#### by Roger Howe

#### Group extensions and cohomology

__\( G \)__to be represented is a semidirect product

__\( G \simeq H\ltimes N, \)__where

__\( N \)__is a normal subgroup of

__\( G, \)__the problem of computing representations of

__\( G \)__from those of

__\( N \)__may involve cyclic central extensions of

__\( H, \)__and these are described by (an appropriate topological version of) second degree 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
__\( A, \)__ on which __\( G \)__ acts by automorphisms (a locally compact
__\( G \)__-module). In the 1976 papers, although __\( G \)__ remains locally compact,
the group __\( A \)__ is allowed to be more general. It can be any topological
group whose topology is equivalent to a separable complete metric
topology (the class of polonais groups). So for example, __\( A \)__ could be
the additive group of the Banach space underlying a (strongly
continuous) representation of __\( G \)__ (in particular, of a unitary
representation of __\( G \)__).

Enlarging the class of allowable __\( A \)__ permitted several important
constructions that made the resulting cohomology theory more flexible,
more applicable, and also better defined, in the sense that it could
be characterized by a set of three easily stated axioms. The first two
axioms ask for standard expected features of cohomology. The first
requires the expected functorial properties of the cohomology groups
__\( H^n (G, A), \)__ including coboundary operators __\( \partial_{n} : H^n (G,
A^{\prime\prime}) \rightarrow H^{n+1} (G, A^{\prime}) \)__ that create a long exact sequence of
cohomology groups out of a short exact sequence
__\[ 1 \rightarrow A^{\prime}
\rightarrow A \rightarrow A^{\prime\prime} \rightarrow 1 \]__
of admissible
__\( G \)__-modules. The second is the normalization property that __\( H^0 (G, A)
= A^G, \)__ the subgroup of __\( G \)__-invariant elements of __\( A. \)__

The third property is another normalization axiom, requiring that __\( H^n
(G, I(A)) = 0 \)__ for __\( n > 0, \)__ where __\( A \)__ is a polonais abelian group, and
__\( I(A) \)__ is the “regular representation” of __\( G \)__ on Borel (or
measurable, modulo the usual condition of equivalence up to measure
zero — either choice leads to a satisfactory theory) functions
from __\( G \)__ to __\( A. \)__ (Note that if __\( G \)__ is infinite, then even if __\( A \)__ is
locally compact, __\( I(A) \)__ will not be, but it will be polonais.) This
axiom allows dimension shifting, and for __\( n > 0, \)__ allows one to
determine the __\( H^n (G, A) \)__ in terms of __\( H^m (G, B) \)__ for other __\( B \)__ and
__\( m < n, \)__ and so transfers uniqueness to higher cohomology. In the first
of the two papers comprising
[11],
Moore considers the two different cases of __\( I(A) \)__ as
described above, and shows that they both satisfy the axioms, and thus
give rise to the same cohomology theory. The third axiom is the
challenging issue, and requires an argument closely related to the
Mackey Imprimitivity Theorem
[e2].

#### 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 __\( G \)__ that is “perfect”, i.e., equal to its own commutator
subgroup, Schur
[e1]
had shown that a
good theory of central extensions exists; in particular, there is a
“universal central extension” __\( \tilde{G}, \)__ that can map onto any
given central extension. The kernel __\( \pi_1 (G) \)__ of the covering map
__\[
1 \rightarrow \pi_1 (G) \rightarrow \tilde{G} \rightarrow G \rightarrow 1
\]__
is the “Schur fundamental group” of __\( G. \)__ Given some abelian group
__\( A, \)__ the central extensions of __\( G \)__ by __\( A \)__ are then classified by
__\[
H^2 (G, A) = \operatorname{Hom}(\pi_1 (G), A).
\]__

For topological groups, one should consider central extensions
__\[
1 \rightarrow A \rightarrow E \rightarrow G \rightarrow 1
\]__
with all
members of the sequence locally compact, and the mappings continuous.
These extensions are classified by __\( H^2_{\mathrm{top}} (G, A), \)__ with
these groups being as defined in Moore’s first papers. It may not be
the case that __\( \pi_1^{\text{top}} (G), \)__ a universal topological
central extension, exists, but if it does, and if __\( G \)__ is perfect, then
one can compare __\( \pi^{\text{top}}_1 (G) \)__ with __\( \pi_1 (G), \)__ the Schur
fundamental group. The former will be a quotient of the latter.

