The Gindikin-Karpelevich Formula and Combinatorics of Crystals
| Speaker: | Jintai Ding |
|---|---|
| Affiliation: | University of Cincinnati |
| Room: | Mathematics and Computer Building (MC) 5168 |
Abstract:
The
Gindikin-Karpelevich
formula
computes
the
constant
of
proportionality
for
the
intertwining
integral
between
two
induced
spherical
representation
of
a
$p$-adic
reductive
group
$G.$
The
right-hand
side
of
this
formula
is
a
product
over
positive
roots
(with
respect
to
the
Langlands
dual
of
$G$)
which
may
also
be
interpreted
as
a
sum
over
a
crystal
graph.
The
original
interpretation
as
a
sum
is
due
to
Brubaker-Bump-Friedberg
and
Bump-Nakasuji
where
$G=GL_{r+1},$
in
which
vertices
of
the
crystal
graph
are
parametrized
by
paths
to
a
specific
vector
in
the
graph
according
to
a
pattern
prescribed
by
a
reduced
expression
of
the
longest
element
of
the
Weyl
group
of
$G$.
I
will
explain
this
rule,
explain
how
it
may
be
translated
into
a
statistic
on
the
Young
tableaux
realization
of
the
same
crystal
graph,
and
discuss
its
generalization
to
the
affine
Kac-Moody
setting
where
the
notion
of
the
longest
element
of
the
Weyl
group
no
longer
makes
sense.
This
is
joint
work
with
Kyu-Hwan
Lee,
Seok-Jin
Kang,
and
Hansol
Ryu.