Celebratio Mathematica

Dick Gross was ap­poin­ted a pro­fess­or at Har­vard in 1985. At that time I was a second-year gradu­ate stu­dent there. Much time has passed since then, but I am cer­tain that some of what fol­lows is true. Since my math­em­at­ic­al in­terests were in rep­res­ent­a­tion the­ory, in the be­gin­ning I did not in­ter­act much with Dick. However, Di­pen­dra Prasad was my con­tem­por­ary at Har­vard and we of­ten dis­cussed math­em­at­ics, in par­tic­u­lar rep­res­ent­a­tion the­ory. As is well known, un­der Dick’s dir­ec­tion, Di­pen­dra wrote an in­flu­en­tial thes­is in that sub­ject that be­came a basis of the Gross–Prasad con­jec­tures. A lot changed in 1993 when Dick took a sab­bat­ic­al in San Diego. There he wrote a pa­per with No­lan Wal­lach on qua­ternion­ic dis­crete series rep­res­ent­a­tions [1] that had an enorm­ous im­pact on my ca­reer. Their work star­ted as an at­tempt to check some cases of the Gross–Prasad con­jec­ture for dis­crete series rep­res­ent­a­tions of real groups, but it ended as something else. Real groups that ad­mit qua­ternion­ic dis­crete series rep­res­ent­a­tions can be de­scribed in terms of def­in­ite cu­bic Jordan al­geb­ras. In this scheme, as one of the most in­ter­est­ing ex­amples, the ex­cep­tion­al group $E_8$ cor­res­ponds to the ex­cep­tion­al Jordan al­gebra, also known as the Al­bert al­gebra. As a side product, Gross and Wal­lach also con­struc­ted min­im­al rep­res­ent­a­tions as ana­lyt­ic con­tinu­ation of qua­ternion­ic dis­crete series rep­res­ent­a­tions. Ex­cep­tion­al groups con­tain dual pairs (a ter­min­o­logy in­tro­duced by Ro­ger Howe, mean­ing a pair of sub­groups such that each group is the cent­ral­izer of the oth­er) where one group is the ex­cep­tion­al $G_2$ and the oth­er is the auto­morph­ism group of the cu­bic Jordan al­gebra. Since the group of auto­morph­isms of the Al­bert al­gebra is the ex­cep­tion­al $F_4$ we get the most ex­cep­tion­al dual pair $(G_2,F_4)$ in $E_8$. There was a hope/ex­pect­a­tion that re­strict­ing min­im­al rep­res­ent­a­tions of ex­cep­tion­al groups to dual pairs would give a cor­res­pond­ence of rep­res­ent­a­tions, gen­er­al­iz­ing the clas­sic­al theta cor­res­pond­ence, where one re­stricts the Weil rep­res­ent­a­tion to sym­plect­ic-or­tho­gon­al dual pairs. The Weil rep­res­ent­a­tion has sev­er­al ex­pli­cit mod­els/real­iz­a­tions that fa­cil­it­ate com­pu­ta­tions, but noth­ing of that sort ex­ists for min­im­al rep­res­ent­a­tions of ex­cep­tion­al groups. This was a ma­jor obstacle. Dick, aware of my work with Kazh­dan on min­im­al rep­res­ent­a­tions, sent me an email in­form­ing me of his work with No­lan. I real­ized that their work was a good start­ing point and, in­deed, Jing-Song Huang, Pavle Pandžić and I soon ob­tained the first ex­ample of ex­cep­tion­al theta cor­res­pond­ence for $G_2$ dual pairs in a joint pa­per [e1].

Dick took an in­terest in our work and soon made a vis­it to Salt Lake City. Utah is rather far from the East Coast (and from al­most any oth­er math­em­at­ics de­part­ment on plan­et earth), there­fore our vis­it­ors typ­ic­ally stay more than a couple of days, which al­lows time for ex­tra­cur­ricular activ­it­ies such as ski­ing. As a young man, Dick traveled around the world and had skied the Alps, but at the time of this vis­it had not skied in two dec­ades. He was game to try again and we went to a ski area in the Big Cot­ton­wood Canyon. It was a cloudy day, mid­week, so there was prac­tic­ally no one else on the moun­tain. It turned out that Dick was an ex­cel­lent ski­er: he had learned, after all, in the era of long skis, where­as mod­ern “shape” skis made ski­ing a more for­giv­ing sport. It took him just a couple of runs to re­gain his skill. It was one of many great ski days we have had to­geth­er, for he has been an act­ive ski­er ever since.

