477–554. Appendix B is the first place I am
aware of where Langlands’ combinatorial lemma is
formulated for general cones.
M. N. Ishida, Polyhedral Laurent series and Brion’s
inequalities, International Journal of Mathematics 1
(1990), 251–265.
J-P. Labesse, La formules de trace d’Arthur-Selberg,
Séminaire Bourbaki, 1984–85, exposé 636.
68
R. P. Langlands, Some lemmas to be applied to the
Eisenstein series, personal notes from around 1965
available at
w
x
|}
|
|
|
}
~
x
}
~ |}
x
{zy
x
w
, Eisenstein series, Proceedings of
Symposia in Pure Mathematics IX, 1965. This was
the Boulder conference. The relevant section is picturesquely called ‘
as the bed of Procrustes’.
69
