Celebratio Mathematica

Benedict H. Gross

Memories of my friendship with Dick Gross

by Nolan R. Wallach

Dur­ing my years at Rut­gers, I had on many oc­ca­sions at­ten­ded or spoken at sem­inars at Prin­ceton. I was aware of Dick but had no in­ter­ac­tion with him. In fact, the main thing that I re­mem­ber about him from that time was that he tutored a movie star on how to give a math­em­at­ics lec­ture. Be­cause of him Jill Clay­burgh gave a pass­able lec­ture on the snake lemma in the 1990 movie It’s my turn.

Dick’s wife, Jill Mesirov, spent the aca­dem­ic year 1993–1994 at CCR La Jolla (a branch of a private cor­por­a­tion whose main cli­ent was NSA). Dick de­cided to go on leave from Har­vard to UC­SD with the plan of work­ing with Har­old Stark. For­tu­nately for me, he and Di­pen­dra Prasad had re­cently dis­covered the Gross–Prasad con­jec­ture about the re­stric­tion of unit­ary rep­res­ent­a­tions of or­tho­gon­al groups to sym­met­ric or­tho­gon­al sub­groups. They had de­rived the con­jec­ture as a con­sequence of part of the Lang­lands pro­gram. Thus, in ad­di­tion to the beauty of the im­plied for­mu­las, the con­jec­tures had a ser­i­ous role to play in num­ber the­ory. Dick came to my of­fice with a ques­tion about the first non­trivi­al spe­cial case of the con­jec­ture for real or­tho­gon­al groups. I was im­me­di­ately taken with the prob­lem and star­ted work­ing on it. Since Har­old was the chair of the UC­SD De­part­ment of Math­em­at­ics he had al­most no time to do math­em­at­ics. After a short time Dick and I were work­ing full-time to­geth­er. I have had deep col­lab­or­a­tions with many of the best math­em­aticians of the twen­ti­eth cen­tury, but I treas­ure that year of work as the one that I en­joyed the most. In ad­di­tion to work­ing with Dick on at­tempt­ing to prove his con­jec­ture in a spe­cial case I was his de facto ment­or on the so called “real case” (or as a num­ber the­or­ist would say the “in­fin­ite prime”). Like many num­ber-the­or­ists Dick was con­vers­ant with the so called \( p \)-ad­ic case (“fi­nite primes”). I had two oth­er great math­em­aticians who had asked me to ex­plain some as­pect or rep­res­ent­a­tion the­ory to them: Har­ish-Chandra, who asked about the re­cent work (at the time) on ap­plic­a­tions of ho­mo­lo­gic­al al­gebra to rep­res­ent­a­tion the­ory, and Pierre De­ligne who asked for an ex­plan­a­tion of Lang­lands’ proof of the Lang­lands quo­tient the­or­em. In my first meet­ing with Har­ish-Chandra I star­ted with a defin­i­tion of an in­ject­ive mod­ule. Har­ish-Chandra stopped me and said, “When I first came to the in­sti­tute I gave a lec­ture on class field the­ory.” He then gave me a lec­ture on class field the­ory and asked me to re­turn in a week. A week later I star­ted as be­fore and he in­ter­rup­ted me with “I will nev­er un­der­stand ho­mo­lo­gic­al al­gebra” and he in­dic­ated that there was no point to con­tinu­ing the les­sons. In De­ligne’s case it was the op­pos­ite: he grasped everything rap­idly and even found an er­ror in Lang­lands’ proof of one of his main lem­mas. Dick was like De­ligne: teach­ing him forced me to have a deep­er un­der­stand­ing of my own work. Dur­ing that year Wiles’ first proof of Fer­mat’s Last The­or­em was sub­mit­ted. Some­how, Dick got a copy of it. One time when I stopped in his of­fice, he looked up from the manuscript, poin­ted at it and said, “There’s a black hole in the pa­per.” We all now know what that was. An­oth­er time, he was do­ing a cal­cu­la­tion with pen­cil and pa­per that in­volved im­mense num­bers. I asked how he could be sure that his cal­cu­la­tions were cor­rect. He said, “I’m a num­ber the­or­ist”.

Our col­lab­or­a­tion led to three pa­pers. Two of them [1], [2] were beau­ti­fully writ­ten by Dick about how the geo­metry of the qua­ternion­ic real forms and the cor­res­pond­ing twis­tor spaces ef­fected the struc­ture of unit­ary rep­res­ent­a­tions re­lated to the qua­ternion­ic dis­crete series. The third gave the solu­tion to the ini­tial prob­lem that Dick posed when he ar­rived at UC­SD [3].

The third pa­per was writ­ten two years after we did our work. This delay was caused by two ma­jor med­ic­al events in our lives. Dur­ing the year after his vis­it to UC­SD Dick was forced to have a ma­jor op­er­a­tion to re­pair part of his ab­do­men that was dam­aged by ra­di­ation ther­apy when he was a child. The next year, it was my turn to have in­vas­ive sur­gery. The day be­fore I was to go un­der the scalpel I re­ceived a call from Dick giv­ing me ad­vice about my up­com­ing sur­gery. This in­cluded, “Don’t be brave, take the morphine”.