Dur­ing our col­lab­or­a­tion over the next couple of years, we for­mu­lated a con­jec­tur­al an­swer for ex­cep­tion­al theta cor­res­pond­ences in terms of Vogan pack­ets, and pro­posed a con­struc­tion of a $G_2$-motive. It was ob­served by Serre that the ex­cep­tion­al group $G_2$ does not act on a her­mitian sym­met­ric do­main, so there is no ob­vi­ous place to look for such a motive (and that place would be motives at­tached to holo­morph­ic forms). Our con­struc­tion con­sists of two steps. The first step uses an ex­cep­tion­al theta cor­res­pond­ence arising from the min­im­al rep­res­ent­a­tion of $E_7$. This cor­res­pond­ence is a func­tori­al lift from $G_2$ to ${\mathrm{PGSp}}_6$, and we con­struc­ted a holo­morph­ic form on ${\mathrm{PGSp}}_6$, as a lift of an ex­pli­cit auto­morph­ic rep­res­ent­a­tion of $G_2$ that we dis­covered. The second step con­structs the motive from the holo­morph­ic form on ${\mathrm{PGSp}}_6$. We could not do the second step. However, this was ac­com­plished re­cently by Arno Kret and Sug Woo Shin [e6], thanks to tre­mend­ous ad­vances made in the sub­ject of Shimura vari­et­ies in re­cent dec­ades. The late nineties were great math­em­at­ic­ally for me as the the­ory of ex­cep­tion­al theta cor­res­pond­ences took its shape. Dick had sev­er­al stu­dents work­ing on the sub­ject and two of them, Hung Yean Loke and Wee Teck Gan, have be­come my most pro­lif­ic col­lab­or­at­ors.

In 2000 Dick spent a sab­bat­ic­al semester in Utah. I joined a pro­ject he had with Wee Teck, de­fin­ing Four­i­er coef­fi­cients for qua­ternion­ic mod­u­lar forms on $G_2$. As men­tioned earli­er, $G_2$ does not act on a her­mitian sym­met­ric do­main, thus it has no holo­morph­ic mod­u­lar forms. It was Dick’s bril­liant idea that there should be a the­ory of Four­i­er coef­fi­cients for qua­ternion­ic mod­u­lar forms — that is, auto­morph­ic rep­res­ent­a­tions whose archimedean com­pon­ents are qua­ternion­ic dis­crete series. More pre­cisely, Four­i­er coef­fi­cients should be num­bers, rather than func­tion­als, as it is cus­tom­ary to con­sider when passing from holo­morph­ic forms to gen­er­al auto­morph­ic rep­res­ent­a­tions. We man­aged to de­vel­op such a the­ory for $G_2$ [2]. It turns out that Four­i­er coef­fi­cients in this case are para­met­rized by cu­bic rings. So it is a nat­ur­al prob­lem to find ex­amples of qua­ternion­ic forms and com­pute their Four­i­er coef­fi­cients. This is where ex­cep­tion­al theta cor­res­pond­ences enter the pic­ture, as they provide ex­amples of qua­ternion­ic mod­u­lar forms, just as clas­sic­al theta series provide ex­amples of Siegel holo­morph­ic mod­u­lar forms. And in­deed, us­ing ex­cep­tion­al theta cor­res­pond­ences, Wee Teck con­struc­ted ex­amples of qua­ternion­ic forms on $G_2$ whose Four­i­er coef­fi­cients count the num­ber of em­bed­dings of cu­bic rings in­to a (fixed) or­der in the Al­bert al­gebra [e2]. In re­cent years Aaron Pol­lack has greatly ex­pan­ded the the­ory of Four­i­er coef­fi­cients for qua­ternion­ic forms, ob­tain­ing some strik­ing res­ults bey­ond $G_2$ [e5]. Moreover, in the case of $G_2$, he was able to write down the stand­ard $L$-func­tion of qua­ternion­ic forms us­ing the Four­i­er coef­fi­cients, gen­er­al­iz­ing the clas­sic­al res­ult of Hecke for mod­u­lar forms of one vari­able.

In the years fol­low­ing Dick served as a Dean of Har­vard Col­lege and our math­em­at­ic­al in­ter­ac­tion slowed down. However, we would still vis­it each oth­er, and dur­ing one of the vis­its he told me, on a ski lift, about his work with Mark Reed­er on simple su­per­cuspid­al rep­res­ent­a­tions. This got my at­ten­tion and, dur­ing an­oth­er con­ver­sa­tion with him in Bom­bay, in Janu­ary 2012, it emerged that epipela­gic rep­res­ent­a­tions (a gen­er­al­iz­a­tion of simple su­per­cuspid­al rep­res­ent­a­tions) be­have well in theta cor­res­pond­ences. More pre­cisely, the theta cor­res­pond­ence for epipela­gic rep­res­ent­a­tions can be de­scribed by a cer­tain mo­ment map. These ob­ser­va­tions were turned in­to a pa­per with Hung Yean Loke and Jia-Jun Ma where we de­scribed clas­sic­al theta cor­res­pond­ences for epipela­gic rep­res­ent­a­tions [e3]. Build­ing on that work, Loke and Ma went on to com­pletely de­term­ine how min­im­al types be­have in theta cor­res­pond­ences for $p$-ad­ic groups [e4]. This is a very nice res­ult in­deed. A cas­u­al read­er may not see the in­flu­ence of Dick Gross in this end res­ult, yet it was he who star­ted this “snow­ball” rolling.

Gordan Sav­in re­ceived PhD from Har­vard in 1988. After two short stints at M.I.T. and Yale, he has been a pro­fess­or of math­em­at­ics at the Uni­versity of Utah since 1993.