Celebratio Mathematica

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].

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.

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]).

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

1. \( m \) con­tains the Lie al­gebra \( n_\lambda \) of the sta­bil­izer of \( \lambda \) un­der \( Ad^\ast N \); and
2. \( m \) 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
3. \( m \) 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.