Schubert Eisenstein Series
YoungJu Choie
Dept of Math
Pohng Mathematical Institute
POSTECH,Pohang, Korea
Talk at ICERM
Jan 30, 2013
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 1 / 12
Where is POSTECH(Pohang university of Science and Technology)?
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 2 / 12
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 3 / 12
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 4 / 12
그림: POSTECH (1986- ) http://postech.ac.kr
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 5 / 12
What is ”Schubert Eisenstein” series?
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 6 / 12
What is Schubert Eisenstein Series?
Schubert Eisenstein series is defined as sums like usual Eisenstein series
but with the summation restricted to elements coming from a particular
Schubert cell.
Let G be a split semisimple algebraic group over a global field F and B be
its Borel subgroup.
The usual Eisenstein series are sums over B(F )\G (F ), that is, over the
integer points in the flag variety X = B\G .Given a Weyl group element w , one may consider the sum restricted to a
single Schubert cell Xw . This is called a Schubert Eisenstein series Ew .
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 7 / 12
Schubert Cell
More precisely, consider the Bruhat decomposition of G
G =⋃
w∈WBwB
where W is the Weyl group.
This gives the decomposition of the flag variety into Schubert cells
X = ∪w∈WYw
where Yw is the image of BwB in X = B\G .The Schubert cell Xw is the Zariski closure of Yw :
Xw :=⋃
u ∈W , u ≤ w
Yu,
where u ≤ w is the Bruhat order.
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 8 / 12
Schubert Eisenstein Series
Define the Schubert Eisenstein series
Ew (g , ν) =∑
γ∈Xw (Z)
fν(γg)
where
fν(bg) = (δ1/2χν)(b) f (g), b ∈ B(A).
a character χν on T (A)/T (F ), ν ∈ T̂ and δ is a modular quasicharacter.
If w0 is the long Weyl group element, Ew0(g , ν) is the usual Eisenstein
series , so automorphic object.
However, in general Schubert Eisenstein series is no longer automorphic!
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 9 / 12
We would like to explore...
· Does SE have meromorphic continuation to all values of the
parameters?
· Do they have some functional equations?
· One may represent SE recursively using Bott-Samelson map if
Bott-Samelson variety is isomorphic to Schubert variety. How to
represent SE when Bott-Samelson map is not isomorphic?
· Is there any arithmetic implication?
· ... ? more connections with others?
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 10 / 12
Affermative answers by explicit computation
in the case when G = GL(3) with Bump
그림: D. Bump (Stanford U)
Now working on G = GL(4) case with Bump ,
More..YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 11 / 12
YoungJu Choie Dept of Math Pohng Mathematical Institute POSTECH,Pohang, Korea ( Talk at ICERM )Schubert Eisenstein Series Jan 30, 2013 12 / 12