# Celebratio Mathematica

## Calvin C. Moore

### The mathematical work of Calvin C. Moore

#### Group extensions and cohomology

Calv­in Moore was a doc­tor­al stu­dent of George Mackey at Har­vard. His early pa­pers [2] deal with group ex­ten­sions and low di­men­sion­al co­homo­logy of to­po­lo­gic­al groups, and come dir­ectly out of his thes­is. They re­flect Mackey’s in­terest in de­scrib­ing rep­res­ent­a­tions of group ex­ten­sions, which of course in­volves ques­tions about the struc­ture of group ex­ten­sions. Even when the group $$G$$ 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

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.

### Works

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[2] C. C. Moore: “Ex­ten­sions and low di­men­sion­al co­homo­logy the­ory of loc­ally com­pact groups, I & II,” Trans. Amer. Math. Soc. 113 (1964), pp. 40–​63; 64–​86. MR 171880 Zbl 0131.​26902 article

[3] C. C. Moore: “De­com­pos­i­tion of unit­ary rep­res­ent­a­tions defined by dis­crete sub­groups of nil­po­tent groups,” Ann. of Math. (2) 82 (1965), pp. 146–​182. MR 181701 Zbl 0139.​30702 article

[4] C. C. Moore: “Er­godi­city of flows on ho­mo­gen­eous spaces,” Amer. J. Math. 88 (1966), pp. 154–​178. MR 193188 Zbl 0148.​37902 article

[5] L. Aus­lander and C. C. Moore: Unit­ary rep­res­ent­a­tions of solv­able Lie groups. Mem. Amer. Math. Soc. 62. Amer. Math. Soc. (Provid­ence, RI), 1966. MR 207910 Zbl 0204.​14202 book

[6] C. C. Moore: “Group ex­ten­sions of $$p$$-ad­ic and ad­el­ic lin­ear groups,” Inst. Hautes Études Sci. Publ. Math. 35 (1968), pp. 157–​222. MR 244258 Zbl 0159.​03203 article

[7] C. C. Moore: “Groups with fi­nite di­men­sion­al ir­re­du­cible rep­res­ent­a­tions,” Trans. Amer. Math. Soc. 166 (1972), pp. 401–​410. MR 302817 Zbl 0236.​22010 article

[8] C. C. Moore and J. A. Wolf: “Square in­teg­rable rep­res­ent­a­tions of nil­po­tent groups,” Trans. Amer. Math. Soc. 185 (1973), pp. 445–​462. MR 338267 Zbl 0302.​43014 article

[9] M. Du­flo and C. C. Moore: “On the reg­u­lar rep­res­ent­a­tion of a nonun­im­od­u­lar loc­ally com­pact group,” J. Func­tion­al Ana­lys­is 21 : 2 (1976), pp. 209–​243. MR 393335 Zbl 0317.​43013 article

[10] C. C. Moore and J. Rosen­berg: “Groups with $$T_{1}$$ prim­it­ive ideal spaces,” J. Func­tion­al Ana­lys­is 22 : 3 (1976), pp. 204–​224. MR 419675 Zbl 0328.​22014 article

[11] C. C. Moore: “Group ex­ten­sions and co­homo­logy for loc­ally com­pact groups,” Trans. Amer. Math. Soc. 221 : 1 (1976), pp. 1–​33; 35–​58. MR 414775 Zbl 0366.​22006 article

[12] C. C. Moore and R. J. Zi­m­mer: “Groups ad­mit­ting er­god­ic ac­tions with gen­er­al­ized dis­crete spec­trum,” In­vent. Math. 51 : 2 (1979), pp. 171–​188. MR 528022 Zbl 0399.​22005 article

[13] R. E. Howe and C. C. Moore: “Asymp­tot­ic prop­er­ties of unit­ary rep­res­ent­a­tions,” J. Func­tion­al Ana­lys­is 32 : 1 (1979), pp. 72–​96. MR 533220 Zbl 0404.​22015 article

[14] C. C. Moore: “The Maut­ner phe­nomen­on for gen­er­al unit­ary rep­res­ent­a­tions,” Pa­cific J. Math. 86 : 1 (1980), pp. 155–​169. MR 586875 Zbl 0446.​22014 article

[15] C. C. Moore: “Ex­po­nen­tial de­cay of cor­rel­a­tion coef­fi­cients for geodes­ic flows,” pp. 163–​181 in Group rep­res­ent­a­tions, er­god­ic the­ory, op­er­at­or al­geb­ras, and math­em­at­ic­al phys­ics (Berke­ley, Cal­if., 1984). Edi­ted by C. C. Moore. Math. Sci. Res. Inst. Publ. 6. Spring­er (New York), 1987. MR 880376 Zbl 0625.​58023 incollection

[16] T. Aus­tin and C. C. Moore: “Con­tinu­ity prop­er­ties of meas­ur­able group co­homo­logy,” Math. Ann. 356 : 3 (2013), pp. 885–​937. MR 3063901 Zbl 1330.​22005 article