by Gordan Savin
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.
