Celebratio Mathematica

Benedict H. Gross

In Celebration of Dick Gross

by Wee Teck Gan

I first met Dick Gross in the spring semester of 1994. At that time, I was a third-year un­der­gradu­ate stu­dent at Cam­bridge Uni­versity, ap­ply­ing for PhD pro­grams in US schools, with the plan to spe­cial­ize in num­ber the­ory. For­tu­nate enough to be ad­mit­ted to Prin­ceton and Har­vard, I took up the in­vit­a­tion to vis­it both schools at the end of March. This was less than two years after An­drew Wiles had an­nounced his proof of Fer­mat’s Last The­or­em, so for an as­pir­ing num­ber the­or­ist, the tempta­tion to simply ac­cept Prin­ceton’s of­fer and study with Wiles was very strong in­deed. At least, that was my ini­tial think­ing when I began my US trip. It was dur­ing my vis­it to Har­vard that I met Dick for the first time. I re­called sit­ting in his of­fice and ex­plain­ing the hard choice I had to make. Dick did not try to con­vince me to pick Har­vard. Rather, he ad­vised me to simply pick the place that felt right or res­on­ated with me, say­ing (para­phrase): “For num­ber the­ory, you can­not go wrong with either place; plus, you can al­ways come here for a postdoc after your PhD in Prin­ceton, or vice versa.” (Na­ively, I had be­lieved that would be the nat­ur­al course of events.) I was sur­prised by Dick’s frank and open ad­vice and the ease of com­mu­nic­a­tion with him, and left his of­fice feel­ing en­er­gized. That con­ver­sa­tion sealed my de­cision, and would serve as a mold for the many con­ver­sa­tions I have had with Dick since: I al­ways come away with re­newed en­ergy and a fresh per­spect­ive, feel­ing in­spired and em­powered.

After his im­port­ant work [2] with Za­gi­er ap­peared, Dick’s re­search spec­trum began to broaden to in­cor­por­ate the rep­res­ent­a­tion the­ory of re­duct­ive Lie groups, so as to provide a rep­res­ent­a­tion the­or­et­ic for­mu­la­tion of the Gross–Za­gi­er the­or­em which is more amen­able to gen­er­al­iz­a­tion to high­er ranks. Be­fore 1990, his work was largely fo­cused on the arith­met­ic of el­lipt­ic curves and mod­u­lar forms. The peri­od of trans­ition to in­clude the rep­res­ent­a­tion the­ory of high­er-rank groups was es­sen­tially com­plete by the time I entered Har­vard in 1994, with the ex­cep­tion­al groups be­ing of par­tic­u­lar in­terest to him. Two high­lights from this peri­od were his Crelle pa­per [3] with No­lan Wal­lach on qua­ternion­ic rep­res­ent­a­tions of qua­ternion­ic groups and his Com­posi­tio pa­per [4] with Gordan Sav­in on ex­cep­tion­al theta cor­res­pond­ences and the pro­posed con­struc­tion of a \( G_2 \)-motive. My cur­rent NUS col­league, Hung Yean Loke, was Dick’s stu­dent at that time and two years ahead of me. I learned from Hung Yean that he was work­ing on branch­ing prob­lems for the min­im­al rep­res­ent­a­tion of qua­ternion­ic ex­cep­tion­al groups. At that time, I did not know any­thing about Lie groups or Lie al­geb­ras, and wondered why a num­ber the­or­ist would need to learn such things at all, in­stead of the more usu­al el­lipt­ic curves, class field the­ory, al­geb­ra­ic geo­metry and mod­u­lar forms. To com­pensate for my ig­nor­ance, I at­ten­ded in my second semester an un­der­gradu­ate course on Lie al­geb­ras giv­en by Shlomo Stern­berg and also spent a week­end read­ing Serre’s de­light­ful little book [e1]. Some­what for­tu­it­ously, in the middle of the semester, Dick ini­ti­ated a learn­ing sem­in­ar on the Lang­lands pro­gram, in which he gave the first few talks, dis­cuss­ing the Satake iso­morph­ism, the Lang­lands dual group and the func­tori­al­ity prin­ciple. Dick has this ma­gic­al abil­ity to make the math­em­at­ics and ideas come alive and ap­pear in­ev­it­able. I was com­pletely mes­mer­ized by the beauty of the math­em­at­ics and his in­cred­ibly clear ex­pos­i­tion. After this sem­in­ar series, I re­ques­ted Dick to be my ad­visor, de­cid­ing that the rep­res­ent­a­tion the­or­et­ic as­pects of num­ber the­ory was what res­on­ated most with me. So it is not an ex­ag­ger­a­tion to say that Dick’s sem­in­ar series had a life-chan­ging im­pact on me.