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 __\( C. \)__ Suppose that __\( G \)__ is an
algebraic group that is connected and simply connected in the sense of
algebraic groups. Moore first shows that in almost all cases, __\( G_k, \)__
the group of __\( k \)__-rational points of __\( G, \)__ is perfect, i.e., connected
in the algebraic sense.

He then considers Chevalley groups — simple, split algebraic
groups __\( G, \)__ defined over a field __\( k, \)__ which he restricts to be a local
field. He asks: is the group __\( G_k \)__ of __\( k \)__-rational points of __\( G, \)__
simply connected as an abstract group or topological group? And if
not, what is __\( \pi_1 (G_k), \)__ and what is __\( \pi_1^{\text{top}} (G_k) \)__?

Steinberg
[e6]
had given an explicit construction by
generators and relations, of the universal central extension
__\( \tilde{G}_k, \)__ using the Bruhat decomposition of __\( G, \)__ relative to a
selected __\( k \)__-split torus __\( H \)__ in __\( G. \)__ Implicit in Steinberg’s work is
the construction of a cocycle
__\[ c : k^{\ast} \times k^{\ast}
\rightarrow \pi_1 (G k ), \]__
such that the values __\( c(x, y) \)__ for __\( x \)__
and __\( y \)__ in __\( k^{\ast}, \)__ generate __\( \pi_1 (G_k). \)__ (Strictly speaking,
there is one cocycle __\( c_{\alpha} \)__ for each long root in the root
system of __\( G \)__; but these are almost independent of __\( \alpha. \)__)

The question thus becomes, what relations must be satisfied by the
generators __\( c(x, y) \)__? Moore gives a list of four types of identities.
(One of the identities is closely related to bilinearity, but somewhat
more complicated. However, Moore shows by careful investigation of
root systems that for all simple groups except symplectic groups, it
can be replaced by bilinearity.) He calls continuous functions __\( c \)__
with values in a given (locally compact, separable) abelian group __\( A \)__
and satisfying his four identities, “Steinberg cocycles” with values
in __\( A, \)__ denoted __\( S^G (k, A). \)__ The subgroup of Steinberg cocycles that
are continuous as functions on __\( k^{\ast} \times k^{\ast} \)__ are called
“topological”, and denoted __\( S_{\text{top}}^G (k,A). \)__ Moore
constructs
embeddings
__\begin{align*}
& \phi : H^2 (G_k, A)\rightarrow S^G(k,A) \quad \text{and}\\
& \phi_{\text{top}}: H^2_{\text{top}} (G_k , A) \rightarrow
S_{\text{top}}^G (k, A).
\end{align*}__
He shows that for __\( G = \operatorname{SL}_2, \)__ these maps
are isomorphisms, and conjectures the same for general groups. This
was established by
H. Matsumoto
[e12].

Steinberg cocycles in fact had been encountered before, in a seemingly
different context, that of class field theory
[e22].
The norm
residue symbol __\( \nu_k : k^{\ast} \times k^{\ast} \rightarrow \mu_k , \)__
where __\( \mu_k \)__ is the group of roots of unity of __\( k, \)__ provides a
continuous Steinberg cocycle. Moore proves that it is essentially the
only one: he shows that for a nonarchimedean local field with roots
of unity __\( \mu_k, \)__ the map __\( f \rightarrow f \circ \nu_k \)__ defines an
isomorphism __\( \operatorname{Hom}(\mu_k , A) \simeq S_{\text{top}}^G (k, A). \)__ It follows that __\( \pi_1^{\text{top}} (G_k )\simeq \mu_k. \)__ In
particular, __\( \pi_1^{\text{top}} (G_k) \)__ exists.

Although the notation above refers to the group __\( G, \)__ in fact, the
relations are almost independent of __\( G. \)__ Only because bilinearity of
the Steinberg cocycles fails for symplectic groups is there any
dependence on the group. Moore notes that this exception permits the
existence of the “metaplectic cover” of __\( \operatorname{Sp}_{2n}, \)__ which intervenes
in the “Weil representation”
[e7]
(aka “oscillator
representation”), which links representation theory with both quantum
mechanics and automorphic forms.

