Celebratio Mathematica

My story, in re­la­tion to Paul, be­gins in Fall 1983, lunch time in IHES, France. In those hal­cy­on days be­fore email, smart­phones and all the rest, con­ver­sa­tion and dis­cus­sion were much more im­port­ant. Alain Connes had just had a con­ver­sa­tion with Pierre De­ligne, a con­ver­sa­tion which roamed around the rep­res­ent­a­tion the­ory of $p$-ad­ic groups, on the one hand, and on “the con­jec­ture” on the oth­er hand. As a res­ult, Alain earn­estly en­cour­aged me to work on the con­jec­ture for $p$-ad­ic groups. I liked the sound of this idea, but there were two is­sues:

• I did not know what a $p$-ad­ic num­ber was.
• I did not know (neither did any­one else) how to for­mu­late the con­jec­ture for $p$-ad­ic groups.

I as­sumed that the right-hand side of the con­jec­ture (which came to be known as the Baum–Connes con­jec­ture) would not change un­der re­for­mu­la­tion (this has turned out broadly speak­ing to be true) and star­ted on the com­plex to­po­lo­gic­al $K$-the­ory of the re­duced $C^{\ast }$-al­gebra of re­duct­ive $p$-ad­ic groups such as ${\mathrm{SL}}_2$. This was, to put it mildly, a steep learn­ing curve.

Some­time in Spring 1989, shortly after the pub­lic­a­tion of his sem­in­al pa­per with Alain Connes [1], Paul called me with his idea for for­mu­lat­ing the BC-con­jec­ture for all second count­able loc­ally com­pact groups.

In his pa­per with Alain Connes on the Chern char­ac­ter for dis­crete groups [1], there may have been in­tim­a­tions of the uni­ver­sal ex­ample, but it did not have a name, and was cer­tainly not centre stage. However, while on his ex­er­cise bike, Paul had the cru­cial idea of de­fin­ing the uni­ver­sal ex­ample $\underset{―}{E}G$ as a con­tract­ible space on which $G$ acts prop­erly, and bring­ing this concept centre stage. It be­came the corner­stone of the left-hand side of what came to be know as the Baum–Connes con­jec­ture. Paul was then able to bring his vast ex­per­i­ence of $K$-ho­mo­logy in­to the sub­ject.

In the con­text of the equivari­ant Chern char­ac­ter, the ex­ten­ded quo­tient $X//\mathrm{\Gamma }$ ap­peared for the first time in [1]. The nota­tion $X//\mathrm{\Gamma }$ was in­flu­enced by an art­icle of Carti­er [e3].

I had already (in some kind of an­ti­cip­a­tion) done some read­ing on af­fine build­ings, and real­ized im­me­di­ately that the af­fine build­ing of $G$ qual­i­fied as uni­ver­sal ex­ample. And so it came about that the BC-con­jec­ture for $p$-ad­ic groups was, for the first time, well-for­mu­lated. With Paul and Nigel Hig­son, we even­tu­ally pub­lished a proof of BC for the $p$-ad­ic gen­er­al lin­ear group; see [3].

I learnt the for­mu­la­tion of the LLC (loc­al Lang­lands con­jec­ture) from the ex­cel­lent ex­pos­it­ory art­icle by Steve Kudla [e1]. At the time, I was try­ing to re­late the BC con­jec­ture for ${\mathrm{GL}}_n$ to the LLC for ${\mathrm{GL}}_n$. The right-hand side of BC is for­mu­lated in terms of to­po­lo­gic­al $K$-the­ory. I wondered wheth­er, thanks to the equivari­ant Chern char­ac­ter, one could re­place the $K$-the­ory by the co­homo­logy of an ex­ten­ded quo­tient, and then strip away the co­homo­logy al­to­geth­er. Would a valu­able geo­met­ric state­ment re­main?

The an­swer is “yes”, and after much ex­per­i­ment­a­tion, I even­tu­ally pub­lished with Jacek Brodzki, a CR note [e2] which, for the first time, re­lated the BC-con­jec­ture with the LLC (loc­al Lang­lands con­jec­ture) for ${\mathrm{GL}}_n$. This was for­mu­lated en­tirely in terms of Lang­lands para­met­ers, a point of view which has turned out to be very fruit­ful; see the works of Mous­saoui et al [e6], [e8], [e10], [e7].

I was pleased with the CR note with Jacek, but had no real thoughts on how it might be de­veloped.

It was Paul who real­ised that [e4] should be re­garded as a re­fine­ment of the Bern­stein pro­gram, and there­fore ap­plic­able to all re­duct­ive $p$-ad­ic groups. Anne-Mar­ie Au­bert joined the team, and then, some time later, Maarten Sol­leveld also joined us.

