by Roger Howe
Group extensions and cohomology
Although he went on to deal with many other subjects, including ergodic theory and harmonic analysis on nilpotent and solvable groups (both discussed further below), Moore also continued to reflect on issues around cohomology theories for topological groups, and published two further papers (or four, depending on how you count [6], [11], [16]) relating to this subject, the last one fairly recently.
In back-to-back
papers
[11],
he revisited and substantially reformulated and
generalized the work of his thesis. A major aspect of the new
treatment was to expand the class of groups under consideration. In
the 1964 papers, the main object of consideration was a separable
locally compact group G, and a separable abelian locally compact group
Enlarging the class of allowable
The third property is another normalization axiom, requiring that
Central extensions and number theory
These results about group cohomology for locally compact groups, with their spaces of Borel or measurable functions, have an air of general topology about them, and may seem rather abstract and removed from the examples that have been of greatest interest in representation theory, such as algebraic groups, especially reductive groups, over local or global fields. However, in [6], a work with a different flavor, Moore investigated central extensions of such groups, and found remarkable connections with number theory, particularly the Artin reciprocity map of class field theory. Along with [3], this research showcases Moore’s pathbreaking work in the setting of adelic groups.
The main object of study in
[6]
is the notion of “fundamental
group” for Chevalley (simple, split algebraic) groups over local
fields. For connected topological groups, there is a natural notion of
fundamental group in the sense of covering spaces. For an abstract
group
For topological groups, one should consider central extensions
There are also notions of “connected” and “simply connected” in
the context of algebraic groups, which equate to the like-named
topological notions when the base field is
He then considers Chevalley groups — simple, split algebraic
groups
Steinberg
[e6]
had given an explicit construction by
generators and relations, of the universal central extension
The question thus becomes, what relations must be satisfied by the
generators
Steinberg cocycles in fact had been encountered before, in a seemingly
different context, that of class field theory
[e22].
The norm
residue symbol
Although the notation above refers to the group
Moore goes on to consider the case of a global field
Ergodic theory
Moore was an early exponent of using representation theory to study
ergodic theory. The possibility for doing this arises from the fact
that ergodic properties of a transformation
Recall the basic framework. Let
The transformation
Guided by these connections, in
[4],
Moore considers the
following properties of a (closed) subgroup
has property if, in every unitary representation of any vector that is fixed by all the operators from is also fixed by all the operators from has property if, in every unitary representation of any finite-dimensional subspace that is invariant by all the operators from is also invariant by all the operators from
Given these definitions, he shows that, if
Focusing on the situation when the large group
Note that, if a one parameter group
A general connected semisimple Lie group
Let
Moore shows:
Let
Harmonic analysis on nilpotent and solvable groups
Moore’s work on representation theory dealt with a number of issues for general groups (see [1], [7], [9], [10], [12]), and also dealt in detail with aspects of representation theory of solvable groups (see [5], [3], [8]). We briefly discuss [5], and then look at [3] in more detail.
Representation theory of solvable groups has been built around the
orbit method, first enunciated by
A. A. Kirillov
for nilpotent groups
[e5].
Kirillov’s main result is that the unitary dual of a (connected,
simply connected) Lie group
Kirillov also gave a reasonably explicit recipe for realizing the
representation
contains the Lie algebra of the stabilizer of under ; and is subordinate to meaning that (which implies, since for a group acting by unipotent operators, the stabilizer of any point is connected, that can be exponentiated to define a one-dimensional unitary character of the group ), and is of maximal dimension subject to (ii).
Condition (iii) implies that
Such a subalgebra m is called a maximal subordinate
subalgebra, or a polarizing subalgebra, or simply a
polarization, for
Kirillov’s theory was gradually adapted to larger and larger classes
of solvable groups, until the representation theory of the general
connected solvable Lie group was described by the orbit method. The
first extension was to exponential solvable groups, by a French team
led by
P. Bernat
[e8],
[e13].
An exponential solvable
group is a connected group
The next major step beyond exponential solvable groups was taken by Moore and Louis Auslander [5]. They studied solvable groups of “type R”, which are in some sense the opposite of exponential. One definition of type R is that the roots of the adjoint action for the Lie algebra on itself are all pure imaginary. Here there were several new conceptual issues to overcome, including the fact that polarizations might not exist, and even if they did, the linear functional might not exponentiate to a character of the polarizing group. Also, the key regularity condition of being type I, in the sense of von Neumann algebras generated by unitary representations of the group [e19], which always holds for exponential groups [e4], can fail for type R groups.
However, [5] establishes several notable results, including a criterion for when such a group is type I, and that, when this happens, a description of the construction of irreducible unitary representations. They also investigate the stronger regularity condition of being CCR (see [5], Chapter 5), and show that a solvable group of type R that is type I is also CCR. By contrast, nonnilpotent exponential groups, although always of type 1, are never CCR.
After [5], the theory was pushed further by J. Brezin [e11], who found a criterion for a general solvable group to be type I. Then Auslander and Kostant [e10], [e16] showed how to introduce symplectic geometry ideas, and also a more general notion of polarization, to create an account that, while noticeably more complex, retains the essential flavor of Kirillov’s account of nilpotent groups. Over roughly the following decade, the theory was tied up nicely in several follow-on papers of Pukanszky; see, for example, [e9], [e17].
Classifying irreducible representations
is only the start of harmonic
analysis. It is also of considerable interest to describe explicitly
the unitary representations that appear “in nature”. In particular,
if
Moore considered coset spaces
Associated to
A standard part of the construction of adeles is that the natural
embeddings
Moore observes that the orbit method applies more or less without
change to all the local groups
The more naive question, to describe the multiplicities of
representations in
As noted, the exponential map
On the other hand, it may happen that
For lattice subgroups, the hoped-for parallel with the abelian case
holds, in the sense that a representation