Moore goes on to consider the case of a global field __\( k, \)__ and its ring
__\( A \)__ of adèles. General considerations on restricted products imply
that __\( \pi^{\text{top}}_1 (G_A) \)__ exists, and is the restricted product
__\( \prod\pi^{\text{top}}_1 (G_\nu), \)__ where __\( G_\nu = G_{k_\nu}, \)__ and
__\( \nu \)__ runs over all places of __\( k. \)__ In this context, Moore proves a
second basic theorem: the sequence
__\[ \pi_1(G_k) \rightarrow
\pi^{\text{top}}_1 (G_A)\rightarrow \mu_k \rightarrow 1 \]__
is exact.
This in particular proves the essential uniqueness of the Artin
reciprocity law of abelian class field theory. It also classifies
topological central extensions of __\( G_A \)__ that split over __\( G_k. \)__ The
primary example of this is the metaplectic cover of __\( \operatorname{Sp}_{2n}, \)__
mentioned above as intervening in the construction of the
Weil/oscillator representation.

#### 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 __\( T \)__ can be reinterpreted
in terms of spectral theory. This equivalence is explained, for
example, in
Halmos’
well-known text
[e3]
on ergodic theory.

Recall the basic framework. Let __\( X \)__ denote a measure space with total
measure equal to 1 (aka, a probability space). For a measurable subset
__\( A \subset X, \)__ denote the measure of __\( A \)__ by __\( \mu(A). \)__ Let __\( T : X
\rightarrow X \)__ be an invertible transformation of __\( X \)__ that preserves
the measure, meaning that __\( \mu(T (A)) = \mu(A) \)__ for any measurable
subset __\( A \)__ of __\( X. \)__ The operators __\( T^n, \)__ for __\( n \in \mathbb{Z} \)__ then
form a group of transformations on __\( X, \)__ and ergodic theory is
concerned with the “long term” behavior of this action, that is,
what happens as __\( n \rightarrow \infty. \)__ There is also interest in
one-parameter groups __\( T_t \)__ of measure-preserving transformations, for
__\( t \in \mathbb{R}, \)__ with __\( T_s \circ T_t =
T_{s+t}. \)__ The definitions and properties below can be adapted to this
case, and indeed, to the action of any group on __\( X \)__ by means of
measure-preserving transformations.

The transformation __\( T \)__ is called *ergodic* provided that
there is no (measurable) __\( T \)__-invariant subset __\( A \)__ in __\( X \)__ with measure
strictly between 0 and __\( 1. \)__ If __\( T \)__ is ergodic, it is called *weakly mixing* if
the transformation __\( T \times T \)__ on __\( X \times X \)__ is also ergodic. The
transformation __\( T \)__ is called *strongly mixing* if a subset __\( A \)__ of __\( X \)__
gets spread out in a uniform way under successive applications of __\( T. \)__
Formally, this is expressed by the requirement that the measure
__\( \mu(T^n (A) \cap B), \)__ of the intersection of a multiply transformed
set with some other set, approaches the expectation dictated by
randomness:
__\[
\lim_{\mu\to\infty} \mu(T^n (A) \cap B) = \mu(A) \cdot
\mu(B).
\]__
As Halmos explains, these properties can all be
reformulated in terms of operator theory. From the transformation __\( T \)__
of __\( X, \)__ define an operator __\( U_T \)__ on the Hilbert space __\( L^2 (X) \)__ of
square integrable measurable functions on __\( X, \)__ by
the recipe
__\[
U_T (f)(x) = f (T^{-1} (x)),
\]__
for __\( f \)__ in __\( L^2 (X) \)__
and __\( x \in X. \)__ It is
straightforward to verify that __\( U_T \)__ is a unitary operator. Similar
definitions apply to a one-parameter group __\( T_s \)__ of transformations.
Using these correspondences, one can check that ergodicity of __\( T \)__ is
equivalent to the statement that the eigenspace of __\( U_T \)__ for the
eigenvalue 1 is one-dimensional, and consists of the constant
functions. Weak mixing corresponds to __\( U_T \)__ not having any
eigenvectors for any eigenvalue, other than the constant functions.
Strong mixing corresponds to the requirement that matrix coefficients
of functions orthogonal to the constant functions (i.e., with integral
zero), should “vanish at infinity”:
__\[
\lim_{n\to\infty} (\xi_{\phi,\psi} (n) = ((U_T)^n (\phi), \psi)) =
0,
\]__
for any two functions __\( \phi \)__ and __\( \psi \)__ in __\( L^2 (X) \)__ that are
orthogonal to the constant functions. Again, there are analogous
definitions for one-parameter groups, and in fact, for any locally
compact group of measure preserving transformations.