In this way, the ABPS (Au­bert–Baum–Ply­men–Sol­leveld) con­jec­ture was born: see Sec­tion 3. The ABPS col­lab­or­a­tion cre­ated a stream of pub­lic­a­tions, all of them around the ABPS con­jec­ture [12], [13], [17], [16], [21], [15], [20], [19], [22], [18].

To seek and find the es­sen­tial sim­pli­cit­ies lurk­ing with­in an ap­par­ently very tech­nic­al and ob­scure sub­ject.

Paul was one of the first to­po­lo­gists to ad­apt thor­oughly to the non­com­mut­at­ive vis­ion of Alain Connes.

One thing I have found work­ing with Paul is his ad­apt­ab­il­ity. On one oc­ca­sion, I was work­ing with Steve Mil­ling­ton, try­ing to prove the BC-con­jec­ture for re­duct­ive ad­el­ic groups. We were thor­oughly stuck, had no real idea how to pro­ceed. Paul ar­rived, and with­in a day or two had suf­fi­ciently ad­ap­ted to the is­sues to make some good sug­ges­tions. This, to­geth­er with a cru­cial re­mark of Vin­cent Laf­forgue, even­tu­ally led to a com­plete proof; see [7], [5].

On an­oth­er oc­ca­sion I was work­ing re­motely with Jam­ila Jawd­at on a rather del­ic­ate $K$-the­ory is­sue in con­nec­tion with the $p$-ad­ic group ${\mathrm{SL}}_n$. Paul gen­er­ously con­cen­trated on these is­sues for sev­er­al days, dur­ing one of his reg­u­lar vis­its to me in Eng­land, and we solved the prob­lem; see [e5].

In gen­er­al, Paul and I like to get away from maths de­part­ments, and prefer to work in quiet cafés, prefer­ably with a good view, and a re­li­able source of food and drink. This routine has served us well over a peri­od of three dec­ades.

In 2003, in the middle of the sum­mer hol­i­days, I re­ceived an email from Ro­ger Ply­men, with whom I was already work­ing, sug­gest­ing that Paul should join our pro­ject. Ro­ger’s email began, “I have had long dis­cus­sions this week with Paul Baum. He has been sup­ply­ing many new ideas.” Soon after, our col­lab­or­a­tion star­ted, mainly via ex­chan­ging emails and files.

We met some­time later at the In­sti­tut Henri Poin­caré in Par­is, which Paul was vis­it­ing. I re­mem­ber that when my moth­er and I ar­rived, Paul was at the café work­ing with Nigel Hig­son. Then we went to his of­fice and had very in­ter­est­ing dis­cus­sions on our work, on the LLC and on the con­struc­tion of su­per­cuspid­al rep­res­ent­a­tions of $p$-ad­ic groups, a ques­tion that has known many de­vel­op­ments over the years but that is not yet solved in its full gen­er­al­ity. I al­ways learn a lot from talk­ing and ex­chan­ging emails with Paul! At this oc­ca­sion, Paul and my moth­er man­aged also to talk to­geth­er (in French!), and I found out that Paul brought his moth­er too on some of his travels and that she liked meet­ing his col­leagues.

The first talk by Paul that I at­ten­ded was at the con­fer­ence in hon­or of Alain Connes, “Géométrie non com­mut­at­ive”, again at IHP, in April 2007. A large part of the talk, en­titled “Geo­met­ric struc­ture in the rep­res­ent­a­tion the­ory of $p$-ad­ic groups”, was on our joint works [8], [9], on the very early stages of what is now known as the “ABPS con­jec­ture” (see Sec­tion 3 be­low). That year, Paul gave talks on the con­jec­ture in sev­er­al places, in­clud­ing Ox­ford, Mo­scow, Göttin­gen, Mel­bourne and Warsaw, and nu­mer­ous talks on oth­er parts of his work. I am amazed at the num­ber of talks that Paul is able to give on many dif­fer­ent sub­jects and in many dif­fer­ent places around the world.

In Ju­ly 2008, I at­ten­ded an­oth­er talk by Paul (with ex­actly the same title) and again at a birth­day con­fer­ence, with al­most the same title “New dir­ec­tions in non-com­mut­at­ive geo­metry”, but this time in Manchester, and in hon­or of Ro­ger. Paul was speak­ing right be­fore me and he presen­ted, with a lot of kind­ness, his talk as an in­tro­duc­tion of mine. At that oc­ca­sion I talked about my paint­ing with Paul, and I dis­covered with in­terest that his fath­er, Mark Baum, was a paint­er.