Dick had many PhD stu­dents in the 1990s, es­pe­cially in the year ahead of me, and he worked very closely with some of them (for ex­ample, Dav­id Pol­lack, Josh Lansky and Seth Pad­ow­itz). For me, however, his su­per­vi­sion was more hands-off. My thes­is prob­lem came about after I told Dick that I had been learn­ing about the rep­res­ent­a­tion the­ory of fi­nite re­duct­ive groups (à la De­ligne–Lusztig) from Carter’s book [1] over the winter break. He men­tioned cas­u­ally that I could con­sider the fi­nite field ana­log of his work with Sav­in, and this be­came the be­gin­ning of my thes­is pro­ject. Dur­ing the course of my thes­is work, I did not re­ceive much dir­ect su­per­vi­sion from him (for ex­ample, we did not have reg­u­lar meet­ings). What I re­ceived was bet­ter: the op­por­tun­ity to work with Dick as a col­lab­or­at­or. In­deed, by the end of my PhD stud­ies, I had writ­ten two pa­pers with him [5], [6], and com­pleted two more [7], [8] over the next couple of years.

This early col­lab­or­a­tion with Dick taught me a few things. Firstly, the pa­pers we wrote to­geth­er then were mostly about the arith­met­ic of ex­cep­tion­al groups and their auto­morph­ic forms. In the grand scheme of things, these are far from be­ing Dick’s most im­port­ant pa­pers, but he ap­proached them with the same en­thu­si­asm, joy and de­gree of care, be­cause at that mo­ment, that was the math­em­at­ics that cap­tured his in­terest and ima­gin­a­tion. I learned from this ex­per­i­ence that I am free to pur­sue my own math­em­at­ic­al taste and in­terest without wor­ry­ing about how it may be per­ceived by oth­ers. Secondly, it gave me the chance to learn firsthand from him how to write pa­pers, clearly a valu­able skill for our pro­fes­sion, and I would be hard-pressed to find a bet­ter in­struct­or for this. Fi­nally, as a con­sequence of our col­lab­or­a­tion, I have over the years re­ceived sev­er­al hand­writ­ten let­ters from Dick (even though he could have used email). These let­ters were per­fectly writ­ten with beau­ti­ful hand­writ­ing and al­most no re­vi­sions, and of course the clar­ity of the line of think­ing in them is crys­tal. It al­ways amazes me to re­ceive such a let­ter from Dick.

Al­most 24 years after my PhD, I am still col­lab­or­at­ing with Dick! Our second bout of col­lab­or­a­tion began when Di­pen­dra Prasad vis­ited me at UC San Diego dur­ing the aca­dem­ic year 2007–08, and we star­ted think­ing about the ex­ten­sion of the Gross–Prasad con­jec­ture to the set­ting of all clas­sic­al groups. As we were for­mu­lat­ing the so-called GGP con­jec­tures, I asked Di­pen­dra for the mo­tiv­a­tion of the re­cipe for the char­ac­ters of com­pon­ent groups in­ter­ven­ing in the ori­gin­al GP con­jec­ture. Di­pen­dra told me that the re­cipe was en­tirely due to Dick and I should check with him. When I fi­nally got the chance to do that, Dick’s re­sponse was (para­phrase): “I used Wald­spur­ger’s and Di­pen­dra’s res­ults in low rank as a guide and came up with the simplest re­cipe I could think of for a char­ac­ter which is trivi­al on the iden­tity com­pon­ent of the cent­ral­izer of the \( L \)-para­met­er.” Any­one who has looked at those char­ac­ters will know that it is not at all ob­vi­ous to guess what they should be from such scant evid­ence, and Dick’s an­swer ba­sic­ally con­firmed what I sus­pec­ted: it was a stroke of geni­us.

Our pa­pers [9], [10] took about 2 years to com­plete and ap­peared in 2012 as part of an Astérisque volume with Wald­spur­ger (who had giv­en a bril­liant proof of the \( p \)-ad­ic case of the GP con­jec­ture). Two years ago, Dick, Di­pen­dra and I fol­lowed up with a pa­per [11] ex­tend­ing the con­jec­tures to the set­ting of \( A \)-pack­ets, and cur­rently we are pre­par­ing an­oth­er pa­per [12] on a twis­ted ver­sion of GGP.

I feel very for­tu­nate that I have been able to work with and learn from Dick over the past 27–28 years. In pre­par­ing for this art­icle, I took a look at Dick’s Math­s­cinet page and noted the large num­ber of col­lab­or­at­ors he has had over many areas, a test­a­ment to his broad math­em­at­ic­al in­terest and strong col­lab­or­at­ive skills. This will not sur­prise those who know him. I was however (pleas­antly) sur­prised that (at this time of writ­ing), I hap­pen to be his most fre­quent col­lab­or­at­or. Suf­fice it to say that if this is the only thing I will be re­cog­nized for pro­fes­sion­ally, I shall be very con­tent.