We both re­covered from our sur­ger­ies. Most of our later in­ter­ac­tions be­fore he moved to UC­SD in­volved see­ing him on vis­its to South­ern Cali­for­nia or in emails. One email in­volved a ques­tion about gen­er­al­ized Whit­taker mod­els for holo­morph­ic dis­crete series. When I answered his query, he then let me know what he really needed: the same ques­tion for qua­ternion­ic dis­crete series for \( G_2 \). This was much more dif­fi­cult and led to my pa­per on the mod­els for qua­ternion­ic and holo­morph­ic dis­crete series. Dick’s ideas, in this dir­ec­tion, led to his work with Wee Teck Gan and Gordan Sav­in. Dick spent an­oth­er year as a vis­it­or at UC­SD in 2007. Dur­ing that year he was deeply im­mersed in work with Wee Teck (a pro­fess­or at UC­SD at the time) and Di­pen­dra (on leave to UC­SD). So, I had little chance to work with him.

Dur­ing his 60th birth­day con­fer­ence, in ad­di­tion to a pic­ture with his former stu­dents, Dick set up for a pic­ture of his teach­ers to be taken with him. It was my great hon­or to be in­cluded in that dis­tin­guished group.

We wrote our most re­cent pa­per in 2011 [4]. He showed me his in­geni­ous meth­od of us­ing the Weyl di­men­sion for­mula to cal­cu­late the Hil­bert poly­no­mi­als of ho­mo­gen­eous pro­ject­ive vari­et­ies. I showed him how his idea could be mod­i­fied to also cal­cu­late the Hil­bert series. Many gradu­ate stu­dents have thanked me for that pa­per since it freed them from the stand­ard (hor­rible) cal­cu­la­tions of the Hil­bert poly­no­mi­als of Grass­man­ni­ans.

In 2016, Dick and Jill moved to San Diego. Jill was ap­poin­ted to a high ad­min­is­trat­ive po­s­i­tion in the UC­SD Med­ic­al School in 2015 (she is now an As­so­ci­ate Vice Chan­cel­lor of the Med­ic­al School) and Dick even­tu­ally be­came a reg­u­lar (1/2 time) mem­ber of the math­em­at­ics de­part­ment. As an emer­it­us fac­ulty mem­ber who had just giv­en up his of­fice, I was as­signed to one of the of­fices for four or five emer­iti. Dick was kind enough to let me be his of­ficemate. Un­for­tu­nately, dur­ing his ten­ure at UC­SD we have had little chance for in­ter­ac­tion. One reas­on is that the prob­lems that led to our sur­ger­ies in the 1990s re­curred. In Au­gust of 2017 I had ma­jor heart sur­gery. The day after the sur­gery I was walk­ing around the halls of the hos­pit­al us­ing a wheel­chair to hold my­self up and pulling an oxy­gen tank when I saw Dick. He came with a book for me — Bar­bar­i­an Days, a Pulitzer Prize-win­ning book about surf­ing. The pan­dem­ic kept me from vis­it­ing Dick dur­ing his latest hos­pit­al­iz­a­tion. In­stead, I sent him a book, Noise, by Kahne­man et al. Dick wrote to thank me with, “Right now I am think­ing slowly” (this was an in­dic­a­tion that he had read Kahne­man’s pre­vi­ous book Think­ing Fast and Slow). I wrote back: “Think­ing slow for you is like oth­ers think­ing fast.” By which I meant, even when Dick thinks slowly, in the sense of Kahne­man, he does it fast.

No­lan Wal­lach is a Pro­fess­or Emer­it­us of Math­em­at­ics at the Uni­versity of Cali­for­nia, San Diego. He has done re­search in Rieman­ni­an geo­metry, al­geb­ra­ic geo­metry, rep­res­ent­a­tion the­ory, ana­lys­is com­bin­at­or­ics, and quantum in­form­a­tion.


[1] B. H. Gross and N. R. Wal­lach: “A dis­tin­guished fam­ily of unit­ary rep­res­ent­a­tions for the ex­cep­tion­al groups of real rank \( = 4 \),” pp. 289–​304 in Lie the­ory and geo­metry: In hon­or of Ber­tram Kostant on the oc­ca­sion of his 65th birth­day (Cam­bridge, MA, May 1993). Edi­ted by J.-L. Bryl­in­ski, R. Bryl­in­ski, V. Guille­min, and V. Kac. Birkhäuser (Bo­ston), 1994. MR 1327538 Zbl 0839.​22006 incollection

[2] 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

[3] B. Gross and N. Wal­lach: “Re­stric­tion of small dis­crete series rep­res­ent­a­tions to sym­met­ric sub­groups,” pp. 255–​272 in The math­em­at­ic­al leg­acy of Har­ish-Chandra: A cel­eb­ra­tion of rep­res­ent­a­tion the­ory and har­mon­ic ana­lys­is (Bal­timore, MD, 9–10 Janu­ary 1998). Edi­ted by R. S. Dor­an and V. S. Varada­ra­jan. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 68. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2000. MR 1767899 Zbl 0960.​22008 incollection

[4] B. H. Gross and N. R. Wal­lach: “On the Hil­bert poly­no­mi­als and Hil­bert series of ho­mo­gen­eous pro­ject­ive vari­et­ies,” pp. 253–​263 in Arith­met­ic geo­metry and auto­morph­ic forms: Fest­s­chrift ded­ic­ated to Steph­en Kudla on the oc­ca­sion of his 60th birth­day. Edi­ted by J. Cog­dell, J. Funke, M. Ra­po­port, and T. Yang. Ad­vanced Lec­tures in Math­em­at­ics 19. In­ter­na­tion­al Press (Somerville, MA), 2011. MR 2906911 Zbl 1310.​14044 incollection