My most vivid memory of math­em­at­ic­al dis­cus­sions with Paul and Ro­ger is our day at the reser­voir in Manchester. Paul and I were vis­it­ing Ro­ger for a few days, and one morn­ing Ro­ger took the two of us (and my moth­er) to a very pleas­ant café, and we were sit­ting there all day talk­ing of math and, at the same time, look­ing at the beau­ti­ful view and en­joy­ing nice food. I re­mem­ber that we were work­ing on the last sec­tion of our ${\mathrm{G}}_2$ pa­per [11], and, at the end of the day, es­sen­tially all the miss­ing tech­nic­al points were achieved. We were look­ing at the rep­res­ent­a­tions in the prin­cip­al series of the $p$-ad­ic group ${\mathrm{G}}_2$ which cor­res­pond, thanks to Alan Roche’s res­ults, to Iwahori-spher­ic­al rep­res­ent­a­tions of the en­do­scop­ic group ${\mathrm{SO}}_4$. We had par­ti­tioned the ex­ten­ded quo­tient $T^{\vee }//W$, where $T^{\vee }$ and $$W\simeq \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$$ are the stand­ard max­im­al tor­us and the Weyl group of ${\mathrm{SO}}_4(\mathbb{C})$, re­spect­ively, in­to four sub­sets $(T^{\vee }//W{)}_i$ cor­res­pond­ing to the four uni­po­tent classes $C_1$, $C_2$, $C_3$, $C_4$ of ${\mathrm{SO}}_4(\mathbb{C})$, and the is­sue was to prove that the co­ordin­ate al­gebra of $(T^{\vee }//W{)}_i$ is “equi­val­ent” to the part $J_{{\mathbf{c}}_i}$ of the Lusztig asymp­tot­ic al­gebra $J$ that was at­tached by him to the two-sided cell ${\mathbf{c}}_i$ cor­res­pond­ing to the uni­po­tent class $C_i$. Res­ults on this type for an ar­bit­rary split $p$-re­duct­ive are still far to reach, even in the case of Iwahori-spher­ic­al rep­res­ent­a­tions. This equi­val­ence is not a Mor­ita equi­val­ence, nor is it spec­trum-pre­serving in gen­er­al. For in­stance, for the trivi­al uni­po­tent class, it is spec­trum-pre­serving with re­spect to a fil­tra­tion of length 2. An equi­val­ence of this kind ap­peared pre­vi­ously in joint work of Paul with Vic­tor Nis­tor [6] and in [e4]. It is one of the in­gredi­ents of the strong form of the ABPS con­jec­ture as stated for split re­duct­ive groups in [12] (based on Paul’s Tak­agi lec­tures, at the Uni­versity of Tokyo in 2012), and is the main sub­ject of [18].

One of the oth­er mys­ter­i­ous as­pects of the ABPS con­jec­ture is its $K$-the­or­et­ic­al fa­cet, which provides a bridge between two dif­fer­ent math­em­at­ic­al worlds: that of BC, and that of the LLC. Paul, Maarten, Ro­ger and I ex­plored this as­pect in [21]. In par­tic­u­lar, by us­ing the Bern­stein de­com­pos­i­tion of the re­duced $C^{\ast }$-al­gebra of a re­duct­ive $p$-ad­ic group, we ob­tained a very pre­cise (still con­jec­tur­al in gen­er­al) de­scrip­tion of the $K$-the­ory of the lat­ter. I was very glad and honored to have the op­por­tun­ity to present our work on the sub­ject at the Fields In­sti­tute in Toronto, for the “Con­fer­ence on geo­metry, rep­res­ent­a­tion the­ory and the Baum–Connes con­jec­ture”, on Ju­ly 20, 2016. A very spe­cial day: Paul’s 80th birth­day! This great day saw also talks by Ro­ger, Maarten, Nan­hua Xi and Paul him­self. That con­fer­ence was the first time we met all four to­geth­er as cowork­ers. It was also a won­der­ful oc­ca­sion for me to meet sev­er­al oth­er coau­thors of Paul, work­ing in many dif­fer­ent do­mains.