Wee Teck Gan was a PhD stu­dent of Dick Gross from 1994–98. He works on the the­ory of auto­morph­ic forms and the Lang­lands pro­gram. After a postdoc at Prin­ceton (1998–2003), he be­came a fac­ulty mem­ber at the Uni­versity of Cali­for­nia, San Diego (2003–2011), be­fore mov­ing back to Singa­pore (where he grew up) in 2011. He is cur­rently the Tan Chin Tu­an Centen­ni­al Pro­fess­or at the Na­tion­al Uni­versity of Singa­pore.


[1] R. W. Carter: Fi­nite groups of Lie type: Con­jugacy classes and com­plex char­ac­ters. Wiley (New York), 1985. MR 794307 Zbl 0567.​20023 book

[2] B. H. Gross and D. B. Za­gi­er: “Hee­gn­er points and de­riv­at­ives of \( L \)-series,” In­vent. Math. 84 (1986), pp. 225–​320. To John Tate. This work ex­pands on a short note pub­lished in C. R. Acad. Sci., Par­is 297 (1983). Part II was pub­lished in Math. Ann. 278 (1987). MR 833192 Zbl 0608.​14019 article

[3] B. H. Gross and N. R. Wal­lach: “On qua­ternion­ic dis­crete series rep­res­ent­a­tions, and their con­tinu­ations,” J. Reine An­gew. Math. 1996 : 481 (1996), pp. 73–​123. MR 1421947 Zbl 0857.​22012 article

[4] B. H. Gross and G. Sav­in: “Motives with Galois group of type \( \mathrm{G}_2 \): An ex­cep­tion­al theta-cor­res­pond­ence,” Com­pos. Math. 114 : 2 (1998), pp. 153–​217. MR 1661756 Zbl 0931.​11015 article

[5] B. H. Gross and W. T. Gan: “Haar meas­ure and the Artin con­duct­or,” Trans. Am. Math. Soc. 351 : 4 (1999), pp. 1691–​1704. MR 1458303 Zbl 0991.​20033 article

[6] B. H. Gross and W. T. Gan: “Com­mut­at­ive sub­rings of cer­tain non-as­so­ci­at­ive rings,” Math. Ann. 314 : 2 (1999), pp. 265–​283. MR 1697445 Zbl 0990.​11018 article

[7] W. T. Gan and B. H. Gross: “In­teg­ral em­bed­dings of cu­bic norm struc­tures,” J. Al­gebra 233 : 1 (November 2000), pp. 363–​397. To Nath­an Jac­ob­son, in me­mori­am. MR 1793601 Zbl 0990.​11017 article

[8] W. T. Gan, B. Gross, and G. Sav­in: “Four­i­er coef­fi­cients of mod­u­lar forms on \( G_2 \),” Duke Math. J. 115 : 1 (2002), pp. 105–​169. MR 1932327 Zbl 1165.​11315 article

[9] W. T. Gan, B. H. Gross, and D. Prasad: “Sym­plect­ic loc­al root num­bers, cent­ral crit­ic­al \( L \) val­ues, and re­stric­tion prob­lems in the rep­res­ent­a­tion the­ory of clas­sic­al groups,” pp. 1–​109 in Sur les con­jec­tures de Gross et Prasad, I [The con­jec­tures of Gross and Prasad, I]. Edi­ted by W. T. Gan, B. H. Gross, D. Prasad, and J.-L. Wald­spur­ger. Astérisque 346. Société Mathématique de France (Par­is), 2012. MR 3202556 Zbl 1280.​22019 ArXiv 0909.​2999 incollection

[10] W. T. Gan, B. H. Gross, and D. Prasad: “Re­stric­tions of rep­res­ent­a­tions of clas­sic­al groups: Ex­amples,” pp. 111–​170 in Sur les con­jec­tures de Gross et Prasad, I [The con­jec­tures of Gross and Prasad, I]. Edi­ted by W. T. Gan, B. H. Gross, D. Prasad, and J.-L. Wald­spur­ger. Astérisque 346. Société Mathématique de France (Par­is), 2012. with French sum­mary. MR 3202557 Zbl 1279.​22023 ArXiv 0909.​2993 incollection

[11] W. T. Gan, B. H. Gross, and D. Prasad: “Branch­ing laws for clas­sic­al groups: The non-tempered case,” Com­pos. Math. 156 : 11 (2020), pp. 2298–​2367. MR 4190046 Zbl 1470.​11126 ArXiv 1911.​02783 article

[12] W. T. Gan, B. H. Gross, and D. Prasad: Twis­ted GGP prob­lems and con­jec­tures, 2022. In pre­par­a­tion. misc