Guided by these connections, in
[4],
Moore considers the
following properties of a (closed) subgroup __\( H \)__ of a (locally compact)
group __\( G \)__:

__\( H \)__has property__\( (E) \)__if, in every unitary representation of__\( G, \)__any vector that is fixed by all the operators from__\( H, \)__is also fixed by all the operators from__\( G. \)____\( H \)__has property__\( (W M ) \)__if, in every unitary representation of__\( G, \)__any finite-dimensional subspace that is invariant by all the operators from__\( H, \)__is also invariant by all the operators from__\( G. \)__

Given these definitions, he shows that, if __\( X\simeq G/\Gamma \)__ is a
homogeneous space for __\( G \)__ with finite __\( G \)__-invariant measure, and __\( H \)__
is a subgroup of __\( G, \)__ then property __\( (E) \)__ for __\( H \)__ implies that the
action of __\( H \)__ on __\( X \)__ is ergodic, and property __\( (W M ) \)__ for __\( H \)__ implies
that the action of __\( H \)__ on __\( X \)__ is weakly mixing. In case __\( H \)__ is a
singly generated subgroup or a one-parameter subgroup, he proves an
analogous statement about strong mixing.

Focusing on the situation when the large group __\( G \)__ is a semisimple Lie
group, which provides a large family of interesting examples,
especially by allowing __\( \Gamma \)__ to be an arithmetic subgroup, Moore
proves an essentially optimal result about ergodicity of one-parameter
group actions.

Note that, if a one parameter group __\( H \subset G \)__ is in fact compact,
in other words, is a copy of the unit circle __\( \mathbb{T} \)__ in the complex plane,
then one knows that any representation of __\( H \)__ will decompose into a
sum of one-dimensional representations. A bit more subtly, if __\( H \)__ is a
winding line in a compact torus, i.e., if __\( \overline{H}, \)__ the closure of __\( H \)__ in
__\( G, \)__ is compact, then a similar conclusion holds. Moore formulates a
general condition on __\( H \)__ that precludes these situations: the notion
of *total noncompactness*.

A general connected semisimple Lie group __\( G \)__ is almost a product of
almost simple groups. Precisely, the adjoint group of __\( G, \)__ i.e., the
quotient __\( G/Z(G), \)__ where __\( Z(G) \)__ is the center of __\( G, \)__ is a product of
simple groups. In other words, given a semisimple Lie group __\( G, \)__ there
__\( Q \)__ is a finite number of simple Lie groups __\( G_j, \)__ for __\( 1 \leq j \leq
r, \)__ such that __\( G \)__ maps onto the product group __\( G \rightarrow
\prod_{j=1}^r G_j , \)__ with discrete, central kernel. Given a subgroup
__\( H \subset G, \)__ we can look at the image __\( H_j \)__ of __\( H \)__ in each one of
the simple factors __\( G_j. \)__ We say that __\( H \)__ is totally noncompact in __\( G \)__
if the closure of each __\( H_j \)__ in __\( G_j \)__ is noncompact. Moore
gives in
[4]
the following sharp description of the spectral
behavior of a totally noncompact one-parameter subgroup.