The first (still rudi­ment­ary) ver­sion of their con­jec­tures ap­peared in [8]. I be­came aware of this pa­per when I was a PhD stu­dent un­der the guid­ance of Eric Op­dam in Am­s­ter­dam. This work was strik­ingly sim­il­ar to what Eric and I had been try­ing to do in the con­text of af­fine Hecke al­geb­ras, so far with lim­ited suc­cess. Ba­sic­ally, the com­mon theme was the in­ter­play of non­com­mut­at­ive geo­metry and rep­res­ent­a­tion the­ory, with the goal to de­rive res­ults about the lat­ter. The ob­vi­ous dif­fer­ence between the at­tempts of Eric and me and [8] was that Anne-Mar­ie, Ro­ger and Paul had man­aged to for­mu­late and write down their con­jec­tures in a clear way, and to already prove a few in­stances. This pa­per, to­geth­er with the closely re­lated work [4], be­came a ma­jor source of in­spir­a­tion for my re­search.

Nev­er­the­less, for some years the ABP trio and I con­tin­ued to work in­de­pend­ently on re­lated top­ics. It was not un­til Paul’s vis­it to Göttin­gen in 2011 that this changed. Paul brought with him some very in­ter­est­ing ques­tions, which set me think­ing. And then, with his usu­al re­lent­less en­thu­si­asm, Paul con­vinced me to join the team. I re­mem­ber well how Paul and I spent sev­er­al days dis­cuss­ing our con­jec­tures. Paul, Ro­ger and Anne-Mar­ie had thought about this much more than I had, but I came in with new ideas from af­fine Hecke al­geb­ras. Paul wanted to know everything! He al­ways strove to make things clear from a broad per­spect­ive, and could per­sist­ently ask ques­tions for hours. Shortly af­ter­wards, these find­ings were pol­ished in ex­tens­ive email con­ver­sa­tions between the four of us. This led to the defin­it­ive for­mu­la­tion of the ABPS con­jec­tures for split groups, pub­lished in [12].

What amazed at that time was Paul’s activ­ity. I thought I vis­ited a reas­on­able num­ber of con­fer­ences and sem­inars, but it was very little com­pared to Paul — at the tender age of 75. He was trav­el­ing the world like an ex­plorer, and just in hol­i­days from teach­ing, seem­ingly non­stop! And every­where he gave talks, in his own un­mis­tak­able style. For ex­ample, at some point Paul ad­dressed an audi­ence in Ko­pen­ha­gen. He star­ted with a story about his fath­er, and drew some com­par­is­ons between Europe and the United States. The lengthy in­tro­duc­tion came to a cli­max with the words “Europe was in ru­ins, again. As usu­al, hahaha!” Need­less to say, this was com­pletely ir­rel­ev­ant and un­re­lated to the math­em­at­ic­al con­tent. But Paul was clearly en­ter­tain­ing every­body in the hall, him­self in­cluded.

In the years 2011–2017, the ABPS con­jec­ture provided ample in­spir­a­tion for new pa­pers. We man­aged to veri­fy it for prin­cip­al series rep­res­ent­a­tions of split groups [15], [20] and for in­ner forms of gen­er­al lin­ear groups. Anne-Mar­ie’s PhD stu­dent Ahmed Mous­saoui found a proof for sym­plect­ic and or­tho­gon­al groups [e8].

To test our con­jec­ture in more com­plic­ated cases, like non­split groups, we em­barked on a deep study of in­ner forms of spe­cial lin­ear groups [16], [19]. To our sur­prise, this un­earthed a counter­example! See ([19], Ex­ample 5.5). Of course this was dis­tress­ing as well as mighty in­ter­est­ing.

Here Paul’s vast ex­per­i­ence with con­jec­tures came in handy. Namely, not long be­fore it had be­come clear that there were ser­i­ous prob­lems with the Baum–Connes con­jec­ture for nonex­act groups. Paul’s solu­tion: modi­fy the con­jec­ture, so that all pre­vi­ous sup­port­ing ex­amples re­main val­id, and the prob­lem­at­ic cases be­come con­firm­ing ex­amples for the re­fined con­jec­ture. For BC this has been done by modi­fy­ing the no­tion of crossed products [14].

For $p$-ad­ic groups, we did something sim­il­ar. We gen­er­al­ized the no­tion of ex­ten­ded quo­tients to twis­ted ex­ten­ded quo­tients, which are rel­ev­ant for non­split groups. In those terms we fi­nally for­mu­lated the ABPS con­jec­tures for all re­duct­ive $p$-ad­ic groups [22], [21], at the same time mak­ing the link with the loc­al Lang­lands cor­res­pond­ence more pre­cise. These con­jec­tures re­main a hard nut to crack. Al­though there cur­rently is good pro­gress to­wards a proof of the ABPS con­jec­tures for depth-zero rep­res­ent­a­tions of ar­bit­rary $p$-ad­ic groups, a gen­er­al proof is not yet in sight.