#### 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.*