Let __\( L^+ = L^2 (\mathbb{R}^+) \)__ be the __\( L^2 \)__ space of the positive
half-line, and let __\( L^{-} = L^2 (R^{-}) \)__ be the __\( L^2 \)__ space of the
negative half line. The one parameter group of multiplication
operators __\( U_t \)__ on __\( L^2 (\mathbb{R}), \)__ defined by
__\[
U_t (f )(x) = e^{2\pi itx} f (x),
\]__
for __\( f \)__ in __\( L^2 (\mathbb{R}), \)__ and __\( x \in
\mathbb{R}, \)__ preserves both of __\( L^{\pm}, \)__ and its infinitesimal
generator __\( 2\pi ix \)__ has purely positive imaginary spectrum on __\( L^+, \)__
and purely negative imaginary spectrum on __\( L^-. \)__

Moore shows:

Let __\( H \)__ be a totally noncompact one-parameter subgroup of the
semisimple Lie group __\( G, \)__ and let __\( \pi \)__ be a unitary representation of
__\( G. \)__ Then the restriction of __\( \pi \)__ to __\( H \)__ is unitarily equivalent to
multiplication by __\( e^{2\pi ix} \)__ on
__\[
m^+ L^+ \oplus m^{-} L^{-},
\]__
where __\( m^{\pm} \)__ are nonnegative integers, or __\( +\infty. \)__ He also shows
by example that all values of __\( m^{\pm} \)__ can occur. (In some sense, the typical case is that both of
__\( m^{\pm} \)__ equal __\( +\infty. \)__) This result implies that the action of __\( H \)__
on any homogeneous probability space for __\( G \)__ will be strongly mixing,
but it is considerably more refined. As noted, strong mixing of a
one-parameter group of transformations is equivalent to the condition
that the matrix coefficients of the group vanish at infinity. Somewhat
later, in
[15]
Moore proved that, for the geodesic flow on
certain homogeneous spaces, appropriate matrix coefficients would
actually decay exponentially as the group parameter goes to infinity.
In between, I had the pleasure of collaborating with Moore on a result
that proves ergodicity, and indeed, strong mixing, for a large class
of actions of fairly general groups. Matrix coefficients can be
studied for a wide class of groups. Given a locally compact group __\( G \)__
and a unitary representation __\( \pi \)__ of __\( G \)__ on the Hilbert space __\( H \)__
with scalar product __\( (\phi, \psi) \)__ for vectors __\( \phi \)__ and __\( \psi \)__ in
__\( H, \)__ we can consider the matrix coefficients
__\[
\xi_{\phi,\psi} (g) = (\pi(g)(\phi), \psi)
\]__
as functions of __\( g \in G. \)__ We will say that
the matrix coefficients of __\( \pi \)__ vanish at infinity if, for fixed
__\( \phi \)__ and __\( \psi, \)__ and any __\( \epsilon > 0, \)__ there is a compact set __\( J
\subset G \)__ such that __\( |\xi_{\phi,\psi} (g)| < \epsilon \)__ for __\( g \)__ not in
__\( J. \)__ The paper
[13]
establishes that, barring some obvious
conditions that would prevent it, for a very large class of
representations of a large class of Lie groups, the matrix
coefficients vanish at infinity. This result has been widely used by
workers studying dynamical systems on homogeneous spaces, and even
beyond (see, e.g.,
[e25],
[e21],
[e24],
[e23]).

#### 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 __\( N \)__ is naturally parametrized by the set
__\( \{n^{\ast} /Ad^{\ast} N \} \)__ of orbits the for action of __\( N \)__ on the
dual space __\( n^{\ast} \)__ of linear functionals on the Lie algebra __\( n \)__ of
__\( N, \)__ by the dual of the standard action __\( AdN \)__ on __\( n \)__ by conjugation.
This is known as the *coadjoint* action of __\( N, \)__ and is denoted
__\( Ad^{\ast} N \)__; its orbits are often referred to as *coadjoint
orbits*.

Kirillov also gave a reasonably explicit recipe for realizing the
representation __\( \rho_{\lambda} \)__ corresponding to the orbit
__\( O_{\lambda} \)__ of an element __\( \lambda \)__ of __\( n^{\ast}. \)__ Given __\( \lambda, \)__
one can find Lie subalgebras __\( m \subset n, \)__ such that

__\( m \)__contains the Lie algebra__\( n_\lambda \)__of the stabilizer of__\( \lambda \)__under__\( Ad^\ast N \)__; and__\( m \)__is*subordinate*to__\( \lambda, \)__meaning that__\( \lambda_{|[m,m]} = 0 \)__(which implies, since for a group acting by unipotent operators, the stabilizer of any point is connected, that__\( \lambda \)__can be exponentiated to define a one-dimensional unitary character__\( \chi_\lambda \)__of the group__\( M = \exp m \)__), and__\( m \)__is of maximal dimension subject to (ii).

