CHAPTER 9: APPLICATIONS OF LOCAL FIELDS
PETE L. CLARK
The topological groups GL
n
(
R
) and GL
n
(
C
) have an inexhaustibly rich structure
and importance in all parts of modern mathematics: analysis, geometry, topology,
representation theory, number theory
....
The serious study of these groups was al
ready begun in the 19th century by Lie and his contemporaries.
Somewhat more recently (say, about 1950) it has been realized that for a non
Archimedean locally compact field
K
, the groups GL
n
(
K
) also have a rich and
useful structure.
We will give some of this structure theory here: namely, we will classify the maxi
mal compact subgroups of GL
n
(
K
) for
K
a nondiscrete locally compact field. This
has immediate applications to the structure of finite subgroups of GL
n
(
Q
), which
are of intrinsic interest and are quite useful in areas like representation theory and
modular and automorphic forms. Moreover, this material (actually, a small piece of
it suffices) can be combined with a beautiful embedding theorem of J.W.S. Cassels
to deduce a celebrated 1960 theorem of A. Selberg: for any field
K
of characteristic
0, a finitely generated subgroup of GL
n
(
K
) is virtually torsionfree: i.e., has a finite
index subgroup without any nontrivial elements of finite order.
1.
General linear groups over locally compact fields
1.1. GL
n
(
K
)
is a locally compact group.
Let
K
be a nondiscrete locally com
pact field, and let
n
be a positive integer.
We consider the group GL
n
(
K
) of
invertible
n
×
n
matrices with coefficients in
K
. We wish to endow GL
n
(
K
) with
a natural locally compact topology. There are in fact two natural ways to do this,
which, happily, lead to the same result.
For any
n
∈
Z
+
, we endow the Cartesian product
K
n
with the product topology,
which of course makes it a locally compact topological group. We will sometimes
refer to this topology on
K
n
and other topologies induced from it as the
analytic
topology
, to distinguish it from the Zariski topology. (However, the reader need
not know what the Zariski topology is in order to read these notes.)
Let
M
n
(
K
) be the ring of
n
×
n
matrices with entries in
K
. As a
K
vector space,
M
n
(
K
)
∼
=
K
n
2
, and we give it the topology pulled back from the analytic topology
on
K
n
2
via the isomorphism.
(Easy exercise: the topology we get on
M
n
(
K
) is
independent of the chosen basis.)
Now GL
n
(
K
) is a subset of
M
n
(
K
).
We claim that in the induced (subspace)
topology it is locally compact, and indeed this is foisted off on the reader in the
1
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PETE L. CLARK
form of the following straightforward exercises.
Exercise 9.1: Let
P
(
t
1
, . . . , t
n
)
∈
K
[
t
1
, . . . , t
n
] be a polynomial, thought of as an
algebraic object.
Then
P
induces a function
P
:
K
n
→
K
in the usual way:
(
x
1
, . . . , x
n
)
→
P
(
x
1
, . . . , x
n
). Show that
P
is continuous for the analytic topolo
gies on
K
n
and
K
.
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 Fall '11
 Staff
 Logic, Geometry, Number Theory, Topology, Compact space, Lie group, GLN, pete l. clark, maximal compact subgroups

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