Condition (iii) implies that
__\begin{align*}
\dim n - \dim m & = \dim m - \dim n_{\lambda} \\
& = \tfrac{1}{2} (\dim n - \dim n_{\lambda}) = \tfrac{1}{2}
\dim O_{\lambda}.
\end{align*}__
(In fact, condition (iii) makes condition (i)
redundant; but it is perhaps desirable to make (i) explicit.)

Such a subalgebra m is called a *maximal subordinate
subalgebra*, or a *polarizing subalgebra*, or simply a
*polarization*, for __\( \lambda. \)__ Given a polarization __\( m \)__ for
__\( \lambda, \)__ the representation __\( \rho_{\lambda} \)__ of __\( N \)__ is realizable as
the induced
representation
__\[
\operatorname{ind}^N_M \chi_{\lambda} .
\]__
Implicit in this
statement, of course, is the fact that this induced representation is
independent of the choice of polarization __\( m \)__ of __\( \lambda, \)__ and
proving this was an important part of Kirillov’s work.

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 __\( S, \)__ with Lie algebra __\( s, \)__
for which
the exponential
map
__\[
\exp: s \to S
\]__
is a diffeomorphism. Here the theory went
through almost unchanged, although a slight refinement was needed in
the definition of polarization.

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 __\( H \subset G \)__ is a subgroup, the question of describing the
representations that appear in the natural action of __\( G \)__ on the
homogeneous space __\( G/H \)__ presents itself in many situations.
Particularly interesting examples are the cases when __\( H \)__ is a discrete
subgroup and __\( G/H \)__ is compact. (We say, “__\( H \)__ is cocompact.”) Then
the representation of __\( G \)__ on __\( L^2 (G/H) \)__ decomposes into a direct sum
of irreducible representations, each of which appears with finite
multiplicity. These multiplicities may have arithmetic significance.
When __\( G \)__ is a reductive algebraic group and __\( H \)__ is an arithmetic
subgroup, we are in the domain of automorphic forms, which have been
important for number theory for several hundred years, and a subject
of particularly intense study since the 1960s.

Moore considered coset spaces __\( G/H \)__ with __\( G = N, \)__ a nilpotent Lie
group (taken to be connected and simply connected), and __\( H = \Gamma \)__ a
discrete cocompact subgroup. In this situation, it can be shown that
__\( N \)__ can be identified with the completion __\( N_{\mathbb{R}} \)__ over
__\( \mathbb{R} \)__ of a unipotent algebraic group __\( N_{\mathbb{Q}} \)__ over
__\( \mathbb{Q}, \)__ the rational numbers. Thus, one can consider at the same
time the completion __\( N_p \)__ of __\( N_{\mathbb{Q}} \)__ over the field of
__\( p \)__-adic numbers for any prime __\( p \)__; and further one can construct
__\( N_{\mathbb{A}}, \)__ the adelic points of __\( N_{\mathbb{Q}}, \)__ as the
restricted direct product of the __\( N_p. \)__ Moore did this; he was thus a
pioneer in investigating the adelic viewpoint, which since has become
standard in the theory of automorphic forms.

Associated to __\( N_{\mathbb{Q}}, \)__ there is a Lie algebra
__\( n_{\mathbb{Q}}, \)__ with all the appropriate functorial properties,
including an exponential map
__\( \exp : n_{\mathbb{Q}}\to N_{\mathbb{Q}}, \)__ that is in fact a
polynomial mapping, with polynomial inverse, and which defines
exponential maps for all the local completions of __\( n_{\mathbb{Q}} \)__ and
__\( N_{\mathbb{Q}}, \)__ and for the adelic groups as well. Further, the dual
space __\( n^{\ast}_{\mathbb{Q}} \)__ localizes to the dual spaces over all
completions of __\( Q, \)__ and these can be assembled into a global dual
__\( n^{\ast}_{\mathbb{A}}. \)__

A standard part of the construction of adeles is that the natural
embeddings __\( n_{\mathbb{Q}}\hookrightarrow n_{\mathbb{Q}_p} \)__ can be
assembled into a global embedding __\( n_{\mathbb{Q}}\hookrightarrow
n_{\mathbb{A}}, \)__ with image that is discrete and cocompact. The
parallel facts hold also for the group __\( N_{\mathbb{Q}} \)__ and the dual
space __\( n^{\ast}_{\mathbb{Q}}. \)__

Moore observes that the orbit method applies more or less without
change to all the local groups __\( N_{\mathbb{Q}_p}, \)__ and that these can
be assembled into an orbital description of the unitary dual of
__\( N_{\mathbb{A}}. \)__ In __\( n^{\ast}_{\mathbb{A}}, \)__ one has a collection of
rational points, __\( n^{\ast}_{\mathbb{Q}} \)__; their __\( Ad^{\ast}
N_{\mathbb{A}} \)__ orbits provide a collection of “rational orbits” in
__\( n^{\ast}_{\mathbb{A}}. \)__ A lovely primary result of
[3]
is that
the Hilbert space __\( L^2 (N_{\mathbb{A}} /N_{\mathbb{Q}}) \)__ decomposes
into a multiplicity-free sum of representations parametrized by the
rational orbits in __\( n^{\ast}_{\mathbb{A}}. \)__

The more naive question, to describe the multiplicities of
representations in __\( L^2 (N_{\mathbb{R}}/\Gamma), \)__ turns out to be
trickier to answer. One might hope for something analogous to the
situation when __\( N \)__ is actually commutative, so that __\( \Gamma \)__ is a
lattice in __\( N, \)__ and the (one-dimensional) representations that appear
are given by the points in the dual lattice __\( \Gamma^{\ast} \subset
N^{\ast}. \)__ This is approximately the right idea, but there are a
number of complications to be addressed before the correct analog can
be formulated.

As noted, the exponential map
__\( \exp : n \to N \)__ is a global diffeomorphism. Its inverse will be
denoted by __\( \log. \)__ Given a discrete subgroup __\( \Gamma \subset N, \)__ the set
__\( \log \Gamma \subset n \)__ need not be closed under addition in __\( n \)__; and
likewise, given a lattice __\( L \subset n, \)__ its image __\( \exp(L) \)__ need not
be a subgroup of __\( N. \)__ There are examples of both failures already in
the three-dimensional Heisenberg group.

On the other hand, it may happen that __\( \log\Gamma \)__ is a lattice. Moore
then calls __\( \Gamma \)__ a lattice subgroup. He also shows that, for any
__\( \Gamma, \)__ there are lattice subgroups __\( \Gamma^{\prime} \subset \Gamma \subset
\Gamma^{\prime\prime}, \)__ such that the index __\( \Gamma^{\prime\prime} /\Gamma^{\prime} \)__ is bounded
independent of __\( \Gamma \)__ (for given __\( N \)__).

For lattice subgroups, the hoped-for parallel with the abelian case
holds, in the sense that a representation __\( \rho_O \)__ corresponding to a
coadjoint orbit __\( O \subset n^{\ast} \)__ appears with positive
multiplicity in __\( L^2 (N_{\mathbb{R}} /\Gamma) \)__ if and only if __\( O \cap
(\log\Gamma)^{\bot} \)__ is nonempty, where __\( (\log\Gamma)^{\bot} \subset
n^{\ast} \)__ is the dual lattice to __\( \log\Gamma. \)__ However, the
multiplicity with which such a representation __\( \rho_O \)__ appears in __\( L^2
(N_{\mathbb{ R}} /\Gamma) \)__ is somewhat tricky to compute. It is not
something relatively straightforward, such as the cardinality of the
intersection __\( O \cap (\log\Gamma)^{\bot}, \)__ or the number of __\( Ad^{\ast}
\Gamma \)__ orbits in this set. The exact description of the multiplicity
was given in
[e15]
and
[e14].
It requires selecting an
appropriate polarization. The paper
[e14]
also related
this multiplicity to the adelic representation on
__\( L^2 (N_{\mathbb{ A}} /N_{\mathbb{Q}} ), \)__ underscoring the
appropriateness of Moore’s consideration of the global situation.